text
stringlengths 2
512
| embedding
sequencelengths 768
768
|
|---|---|
Matrix Denoising with Doubly Heteroscedastic Noise:
Fundamental Limits and Optimal Spectral Methods
Yihan Zhang
Institute of Science and Technology Austria
[email protected]
Marco Mondelli
Institute of Science and Technology Austria
[email protected]
Abstract
We study the matrix denoising problem of estimating the singular vectors of a
rank-1 signal corrupted by noise with both column and row correlations. Existing
works are either unable to pinpoint the exact asymptotic estimation error or, when | [
-0.011110450141131878,
0.23494981229305267,
-0.016464726999402046,
0.19671818614006042,
-0.17327246069908142,
0.22296208143234253,
-0.12856364250183105,
-0.4026138186454773,
0.025135979056358337,
-0.3536531925201416,
-0.20055367052555084,
0.3331570327281952,
-0.4114185869693756,
0.2192540019750595,
0.1623125821352005,
0.3727922737598419,
0.3444685935974121,
0.1888130158185959,
-0.41609176993370056,
-0.6684750914573669,
0.06335486471652985,
0.11059314757585526,
0.44645997881889343,
0.028189582750201225,
-0.0677432119846344,
0.23557676374912262,
0.24348674714565277,
0.18606583774089813,
-0.38616207242012024,
0.29034510254859924,
0.003952813800424337,
0.9732127785682678,
-0.09946002066135406,
-0.09315285831689835,
0.004022567067295313,
-0.2071642130613327,
0.7280060052871704,
-0.05134750157594681,
0.1601676046848297,
-0.050183285027742386,
0.5617221593856812,
-0.26662519574165344,
-0.12296383082866669,
0.48359939455986023,
-0.313060998916626,
-0.3323446810245514,
-1.6800312995910645,
0.10147849470376968,
-0.5552982687950134,
-0.7187741994857788,
-0.06748712062835693,
0.2298164963722229,
0.23693180084228516,
0.7909778356552124,
-0.4967917203903198,
0.1987770050764084,
-0.6996515989303589,
-0.7415703535079956,
0.6811249256134033,
-0.19493146240711212,
-0.31631848216056824,
0.35939618945121765,
-0.13779988884925842,
-0.03435060381889343,
0.36950767040252686,
0.38408735394477844,
0.32729172706604004,
-0.4098082482814789,
-0.4900265038013458,
-0.1378410905599594,
0.016107920557260513,
-0.10156956315040588,
-0.3608861565589905,
0.10653464496135712,
0.008529826067388058,
-0.20477433502674103,
-0.06459872424602509,
0.2526324987411499,
0.18232129514217377,
-0.11037787050008774,
0.07615844160318375,
-0.1443062722682953,
0.836609959602356,
-0.5655615329742432,
0.7840774655342102,
0.08057991415262222,
-0.38413161039352417,
0.6390379667282104,
-0.7914687991142273,
-0.06621342152357101,
-0.19641029834747314,
-0.4073115289211273,
0.39182594418525696,
0.3833688795566559,
0.2903122901916504,
0.5985303521156311,
0.25654172897338867,
0.16434378921985626,
0.5631820559501648,
0.476345032453537,
-0.32438766956329346,
-0.035553429275751114,
0.16455893218517303,
-0.3377981185913086,
-0.6762164235115051,
-0.3730217516422272,
-0.7323117256164551,
-0.032093144953250885,
-0.26508474349975586,
0.41415277123451233,
-0.5302984118461609,
0.6086825132369995,
0.291415274143219,
-0.5951090455055237,
-0.15522456169128418,
-0.021434197202324867,
0.30578988790512085,
0.05654425546526909,
-0.0022787353955209255,
0.6656748056411743,
0.10868621617555618,
1.4534896612167358,
-0.010032515041530132,
0.5130459070205688,
0.4322035014629364,
0.267818808555603,
0.4666290879249573,
0.47892192006111145,
-0.41112053394317627,
0.5766581296920776,
0.04131750017404556,
0.8932780027389526,
-0.21155966818332672,
0.20867031812667847,
-0.1660456657409668,
-0.5215641260147095,
-0.3478897511959076,
-0.3341737985610962,
0.14272142946720123,
0.03482051193714142,
-0.5113054513931274,
-0.36144766211509705,
-0.3885045647621155,
0.285005658864975,
0.5027405619621277,
-0.09888570010662079,
0.04749085009098053,
-0.41468754410743713,
0.10987050831317902,
-0.06526345759630203,
-0.24836288392543793,
0.24990537762641907,
-1.0183111429214478,
-0.12049754709005356,
-0.11663685739040375,
0.6170065402984619,
0.13619956374168396,
-0.4253545105457306,
0.0368945375084877,
0.18204081058502197,
0.07531558722257614,
-0.30886131525039673,
-0.8016242980957031,
-0.30817529559135437,
0.17206454277038574,
-1.0610663890838623,
-0.06796705722808838,
0.44591081142425537,
-0.16068612039089203,
-0.19886870682239532,
-0.4436797499656677,
-0.41950103640556335,
-0.25463125109672546,
-0.4770910441875458,
0.1687430441379547,
0.4850117564201355,
0.21428421139717102,
-0.10377652198076248,
0.22696131467819214,
0.30427756905555725,
-0.7689359784126282,
-0.15898744761943817,
0.2648342549800873,
-0.182602196931839,
0.2439645677804947,
0.48681503534317017,
0.281561940908432,
0.322620689868927,
0.37472712993621826,
0.19369402527809143,
-0.44671982526779175,
0.05679868534207344,
0.3977387845516205,
-1.0319576263427734,
1.167910099029541,
0.18406426906585693,
-0.4055958688259125,
0.9791600704193115,
0.09914468973875046,
-0.03798838332295418,
-0.4030994772911072,
-0.2754305899143219,
0.41444310545921326,
0.17365093529224396,
-0.37463414669036865,
0.3521297574043274,
0.14319254457950592,
-0.14941027760505676,
-0.4932818114757538,
0.5696241855621338,
0.5626515746116638,
0.1369456797838211,
0.06171289086341858,
-0.31493985652923584,
0.04320988431572914,
-0.01587858237326145,
-0.20601341128349304,
0.3157575726509094,
0.15639984607696533,
0.18206168711185455,
1.2335678339004517,
-0.2694764733314514,
0.2907491624355316,
0.5885412096977234,
-0.1139722391963005,
0.8957604169845581,
0.47659412026405334,
0.05995086207985878,
0.8039345145225525,
-0.25044384598731995,
-0.13856706023216248,
-0.4083024859428406,
0.37805381417274475,
-0.3301253616809845,
-0.05349422246217728,
-0.0005528198089450598,
0.3316859006881714,
-0.2890911400318146,
0.12900424003601074,
0.07703325897455215,
0.029778340831398964,
0.5202839374542236,
-0.3436846137046814,
0.14734983444213867,
-0.12503336369991302,
-0.1609780341386795,
-0.4426017701625824,
0.16400699317455292,
-0.5763194561004639,
0.011583095416426659,
0.236427441239357,
0.09150415658950806,
-0.21794071793556213,
-0.67563396692276,
-0.003913114778697491,
0.3651498556137085,
0.041120801120996475,
-0.043318115174770355,
0.37885308265686035,
0.1521008461713791,
-0.05675728991627693,
-0.1849963665008545,
-0.16387112438678741,
-0.07045664638280869,
-0.26008912920951843,
-0.18308697640895844,
-0.13613860309123993,
-0.5506133437156677,
0.699690580368042,
-0.02900528721511364,
0.32919588685035706,
-0.21145828068256378,
-0.5253312587738037,
0.49715372920036316,
-0.46184423565864563,
-0.6630075573921204,
-0.4431150555610657,
0.6107194423675537,
-1.0025100708007812,
-0.2660142779350281,
0.003220274578779936,
0.7211582660675049,
0.36188387870788574,
0.21470916271209717,
-0.7509734630584717,
0.03932269662618637,
-0.21889765560626984,
-0.4953267276287079,
-0.153963103890419,
-0.029238449409604073,
0.029630903154611588,
0.6244574189186096,
0.11096464842557907,
-0.13236667215824127,
0.1425989717245102,
0.8705742955207825,
0.07963421195745468,
0.039835669100284576,
0.27399295568466187,
-0.09812210500240326,
-0.23277045786380768,
0.2657238245010376,
-0.3048689663410187,
0.299174427986145,
0.18532830476760864,
-0.1365688294172287,
-0.6661547422409058,
0.49333909153938293,
-5.752726078033447,
0.2563088536262512,
-0.5734378099441528,
0.4201498329639435,
-0.19127380847930908,
-0.08351612836122513,
0.4935729205608368,
-0.14867933094501495,
0.23751486837863922,
-0.1275428831577301,
0.05532117560505867,
0.3939991593360901,
-0.41842934489250183,
0.3972698450088501,
0.343183696269989,
0.23182760179042816,
0.11563534289598465,
0.16662998497486115,
0.36554965376853943,
0.34869271516799927,
-0.334542840719223,
-0.21993249654769897,
0.007862026803195477,
0.34862038493156433,
0.2787288725376129,
0.2527247965335846,
-0.7824224829673767,
0.7310048937797546,
-0.2497754842042923,
-0.3956330120563507,
0.020774833858013153,
-0.5755307078361511,
-0.4342283606529236,
-0.10560353845357895,
-0.024341918528079987,
0.7426250576972961,
0.4276772737503052,
0.09272988885641098,
0.3122076094150543,
-0.11711324006319046,
0.4367738962173462,
0.6854677200317383,
0.3529432713985443,
-0.20643416047096252,
0.5437456965446472,
-0.27369746565818787,
-0.28674137592315674,
0.4170740246772766,
-0.17174215614795685,
0.6447171568870544,
0.004704437684267759,
-0.5771872401237488,
0.18443024158477783,
0.3498014807701111,
-0.11226445436477661,
-0.33001288771629333,
0.4704021215438843,
0.11194614320993423,
-0.30208879709243774,
0.7165594100952148,
0.15991555154323578,
-0.49961620569229126,
0.23524968326091766,
-0.5952485203742981,
-0.7717292904853821,
0.29880475997924805,
-0.7107008099555969,
-0.3655780255794525,
-0.04747334495186806,
-0.05324378237128258,
-0.8819102644920349,
0.20859190821647644,
0.41508495807647705,
-0.5721479654312134,
0.2444494366645813,
-0.5500245094299316,
-0.05203741043806076,
-0.3202815651893616,
-0.1841793954372406,
0.5136164426803589,
-0.32338637113571167,
-0.4996389150619507,
-0.12949486076831818,
0.10423827171325684,
0.11965912580490112,
-0.1878894716501236,
0.06545493751764297,
-0.413423627614975,
-0.4370339810848236,
-0.064669169485569,
0.28182896971702576,
-0.1515929251909256,
0.44191208481788635,
0.4573027193546295,
0.17125076055526733,
0.47070029377937317,
0.29342120885849,
0.20741675794124603,
0.6478175520896912,
0.1864505112171173,
0.73613440990448,
-0.1517200767993927,
-0.0742843747138977,
-1.0708924531936646,
-0.1407918930053711,
0.18675218522548676,
-0.16595511138439178,
0.5333982110023499,
0.3865598440170288,
-0.28783196210861206,
-0.19296063482761383,
0.35878947377204895,
-0.5050780177116394,
-0.47192493081092834,
0.6709277033805847,
0.06393994390964508,
-0.5467854738235474,
-0.15316425263881683,
0.4106713533401489,
-0.19970299303531647,
0.23953047394752502,
-0.07675410807132721,
0.3157466948032379,
0.0902043879032135,
0.438188761472702,
-0.33919402956962585,
-0.020222656428813934,
0.46287593245506287,
-0.5287419557571411,
-0.1191416084766388,
0.2667490839958191,
0.5844376683235168,
-0.23248720169067383,
0.1628938913345337,
0.5558721423149109,
-0.6003542542457581,
-0.3590570092201233,
-0.06943339109420776,
0.6072268486022949,
-0.3883421719074249,
0.8620442152023315,
0.03383348137140274,
-0.680314302444458,
0.051237691193819046,
0.20156484842300415,
0.2659085690975189,
-0.41880351305007935,
-0.31638002395629883,
-0.46332868933677673,
0.14844445884227753,
0.04230117425322533,
-0.5655865669250488,
0.5162988901138306,
-0.31989336013793945,
-0.06750258803367615,
0.23531247675418854,
0.3037774860858917,
0.5172474384307861,
0.07859703153371811,
-0.7314696907997131,
0.5918770432472229,
0.1294156163930893,
-0.5507903695106506,
-0.13962869346141815,
0.045944008976221085,
-0.37319549918174744,
-0.31401821970939636,
-0.28728920221328735,
-0.3789336383342743,
0.4730199873447418,
-0.35094159841537476,
-0.13904498517513275,
-0.2992785573005676,
-0.6602189540863037,
0.09456298500299454,
0.09159494191408157,
0.1642526388168335,
-0.0813046395778656,
0.04549869894981384,
-0.04697243869304657,
0.1922532320022583,
-0.21204733848571777,
-0.11502402275800705,
0.6084604859352112,
0.8605835437774658,
0.12580858170986176,
0.19548280537128448,
-0.16208422183990479,
-0.7642789483070374,
-0.4862135946750641,
-0.030488884076476097,
0.4369363784790039,
0.5905080437660217,
0.17624631524085999,
0.37605756521224976,
-0.6294404864311218,
0.7403172254562378,
0.29779621958732605,
-0.7004125118255615,
-0.08196976035833359,
-0.1933150589466095,
0.5205219388008118,
-0.3879697620868683,
0.8700122833251953,
0.23549818992614746,
-0.4235478937625885,
0.01297783199697733,
-0.11658626049757004,
-0.4491291344165802,
-0.13627950847148895,
-0.5823873281478882,
-0.5203951597213745,
-0.1205497682094574,
-0.3178568184375763,
-0.3170064091682434,
-0.10308519005775452,
0.38296693563461304,
-0.15503734350204468,
-1.0838088989257812,
0.38704240322113037,
0.2775145173072815,
-0.17559345066547394,
0.016447538509964943,
0.39310240745544434,
-0.08522983640432358,
0.16329436004161835,
0.40819185972213745,
-0.41944190859794617,
-0.055328618735075,
-0.18576475977897644,
-0.294179767370224,
-0.20605862140655518,
0.06620953977108002,
0.5276350975036621,
-0.5500794053077698,
-0.04700278490781784,
0.1830328404903412,
0.2930486798286438,
-0.6599817276000977,
-0.1603533774614334,
-0.07547077536582947,
0.205404594540596,
-0.1443592756986618,
-0.6380285620689392,
-0.46795815229415894,
-0.013436054810881615,
0.064938984811306,
0.16849829256534576,
-0.6319620013237,
-0.2775641679763794,
-0.2028752714395523,
-0.3286798596382141,
0.41373303532600403,
-0.10363209992647171,
0.4126497805118561,
0.06573895364999771,
-0.12654079496860504,
-0.7467417120933533,
-0.08778925240039825,
0.30980002880096436,
-0.01556719932705164,
-0.2356465458869934,
0.07110843807458878,
-0.4134540557861328,
-0.21533282101154327,
0.10175488144159317,
-0.014854797162115574,
-0.7259693741798401,
0.3725404739379883,
0.15228703618049622,
-0.6785625219345093,
-0.12430576980113983,
0.2550269365310669,
0.610991358757019,
-0.14193688333034515,
-0.07747918367385864,
-0.1949886828660965,
0.3032435476779938,
-0.30223774909973145,
-0.040857601910829544,
-0.7988312244415283,
-0.32453492283821106,
-0.24860365688800812,
-0.3978240489959717,
0.049570631235837936,
-0.0971570760011673,
0.2544940114021301,
0.4737483859062195,
0.21563251316547394,
0.4287559688091278,
-0.3995305895805359,
0.2402130514383316,
-0.6132065057754517,
-0.35685956478118896,
-0.4008384644985199,
0.29361197352409363,
-0.08138737827539444,
0.01669088937342167,
0.08894043415784836,
-0.2773703634738922,
-0.3508441746234894,
0.4733680486679077,
0.2963251769542694,
0.04900137707591057,
0.17601412534713745,
0.3251346945762634,
0.40082502365112305,
0.1535203456878662,
-0.21251443028450012,
-0.17200440168380737,
0.2656957507133484,
-0.1806585043668747,
0.13850802183151245,
0.7628379464149475,
-0.2988147735595703,
-0.0434817299246788,
0.3490268588066101,
-0.12589114904403687,
0.3735201358795166,
0.4278661906719208,
-0.06035934016108513,
0.24614758789539337,
0.6556370854377747,
-0.3520101308822632,
-0.5786155462265015,
0.6587966084480286,
-0.22338610887527466,
0.060704465955495834,
0.12429602444171906,
0.266948401927948,
0.28182125091552734,
0.6125726103782654,
-0.6377881169319153,
0.23990841209888458,
0.33752185106277466,
0.2605356276035309,
0.4399181008338928,
-0.9649989008903503,
-0.16297316551208496,
-0.05930011719465256,
0.4884170889854431,
0.9590081572532654,
-1.222274899482727,
0.4818962812423706,
-0.1280614584684372,
0.14410527050495148,
0.1808837652206421,
0.24115313589572906,
-0.307729572057724,
0.20194534957408905,
-0.09725423902273178,
0.2841395139694214,
-0.28686049580574036,
-0.38480159640312195,
-0.6054216027259827,
-0.12342865020036697,
0.009480026550590992,
-0.3300534784793854,
-0.41984033584594727,
0.04091275855898857,
0.5793768167495728,
0.3496861159801483,
-0.4205232262611389,
0.07353050261735916,
-0.4044046998023987,
-0.10084529966115952,
0.5874797701835632,
-0.16384704411029816,
0.06794648617506027,
-0.4580349922180176,
-0.4364857077598572,
-0.030791549012064934,
-0.17894209921360016,
0.5338842272758484,
0.19688710570335388,
-0.8400170207023621,
0.20849938690662384,
0.004638360347598791,
-0.16016846895217896,
0.15966035425662994,
-0.040280550718307495,
0.1953224241733551,
-0.1333247423171997,
-0.14712966978549957,
0.06006758287549019,
-0.6052685976028442,
0.15944461524486542,
0.5185195207595825,
0.10194401443004608,
-0.3161519467830658,
-0.03348071873188019,
-0.493207722902298,
-0.12746581435203552,
0.5034940242767334,
-0.20217376947402954,
-0.1914653480052948,
-0.7699783444404602,
0.775296688079834,
0.6628093123435974,
0.054388828575611115,
0.24744251370429993,
-0.03941386193037033,
0.09015434235334396,
0.6665013432502747,
-0.40840283036231995,
-0.4729721248149872,
-0.539863109588623,
0.22063952684402466,
-1.0210171937942505,
-0.009221291169524193,
0.4304967224597931,
-0.07784143090248108,
-0.5383496284484863,
-0.0766417533159256,
-0.04923277720808983,
0.2871988117694855,
0.5343298316001892,
0.03043450228869915,
-0.07855571061372757,
-0.3915955424308777,
-0.20107418298721313,
-0.505213737487793,
0.49412739276885986,
0.004973494913429022,
0.2474188506603241,
0.18788985908031464,
0.2427811175584793,
-0.059541501104831696,
-0.24917691946029663,
0.05967333912849426,
0.15625284612178802,
-0.10571124404668808,
-0.5122657418251038,
0.7147164940834045,
-0.3426826298236847,
0.48514682054519653,
0.28818753361701965,
-0.007869507186114788,
0.5305665731430054,
-0.5212107300758362,
-0.47418779134750366,
0.42393139004707336,
-0.42652347683906555,
0.489030122756958,
-0.23954902589321136,
0.9261044263839722,
-0.4936170279979706,
-0.42176997661590576,
-0.3960981070995331,
0.6010399460792542,
0.21348753571510315,
-0.22769300639629364,
0.18129277229309082,
-0.6003263592720032,
-0.22920826077461243,
0.4829884469509125,
-0.3277275264263153,
-0.6699910163879395,
-0.037693001329898834,
-0.1978628784418106
] |
works are either unable to pinpoint the exact asymptotic estimation error or, when
they do so, the resulting approaches (e.g., based on whitening or singular value
shrinkage) remain vastly suboptimal. On top of this, most of the literature has
focused on the special case of estimating the left singular vector of the signal
when the noise only possesses row correlation (one-sided heteroscedasticity). In
contrast, our work establishes the information-theoretic and algorithmic limits of | [
0.10198456794023514,
-0.07098908722400665,
-0.10941815376281738,
0.17469044029712677,
-0.20277489721775055,
0.18335001170635223,
-0.08232486248016357,
-0.40242359042167664,
-0.027722947299480438,
-0.3925609588623047,
-0.1938684731721878,
0.42303913831710815,
0.030774664133787155,
0.19402751326560974,
-0.4012101888656616,
0.4081006944179535,
0.4491458535194397,
-0.1867743730545044,
-0.3209879398345947,
-0.39574575424194336,
-0.11773523688316345,
0.21490035951137543,
0.11636950075626373,
0.0935370922088623,
-0.11027359217405319,
0.01680084317922592,
0.22569867968559265,
-0.09012704342603683,
-0.2710250914096832,
-0.016011130064725876,
-0.28156718611717224,
0.5556613206863403,
-0.2046326845884323,
-0.34086376428604126,
-0.010716181248426437,
0.12126922607421875,
0.007241602521389723,
0.3101658523082733,
0.23829665780067444,
0.28599730134010315,
0.1553773581981659,
0.04022275283932686,
-0.04217606037855148,
0.28508809208869934,
-0.3638077676296234,
-0.6456391215324402,
-1.5326645374298096,
0.201498344540596,
-0.32720229029655457,
0.04266251251101494,
-0.01864386536180973,
0.28582096099853516,
0.5657514333724976,
0.7132120728492737,
-0.35899800062179565,
0.1370498538017273,
0.12594257295131683,
-0.23257748782634735,
0.41337358951568604,
-0.42362236976623535,
0.28352880477905273,
0.02496812865138054,
0.020710403099656105,
0.10147888213396072,
0.08918890357017517,
0.35299280285835266,
0.18070578575134277,
-0.26805081963539124,
-0.846982479095459,
0.062499597668647766,
-0.04839670658111572,
-0.32503509521484375,
-0.13565407693386078,
0.1288309544324875,
-0.28259071707725525,
0.040958303958177567,
0.3807083070278168,
-0.05011601746082306,
-0.08205008506774902,
-0.21488022804260254,
-0.09808985143899918,
0.21748031675815582,
0.1769416481256485,
-0.28068768978118896,
0.6879643201828003,
0.2332196980714798,
-0.18094418942928314,
0.510020911693573,
-0.7129655480384827,
0.16084954142570496,
0.30671149492263794,
-0.44142720103263855,
0.10373488068580627,
0.17382706701755524,
0.2347726672887802,
0.5530081391334534,
0.5906738042831421,
0.050405316054821014,
0.11356762796640396,
0.315189927816391,
-0.24817050993442535,
0.053156547248363495,
0.4997802674770355,
-0.13398562371730804,
-0.08827479183673859,
-0.570354163646698,
-0.1280536949634552,
-0.1291002631187439,
-0.25669559836387634,
0.2950209081172943,
-0.57581627368927,
0.6082936525344849,
-0.04164260998368263,
-0.18679043650627136,
-0.4193473160266876,
-0.1804388165473938,
-0.15113775432109833,
0.1830531805753708,
-0.24054022133350372,
0.3789684474468231,
-0.07990501075983047,
1.387617826461792,
-0.13695020973682404,
0.03406878560781479,
0.13240312039852142,
-0.08933913707733154,
0.67628014087677,
0.5783507823944092,
-0.25613927841186523,
0.1717493236064911,
-0.04350527003407478,
0.8991203308105469,
-0.09347186237573624,
0.09660657495260239,
-0.2412450760602951,
-0.3700510561466217,
0.3065982460975647,
0.40307745337486267,
-0.3048173785209656,
-0.043062057346105576,
-0.5050429701805115,
-0.6203889846801758,
-0.21251463890075684,
0.291000634431839,
0.1816994696855545,
-0.41624951362609863,
0.08601617068052292,
-0.049357347190380096,
-0.06777799874544144,
-0.11908253282308578,
-0.10634279251098633,
0.40181005001068115,
-0.8337227702140808,
-0.1548006236553192,
-0.08855780959129333,
0.4002702534198761,
0.37291091680526733,
-0.2565711438655853,
-0.21029800176620483,
0.30963531136512756,
0.074672631919384,
-0.11121103912591934,
-0.2259521484375,
-0.31810933351516724,
0.16399472951889038,
-0.836650550365448,
0.15277796983718872,
0.6376853585243225,
-0.26399123668670654,
-0.030341260135173798,
0.20838317275047302,
-0.1918189525604248,
0.014156586490571499,
0.030506093055009842,
-0.24661573767662048,
0.25838804244995117,
0.17810457944869995,
0.17528174817562103,
0.42890462279319763,
0.36244329810142517,
-0.6246798038482666,
0.1078539714217186,
0.1505967378616333,
-0.23862224817276,
0.006047192495316267,
0.24393364787101746,
0.2043725699186325,
-0.153443843126297,
-0.28708595037460327,
-0.144603431224823,
-0.2893025279045105,
-0.33204466104507446,
-0.13827502727508545,
-0.8500648736953735,
1.1153326034545898,
0.20180641114711761,
-0.03333982452750206,
1.385824203491211,
-0.0928652212023735,
0.1725492924451828,
0.15252147614955902,
-0.16864384710788727,
-0.26438233256340027,
0.10977146029472351,
-0.6863144040107727,
0.434757262468338,
-0.014076088555157185,
0.2063843160867691,
-0.4643794894218445,
-0.030365606769919395,
0.7226505279541016,
-0.2187422811985016,
-0.13071274757385254,
-0.06814947724342346,
-0.10707900673151016,
0.3362685739994049,
-0.30168408155441284,
0.04440390691161156,
0.11591845005750656,
-0.15361741185188293,
0.9367818236351013,
-0.139556422829628,
0.7236201763153076,
0.28745585680007935,
0.15422339737415314,
0.5234225392341614,
0.6741393804550171,
0.49371862411499023,
0.7008927464485168,
-0.4725217819213867,
0.02334611304104328,
-0.15806664526462555,
0.2144555151462555,
-0.16507108509540558,
-0.24436317384243011,
0.2345873862504959,
0.07829787582159042,
-0.15709885954856873,
0.08791627734899521,
-0.02378137595951557,
-0.18178442120552063,
0.21058033406734467,
-0.2790723145008087,
0.39147859811782837,
0.12266110628843307,
-0.5273685455322266,
-0.07494419813156128,
0.1681542694568634,
-0.4218231737613678,
0.1858002245426178,
0.02697879821062088,
0.5758964419364929,
-0.03840702772140503,
-0.6620877385139465,
0.15439796447753906,
0.019048474729061127,
-0.13578221201896667,
-0.41002678871154785,
0.3806915581226349,
0.21867643296718597,
0.2163994014263153,
-0.02810617722570896,
-0.011294106021523476,
-0.1869913786649704,
-0.4178352952003479,
-0.35010087490081787,
-0.6937172412872314,
-0.33874818682670593,
0.23792526125907898,
0.25934943556785583,
0.41076159477233887,
-0.10137250274419785,
-0.4130764901638031,
0.11182703077793121,
-0.46440473198890686,
-0.7307001352310181,
0.04130789265036583,
0.17861011624336243,
-0.45073169469833374,
-0.47290319204330444,
0.41841378808021545,
0.516535222530365,
0.552038311958313,
-0.15281429886817932,
-0.34891045093536377,
-0.20593352615833282,
-0.1684684455394745,
-0.11841201782226562,
0.054687388241291046,
-0.002300701569765806,
-0.40054187178611755,
0.4002441465854645,
-0.31255197525024414,
-0.5799943804740906,
0.20674802362918854,
0.7979512810707092,
-0.04347674176096916,
-0.10527002811431885,
-0.02063247561454773,
-0.3575931191444397,
-0.48998692631721497,
0.01700431853532791,
0.09433624893426895,
0.024313243106007576,
-0.25282853841781616,
-0.09043999761343002,
-0.3894098699092865,
0.39947250485420227,
-6.230367183685303,
0.04590434953570366,
-0.31697091460227966,
0.3812812566757202,
-0.31949448585510254,
-0.06643617898225784,
0.7087276577949524,
-0.04198276624083519,
0.08576152473688126,
-0.18494375050067902,
0.19217632710933685,
0.17779098451137543,
-0.10587484389543533,
0.6245360374450684,
0.38657331466674805,
-0.2089749127626419,
-0.15057803690433502,
0.18505176901817322,
0.3693591058254242,
0.15259569883346558,
0.11113335192203522,
-0.17945924401283264,
0.09453275799751282,
0.35383862257003784,
0.08094082772731781,
0.37127745151519775,
-0.812505304813385,
0.2542478144168854,
-0.38711899518966675,
-0.24077874422073364,
-0.028384367004036903,
-0.7084164619445801,
-0.4934537708759308,
-0.18038010597229004,
0.6025673151016235,
0.24165359139442444,
0.22184647619724274,
0.05428428575396538,
-0.017811788246035576,
-0.04642230644822121,
0.3580097258090973,
0.6316661238670349,
-0.2965502142906189,
0.05100551247596741,
0.2434544414281845,
0.02573424205183983,
-0.2646853029727936,
0.18217673897743225,
-0.1249593123793602,
0.36656099557876587,
-0.4483609199523926,
-0.10463892668485641,
0.35428085923194885,
-0.015864655375480652,
0.10927829891443253,
-0.5266703963279724,
0.2490832805633545,
0.10743066668510437,
-0.08194921165704727,
-0.00604677852243185,
0.10969014465808868,
-0.42716699838638306,
0.07330837100744247,
-0.6501731276512146,
-0.45114773511886597,
0.048150524497032166,
-0.3595251441001892,
-0.5715449452400208,
0.29620084166526794,
-0.12763239443302155,
-0.2945522665977478,
