text
stringlengths 2
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V
2�
E
�
B
�
where V ∼ Q.
Proposition G.3. Let
(W1, W2) ∼ N
��
0d
0d
�
,
�
σ2
1
ρ
ρ
σ2
2
�
⊗ Id
�
.
Let B ∈ Rd×d denote a sequence (indexed by d) of deterministic matrices such that the empirical
spectral distribution of 1
dB converges to the law of a random variable B. Then
lim
d→∞
1
dE
�
W ⊤
1 BW2
�
= ρE
�
B
�
.
Proposition G.4 (Nishimori identity). Let (X, Y ) be two random variables. Let k ≥ 1 and
X1, · · · , Xk be k i.i.d. samples (given Y ) from the distribution law(X | Y ). Denote by ⟨·⟩, E[·] | [
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] |
the expectations with respect to law(X | Y ) and law(X, Y ), respectively. Then for all continuous
bounded function f, it holds that
E[⟨f(Y, X1, · · · , Xk)⟩] = E[⟨f(Y, X1, · · · , Xk−1, X)⟩].
Proposition G.5 (Conditional distribution of Gaussians). Let d ≥ 2 and 1 ≤ p ≤ d − 1 be integers.
Let
�
G1
G2
�
∼ N
��
µ1
µ2
�
,
�Σ1,1
Σ1,2
Σ⊤
1,2
Σ2,2
��
be a d-dimensional Gaussian random vector, where G1 ∈ Rp, G2 ∈ Rd−p, µ1 ∈ Rp, µ2 ∈ | [
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] |
Σ⊤
1,2
Σ2,2
��
be a d-dimensional Gaussian random vector, where G1 ∈ Rp, G2 ∈ Rd−p, µ1 ∈ Rp, µ2 ∈
Rd−p, Σ1,1 ∈ Rp×p, Σ1,2 ∈ Rp×(d−p), Σ2,2 ∈ R(d−p)×(d−p). Then for any g2 ∈ Rd−p, the
distribution of G1 conditioned on G2 = g2 is given by G1 | {G2 = g2} ∼ N(µ′
1, Σ′
1) where
µ′
1 = µ1 + Σ1,2Σ−1
2,2(g2 − µ2) ∈ Rp,
Σ′
1 = Σ1,1 − Σ1,2Σ−1
2,2Σ⊤
1,2 ∈ Rp×p
and Σ−1
2,2 denotes the generalized inverse of Σ2,2.
49
Proposition G.6 (Chain rule of derivatives). Let A ∈ Rn×n and f : Rn → Rn. Let x ∈ Rn and | [
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] |
49
Proposition G.6 (Chain rule of derivatives). Let A ∈ Rn×n and f : Rn → Rn. Let x ∈ Rn and
�x := Ax. Then for any i, j ∈ [n],
∂
∂xi
(Af(Ax))i =
n
�
j=1
n
�
k=1
Ai,jAk,i
∂f(�x)j
∂�xk
,
where f(�x)j ∈ R denotes the j-th (j ∈ [n]) output of f(�x) ∈ Rn.
Proof. The proof follows from elementary applications of the chain rule for derivatives. Writing
A = [a1
· · ·
an]⊤ and f = [f1
· · ·
fn]⊤, we have
∂
∂xi
fj(Ax) =
∂
∂xi
fj(⟨a1, x⟩, · · · , ⟨an, x⟩) =
n
�
k=1
∂kfj(Ax)∂⟨ak, x⟩
∂xi
=
n
�
k=1
Ak,i∂kfj(�x), | [
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] |
fj(Ax) =
∂
∂xi
fj(⟨a1, x⟩, · · · , ⟨an, x⟩) =
n
�
k=1
∂kfj(Ax)∂⟨ak, x⟩
∂xi
=
n
�
k=1
Ak,i∂kfj(�x),
where ∂kfj denotes the partial derivative of fj : Rn → R with respect to its k-th argument. Then,
∂
∂xi
(Af(Ax))i =
n
�
j=1
Ai,j
∂
∂xi
fj(Ax) =
n
�
j=1
n
�
k=1
Ai,jAk,i∂kfj(�x),
as claimed.
Proposition G.7 (Stein’s lemma [65]). Let W ∼ N(0, σ2) and let f : R → R be such that both
expectations below exist. Then E[Wf(W)] = σ2E[f ′(W)]. | [
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0.6492809057235718,
0.2183012217283249,
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0.5628615617752075,
-0.12645530700683594,
-0.41179168224334717,
-0.29356974363327026,
0.157976433634758
] |
expectations below exist. Then E[Wf(W)] = σ2E[f ′(W)].
Proposition G.8 (Gaussian Poincaré inequality [15, Theorem 3.20]). Let X ∼ N(0n, In) and
f : Rn → R a differentiable function. Then
Var[f(X)] ≤ E
�
∥∇f(X)∥2
2
�
.
Proposition G.9 (Bounded difference inequality [15, Corollary 3.2]). Let U ⊂ R and f : Un → R a
function such that there exist c = (c1, · · · , cn) ∈ Rn
≥0 satisfying for all i ∈ [n],
sup
(x1,··· ,xn,x′
i)∈Un+1|f(x1, · · · , xi−1, xi, xi+1, · · · , xn) − f(x1, · · · , xi−1, x′ | [
-0.15420497953891754,
-0.33474811911582947,
0.4981900155544281,
0.06184995919466019,
0.1639649122953415,
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0.9417360424995422,
0.0824471116065979,
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] |
sup
(x1,··· ,xn,x′
i)∈Un+1|f(x1, · · · , xi−1, xi, xi+1, · · · , xn) − f(x1, · · · , xi−1, x′
i, xi+1, · · · , xn)| ≤ ci.
Then if X ∈ Un is a random vector consisting of independent elements, we have Var[f(X)] ≤
∥c∥2
2/4.
Proposition G.10 ([59, Lemma 3.2]). If f, g are differentiable convex functions, then for any a ∈ R
and a′ > 0,
|f ′(a) − g′(a)| ≤ g′(a + a′) − g′(a − a′) + B/a′,
where
B := |f(a + a′) − g(a + a′)| + |f(a − a′) − g(a − a′)| + |f(a) − g(a)|. | [
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0.1525120884180069,
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0.6154226660728455
] |
where
B := |f(a + a′) − g(a + a′)| + |f(a − a′) − g(a − a′)| + |f(a) − g(a)|.
Definition G.1 (Monotone conjugate). Let f : R≥0 → R be a non-decreasing convex function. Its
monotone conjugate f ∗ is defined as
f ∗(x) = sup
y≥0
xy − f(y).
