Minimum Rank Solutions to the Matrix Approximation Problems in the Spectral Norm

Abstract

In this paper, we discuss the following two minimum rank matrix approximation problems in the spectral norm: $\min\nolimits_X {\rm rank}(X)$, subject to $ \|A-BXC\|_2=\theta$, where $ \theta \triangleq \min\nolimits_{Y}\|A-BYC\|_2$; $\min\nolimits_X {\rm rank}(X)$, subject to $ \|A-BXC\|_2 \theta$. We solve the first problem by applying the norm-preserving dilation theorem and the restricted singular value decomposition (R-SVD) to characterize three expressions of the minimum rank, and we derive general forms of minimum rank solutions. We also characterize the expressions of the minimum rank of the second problem, which is a generalization known results in the literature.