-0.0917978435754776,
0.4565763473510742,
-0.35011041164398193,
0.2600807249546051,
-0.40792661905288696,
0.08256901055574417,
-0.0017820231150835752,
-0.02195257879793644,
0.21800300478935242,
-0.7822232246398926,
-0.5495809316635132,
0.3522036075592041,
-0.23152965307235718,
0.03416096419095993,
-0.4570470452308655,
-0.18665747344493866,
0.04831928387284279,
-0.2902364134788513,
0.4074443578720093,
0.6777036786079407,
-0.1325572431087494,
0.16280697286128998,
0.42372941970825195,
0.5941920280456543,
0.6729696393013,
0.5315786600112915,
0.36806419491767883,
0.429514616727829,
-0.012824657373130322,
0.41063451766967773,
-0.4071553945541382,
-0.34418100118637085,
-0.7676900029182434,
0.007877944968640804,
0.3349393308162689,
-0.8106896877288818,
0.24869273602962494,
0.6227991580963135,
-0.7637715935707092,
0.17040491104125977,
0.13879238069057465,
0.014961942099034786,
-0.4779859185218811,
0.5004932284355164,
0.21587291359901428,
-0.22151853144168854,
-0.08010238409042358,
0.055945832282304764,
-0.3066321015357971,
0.1385958343744278,
-0.1337762475013733,
-0.26444000005722046,
0.1038239598274231,
0.2647787034511566,
0.01046792883425951,
0.09351325780153275,
0.6548379063606262,
-0.6358793377876282,
-0.18683718144893646,
0.28033173084259033,
0.49072831869125366,
-0.5301846861839294,
0.425711065530777,
0.1887732446193695,
-0.8335615396499634,
0.07313686609268188,
0.5970845222473145,
0.2783052325248718,
-0.4691931903362274,
0.7296583652496338,
-0.0373593345284462,
-0.2512090504169464,
0.03804782032966614,
-0.3248438835144043,
0.11584550887346268,
-0.25157323479652405,
-0.37093985080718994,
-0.18339699506759644,
-0.115694560110569,
0.0965169295668602,
-0.42620596289634705,
0.01735536754131317,
-0.1295546591281891,
-0.022829463705420494,
0.14247608184814453,
0.24459043145179749,
0.15442556142807007,
0.036167386919260025,
0.009155101142823696,
-0.05276760831475258,
0.1382603943347931,
-0.896895706653595,
-0.29679030179977417,
0.2539947032928467,
-0.3746935725212097,
-0.6484785079956055,
0.0390126034617424,
0.0037562681827694178,
0.16926084458827972,
-0.06932855397462845,
-0.8400196433067322,
-0.1067177951335907,
-0.3524833023548126,
0.12571486830711365,
0.19519376754760742,
-0.027847927063703537,
-0.17500212788581848,
-0.024670839309692383,
-0.35253801941871643,
0.15596142411231995,
-0.34513434767723083,
-0.18170848488807678,
0.6038717031478882,
0.7953197956085205,
-0.16061560809612274,
0.18233591318130493,
0.126750648021698,
-0.44113776087760925,
0.07896914333105087,
0.06265675276517868,
0.1838623434305191,
0.3533381223678589,
0.06407497823238373,
-0.11272701621055603,
-0.010943155735731125,
0.46439245343208313,
-0.13335822522640228,
-0.3154504597187042,
-0.3356129229068756,
0.1326368749141693,
0.21251225471496582,
-0.43270742893218994,
0.19739842414855957,
0.31633374094963074,
-0.3704532980918884,
0.09173673391342163,
0.025796975940465927,
-0.344059020280838,
-0.45927491784095764,
-0.7999804615974426,
-0.2309880405664444,
-0.18486103415489197,
-0.0073762740939855576,
-0.055050525814294815,
-0.08518347144126892,
0.5304009318351746,
-0.1685289740562439,
-0.1783803254365921,
0.4264349639415741,
0.2654215693473816,
-0.04693453758955002,
-0.45709043741226196,
0.17436562478542328,
0.017549730837345123,
0.24174469709396362,
0.014483988285064697,
-0.5874846577644348,
0.2650447189807892,
-0.2501160502433777,
-0.35019636154174805,
-0.1260487139225006,
-0.33727189898490906,
0.23607631027698517,
-0.02753298729658127,
-0.17588672041893005,
-0.1728900820016861,
0.35350310802459717,
-0.28261420130729675,
-0.20085284113883972,
0.4452378451824188,
0.6423812508583069,
-0.40286383032798767,
0.2805274426937103,
-0.14742986857891083,
0.11460278183221817,
0.30776259303092957,
-0.6099740266799927,
-0.5904015898704529,
-0.22529150545597076,
0.08318191766738892,
0.017142483964562416,
-0.13507740199565887,
0.3160180151462555,
0.6934385895729065,
0.26277580857276917,
-0.20788772404193878,
-0.8497504591941833,
0.07931111007928848,
0.38991641998291016,
-0.16733837127685547,
-0.6732432842254639,
-0.08590764552354813,
-0.533312976360321,
0.21108517050743103,
0.1400928497314453,
-0.362491250038147,
-0.701187252998352,
0.4339294731616974,
-0.3174116611480713,
-0.10279706865549088,
-0.14896225929260254,
0.53775954246521,
0.167587012052536,
0.08215026557445526,
0.2509651780128479,
-0.22206330299377441,
0.3018544316291809,
-0.12992116808891296,
0.03985012322664261,
-0.05166464298963547,
-0.601463258266449,
-0.18404521048069,
-0.46965616941452026,
-0.11682373285293579,
-0.03143167495727539,
0.7243485450744629,
-0.15131263434886932,
-0.0041085039265453815,
-0.268828809261322,
-0.3577631413936615,
0.0765649601817131,
-0.2098570317029953,
-0.2725694179534912,
-0.27916961908340454,
-0.0552995540201664,
0.40930426120758057,
0.2305234670639038,
0.0781029611825943,
-0.3483193516731262,
-0.2800765037536621,
0.09376557916402817,
0.36608821153640747,
-0.6262672543525696,
0.5029860138893127,
0.2075386941432953,
0.021260971203446388,
0.2112317830324173,
0.0662122294306755,
0.17949411273002625,
0.09665540605783463,
0.119181327521801,
0.01783967949450016,
0.862350583076477,
-0.044133346527814865,
-0.10308796167373657,
0.3615507483482361,
-0.17241628468036652,
0.9281774163246155,
0.18302510678768158,
-0.12171579152345657,
0.012828941456973553,
0.48430904746055603,
-0.30261150002479553,
-0.5697779059410095,
0.4525846540927887,
0.06487718969583511,
0.2796783149242401,
-0.00756516819819808,
0.4556253254413605,
0.3985455632209778,
0.6161581873893738,
-0.21134185791015625,
0.19243748486042023,
-0.1342700719833374,
-0.17922896146774292,
0.5198668837547302,
-0.454873263835907,
-0.2536337971687317,
-0.026148496195673943,
0.32325151562690735,
0.18163160979747772,
-0.14395681023597717,
0.5449929237365723,
-0.12304824590682983,
0.2825181782245636,
0.4553818106651306,
0.37393918633461,
-0.748732328414917,
0.23385119438171387,
0.1260007917881012,
0.11487777531147003,
0.12648475170135498,
-0.020866254344582558,
0.046720992773771286,
-0.04821543022990227,
0.6731706857681274,
-0.2785661518573761,
-0.1491997241973877,
0.41335761547088623,
0.3806791603565216,
0.8423882722854614,
-0.45163094997406006,
-0.142160102725029,
-0.11713211238384247,
-0.2127147614955902,
0.7124603986740112,
0.15294592082500458,
-0.035848796367645264,
-0.19849365949630737,
-0.04847155138850212,
0.0013821476604789495,
-0.470935583114624,
0.33036723732948303,
-0.02777749113738537,
-0.37200334668159485,
0.47128012776374817,
0.35428524017333984,
-0.18893815577030182,
-0.0930325984954834,
0.24257804453372955,
0.011031391099095345,
-0.2972267270088196,
-0.31266355514526367,
0.37330201268196106,
-0.3959585130214691,
-0.001754849567078054,
0.4437366724014282,
-0.07536453753709793,
0.07171415537595749,
-0.18769878149032593,
-0.7953563928604126,
-0.5864382386207581,
0.3177444636821747,
0.03645264357328415,
-0.19155634939670563,
-0.3616488575935364,
0.7518115639686584,
0.050320930778980255,
-0.06985501199960709,
0.050359733402729034,
0.18725213408470154,
-0.22511626780033112,
0.9290611147880554,
-0.5687856078147888,
0.48281803727149963,
-0.27445241808891296,
0.18333035707473755,
-0.6448562145233154,
0.07997307926416397,
0.20268532633781433,
0.08385838568210602,
-0.3978537619113922,
-0.20192359387874603,
-0.12554875016212463,
-0.4519153833389282,
0.05449202284216881,
-0.3255440592765808,
0.1533636897802353,
-0.4386025071144104,
-0.24356922507286072,
-0.4139062762260437,
0.8379230499267578,
-0.023068610578775406,
0.08807370066642761,
-0.1494680643081665,
0.386447548866272,
0.10178081691265106,
-0.3255871832370758,
0.19059258699417114,
-0.012862509116530418,
0.2779631018638611,
-0.44033563137054443,
0.5372949838638306,
-0.1526065468788147,
-0.03130442276597023,
-0.006615683436393738,
-0.2149394154548645,
0.3695368468761444,
-0.44498181343078613,
-0.3286844789981842,
-0.09234706312417984,
-0.3186482787132263,
0.570787787437439,
0.03475014492869377,
0.3837195336818695,
-0.2372053563594818,
-0.17006395757198334,
-0.2080419510602951,
0.676157534122467,
-0.1730046272277832,
0.17256596684455872,
-0.3124096989631653,
-0.2773163616657257,
-0.6102350354194641,
0.30497419834136963,
0.11060057580471039,
0.039547063410282135,
0.014630724675953388,
0.05396923050284386
] |
contrast, our work establishes the information-theoretic and algorithmic limits of
matrix denoising with doubly heteroscedastic noise. We characterize the exact
asymptotic minimum mean square error, and design a novel spectral estimator
with rigorous optimality guarantees: under a technical condition, it attains positive
correlation with the signals whenever information-theoretically possible and, for
one-sided heteroscedasticity, it also achieves the Bayes-optimal error. Numerical | [
0.002928248606622219,
-0.04070032015442848,
0.0004898083861917257,
0.11757022142410278,
-0.20508430898189545,
0.4052019417285919,
-0.07350008934736252,
-0.6545189619064331,
-0.08046606928110123,
-0.6584310531616211,
-0.10916861146688461,
0.6453147530555725,
-0.22182048857212067,
0.10613565146923065,
-0.032830819487571716,
0.36291587352752686,
0.4566410481929779,
0.12530609965324402,
-0.2768714427947998,
-0.6451574563980103,
0.02450590580701828,
0.10539393872022629,
0.3571849763393402,
0.21329039335250854,
0.17843163013458252,
-0.024816909804940224,
0.5389761328697205,
0.20910468697547913,
0.04716368019580841,
-0.03716017305850983,
-0.09922084957361221,
0.588423490524292,
-0.3224141597747803,
-0.22625726461410522,
0.5024254322052002,
0.23847217857837677,
0.374532550573349,
0.14531660079956055,
0.09466133266687393,
0.02191815711557865,
0.5258058905601501,
0.11105062067508698,
-0.10331809520721436,
0.2920379042625427,
-0.533433198928833,
-0.9061430096626282,
-1.4238512516021729,
-0.008365224115550518,
-0.08975367248058319,
-0.7969328761100769,
0.027027327567338943,
0.4088851511478424,
0.610357403755188,
0.3998492956161499,
-0.3439096212387085,
0.30120140314102173,
-0.2678171992301941,
-0.3273363709449768,
0.553771436214447,
-0.21862804889678955,
-0.05800040066242218,
-0.3678363263607025,
0.0029069934971630573,
0.1337900608778,
0.07299914956092834,
0.645106852054596,
0.4570142924785614,
-0.3586013615131378,
-0.5887126922607422,
-0.0005214391276240349,
0.02202264964580536,
-0.23633567988872528,
-0.3620968461036682,
0.06343435496091843,
-0.49955374002456665,
-0.006921194028109312,
-0.1107480600476265,
0.05579546093940735,
0.05637767165899277,
0.07493101805448532,
0.0906265377998352,
-0.1463904082775116,
0.3259678781032562,
-0.3635350167751312,
0.49504759907722473,
0.4521825313568115,
0.14922140538692474,
0.5230161547660828,
-0.9793029427528381,
0.4245513081550598,
-0.2543160617351532,
-0.15759426355361938,
0.06485560536384583,
0.2618640065193176,
0.45912498235702515,
0.8119608163833618,
0.31977036595344543,
-0.07792406529188156,
0.32346174120903015,
0.3699411451816559,
-0.2437180131673813,
-0.3037142753601074,
-0.08611398935317993,
-0.20853406190872192,
-0.23289361596107483,
-0.7993447780609131,
-0.5399729609489441,
-0.17713496088981628,
-0.5131409764289856,
0.11052802205085754,
-0.40766996145248413,
0.7789282202720642,
0.2114693969488144,
-0.2092539370059967,
-0.05909879505634308,
-0.02327883429825306,
-0.16240954399108887,
0.19172526895999908,
-0.2688620984554291,
0.5147148966789246,
-0.12679685652256012,
1.3980368375778198,
-0.13608060777187347,
0.4307118058204651,
0.061910875141620636,
0.11689239740371704,
0.6994895339012146,
0.5832475423812866,
-0.4935964047908783,
-0.09122156351804733,
-0.0015305646229535341,
0.8325814604759216,
-0.00929749384522438,
0.3074720799922943,
-0.2620864808559418,
-0.5972859263420105,
-0.2129344791173935,
0.13487166166305542,
-0.32271695137023926,
-0.2477334886789322,
-0.3722308576107025,
-0.6720362305641174,
0.051844991743564606,
0.18226584792137146,
0.345797598361969,
-0.2041616290807724,
0.05199360474944115,
-0.28716546297073364,
-0.010669905692338943,
-0.027047935873270035,
-0.1718260794878006,
0.6440433859825134,
-1.0191686153411865,
-0.1818874329328537,
-0.43054652214050293,
0.1536916196346283,
0.06513752043247223,
0.12680961191654205,
-0.2679627239704132,
0.4664844274520874,
0.099936343729496,
0.03704581782221794,
-0.2873137891292572,
-0.37460073828697205,
0.012981999665498734,
-0.9512283205986023,
0.214503213763237,
0.45766472816467285,
-0.18985308706760406,
-0.04246515408158302,
-0.015860265120863914,
-0.4543568789958954,
-0.195214182138443,
-0.13125506043434143,
0.24105533957481384,
0.21459856629371643,
0.2636864185333252,
-0.023340405896306038,
0.22944042086601257,
0.33232203125953674,
-0.7582998871803284,
0.28511205315589905,
0.041516415774822235,
-0.3046550154685974,
0.010939066298305988,
0.10314253717660904,
0.3482232689857483,
-0.18339265882968903,
0.02401822805404663,
-0.11247711628675461,
-0.2732042968273163,
-0.01722763106226921,
0.2160954773426056,
-1.002411127090454,
1.0710432529449463,
0.21656498312950134,
-0.22328445315361023,
0.7510277628898621,
-0.04172196984291077,
0.18182402849197388,
-0.055917996913194656,
-0.3294779062271118,
0.15513238310813904,
-0.10260818153619766,
-0.44732287526130676,
0.35445651412010193,
-0.04810196906328201,
0.1756659597158432,
-0.6995986700057983,
0.1253208965063095,
0.5774956345558167,
0.06822021305561066,
0.04069601371884346,
-0.35712549090385437,
0.25653257966041565,
0.04271043464541435,
-0.16076670587062836,
0.31425154209136963,
0.0662810206413269,
0.03756776079535484,
0.6926976442337036,
-0.4536053240299225,
0.6283457279205322,
0.476837694644928,
0.3710668981075287,
0.771522045135498,
0.7777723670005798,
0.23330596089363098,
0.653515636920929,
-0.7701961994171143,
-0.07845764607191086,
-0.14938506484031677,
0.05743187665939331,
0.015890706330537796,
-0.3039531111717224,
0.3500133752822876,
0.2635508179664612,
-0.26355111598968506,
0.038832541555166245,
-0.15743090212345123,
-0.20744551718235016,
0.40244248509407043,
-0.32984215021133423,
0.3436567485332489,
-0.25691771507263184,
-0.4914981722831726,
-0.09761607646942139,
-0.026167623698711395,
-0.6792536973953247,
-0.09028247743844986,
0.03560112789273262,
0.19610200822353363,
0.314311683177948,
-0.6297429203987122,
-0.0708380788564682,
-0.2567031681537628,
-0.12457241863012314,
-0.3234561085700989,
0.3163664638996124,
0.19787757098674774,
0.1322273313999176,
-0.051203403621912,
0.07590098679065704,
-0.4517030417919159,
-0.2928221821784973,
-0.1469467580318451,
-0.7200900912284851,
-0.4673703610897064,
0.37685930728912354,
0.3651019334793091,
0.30747804045677185,
-0.4289851188659668,
-0.3593287169933319,
-0.0065046194940805435,
-0.22074787318706512,
-0.8598341345787048,
0.011163461022078991,
0.47479936480522156,
-0.8497986793518066,
-0.38312509655952454,
-0.011636598035693169,
0.7790162563323975,
0.0788213461637497,
0.1037197932600975,
-0.39141884446144104,
-0.05125012993812561,
-0.13718783855438232,
-0.30950430035591125,
-0.09829879552125931,
-0.24826687574386597,
-0.3980490267276764,
0.40342849493026733,
-0.1602560132741928,
-0.4479677081108093,
0.32236549258232117,
0.8907480239868164,
-0.23397144675254822,
0.04230625554919243,
0.2393673211336136,
-0.09793121367692947,
-0.24096785485744476,
0.16132481396198273,
0.00832012016326189,
0.01448124647140503,
-0.11355104297399521,
-0.3435690402984619,
-0.7064191102981567,
0.6992704272270203,
-6.075658321380615,
0.015255573205649853,
-0.3511667251586914,
0.42682120203971863,
-0.3000842332839966,
0.06506938487291336,
0.6619384288787842,
-0.13364385068416595,
0.14679844677448273,
-0.011973252519965172,
-0.05104687064886093,
0.1602109968662262,
-0.451009064912796,
0.5158994793891907,
0.5366553664207458,
-0.32515910267829895,
0.1427282989025116,
-0.023551488295197487,
0.4739401638507843,
0.19632771611213684,
-0.3850007653236389,
-0.337556391954422,
-0.25358307361602783,
-0.09008797258138657,
0.12330110371112823,
0.11514391750097275,
-0.6176921725273132,
0.3189687132835388,
-0.4631495177745819,
-0.19776760041713715,
-0.08263861387968063,
-0.6876490116119385,
-0.1866450160741806,
0.020538657903671265,
0.4987565577030182,
0.42646780610084534,
0.3790915310382843,
-0.11285315454006195,
0.14796513319015503,
-0.3574518859386444,
0.4053654968738556,
0.871008038520813,
-0.34275344014167786,
0.059014737606048584,
0.33843111991882324,
0.04140470176935196,
0.032431576400995255,
0.16686491668224335,
-0.13784591853618622,
0.63619065284729,
-0.21574528515338898,
-0.036725062876939774,
0.15741947293281555,
0.15110376477241516,
-0.24139253795146942,
-0.24359284341335297,
-0.17735114693641663,
-0.07753191888332367,
-0.11116808652877808,
0.1739981472492218,
0.1395300328731537,
-0.551256000995636,
0.2269095927476883,
-0.31261032819747925,
-0.3800850510597229,
0.0795576199889183,
-0.44493716955184937,
-0.5102311372756958,
0.1521703600883484,
-0.05674309656023979,
-0.13756132125854492,
-0.008449693210422993,
0.2825559079647064,
-0.28517019748687744,
0.17652681469917297,
-0.3087887465953827,
0.23829221725463867,
-0.013754408806562424,
0.07193701714277267,
0.0695088729262352,
-0.3083541989326477,
-0.662214994430542,
0.3578537702560425,
0.15677234530448914,
0.15849055349826813,
-0.339136004447937,
-0.049779973924160004,
-0.07528151571750641,
-0.4432549476623535,
-0.08325251936912537,
0.332683801651001,
-0.4129107892513275,
0.18903817236423492,
0.6512599587440491,
0.2742246687412262,
0.7534830570220947,
0.5285255312919617,
0.007198623847216368,
0.446284681558609,
0.06369181722402573,
0.6024748682975769,
-0.12942923605442047,
-0.26467302441596985,
-0.5548774600028992,
0.245774507522583,
0.568942129611969,
-0.2977791428565979,
0.47236284613609314,
0.532741904258728,
-0.4153389632701874,
-0.10925965756177902,
0.0858103334903717,
-0.2153586745262146,
-0.4957132339477539,
0.6183116436004639,
0.003977453801780939,
-0.2986410856246948,
-0.16208934783935547,
0.555392324924469,
0.0020434472244232893,
0.08620383590459824,
0.17156246304512024,
-0.11230531334877014,
0.24006159603595734,
0.4478173851966858,
-0.3270203769207001,
0.09893897920846939,
0.8319609761238098,
-0.49671098589897156,
-0.16686049103736877,
0.2799898087978363,
0.2998841106891632,
-0.5121570825576782,
0.2862588167190552,
0.4341963827610016,
-0.4979487657546997,
0.03416997566819191,
0.2746396064758301,
0.2990832030773163,
-0.38863837718963623,
0.882996678352356,
0.16311819851398468,
-0.37948787212371826,
-0.08645040541887283,
0.0005132809747010469,
0.0606197789311409,
-0.20013375580310822,
-0.1426180750131607,
-0.5342422723770142,
-0.29748597741127014,
-0.13414452970027924,
-0.5471625328063965,
0.227542445063591,
-0.24192385375499725,
0.07934711873531342,
0.04400898516178131,
0.38236203789711,
0.12092427164316177,
0.33690014481544495,
-0.33509930968284607,
0.3995450735092163,
0.13669469952583313,
-0.7958802580833435,
-0.21881070733070374,
-0.014408975839614868,
-0.027805935591459274,
-0.7185394167900085,
-0.2405257225036621,
-0.2126384675502777,
0.593009352684021,
-0.13695862889289856,
-0.5421008467674255,
-0.1985657513141632,
-0.5316510200500488,
0.18773441016674042,
0.01535357628017664,
0.06672285497188568,
-0.13835568726062775,
0.00897576566785574,
-0.34576964378356934,
0.3248730003833771,
-0.33126822113990784,
-0.42346426844596863,
0.8952066898345947,
1.1039352416992188,
0.2138555645942688,
0.27802443504333496,
0.021538419649004936,
-0.6802897453308105,
-0.21733830869197845,
0.15699660778045654,
0.2603103816509247,
0.37081626057624817,
-0.05297277495265007,
0.2813246250152588,
-0.1892397105693817,
0.20425613224506378,
0.057808954268693924,
-0.3495085537433624,
-0.18838560581207275,
-0.09360744059085846,
0.49526745080947876,
-0.43641895055770874,
0.3853774964809418,
0.11680018901824951,
-0.16970978677272797,
-0.10816457867622375,
0.12644152343273163,
-0.49655768275260925,
-0.4550642669200897,
-0.8124091625213623,
-0.6132400035858154,
-0.2614002823829651,
-0.22908222675323486,
-0.3792897164821625,
-0.02153146266937256,
0.4155389368534088,
-0.12630252540111542,
-0.11222852021455765,
0.45691269636154175,
0.4091966152191162,
-0.5006827712059021,
-0.12243962287902832,
0.142086461186409,
-0.0016740397550165653,
0.14808210730552673,
0.1392933875322342,
-0.4656528830528259,
0.11320187896490097,
-0.009205355308949947,
-0.3864043056964874,
-0.08396010845899582,
-0.12141365557909012,
0.13247467577457428,
-0.30745118856430054,
-0.22462521493434906,
-0.2919541895389557,
0.371383935213089,
-0.5351499319076538,
-0.2802491784095764,
0.1220710426568985,
0.6172195076942444,
-0.20540601015090942,
-0.20875541865825653,
-0.1581435650587082,
0.4697220027446747,
0.1899625062942505,
0.015842504799365997,
-0.8455647230148315,
-0.28167489171028137,
-0.14420345425605774,
0.0775497779250145,
0.11410000175237656,
0.05408571660518646,
0.4911956787109375,
0.2820567488670349,
0.03305350989103317,
-0.587508499622345,
-0.05141323059797287,
0.5097193717956543,
-0.26459062099456787,
-0.5773112177848816,
-0.2110966593027115,
-0.560943067073822,
0.10631793737411499,
0.30040737986564636,
-0.13947686553001404,
-0.5700423121452332,
0.15466469526290894,
0.07355747371912003,
-0.11045534908771515,
-0.2200651466846466,
0.17668889462947845,
0.5498532652854919,
-0.0037723819259554148,
-0.01929561421275139,
-0.09118711948394775,
0.6539586186408997,
-0.2544054090976715,
-0.0153691740706563,
-0.09647424519062042,
-0.3972523808479309,
-0.18013504147529602,
-0.42018264532089233,
0.13139860332012177,
0.23912395536899567,
0.556512176990509,
-0.05420254170894623,
0.29992374777793884,
0.06894462555646896,
-0.3218914866447449,
0.09564787894487381,
-0.5803848505020142,
-0.20596212148666382,
0.0298292338848114,
0.22034002840518951,
0.4743987023830414,
0.3815394639968872,
-0.019545074552297592,
-0.15198877453804016,
-0.4216209053993225,
0.5116136074066162,
0.2852098047733307,
-0.3626513183116913,
0.2690456211566925,
0.09317243099212646,
0.6239808201789856,
0.21864795684814453,
-0.10340196639299393,
0.014476056210696697,
0.30669888854026794,
0.2538253366947174,
-0.05432397127151489,
0.6499165892601013,
-0.06884486973285675,
0.14033053815364838,
0.6495224833488464,
-0.050965383648872375,
0.8947712779045105,
0.4174444377422333,
-0.21810127794742584,
-0.0631520226597786,
0.5992589592933655,
-0.414842814207077,
-0.8040891885757446,
0.6090281009674072,
-0.1435682773590088,
0.4785492718219757,
-0.26979097723960876,
0.274821937084198,
0.444580614566803,
0.8068851828575134,
-0.5225867033004761,
0.3249772787094116,
0.06918899714946747,
0.29935652017593384,
0.37316083908081055,
-0.5326821208000183,
-0.4047333002090454,
0.03470669314265251,
0.7465171813964844,
0.4946533739566803,
-0.6596207618713379,
0.338549941778183,
0.1976218968629837,
-0.07657139003276825,
0.2830944359302521,
0.19519008696079254,
-0.8157899975776672,
0.44947636127471924,
0.25839585065841675,
-0.2441108375787735,
-0.3648357391357422,
-0.02226751111447811,
0.024822132661938667,
-0.2057856172323227,
0.4948122501373291,
-0.2062215954065323,
-0.0646614283323288,
0.4056881070137024,
0.4732293486595154,
0.6887706518173218,
-0.3977937698364258,
-0.04045813903212547,
-0.31517767906188965,
-0.23735494911670685,
0.5974457859992981,
0.3987867832183838,
0.06414137035608292,
-0.11712861806154251,
-0.21257959306240082,
0.2480045109987259,
-0.30868402123451233,
0.3235619068145752,
0.11009156703948975,
-0.5957063436508179,
-0.10013429075479507,
0.26264917850494385,
-0.20285072922706604,
0.06936650723218918,
0.2788330912590027,
0.4416358172893524,
-0.25645044445991516,
-0.4833879768848419,
-0.09582365304231644,
-0.28862854838371277,
-0.07148349285125732,
0.19910934567451477,
0.19848448038101196,
-0.0959167405962944,
0.34519824385643005,
-0.616851270198822,
-0.5078041553497314,
0.08883167058229446,
-0.04086882248520851,
0.05056300759315491,
-0.48198553919792175,
0.9879816174507141,
0.3769012987613678,
-0.1396055519580841,
0.37667328119277954,
-0.07822434604167938,
-0.15269218385219574,
0.7659876346588135,
-0.2574620544910431,
0.06676790118217468,
-0.6144912838935852,
0.11674411594867706,
-0.7919224500656128,
-0.05626992881298065,
0.4939880073070526,
-0.19011715054512024,
-0.5278194546699524,
-0.06878267973661423,
0.08927404135465622,
0.1469065248966217,
0.2563425600528717,
-0.28311625123023987,
-0.16215941309928894,
-0.4562518000602722,
-0.23067519068717957,
-0.34958067536354065,
0.6098544001579285,
0.010608627460896969,
0.1291089504957199,
-0.04252780228853226,
0.4419926404953003,
-0.023586174473166466,
-0.10223475843667984,
0.1542595773935318,
-0.021042246371507645,
0.033746540546417236,
-0.5685637593269348,
0.5328364372253418,
-0.255197137594223,
-0.08574739098548889,
-0.015490787103772163,
-0.14060094952583313,
0.5193250179290771,
-0.26602622866630554,
-0.5381226539611816,
-0.28702595829963684,
-0.11166402697563171,
0.6754484176635742,
-0.13598980009555817,
0.3530609607696533,
-0.3664010465145111,
-0.20707722008228302,
-0.4084527790546417,
0.7037827372550964,
0.03613783046603203,
-0.13649670779705048,
-0.2831922769546509,
0.2637787461280823,
-0.4838695526123047,
0.3214108943939209,
-0.21647001802921295,
-0.5877710580825806,
-0.13882499933242798,
-0.31907764077186584
] |
one-sided heteroscedasticity, it also achieves the Bayes-optimal error. Numerical
experiments demonstrate the significant advantage of our theoretically principled
method with the state of the art. The proofs draw connections with statistical
physics and approximate message passing, departing drastically from standard
random matrix theory techniques.
