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.