Abstract-We present novel techniques for analyzing the problem of low-rank matrix recovery. The methods are both considerably simpler and more general than previous approaches. It is shown that an unknown n × n matrix of rank r can be efficiently reconstructed from only O(nrν ln 2 n) randomly sampled expansion coefficients with respect to any given matrix basis. The number ν quantifies the "degree of incoherence" between the unknown matrix and the basis. Existing work concentrated mostly on the problem of "matrix completion" where one aims to recover a low-rank matrix from randomly selected matrix elements. Our result covers this situation as a special case. The proof consists of a series of relatively elementary steps, which stands in contrast to the highly involved methods previously employed to obtain comparable results. In cases where bounds had been known before, our estimates are slightly tighter. We discuss operator bases which are incoherent to all lowrank matrices simultaneously. For these bases, we show that O(nrν ln n) randomly sampled expansion coefficients suffice to recover any low-rank matrix with high probability. The latter bound is tight up to multiplicative constants.
Besides matrix completion, Gross analyzed the problem of efficiently recovering a low-rank matrix from a fraction of observations in any basis REF .
Recovering low-rank matrices from few coefficients in any basis
IEEE Trans. on Information Theory, vol. 57, pages 1548 - 1566 (2011)
Computer Science Mathematics Physics