1
Introduction
Matrix denoising is a central primitive in statistics and machine learning, and the problem is to | [
-0.17276827991008759,
0.13172709941864014,
-0.156040757894516,
0.23118956387043,
-0.1899864375591278,
-0.14100177586078644,
-0.09934858977794647,
-0.4196460545063019,
0.052123118191957474,
-0.6265851855278015,
0.012861710973083973,
0.3972685933113098,
-0.40711909532546997,
0.17071297764778137,
-0.12651215493679047,
-0.004097531083971262,
0.3910250663757324,
-0.014933872036635876,
-0.1318402737379074,
-0.5929505228996277,
0.2015690803527832,
0.2560350000858307,
0.00000227088412430021,
0.036332011222839355,
0.07926154881715775,
-0.10464850068092346,
0.305656373500824,
-0.07996892184019089,
0.07542214542627335,
0.05189237371087074,
-0.29012617468833923,
0.26414310932159424,
-0.11171586066484451,
-0.1428121030330658,
0.058120571076869965,
0.09493301063776016,
0.15304213762283325,
0.2762393355369568,
0.22981972992420197,
-0.23194186389446259,
0.6737687587738037,
-0.1108870804309845,
0.07623693346977234,
0.28973811864852905,
-0.5480636954307556,
-0.5303179621696472,
-1.2888575792312622,
0.19208674132823944,
-0.2611579895019531,
-0.6637626886367798,
-0.18757732212543488,
0.23570677638053894,
0.18499428033828735,
0.48577964305877686,
-0.305376797914505,
0.07367545366287231,
-0.035271719098091125,
-0.5068133473396301,
0.4999082088470459,
-0.3727060854434967,
0.018116695806384087,
0.10233934968709946,
-0.1598026603460312,
-0.030482187867164612,
0.20671194791793823,
0.8135178089141846,
0.25581735372543335,
0.11196459084749222,
-0.5390497446060181,
-0.15238498151302338,
-0.029987871646881104,
-0.6577353477478027,
-0.48169904947280884,
0.0369926318526268,
-0.2446659654378891,
-0.2829999625682831,
0.14125117659568787,
0.025894640013575554,
0.07127431780099869,
-0.25807875394821167,
-0.2544468939304352,
0.05213988944888115,
0.3164731562137604,
-0.3288508653640747,
0.601300060749054,
0.165848508477211,
-0.025072624906897545,
0.5327100157737732,
-0.7797036170959473,
0.14034529030323029,
-0.1331840306520462,
-0.2597205638885498,
0.07263688743114471,
-0.05947935953736305,
0.108485147356987,
0.5533872842788696,
0.4218422472476959,
0.0160510316491127,
0.8206795454025269,
0.24121636152267456,
-0.3082508146762848,
-0.34686279296875,
0.010471343994140625,
-0.17776426672935486,
0.034285835921764374,
-0.4888369143009186,
-0.16175177693367004,
-0.1852959543466568,
-0.26274481415748596,
0.18169817328453064,
-0.38420984148979187,
0.3766014873981476,
0.06525202840566635,
-0.15782342851161957,
-0.21678826212882996,
0.19784541428089142,
0.07503858208656311,
0.04366809129714966,
-0.2312251180410385,
0.35522961616516113,
-0.2335236817598343,
1.156021237373352,
-0.15671057999134064,
0.6560596823692322,
0.16304223239421844,
-0.008452268317341805,
0.45555898547172546,
0.9872363209724426,
-0.36927661299705505,
0.12961634993553162,
-0.08160994201898575,
1.086754560470581,
-0.1133679449558258,
-0.0030158909503370523,
0.12445829063653946,
-0.45598316192626953,
-0.5328057408332825,
0.11065544933080673,
-0.4364936947822571,
-0.12109734863042831,
-0.31499096751213074,
-0.4490865170955658,
0.13908393681049347,
0.3023754954338074,
0.2242603898048401,
-0.1425733119249344,
-0.054483477026224136,
-0.28062957525253296,
-0.07542099058628082,
0.041551653295755386,
-0.14737354218959808,
0.4292238652706146,
-0.6616082191467285,
0.11873374134302139,
-0.15385423600673676,
0.42146214842796326,
0.14742015302181244,
0.06753496080636978,
-0.4849507808685303,
0.6903293132781982,
-0.06013155356049538,
-0.09814538806676865,
-0.3130408525466919,
-0.6582381129264832,
0.08163861185312271,
-0.8599585890769958,
0.6942526698112488,
0.31479179859161377,
0.07276061177253723,
-0.018345484510064125,
-0.44298964738845825,
-0.508625864982605,
-0.3820999562740326,
-0.5383064150810242,
-0.019292451441287994,
0.19457222521305084,
0.2083231508731842,
0.029005270451307297,
0.4211733639240265,
0.3033575415611267,
-0.7823716402053833,
0.1854838877916336,
0.009522663429379463,
-0.4490966200828552,
0.2579558789730072,
0.027810396626591682,
0.27619045972824097,
-0.14472569525241852,
-0.04037308692932129,
-0.07714822143316269,
-0.03953547030687332,
0.29613280296325684,
0.416469007730484,
-1.105809211730957,
1.4832985401153564,
0.3172168433666229,
-0.45279255509376526,
0.9474863409996033,
-0.17062778770923615,
0.2599441409111023,
0.049380991607904434,
0.012873312458395958,
0.26915353536605835,
-0.03902950882911682,
-0.42384225130081177,
0.6001673936843872,
0.05840776115655899,
0.05143418163061142,
-0.6913097500801086,
0.16594772040843964,
0.5007357001304626,
0.3872213363647461,
0.06198059767484665,
-0.3305988907814026,
0.1352464109659195,
0.20231768488883972,
-0.1290256381034851,
0.10367891937494278,
0.3843183219432831,
-0.005118540953844786,
0.7950745224952698,
-0.3070301413536072,
0.6978332996368408,
0.5549347996711731,
0.4039967954158783,
0.6870684027671814,
1.1979862451553345,
0.1624804139137268,
0.8746527433395386,
-0.7422054409980774,
-0.2560596466064453,
-0.11727829277515411,
0.6948931813240051,
-0.16176073253154755,
0.10582217574119568,
0.22532083094120026,
0.279714971780777,
-0.544625461101532,
0.1097031757235527,
0.045763321220874786,
-0.07083479315042496,
0.31843772530555725,
-0.3285982310771942,
0.1895458996295929,
-0.14562255144119263,
-0.3623144030570984,
-0.5618951916694641,
-0.061171118170022964,
-0.26434341073036194,
-0.2636992335319519,
-0.16365809738636017,
0.38714179396629333,
0.10785834491252899,
-0.8804810643196106,
0.02696615271270275,
0.10854297876358032,
-0.19000162184238434,
-0.28907886147499084,
0.41129374504089355,
0.41110625863075256,
-0.19479405879974365,
-0.2985094487667084,
-0.3364298641681671,
-0.2536661922931671,
-0.3325071632862091,
0.09638599306344986,
-0.8919536471366882,
-0.5105404853820801,
0.6059743762016296,
0.46978697180747986,
0.3072575330734253,
-0.293896347284317,
-0.526587963104248,
0.13065725564956665,
-0.057806700468063354,
-0.5692803859710693,
0.0005297795287333429,
0.4536319971084595,
-0.886446475982666,
-0.41485533118247986,
0.07318872958421707,
0.5109877586364746,
0.14757244288921356,
-0.32986220717430115,
-0.27559831738471985,
-0.28920474648475647,
-0.06956097483634949,
-0.36568760871887207,
-0.3299214541912079,
-0.3059481680393219,
-0.23123006522655487,
0.6222327351570129,
-0.22373682260513306,
-0.7196296453475952,
0.817430317401886,
0.6222260594367981,
-0.03829816356301308,
-0.23370784521102905,
0.35544365644454956,
-0.10939863324165344,
-0.2838353216648102,
0.08100420981645584,
-0.11512359976768494,
0.1108342632651329,
0.05412221699953079,
-0.2257189303636551,
-0.25201815366744995,
0.5937519073486328,
-6.059581279754639,
-0.1951763927936554,
-0.3746716380119324,
0.07542013376951218,
-0.5626260638237,
-0.08432624489068985,
0.4662911891937256,
0.021059850230813026,
0.16435803472995758,
-0.2305477261543274,
0.40350019931793213,
0.006797363981604576,
-0.4289642572402954,
0.4857863485813141,
0.4401435852050781,
0.0993211567401886,
0.23136673867702484,
-0.05906888470053673,
0.3245527446269989,
0.358733594417572,
-0.4134739339351654,
-0.054312869906425476,
0.000621575687546283,
0.07896581292152405,
-0.030624384060502052,
0.14155972003936768,
-0.8418816328048706,
0.7485957741737366,
-0.38575807213783264,
-0.060613714158535004,
0.14844867587089539,
-0.5665812492370605,
-0.16913248598575592,
-0.6331090331077576,
0.23043985664844513,
0.418141633272171,
0.5130265951156616,
0.08600125461816788,
-0.04539281502366066,
-0.30918657779693604,
0.21989184617996216,
0.20683017373085022,
0.28776708245277405,
0.08845560997724533,
0.37703466415405273,
-0.11286760121583939,
-0.005629887804389,
0.1922072321176529,
-0.27130305767059326,
0.6035638451576233,
-0.43956270813941956,
0.12897422909736633,
0.1879470944404602,
-0.060676947236061096,
-0.0659467801451683,
-0.3074750304222107,
0.09842564910650253,
0.08443185687065125,
-0.2865007817745209,
0.13883285224437714,
0.3368465304374695,
-0.5289201140403748,
0.41934624314308167,
-0.36832454800605774,
-0.38973453640937805,
0.13485504686832428,
-0.3977232277393341,
-0.38338157534599304,
0.24150805175304413,
0.18196150660514832,
-0.21285556256771088,
0.21476459503173828,
0.28028008341789246,
-0.3817008137702942,
0.1195303350687027,
-0.310659259557724,
0.052105531096458435,
-0.004489260260015726,
0.1531187891960144,
0.27366915345191956,
-0.17427074909210205,
-0.6214908957481384,
-0.21248923242092133,
0.16617944836616516,
0.3650473952293396,
-0.001474672695621848,
-0.06735280901193619,
0.00823192484676838,
-0.6987037062644958,
0.2746070921421051,
0.65205979347229,
-0.016677694395184517,
0.2999321520328522,
0.664413571357727,
0.41663292050361633,
0.8837512731552124,
0.5424781441688538,
0.0958438590168953,
0.29985761642456055,
-0.010125386528670788,
0.8466804623603821,
-0.19740606844425201,
-0.000722236349247396,
-0.557153582572937,
0.14261166751384735,
0.05646892264485359,
-0.48972630500793457,
0.38852304220199585,
0.5115694403648376,
-0.3524048924446106,
-0.05250035971403122,
0.11520340293645859,
-0.34310203790664673,
-0.3627792298793793,
0.4554722011089325,
0.23526941239833832,
-0.3690512180328369,
-0.17590069770812988,
0.6076220273971558,
-0.32956671714782715,
0.10251256078481674,
0.3148866593837738,
0.052893079817295074,
0.14255787432193756,
-0.013014017604291439,
-0.08791113644838333,
0.19419419765472412,
0.8730136752128601,
-0.27258071303367615,
-0.21823833882808685,
-0.01354278065264225,
0.6197713017463684,
-0.38643163442611694,
0.06825768947601318,
0.45491403341293335,
-0.33958059549331665,
0.08873725682497025,
0.28172674775123596,
0.29392102360725403,
-0.256517231464386,
1.0806828737258911,
-0.01219881046563387,
-0.19081352651119232,
-0.09239136427640915,
-0.1345977634191513,
0.15256905555725098,
-0.2922520339488983,
-0.20971620082855225,
-0.33001798391342163,
-0.2109164297580719,
-0.007890252396464348,
-0.4394972026348114,
0.5515664219856262,
-0.2836255133152008,
-0.2217361330986023,
0.11129366606473923,
0.2509630620479584,
0.2841176688671112,
0.10288451611995697,
-0.16979794204235077,
0.4651953876018524,
-0.10937326401472092,
-0.6129125356674194,
-0.16397720575332642,
0.2219133824110031,
-0.01981073059141636,
-0.5261931419372559,
-0.3461340069770813,
-0.2361651360988617,
0.3577219843864441,
-0.23145079612731934,
-0.3580172657966614,
0.1339925229549408,
-0.42065170407295227,
0.24123841524124146,
0.11263327300548553,
0.1260969340801239,
-0.1458955854177475,
-0.16279517114162445,
-0.4719603657722473,
0.20796869695186615,
-0.5612449645996094,
-0.18030774593353271,
0.949400007724762,
0.9218243956565857,
-0.3921835720539093,
0.40542352199554443,
0.25684550404548645,
-0.7766939401626587,
-0.28137674927711487,
0.1581209897994995,
0.1320948451757431,
0.6879122257232666,
0.09559524804353714,
0.26580390334129333,
-0.0027058804407715797,
0.1872432678937912,
0.17893844842910767,
-0.448257714509964,
-0.37981656193733215,
-0.17686080932617188,
0.3884981870651245,
-0.19535717368125916,
0.212185338139534,
0.7666268944740295,
-0.28207534551620483,
-0.17669332027435303,
-0.1346636861562729,
-0.19972294569015503,
-0.11506761610507965,
-0.601335883140564,
-0.6315073370933533,
-0.3218131363391876,
-0.3302987515926361,
-0.2997494637966156,
0.026034682989120483,
0.23133760690689087,
0.05788658931851387,
-0.43250852823257446,
0.48642003536224365,
0.5012894868850708,
-0.28065675497055054,
-0.009344914928078651,
0.07761450856924057,
0.17484131455421448,
0.4868793189525604,
-0.005309263709932566,
-0.6623518466949463,
0.06511920690536499,
-0.11222436279058456,
-0.5823561549186707,
-0.34438061714172363,
-0.18935918807983398,
-0.09370426833629608,
-0.321938157081604,
-0.3071569800376892,
-0.10392782837152481,
0.6423875689506531,
-0.6698961853981018,
-0.29872098565101624,
0.017404820770025253,
0.5100783109664917,
-0.07779236137866974,
-0.2830537259578705,
0.017719009891152382,
-0.06862533837556839,
0.28990060091018677,
-0.09819657355546951,
-0.5864701867103577,
-0.0733073428273201,
-0.31854334473609924,
0.20067432522773743,
0.28067150712013245,
0.016228944063186646,
0.6069281101226807,
0.3059639632701874,
0.20880654454231262,
-0.7049079537391663,
-0.08230757713317871,
0.05148378759622574,
-0.13676030933856964,
-0.7407029271125793,
-0.3201451599597931,
-0.48101839423179626,
0.004412967246025801,
0.1705465167760849,
-0.06937720626592636,
-0.4395492374897003,
0.25887027382850647,
-0.14191654324531555,
-0.08995324373245239,
-0.3379667401313782,
0.11519082635641098,
0.4881461262702942,
-0.11620589345693588,
0.11570710688829422,
-0.357388436794281,
0.5436273217201233,
-0.2544352412223816,
0.10078874230384827,
-0.1564846634864807,
-0.36133721470832825,
-0.25967109203338623,
-0.5197858214378357,
0.011259960010647774,
0.33749085664749146,
0.2899080514907837,
0.011953862383961678,
0.14507617056369781,
0.4021245241165161,
-0.6552196741104126,
0.20397338271141052,
-0.31244444847106934,
-0.37150251865386963,
0.03314846009016037,
0.07727014273405075,
0.30278879404067993,
0.2110780030488968,
-0.08114563673734665,
-0.25058266520500183,
-0.14852400124073029,
0.3772887885570526,
0.1601778119802475,
-0.45027944445610046,
0.440805584192276,
0.3232402205467224,
0.3502100110054016,
0.2565588653087616,
-0.34110477566719055,
-0.028014304116368294,
0.3022766411304474,
0.1864323914051056,
0.08824145048856735,
0.35289934277534485,
-0.2746078670024872,
-0.14134985208511353,
0.5536706447601318,
-0.3357880711555481,
0.4513424038887024,
0.6102423071861267,
-0.2752252519130707,
0.1411007046699524,
0.668536365032196,
-0.587571382522583,
-0.6190832257270813,
0.5226361155509949,
-0.31858474016189575,
-0.05452360957860947,
-0.5794003009796143,
0.2792164981365204,
0.4548646807670593,
0.41656607389450073,
-0.7011659145355225,
0.24090343713760376,
-0.217799574136734,
0.41224828362464905,
0.4797685146331787,
-0.8148760199546814,
0.1059875637292862,
-0.20716510713100433,
0.7783472537994385,
0.3104299008846283,
-0.8350287079811096,
0.05478866770863533,
-0.2943497598171234,
0.09754962474107742,
0.40299758315086365,
-0.0971403643488884,
-0.5221940875053406,
0.6168155670166016,
0.513906717300415,
-0.04193880781531334,
-0.004724567756056786,
-0.16121132671833038,
0.03486829996109009,
-0.2942962944507599,
0.04155401140451431,
-0.3948071002960205,
-0.03616814687848091,
0.328823983669281,
0.6483266949653625,
0.47810396552085876,
-0.23766590654850006,
-0.09616832435131073,
-0.4150644540786743,
0.12221293896436691,
0.7079877257347107,
0.26593342423439026,
-0.041918639093637466,
-0.15619254112243652,
-0.4398365020751953,
0.18144874274730682,
-0.44282564520835876,
0.3264561891555786,
0.012231043539941311,
-0.5354034900665283,
-0.0913383737206459,
0.6318132281303406,
0.026931146159768105,
-0.29416757822036743,
0.38086962699890137,
0.11322454363107681,
-0.15842373669147491,
-0.8510861992835999,
0.0663565993309021,
-0.3714672923088074,
0.40012794733047485,
0.22214794158935547,
-0.3066318929195404,
-0.09556623548269272,
-0.18872229754924774,
-0.8209428787231445,
-0.17825546860694885,
0.5153453946113586,
0.2614890933036804,
-0.280463844537735,
-0.5731489062309265,
0.8801532983779907,
0.507427990436554,
0.025223301723599434,
0.08519276976585388,
-0.23435921967029572,
0.24520085752010345,
0.40777212381362915,
-0.3058479428291321,
0.09658870100975037,
-0.3361191153526306,
0.3683876097202301,
-0.6240362524986267,
-0.03222232684493065,
0.40772658586502075,
-0.007495389319956303,
-0.38452261686325073,
-0.10883942246437073,
-0.07313673943281174,
0.017764756456017494,
0.18432511389255524,
-0.15369713306427002,
0.039373718202114105,
-0.26769110560417175,
-0.5501604676246643,
-0.3401671051979065,
0.5878945589065552,
0.14898580312728882,
0.2145605981349945,
-0.04511129856109619,
0.2168959528207779,
0.02869962528347969,
0.13002949953079224,
0.37018996477127075,
0.1582310050725937,
-0.11955777555704117,
-0.29060429334640503,
0.5159181356430054,
-0.05539492517709732,
-0.1766035556793213,
0.24557460844516754,
-0.41119644045829773,
0.2335253804922104,
-0.23351866006851196,
-0.3415583670139313,
0.2093430906534195,
-0.3422780930995941,
0.65707927942276,
-0.39448660612106323,
0.6956704258918762,
-0.35128217935562134,
-0.3487800061702728,
-0.38737237453460693,
0.47492173314094543,
0.13956034183502197,
0.1460193693637848,
-0.28848007321357727,
-0.19283714890480042,
-0.5244771838188171,
0.4288695156574249,
-0.08283493667840958,
-0.34284132719039917,
-0.3799004852771759,
-0.13796894252300262
] |
Matrix denoising is a central primitive in statistics and machine learning, and the problem is to
recover a signal X ∈ Rn×d from an observation A = X + W corrupted by additive noise W. This
finds applications across multiple domains of sciences, e.g., imaging [21, 60], biology [13, 42] and
astronomy [67, 5]. When X has low rank and W i.i.d. entries, A is the standard model for principal
component analysis, typically referred to as the Johnstone spiked covariance model [38]. When | [
0.054392002522945404,
-0.051499638706445694,
0.0685233399271965,
0.07521393895149231,
-0.5585178136825562,
0.010524860583245754,
-0.2783120274543762,
-0.6179121136665344,
0.04216598719358444,
0.06930272281169891,
-0.10102586448192596,
0.41388097405433655,
-0.713725745677948,
0.30248895287513733,
-0.1017141044139862,
0.4337255656719208,
0.5791105031967163,
0.18299664556980133,
-0.33470869064331055,
-0.6839524507522583,
0.2831965684890747,
-0.017352337017655373,
-0.03983854874968529,
0.15953707695007324,
0.16318392753601074,
0.3181455135345459,
0.38126739859580994,
0.10136216878890991,
-0.04283008351922035,
0.3118494153022766,
-0.2449311763048172,
0.3930349349975586,
0.07177600264549255,
0.24730272591114044,
0.13987238705158234,
0.24492189288139343,
0.23515717685222626,
0.07315398007631302,
0.3610859811306,
-0.12445223331451416,
0.6213769912719727,
-0.3727166950702667,
0.057032518088817596,
0.400110125541687,
-0.693314790725708,
-0.7359732389450073,
-1.457537293434143,
0.379793643951416,
-0.4512340724468231,
-0.6694426536560059,
-0.624636709690094,
0.43321657180786133,
0.01295486930757761,
0.6847096085548401,
-0.36635586619377136,
0.24244838953018188,
-0.2914940416812897,
-0.4337431788444519,
0.7463021278381348,
-0.6016483306884766,
0.16296322643756866,
-0.0540609285235405,
-0.05754099041223526,
0.2547192871570587,
0.25957316160202026,
0.6159372925758362,
-0.19979824125766754,
-0.2166738212108612,
-0.7310317158699036,
-0.40682315826416016,
0.08165150880813599,
0.08008406311273575,
-0.4138983488082886,
-0.15956568717956543,
-0.3490673303604126,
0.28086206316947937,
-0.09054811298847198,
-0.03554879501461983,
0.12142408639192581,
-0.15040184557437897,
-0.07614394277334213,
0.22908440232276917,
0.3352169990539551,
-0.4505574703216553,
0.5662808418273926,
0.09701623022556305,
-0.2955275774002075,
0.5074167251586914,
-0.4365156590938568,
0.29772576689720154,
-0.31762585043907166,
0.04273121431469917,
0.14955683052539825,
0.2676342725753784,
0.147576704621315,
0.5966132879257202,
0.09690948575735092,
-0.42004674673080444,
1.0260735750198364,
0.08333233743906021,
-0.3030235767364502,
-0.23123376071453094,
0.1775188446044922,
-0.5261414051055908,
-0.5069729685783386,
-0.38095739483833313,
-0.6480892300605774,
-0.06439109146595001,
-0.5991842746734619,
0.38544684648513794,
-0.6819339394569397,
0.2875763475894928,
0.08560477942228317,
-0.627204954624176,
0.011188755743205547,
-0.08612924069166183,
-0.04742586985230446,
0.44505611062049866,
-0.13990844786167145,
0.7689129710197449,
-0.2032260149717331,
1.1752541065216064,
-0.17871487140655518,
0.5472225546836853,
-0.0517052598297596,
0.20873978734016418,
0.39768096804618835,
0.9122563600540161,
-0.7162743806838989,
0.07949023693799973,
0.08164957165718079,
0.8030319213867188,
0.4534566104412079,
0.6480237245559692,
-0.19418272376060486,
-0.5738984942436218,
-0.4879222810268402,
-0.3458005487918854,
-0.290193647146225,
-0.3637740910053253,
-0.3947658836841583,
-0.4513567388057709,
0.05340707674622536,
-0.006525851786136627,
0.5335821509361267,
-0.29579952359199524,
-0.005604020319879055,
-0.4378754198551178,
0.13639803230762482,
0.12743130326271057,
-0.17680345475673676,
0.4859444797039032,
-0.6182824969291687,
-0.4093170166015625,
-0.27048736810684204,
0.1959930807352066,
0.395826131105423,
-0.08538969606161118,
-0.43863925337791443,
0.42956581711769104,
0.030435562133789062,
-0.20628094673156738,
-0.3205052614212036,
-0.48031923174858093,
0.40729057788848877,
-1.1283848285675049,
0.414996862411499,
0.4369064271450043,
-0.2039201557636261,
-0.046113207936286926,
-0.18783937394618988,
-0.614311695098877,
-0.06743545830249786,
0.04040718451142311,
-0.049163904041051865,
0.18482647836208344,
0.09090057760477066,
-0.20666365325450897,
0.434857577085495,
-0.035519860684871674,
-0.7658453583717346,
0.12121514230966568,
0.38238802552223206,
-0.048425521701574326,
0.1908046007156372,
0.47771573066711426,
0.2930280566215515,
-0.24311770498752594,
0.020911214873194695,
-0.07585860043764114,
-0.30222535133361816,
0.22749699652194977,
0.12496121972799301,
-1.0543620586395264,
1.1097568273544312,
0.2065264731645584,
-0.6171454787254333,
0.9356614351272583,
-0.09757870435714722,
0.05136653780937195,
-0.02281017415225506,
-0.1575489193201065,
0.19199419021606445,
-0.11886400729417801,
-0.26222938299179077,
0.3107728660106659,
0.09610020369291306,
0.3518866300582886,
-0.732672929763794,
-0.013280289247632027,
0.38255685567855835,
0.23226359486579895,
-0.11953789740800858,
0.011146657168865204,
0.4210606515407562,
0.10634700208902359,
-0.6062566041946411,
0.35408687591552734,
0.4743976593017578,
0.01849415712058544,
0.8434723019599915,
-0.544499397277832,
-0.1678602546453476,
0.3158116638660431,
0.2442598044872284,
0.969698965549469,
1.0388840436935425,
0.467462420463562,
0.9855978488922119,
-0.4844881296157837,
-0.08885177224874496,
-0.3249923288822174,
0.2773289978504181,
0.09153813868761063,
-0.07375375181436539,
0.24365419149398804,
0.38193538784980774,
0.12675213813781738,
0.1629800796508789,
-0.2664610743522644,
-0.12779013812541962,
0.40575775504112244,
-0.39214104413986206,
-0.202116921544075,
-0.03326190263032913,
-0.38981887698173523,
-0.39633142948150635,
-0.22076332569122314,
-0.5284060835838318,
0.13099275529384613,
0.04517684876918793,
0.1165681704878807,
-0.22619326412677765,
-0.6737334132194519,
0.33065932989120483,
0.05217510089278221,
-0.10920289903879166,
-0.3884866237640381,
0.12156455963850021,
0.5242511630058289,
0.029504843056201935,
-0.42229989171028137,
-0.6053536534309387,
-0.5010597705841064,
-0.35924893617630005,
-0.043385256081819534,
-0.40169310569763184,
0.008625277318060398,
0.9182125329971313,
0.29153990745544434,
0.12502838671207428,
-0.480300635099411,
-0.44606783986091614,
0.32671716809272766,
-0.6401779055595398,
-1.1874209642410278,
0.008635510690510273,
0.2940814793109894,
-0.9091680645942688,
-0.3752621114253998,
-0.10332728922367096,
0.4089633524417877,
0.16323517262935638,
0.08529747277498245,
-0.6944242715835571,
-0.39785176515579224,
-0.24005715548992157,
-0.2791180908679962,
-0.08636225759983063,
-0.3337869942188263,
-0.37655338644981384,
0.4098389744758606,
0.0679134652018547,
-0.33671581745147705,
0.06658061593770981,
0.7806666493415833,
-0.012352890335023403,
0.07239452004432678,
0.2510641813278198,
-0.004581760615110397,
-0.5436367988586426,
0.17389383912086487,
-0.07493995130062103,
0.25136569142341614,
-0.01496412605047226,
-0.007655547931790352,
-0.32750529050827026,
0.7461647987365723,
-5.799209117889404,
0.25275659561157227,
-0.1934814602136612,
0.006185847334563732,
0.1411880999803543,
0.14925828576087952,
0.180857852101326,
0.022671399638056755,
0.1919594407081604,
-0.09848880767822266,
0.007449038792401552,
-0.1487780064344406,
-0.6417383551597595,
0.6938526630401611,
0.5577311515808105,
-0.0528789646923542,
0.2936488091945648,
0.0802360400557518,
0.28445813059806824,
0.5371910929679871,
-0.12942184507846832,
-0.14989572763442993,
-0.07573375850915909,
-0.11361043900251389,
-0.5081307888031006,
0.2834559977054596,
-0.4597962498664856,
0.3613918423652649,
-0.193428173661232,
-0.5202491879463196,
-0.16625544428825378,
-0.5539131760597229,
-0.04674869775772095,
-0.18609608709812164,
-0.0026545412838459015,
0.5828506350517273,
0.221255362033844,
0.023093977943062782,
-0.02927326038479805,
-0.5817978978157043,
0.42039763927459717,
0.7633923292160034,
0.13524989783763885,
-0.33730363845825195,
0.5118646621704102,
-0.05048249289393425,
-0.020154030993580818,
0.3827292323112488,
-0.2513534426689148,
0.8774588108062744,
-0.3756411075592041,
-0.09493543207645416,
0.04263987019658089,
-0.09294619411230087,
-0.3703964650630951,
-0.2644585072994232,
-0.018640516325831413,
0.27434688806533813,
-0.16739490628242493,
0.3750883936882019,
0.11739253997802734,
-0.2642627954483032,
0.2670891582965851,
-0.23048552870750427,
-0.8767126202583313,
0.19599351286888123,
-0.5874254703521729,
-0.22602128982543945,
0.593261182308197,
0.02236068807542324,
-0.4974779784679413,
0.07403525710105896,
0.06659244745969772,
-0.3363206386566162,
