New conditions on robust and stable recovery of non-strictly k-ranked matrices via Schatten p-norm minimization

The reconstruction of low-rank matrices is of crucial theoretical significance, and has found broad practical applications. In this paper, we investigate a high-order RIP condition concerning [Formula: see text] and [Formula: see text] for the robustness and stability of non-strictly low-rank matrix recovery via Schatten [Formula: see text]-norm minimization ([Formula: see text]). Specifically, the sufficient condition and error estimation for matrix recovery are proposed in [Formula: see text] bounded noise setting ([Formula: see text]) and Dantzig Selector noise setting for various realistic problems. In addition, we derive a general estimation bound under the proposed RIP condition via the weighted Schantten [Formula: see text]-norm minimization method. In the numerical experiments, we use the alternating direction method of multipliers (ADMM) algorithm to implement the two methods and compare their recovery efficiency on low-scale matrices. We apply the weighted Schatten [Formula: see text]-norm minimization method to denoise both color and grayscale images, and demonstrate the effects of the adjustable parameter on denoising performance.