Proposition G.11 ([49, Proposition C.1]). Let I ⊂ R be an interval and (fn : I → R)n be a
sequence of convex functions converging pointwise to f. Then for all t ∈ I such that the following
quantities exist,
lim
s↑t f ′(s) ≤ lim inf
n→∞ lim
s↑t f ′
n(s) ≤ lim sup
n→∞ lim | [
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] |
quantities exist,
lim
s↑t f ′(s) ≤ lim inf
n→∞ lim
s↑t f ′
n(s) ≤ lim sup
n→∞ lim
s↓t f ′
n(s) ≤ lim
s↓t f ′(s).
Proposition G.12 ([49, Proposition C.6]). Let f, g: R≥0 → R be strictly convex differentiable
functions and
C :=
�
(q1, q2) ∈ R2
≥0 : q2 = f ′(q1), q1 = g′(q2)
�
.
Then
sup
(q1,q2)∈C
f(q1) + g(q2) − q1q2 =
sup
q1,q2≥0
q1q2 − f ∗(q2) − g∗(q1) = sup
q1≥0
inf
q2≥0 f(q1) + g(q2) − q1q2,
and sup(q1,q2)∈C and supq1,q2≥0 are achieved at the same (q∗
1, q∗
2).
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] |
Entrywise error bounds for low-rank approximations
of kernel matrices
Alexander Modell
Department of Mathematics
Imperial College London, U.K.
[email protected]
Abstract
In this paper, we derive entrywise error bounds for low-rank approximations of kernel
matrices obtained using the truncated eigen-decomposition (or singular value decomposition).
While this approximation is well-known to be optimal with respect to the spectral and | [
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] |
While this approximation is well-known to be optimal with respect to the spectral and
Frobenius norm error, little is known about the statistical behaviour of individual entries.
Our error bounds fill this gap. A key technical innovation is a delocalisation result for the
eigenvectors of the kernel matrix corresponding to small eigenvalues, which takes inspiration
from the field of Random Matrix Theory. Finally, we validate our theory with an empirical | [
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from the field of Random Matrix Theory. Finally, we validate our theory with an empirical
study of a collection of synthetic and real-world datasets.
1
Introduction
Low-rank approximations of kernel matrices play a central role in many areas of machine learning.
Examples include kernel principal component analysis [Schölkopf et al., 1998], spectral clustering
[Ng et al., 2001, Von Luxburg, 2007] and manifold learning [Roweis and Saul, 2000, Belkin and | [
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[Ng et al., 2001, Von Luxburg, 2007] and manifold learning [Roweis and Saul, 2000, Belkin and
Niyogi, 2001, Coifman and Lafon, 2006], where they serve as a core component of the algorithms,
and support vector machines [Cortes and Vapnik, 1995, Fine and Scheinberg, 2001] and Gaussian
process regression [Williams and Rasmussen, 1995, Ferrari-Trecate et al., 1998] where they serve to
dramatically speed up computation times. | [
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] |
dramatically speed up computation times.
In this paper, we derive entrywise error bounds for low-rank approximations of kernel matrices
obtained using the truncated eigen-decomposition (or singular value decomposition). Entrywise
error bounds are important for a number of reasons. The first is practical: in applications where
individual errors carry a high cost, such as system control and healthcare, we should seek methods | [
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] |
individual errors carry a high cost, such as system control and healthcare, we should seek methods
with low entrywise error. The second is theoretical: good entrywise error bounds can lead to
improved analyses of learning algorithms which use them.
For this reason, a wealth of literature has emerged establishing entrywise error bounds for a
variety of matrix estimation problems, such as covariance estimation [Fan et al., 2018, Abbe et al., | [
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] |
2022], matrix completion [Candes and Recht, 2012, Chi et al., 2019], phase synchronisation [Zhong
and Boumal, 2018, Ma et al., 2018], reinforcement learning [Lei, 2019], community detection
[Balakrishnan et al., 2011, Lyzinski et al., 2014, Eldridge et al., 2018, Lei, 2019, Mao et al., 2021]
and graph inference [Cape et al., 2019, Rubin-Delanchy et al., 2022] to name a few.
1.1
Contributions
• Our main result (Theorem 1) is an entrywise error bound for the low-rank approximation | [
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] |
• Our main result (Theorem 1) is an entrywise error bound for the low-rank approximation
of a kernel matrix. Under regularity conditions, we find that for kernels with polynomial
eigenvalue decay, λi = O(i−α), we require a polynomial-rank approximation, d = Ω(n1/α), to
1
arXiv:2405.14494v1 [math.ST] 23 May 2024
achieve entrywise consistency. For kernels with exponential eigenvalue decay, λi = O(eβiγ),
we require a (poly)log-rank approximation, d > log1/γ(n1/β). | [
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] |
we require a (poly)log-rank approximation, d > log1/γ(n1/β).
• The main technical contribution of this paper is to establish a delocalisation result for the
eigenvectors of the kernel matrix corresponding to small eigenvalues (Theorem 2), the proof
of which draws on ideas from the Random Matrix Theory literature. To our knowledge, this
is the first such result for a random matrix with non-zero mean and dependent entries. | [
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] |
is the first such result for a random matrix with non-zero mean and dependent entries.
• Along the way, we prove a novel concentration inequality for the distance between a random
vector (with a potentially non-zero mean) and a subspace (Lemma 1), which may be of
independent interest.
• We complement our theory with an empirical study on the entrywise errors of low-rank
approximations of the kernel matrices on a collection of synthetic and real datasets. As an | [
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] |
approximations of the kernel matrices on a collection of synthetic and real datasets. As an
aside, we discover an unexpected double-decent phenomenon in the quantiles of the entrywise
errors in some of our datasets which our theory cannot explain.
1.2
Related work
A complementary result to ours is a deterministic error bound by Udell and Townsend [2019]
who use the Johnson–Lindenstrauss lemma to upper bound the best-possible entrywise error of | [
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] |
who use the Johnson–Lindenstrauss lemma to upper bound the best-possible entrywise error of
a low-rank approximation of a matrix in terms of its spectral norm. While their result is of a
fundamentally different nature to ours, their informal conclusion that “nice kernel matrices are of
approximately log-rank”1 aligns with ours for kernels with exponential eigenvalue decay. However,
we find that the story is more nuanced in general. | [
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] |
we find that the story is more nuanced in general.
Our proof strategy draws heavily on ideas from the Random Matrix Theory literature, where
delocalisation results have been established for certain classes of zero-mean random matrices with
independent entries [Erdős et al., 2009b,a, Tao and Vu, 2011, Rudelson and Vershynin, 2015, Vu and
Wang, 2015]. In addition, our result is made possible by recent relative eigenvalue concentration | [
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] |
Wang, 2015]. In addition, our result is made possible by recent relative eigenvalue concentration
bounds for kernel matrices [Braun, 2006, Valdivia, 2018, Barzilai and Shamir, 2023], which improve
upon classical absolute concentration bounds [Rosasco et al., 2010] which would not provide
sufficient control for our purposes.