0.32224026322364807,
-0.559604823589325,
0.5057955980300903,
-0.39529168605804443,
0.041166629642248154,
0.5569354295730591,
-0.433682382106781,
-0.737389087677002,
0.21597857773303986,
-0.15205194056034088,
-0.13400280475616455,
0.05926216393709183,
-0.2190369963645935,
0.2508496046066284,
-0.3256620764732361,
-0.09590037167072296,
0.36606693267822266,
-0.0013653431087732315,
0.36605504155158997,
0.6662957668304443,
0.3006039261817932,
0.3324184715747833,
0.6107569336891174,
0.11308282613754272,
0.6227430105209351,
0.1552899181842804,
0.5303964614868164,
-0.27817922830581665,
0.0708867534995079,
-0.808516800403595,
0.1646157205104828,
0.007224728353321552,
-0.4367104470729828,
0.5076478123664856,
0.41445088386535645,
-0.1125359833240509,
-0.09190095961093903,
0.6527644991874695,
-0.09667717665433884,
-0.4294891953468323,
0.854149580001831,
0.2894553244113922,
-0.3832490146160126,
-0.19239909946918488,
0.6928441524505615,
-0.13098932802677155,
0.01798180676996708,
0.20180949568748474,
-0.038925349712371826,
0.2133764922618866,
0.33990713953971863,
-0.3111702501773834,
0.12859635055065155,
0.4646705389022827,
-0.42750468850135803,
-0.5907034277915955,
0.33151471614837646,
0.3536311388015747,
-0.38654625415802,
0.34252217411994934,
0.2812804877758026,
-0.7976574301719666,
-0.11130103468894958,
0.5509061217308044,
0.24270588159561157,
-0.4443460702896118,
1.0603727102279663,
0.12167636305093765,
-0.33547019958496094,
-0.19176633656024933,
0.044883161783218384,
0.06408325582742691,
-0.12637382745742798,
-0.0747816190123558,
-0.1555665135383606,
0.010598436929285526,
-0.23672091960906982,
-0.5862827301025391,
0.3838701844215393,
-0.4184345304965973,
0.05843503028154373,
-0.04128773510456085,
0.14427408576011658,
0.11816376447677612,
0.6108752489089966,
-0.6217828392982483,
0.2818381190299988,
0.3972347378730774,
-0.7783252596855164,
-0.12373463064432144,
-0.3448573350906372,
0.2907908260822296,
-0.8040108680725098,
-0.12063194066286087,
0.1567026823759079,
0.38656729459762573,
-0.3884033262729645,
-0.21427303552627563,
-0.05055944249033928,
-0.45597389340400696,
0.2978397309780121,
0.3678586781024933,
0.13040004670619965,
-0.08566571027040482,
-0.022252537310123444,
-0.34796881675720215,
0.34028273820877075,
-0.12599435448646545,
-0.5654418468475342,
0.8059321045875549,
1.0959798097610474,
0.050378505140542984,
0.08808164298534393,
0.23076072335243225,
-0.6452240943908691,
-0.2415502369403839,
0.4946252703666687,
0.038583721965551376,
0.2735457718372345,
-0.13665200769901276,
0.15417766571044922,
0.05829045921564102,
0.4094034433364868,
0.39920130372047424,
-0.13691318035125732,
-0.208034947514534,
-0.25244802236557007,
0.546166718006134,
-0.29566672444343567,
0.5762737989425659,
0.3180094063282013,
-0.18883025646209717,
-0.04853871837258339,
-0.13508731126785278,
-0.23236887156963348,
-0.35035523772239685,
-0.467513769865036,
-0.38903099298477173,
-0.34454840421676636,
-0.5049651861190796,
-0.08089683204889297,
0.14422769844532013,
0.35791170597076416,
-0.5219966769218445,
-0.014289060607552528,
0.19605706632137299,
0.15771517157554626,
-0.5538777112960815,
0.7810875773429871,
0.491858571767807,
0.17200085520744324,
0.04833566024899483,
-0.08516260981559753,
-0.5071343779563904,
0.4171838164329529,
0.12235556542873383,
-0.28862887620925903,
-0.25470981001853943,
-0.2368190735578537,
0.45277175307273865,
-0.3352549374103546,
-0.6583921313285828,
-0.2783326208591461,
0.6824115514755249,
-0.36448657512664795,
-0.5591402649879456,
0.2463352084159851,
0.8035798072814941,
0.00023523496929556131,
-0.22853092849254608,
0.12129580974578857,
-0.22927233576774597,
0.05952879413962364,
-0.026457760483026505,
-0.34963956475257874,
-0.18849097192287445,
-0.13662776350975037,
-0.05103623867034912,
0.11670304834842682,
0.3256602883338928,
0.6075196266174316,
0.1711655706167221,
0.22515249252319336,
-0.8681007623672485,
0.07645222544670105,
0.28749197721481323,
-0.061284396797418594,
-0.5873041749000549,
0.07053264230489731,
-0.303900808095932,
-0.018207505345344543,
0.21963369846343994,
-0.12393146753311157,
-0.3221452534198761,
0.37067827582359314,
-0.010330475866794586,
0.06101539358496666,
-0.42450767755508423,
0.2361885905265808,
0.6331436634063721,
0.0599198192358017,
0.02149832807481289,
-0.12720520794391632,
0.30383652448654175,
-0.1350603550672531,
0.14584243297576904,
-0.5925173163414001,
-0.37700414657592773,
-0.03572232276201248,
-0.45848679542541504,
-0.1779935359954834,
-0.004126188810914755,
0.5341154932975769,
0.047642555087804794,
-0.08517899364233017,
0.1851755529642105,
-0.6204668283462524,
0.27137014269828796,
-0.2854223847389221,
-0.6324775815010071,
-0.3248930275440216,
-0.004977295640856028,
0.36856213212013245,
0.5034313201904297,
-0.39969372749328613,
0.008474372327327728,
-0.32030871510505676,
0.2851930558681488,
0.02259555086493492,
-0.3875199556350708,
0.19103577733039856,
-0.14246836304664612,
0.39487501978874207,
-0.031351491808891296,
-0.6350190043449402,
-0.031256262212991714,
0.14970332384109497,
-0.020779339596629143,
0.06944558024406433,
0.17253229022026062,
-0.3482857942581177,
-0.13104170560836792,
0.5260475873947144,
-0.17759324610233307,
0.9174420833587646,
0.5749955177307129,
-0.26012828946113586,
0.3724554181098938,
0.45009374618530273,
-0.28271833062171936,
-0.6792537569999695,
0.752548098564148,
-0.11289136111736298,
0.2228080779314041,
-0.24575383961200714,
0.3807430565357208,
0.08251255005598068,
0.2590259909629822,
-0.6515060663223267,
0.4213033616542816,
-0.3466798663139343,
0.07725698500871658,
0.730843722820282,
-0.517989456653595,
0.09178096055984497,
0.10709904879331589,
0.3395313322544098,
0.7789545059204102,
-0.6627461910247803,
0.2158806324005127,
0.154584139585495,
0.0344979465007782,
0.24406425654888153,
-0.10824679583311081,
-0.4617009162902832,
0.3432264029979706,
0.29584240913391113,
-0.21374422311782837,
0.1353551596403122,
-0.041678667068481445,
-0.06516210734844208,
-0.08796309679746628,
0.2982195019721985,
-0.34299495816230774,
-0.10935665667057037,
0.17179669439792633,
0.4181998670101166,
0.35572323203086853,
-0.4280255138874054,
-0.15706892311573029,
-0.6585984230041504,
-0.27012088894844055,
0.9187126755714417,
0.1476697325706482,
0.08815177530050278,
0.1388585865497589,
-0.24810217320919037,
0.2846315801143646,
-0.12074675410985947,
0.47819939255714417,
-0.05692387744784355,
-0.7561561465263367,
0.1525430530309677,
0.797448456287384,
-0.37750354409217834,
-0.08045832812786102,
0.48161229491233826,
0.23118160665035248,
-0.029359953477978706,
-0.9206910729408264,
0.00642190920189023,
-0.7630128264427185,
0.21239766478538513,
0.2221164107322693,
0.027522608637809753,
0.045870520174503326,
0.2468322366476059,
-0.514847993850708,
-0.32457512617111206,
0.34655511379241943,
0.2768998444080353,
0.18102732300758362,
-0.44010281562805176,
0.9196496605873108,
0.5930087566375732,
0.13923880457878113,
0.19812370836734772,
0.2647115886211395,
0.13728934526443481,
0.38919705152511597,
-0.3526337146759033,
0.02683819830417633,
-0.4840790033340454,
0.12306617945432663,
-0.506866991519928,
-0.04713573679327965,
0.46165838837623596,
-0.45719385147094727,
-0.4283044934272766,
-0.15037304162979126,
0.1388101577758789,
-0.1863875836133957,
0.498198539018631,
0.044341836124658585,
0.11921315640211105,
-0.18122509121894836,
-0.707230806350708,
-0.2450069785118103,
0.8118540644645691,
0.16802087426185608,
-0.17927834391593933,
0.42641404271125793,
0.6669967174530029,
-0.02621646784245968,
-0.44835513830184937,
0.28761738538742065,
-0.03365262970328331,
-0.25146645307540894,
-0.6591355800628662,
0.6672095060348511,
-0.435515433549881,
-0.2694130837917328,
-0.08781664073467255,
-0.20561207830905914,
0.44741830229759216,
-0.4685656726360321,
-0.4279650151729584,
0.1825721561908722,
-0.4411918520927429,
0.0397258996963501,
0.004324130713939667,
0.9470398426055908,
-0.2402527630329132,
0.052622176706790924,
-0.3676837384700775,
0.4077572524547577,
-0.11563321202993393,
-0.48492783308029175,
-0.4782731533050537,
-0.026530301198363304,
-0.6512792706489563,
0.4631865620613098,
-0.10337141901254654,
-0.1753385066986084,
-0.4418503940105438,
-0.13571563363075256
] |
component analysis, typically referred to as the Johnstone spiked covariance model [38]. When
n, d are both large and proportional, which corresponds to the most sample-efficient regime, its
Bayes-optimal limits are well understood [48], and it has been established how to achieve them
efficiently [53]. Minimax/non-asymptotic guarantees are also available in special cases, such as
sparse PCA [17], Gaussian mixtures [69] and certain joint scalings of (n, d) [54]. | [
-0.03025408834218979,
-0.4393158257007599,
0.19519872963428497,
0.34050193428993225,
-0.18859313428401947,
0.2064838409423828,
0.1624588966369629,
-0.5335655212402344,
-0.04395497962832451,
-0.01554141379892826,
-0.2917514145374298,
0.8236665725708008,
-0.34752562642097473,
-0.09040296822786331,
-0.19422495365142822,
0.5727177858352661,
0.44793564081192017,
-0.02289867028594017,
-0.20334471762180328,
-0.4833362400531769,
-0.13074153661727905,
0.059473197907209396,
0.00136537104845047,
-0.031001774594187737,
-0.10858052968978882,
0.33976641297340393,
0.46698233485221863,
0.04167428985238075,
-0.14835719764232635,
0.026305966079235077,
-0.22093476355075836,
0.788424551486969,
-0.08222412317991257,
-0.16647778451442719,
0.3746289014816284,
0.38487035036087036,
0.0741746723651886,
0.11365752667188644,
0.04258802533149719,
-0.16350606083869934,
0.48756644129753113,
-0.3227560520172119,
0.0007412363192997873,
0.19149132072925568,
-0.8391560912132263,
-0.7493532299995422,
-1.445571780204773,
0.2950022220611572,
-0.389904260635376,
-0.3325248956680298,
-0.07101956754922867,
0.7489806413650513,
0.3013877868652344,
0.2504364252090454,
-0.20017573237419128,
0.3001171946525574,
0.05222396180033684,
-0.3589170277118683,
0.7166848182678223,
-0.3834764063358307,
0.0021445797756314278,
0.3543826639652252,
-0.26192402839660645,
0.290396124124527,
-0.0019274865044280887,
0.367900550365448,
-0.11951969563961029,
-0.11576754599809647,
-0.432953417301178,
-0.5734236836433411,
0.23213143646717072,
0.15433695912361145,
-0.3006592392921448,
-0.3003495931625366,
-0.5852424502372742,
0.20876631140708923,
-0.0503571443259716,
0.033935774117708206,
-0.04907349869608879,
-0.303877055644989,
-0.41652026772499084,
0.09208667278289795,
0.178740993142128,
-0.37373945116996765,
0.5106911063194275,
0.08972460776567459,
-0.0026387893594801426,
0.40642666816711426,
-0.20971918106079102,
0.423530638217926,
-0.008754024282097816,
-0.16574156284332275,
0.16229793429374695,
0.3269057869911194,
0.538221538066864,
0.16852331161499023,
0.001579718547873199,
0.0886538103222847,
0.6214650273323059,
0.1519624888896942,
-0.1378166377544403,
0.274710088968277,
-0.10518363118171692,
-0.1949903666973114,
-0.3099801540374756,
-0.11953262239694595,
-0.4459608197212219,
-0.4435212314128876,
-0.6667636036872864,
0.3635637164115906,
-0.713218092918396,
0.08979598432779312,
-0.1675550788640976,
-0.42448315024375916,
-0.4682515561580658,
0.26509785652160645,
0.17624157667160034,
0.4092051684856415,
-0.3894044756889343,
0.5683583617210388,
-0.012392858043313026,
0.980918288230896,
-0.162049800157547,
0.42062199115753174,
0.1559251844882965,
0.19511541724205017,
0.9700901508331299,
0.7404909729957581,
-0.25159743428230286,
-0.2293866127729416,
-0.03742842748761177,
0.9098761677742004,
0.008826608769595623,
-0.019754188135266304,
-0.1453436017036438,
-0.7750229835510254,
-0.04150175303220749,
-0.26082175970077515,
-0.08329910784959793,
0.003766304813325405,
-0.539828360080719,
-0.6597760915756226,
0.06589599698781967,
-0.262077271938324,
0.017249388620257378,
-0.34602320194244385,
0.13052205741405487,
-0.26076740026474,
-0.09628544002771378,
0.029384026303887367,
-0.003752628806978464,
0.40059277415275574,
-0.6307949423789978,
-0.5485185980796814,
-0.08870955556631088,
0.2923951745033264,
0.020407166332006454,
0.19955842196941376,
-0.6161049604415894,
0.25050410628318787,
-0.23952040076255798,
0.44067156314849854,
-0.2968027591705322,
-0.681469738483429,
0.173811137676239,
-0.7856583595275879,
0.5528317093849182,
0.48377954959869385,
-0.04029589891433716,
-0.3921305239200592,
-0.09531396627426147,
-0.1729268580675125,
0.18763373792171478,
0.09437236934900284,
0.11559847742319107,
0.13913503289222717,
0.5785757303237915,
0.047195110470056534,
0.5497159957885742,
-0.2147941291332245,
-0.6324726939201355,
-0.1408519744873047,
0.3098248541355133,
-0.501326322555542,
0.5177136659622192,
0.31188565492630005,
0.3661406934261322,
-0.4993191659450531,
-0.36654549837112427,
-0.09611065685749054,
0.10357438772916794,
-0.24995344877243042,
-0.060980282723903656,
-1.044157862663269,
1.0139864683151245,
0.17960873246192932,
-0.32889145612716675,
1.0211741924285889,
0.06412500143051147,
-0.1836838573217392,
-0.02222013846039772,
0.021983778104186058,
0.30604419112205505,
-0.11195380985736847,
-0.36003953218460083,
-0.10459785908460617,
0.4128161072731018,
0.14140480756759644,
-0.6833404898643494,
-0.08796394616365433,
0.32943296432495117,
-0.15225230157375336,
-0.27114954590797424,
-0.29194408655166626,
0.11764682084321976,
-0.08103475719690323,
-0.7421243190765381,
0.6319262385368347,
0.5381876826286316,
-0.14101561903953552,
0.767964243888855,
-0.3842175006866455,
-0.17088963091373444,
0.45387253165245056,
-0.05503252148628235,
0.8650116324424744,
0.8443840146064758,
0.12696515023708344,
0.5268060564994812,
-0.3247326612472534,
-0.0399298258125782,
-0.3925797641277313,
0.2887389659881592,
-0.141021266579628,
-0.2338811457157135,
0.3548428416252136,
0.04794778674840927,
0.13413724303245544,
0.019616497680544853,
-0.04510499909520149,
-0.2468084692955017,
0.4298366904258728,
-0.4123464822769165,
0.093863345682621,
-0.07054679840803146,
-0.4306757152080536,
-0.3102867901325226,
0.16925935447216034,
-0.04576348513364792,
0.40358617901802063,
-0.15229260921478271,
0.15996386110782623,
0.04566665366292,
-0.7780845165252686,
0.15905657410621643,
-0.06422322243452072,
0.1743626892566681,
-0.566922664642334,
-0.13929542899131775,
0.4178144931793213,
-0.3155822157859802,
-0.1257484257221222,
-0.2832970917224884,
-0.5496601462364197,
-0.08275347203016281,
-0.22358621656894684,
-0.6415777802467346,
-0.5034511685371399,
0.44785037636756897,
0.5236031413078308,
-0.12324199825525284,
-0.3441162109375,
-0.23365168273448944,
0.5340206027030945,
-0.1984361857175827,
-0.9038739204406738,
0.3193208575248718,
0.3132755756378174,
-0.8990640640258789,
-0.3003091812133789,
0.3627951741218567,
0.1850670874118805,
0.4860977828502655,
0.15804213285446167,
-0.8121190071105957,
0.060988280922174454,
-0.09928922355175018,
0.08126683533191681,
-0.1003793329000473,
-0.271449476480484,
-0.47262632846832275,
0.40919968485832214,
-0.13609129190444946,
-0.406554639339447,
0.044830672442913055,
0.4885385036468506,
0.1926613748073578,
0.08616246283054352,
0.29956313967704773,
0.05550752952694893,
-0.5281034708023071,
0.46775275468826294,
0.07347062975168228,
-0.06143130362033844,
0.008171691559255123,
-0.38859957456588745,
-0.3862723112106323,
0.5168148875236511,
-5.843936920166016,
-0.4583777189254761,
-0.4883481562137604,
-0.2505173683166504,
-0.08374857157468796,
0.4629548490047455,
0.5325273871421814,
-0.2952377200126648,
-0.04363711550831795,
0.216189444065094,
0.04618975520133972,
0.008941362611949444,
-0.2702465057373047,
0.6972983479499817,
0.8283745050430298,
-0.3878980576992035,
0.4193188548088074,
-0.07687707990407944,
0.3555926978588104,
0.6728547215461731,
0.3595280647277832,
-0.13516677916049957,
-0.38334041833877563,
0.40704333782196045,
-0.4239380359649658,
0.15811695158481598,
-0.37875914573669434,
0.29546016454696655,
-0.10791094601154327,
-0.42019566893577576,
-0.2395886331796646,
-0.6573925018310547,
-0.26797419786453247,
0.09297578781843185,
-0.010718518868088722,
0.27672818303108215,
0.3842651844024658,
0.03741844743490219,
0.2005877047777176,
-0.138344869017601,
0.48827093839645386,
0.5151022672653198,
0.16057978570461273,
0.15553078055381775,
0.629714846611023,
-0.26734215021133423,
-0.1640794277191162,
0.11975421756505966,
-0.5791085958480835,
0.5925381183624268,
-0.02399788796901703,
-0.011960814706981182,
0.043838582932949066,
0.15844100713729858,
-0.5164721608161926,
-0.25064969062805176,
0.060381930321455,
0.3239382803440094,
-0.41210755705833435,
0.43218621611595154,
-0.09884187579154968,
-0.20829302072525024,
0.14031681418418884,
-0.32113227248191833,
-0.5527985095977783,
0.19430367648601532,
-0.5266336798667908,
-0.5315638184547424,
0.10217513889074326,
0.39853188395500183,
-0.08125746250152588,
0.03435201942920685,
-0.09099350869655609,
-0.19435034692287445,
0.5195841193199158,
-0.2606038451194763,
0.49337631464004517,
-0.34593823552131653,
0.1843261569738388,
0.4203225076198578,
-0.5891203284263611,
-0.629446268081665,
0.8046644330024719,
-0.2915935814380646,
-0.1881035417318344,
0.28872567415237427,
-0.13488605618476868,
0.3446345031261444,
-0.3541075587272644,
-0.18935665488243103,
0.3758946657180786,
-0.3512106239795685,
0.4816964864730835,
0.611589789390564,
0.6041353940963745,
0.6318739652633667,
0.6989173889160156,
0.2850756347179413,
0.6644659042358398,
-0.19799047708511353,
0.7106565833091736,
-0.42165887355804443,
-0.04607637971639633,
-0.8091280460357666,
-0.18818937242031097,
0.04446512833237648,
-0.4854406714439392,
-0.032862335443496704,
0.5812462568283081,
-0.2584887146949768,
0.12500782310962677,
0.3777884840965271,
0.11214699596166611,
0.03324989229440689,
0.44908013939857483,
0.11214815080165863,
-0.18187882006168365,
-0.40059706568717957,
0.48253321647644043,
0.25475069880485535,
-0.041826747357845306,
-0.14474216103553772,
-0.07009296864271164,
0.2591613531112671,
0.11027459800243378,
-0.542884111404419,
0.14194461703300476,
0.5103079080581665,
-0.43743059039115906,
-0.08133258670568466,
0.35016101598739624,
0.18299438059329987,
0.038488950580358505,
0.3610921800136566,
0.5686348676681519,
-0.8121801018714905,
-0.07561646401882172,
0.61648029088974,
0.26236894726753235,
-0.016681257635354996,
0.84642493724823,
0.21776288747787476,
-0.2930337190628052,
-0.16216883063316345,
0.1351512223482132,
-0.29205217957496643,
-0.4238024950027466,
-0.0616707019507885,
-0.13048049807548523,
-0.25413432717323303,
-0.19981858134269714,
-0.2618646025657654,
0.1853947639465332,
-0.44798049330711365,
-0.1283060610294342,
0.15609312057495117,
0.16392411291599274,
-0.0982169434428215,
0.5836739540100098,
-0.7948038578033447,
0.14152465760707855,
0.6590788960456848,
-1.2074733972549438,
-0.20783764123916626,
-0.19945037364959717,
0.23109835386276245,
-0.8908751010894775,
-0.3253322243690491,
0.3209356367588043,
0.8499142527580261,
-0.3011426627635956,
-0.4276222586631775,
0.09696677327156067,
-0.5732033252716064,
0.4481874108314514,
0.5318904519081116,
0.2648889422416687,
-0.2574766278266907,
-0.17486779391765594,
-0.10569974780082703,
0.030780257657170296,
0.13959522545337677,
-0.3455193042755127,
0.70237135887146,
0.943906307220459,
0.2822747826576233,
0.08201788365840912,
0.5986156463623047,
-0.2537612020969391,
0.3934245705604553,
0.23014317452907562,
-0.0061683994717895985,
0.3146263360977173,
-0.30717310309410095,
-0.020554490387439728,
-0.1536591351032257,
0.42780983448028564,
0.6646842360496521,
-0.422397255897522,
0.01187127735465765,
-0.08526782691478729,
0.4014018774032593,
-0.12473992258310318,
0.3596975803375244,
0.40065935254096985,
-0.19480329751968384,
-0.22887937724590302,
-0.35000115633010864,
-0.4602340757846832,
-0.3121458888053894,
-0.4719003736972809,
-0.09407103806734085,
-0.23207654058933258,
-0.08368335664272308,
-0.26729026436805725,
-0.06252317875623703,
0.14455853402614594,
-0.34045732021331787,
-0.02197198010981083,
0.23666995763778687,
-0.15231163799762726,
-0.3672637641429901,
0.01902186870574951,
0.32958558201789856,
0.08415081351995468,
-0.13517366349697113,
0.33769872784614563,
-0.8268314599990845,
0.16501440107822418,
0.4737553000450134,
-0.2801312208175659,
-0.18372571468353271,
-0.2703445851802826,
-0.11836817860603333,
0.07718100398778915,
-0.45173314213752747,
-0.6400821805000305,
0.6146461367607117,
0.10931788384914398,
-0.5331262946128845,
-0.003433691803365946,
0.6075108647346497,
-0.09797745943069458,
0.06756856292486191,
-0.1429724395275116,
0.044025175273418427,
0.27875176072120667,
-0.2640479505062103,
-0.47698524594306946,
0.25916236639022827,
0.05765649303793907,
-0.3234995901584625,
-0.2504717707633972,
-0.2772969603538513,
0.8780951499938965,
-0.03079148195683956,
0.24068234860897064,
-0.8744064569473267,
0.07188952714204788,
0.5760803818702698,
-0.3429482877254486,
-0.6467804312705994,
0.08792497962713242,
-0.5400794148445129,
0.16524215042591095,
0.26623454689979553,
-0.15578007698059082,
-0.6877078413963318,
0.6400530934333801,
-0.24518601596355438,
0.3505682051181793,
-0.3273853659629822,
0.39749273657798767,
0.45972537994384766,
0.281609445810318,
-0.21571332216262817,
-0.00730251194909215,
0.22873030602931976,
0.056585103273391724,
0.27509331703186035,
-0.49497851729393005,
-0.30292659997940063,
0.2820146381855011,
-0.24466510117053986,
0.04963162541389465,
-0.07989967614412308,
0.19061918556690216,
0.111384816467762,
-0.5237176418304443,
0.06573399156332016,
-0.3773307502269745,
0.3503177762031555,
-0.634756326675415,
-0.6912565231323242,
-0.10923238098621368,
-0.0917796939611435,
0.34175294637680054,
0.40566933155059814,
-0.5091647505760193,
0.07009831815958023,
-0.2161320000886917,
0.2149818390607834,
0.29856008291244507,
-0.7426841259002686,
0.12067820876836777,
-0.09646254032850266,
0.28040772676467896,
-0.2827848494052887,
-0.45534747838974,
-0.05981704592704773,
0.24000129103660583,
0.14865374565124512,
0.4280324876308441,
0.27281323075294495,
-0.13650310039520264,
-0.3230364918708801,
0.5981987714767456,
-0.24246764183044434,
1.2295863628387451,
0.5918554663658142,
-0.08687098324298859,
0.5886391401290894,
0.46670737862586975,
-0.18444755673408508,
-0.5938919186592102,
0.47815707325935364,
0.02324289083480835,
0.7291432619094849,
0.07046560198068619,
0.006829237099736929,
0.32984963059425354,
0.16441543400287628,
-0.24075376987457275,
0.27298858761787415,
-0.6188206076622009,
-0.11088778078556061,
0.6520773768424988,
-0.8053407669067383,
-0.28765180706977844,
0.05071762576699257,
0.2956697642803192,
0.2512654662132263,
-0.036401208490133286,
0.4290109872817993,
-0.3650122880935669,
0.3521583378314972,
0.3416462242603302,
-0.10808337479829788,
-0.917371392250061,
0.34980109333992004,
0.09811718016862869,
-0.4928942024707794,
0.5634231567382812,
-0.24175667762756348,
0.12740859389305115,
-0.37136948108673096,
0.456467866897583,
-0.19463682174682617,
-0.12110026925802231,
0.1492491215467453,
0.23890535533428192,
1.070777177810669,
-0.2218596339225769,
-0.23299044370651245,
-0.4453730881214142,
-0.5131949186325073,
0.8446769118309021,
0.1883828341960907,
0.1562248170375824,
-0.11939040571451187,
-0.15709422528743744,
0.4302065968513489,
-0.18173395097255707,
0.7063053250312805,
-0.1254122257232666,
-1.109297752380371,
-0.03886137902736664,
0.7201794981956482,
0.0015736696077510715,
0.16295967996120453,
0.41074973344802856,
0.27480897307395935,
0.06226158142089844,
-0.8179548978805542,
-0.06966585665941238,
-0.25369343161582947,
0.12361573427915573,
0.12759281694889069,
0.26134175062179565,
-0.24230219423770905,
0.4625042974948883,
-0.42230871319770813,
-0.35887157917022705,
0.2676052451133728,
-0.27390316128730774,
-0.10815773159265518,
-0.2943095266819,
0.499194860458374,
0.48487135767936707,
0.1237606629729271,
-0.008177789859473705,
0.07439237833023071,
-0.08594703674316406,
0.5389569997787476,
-0.17225250601768494,
-0.10053537040948868,
-0.35202735662460327,
0.2780342996120453,
-0.7350190281867981,
-0.17963120341300964,
0.39772269129753113,
-0.4574918746948242,
-0.22227507829666138,
0.15613527595996857,
0.198160320520401,
-0.04971427842974663,
0.5346564054489136,
-0.3578919470310211,
0.11153506487607956,
-0.023653946816921234,
-0.452914297580719,
-0.4185106158256531,
1.212602972984314,
-0.03446000814437866,
0.0015371674671769142,
0.21789468824863434,
0.14116564393043518,
-0.05392433702945709,
-0.3728860318660736,
0.5444313883781433,
-0.41538330912590027,
-0.4559951722621918,
-0.38445034623146057,
1.0128612518310547,
0.14390763640403748,
-0.15021392703056335,
0.0494249053299427,
-0.0008854972547851503,
0.585547685623169,
-0.21110786497592926,
-0.5452550649642944,
-0.19564570486545563,
-0.6558150053024292,
-0.1542680859565735,
-0.13086220622062683,
0.8841471672058105,
-0.3424595892429352,
-0.07609985023736954,
-0.16201716661453247,
0.4795168340206146,
-0.19569575786590576,
-0.36063066124916077,
-0.5386953353881836,
-0.09935082495212555,
-0.9160096049308777,
-0.1872645914554596,
-0.1875259131193161,
-0.28865763545036316,
-0.49689340591430664,
-0.28637877106666565
] |
sparse PCA [17], Gaussian mixtures [69] and certain joint scalings of (n, d) [54].
