S-goodness for Low-rank Matrix Recovery, Translated from Sparse Signal Recovery

Low-rank matrix recovery (LMR) is a rank minimization problem subject to linear equality constraints, and it arises in many fields such as signal and image processing, statistics, computer vision, system identification and control. This class of optimization problems is generally NP-hard. A popular approach replaces the rank function with the nuclear norm of the matrix variable. In this paper, we extend and characterize the concept of s-goodness for a sensing matrix in sparse signal recovery (proposed by Juditsky and Nemirovski [Math Program, 2011]) to linear transformations in LMR. Utilizing the two characteristic s-goodness constants, γs and γ̂s, of a linear transformation, we derive necessary and sufficient conditions for a linear transformation to be s-good. Moreover, we establish the equivalence of s-goodness and the null space properties. Therefore, s-goodness is a necessary and sufficient condition for exact s-rank matrix recovery via the nuclear norm minimization.