1.3
Big-O notation and frequent events
We use the standard big-O notation where an = O(bn) (resp.
an = Ω(bn)) means that for | [
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We use the standard big-O notation where an = O(bn) (resp.
an = Ω(bn)) means that for
sufficiently large n, an ≤ Cbn (resp. an ≥ Cbn) for some constant C which doesn’t depend on the
parameters of the problem. We will occasionally write an ≲ bn to mean that an = O(bn).
In addition, we say that an event En holds with overwhelming probability if for every c > 0,
P(En) ≥ 1 − O(n−c), where the hidden constant is allowed to depend on c.
2
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] |
P(En) ≥ 1 − O(n−c), where the hidden constant is allowed to depend on c.
2
Setup
We begin by describing the setup of our problem. We suppose that we observe n, p-dimensional
data points {xi}n
i=1, which we assume were drawn i.i.d. from some probability distribution ρ,
supported on a set X. Given a symmetric kernel k : X × X → R, we construct the n × n kernel
matrix K, with entries
K(i, j) := k(xi, xj). | [
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] |
matrix K, with entries
K(i, j) := k(xi, xj).
1Udell and Townsend [2019] refer to the class of kernel matrices as latent variable models, and this quote is
paraphrased from the statement “nice latent variables models are of log-rank”.
2
We will assume throughout that the kernel is positive-definite, continuous and bounded. Let �Kd
denote the “best” rank-d approximation of K, in the sense that �Kd satisfies
�Kd :=
arg min
K′:rank(K′)=d
∥K − K′∥ξ ,
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�Kd :=
arg min
K′:rank(K′)=d
∥K − K′∥ξ ,
(1)
where ∥·∥ξ is a rotation-invariant norm, such as the spectral or Frobenius norm. Then by the
Eckart-Young-Mirsky theorem [Eckart and Young, 1936, Mirsky, 1960], �Kd is given by the truncated
eigen-decomposition of K, i.e.
�Kd =
d
�
i=1
�λi�ui�u⊤
i
where {�λi}n
i=1 are the eigenvalues of K (counting multiplicities) in decreasing order, and {�ui}n
i=1
are corresponding eigenvectors. | [
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] |
i=1
are corresponding eigenvectors.
We now introduce some population quantities which will form the basis of our theory. Let L2
ρ
denote the Hilbert space of real-valued square integrable functions with respect to ρ, with the inner
product defined as ⟨f, g⟩ρ =
�
f(x)g(x)dρ(x). We define the integral operator K : L2
ρ → L2
ρ by
(Kf) (x) =
�
k(x, y)f(y)dρ(y)
which is the infinite sample limit of 1
nK. The operator K is self-adjoint and compact [Hirsch and | [
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1.1483362913131714,
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0.2292696237564087,
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0.3278891444206238,
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0.7217957377433777,
0.07576331496238708,
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0.32026785612106323,
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0.49240341782569885,
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0.023435786366462708,
0.0592050775885582,
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0.5460213422775269,
0.4493233859539032,
0.31306514143943787,
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0.623958170413971,
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0.17722348868846893,
0.21332187950611115,
0.050198305398225784,
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0.3622295558452606,
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0.5805534720420837,
0.0005511914496310055,
0.08457960933446884,
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-0.36971256136894226,
-0.34005966782569885,
-0.11138124018907547
] |
which is the infinite sample limit of 1
nK. The operator K is self-adjoint and compact [Hirsch and
Lacombe, 1999], so by the spectral theorem for compact operators, there exists a sequence of
eigenvalues {λi}∞
i=1 (counting multiplicities) in decreasing order, with corresponding eigenfunctions
{ui}∞
i=1 which are orthonormal in L2
ρ, such that
Kui = λiui.
Moreover, by the classical Mercer’s theorem [Mercer, 1909, Steinwart and Scovel, 2012], the kernel
k can be decomposed into
k(x, y) =
∞
�
i=1 | [
-0.3728669285774231,
-0.5351851582527161,
0.09728914499282837,
-0.07441217452287674,
-0.07007630169391632,
0.3136633634567261,
-0.03987538069486618,
-0.4127368927001953,
0.03109804540872574,
-0.21677105128765106,
-0.4233511686325073,
0.4298214316368103,
-0.36023616790771484,
0.4113802909851074,
0.456584632396698,
0.6246383786201477,
0.6345354914665222,
0.16179820895195007,
-0.5103524923324585,
-0.7360873818397522,
0.3320213556289673,
0.04414701834321022,
0.3066261112689972,
0.1996375322341919,
-0.04677001014351845,
0.012315862812101841,
0.1679726392030716,
0.09522782266139984,
0.7119908332824707,
0.39311912655830383,
-0.6376142501831055,
0.6895411014556885,
0.027034223079681396,
-0.11228612065315247,
0.3379727005958557,
0.035023462027311325,
0.5264976620674133,
-0.038614798337221146,
0.6081632971763611,
-0.03331582248210907,
0.5111185312271118,
-0.14227668941020966,
-0.15673230588436127,
0.29160168766975403,
-0.6660045981407166,
-0.7201414704322815,
-1.2884392738342285,
0.007492373697459698,
0.11176743358373642,
-0.4913022220134735,
-0.28416430950164795,
0.13003166019916534,
0.35094842314720154,
0.09461791813373566,
-0.16386641561985016,
0.2934240996837616,
-0.31127575039863586,
-0.5864295959472656,
0.7923188805580139,
-0.26174983382225037,
0.012730167247354984,
-0.20198194682598114,
0.2531804144382477,
0.6210700273513794,
-0.3350721001625061,
1.1658176183700562,
-0.023365793749690056,
0.014827318489551544,
-0.5336493253707886,
-0.524558424949646,
0.2573651075363159,
0.0453299880027771,
-0.09631241858005524,
0.18791811168193817,
-0.2437371015548706,
0.06974317878484726,
-0.25875863432884216,
-0.18335260450839996,
0.3163742125034332,
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-0.5105749368667603,
-0.2132652848958969,
0.5690959692001343,
-0.30655717849731445,
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0.08909007906913757,
-0.27967721223831177,
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-0.7413907647132874,
0.4135129153728485,
0.03498797118663788,
-0.03977164998650551,
0.16981345415115356,
0.1291600465774536,
0.21229811012744904,
1.0334640741348267,
0.18118570744991302,
-0.3649581968784332,
0.6344531178474426,
0.05337133631110191,
-0.39018765091896057,
-0.27273544669151306,
-0.4034869372844696,
-0.691912829875946,
-0.6412975788116455,
-0.38417667150497437,
-0.3937564194202423,
-0.20195670425891876,
-0.21983188390731812,
