Restricted $p$ -Isometry Properties of Nonconvex Matrix Recovery

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 <; p ≤ 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.