@JuSuNaKiYe09], active user detection in multi-user communication [@FletcherRanganGoyal2009; @Boljanovic2017], and diffuse optical tomography using multiple illumination patterns [@LeeKimBreslerYe2011]. In MMV, thanks to the dependencies among different signals, the number of sparse coefficients that can be successfully estimated increases with the number of measurements. This property was evaluated rigorously for noiseless measurements using $l_0$ minimization [@DuarteWakinBaronSarvothamBaraniuk2013], if the underlying signals share the same sparse supports. A non-rigorous replica analysis of MMV with measurement noise also shows the benefit of having more signal vectors [@ZhuBaronKrzakala2017IEEE; @ZhuDissertation2017].

[**Related work:**]{} There are many estimation approaches for MMV problems. These include greedy algorithms such as SOMP [@tropp2006ass; @chen2006trs], $l_1$ convex relaxation [@malioutov2005ssr; @tropp2006ass2], and M-FOCUSS [@cotter2005ssl]. REduce MMV and BOost (ReMBo) has been shown to outperform conventional methods [@Mishali08rembo], and subspace methods have also been used to solve MMV problems [@LeeBreslerJunge2012; @YeKimBresler2015]. However, these algorithms cannot handle the case of $J$ [*different*]{} measurement matrices. Statistical approaches [@ZinielSchniter2011] often achieve the oracle minimum mean squared error (MMSE). However, when running estimation algorithms for MMV problems, one might be interested in minimizing some other error. For example, if estimating the underlying signal is important, one could use the mean squared error (MSE) metric; when there might be 