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-0.7131644487380981,
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-0.05688542500138283,
-0.44009020924568176,
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-0.03638127073645592,
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-0.06437689065933228,
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-0.4183347523212433,
-0.2961316704750061,
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-0.23290324211120605,
-0.5260294675827026,
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-0.4423829913139343,
-0.5727444887161255,
-0.23694057762622833,
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-0.07837758213281631,
-0.4709610939025879,
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-0.11341049522161484,
-0.017096785828471184,
0.39638346433639526,
-0.6828933358192444,
-0.006065745837986469,
-0.5475115180015564,
-0.1302410066127777,
0.09074230492115021,
0.36788487434387207,
-0.07159408926963806,
0.06546548753976822,
-0.015245029702782631,
-0.23753571510314941,
0.005550634115934372,
-0.6416902542114258,
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-0.7911332249641418,
0.6625098586082458,
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-0.03016977198421955,
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-0.129706010222435,
-0.5425204038619995,
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-0.5822641849517822,
-0.26768848299980164,
0.2696666121482849,
-0.08163894712924957,
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-0.9190263748168945,
-0.13372786343097687,
-0.14608389139175415,
-0.11303022503852844,
0.22539319097995758,
0.34558871388435364,
0.13057489693164825,
0.2894617021083832,
0.3941851854324341,
0.1637471467256546,
-0.2578231990337372,
0.02904673106968403,
-0.10807440429925919,
-0.8715776205062866,
1.1400442123413086,
0.2920101284980774,
-0.5126664638519287,
-0.1925881803035736,
-0.041897259652614594,
0.13467560708522797,
0.18994872272014618,
-0.007252975832670927,
-0.027191221714019775,
0.15866251289844513,
-0.41026410460472107,
0.4770112633705139,
-0.3563585579395294,
0.15892265737056732,
-0.4730292856693268,
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] |
k can be decomposed into
k(x, y) =
∞
�
i=1
λiui(x)ui(y)
(2)
where the series converges absolutely and uniformly in x, y.
3
Entrywise error bounds
This section is devoted to our main theoretical result. We begin by discussing our assumptions,
before presenting our main theorem and giving some special cases of kernels which fit within our
framework.
In our asymptotics, we will assume that k and ρ are fixed, and that the number of sample n goes | [
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] |
In our asymptotics, we will assume that k and ρ are fixed, and that the number of sample n goes
to infinity. This places us in the so-called low-dimensional regime, in which the dimension p of the
input space is considered fixed.
We shall assume that the eigenvalues of the kernel exhibit either polynomial decay, i.e. λi = O(i−α)
for some α > 1, or exponential decay, i.e. λi = O(e−βiγ) for some β > 0 and 0 < γ ≤ 1. We | [
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] |
for some α > 1, or exponential decay, i.e. λi = O(e−βiγ) for some β > 0 and 0 < γ ≤ 1. We
will refer to these two hypotheses as (P) and (E) respectively. We also assume a corresponding
hypothesis on the supremum-norm growth of the eigenfunctions. Under (P), we assume that
∥ui∥∞ = O(ir) with α > 2r + 1, and under (E), we assume that ∥ui∥∞ = O(esiγ) with β > 2s.
Our eigenvalue decay hypothesis is commonplace in the kernel literature [Braun, 2006, Ostrovskii | [
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] |
Our eigenvalue decay hypothesis is commonplace in the kernel literature [Braun, 2006, Ostrovskii
and Rudi, 2019, Xu, 2018, Lei, 2021], and can be related to the smoothness of the kernel. For
example, the decay of the eigenvalues is directly implied by a Hölder or Sobolev-type smoothness
hypothesis on the kernel (see, for example, Nicaise [2000], Belkin [2018], Section 2.2 of Xu [2018],
3
λi
∥ui∥∞
(P)
O (i−α)
O (ir)
α > 2r + 1
(E)
O
�
e−βiγ�
O
�
esiγ�
β > 2s, 0 < γ ≤ 1
∆i
Γi
(R)
O(λa
i )
O(λb
i) | [
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0.4214603900909424,
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] |
(P)
O (i−α)
O (ir)
α > 2r + 1
(E)
O
�
e−βiγ�
O
�
esiγ�
β > 2s, 0 < γ ≤ 1
∆i
Γi
(R)
O(λa
i )
O(λb
i)
1 ≤ a < b/16
Table 1: Summary of the hypotheses (P), (E) and (R).
Section 5 of Valdivia [2018], Scetbon and Harchaoui [2021] and Proposition 2 in this paper). We
don’t consider a finite-rank (say, D) hypothesis, since in this case the maximum entrywise is
trivially zero whenever d ≥ D.
Our hypothesis on the supremum norm of the eigenfunctions is necessary to control the deviation of | [
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] |
Our hypothesis on the supremum norm of the eigenfunctions is necessary to control the deviation of
the sample eigenvectors from their corresponding population eigenfunctions, and is a requirement
of eigenvalue bounds we employ. In the literature, it is common to see much stronger assumptions,
such as uniformly bounded eigenfunctions [Williamson et al., 2001, Lafferty et al., 2005, Braun,
2006], which do not hold for many commonly-used kernels (see Mendelson and Neeman [2010], | [
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] |
2006], which do not hold for many commonly-used kernels (see Mendelson and Neeman [2010],
Steinwart and Scovel [2012], Zhou [2002] and Barzilai and Shamir [2023] for discussion). This
assumption is reminiscent of the incoherence assumption [Candes and Recht, 2012] in the low-rank
matrix estimation literature — a supremum norm bound on population eigenvectors — which
governs the hardness of compressed sensing and eigenvector estimation problems [Candès and Tao, | [
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] |
governs the hardness of compressed sensing and eigenvector estimation problems [Candès and Tao,
2010, Keshavan et al., 2010, Chi et al., 2019, Abbe et al., 2020, Chen et al., 2021].
In addition, we introduce a regularity hypothesis, which we will refer to as (R), which relates to
the following two quantities:
∆i = max
j≥i {λj − λj+1}
which measures the largest eigengap after a certain point in the spectrum, and
Γi =
∞
�
j=i+1
��
uj(x)dρ(x)
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] |
Γi =
∞
�
j=i+1
��
uj(x)dρ(x)
�2
which measures the squared residual after projecting the unit-norm constant function onto the first
i eigenfunctions. Under (R), we assume that ∆i = Ω (λa
i ) and Γi = O
�
λb
i
�
with 1 ≤ a < b/16 ≤ ∞.