However, in most applications, noise is highly structured and correlated, thereby calling for more
realistic assumptions on W than having i.i.d. entries. A recent line of work addresses this concern
by studying matrix denoising with heteroscedastic noise [1, 66, 29, 40, 23], resting on two basic
ideas: whitening and singular value shrinkage. Whitening refers to multiplying the data matrix by | [
0.13035106658935547,
-0.1265375167131424,
-0.032605089247226715,
-0.2511976659297943,
-0.21057918667793274,
-0.017114590853452682,
-0.24153079092502594,
-0.4219585359096527,
-0.28302666544914246,
-0.3220566511154175,
-0.16410361230373383,
0.05847584828734398,
-0.6699129343032837,
0.4198119640350342,
-0.3444521725177765,
0.5958693027496338,
0.5956477522850037,
0.22791625559329987,
-0.4586901068687439,
-0.5994336009025574,
0.4679894745349884,
0.06759146600961685,
0.08478916436433792,
0.2317795753479004,
-0.11255167424678802,
0.2758610248565674,
0.4430238604545593,
0.3026340901851654,
-0.15614978969097137,
0.0748361125588417,
-0.42321068048477173,
0.525112509727478,
0.47611480951309204,
0.2883816659450531,
0.7165025472640991,
-0.043293070048093796,
0.01948966085910797,
0.09073713421821594,
0.15825626254081726,
-0.06843769550323486,
0.35804909467697144,
0.0574074424803257,
-0.057636842131614685,
0.19325287640094757,
-0.6944367289543152,
-0.7471697926521301,
-1.0922377109527588,
0.5954316258430481,
-0.2265157699584961,
-0.41509899497032166,
-0.5561391711235046,
0.4313232898712158,
0.7661747336387634,
0.7162761688232422,
-0.2523557245731354,
-0.14916162192821503,
-0.13847029209136963,
-0.36258265376091003,
0.6854437589645386,
-0.23761670291423798,
-0.05981604382395744,
-0.1869012862443924,
-0.1383853554725647,
-0.11891913414001465,
0.1373777985572815,
0.46242839097976685,
-0.04610797390341759,
-0.4133562445640564,
-0.5472337007522583,
-0.0960317850112915,
0.07068443298339844,
-0.2799381911754608,
-0.06085522472858429,
0.09503383189439774,
-0.41351550817489624,
0.4745800793170929,
0.18033403158187866,
-0.10437072068452835,
-0.048170410096645355,
-0.04243770241737366,
-0.4383126199245453,
0.07380898296833038,
0.26281288266181946,
-0.20155753195285797,
0.6270328164100647,
0.1029236689209938,
-0.36707451939582825,
0.6678829193115234,
-0.4630468487739563,
-0.009827020578086376,
-0.0656716451048851,
-0.02724197879433632,
0.2683560848236084,
0.5629625916481018,
-0.009690479375422001,
0.5959843993186951,
0.15119439363479614,
-0.4275379478931427,
0.766058623790741,
0.30616816878318787,
-0.27557799220085144,
0.11369751393795013,
0.5934110283851624,
-0.34523406624794006,
-0.4245571196079254,
-0.4966614544391632,
-0.4391697645187378,
-0.09114861488342285,
-0.22908593714237213,
0.24010533094406128,
-0.4651729464530945,
0.241362527012825,
-0.14150302112102509,
-0.3042486608028412,
-0.0876474529504776,
0.3674660623073578,
0.006516104564070702,
0.22697223722934723,
-0.3119373619556427,
0.2553102672100067,
-0.18042434751987457,
1.5369534492492676,
-0.08880141377449036,
0.6731020212173462,
-0.11954565346240997,
0.08802741765975952,
0.6421193480491638,
0.7883363962173462,
-0.5353475213050842,
0.1602386236190796,
0.07691283524036407,
1.0221679210662842,
0.05860473960638046,
0.1592237502336502,
-0.03093930520117283,
-0.5253846049308777,
-0.6150044798851013,
0.09607576578855515,
-0.12161406129598618,
-0.5628742575645447,
-0.32177138328552246,
-0.58939528465271,
0.30170565843582153,
0.22216160595417023,
0.1949233114719391,
-0.3773413896560669,
0.013656108640134335,
-0.30530697107315063,
-0.17570392787456512,
0.12097998708486557,
-0.17955148220062256,
0.5438119173049927,
-0.6075292229652405,
-0.588444173336029,
-0.13581439852714539,
-0.011089351028203964,
0.03434482589364052,
0.011555749922990799,
-0.2924374043941498,
0.21832452714443207,
-0.06720027327537537,
-0.057468656450510025,
-0.5429143905639648,
-0.6951952576637268,
0.21044795215129852,
-0.7976101636886597,
0.44930940866470337,
0.35796424746513367,
0.021201852709054947,
0.2951105237007141,
0.08280418068170547,
-0.2681019604206085,
-0.245966836810112,
0.04152838513255119,
-0.002703930251300335,
0.4546257555484772,
0.1724456548690796,
-0.027202464640140533,
0.26802724599838257,
0.009098273701965809,
-0.8719327449798584,
0.007754424121230841,
0.44835352897644043,
-0.3269161283969879,
0.21811603009700775,
0.41417112946510315,
0.23094648122787476,
0.02398436702787876,
0.05042760446667671,
-0.34346824884414673,
-0.45913001894950867,
0.15991227328777313,
0.0669032409787178,
-1.1119811534881592,
1.1998307704925537,
0.49266883730888367,
-0.15535955131053925,
0.9169414043426514,
0.18486468493938446,
-0.05138994753360748,
-0.0032127914018929005,
-0.19054514169692993,
0.1671871691942215,
-0.14473436772823334,
-0.3134732246398926,
0.2960270643234253,
0.1357552409172058,
0.4695815443992615,
-0.8374575972557068,
-0.13782952725887299,
0.5065780878067017,
-0.3606102764606476,
0.07148427516222,
0.006572227459400892,
0.5293557643890381,
0.07497935742139816,
-0.407048761844635,
0.23976553976535797,
0.5036606192588806,
-0.3492318391799927,
0.8805561065673828,
-0.41092127561569214,
0.09198427945375443,
0.6627464294433594,
0.1303359568119049,
1.2127748727798462,
0.7837294340133667,
0.044427528977394104,
0.8698500990867615,
-0.27743709087371826,
0.09281137585639954,
-0.284537136554718,
0.09693651646375656,
-0.050068166106939316,
-0.030679235234856606,
0.17137554287910461,
0.36417919397354126,
0.009735113009810448,
0.4379170835018158,
-0.3160169720649719,
-0.07487129420042038,
-0.16350743174552917,
-0.33490586280822754,
-0.3032839000225067,
0.30683669447898865,
-0.33857303857803345,
-0.2032482624053955,
-0.1337854415178299,
-0.2106153964996338,
0.37418997287750244,
0.36309385299682617,
0.32686859369277954,
0.37546926736831665,
-0.9125497341156006,
-0.031125059351325035,
-0.07486013323068619,
-0.04539617896080017,
-0.650783896446228,
0.3755783438682556,
0.33570945262908936,
-0.4956881105899811,
-0.27239567041397095,
-0.35850200057029724,
-0.6653241515159607,
-0.09617412835359573,
0.1161734014749527,
-0.716712474822998,
-0.3633759617805481,
0.8417085409164429,
0.07492591440677643,
0.2412748634815216,
-0.2294209748506546,
-0.5119938850402832,
-0.02942105196416378,
-0.674527645111084,
-1.1992366313934326,
0.09328644722700119,
0.04150586575269699,
-0.7431202530860901,
-0.6828542351722717,
0.10921187698841095,
0.44944560527801514,
0.3369070291519165,
-0.2609657943248749,
-0.5253421664237976,
-0.1414482742547989,
-0.16649489104747772,
-0.10767769068479538,
-0.06995353102684021,
-0.33823567628860474,
-0.5467310547828674,
0.7205295562744141,
-0.24085161089897156,
-0.21268630027770996,
0.2591860890388489,
1.0869406461715698,
0.07814127951860428,
-0.21269568800926208,
0.27267056703567505,
-0.21274879574775696,
-0.2937999367713928,
-0.13432756066322327,
-0.05059410259127617,
0.23719070851802826,
-0.6092060804367065,
-0.210971400141716,
-0.38221272826194763,
0.3701416552066803,
-5.696159839630127,
0.27910107374191284,
-0.4746450185775757,
0.2372865527868271,
-0.5313095450401306,
-0.0750422403216362,
0.4089481830596924,
-0.03277967870235443,
0.2793149948120117,
0.19299042224884033,
0.09752503782510757,
0.030110621824860573,
-0.2603951096534729,
0.7658858895301819,
0.6577188968658447,
0.1383861005306244,
0.30563709139823914,
-0.19664447009563446,
0.5081846117973328,
0.25098755955696106,
0.09932978451251984,
0.19783562421798706,
-0.13583895564079285,
-0.36558520793914795,
0.0028373529203236103,
0.10886431485414505,
-0.35070517659187317,
-0.060835301876068115,
-0.5132756233215332,
-0.7715319395065308,
-0.2738797664642334,
-0.7819021940231323,
-0.14165815711021423,
-0.1483668088912964,
0.19792303442955017,
0.38826459646224976,
0.28175750374794006,
0.12466274946928024,
-0.07824953645467758,
-0.41969525814056396,
0.6326108574867249,
0.9693347215652466,
0.13142210245132446,
-0.1266290694475174,
0.685610294342041,
0.265518456697464,
-0.2638634443283081,
-0.3185957670211792,
-0.12705601751804352,
0.5738765597343445,
-0.2945767343044281,
-0.04312412813305855,
0.28597477078437805,
0.2660600543022156,
-0.07333209365606308,
0.10049020498991013,
-0.03071352280676365,
0.24022580683231354,
-0.5235424637794495,
-0.003672591643407941,
0.08975886553525925,
-0.24393735826015472,
0.19735988974571228,
-0.39563417434692383,
-0.5642449855804443,
0.28060686588287354,
-0.4626583456993103,
-0.0589471310377121,
0.6241165995597839,
-0.12006045132875443,
-0.12208381295204163,
-0.6260740756988525,
0.1944476217031479,
-0.1944946050643921,
0.4376632869243622,
-0.5587427616119385,
0.46803370118141174,
-0.04368775337934494,
-0.1643380969762802,
0.2828559875488281,
-0.25877976417541504,
-0.507206916809082,
0.1379779577255249,
-0.02218799479305744,
0.09542266279459,
-0.13289663195610046,
0.00493615260347724,
0.26905253529548645,
-0.32110047340393066,
0.22568228840827942,
0.33043956756591797,
0.07677234709262848,
0.1662982702255249,
0.6434895396232605,
0.905798077583313,
0.3074086606502533,
0.7838367223739624,
0.25589480996131897,
0.7776799201965332,
0.2567135989665985,
0.7320108413696289,
-0.2800074517726898,
-0.6076155304908752,
-0.9930203557014465,
0.004492633044719696,
0.2595803141593933,
-0.4558102488517761,
0.3340982496738434,
0.6611964106559753,
-0.25491511821746826,
0.10972785204648972,
0.45418018102645874,
-0.3223142921924591,
-0.28629687428474426,
0.7588673233985901,
-0.03760116174817085,
-0.21517814695835114,
0.08298292011022568,
0.24667786061763763,
-0.2209511548280716,
0.032909806817770004,
0.39264699816703796,
-0.30926424264907837,
0.3585846424102783,
0.07134942710399628,
-0.4089146554470062,
-0.27786481380462646,
0.7461742162704468,
-0.6776350736618042,
-0.5699885487556458,
0.5282445549964905,
0.4159601926803589,
-0.6025113463401794,
0.12839758396148682,
0.203209787607193,
-0.762248158454895,
-0.11547966301441193,
0.46153658628463745,
0.27047616243362427,
-0.3804386258125305,
0.6041831374168396,
-0.09083274751901627,
-0.20589564740657806,
0.2332761138677597,
-0.14993742108345032,
0.07052301615476608,
-0.14517462253570557,
-0.20832692086696625,
-0.16629067063331604,
-0.01880352757871151,
-0.00801275297999382,
-0.6513946652412415,
0.7183383107185364,
0.012172270566225052,
-0.15794207155704498,
0.21259717643260956,
0.558821439743042,
0.0972786545753479,
0.3618939220905304,
-0.5156185030937195,
0.16383415460586548,
0.4945134222507477,
-0.7474625706672668,
-0.06442335993051529,
-0.21631528437137604,
-0.5142314434051514,
-0.8904827237129211,
-0.3094201385974884,
0.0012645595706999302,
0.40127673745155334,
-0.026223234832286835,
-0.49158844351768494,
0.08509562164545059,
-0.0852590948343277,
0.31598973274230957,
0.11715758591890335,
0.5673646926879883,
-0.24542143940925598,
-0.29871803522109985,
-0.20978161692619324,
-0.06214139610528946,
-0.3038104474544525,
-0.5588038563728333,
0.9760006666183472,
0.8193342685699463,
0.5317596793174744,
0.3320055902004242,
0.4000498950481415,
-0.5211650133132935,
-0.33191290497779846,
0.12584693729877472,
-0.16680501401424408,
0.508415937423706,
-0.34191107749938965,
0.059629887342453,
-0.2537917196750641,
0.8892636299133301,
0.2240237146615982,
-0.797067403793335,
-0.3778529465198517,
-0.09203480929136276,
0.29320746660232544,
-0.7419390678405762,
0.32587307691574097,
0.27312925457954407,
-0.15930427610874176,
-0.4871255159378052,
-0.4324348270893097,
-0.6978244781494141,
-0.5399763584136963,
-0.5524028539657593,
-0.14993436634540558,
-0.48360252380371094,
-0.5326572060585022,
-0.3071395754814148,
0.2637283205986023,
0.11757564544677734,
-0.2665875554084778,
-0.2370636910200119,
-0.06401921063661575,
0.23666758835315704,
-0.7237128019332886,
0.476104736328125,
0.11770609766244888,
0.00390706118196249,
0.19206108152866364,
-0.17583346366882324,
-0.6691365838050842,
0.16408255696296692,
-0.06369255483150482,
-0.3690980076789856,
-0.2323620468378067,
-0.43174001574516296,
0.21808750927448273,
-0.46126827597618103,
-0.16226999461650848,
-0.7075544595718384,
0.3267984986305237,
0.1506006270647049,
0.29771438241004944,
0.5756229162216187,
0.7756384611129761,
-0.22004064917564392,
-0.19723166525363922,
-0.14121130108833313,
0.34123873710632324,
0.001308455248363316,
-0.34066152572631836,
-0.5841359496116638,
-0.18144087493419647,
0.17179858684539795,
-0.1781839281320572,
-0.007179349195212126,
0.2340150773525238,
0.949623167514801,
0.253586083650589,
0.11381600052118301,
-1.0076936483383179,
0.06305079907178879,
0.413906991481781,
-0.33062034845352173,
-0.7563794851303101,
0.2647189795970917,
-0.3677467405796051,
0.41709744930267334,
0.1687169224023819,
-0.21198171377182007,
-0.17658521234989166,
0.5070592761039734,
-0.3354525864124298,
-0.000400462478864938,
-0.13722757995128632,
0.18248458206653595,
0.39533793926239014,
-0.0315692164003849,
0.11473902314901352,
-0.294182151556015,
0.3347232937812805,
0.06858118623495102,
0.1182580441236496,
-0.5347739458084106,
-0.7239041328430176,
0.057364437729120255,
-0.4550533890724182,
-0.07026199996471405,
0.22967010736465454,
0.29799917340278625,
-0.3173002600669861,
-0.24800056219100952,
0.07825329154729843,
-0.4770352840423584,
-0.1769535392522812,
-0.4333741366863251,
-0.5378007888793945,
-0.1430259793996811,
-0.2724289298057556,
0.24812859296798706,
0.635854959487915,
-0.014985857531428337,
-0.1670995056629181,
-0.08310244977474213,
0.4267425239086151,
-0.08462207019329071,
-0.7374663949012756,
0.3386511206626892,
-0.08492468297481537,
0.38209253549575806,
-0.04611814394593239,
-0.44165003299713135,
-0.4277779757976532,
0.12833580374717712,
-0.05006794631481171,
0.1836400330066681,
0.3284389078617096,
-0.5621330738067627,
0.10251186788082123,
0.7593256831169128,
0.08460855484008789,
1.0626282691955566,
0.0480329804122448,
-0.11503902822732925,
-0.010332413949072361,
0.2107500433921814,
0.1364988386631012,
-0.5367234945297241,
0.5002649426460266,
0.011986094526946545,
0.11983571946620941,
-0.05587952211499214,
0.0449410118162632,
0.3321807086467743,
0.5892989635467529,
-0.5986682176589966,
0.19912047684192657,
-0.20315691828727722,
0.16468200087547302,
0.521177351474762,
-0.39034461975097656,
-0.2609110176563263,
0.22646857798099518,
0.8879169225692749,
0.8939995765686035,
-0.7697184085845947,
0.6414200663566589,
0.1402171552181244,
0.059144116938114166,
0.3922859728336334,
-0.04917731508612633,
-0.2657105326652527,
0.34513619542121887,
0.08797609061002731,
-0.06777188181877136,
0.08831071108579636,
0.00984288565814495,
-0.09198246151208878,
-0.10279601067304611,
0.4493439495563507,
-0.428993821144104,
0.2594640254974365,
0.33761096000671387,
0.34171411395072937,
0.5227013230323792,
-0.575563907623291,
0.38388949632644653,
-0.641717255115509,
-0.3055315613746643,
0.4413266181945801,
0.2775636911392212,
0.16302210092544556,
-0.5707612037658691,
0.03631561994552612,
0.3025316298007965,
-0.2737695574760437,
0.26137763261795044,
-0.07061748206615448,
-0.586391270160675,
0.07014269381761551,
0.534813404083252,
-0.03760867938399315,
0.08951554447412491,
0.47942492365837097,
0.1958264857530594,
0.005319382529705763,
-0.8084924817085266,
0.33962324261665344,
-0.49589747190475464,
0.36826613545417786,
0.38921311497688293,
0.18914082646369934,
0.3603302836418152,
0.14257293939590454,
-0.7954014539718628,
-0.5828749537467957,
-0.021764429286122322,
0.12661214172840118,
-0.1398264765739441,
-0.3308960199356079,
0.854619562625885,
0.4672388434410095,
-0.005022678058594465,
0.586753249168396,
0.16956110298633575,
0.21179218590259552,
0.5886112451553345,
-0.45479366183280945,
0.23971371352672577,
-0.7028535604476929,
-0.3069360852241516,
-1.0412935018539429,
0.07092953473329544,
0.8014247417449951,
-0.1696079820394516,
-0.4263750910758972,
-0.06279870867729187,
0.27131882309913635,
0.005241850391030312,
0.06475773453712463,
-0.37208691239356995,
-0.23612160980701447,
-0.4455093741416931,
-0.5427140593528748,
-0.3691803514957428,
0.8925168514251709,
-0.05380995199084282,
0.4080059230327606,
0.37337052822113037,
0.883674144744873,
-0.2017514407634735,
0.07552476972341537,
0.49775367975234985,
-0.09558751434087753,
-0.07624392211437225,
-0.08523771911859512,
0.6563466191291809,
-0.05395953729748726,
0.08391018956899643,
0.03339061141014099,
-0.087821826338768,
0.4926077723503113,
-0.3366239666938782,
-0.16374889016151428,
-0.2929244339466095,
-0.6421480774879456,
-0.014262145385146141,
-0.286117821931839,
0.8984804153442383,
-0.2919074594974518,
-0.45292437076568604,
-0.8569293022155762,
0.3924548923969269,
-0.06778105348348618,
-0.3757595419883728,
-0.3688754141330719,
-0.039834022521972656,
-0.44032415747642517,
0.34601065516471863,
-0.18577617406845093,
-0.16104134917259216,
-0.32584160566329956,
-0.18750663101673126
] |
ideas: whitening and singular value shrinkage. Whitening refers to multiplying the data matrix by
the square root of the inverse covariance, in order to reduce the model to one with i.i.d. noise; and
singular value shrinkage retains the singular vectors of the data while deflating the singular values to
correct for the noise. Though the exact asymptotic performance of these algorithms has been derived
[66, 29, 40, 23], their optimality is yet to be determined from a Bayesian standpoint. In fact, we will | [
0.006064620800316334,
-0.004590221215039492,
-0.07696246355772018,
-0.009579046629369259,
-0.1730508804321289,
0.4136766791343689,
-0.12243042141199112,
-0.3679742217063904,
-0.21569442749023438,
-0.23310315608978271,
0.0391596257686615,
0.40722817182540894,
-0.5499229431152344,
0.2670421004295349,
-0.4193200469017029,
0.3686562478542328,
0.44694238901138306,
-0.07239034026861191,
-0.30431532859802246,
-0.02192625217139721,
0.11722631007432938,
0.10095246136188507,
-0.2760319411754608,
-0.2097621113061905,
-0.18588459491729736,
-0.04924009367823601,
0.2895919382572174,
-0.10978872328996658,
-0.211069256067276,
0.21215671300888062,
-0.22911790013313293,
0.4917828440666199,
-0.16541707515716553,
-0.1509716510772705,
0.44108283519744873,
0.26114410161972046,
0.13898298144340515,
0.07575684785842896,
0.01960400678217411,
0.07242197543382645,
0.15735793113708496,
-0.10421006381511688,
0.021953776478767395,
0.340838760137558,
-0.5902553796768188,
-0.5455917716026306,
-1.4208165407180786,
0.16258342564105988,
-0.3749738037586212,
-0.35034361481666565,
-0.5014110803604126,
0.40118491649627686,
0.2689189612865448,
0.7721758484840393,
-0.4615781009197235,
0.17266885936260223,
0.2896408140659332,
0.04405955970287323,
0.2848357558250427,
-0.7691853642463684,
0.025007838383316994,
-0.1526026725769043,
-0.2139778882265091,
0.07619541883468628,
-0.21589435636997223,
0.17371311783790588,
-0.03164473921060562,
0.26101040840148926,
-0.8789656758308411,
-0.5252767205238342,
-0.04289388656616211,
-0.3153274953365326,
-0.6317334771156311,
0.27170389890670776,
-0.184332013130188,
-0.06062927842140198,
0.1714225858449936,
-0.3622259497642517,
0.07622665911912918,
-0.331353098154068,
-0.23497112095355988,
0.17920322716236115,
0.265971839427948,
-0.5200923085212708,
0.6902350783348083,
-0.16359089314937592,
-0.09258406609296799,
0.7118414044380188,
-0.8985072374343872,
0.04781269654631615,
-0.020203951746225357,
-0.19961273670196533,
0.5898703336715698,
0.49926039576530457,
0.3984951376914978,
0.5255224108695984,
0.6365881562232971,
-0.03660004958510399,
0.5849723815917969,
-0.045131102204322815,
-0.07618559151887894,
-0.3578321933746338,
0.09080474823713303,
-0.03814247250556946,
-0.3431963622570038,
-0.6025955080986023,
-0.24294091761112213,
-0.32013508677482605,
0.01865038275718689,
0.2535349428653717,
-0.839692234992981,
0.553653359413147,
0.30667644739151,
0.08458776026964188,
-0.49127084016799927,
0.22410455346107483,
-0.3930583596229553,
0.19797925651073456,
-0.40366172790527344,
0.5271942019462585,
-0.11912210285663605,
1.419898271560669,
-0.029335521161556244,
0.4548948109149933,
0.419428288936615,
0.18852922320365906,
0.3380569815635681,
0.9676839709281921,
-0.49644985795021057,
-0.16618631780147552,
0.21007569134235382,
0.5919807553291321,
-0.18178604543209076,
-0.38057807087898254,
-0.021274898201227188,
-0.5738211274147034,
-0.10997463017702103,
0.08578751981258392,
-0.13816474378108978,
-0.29051122069358826,
-0.19674864411354065,
-0.027739260345697403,
0.1295318752527237,
0.26055029034614563,
0.5319643020629883,
-0.7094954252243042,
0.04926362261176109,
-0.034982722252607346,
-0.06262636184692383,
0.10535900294780731,
-0.17938697338104248,
0.2835041880607605,
-0.49142953753471375,
-0.022131210193037987,
-0.0309952050447464,
0.340455025434494,
-0.3189636468887329,
-0.3054501414299011,
-0.3630504608154297,
0.6209278106689453,
0.19091548025608063,
-0.17476199567317963,
-0.6387197971343994,
-0.616773784160614,
0.18566420674324036,
-0.7531155347824097,
0.47582387924194336,
0.29672771692276,
0.1412985920906067,
-0.02001575380563736,
-0.39790427684783936,
-0.1676255315542221,
0.34423941373825073,
-0.3744577169418335,
0.34606456756591797,
0.42337384819984436,
-0.12683641910552979,
0.03718683496117592,
0.31364330649375916,
0.33486834168434143,
-0.8506626486778259,
-0.10690009593963623,
-0.02372417040169239,
-0.21665886044502258,
-0.5175100564956665,
0.05658455565571785,
0.08400740474462509,
0.06651746481657028,
0.32968655228614807,
-0.046477314084768295,
-0.7684406042098999,
0.22466742992401123,
0.07038175314664841,
-0.6893652677536011,
1.149228572845459,
0.1942407637834549,
-0.6451684832572937,
0.7882104516029358,
-0.038562607020139694,
0.21124866604804993,
-0.3255693018436432,
-0.2955578565597534,
-0.11807624995708466,
0.36750006675720215,
-0.3937787413597107,
0.24652937054634094,
-0.08387861400842667,
0.11873099207878113,
-0.5543428063392639,
0.1532057672739029,
0.6051095128059387,
-0.20298047363758087,
-0.11166703701019287,
-0.056465644389390945,
0.3198925852775574,
0.1303403526544571,
-0.38801974058151245,
0.18349716067314148,
0.1268923282623291,
-0.12405867874622345,
0.6017822623252869,
-0.0975155234336853,
-0.0847577229142189,
0.2686934471130371,
0.44352659583091736,
0.696338415145874,
0.8636560440063477,
0.12606477737426758,
0.8784959316253662,
-0.4448813498020172,
0.2540935277938843,
0.21281011402606964,
0.20431740581989288,
-0.16081084311008453,
-0.12444388121366501,
0.24286426603794098,
0.3806832730770111,
-0.1302054226398468,
0.5812352299690247,
-0.577738344669342,
-0.2567097842693329,
-0.06562802940607071,
-0.08418115973472595,
-0.008602514863014221,
0.10518346726894379,
-0.037212908267974854,
-0.16162872314453125,
0.11546511203050613,
-0.5360087156295776,
0.08673886209726334,
-0.09567215293645859,
0.6031960248947144,
0.08285193145275116,
-0.8220705986022949,
-0.008760632015764713,
0.12426544725894928,
-0.19856996834278107,
-0.07843916863203049,
0.3760409355163574,
0.6306293606758118,
-0.39374756813049316,
-0.10379064828157425,
-0.1858261376619339,
-0.48697513341903687,
-0.3868376612663269,
-0.08407386392354965,
-0.5416499972343445,
-0.07158549129962921,
0.49618610739707947,
0.3629037141799927,
0.42757952213287354,
-0.10595280677080154,
-0.7157531976699829,
0.15226340293884277,
-0.7792445421218872,
-1.1345270872116089,
0.4115787148475647,
0.36506858468055725,
-0.9007598757743835,
-0.5546373128890991,
-0.11022456735372543,
0.7265782356262207,
0.46613016724586487,
-0.053329240530729294,
-0.26355472207069397,
-0.33072584867477417,
0.06155465543270111,
-0.23070499300956726,
-0.7897568345069885,
0.13100188970565796,
-0.27835607528686523,
0.7119089365005493,
0.024747706949710846,
-0.21294763684272766,
0.11975502967834473,
0.7853903770446777,
-0.20787858963012695,
-0.3269897401332855,
0.5235629081726074,
0.09541009366512299,
-0.5204922556877136,
-0.0065940506756305695,
0.2481439709663391,
-0.024456551298499107,
-0.2291925549507141,
-0.19672009348869324,
-0.4713611304759979,
0.5728371739387512,
-6.025332927703857,
-0.07205391675233841,
-0.3888140022754669,
0.3559698164463043,
-0.5611916780471802,
-0.5068361759185791,
0.4877474308013916,
0.03807349130511284,
0.2062491923570633,
-0.23557813465595245,
-0.10598661005496979,
0.07860117405653,
-0.41275277733802795,
0.8355002999305725,
0.382990300655365,
0.31679922342300415,
0.6460213661193848,
-0.2100648283958435,
0.2599875032901764,
0.23470176756381989,
-0.06457608193159103,
-0.04023182764649391,
-0.17921946942806244,
0.100030317902565,
-0.08018183708190918,
0.5113505125045776,
-0.7460874319076538,
0.5241876244544983,
-0.7107611298561096,
-0.38721930980682373,
0.02164890244603157,
-0.3432362675666809,
0.07051024585962296,
-0.3697604835033417,
0.3843265175819397,
0.498857706785202,
0.1480407416820526,
-0.11426936835050583,
-0.08583033829927444,
-0.2837176024913788,
0.36905747652053833,
0.8332430720329285,
-0.1426658034324646,
0.21123449504375458,
0.6453712582588196,
-0.18431223928928375,
-0.10832483321428299,
0.022743579000234604,
-0.3429160416126251,
0.3880235254764557,
-0.038918375968933105,
-0.008379638195037842,
0.31178247928619385,
0.15532314777374268,
-0.08324284106492996,
-0.250595360994339,
0.004496896639466286,
0.17551499605178833,
-0.27217498421669006,
0.12676213681697845,
0.4250310957431793,
-0.2025948315858841,
-0.16888539493083954,
-0.22415944933891296,
-0.18426597118377686,
-0.0815737321972847,
-0.3957271873950958,
-0.48184576630592346,
0.5814527273178101,
0.011603295803070068,
-0.3604191839694977,
0.046152688562870026,
