Near-optimal matrix recovery from random linear measurements

In matrix recovery from random linear measurements, one is interested in recovering an unknown M -by-N matrix X0 from n < MN measurements yi = Tr(Ai X0) where each Ai is an M -by-N measurement matrix with i.i.d random entries, i = 1, . . . , n. We present a novel matrix recovery algorithm, based on approximate message passing, which iteratively applies an optimal singular value shrinker – a nonconvex nonlinearity tailored specifically for matrix estimation. Our algorithm often converges exponentially fast, offering a significant speedup over previously suggested matrix recovery algorithms, such as iterative solvers for Nuclear Norm Minimization (NNM). It is well known that there is recovery tradeoff between the information content of the object X0 to be recovered (specifically, its matrix rank r) and the number of linear measurements n from which recovery is to be attempted. The precise tradeoff between r and n, beyond which recovery by a given algorithm becomes possible, traces the so-called phase transition curve of that algorithm in the (r, n) plane. The phase transition curve of our algorithm is noticeably better than that of NNM. Interestingly, it is close to the information-theoretic lower bound for the minimal number of measurements needed for matrix recovery, making it not only the fastest algorithm known, but also near-optimal in terms of the matrices it successfully recovers. ∗To whom correspondence should be addressed. E-mail: [email protected] 1 ar X iv :1 70 5. 09 95 8v 1 [ cs .I T ] 2 8 M ay 2 01 7