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The low-rank matrix recovery (LMR) is a rank minimization problem subject tolinear equality constraints, and it arises in many fields such as signal andimage processing, statistics, computer vision, system identification andcontrol. This class of optimization problems is $\N\P$-hard and a popularapproach replaces the rank function with the nuclear norm of the matrixvariable. In this paper, we extend the concept of $s$-goodness for a sensingmatrix in sparse signal recovery (proposed by Juditsky and Nemirovski [MathProgram, 2011]) to linear transformations in LMR. Then, we givecharacterizations of $s$-goodness in the context of LMR. Using the twocharacteristic $s$-goodness constants, ${\gamma}_s$ and $\hat{\gamma}_s$, of alinear transformation, not only do we derive necessary and sufficientconditions for a linear transformation to be $s$-good, but also providesufficient conditions for exact and stable $s$-rank matrix recovery via thenuclear norm minimization under mild assumptions. Moreover, we give computableupper bounds for one of the $s$-goodness characteristics which leads toverifiable sufficient conditions for exact low-rank matrix recovery.

Sufficient Conditions for Low-rank Matrix Recovery, Translated from Sparse Signal Recovery