0.08941421657800674,
-0.5079100728034973,
0.2673846483230591,
-0.34921398758888245,
0.0790429636836052,
0.04417014122009277,
0.06538394838571548,
0.6290831565856934,
-0.3383565843105316,
-0.7831400632858276,
0.11240549385547638,
-0.20135235786437988,
-0.14720594882965088,
-0.0924011841416359,
-0.023203063756227493,
0.2485198676586151,
-0.3009628355503082,
0.31764957308769226,
0.6497674584388733,
-0.03464384749531746,
0.32061275839805603,
0.20045121014118195,
0.5036861896514893,
0.580082356929779,
0.273220032453537,
0.15207946300506592,
0.23745335638523102,
0.10202540457248688,
0.32719290256500244,
-0.2823482155799866,
-0.04087322577834129,
-0.638416051864624,
0.24228379130363464,
0.23025910556316376,
-0.1639380007982254,
0.5168489813804626,
0.6088185906410217,
-0.7119203209877014,
-0.23465414345264435,
0.5520533323287964,
-0.0660969689488411,
0.05374223738908768,
0.4808133840560913,
-0.025400761514902115,
-0.199066624045372,
0.06326126307249069,
0.2553926408290863,
-0.18736274540424347,
0.08411922305822372,
0.12160228937864304,
-0.2747376561164856,
-0.08936254680156708,
-0.2314867228269577,
-0.2764400839805603,
-0.22031152248382568,
0.6133330464363098,
-0.03756173700094223,
0.34008994698524475,
0.4281751811504364,
0.3802906274795532,
-0.6874704957008362,
0.21824952960014343,
0.5303774476051331,
-0.4924637973308563,
-0.10627055913209915,
0.36453843116760254,
-0.03270817548036575,
-0.2592156231403351,
0.8060922026634216,
0.3020603060722351,
-0.05192256346344948,
-0.20885871350765228,
0.148323655128479,
0.330814391374588,
-0.361009806394577,
-0.4568929970264435,
-0.22135762870311737,
0.023297511041164398,
0.0014865008415654302,
-0.42777517437934875,
-0.0442463643848896,
-0.3926841616630554,
0.000939294055569917,
0.23096585273742676,
0.3352311849594116,
-0.19337677955627441,
0.29575395584106445,
-0.11392475664615631,
0.0635979175567627,
0.16619989275932312,
-0.7373344898223877,
-0.4382511079311371,
0.1217355951666832,
-0.09891784936189651,
-0.7373654246330261,
-0.41150495409965515,
0.020176079124212265,
0.3599480390548706,
-0.1455857902765274,
-0.43573740124702454,
-0.7748265266418457,
-0.13661277294158936,
0.2750539183616638,
0.08243826776742935,
0.2753567397594452,
-0.19800852239131927,
-0.19976437091827393,
-0.2665269374847412,
0.07753829658031464,
-0.7691617012023926,
-0.14166705310344696,
0.558932363986969,
0.9159026145935059,
-0.0761328637599945,
0.16832636296749115,
0.3113950788974762,
-0.41426438093185425,
-0.3681514859199524,
0.25695666670799255,
-0.09779863059520721,
0.20016969740390778,
0.1302441656589508,
0.19968752562999725,
-0.21522319316864014,
0.12588046491146088,
0.17618690431118011,
-0.2098122239112854,
-0.4563688039779663,
-0.13066790997982025,
0.3850405216217041,
-0.3287796974182129,
0.3194829225540161,
0.30346620082855225,
-0.515622615814209,
-0.07048922777175903,
-0.003494128119200468,
-0.24673697352409363,
-0.22244034707546234,
-0.23069973289966583,
-0.5582116842269897,
-0.6427741646766663,
-0.5660719275474548,
0.1115470603108406,
0.33420422673225403,
0.3304855525493622,
-0.40447211265563965,
-0.4278716742992401,
0.11387821286916733,
0.5516045093536377,
-0.5807482004165649,
-0.06563890725374222,
0.3902779817581177,
0.3496539890766144,
0.1891142725944519,
-0.03716059401631355,
-0.2788269519805908,
0.2488904446363449,
-0.036274876445531845,
0.14652352035045624,
0.14263737201690674,
-0.36288419365882874,
0.45294103026390076,
0.013862485997378826,
-0.032667648047208786,
-0.1078462153673172,
0.4022909700870514,
-0.017562903463840485,
-0.25461727380752563,
0.3648173213005066,
0.3015502393245697,
-0.47498762607574463,
-0.16735036671161652,
0.23045825958251953,
0.10974951088428497,
0.27507585287094116,
-0.11911273747682571,
-0.7172566652297974,
-0.04942205548286438,
-0.17144140601158142,
-0.07749595493078232,
-0.22579914331436157,
-0.03164096176624298,
0.33442622423171997,
0.31528186798095703,
-0.02791113220155239,
-0.8359955549240112,
-0.12462054193019867,
0.2747598886489868,
-0.2535936236381531,
-0.049275752156972885,
0.007980158552527428,
-0.46562880277633667,
-0.1511405110359192,
0.15171553194522858,
-0.10328151285648346,
-0.6776874661445618,
0.6366783380508423,
-0.27949097752571106,
0.1765223890542984,
0.1507074385881424,
0.28035157918930054,
0.15021036565303802,
-0.14736264944076538,
0.17030160129070282,
-0.5934133529663086,
0.11069938540458679,
-0.4928604066371918,
-0.3239763677120209,
-0.10238651186227798,
-0.15195994079113007,
-0.1678219735622406,
-0.13007812201976776,
0.04187703877687454,
0.16919490694999695,
0.7724796533584595,
0.07905487716197968,
-0.0331534668803215,
0.0005937047535553575,
-0.6701501607894897,
0.16868968307971954,
-0.03333332762122154,
-0.5100095272064209,
-0.5696985125541687,
0.10332421213388443,
0.17271748185157776,
0.0304412841796875,
0.05054172873497009,
-0.3752020001411438,
-0.07419495284557343,
0.004276311956346035,
-0.11539245396852493,
-0.20754219591617584,
0.23823583126068115,
0.20030426979064941,
0.4878612756729126,
-0.10876775532960892,
-0.20187626779079437,
-0.2052948921918869,
0.6710834503173828,
0.07003713399171829,
0.13837766647338867,
0.4675595164299011,
0.010323897935450077,
0.19763636589050293,
0.3973939120769501,
-0.07161638885736465,
0.9032248258590698,
0.3037180006504059,
-0.37606507539749146,
0.04444466903805733,
0.3726062476634979,
-0.5856307744979858,
-0.3674437403678894,
0.2589857280254364,
-0.08792585879564285,
0.0938190370798111,
-0.3123454451560974,
0.7958666682243347,
0.33397987484931946,
0.7110053300857544,
-0.44150304794311523,
0.26756787300109863,
-0.11588558554649353,
-0.06984744220972061,
0.9724904298782349,
-0.33987537026405334,
0.15000377595424652,
-0.49688446521759033,
0.9524093270301819,
0.5922998785972595,
-0.7619215846061707,
0.16756555438041687,
0.1344040334224701,
0.32703813910484314,
0.4526371657848358,
0.2949821949005127,
-0.3415408134460449,
0.9392758011817932,
0.2788056433200836,
-0.3085416257381439,
-0.1383463442325592,
-0.0911831483244896,
0.14249972999095917,
-0.20146922767162323,
0.5579879283905029,
-0.16674867272377014,
0.2623071074485779,
0.41043946146965027,
0.5710492730140686,
0.643086314201355,
-0.7039187550544739,
-0.32285773754119873,
-0.5562523007392883,
0.007495796773582697,
0.5072231888771057,
0.2088065892457962,
-0.15842533111572266,
0.024168653413653374,
-0.378437876701355,
0.19775421917438507,
0.058102235198020935,
0.2523726224899292,
0.13937336206436157,
-0.43704694509506226,
0.6969460248947144,
0.6735684871673584,
0.04238300397992134,
0.2159227579832077,
0.11461443454027176,
-0.13693897426128387,
-0.38266751170158386,
-0.41806453466415405,
-0.04479574039578438,
-0.48878467082977295,
0.08821495622396469,
0.25317642092704773,
-0.09126781672239304,
-0.128984272480011,
-0.34398889541625977,
-0.6648932099342346,
-0.6395078897476196,
0.41361862421035767,
0.034818366169929504,
0.08126959204673767,
-0.36421895027160645,
0.8807974457740784,
0.2078152596950531,
0.19210022687911987,
-0.13151025772094727,
0.17250773310661316,
0.21895141899585724,
0.7275820970535278,
-0.30084022879600525,
0.22647778689861298,
-0.6088187098503113,
-0.10851670056581497,
-0.49582210183143616,
-0.19780637323856354,
0.38221871852874756,
-0.3052135705947876,
-0.318331241607666,
0.16968503594398499,
-0.1778147667646408,
0.01717136986553669,
0.1719241440296173,
-0.13061882555484772,
-0.12239131331443787,
-0.09779627621173859,
-0.23680491745471954,
-0.061034053564071655,
0.6759751439094543,
-0.12326814234256744,
0.0968632623553276,
-0.0007803532644174993,
0.4378450810909271,
0.21284961700439453,
0.11120051890611649,
0.47013992071151733,
-0.26002252101898193,
0.04054723680019379,
-0.2430138885974884,
0.6984716057777405,
-0.717053234577179,
-0.08355501294136047,
0.1665569245815277,
-0.08243100345134735,
0.13833601772785187,
-0.42115679383277893,
-0.29538023471832275,
-0.046216148883104324,
-0.5341519117355347,
0.5393096208572388,
-0.1482415497303009,
0.578235924243927,
-0.11395099014043808,
0.01970631629228592,
-0.062121108174324036,
0.5725182294845581,
-0.16491490602493286,
0.11620228737592697,
-0.25712278485298157,
-0.04897576570510864,
-0.1326129138469696,
0.08882429450750351,
0.10648403316736221,
-0.42833614349365234,
-0.5414050221443176,
-0.36939388513565063
] |
prove that whitening and shrinkage are not the correct way to approach Bayes optimality.
Preprint. Under review.
arXiv:2405.13912v1 [math.ST] 22 May 2024
Main contributions.
We focus on the prototypical model A = X + W, where X = λ
nu∗v∗⊤ is a
rank-1 signal, λ is the signal-to-noise ratio (SNR), and W = Ξ1/2�
WΣ1/2 is doubly heterogeneous
noise. Here u∗, v∗ follow i.i.d. priors; �
W contains i.i.d. Gaussian entries; the covariance matrices | [
-0.1329822987318039,
-0.2078491598367691,
0.29808828234672546,
-0.08694356679916382,
-0.1687515377998352,
0.2347637414932251,
0.20666925609111786,
-0.735339343547821,
-0.22419233620166779,
-0.5295178890228271,
-0.1676347851753235,
0.3231945037841797,
-0.18935628235340118,
0.382404088973999,
0.2951747179031372,
0.1842581033706665,
0.867260754108429,
-0.304726779460907,
-0.263802170753479,
-0.6087139248847961,
0.23334813117980957,
-0.16347897052764893,
0.21730811893939972,
0.10381121188402176,
0.011411793529987335,
-0.19443976879119873,
0.5875343084335327,
0.021609768271446228,
0.6931915283203125,
0.32318878173828125,
-0.5490298271179199,
0.49577444791793823,
-0.3381953239440918,
-0.10276374220848083,
0.07646804302930832,
0.47752684354782104,
0.3950934112071991,
0.3428136706352234,
-0.0023757158778607845,
0.29873305559158325,
0.18581901490688324,
-0.02390642836689949,
-0.01206096913665533,
0.33797624707221985,
-0.7063294053077698,
-0.918243408203125,
-1.3735250234603882,
-0.15759329497814178,
-0.24688191711902618,
-0.7163527011871338,
0.010965635068714619,
0.5332817435264587,
0.028953546658158302,
0.5209231376647949,
-0.5255720615386963,
-0.08321985602378845,
-0.02326696366071701,
-0.50846928358078,
0.4338131844997406,
-0.5696951746940613,
-0.04036590829491615,
-0.24164573848247528,
0.19020281732082367,
0.14348524808883667,
0.24969962239265442,
0.4190741777420044,
0.27710244059562683,
-0.1411162167787552,
-0.5304832458496094,
-0.2824966609477997,
0.16655606031417847,
-0.08364320546388626,
-0.1980687826871872,
-0.15774713456630707,
0.06748945266008377,
-0.16355568170547485,
0.11093568801879883,
-0.2346535623073578,
0.2095351368188858,
-0.18251316249370575,
-0.08357109129428864,
-0.09367737919092178,
0.2663339674472809,
-0.5173360705375671,
0.4347086548805237,
0.5488757491111755,
0.06307713687419891,
0.402936726808548,
-0.8887064456939697,
0.3738687038421631,
0.11845195293426514,
-0.18156801164150238,
0.1978289932012558,
0.18161273002624512,
0.4283832311630249,
0.8047940731048584,
0.07518120110034943,
0.04835272580385208,
0.5813877582550049,
-0.21249109506607056,
-0.47561198472976685,
0.1941947638988495,
0.1979854553937912,
-0.5621444582939148,
-0.20443332195281982,
-0.6957730054855347,
-0.21633420884609222,
-0.5101751685142517,
-0.8659008741378784,
0.12557730078697205,
-0.3672468662261963,
0.39753690361976624,
0.2831547260284424,
-0.33484986424446106,
-0.30113983154296875,
0.07879030704498291,
-0.09120555967092514,
-0.2951989769935608,
0.01988924853503704,
0.5715754628181458,
-0.2715104818344116,
1.1634186506271362,
-0.13809502124786377,
0.8159016370773315,
0.30427342653274536,
0.05271720141172409,
0.35560178756713867,
0.9691861271858215,
-0.21666422486305237,
-0.056615471839904785,
-0.17581239342689514,
0.7310613989830017,
-0.2542926073074341,
0.5134967565536499,
-0.10275533050298691,
-0.10987669974565506,
-0.4531041085720062,
-0.20925664901733398,
-0.45686811208724976,
0.2700284421443939,
-0.3170335590839386,
-0.4311192035675049,
-0.022186072543263435,
0.1557098925113678,
0.7330872416496277,
-0.30409571528434753,
-0.28924402594566345,
-0.2781093716621399,
-0.3747079074382782,
-0.0005630330997519195,
-0.5624160766601562,
0.27955758571624756,
-0.8141356110572815,
0.13640762865543365,
-0.3437354266643524,
0.08841007202863693,
0.15127576887607574,
0.2895472049713135,
-0.4785165786743164,
0.2619627118110657,
0.3025452494621277,
-0.17798270285129547,
-0.1913394331932068,
-0.44790226221084595,
0.23867079615592957,
-0.8485170006752014,
0.4963108003139496,
0.14110124111175537,
-0.017733974382281303,
-0.21757625043392181,
0.10231713950634003,
-0.33758968114852905,
-0.08146291226148605,
-0.15971951186656952,
-0.22811393439769745,
-0.2099555879831314,
0.30065733194351196,
0.05013292282819748,
0.4259161949157715,
0.23987823724746704,
-0.8384503722190857,
-0.04956062510609627,
0.16138456761837006,
-0.40105631947517395,
0.15023872256278992,
0.29096519947052,
0.3674109876155853,
0.024033479392528534,
-0.10437659174203873,
0.2115892469882965,
0.0801406130194664,
0.055214084684848785,
0.1591261327266693,
-0.6701274514198303,
1.2194671630859375,
0.3277192711830139,
-0.4747083783149719,
1.2013213634490967,
-0.16933926939964294,
0.2747921049594879,
0.3416030704975128,
-0.25489112734794617,
0.20170092582702637,
0.2725755274295807,
-0.3456157445907593,
0.12024429440498352,
0.06990673393011093,
-0.00023304886417463422,
-0.494864284992218,
0.13447606563568115,
0.35282203555107117,
-0.14681343734264374,
0.05549107491970062,
-0.13459815084934235,
0.0077406009659171104,
-0.17432968318462372,
-0.06669095903635025,
0.26270315051078796,
0.1668672263622284,
-0.32792824506759644,
0.8564523458480835,
-0.22658796608448029,
0.36659005284309387,
0.4145843982696533,
0.4442969560623169,
0.9898353815078735,
1.2609297037124634,
0.3089715540409088,
0.5841655731201172,
-0.6013619899749756,
-0.12338875234127045,
-0.2903610169887543,
0.07998392730951309,
0.2298155575990677,
-0.05689413100481033,
0.4422415792942047,
-0.007402189541608095,
-0.11089735478162766,
0.1282283216714859,
-0.2492259442806244,
0.09477170556783676,
0.3900642693042755,
-0.21493691205978394,
0.2936961054801941,
-0.04639650881290436,
-0.5343051552772522,
0.451774537563324,
0.11774938553571701,
-0.178624227643013,
-0.27645280957221985,
-0.19626660645008087,
0.18472930788993835,
-0.06675787270069122,
-0.5546615123748779,
-0.1927293986082077,
0.15331728756427765,
-0.26363086700439453,
-0.40055540204048157,
0.3768197000026703,
0.5822280645370483,
-0.06515134125947952,
-0.3987715244293213,
-0.10942987352609634,
-0.28055280447006226,
-0.2607247233390808,
-0.43898627161979675,
-0.6076465845108032,
-0.267490953207016,
0.7761772871017456,
0.4749058783054352,
0.5320550799369812,
-0.28229624032974243,
-0.2691552937030792,
-0.17834311723709106,
-0.24037101864814758,
-1.0621081590652466,
0.2113059014081955,
0.36745449900627136,
-0.5973970890045166,
-0.5201180577278137,
0.46229150891304016,
0.8766676187515259,
0.28816378116607666,
0.551818311214447,
-0.5287829041481018,
-0.4676593542098999,
-0.16183219850063324,
-0.13396848738193512,
0.05015384033322334,
-0.25988611578941345,
-0.3963741064071655,
0.6248442530632019,
-0.2815241515636444,
-0.5760645866394043,
0.07167398184537888,
0.8174024224281311,
0.07342729717493057,
-0.18617890775203705,
0.1706942766904831,
0.030805636197328568,
-0.6739183664321899,
-0.03285836800932884,
0.005947420839220285,
0.4286242723464966,
0.08567159622907639,
0.0628843829035759,
-0.3356647491455078,
0.4497098922729492,
-5.887241363525391,
-0.2850798964500427,
-0.23351754248142242,
-0.054827913641929626,
-0.24950112402439117,
-0.03615567088127136,
0.7025469541549683,
-0.19132530689239502,
0.2338474839925766,
-0.04776967316865921,
-0.22684375941753387,
0.14417579770088196,
-0.7405269742012024,
0.5953710675239563,
0.36921313405036926,
0.018510838970541954,
0.048242565244436264,
0.1425497978925705,
0.36194825172424316,
0.2019115537405014,
-0.07658042758703232,
-0.12611742317676544,
-0.010034780949354172,
-0.003619055263698101,
0.1435815542936325,
0.2744963467121124,
-0.9271211624145508,
0.5477995872497559,
-0.36660048365592957,
-0.21135251224040985,
-0.18270985782146454,
-0.822283148765564,
-0.0866740494966507,
-0.4223766624927521,
0.5779473185539246,
0.427991658449173,
0.09362390637397766,
0.1965673416852951,
0.3557467460632324,
-0.022369304671883583,
0.5996952056884766,
0.9021723866462708,
-0.13913647830486298,
-0.36611899733543396,
-0.11825089156627655,
-0.2489130198955536,
0.16055920720100403,
0.5363958477973938,
-0.24933601915836334,
0.8142295479774475,
-0.15151773393154144,
0.2232692539691925,
0.060262471437454224,
-0.07567234337329865,
-0.2758253514766693,
-0.21842317283153534,
0.03336392715573311,
0.43909385800361633,
-0.11726035922765732,
0.17044000327587128,
0.08581266552209854,
-0.518924355506897,
0.17105969786643982,
-0.5305277109146118,
-0.3650778830051422,
0.21073390543460846,
-0.5512020587921143,
-0.23207131028175354,
0.4229746162891388,
-0.1333838403224945,
-0.06604725867509842,
0.012868969701230526,
0.1173030287027359,
-0.6320968270301819,
0.13235075771808624,
-0.618651807308197,
-0.0847054272890091,
-0.13236480951309204,
-0.2622819244861603,
0.2669150233268738,
-0.20268473029136658,
-0.6571469306945801,
0.04319894686341286,
0.060723550617694855,
0.18506759405136108,
-0.45870810747146606,
0.04347094148397446,
-0.18392574787139893,
-0.3378945589065552,
-0.053784146904945374,
0.36742642521858215,
-0.03794559836387634,
0.155379518866539,
0.4202615022659302,
0.2862984538078308,
0.4358096718788147,
0.6279821991920471,
0.3500482439994812,
0.23175714910030365,
-0.24406148493289948,
0.4036799371242523,
-0.3653242588043213,
-0.13285134732723236,
-0.8134454488754272,
0.1326679140329361,
0.5958094596862793,
-0.49125638604164124,
0.21802598237991333,
0.7736561894416809,
0.010174354538321495,
0.11527743935585022,
-0.21643830835819244,
-0.446922242641449,
-0.20895791053771973,
0.721869707107544,
0.13754671812057495,
-0.6900067925453186,
0.04735829681158066,
0.6062342524528503,
-0.5626514554023743,
0.3305477797985077,
-0.018547171726822853,
-0.26030176877975464,
0.4315618574619293,
0.4082442820072174,
-0.31275755167007446,
-0.09928508847951889,
0.7544239163398743,
-0.48532038927078247,
-0.1923207938671112,
0.2644043266773224,
0.43501025438308716,
-0.7171045541763306,
0.21880832314491272,
0.7772549390792847,
-0.5796322822570801,
-0.2155260145664215,
0.24842947721481323,
0.501451849937439,
-0.10149718075990677,
0.49958133697509766,
0.2761373817920685,
-0.15573666989803314,
-0.22537164390087128,
-0.23974637687206268,
0.24191780388355255,
-0.037667084485292435,
-0.21114352345466614,
-0.4703899323940277,
-0.23341169953346252,
0.008468534797430038,
-0.4963234066963196,
0.03928142786026001,
-0.1551438868045807,
-0.22264334559440613,
0.014579878188669682,
0.11579714715480804,
0.0839802548289299,
0.08339573442935944,
-0.1735885590314865,
0.26307299733161926,
0.3875291645526886,
-0.8417733311653137,
-0.39666593074798584,
0.12656477093696594,
0.00894144456833601,
-0.49184077978134155,
0.05654117092490196,
-0.1756712645292282,
0.18287476897239685,
-0.4848148226737976,
-0.7631936073303223,
0.2666049301624298,
-0.6757798194885254,
-0.0442536398768425,
0.6176296472549438,
0.24333029985427856,
-0.24327421188354492,
-0.37954646348953247,
-0.20182660222053528,
0.3779304623603821,
-0.40969008207321167,
-0.21138304471969604,
1.1369366645812988,
1.0642632246017456,
0.009843357838690281,
-0.02371365949511528,
0.1007937341928482,
-0.6072975397109985,
0.0032234841492027044,
-0.035166479647159576,
0.03879409655928612,
0.5557403564453125,
0.15062783658504486,
0.08760768920183182,
-0.13398900628089905,
0.2131197154521942,
0.09758780151605606,
-0.19591230154037476,
-0.4699886441230774,
-0.16303646564483643,
0.42349475622177124,
-0.10336508601903915,
0.686070442199707,
0.49182477593421936,
-0.22701384127140045,
-0.20472081005573273,
0.3711027204990387,
0.12040719389915466,
-0.3983701467514038,
-0.20746034383773804,
-0.5242751240730286,
-0.1898491531610489,
-0.2531999349594116,
-0.10476484894752502,
0.1633589267730713,
0.37153565883636475,
-0.28609320521354675,
-0.21077373623847961,
0.2606362998485565,
0.31505346298217773,
-0.5576985478401184,
-0.3151610493659973,
-0.09191644936800003,
-0.12883031368255615,
0.46672090888023376,
-0.16886666417121887,
-0.3838515877723694,
0.4263629913330078,
-0.04957257956266403,
-0.15950170159339905,
-0.1725768893957138,
-0.3707506060600281,
0.33007851243019104,
-0.24237634241580963,
-0.5770120620727539,
-0.7045388221740723,
0.1629408597946167,
-0.4361153841018677,
0.012654895894229412,
0.12331624329090118,
0.3611227571964264,
-0.19851398468017578,
-0.5587888956069946,
-0.44423261284828186,
-0.0187689196318388,
0.4698750674724579,
0.3406286835670471,
-0.28343042731285095,
-0.11214170604944229,
-0.30548781156539917,
-0.032740551978349686,
-0.020749632269144058,
0.12284953147172928,
0.7023066878318787,
0.44847366213798523,
-0.0494341105222702,
-0.49713873863220215,
-0.13936160504817963,
0.2054435759782791,
-0.17752432823181152,
-0.31080639362335205,
-0.44161659479141235,
-0.4665350317955017,
-0.07265233993530273,
0.43118351697921753,
-0.4668571949005127,
-0.5326308608055115,
0.2528442144393921,
0.25668641924858093,
-0.3412254750728607,
-0.041662588715553284,
0.18600770831108093,
0.23943740129470825,
-0.030957147479057312,
0.11936052143573761,
-0.0823996290564537,
0.2948707044124603,
-0.2648969292640686,
0.4006810486316681,
-0.25315871834754944,
-0.5186111927032471,
-0.258830726146698,
-0.5195972919464111,
0.1589696854352951,
0.001087386510334909,
0.5816826820373535,
-0.19203291833400726,
0.1945926696062088,
0.24515438079833984,
-0.30710211396217346,
0.285553514957428,
-0.5860038995742798,
-0.40732884407043457,
-0.0427597239613533,
0.30844128131866455,
0.3847177028656006,
0.28904250264167786,
0.003786603920161724,
-0.19157640635967255,
-0.3159230351448059,
-0.07904276251792908,
0.12376699596643448,
-0.1171288713812828,
0.2521996796131134,
0.49279189109802246,
0.21993489563465118,
0.3614239990711212,
-0.6842722296714783,
0.1941576898097992,
0.1656615138053894,
0.2240748256444931,
0.08424299210309982,
0.4608655273914337,
-0.40389394760131836,
0.17455050349235535,
0.45895105600357056,
-0.3924449682235718,
0.4246414601802826,
0.17245200276374817,
-0.1429745852947235,
0.31621217727661133,
0.5124039053916931,
-0.5300764441490173,
-0.9992371797561646,
0.25951987504959106,
-0.35530295968055725,
0.33033332228660583,
-0.5261400938034058,
0.6108599305152893,
0.27493759989738464,
0.5192861557006836,
-0.15869145095348358,
0.48044320940971375,
-0.08791926503181458,
0.15319304168224335,
0.49973827600479126,
-0.40797892212867737,
-0.12248195707798004,
-0.11706990748643875,
0.8577517867088318,
0.8138889670372009,
-0.500349760055542,
0.06414967775344849,
0.22176995873451233,
0.0238350722938776,
0.19996525347232819,
0.1974841207265854,
-0.4201117157936096,
0.5394482016563416,
0.43718451261520386,
-0.048552241176366806,
0.2893740236759186,
0.014409094117581844,
-0.035193610936403275,
0.13590142130851746,
0.5221644639968872,
-0.21532094478607178,
-0.09740938991308212,
0.25110945105552673,
0.7098045349121094,
0.7137548327445984,
-0.01754089444875717,
-0.26156824827194214,
-0.604753315448761,
-0.1540834754705429,
0.7202181220054626,
0.5674728155136108,
-0.06481800228357315,
-0.19484290480613708,
0.04375714808702469,
-0.017911545932292938,
-0.1365179717540741,
0.4744717478752136,
-0.11848485469818115,
-0.24074797332286835,
-0.34037768840789795,
0.24232254922389984,
-0.5632168650627136,
-0.36494943499565125,
-0.04501430317759514,
-0.0904328003525734,
-0.3544832170009613,
-0.48011884093284607,
0.13934068381786346,
-0.5621103048324585,
-0.22868099808692932,
0.03651602566242218,
-0.19300350546836853,
-0.17540961503982544,
-0.05809076130390167,
-0.8155165314674377,
-0.35146692395210266,
0.28881239891052246,
-0.06509927660226822,
-0.028345711529254913,
-0.24537265300750732,
0.8104158043861389,
0.4142034947872162,
0.22942426800727844,
0.19589965045452118,
-0.19692374765872955,
-0.3295377194881439,
1.002144694328308,
-0.29921165108680725,
0.2568492889404297,
-0.5321112871170044,
0.36969509720802307,
-0.3525717854499817,
-0.35289016366004944,
0.5327281951904297,
-0.11620000749826431,
-0.14948374032974243,
0.03509734198451042,
0.1602531522512436,
0.041274286806583405,
0.532383918762207,
-0.03385560214519501,
0.10898081958293915,
-0.3410310447216034,
-0.668012797832489,
-0.9286768436431885,
0.5495583415031433,
0.14909736812114716,
-0.018556205555796623,
0.16337724030017853,
0.487967848777771,
-0.05110001191496849,
-0.14463064074516296,
0.06058931723237038,
0.16680824756622314,
-0.15999311208724976,
-0.38959676027297974,
0.8510952591896057,
-0.3174021542072296,
-0.17907272279262543,
0.4824952185153961,
0.3016735315322876,
0.33304017782211304,
-0.6071493029594421,
-0.7952221035957336,
0.08384351432323456,
-0.1731611043214798,
0.811623215675354,
0.026815926656126976,
0.22402481734752655,
-0.192278191447258,
-0.45587173104286194,
-0.6649907827377319,
0.807953953742981,
-0.08500923961400986,
0.0895954817533493,
-0.31717154383659363,
-0.18847577273845673,
-0.5951095819473267,
0.5886374711990356,
-0.14483973383903503,
-0.13184812664985657,
-0.36452874541282654,
-0.5000869631767273
] |
W contains i.i.d. Gaussian entries; the covariance matrices
Ξ, Σ capture column and row correlations; and we consider the typical high-dimensional regime in
which n, d are both large and proportional. Our main results are summarized below.