A sufficient condition for (R) to hold, is that the first eigenfunction is constant. This holds, for
example, when k is a dot-product kernel and ρ is a uniform distribution on a hypersphere. In such | [
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] |
example, when k is a dot-product kernel and ρ is a uniform distribution on a hypersphere. In such
scenarios, Γi = 0 for all i ≥ 1 and it is not necessary to make any assumptions on the eigengap
quantity ∆i. We remark that (R) permits repeated eigenvalues in the spectrum of K, which
occur for many commonly-used kernels, but which are often precluded in the literature [Hall and
Horowitz, 2007, Meister, 2011, Lei, 2014, 2021]. | [
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] |
Horowitz, 2007, Meister, 2011, Lei, 2014, 2021].
The hypotheses (P), (E) and (R) are summarised in Table 1. We are now ready to state our main
theorem.
Theorem 1. Suppose that k is a symmetric, positive-definite, continuous and bounded kernel and
ρ is a probability measure which satisfy (R) and one of either (P) or (E). If the hypothesis (P)
holds and d = Ω
�
n1/α�
, then
��� �Kd − K
���
max = O
�
n− α−1
α log(n)
�
(3)
with overwhelming probability. If the hypothesis (E) holds and d > log1/γ(n1/β), then | [
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] |
�
(3)
with overwhelming probability. If the hypothesis (E) holds and d > log1/γ(n1/β), then
��� �Kd − K
���
max = O
�
n−1�
with overwhelming probability.
4
3.1
Special cases
In this section, we provide some examples of kernels which satisfy the assumptions of Theorem 1.
Proofs of the propositions in this section are given in Section A of the appendix. We start with a
canonical example of a radial basis kernel.
Proposition 1. Suppose k(x, y) = exp
�
−∥x − y∥2/2ω2�
is a radial basis kernel, and ρ ∼ | [
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] |
Proposition 1. Suppose k(x, y) = exp
�
−∥x − y∥2/2ω2�
is a radial basis kernel, and ρ ∼
N(0, σ2Ip) is a isotropic Gaussian distribution on Rp. Then the hypotheses (E) and (R) are
satisfied with
β = log
�1 + υ + √1 + 2υ
υ
�
,
γ = 1
where υ := 2σ2/ω2.
For this example, the eigenvalues and eigenfunctions were explicitly calculated in Zhu et al. [1997]
(see also Shi et al. [2008] and Shi et al. [2009]), and we are able to verify the assumptions by direct
calculation. | [
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] |
calculation.
For our second example, we consider the case that ρ is the uniform distribution on a hypersphere
Sp−1, and k is a dot-product kernel. In this setting, we are able to replace our assumptions with
a smoothness hypotheses on the kernel. Note that this class of kernels includes those which are
functions of Euclidean distance, since on the sphere we have the identity ∥x − y∥2 = 2 − 2 ⟨x, y⟩.
Proposition 2. Suppose that
k(x, y) = f(⟨x, y⟩) ≡
∞
�
i=0
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] |
Proposition 2. Suppose that
k(x, y) = f(⟨x, y⟩) ≡
∞
�
i=0
bi (⟨x, y⟩)i
is a dot-product kernel and ρ is the uniform distribution on the hypersphere Sp−1 with p ≥ 3. If
there exists a > (p2 − 4p + 5)/2 such that bi = O(i−a), then (P) and (R) are satisfied with
α = 2a + p − 3
p − 2
.
Alternatively, if there exists 0 < r < 1 such that bi = O(ri), then (E) and (R) are satisfied with
β = (p − 1)!
C
log(1/r),
γ =
1
p − 1
for some universal constant C > 0. | [
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] |
β = (p − 1)!
C
log(1/r),
γ =
1
p − 1
for some universal constant C > 0.
In this example, the eigenfunctions posses the property that they do not depend on the choice
of kernel, and are made up of spherical harmonics [Smola et al., 2000]. In particular, the first
eigenfunction is constant, and therefore (R) is satisfied automatically. The eigenvalue bounds
are derived in Scetbon and Harchaoui [2021], and we make use of a supremum norm bound for
spherical harmonics in Minh et al. [2006].
4
Proof of Theorem 1 | [
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-0.3145008981227875,
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0.16193944215774536,
0.385077565908432,
0.13891807198524475,
0.0606229342520237,
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0.5977160334587097,
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-0.6105746626853943,
0.5945491194725037,
0.0986097902059555,
0.32392436265945435,
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0.009264945983886719,
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0.29177340865135193
] |
spherical harmonics in Minh et al. [2006].
4
Proof of Theorem 1
In this section, we outline the proof of Theorem 1. Without loss of generality, we will assume that
k is upper bounded by one. We cover the main details here, and defer some of the technical details
to the appendix. By the Eckart-Young-Mirsky theorem, we have that
��� �Kd − K
���
max :=
max
1≤i,j≤n
��� �Kd(i, j) − K(i, j)
���=
��� max
1≤i,j≤n
n
�
l=d+1
�λl�ul(i)�ul(j)
���
≤
n
�
l=d+1
���λl
��· max
d<l≤n ∥�ul∥2
∞ .
5 | [
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] |
���=
��� max
1≤i,j≤n
n
�
l=d+1
�λl�ul(i)�ul(j)
���
≤
n
�
l=d+1
���λl
��· max
d<l≤n ∥�ul∥2
∞ .
5
Using a concentration bound due to Valdivia [2018], we are able to show that
n
�
l=d+1
���λl
��=
�
O
�
n1/α log(n)
�
under (P) with d = Ω
�
n1/α�
O (1)
under (E) with d > log1/γ(n1/β).
(4)
with overwhelming probability, the details of which are given in Section B of the appendix. Then,
the proof boils down to showing the following result, which we state as an independent theorem. | [
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] |
the proof boils down to showing the following result, which we state as an independent theorem.
Theorem 2. Assume the setting of Theorem 1, then simultaneously for all d + 1 ≤ l ≤ n,
∥�ul∥∞ = O
�
n−1/2�
(5)
with overwhelming probability.
When a unit eigenvector satisfies (5) (up to log factors), it is said to be completely delocalised.
There is a now expansive literature in the field of Random Matrix Theory proving the delocalisation | [
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] |
There is a now expansive literature in the field of Random Matrix Theory proving the delocalisation
of the eigenvectors of certain mean-zero random matrices with independent entries [Erdős et al.,
2009b,a, Tao and Vu, 2011, Rudelson and Vershynin, 2015, Vu and Wang, 2015]. Theorem 2 may
be of independent interest since to our knowledge, it is the first eigenvector delocalisation result
for a random matrix with non-zero mean and dependent entries. | [
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] |
for a random matrix with non-zero mean and dependent entries.
