Restricted $p$ -Isometry Properties of Nonconvex Matrix Recovery

摘要

Recently, a nonconvex relaxation of low-rank matrix recovery (LMR), called the Schatten- $p$ quasi-norm minimization ($0< p< 1$), was introduced instead of the previous nuclear norm minimization in order to approximate the problem of LMR closer. In this paper, we introduce a notion of the restricted $p$-isometry constants ( $0< pleq 1$) and derive a $p$ -RIP condition for exact reconstruction of LMR via Schatten-$p$ quasi-norm minimization. In particular, we determine how many random, Gaussian measurements are needed for the $p$-RIP condition to hold with high probability, which gives a theoretical result that it needs fewer measurements with small $p$ for exact recovery via Schatten- $p$ quasi-norm minimization than when $p=1$.