1. We design an efficient spectral estimator to recover u∗, v∗, and we provide a precise asymp-
totic analysis of its performance, see Theorem 5.1. This estimator is given by the top singular
vectors of a matrix obtained by carefully pre-processing A, see (5.3). | [
0.050569143146276474,
-0.47831231355667114,
0.25020089745521545,
-0.07435595989227295,
-0.12667694687843323,
0.18910524249076843,
0.3694501519203186,
-0.8951903581619263,
0.2192826122045517,
-0.6821157932281494,
-0.20034907758235931,
0.45331189036369324,
-0.10733465850353241,
0.17449599504470825,
0.038009095937013626,
0.7492669224739075,
0.7907223105430603,
0.4349111318588257,
-0.14325779676437378,
-0.8706026077270508,
0.2295898050069809,
0.14222407341003418,
0.6342808604240417,
-0.03548196703195572,
0.12690795958042145,
0.0978890210390091,
0.23534901440143585,
0.0037817414849996567,
0.2571254074573517,
0.3763200342655182,
-0.26767295598983765,
0.6554909348487854,
-0.09296754002571106,
-0.10514979064464569,
0.3480680584907532,
0.15720942616462708,
0.4646088182926178,
0.39793097972869873,
0.4249573051929474,
-0.3029363751411438,
0.07067189365625381,
-0.2489481419324875,
0.11905140429735184,
0.37981635332107544,
-0.5584942102432251,
-0.7892264723777771,
-1.188211441040039,
-0.1550806164741516,
-0.14694024622440338,
-0.43854209780693054,
-0.1549036204814911,
0.3356989026069641,
0.366665244102478,
0.5599645972251892,
-0.3877965807914734,
0.07633985579013824,
-0.2507514953613281,
-0.22752471268177032,
0.7499139904975891,
-0.08069953322410583,
0.22774745523929596,
-0.21920818090438843,
0.06015550345182419,
0.4634243845939636,
0.15603181719779968,
0.6283479332923889,
0.10054737329483032,
-0.14349932968616486,
-0.672990620136261,
-0.36989060044288635,
0.24795375764369965,
-0.015632888302206993,
-0.2586487829685211,
-0.05728751793503761,
-0.3328540325164795,
0.25752416253089905,
-0.37546512484550476,
-0.251186728477478,
0.17125838994979858,
-0.11929787695407867,
-0.3442564308643341,
-0.04934847727417946,
0.3117295205593109,
-0.5039022564888,
0.2854754626750946,
0.1352456957101822,
0.1042005866765976,
0.26181089878082275,
-0.7314593195915222,
0.3252617418766022,
0.053791943937540054,
-0.29361021518707275,
-0.05955386534333229,
0.02596665918827057,
-0.110499806702137,
0.8298749327659607,
0.20029804110527039,
-0.32698869705200195,
0.8627490401268005,
-0.22059088945388794,
-0.527292013168335,
-0.23417116701602936,
-0.21477599442005157,
-0.5991285443305969,
-0.44658517837524414,
-0.523110032081604,
-0.566487729549408,
-0.15810155868530273,
-0.24494203925132751,
0.1880749762058258,
-0.7616150975227356,
0.5941380858421326,
0.2062862515449524,
-0.39080777764320374,
-0.059089262038469315,
-0.05235189199447632,
0.2840914726257324,
0.007515681907534599,
-0.12086258083581924,
0.6906781792640686,
-0.26611313223838806,
1.0932248830795288,
0.07257982343435287,
0.5452110767364502,
0.2830931544303894,
-0.2979278564453125,
0.8666276931762695,
0.6601110696792603,
-0.26082703471183777,
-0.38359490036964417,
0.37046587467193604,
0.9690365195274353,
0.36924123764038086,
0.3838973939418793,
-0.19140857458114624,
-0.5545527935028076,
-0.5983371138572693,
-0.13159558176994324,
-0.5422213077545166,
-0.2813064754009247,
-0.1985938847064972,
-0.4274947941303253,
-0.35849952697753906,
-0.08030833303928375,
0.29867464303970337,
-0.20119155943393707,
-0.36469560861587524,
-0.3571787476539612,
0.04845043271780014,
0.08217975497245789,
0.14815667271614075,
0.7333084344863892,
-0.7017776370048523,
0.10989812761545181,
-0.3853919208049774,
-0.13686983287334442,
0.10077884048223495,
0.20898635685443878,
-0.5548707842826843,
0.4471903145313263,
0.08628375828266144,
0.10659996420145035,
0.23980920016765594,
-0.3664020597934723,
0.14941716194152832,
-0.6507872939109802,
0.4518234431743622,
0.027943184599280357,
-0.1736138015985489,
-0.21536235511302948,
-0.037058427929878235,
-0.10871952027082443,
-0.13825246691703796,
0.3065388798713684,
-0.09406859427690506,
-0.10635106265544891,
0.29272064566612244,
-0.1574334353208542,
0.5963186025619507,
0.1626821607351303,
-0.7769331336021423,
0.24419005215168,
0.09137862175703049,
-0.36485815048217773,
0.5253897905349731,
0.36101964116096497,
0.509641170501709,
0.45058971643447876,
0.1812092661857605,
0.2033408284187317,
-0.24598155915737152,
-0.15198089182376862,
-0.056347355246543884,
-1.0561951398849487,
0.8011685013771057,
0.45744529366493225,
-0.4749389588832855,
0.2093370258808136,
-0.08287429809570312,
-0.12941879034042358,
0.21018251776695251,
-0.027162525802850723,
0.26370343565940857,
-0.1235930472612381,
-0.33531057834625244,
0.1721162647008896,
0.07628173381090164,
0.06573524326086044,
-0.7203583121299744,
-0.15168145298957825,
0.41349270939826965,
0.0073791127651929855,
-0.13807491958141327,
-0.48973315954208374,
0.37617889046669006,
-0.22073684632778168,
-0.07118894159793854,
0.21704737842082977,
0.7347338795661926,
0.08513777703046799,
1.2777186632156372,
-0.4848741888999939,
0.3265783488750458,
0.417672336101532,
-0.0421328991651535,
0.7950800657272339,
0.6080157160758972,
0.15490588545799255,
0.7919726371765137,
-1.112100601196289,
-0.5646540522575378,
-0.4181543290615082,
0.00771963968873024,
0.1963292509317398,
-0.03810684010386467,
0.2813812792301178,
-0.016039198264479637,
-0.016614217311143875,
0.6724042296409607,
-0.46272772550582886,
0.31902381777763367,
0.5851925015449524,
-0.23334068059921265,
0.11264628171920776,
0.12507271766662598,
-0.22436995804309845,
0.03613092750310898,
-0.14335423707962036,
-0.3403342068195343,
-0.342803418636322,
-0.0008249703678302467,
0.2504708170890808,
0.22257724404335022,
-0.7222458720207214,
0.0061369710601866245,
-0.0559975691139698,
0.15649114549160004,
-0.31598547101020813,
0.28887128829956055,
0.5713866353034973,
-0.2086372673511505,
-0.28106579184532166,
0.10235375910997391,
-0.3660842776298523,
-0.23281560838222504,
-0.20761694014072418,
-0.9848232269287109,
-0.3289409279823303,
0.8573382496833801,
0.4338665008544922,
0.4439939856529236,
-0.25480371713638306,
-0.35243654251098633,
-0.05034444481134415,
-0.40304479002952576,
-0.9098501205444336,
0.42304477095603943,
0.23035624623298645,
-1.001616358757019,
-0.6196577548980713,
0.261851966381073,
0.724500298500061,
0.4377877712249756,
0.030951453372836113,
-0.39982038736343384,
-0.48254626989364624,
-0.0738227441906929,
-0.1282021552324295,
-0.3292315602302551,
-0.44643884897232056,
-0.7396627068519592,
0.5426835417747498,
-0.24028357863426208,
-0.7756054997444153,
0.010100922547280788,
0.5137622952461243,
-0.25742799043655396,
0.13711020350456238,
0.05372941121459007,
0.09331649541854858,
-0.6949587464332581,
0.12060650438070297,
-0.5128249526023865,
-0.1263047605752945,
0.025477126240730286,
-0.414053738117218,
-0.4603365361690521,
0.8660330176353455,
-5.849216461181641,
0.08080116659402847,
-0.19975513219833374,
0.38455769419670105,
-0.011929383501410484,
-0.1183750331401825,
0.3148764967918396,
0.005539524368941784,
0.012060191482305527,
-0.12567968666553497,
0.0052758147940039635,
-0.02152349427342415,
-0.3409288227558136,
0.6405771970748901,
0.3523607552051544,
-0.09537883847951889,
0.481179416179657,
0.26962709426879883,
0.3943544626235962,
0.06244967505335808,
-0.12804697453975677,
-0.01311083510518074,
-0.14855971932411194,
-0.32649698853492737,
0.2597266137599945,
0.1318562775850296,
-0.3566535711288452,
0.5636845827102661,
-0.4844644069671631,
-0.6578105688095093,
-0.5113381147384644,
-0.48189443349838257,
0.05567058548331261,
-0.2870316803455353,
0.3980923593044281,
0.028942205011844635,
0.3035275936126709,
-0.4866744875907898,
0.23394861817359924,
-0.5689437985420227,
0.17505693435668945,
0.6499003171920776,
-0.3626590371131897,
-0.560203492641449,
-0.09032397717237473,
0.11511391401290894,
0.041413191705942154,
0.127232164144516,
-0.0429590567946434,
0.7718895673751831,
-0.21822312474250793,
-0.15583865344524384,
-0.009953506290912628,
-0.08424913883209229,
-0.07530803233385086,
-0.15828917920589447,
-0.1516706794500351,
0.44999897480010986,
-0.23303689062595367,
0.46951332688331604,
0.2172410488128662,
-0.43981486558914185,
0.2697605490684509,
-0.4566555917263031,
-0.39497143030166626,
0.31354084610939026,
-0.6328095197677612,
-0.2675485908985138,
0.37423619627952576,
-0.2223743349313736,
-0.10137983411550522,
-0.14777420461177826,
0.14131902158260345,
-0.38826242089271545,
0.3023194968700409,
-0.6574032306671143,
0.2600324749946594,
-0.032390788197517395,
0.07518130540847778,
-0.038604047149419785,
-0.3890739977359772,
-0.6881467700004578,
0.06663365662097931,
-0.042518485337495804,
-0.04896854981780052,
-0.07722695916891098,
0.19385051727294922,
-0.1947057694196701,
-0.2905106246471405,
-0.0024952003732323647,
0.3262577950954437,
-0.2672058939933777,
0.2815268337726593,
0.5093297958374023,
0.43279901146888733,
0.4578343331813812,
0.6077210307121277,
-0.15263396501541138,
0.557134211063385,
-0.044147416949272156,
0.6461136341094971,
-0.49701324105262756,
-0.16716833412647247,
-0.23618191480636597,
0.16152343153953552,
0.706038773059845,
-0.4349016845226288,
0.2656644582748413,
1.0436294078826904,
-0.1621285229921341,
-0.06357088685035706,
-0.18602940440177917,
-0.3972441256046295,
-0.27401015162467957,
0.7336256504058838,
-0.4337991178035736,
-0.6698135137557983,
-0.02592003345489502,
0.8867293000221252,
-0.11160487681627274,
0.26958996057510376,
0.19452498853206635,
-0.1891026794910431,
-0.21309298276901245,
0.001277333009056747,
-0.04144525155425072,
0.32951807975769043,
0.48774266242980957,
-0.6366376280784607,
-0.44806334376335144,
0.41628602147102356,
0.13350266218185425,
-0.26388052105903625,
0.15508919954299927,
0.6363531351089478,
-0.13965141773223877,
0.1676723212003708,
0.19509321451187134,
0.2976265251636505,
-0.3349720239639282,
0.5294175744056702,
0.3571321368217468,
-0.3192850947380066,
-0.4381294548511505,
0.11967296153306961,
0.3110523223876953,
-0.09855348616838455,
-0.22486594319343567,
-0.3789202868938446,
-0.003790434915572405,
0.21098458766937256,
-0.4491086006164551,
0.5422953367233276,
-0.19910414516925812,
-0.12873074412345886,
0.2460019886493683,
0.2687057852745056,
-0.09705322235822678,
0.4426361918449402,
-0.5327439308166504,
0.47971904277801514,
0.2832725942134857,
-1.0126092433929443,
-0.09470631182193756,
0.27925926446914673,
-0.03177323192358017,
-0.6743673086166382,
-0.003453858895227313,
0.16993321478366852,
0.5270286202430725,
-0.029753917828202248,
-0.17139944434165955,
0.07949855178594589,
-0.1436622589826584,
0.013406758196651936,
0.13951821625232697,
0.12208625674247742,
0.2362975925207138,
0.27352163195610046,
-0.08738590776920319,
0.4196460247039795,
-0.09918633848428726,
-0.24610430002212524,
0.6930044889450073,
0.7609221339225769,
-0.1807270050048828,
-0.018141724169254303,
0.22248685359954834,
-0.8973285555839539,
0.16502520442008972,
0.12905815243721008,
0.03853625804185867,
0.3372969627380371,
0.040327053517103195,
0.22938597202301025,
-0.012952825054526329,
0.40455150604248047,
0.40731310844421387,
-0.3471710979938507,
-0.17081715166568756,
-0.0953449010848999,
0.6658371686935425,
-0.38820287585258484,
0.35039424896240234,
-0.13470163941383362,
-0.164255753159523,
-0.3707248866558075,
-0.11132150143384933,
0.18967102468013763,
-0.2425958812236786,
-0.5601061582565308,
-0.31409257650375366,
-0.5554136633872986,
-0.025381192564964294,
-0.31548213958740234,
-0.22900566458702087,
0.3876541554927826,
-0.41582390666007996,
0.028354739770293236,
-0.49215227365493774,
0.48264992237091064,
-0.43851837515830994,
0.28181692957878113,
0.24850568175315857,
0.2516431510448456,
0.06244388595223427,
-0.41778305172920227,
-0.7378819584846497,
-0.02452368475496769,
0.11722704768180847,
-0.30478042364120483,
-0.1464691013097763,
-0.0845712423324585,
0.24559053778648376,
-0.23151424527168274,
-0.5578282475471497,
-0.608103334903717,
0.22406792640686035,
-0.5449701547622681,
-0.19487693905830383,
-0.08312293142080307,
0.7109196782112122,
-0.12305188179016113,
-0.47536519169807434,
-0.024346226826310158,
0.20834925770759583,
0.2920639216899872,
-0.020410995930433273,
-0.4230165183544159,
-0.2863747775554657,
-0.005170328076928854,
-0.012518196366727352,
0.09516799449920654,
-0.34908565878868103,
0.9093043208122253,
0.14696726202964783,
-0.0003303464036434889,
-0.6684624552726746,
0.1261776089668274,
0.43344515562057495,
-0.11506955325603485,
-0.4364664554595947,
-0.5162864923477173,
-0.09340598434209824,
-0.020871493965387344,
0.43676719069480896,
-0.49705231189727783,
-0.23250746726989746,
0.1836523562669754,
-0.36483076214790344,
-0.09927601367235184,
0.10780321061611176,
-0.3207334876060486,
0.5240108370780945,
0.3488032817840576,
0.18426689505577087,
-0.24167077243328094,
0.23419465124607086,
0.2792166769504547,
0.15951929986476898,
-0.48744750022888184,
-0.5067171454429626,
-0.12159450352191925,
-0.29720720648765564,
0.07787488400936127,
0.27045053243637085,
0.1075073778629303,
0.23191407322883606,
-0.21145951747894287,
0.28957271575927734,
-0.3284575045108795,
-0.2872391641139984,
-0.8265852928161621,
-0.7474066019058228,
0.02572188340127468,
-0.02223336696624756,
0.524453341960907,
0.2249937504529953,
-0.2608643174171448,
-0.19883765280246735,
-0.4701242744922638,
-0.31617435812950134,
0.2629072368144989,
-0.2257501780986786,
0.5666998624801636,
0.1606832593679428,
0.34737199544906616,
-0.012645785696804523,
-0.3218185007572174,
-0.28045517206192017,
0.47201764583587646,
0.13180695474147797,
0.34120091795921326,
0.18446649610996246,
-0.4984983205795288,
-0.1766391396522522,
0.4586937725543976,
-0.19074293971061707,
0.653157651424408,
0.4494421184062958,
0.11840534955263138,
-0.0418345108628273,
0.6997945308685303,
-0.22839489579200745,
-0.8088741302490234,
0.7236888408660889,
-0.2782529294490814,
0.31853461265563965,
-0.36583513021469116,
0.15219491720199585,
0.43084490299224854,
0.6925815343856812,
-0.7441607713699341,
0.18324092030525208,
0.02778070978820324,
0.24066942930221558,
0.26051634550094604,
-0.13637951016426086,
-0.14800694584846497,
0.2803898751735687,
0.9957256317138672,
0.9401007890701294,
0.034883882850408554,
0.423057496547699,
0.23464004695415497,
-0.0926588699221611,
-0.18774256110191345,
-0.1505957692861557,
-0.25296998023986816,
-0.1572805494070053,
0.13704459369182587,
-0.2474389672279358,
-0.0038192421197891235,
-0.15789370238780975,
-0.2574312686920166,
0.3163245916366577,
0.514417827129364,
-0.428882896900177,
-0.1709805428981781,
0.41277560591697693,
0.554345428943634,
0.33401310443878174,
-0.12454712390899658,
-0.05086011439561844,
-0.509311854839325,
-0.25344380736351013,
0.341055303812027,
0.19331392645835876,
-0.13334383070468903,
-0.10527803003787994,
0.10087098926305771,
0.3372924029827118,
-0.35987555980682373,
0.20998671650886536,
0.14318962395191193,
-0.316649466753006,
-0.015465927310287952,
0.29119327664375305,
-0.30826535820961,
0.003486019792035222,
0.6317925453186035,
0.5272459387779236,
-0.19250191748142242,
-0.5209705233573914,
-0.13982677459716797,
-0.20867331326007843,
0.07351312786340714,
-0.4296884834766388,
0.2443971484899521,
0.1873602271080017,
0.15320059657096863,
-0.7007873058319092,
-0.618547797203064,
0.13324148952960968,
0.18726062774658203,
0.3112131357192993,
-0.6377056241035461,
1.149204134941101,
0.7360109686851501,
0.14413119852542877,
0.4401393532752991,
-0.03505760431289673,
-0.07836475968360901,
0.7517150044441223,
-0.31841519474983215,
0.12378590553998947,
-0.5445372462272644,
0.2554057538509369,
-0.6018018126487732,
0.13025282323360443,
0.9017194509506226,
0.05471594259142876,
-0.1422395557165146,
-0.07421199977397919,
0.0684560090303421,
-0.20659182965755463,
0.4094104468822479,
-0.20594754815101624,
0.062381844967603683,
-0.18452811241149902,
-0.7622995972633362,
-0.7256160974502563,
0.5474731922149658,
0.3983432948589325,
-0.19871650636196136,
-0.14143908023834229,
0.5473582148551941,
0.2530634105205536,
-0.5610002875328064,
0.2755832076072693,
-0.30722859501838684,
0.007679654750972986,
-0.5708377361297607,
0.7386358976364136,
-0.6310091018676758,
0.31945544481277466,
0.3104071617126465,
0.30292680859565735,
0.30970096588134766,
-0.5295361280441284,
-0.5748472809791565,
-0.2762591540813446,
-0.48339030146598816,
0.7289100289344788,
-0.2271733283996582,
0.386540025472641,
-0.02064000442624092,
-0.15133564174175262,
-0.6620164513587952,
0.5797632336616516,
0.0006101667531765997,
-0.17638707160949707,
-0.23394834995269775,
-0.06562141329050064,
-0.6098524332046509,
0.6122399568557739,
0.07139424979686737,
-0.3477781116962433,
-0.2827138900756836,
-0.09699464589357376
] |
vectors of a matrix obtained by carefully pre-processing A, see (5.3).
2. When the priors of u∗, v∗ are standard Gaussian, we show in Corollary 5.2 that the spectral
estimator above is optimal in the following sense: (i) under a technical condition, it achieves
the optimal weak recovery threshold, namely its mean square error is non-trivial as soon
as this is information-theoretically possible; (ii) it achieves the Bayes-optimal error for | [
-0.17633366584777832,
-0.45110151171684265,
0.4138011038303375,
-0.09668873250484467,
-0.49817484617233276,
0.45358535647392273,
0.26703715324401855,
-0.4758678674697876,
-0.1296229362487793,
-0.7852433919906616,
-0.28897789120674133,
0.5262942314147949,
-0.5437855124473572,
-0.06056007370352745,
0.1402004212141037,
0.3700220286846161,
0.33111897110939026,
-0.03730655834078789,
-0.545224666595459,
-0.7428045272827148,
0.09409629553556442,
0.08793775737285614,
0.8494389653205872,
0.009866997599601746,
0.17541050910949707,
-0.35726749897003174,
0.7917402982711792,
0.45959457755088806,
0.1471276581287384,
0.05971602350473404,
-0.37413039803504944,
0.4009823203086853,
-0.008979476988315582,
0.013653152622282505,
0.36462026834487915,
0.18692471086978912,
0.42965927720069885,
0.13214105367660522,
0.24795463681221008,
-0.16760148108005524,
0.3702464997768402,
-0.07573727518320084,
-0.12513571977615356,
0.6360284090042114,
-0.3541702628135681,
-0.7079920172691345,
-1.3779863119125366,
-0.20292073488235474,
-0.0805610939860344,
-0.7006614804267883,
-0.3380897045135498,
0.42829132080078125,
0.0671548843383789,
0.6514958739280701,
-0.06384579092264175,
0.6079819202423096,
-0.4826638400554657,
-0.3312458097934723,
0.547491192817688,
-0.03337918967008591,
0.07226933538913727,
-0.15861541032791138,
-0.024778999388217926,
0.6076893210411072,
-0.14717057347297668,
0.6384953856468201,
0.4215182960033417,
-0.03675070032477379,
-0.41252055764198303,
-0.8610920906066895,
-0.03617550805211067,
0.033143024891614914,
-0.2815195322036743,
0.42677101492881775,
-0.25768613815307617,
-0.24783360958099365,
-0.16547925770282745,
-0.31637972593307495,
0.08824575692415237,
-0.36497360467910767,
-0.0936204195022583,
0.36291491985321045,
0.31608596444129944,
-0.09252402186393738,
0.1823451668024063,
0.0783245638012886,
0.1222572922706604,
0.2470572292804718,
-0.4863012135028839,
0.5195936560630798,
-0.049986839294433594,
-0.00850618351250887,
-0.13035018742084503,
0.2333904355764389,
0.30041608214378357,
0.6063950061798096,
-0.1748107671737671,
-0.3045048117637634,
0.8466977477073669,
0.11492688208818436,
-0.6532592177391052,
-0.2609182894229889,
-0.05357670783996582,
-0.5038867592811584,
-0.4206635057926178,
-0.8188643455505371,
-0.405230849981308,
-0.003380189184099436,
-0.31464290618896484,
0.1408616155385971,
-0.7046851515769958,
0.671978771686554,
0.20528027415275574,
-0.42937466502189636,
-0.1127866730093956,
0.035055894404649734,
0.05090295895934105,
-0.24412934482097626,
-0.49115172028541565,
0.5003584623336792,
-0.11480281502008438,
0.9818130135536194,
-0.024295173585414886,
0.1195145845413208,
0.08190950006246567,
-0.21090710163116455,
0.7345236539840698,
0.8031368851661682,
-0.3714632987976074,
-0.14833250641822815,
0.09709072858095169,
0.7452313303947449,
0.11667656898498535,
0.2282194346189499,
-0.41323545575141907,
-0.9096062183380127,
-0.5608681440353394,
-0.2300613522529602,
-0.8356643915176392,
-0.4472692906856537,
-0.5332244634628296,
-0.5788633823394775,
-0.0335082970559597,
-0.012419427745044231,
0.7853766083717346,
-0.24027051031589508,
0.18241190910339355,
-0.24700328707695007,
-0.2523316442966461,
0.3014233112335205,
0.0013867567759007215,
0.5521697402000427,
-0.4536917507648468,
0.0830266922712326,
-0.48196980357170105,
-0.18747267127037048,
0.6131491661071777,
0.29078400135040283,
-0.7016974091529846,
0.3408827483654022,
0.09061657637357712,
0.3010280430316925,
0.031697649508714676,
-0.7028040289878845,
-0.11149394512176514,
-0.7675384879112244,
0.4761981666088104,
-0.05489591136574745,
0.3649798035621643,
-0.284641832113266,
-0.24382807314395905,
-0.5417003631591797,
0.05945191904902458,
0.22143830358982086,
-0.24994413554668427,
-0.2376406341791153,
0.3945681154727936,
-0.11940424889326096,
0.4715458154678345,
-0.18492159247398376,
-0.6964842677116394,
0.2354402095079422,
0.39620938897132874,
0.24374809861183167,
0.21993882954120636,
0.04201209917664528,
0.5177637338638306,
0.0656903013586998,
0.506382167339325,
-0.08978118747472763,
0.038337342441082,
-0.03018321469426155,
-0.24102923274040222,
-0.8253749012947083,
1.1414049863815308,
0.42697322368621826,
-0.8745357394218445,
0.6775491237640381,
-0.03604524955153465,
0.45883092284202576,
0.23393861949443817,
-0.015207372605800629,
0.21015365421772003,
-0.015187854878604412,
-0.2198023498058319,
0.13331255316734314,
0.18692253530025482,
0.3908648192882538,
-0.7641714811325073,
0.3881102204322815,
0.4326009452342987,
0.015138104557991028,
-0.1540493220090866,
-0.183376282453537,
0.6494075059890747,
0.033682942390441895,
-0.03613082692027092,
0.23645882308483124,
0.4600415527820587,
-0.1844523400068283,
1.2584643363952637,
-0.2612042725086212,
0.2595897912979126,
0.22574424743652344,
0.17522193491458893,
1.2245436906814575,
0.6452624201774597,
0.2322578728199005,
0.27219298481941223,
-0.4721359610557556,
-0.15377402305603027,
0.03978259861469269,
0.07656245678663254,
0.3134298622608185,
-0.40821516513824463,
0.5322069525718689,
0.2666700482368469,
-0.2519182562828064,
0.22330664098262787,
-0.37032631039619446,
0.18146361410617828,
0.9154859781265259,
-0.04966771602630615,
0.04238530248403549,
-0.36649367213249207,
-0.39510536193847656,
-0.26978740096092224,
-0.3239399492740631,
-0.37356558442115784,
-0.5410327911376953,
-0.18721410632133484,
-0.39188870787620544,
0.040911197662353516,
-0.36176201701164246,
-0.16839198768138885,
0.29878246784210205,
-0.17540252208709717,
-0.4701007604598999,
0.46939170360565186,
0.6193109154701233,
-0.31817683577537537,
0.12564565241336823,
0.11067567020654678,
-0.23844332993030548,
-0.4571719169616699,
-0.159306138753891,
-0.46869736909866333,
-0.27764949202537537,
0.33494821190834045,
0.8076725602149963,
0.35961276292800903,
-0.3831113576889038,
-0.7233770489692688,
-0.103580042719841,
-0.40097084641456604,
-0.7967212796211243,
0.4276615381240845,
0.2601518929004669,
-0.8165004849433899,
-0.26817235350608826,
0.18369160592556,
0.9677815437316895,
0.4890570044517517,
0.10640527307987213,
-0.43235594034194946,
-0.4948309063911438,
-0.12839536368846893,
-0.1897352635860443,
-0.4838124215602875,
-0.5104429125785828,
-0.6649781465530396,
0.6904845833778381,
-0.5265280604362488,
-0.7388593554496765,
0.2154097855091095,
0.752633810043335,
-0.17691472172737122,
-0.1206093430519104,
0.24643808603286743,
-0.14612415432929993,
-0.7024019956588745,
-0.13861002027988434,
-0.45922887325286865,
0.09414030611515045,
-0.02461673691868782,
-0.5177737474441528,
-0.6810390949249268,
0.5675752758979797,
-5.654379844665527,
0.021860036998987198,
-0.3454975485801697,
0.3942347764968872,
0.23679833114147186,
-0.13986113667488098,
0.19666562974452972,
-0.17040757834911346,
0.4309752285480499,
-0.26342323422431946,
0.02504860982298851,
-0.030949270352721214,
-0.6170167922973633,
0.5189498662948608,
0.2644534409046173,
-0.0014248163206502795,
0.24597108364105225,
0.2834738790988922,
0.37616682052612305,
0.3545510172843933,
0.30533668398857117,
-0.40027284622192383,
-0.6258143186569214,
-0.3490058183670044,
0.07155292481184006,
0.17094682157039642,
-0.2402719408273697,
0.5948945879936218,
-0.42968225479125977,
-0.4880198538303375,
-0.28980186581611633,
-0.4868299663066864,
-0.1108432412147522,
-0.1582503318786621,
0.16562196612358093,
0.35891416668891907,
0.354320764541626,
-0.383564293384552,
0.35757189989089966,
-0.8086660504341125,
0.3908904194831848,
0.5808727145195007,
-0.19952113926410675,
-0.5807496309280396,
0.11962036788463593,
-0.08571448177099228,
-0.30074724555015564,
0.363153874874115,
0.2720179855823517,
0.43436187505722046,
0.07001391053199768,
-0.07200068235397339,
-0.06394682079553604,
0.029276739805936813,
-0.38935813307762146,
-0.21994759142398834,
0.32245129346847534,
-0.07633984833955765,
-0.09859899431467056,
0.34850186109542847,
0.15653164684772491,
-0.4090331494808197,
0.20461341738700867,
-0.15479587018489838,
-0.5229019522666931,
0.16904599964618683,
-0.3207445740699768,
-0.13979551196098328,
0.261822372674942,
-0.17366492748260498,
-0.23093537986278534,
-0.06886862963438034,