To prove Theorem 2, we take inspiration from a proof strategy employed in Tao and Vu [2011] (see
also Erdős et al. [2009b]) which makes use of an identity relating with eigenvalues and eigenvectors
of a matrix with that of its principal minor. The non-zero mean, and dependence between the
entries of a kernel matrix present new challenges which require novel technical insights and tools
and make up the bulk of our technical contribution. | [
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0.7249284982681274,
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0.5242968797683716,
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0.4920049011707306,
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0.0443708710372448,
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] |
and make up the bulk of our technical contribution.
Proof of Theorem 2. By symmetry and a union bound, to prove Theorem 2, it suffices to establish
this bound for the first coordinate of �ul for some an arbitrary index d < l ≤ n. We shall let �K
denote the bottom right principal minor of K , that is the n − 1 × n − 1 matrix such that
K =
�
a
b⊤
b
�K
�
where and let a = k(x1, x1) and b = (k(x1, x2), . . . , k(x1, xn))⊤. We will denote the ordered-by- | [
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0.4422823190689087,
0.3164047598838806,
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0.553558886051178,
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0.22281019389629364,
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-0.6678292751312256,
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0.7115412950515747,
0.526004433631897,
0.08429588377475739,
0.02954397164285183,
-0.2988525331020355,
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-0.06895099580287933,
-0.34975120425224304,
0.09693597257137299,
0.21089163422584534,
0.4463688135147095,
0.07870832085609436,
0.04967942461371422,
-0.3934823274612427,
-0.09558717906475067,
0.32871201634407043,
-0.06528511643409729,
-0.15978626906871796,
0.2878535985946655,
-0.29658427834510803,
-0.01214641984552145,
0.9218328595161438,
-0.379277765750885,
0.24342738091945648,
0.33097973465919495,
0.12082423269748688,
0.47030526399612427,
0.3792072832584381,
-0.32151439785957336,
-0.7895068526268005,
0.22163568437099457,
-0.46859556436538696,
0.3149329721927643,
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0.17785732448101044,
0.11533112078905106,
0.5945488214492798,
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0.16245988011360168,
0.1613377034664154,
0.11090774834156036,
0.01849248819053173,
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0.2651968002319336,
0.3730314373970032,
0.6403007507324219,
0.8784687519073486,
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0.25650158524513245,
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0.7299314737319946,
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0.034966304898262024,
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1.0814204216003418,
0.8019152283668518,
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0.11625941842794418,
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0.3166390061378479,
0.5684636831283569,
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0.4124806225299835,
-0.26976385712623596,
0.19941231608390808,
0.3216742277145386,
0.24429018795490265,
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-0.2387418895959854,
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0.34038233757019043,
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0.19949108362197876,
0.5998001098632812,
0.01889161393046379,
-0.2947632670402527,
0.2915481925010681,
0.41768842935562134,
-0.02148250862956047,
0.486258327960968,
0.29727402329444885,
0.020800555124878883,
-0.4595804810523987,
0.4091047942638397,
-0.12561997771263123,
0.13945095241069794,
0.4523219168186188,
0.7945748567581177,
0.4442313611507416,
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0.38541969656944275,
0.06799791753292084,
0.3736242353916168,
-0.39544162154197693,
-0.27015379071235657,
-0.1842663288116455,
0.5558046102523804,
0.04410513862967491,
0.04615171253681183,
0.14026476442813873,
0.21574470400810242,
-0.12858439981937408,
-0.06319177895784378,
-0.0801258534193039,
-0.547248363494873,
0.06946961581707001,
-0.09030470252037048
] |
where and let a = k(x1, x1) and b = (k(x1, x2), . . . , k(x1, xn))⊤. We will denote the ordered-by-
magnitude eigenvalues and corresponding eigenvectors of �K by {�λl}n−1
l=1 and {�ul}n−1
l=1 respectively.
By Lemma 41 of Tao and Vu [2011], we have the following remarkable identity:
�ul(1)2 =
1
1 + �n−1
j=1
��λj − �λi
�−2�
�u⊤
j b
�2 .
(6)
In addition, Cauchy’s interlacing theorem tells us that the eigenvalues of K and �K interlace, i.e. | [
-0.3490849733352661,
-0.25161808729171753,
0.41912946105003357,
-0.5377107858657837,
-0.32104212045669556,
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-0.5228627324104309,
-0.04388497769832611,
-0.28520235419273376,
-0.3553612530231476,
0.6442998647689819,
-0.35037726163864136,
0.2327963262796402,
0.35555708408355713,
0.8275960683822632,
0.7273908257484436,
0.37899845838546753,
-0.8917676210403442,
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0.774065375328064,
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0.14835606515407562,
0.1126292496919632,
0.12323712557554245,
0.19596098363399506,
0.27204766869544983,
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0.35777944326400757,
0.5262191295623779,
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0.6830125451087952,
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0.29445114731788635,
0.22262641787528992,
0.8015767931938171,
0.08229146152734756,
0.30106088519096375,
0.30535024404525757,
0.522120475769043,
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0.6528132557868958,
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0.4830629229545593,
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1.2243127822875977,
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] |
In addition, Cauchy’s interlacing theorem tells us that the eigenvalues of K and �K interlace, i.e.
�λi ≤ �λi ≤ �λi+1 for all 1 ≤ i ≤ n − 1. By (4) we have that |�λi| = O(1) for all d + 1 ≤ i ≤ n with
overwhelming probability and so by Cauchy’s interlacing theorem, we can find a set of indices
J ⊂ {d + 1, . . . , n − 1} with |J| ≥ (n − d)/2 such that |�λj − �λi| = O (1) for all j ∈ J. Combining
this observation with (6), we have that
n−1
�
j=1
��λj − �λi
�−2�
�u⊤
j b
�2 ≥
�
j∈J
��λj − �λi
�−2�
�u⊤
j b | [
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] |
n−1
�
j=1
��λj − �λi
�−2�
�u⊤
j b
�2 ≥
�
j∈J
��λj − �λi
�−2�
�u⊤
j b
�2 ≳ ∥πJ(b)∥2 .
(7)
where πJ denotes the orthogonal projection onto the subspace spanned by {�uj}j∈J. So, to establish
(5), it suffices to show that
∥πJ(b)∥2 = Ω(n)
(8)
6
with overwhelming probability. We condition on x1, so that b is a vector of independent random
variables and denote its conditional mean by ¯b, which is a constant vector whose entries are less
than one. | [
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than one.
To obtain the lower bound (8), we prove a novel concentration inequality for the distance between
a random vector and a subspace, which may be of independent interest. Our lemma generalises
a similar result in Tao and Vu [2011] which holds only for random vectors with zero mean. The
proof is provided in Section C of the appendix.
