New Bounds for Restricted Isometry Constants in Low-rank Matrix Recovery

In this paper, we establish new bounds for restricted isometry constants (RIC) in low-rank matrix recovery. Let A be a linear transformation from Rm×n into Rp, and r the rank of recovered matrix X ∈ Rm×n. Our main result is that if the condition on RIC satisfies δ2r+k + 2( r k ) δmax{r+ 3 2 k,2k} < 1 for a given positive integer k ≤ m − r, then r-rank matrix can be exactly recovered via nuclear norm minimization problem in noiseless case, and estimated stably in the noise case. Taking different k, we obtain some improved and new RIC bounds such as δ 7 3 r + 2 √ 3δ1.5r < 1, δ2.5r + 2 √ 2δ1.75r < 1, δ2r+1 + 2 √ rδr+2 < 1, δ2r+2 + √ 2rδr+3 < 1, or δ2r+4+ √ rδr+7 < 1. To the best of our knowledge, these are the first such conditions on RIC.