0.1817532479763031,
-0.3458698093891144,
0.5868985652923584,
-0.26780766248703003,
0.04689064249396324,
-0.09830965846776962,
-0.1902015656232834,
-0.08990995585918427,
-0.5472524166107178,
-1.152376651763916,
0.18395081162452698,
0.13543860614299774,
0.25449874997138977,
-0.2452736347913742,
0.015012279152870178,
-0.08916708827018738,
-0.4513317346572876,
-0.2617265284061432,
0.727134644985199,
-0.992500364780426,
0.5754015445709229,
0.5998772978782654,
0.45973438024520874,
0.3387417197227478,
0.6833521723747253,
-0.2241058498620987,
0.5563232898712158,
0.2102842628955841,
0.46959030628204346,
-0.03927047550678253,
-0.04114992916584015,
-0.5773026943206787,
0.13704317808151245,
0.5742273330688477,
-0.5914955139160156,
0.4069487750530243,
0.6908913254737854,
-0.054737240076065063,
0.08580369502305984,
0.05240511894226074,
-0.40038061141967773,
-0.17245939373970032,
0.345388799905777,
-0.05982344597578049,
-0.41693028807640076,
-0.09652046114206314,
0.9327576756477356,
0.10426530241966248,
-0.1271980106830597,
0.05914665013551712,
0.0012915992410853505,
-0.16867610812187195,
-0.1844242662191391,
-0.182758629322052,
0.24110610783100128,
0.5889323353767395,
-0.3064476251602173,
-0.3132607936859131,
0.15525618195533752,
0.5614093542098999,
-0.1733972430229187,
0.12603545188903809,
0.1994367092847824,
-0.36064714193344116,
0.4550543427467346,
0.29930102825164795,
0.12934990227222443,
-0.539302408695221,
0.38244491815567017,
0.3437131643295288,
-0.14414197206497192,
-0.5635740756988525,
-0.11851692199707031,
0.18644772469997406,
0.21834607422351837,
0.1714121550321579,
-0.42342931032180786,
-0.2662157118320465,
0.03092009201645851,
-0.5927022099494934,
0.704214334487915,
-0.2670719027519226,
0.010861369781196117,
-0.05643298849463463,
0.1511322557926178,
0.10057072341442108,
0.2565822899341583,
-0.7138475179672241,
0.5012506246566772,
0.38266053795814514,
-0.832116425037384,
-0.08797892928123474,
-0.24451324343681335,
0.2699732482433319,
-0.08130425214767456,
-0.4107925295829773,
0.3743645250797272,
0.6785060167312622,
0.02734002098441124,
-0.0866679847240448,
0.09417542815208435,
-0.34941521286964417,
0.24299027025699615,
-0.11169392615556717,
0.028371354565024376,
-0.018971389159560204,
0.0331333689391613,
-0.12979930639266968,
0.31848806142807007,
-0.13537046313285828,
-0.22262312471866608,
0.4703690707683563,
0.777906596660614,
-0.004013819620013237,
-0.07533764094114304,
0.1656729131937027,
-0.9253889918327332,
-0.4709586203098297,
0.06542888283729553,
-0.007606559433043003,
0.45141783356666565,
-0.4945200979709625,
0.27302446961402893,
0.032771673053503036,
0.7494593858718872,
0.9433157444000244,
-0.2978237271308899,
-0.1954682469367981,
-0.40892109274864197,
0.6273788809776306,
-0.4138127267360687,
0.6513385772705078,
0.17702127993106842,
-0.06334336847066879,
-0.5010960698127747,
0.12081680446863174,
0.1526910364627838,
-0.30625462532043457,
-0.7080156207084656,
-0.38480207324028015,
-0.05262145400047302,
-0.30146944522857666,
-0.2689497470855713,
0.07597323507070541,
0.07981189340353012,
-0.27923622727394104,
0.026895228773355484,
-0.39606285095214844,
0.14564889669418335,
-0.621316134929657,
-0.23457379639148712,
0.22897648811340332,
0.07823561877012253,
0.2803572118282318,
0.16672706604003906,
-0.8398669958114624,
0.3698972761631012,
-0.04818924516439438,
0.4052180051803589,
-0.3623102605342865,
-0.11906600743532181,
0.53739333152771,
-0.05111923813819885,
-0.7275238633155823,
-0.6136566996574402,
0.2016141712665558,
-0.8473921418190002,
-0.46146145462989807,
0.08098091185092926,
0.8582409024238586,
-0.144195556640625,
-0.13616600632667542,
0.14717283844947815,
0.3636628985404968,
-0.03430166840553284,
-0.19486548006534576,
-0.34149497747421265,
0.32186415791511536,
0.0022895175497978926,
0.09278364479541779,
-0.03245200961828232,
-0.06290354579687119,
0.6260833144187927,
0.3605934977531433,
0.07084321975708008,
-0.5678496956825256,
0.18370464444160461,
0.5331572890281677,
0.31158125400543213,
-0.1105908676981926,
-0.5340017080307007,
-0.27500399947166443,
-0.30991366505622864,
0.44156020879745483,
-0.014178184792399406,
0.011858122423291206,
0.6445074677467346,
0.36606815457344055,
-0.32341939210891724,
-0.18067532777786255,
-0.251975417137146,
0.6418318152427673,
-0.17299483716487885,
-0.1467447727918625,
-0.1786108911037445,
0.540942370891571,
-0.01902758702635765,
-0.000704011123161763,
-0.6321075558662415,
-0.3293370008468628,
-0.0006396626704372466,
-0.8791234493255615,
-0.08080584555864334,
0.22966504096984863,
0.5793742537498474,
0.38484427332878113,
0.015815410763025284,
0.6658484935760498,
-0.20425212383270264,
-0.1225215271115303,
-0.7030519843101501,
-0.7018133401870728,
-0.3550006151199341,
0.30503565073013306,
0.29434239864349365,
0.23371614515781403,
-0.1636500060558319,
-0.5000661015510559,
-0.8635340332984924,
-0.1471838504076004,
0.10465555638074875,
-0.4246930778026581,
-0.016411038115620613,
0.19227354228496552,
0.5575914978981018,
-0.3464563488960266,
-0.777851402759552,
0.02272588200867176,
0.19147838652133942,
-0.2549148201942444,
0.5168048739433289,
0.2637297511100769,
-0.39683613181114197,
0.08713264018297195,
0.4222172200679779,
-0.05229170247912407,
0.13046707212924957,
0.6489236950874329,
-0.06874513626098633,
0.09738907217979431,
0.2626951038837433,
-0.6006456017494202,
-0.7454449534416199,
0.5713298320770264,
-0.1781676560640335,
-0.011994528584182262,
-0.4431981146335602,
0.2854900360107422,
0.38484811782836914,
0.11032619327306747,
-0.8874242901802063,
0.4352361261844635,
0.4817548394203186,
0.07766084372997284,
0.5992797613143921,
-0.35995984077453613,
0.08231145888566971,
0.014596112072467804,
0.8170399069786072,
0.7810366153717041,
-0.32201218605041504,
0.3584010899066925,
0.15105511248111725,
-0.14160877466201782,
0.04199986532330513,
-0.18151450157165527,
-0.1924750953912735,
-0.09079153835773468,
0.43872207403182983,
-0.4401261806488037,
-0.231621652841568,
-0.10888355225324631,
0.09718596190214157,
0.07081621885299683,
0.511199414730072,
-0.23279592394828796,
-0.33500009775161743,
0.23235619068145752,
0.5290365219116211,
0.4725411534309387,
-0.17744535207748413,
-0.2929632365703583,
-0.22345425188541412,
-0.43187519907951355,
1.0468742847442627,
0.076802097260952,
-0.09683185815811157,
-0.1055968627333641,
-0.3331775367259979,
0.5580361485481262,
-0.12246732413768768,
0.29636552929878235,
0.39760178327560425,
-0.05470335856080055,
-0.07154607772827148,
0.23133544623851776,
-0.24915698170661926,
-0.2766518294811249,
0.9973257780075073,
0.056616537272930145,
0.18690623342990875,
-0.5787572860717773,
-0.3278331160545349,
-0.5050114989280701,
0.3610702157020569,
-0.20293821394443512,
0.018186064437031746,
-0.10666658729314804,
0.4935331642627716,
-0.5193173289299011,
-0.2580736577510834,
0.31981274485588074,
0.37444865703582764,
0.1177380308508873,
-0.9061430096626282,
1.3498516082763672,
0.7108285427093506,
0.41952183842658997,
0.505146861076355,
-0.035964980721473694,
0.2445284128189087,
1.0897427797317505,
-0.34001943469047546,
-0.2525538504123688,
-0.5858863592147827,
0.052576299756765366,
-0.7957090735435486,
0.16497819125652313,
1.0163899660110474,
0.056627288460731506,
-0.20123830437660217,
-0.2796974182128906,
0.3939361870288849,
0.19160965085029602,
0.16542382538318634,
-0.2784814238548279,
-0.1728593111038208,
-0.4018970727920532,
-0.27527692914009094,
-0.4149979054927826,
0.37653249502182007,
0.6776450872421265,
-0.29019734263420105,
-0.1512068659067154,
0.6215975284576416,
0.1398160755634308,
-0.23793846368789673,
0.22814053297042847,
-0.005720547400414944,
0.47727906703948975,
-0.48091599345207214,
0.553202748298645,
-0.5009950399398804,
-0.06277794390916824,
0.4924301505088806,
0.3658809959888458,
0.49469709396362305,
-0.32050055265426636,
-0.9365816116333008,
-0.4286527931690216,
-0.12050188332796097,
0.5717295408248901,
0.03338199481368065,
0.27343130111694336,
-0.07418245822191238,
-0.162360280752182,
-0.7838410139083862,
0.6571813821792603,
0.03970084339380264,
-0.15356972813606262,
-0.16589981317520142,
0.2587253749370575,
-0.3606560230255127,
0.5697019696235657,
-0.13114580512046814,
-0.4785773754119873,
-0.7240503430366516,
-0.29411301016807556
] |
as this is information-theoretically possible; (ii) it achieves the Bayes-optimal error for
u∗ (resp. v∗) when Ξ (resp. Σ) is the identity. These optimality guarantees follow from
rigorously obtaining the asymptotic minimum mean square error (MMSE) for the estimation
of the whitened signals Ξ−1/2u∗ and Σ−1/2v∗, see Theorem 4.2.
Our spectral estimator only involves matrix multiplication and computing principal singular vectors. | [
-0.10381801426410675,
-0.360792875289917,
0.31785115599632263,
-0.30547964572906494,
-0.31002718210220337,
0.24832552671432495,
0.2759808301925659,
-0.7398487329483032,
0.045286279171705246,
-0.714417576789856,
-0.2925715148448944,
0.6374086141586304,
-0.2459782212972641,
0.0017908578738570213,
0.22105772793293,
0.37476518750190735,
0.9115952253341675,
0.10465335100889206,
-0.47787413001060486,
-0.7927879691123962,
0.3954862356185913,
0.35377398133277893,
0.40269792079925537,
-0.00007917208131402731,
-0.004161741118878126,
0.12231916934251785,
0.5547747611999512,
0.17736949026584625,
0.47430557012557983,
0.0940525084733963,
-0.1647576540708542,
0.4116394519805908,
-0.3911193013191223,
-0.09138258546590805,
0.6945192217826843,
0.31684303283691406,
0.6541644930839539,
0.1754753589630127,
0.29518234729766846,
0.21936647593975067,
0.13519546389579773,
-0.23666684329509735,
0.20622964203357697,
0.22334493696689606,
-0.31899604201316833,
-0.8711435198783875,
-1.217357873916626,
-0.05633312463760376,
-0.14759257435798645,
-0.7559040188789368,
0.023577863350510597,
0.2389470338821411,
0.22739580273628235,
0.5450228452682495,
-0.2581595182418823,
0.334717720746994,
-0.272495836019516,
-0.254366934299469,
0.698154628276825,
-0.12919436395168304,
-0.08037997782230377,
-0.2937767803668976,
-0.024773260578513145,
0.24924927949905396,
0.010823617689311504,
0.8776338696479797,
-0.010455334559082985,
-0.26476535201072693,
-0.1654997617006302,
-0.46405065059661865,
0.15694549679756165,
-0.2795715034008026,
-0.32602372765541077,
0.12636934220790863,
-0.37048837542533875,
0.02310517057776451,
-0.07134529948234558,
-0.213864266872406,
-0.0040504527278244495,
0.0059326025657355785,
-0.24698986113071442,
0.10675808787345886,
0.37278562784194946,
-0.26365187764167786,
0.37564733624458313,
0.2898884117603302,
0.024100491777062416,
0.6455926299095154,
-0.6405448913574219,
0.5441531538963318,
-0.03774755075573921,
-0.2948742210865021,
-0.007648067083209753,
0.19546723365783691,
0.4205831289291382,
0.9436017274856567,
-0.1026422306895256,
-0.33639881014823914,
0.8326178789138794,
0.09402421116828918,
-0.6846636533737183,
-0.10777178406715393,
-0.11191418766975403,
-0.48607686161994934,
-0.2809990346431732,
-0.6402832269668579,
-0.6584817171096802,
-0.2158232182264328,
-0.49550771713256836,
0.186432883143425,
-0.6680024266242981,
0.5621157288551331,
0.2296646237373352,
-0.5637246370315552,
-0.07055527716875076,
-0.2223709225654602,
-0.018636111170053482,
-0.08341018110513687,
-0.16839200258255005,
0.6995853185653687,
-0.12558801472187042,
1.2561125755310059,
-0.04009896516799927,
0.4367273151874542,
0.25517064332962036,
-0.07846429944038391,
0.7585638165473938,
0.9133679866790771,
-0.4290885031223297,
0.04597631096839905,
-0.020567115396261215,
0.9786266088485718,
0.19210419058799744,
0.5249158143997192,
-0.2832704186439514,
-0.7063250541687012,
-0.4444481432437897,
-0.37755292654037476,
-0.8655703663825989,
-0.14921879768371582,
-0.20232096314430237,
-0.41866758465766907,
0.003671468235552311,
0.16446945071220398,
0.5283768773078918,
-0.12639759480953217,
-0.14671415090560913,
-0.3039263188838959,
-0.06770837306976318,
0.022678228095173836,
-0.10773475468158722,
0.5175569653511047,
-0.7902604937553406,
-0.04853227734565735,
-0.5802355408668518,
-0.13076259195804596,
0.32151085138320923,
0.4094379246234894,
-0.6574150919914246,
0.4349777400493622,
0.14686435461044312,
0.01961774379014969,
-0.019084500148892403,
-0.4493328928947449,
0.06750708818435669,
-0.5868901610374451,
0.6207235455513,
0.3017284572124481,
0.24016158282756805,
-0.3133150041103363,
0.13791528344154358,
-0.4059962332248688,
-0.5178807377815247,
0.059655677527189255,
-0.12492924928665161,
-0.11625481396913528,
0.16676294803619385,
0.1259770542383194,
0.24596449732780457,
-0.07822167128324509,
-0.7205284237861633,
0.07263102382421494,
0.019092252478003502,
-0.24021238088607788,
0.49248409271240234,
0.1280706524848938,
0.20164774358272552,
0.22657276690006256,
-0.051499590277671814,
-0.01693229004740715,
-0.312485009431839,
0.11256249994039536,
0.17779093980789185,
-1.1419320106506348,
0.8243163824081421,
0.18005315959453583,
-0.5310177206993103,
0.566823422908783,
0.2197202742099762,
0.09873559325933456,
0.08666275441646576,
0.04545239731669426,
-0.059409160166978836,
0.19770435988903046,
-0.5347591042518616,
0.3974275290966034,
0.1062530130147934,
0.31941837072372437,
-0.5759328603744507,
0.05861031636595726,
0.4540274739265442,
0.027736136689782143,
0.06910566985607147,
-0.2336423248052597,
0.3709976077079773,
0.06982570886611938,
-0.13647180795669556,
0.21844100952148438,
0.6628298163414001,
0.052244752645492554,
1.0166294574737549,
-0.41282281279563904,
0.46851861476898193,
0.39144110679626465,
0.039344608783721924,
0.8071371912956238,
0.5618932247161865,
0.2806128263473511,
0.7001425623893738,
-0.7395265698432922,
-0.3624410033226013,
-0.5703645944595337,
0.2207082211971283,
0.00814596563577652,
-0.32053521275520325,
0.4597119390964508,
0.26470962166786194,
-0.1802445948123932,
0.32489004731178284,
-0.11492092907428741,
0.19556406140327454,
0.4065355360507965,
-0.06467492878437042,
-0.012977655977010727,
-0.15590542554855347,
-0.27588462829589844,
0.006248236633837223,
-0.3861103653907776,
-0.23401600122451782,
-0.23786188662052155,
-0.0012105583446100354,
0.25993800163269043,
-0.04952309653162956,
-0.7597397565841675,
0.1273152232170105,
-0.10249505192041397,
-0.2715131938457489,
-0.4293127655982971,
0.4273410439491272,
0.42727234959602356,
0.0004726303741335869,
-0.3316101133823395,
0.0845332220196724,
-0.19658450782299042,
-0.5751012563705444,
-0.14927884936332703,
-1.0339428186416626,
-0.04162372276186943,
0.436335027217865,
0.5395136475563049,
0.3619368374347687,
-0.5971910357475281,
-0.5873849391937256,
0.07732489705085754,
-0.3270692825317383,
-0.7381565570831299,
0.11825844645500183,
0.508860170841217,
-0.9065740704536438,
-0.19964733719825745,
0.17962028086185455,
0.6384422183036804,
0.3574760854244232,
0.3463343679904938,
-0.6506680846214294,
-0.3151884377002716,
-0.005651815328747034,
-0.3411913514137268,
-0.4114261269569397,
-0.3296111822128296,
-0.48360469937324524,
0.7517205476760864,
-0.021795975044369698,
-0.46934473514556885,
0.16632896661758423,
0.7478824257850647,
-0.21587833762168884,
-0.14278165996074677,
0.2844974398612976,
-0.27874138951301575,
-0.5509935617446899,
0.01503877155482769,
-0.009920069016516209,
-0.005804813466966152,
0.03651800751686096,
-0.18723368644714355,
-0.4577723443508148,
0.6111426949501038,
-5.802536487579346,
-0.18224814534187317,
-0.34315311908721924,
0.5059580206871033,
-0.1889210343360901,
-0.09769773483276367,
0.6465606093406677,
-0.29999542236328125,
0.3365563452243805,
-0.08379976451396942,
0.035844627767801285,
0.2072376161813736,
-0.5339357852935791,
0.47737449407577515,
0.29304394125938416,
-0.11120806634426117,
0.1807027906179428,
0.11874198168516159,
0.5526162981987,
0.20016954839229584,
-0.08682200312614441,
-0.2746834456920624,
-0.15815076231956482,
-0.1089307889342308,
0.01666993275284767,
0.012340662069618702,
-0.5636869668960571,
0.4733465313911438,
-0.2569064795970917,
-0.5503520965576172,
-0.039046578109264374,
-0.7578014135360718,
0.2619371712207794,
-0.04199320822954178,
0.41248998045921326,
0.1615288257598877,
0.4644961953163147,
-0.4877936840057373,
0.09901049733161926,
-0.7241299152374268,
0.4241566061973572,
0.8183147311210632,
-0.4920356869697571,
-0.3584141731262207,
0.11031109094619751,
-0.13715633749961853,
-0.001592851709574461,
0.32150179147720337,
0.10562314093112946,
0.9326149225234985,
-0.3010549545288086,
-0.1593298614025116,
-0.24029618501663208,
0.013721756637096405,
-0.5372588038444519,
-0.19207610189914703,
-0.1337219476699829,
0.25363337993621826,
-0.31039029359817505,
0.28103363513946533,
0.19238415360450745,
-0.42322954535484314,
0.06392661482095718,
-0.3081952631473541,
-0.3552541136741638,
0.2799370586872101,
-0.5154945850372314,
-0.12187017500400543,
0.33054065704345703,
-0.11397663503885269,
-0.2732793986797333,
-0.28023219108581543,
0.2595880329608917,
-0.36585354804992676,
0.33740782737731934,
-0.4754268229007721,
0.330429345369339,
-0.1561025232076645,
-0.12663818895816803,
0.24370470643043518,
-0.21384797990322113,
-0.7775048017501831,
0.12744954228401184,
0.19490806758403778,
0.18312028050422668,
-0.11487340182065964,
0.05407217517495155,
0.034485019743442535,
-0.41269582509994507,
-0.16506464779376984,
0.5802858471870422,
-0.17990118265151978,
0.241209015250206,
0.5071508884429932,
0.11306898295879364,
0.618939220905304,
0.5292544364929199,
-0.09071416407823563,
0.5619439482688904,
-0.05116619914770126,
0.6284996867179871,
-0.31291231513023376,
-0.10803677141666412,
-0.7151245474815369,
0.10289875417947769,
0.7008364200592041,
-0.5227542519569397,
0.42257970571517944,
0.8222904801368713,
-0.3374079763889313,
-0.0012901145964860916,
-0.1256563365459442,
-0.5592203140258789,
-0.18452788889408112,
0.7851114273071289,
-0.11718790978193283,
-0.41249915957450867,
0.029228458181023598,
0.7084946632385254,
0.1103539913892746,
0.21821007132530212,
0.10226123780012131,
-0.11346730589866638,
-0.0012910196091979742,
-0.008983130566775799,
-0.19965332746505737,
0.13171672821044922,
1.0663923025131226,
-0.6924946308135986,
-0.2871761918067932,
0.40071621537208557,
0.38107532262802124,
-0.5595637559890747,
0.26814892888069153,
0.4621175229549408,
-0.39847177267074585,
0.20184960961341858,
0.4008863568305969,
0.7017117738723755,
-0.2532850205898285,
0.9375455379486084,
0.31338146328926086,
-0.33011382818222046,
-0.22142527997493744,
-0.08499445766210556,
0.30123329162597656,
-0.038055017590522766,
-0.3499971032142639,
-0.7327985167503357,
-0.12019124627113342,
-0.0034595124889165163,
-0.5922816395759583,
0.44991904497146606,
-0.14624613523483276,
-0.00022475105652119964,
0.24692893028259277,
-0.00896180048584938,
-0.06942784786224365,
0.41683968901634216,
-0.3632042109966278,
0.24438270926475525,
0.48626384139060974,
-0.7933330535888672,
-0.06137319654226303,
0.03624933958053589,
-0.02632559835910797,
-0.7020978927612305,
-0.32157692313194275,
-0.18195760250091553,
0.5746252536773682,
-0.1615891456604004,
-0.5463355183601379,
0.12071707099676132,
-0.49389415979385376,
0.1040535643696785,
0.15334726870059967,
-0.10843227803707123,
-0.12286119908094406,
0.011451193131506443,
-0.45646631717681885,
0.27588480710983276,
-0.28166472911834717,
-0.45149314403533936,
0.815504789352417,
0.9834446310997009,
-0.23011136054992676,
0.05069131404161453,
-0.1248934343457222,
-0.6820869445800781,
-0.05130041763186455,
0.07928118854761124,
0.10688052326440811,
0.3461821377277374,
-0.36751431226730347,
0.257707804441452,
-0.16218577325344086,
0.4809340238571167,
0.34967517852783203,
-0.2589317560195923,
-0.36610132455825806,
-0.029428310692310333,
0.7276701927185059,
-0.5906369686126709,
0.395891010761261,
-0.04336007311940193,
-0.21562781929969788,
-0.10945125669240952,
0.4126197397708893,
-0.31986576318740845,
-0.5617954134941101,
-0.6320644021034241,
-0.6380504369735718,
-0.30517011880874634,
-0.20344288647174835,
-0.3456927537918091,
0.15707360208034515,
0.27319443225860596,
-0.28166642785072327,
0.06290819495916367,
-0.09520258754491806,
0.5230863094329834,
-0.6142899394035339,
0.2732295095920563,
0.32851889729499817,
-0.0011041867546737194,
0.23773446679115295,
-0.09608535468578339,
-0.8338080644607544,
0.380693256855011,
0.06571884453296661,
-0.17479366064071655,
-0.14865639805793762,
-0.2959350645542145,
0.29962339997291565,
-0.16078291833400726,
-0.3472204804420471,
-0.7064294815063477,
0.19759336113929749,
-0.6316310167312622,
-0.07944163680076599,
-0.09536496549844742,
0.8274858593940735,
-0.1770641803741455,
-0.1281771957874298,
-0.12616916000843048,
0.3412263095378876,
0.5348026752471924,
-0.0004972114693373442,
-0.6276474595069885,
-0.032639823853969574,
0.0377071276307106,
-0.10855741053819656,
0.3154750466346741,
-0.008818295784294605,
0.6068079471588135,
0.2568094730377197,
0.011137891560792923,
-0.5971843004226685,
-0.09865575283765793,
0.3520943224430084,
-0.1297852247953415,
-0.5861815214157104,
-0.14454679191112518,
-0.4331769347190857,
-0.22009485960006714,
0.4229448735713959,
-0.013469378463923931,
-0.48192325234413147,
0.15951132774353027,
0.20623348653316498,
0.04937000572681427,
-0.4324323236942291,
0.03687603771686554,
0.4127681255340576,
-0.2185221165418625,
0.1582001894712448,
-0.3746128976345062,
0.6173684597015381,
0.07653547078371048,
0.12912343442440033,
-0.36402860283851624,
-0.6887755393981934,
-0.25896623730659485,
-0.5280720591545105,
0.038358740508556366,
0.30538010597229004,
0.6264200210571289,
-0.07846593856811523,
-0.1380535364151001,
0.36492598056793213,
-0.1835019588470459,
-0.1851225048303604,
-1.0099177360534668,
-0.36028817296028137,
-0.2708541452884674,
0.36258041858673096,
0.30450063943862915,
0.3514000177383423,
-0.18183472752571106,
-0.28474900126457214,
-0.3121422231197357,
-0.02566288784146309,
0.17030389606952667,
-0.2668476998806,
0.13275592029094696,
0.1135440543293953,
0.44731709361076355,
0.024898361414670944,
-0.5421302914619446,
-0.2571012079715729,
0.24569274485111237,
0.25687751173973083,
0.08569712191820145,
0.33859384059906006,
-0.41203030943870544,
-0.03088689036667347,
0.6243085265159607,
0.10090581327676773,
0.5767219662666321,
0.5635595321655273,
-0.19863954186439514,
0.16768930852413177,
0.6997132301330566,
-0.39260679483413696,
-0.9196351766586304,
0.6920218467712402,
-0.32802364230155945,
0.397067666053772,
-0.1513354480266571,
0.27961266040802,
0.16152793169021606,
0.5423271059989929,
-0.6666033267974854,
0.44038495421409607,
0.15681931376457214,
0.32771849632263184,
0.4571681618690491,
-0.5730575919151306,
-0.11656726151704788,
0.07236924767494202,
0.7862239480018616,
0.8620238304138184,
-0.444777250289917,
0.09622514247894287,
0.08986552804708481,
0.04041258245706558,
0.09650939702987671,
-0.037602752447128296,
-0.39137768745422363,
0.11111090332269669,
0.1083039939403534,
-0.371101051568985,
-0.04248179495334625,
-0.024442149326205254,
-0.12890055775642395,
-0.009706488810479641,
0.2753399908542633,
-0.4960189163684845,
-0.0992751270532608,
0.32845601439476013,
0.5345615148544312,
0.26709291338920593,
-0.368677020072937,
-0.12897196412086487,
-0.669684112071991,
-0.41135185956954956,
0.521686851978302,
0.4878319501876831,
-0.08235017210245132,
0.016352472826838493,
-0.007774012628942728,
0.5533145666122437,
-0.13486835360527039,
0.336309015750885,
0.35954973101615906,
-0.3918045163154602,
-0.06405088305473328,
0.2117999941110611,
-0.46931102871894836,
-0.3349105715751648,
0.40608981251716614,
0.3852105736732483,
-0.16406577825546265,
-0.6945193409919739,
-0.4268207550048828,
-0.4069860279560089,
0.13874965906143188,
-0.23837627470493317,
-0.10118867456912994,
0.09737631678581238,
0.23308254778385162,
-0.5174139142036438,
-0.5356162190437317,
0.016218969598412514,
0.33810433745384216,
0.10690929740667343,
-0.6134994626045227,
0.9199342131614685,
0.683707594871521,
0.11784505099058151,
0.5056421160697937,
-0.04107465222477913,
0.22267881035804749,
0.6877741813659668,
-0.33514147996902466,
0.3223036527633667,
-0.43926507234573364,
0.20325075089931488,
-0.6062940955162048,
-0.28325799107551575,
0.7467917799949646,
-0.029254255816340446,
-0.13961079716682434,
-0.19836413860321045,
-0.08047760277986526,
0.18936097621917725,
0.4170669913291931,
0.013162542134523392,
-0.05898211523890495,
-0.13424360752105713,
-0.7484229803085327,
-0.6490654945373535,
0.76670241355896,
0.4444234073162079,
0.15002666413784027,
-0.15906113386154175,
0.7021582722663879,
0.2824215292930603,
-0.02203875035047531,
0.23965443670749664,
0.0906573086977005,
0.08736363798379898,
-0.4166480302810669,
0.7584339380264282,
-0.481545627117157,
0.013224414549767971,
-0.09768297523260117,
0.07896503061056137,
0.5356380939483643,
-0.619502604007721,
-0.3690025210380554,
-0.16809576749801636,
-0.3746470808982849,
0.7019407749176025,
-0.05109665170311928,
0.3484286665916443,
-0.15830376744270325,
-0.44091635942459106,
-0.5638808012008667,
0.6762526035308838,
0.03477872535586357,
-0.41503211855888367,
-0.240572988986969,
0.11285796016454697,
-0.39004263281822205,
0.8218206763267517,
0.08782368153333664,
-0.36627447605133057,
-0.38059863448143005,
-0.16196782886981964
] |
End of preview. Expand
in Dataset Viewer.
README.md exists but content is empty.
- Downloads last month
- 36