Lemma 1. Let a ∈ Rn be a random vector with mean ¯a := Ea whose entries are independent and | [
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] |
Lemma 1. Let a ∈ Rn be a random vector with mean ¯a := Ea whose entries are independent and
are bounded in [0, 1] almost surely. Let H be a subspace of dimension q ≥ 4 and πH the orthogonal
projection onto H. If H is such that ∥πH(¯a)∥ ≤ 2q1/4, then for any t ≥ 8
P
����∥πH (a)∥ − q1/2��� ≥ t
�
≤ 4 exp
�
−t2/32
�
.
In particular, one has
∥πH (a)∥ = q1/2 + O
�
log1/2(n)
�
(9)
with overwhelming probability.
Returning to the main thread, we claim for the moment that ∥πJ(¯b)∥ ≤ 2|J|1/4. Then by Lemma 1 | [
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] |
Returning to the main thread, we claim for the moment that ∥πJ(¯b)∥ ≤ 2|J|1/4. Then by Lemma 1
we have that, conditional on x1,
∥πJ(b)∥2 ≳ |J| ≥ (n − d)/2 = Ω(n)
with overwhelming probability. This holds for all x1 ∈ X and therefore establishes (8). To complete
the proof, then, it remains to prove our claim, and it is here where we require the regularity
hypothesis (R). The proof of the claim is quite involved, so we defer the details to Section D of the | [
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] |
appendix, given which, the proof of Theorem 1 is complete.
5
Experiments
5.1
Datasets and setup
To see how our theory translates into practice, we examine the maximum entrywise error of the
low-rank approximations of kernel matrices derived from a synthetic dataset and a collection of
five real-world data sets, which are summarised in the following table.
Dataset
Description
# Instances
# Dimensions
GMM
simulated data from a Gaussian mixture model
1,000
10
Abalone
physical measurements of Abalone trees | [
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] |
1,000
10
Abalone
physical measurements of Abalone trees
4,177
7
Wine Quality
physicochemical measurements of wines
6,497
11
MNIST
handwritten digits
10,000
784
20 Newsgroups
tf-idf vectors from newsgroup messages
11,314
21,108
Zebrafish
gene expression in zebrafish embryo cells
6,346
5,434
Additional details about the dataset are provided in Section E of the appendix.
7
0
200
400
600
800
1000
0.0
0.2
0.4
0.6
0.8
1.0
maximum
entrywise error
GMM
0
1000
2000
3000
4000
0.0
0.2
0.4
0.6
0.8
1.0
Abalone
0
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] |
0.8
1.0
maximum
entrywise error
GMM
0
1000
2000
3000
4000
0.0
0.2
0.4
0.6
0.8
1.0
Abalone
0
2000
4000
6000
0.0
0.2
0.4
0.6
0.8
1.0
Wine Quality
0
2000
4000
6000
8000 10000
rank
0.0
0.2
0.4
0.6
0.8
1.0
maximum
entrywise error
MNIST
0
2500
5000
7500
10000
rank
0.0
0.2
0.4
0.6
0.8
1.0
20 Newsgroups
0
2000
4000
6000
rank
0.0
0.2
0.4
0.6
0.8
1.0
Zebrafish
= 1/2
= 3/2
= 5/2
=
Figure 1: The maximum entrywise error against rank for low-rank approximations of kernel | [
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] |
= 5/2
=
Figure 1: The maximum entrywise error against rank for low-rank approximations of kernel
matrices constructed from a collection of datasets. The kernel matrices are constructed using
Matérn kernels with a range of smoothness parameters, each of which is represented by a line in
each plot. Details of the experiment are provided in Section 5.
For the purpose of our experiment, we employ kernels in the class of Matérn kernels, of the form
kν(x, y) = 21−ν
Γ(ν)
�√
2ν ∥x − y∥
ω
�ν
Kν
�√
2ν ∥x − y∥
ω
� | [
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] |
kν(x, y) = 21−ν
Γ(ν)
�√
2ν ∥x − y∥
ω
�ν
Kν
�√
2ν ∥x − y∥
ω
�
where Γ denotes the gamma function, and Kν is the modified Bessel function of the second kind.
The class of Matérn kernels is a generalisation of the radial basis kernel, with an additional
parameter ν which governs the smoothness of the resulting kernel. When ν = 1/2, we obtain the
non-differentiable exponential kernel, and in the ν → ∞ limit, we obtain the infinitely-differentiable | [
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radial basis kernel. For the intermediate values ν = 3/2 and ν = 5/2, we obtain, respectively, once
and twice-differentiable functions.
The optimal choice of the bandwidth parameter is problem-dependent, and in supervised settings is
typically chosen using cross-validation. In unsupervised settings, it is necessary to rely on heuristics,
and for this experiment, we use the popular median heuristic [Flaxman et al., 2016, Mooij et al., | [
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] |
and for this experiment, we use the popular median heuristic [Flaxman et al., 2016, Mooij et al.,
2016, Mu et al., 2016, Garreau et al., 2017], which has been shown to perform well in practice.
For each dataset, we construct four kernel matrices using Matérn kernels with smoothness parame-
ters ν = 1
2, 3
2, 5
2, ∞, each time selecting the bandwidth using the median heuristic. For each kernel,
we compute the best rank-d low-rank approximation of the kernel matrix using the svds function | [
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] |
we compute the best rank-d low-rank approximation of the kernel matrix using the svds function
in the SciPy library for Python [Virtanen et al., 2020]. We do this for a range of ranks d from 1 to
n, where n is the number of instances in the dataset, and record the entrywise errors.
5.2
Interpretation of the results
Figure 1 shows the maximum entrywise errors for each dataset and kernel. For comparison, the
Frobenius norm errors are plotted in Figure 3 in Section E of the appendix. | [
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] |
Frobenius norm errors are plotted in Figure 3 in Section E of the appendix.
As predicted by our theory, for the four “low-dimensional” datasets, GMM, Abalone, Wine Quality
and MNIST, the maximum entrywise decays rapidly as we increase the rank of the approximation,
8
0
2000 4000 6000 8000 10000
rank
0.00
0.01
0.02
0.03
0.04
99.9th percentile
entrywise error
MNIST
0
2500
5000
7500 10000
rank
0.00
0.01
0.02
0.03
0.04
20 Newsgroups
0
2000
4000
6000
rank
0.00
0.01
0.02
0.03
0.04
Zebrafish
= 1/2
= 3/2
= 5/2
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0.04
20 Newsgroups
0
2000
4000
6000
rank
0.00
0.01
0.02
0.03
0.04
Zebrafish
= 1/2
= 3/2
= 5/2
=
Figure 2: The 99.95th percentile of the entrywise error against rank for low-rank approximations of
kernel matrices constructed from a collection of datasets. The error curves exhibit a double-descent
shape which is not explained by our theory. Details of the experiment are provided in Section 5.
with the exception of the highly non-smooth v = 1
2 kernel, for which the maximum entrywise error | [
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] |
with the exception of the highly non-smooth v = 1
2 kernel, for which the maximum entrywise error
decays much more slowly. In addition, the decay rates of the maximum entrywise error are in
order of the smoothness of the kernels.
For the “high-dimensional” datasets, 20 Newsgroups and Zebrafish, a different story emerges. Even
for the smooth radial basis kernel (ν = ∞), the maximum entrywise error decays very slowly. This | [
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] |
for the smooth radial basis kernel (ν = ∞), the maximum entrywise error decays very slowly. This
would suggests that our theory does potentially not carry over to the high-dimensional setting, and
that caution should be taken when employing low-rank approximations for such data. Interestingly,
the 20 Newsgroups dataset exhibits a sharp drop in maximum entrywise error between d = 2500
and d = 3000 which is not seen in the decay of the Frobenius norm error (Figure 3 in Section E).
5.3 | [
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] |
5.3
An unexplained double-decent curve
When we plot the upper quantiles of the entrywise errors for the MNIST, 20 Newsgroups and
Zebrafish datasets, a curious phenomenon is observed. Figure 2 shows the 99.95th percentile of the
entrywise errors for these datasets. The error curves exhibit a double-descent shape, which is not
observed in the maximum or Frobenius norm error curves. This phenomenon is not explained by | [
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observed in the maximum or Frobenius norm error curves. This phenomenon is not explained by
our theory, and we pose its explanation as an open problem to the machine-learning community.
6
Limitations and open problems
To conclude, we discuss some of the limitations of our theory, as well as some of the open problems.
6.1
Limitations of our theory
Positive semi-definite kernels.
One significant limitation of theory is the assumption that the | [
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Positive semi-definite kernels.
One significant limitation of theory is the assumption that the
kernel is positive semi-definite and continuous. This condition is known as Mercer’s condition in
the literature and ensures that the spectral decomposition of the kernel (2) converges uniformly,
however we don’t actually require such a strong notion of convergence for our theory. Valdivia
[2018, Lemma 22] show that the decomposition converges almost surely under a much weaker | [
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[2018, Lemma 22] show that the decomposition converges almost surely under a much weaker
condition which is implied by our hypotheses (P) and (E). The only other places we need this
assumption is to make use of results in Rosasco et al. [2010] and Tang et al. [2013]. These results
make heavy use of reproducing kernel Hilbert space technology though it seems plausible that they
could be generalised to the indefinite setting using the framework of Krein spaces [Ong et al., 2004,
Lei, 2021]. | [
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] |
Lei, 2021].
Low-dimensional setting.
In our asymptotics, we explicitly assume that the dimension of the
9
input space remains fixed as the number of sample increases, which places us in the so-called
low-dimensional setting. We do not consider the high-dimensional setting, however our empirical
experiments suggest that our conclusions may not carry over.
Verification of the assumptions.
While there is a established literature studying the eigenvalue | [
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] |
Verification of the assumptions.
While there is a established literature studying the eigenvalue
decay of kernels under general probability measures [Kühn, 1987, Cobos and Kühn, 1990, Ferreira
and Menegatto, 2013, Belkin, 2018, Li et al., 2024], except in very specialised settings (such as
Propositions 1 and 2), control of the eigenfunctions is typically out of reach. This makes verifying
the assumptions of our theory under general probability distributions quite challenging. This is a | [
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] |
the assumptions of our theory under general probability distributions quite challenging. This is a
widespread limitation of many theoretical analyses in the kernel literature, and for an extended
discussion, we refer the reader to Barzilai and Shamir [2023].
6.2
Open problems
Double-descent.
The so-call double descent phenomenon has been observed in many areas
of machine-learning [Belkin et al., 2019], including in the Frobenius norm error of low-rank | [
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of machine-learning [Belkin et al., 2019], including in the Frobenius norm error of low-rank
approximations obtained using the Nystöm method [Derezinski et al., 2020]. To our knowledge,
we are the first to observe this in the quantiles of the entrywise error of the truncated eigen-
decomposition, and finding a theoretical explanation for this remains an open question.
Randomised low-rank approximations. While the truncated spectral decomposition provides the | [
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] |
Randomised low-rank approximations. While the truncated spectral decomposition provides the
“ideal” low-rank approximation, it requires computing the whole kernel matrix which can be
prohibitive for very large datasets. Randomised low-rank approximations, such as the randomised
SVD [Halko et al., 2011], the Nyström method [Williams and Seeger, 2000, Drineas et al., 2005] and
random Fourier features [Rahimi and Recht, 2007, 2008], have emerged as efficient alternatives, and | [
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random Fourier features [Rahimi and Recht, 2007, 2008], have emerged as efficient alternatives, and
there is an extensive body of literature examining their statistical performance [Drineas et al., 2005,
Rahimi and Recht, 2007, Belabbas and Wolfe, 2009, Boutsidis et al., 2009, Kumar et al., 2009a,b,
Gittens, 2011, Gittens and Mahoney, 2013, Altschuler et al., 2016, Derezinski et al., 2020]. However,
their primary focus is on classical error metrics such as the spectral and Frobenius norm errors | [
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] |
their primary focus is on classical error metrics such as the spectral and Frobenius norm errors
and an entrywise analysis would presumably provide greater insights into these approximations,
particularly given recently observed multiple-descent phenomena [Derezinski et al., 2020].
Acknoledgements
The author thanks Nick Whiteley, Yanbo Tang and Mahmoud Khabou for helpful discussions and
Annie Gray for providing code to preprocess the 20 Newsgroups and Zebrafish datasets.
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] |
Annie Gray for providing code to preprocess the 20 Newsgroups and Zebrafish datasets.
References
Emmanuel Abbe, Jianqing Fan, Kaizheng Wang, and Yiqiao Zhong. Entrywise eigenvector analysis
of random matrices with low expected rank. Annals of statistics, 48(3):1452, 2020.
Emmanuel Abbe, Jianqing Fan, and Kaizheng Wang. An ℓp theory of pca and spectral clustering.
The Annals of Statistics, 50(4):2359–2385, 2022.
Jason Altschuler, Aditya Bhaskara, Gang Fu, Vahab Mirrokni, Afshin Rostamizadeh, and Morteza | [
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] |
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] |
descent curve for column subset selection and the nystrom method.
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] |
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