Singular vector and singular subspace distribution for the matrix denoising model

Abstract

In this paper, we study the matrix denoising model Y = S + X, where S is a low rank deterministic signal matrix and X is a random noise matrix, and both are M × n. In the scenario that M and n are comparably large and the signals are supercritical, we study the fluctuation of the outlier singular vectors of Y, under fully general assumptions on the structure of S and the distribution of X. More specifically, we derive the limiting distribution of angles between the principal singular vectors of Y and their deterministic counterparts, the singular vectors of S. Further, we also derive the distribution of the distance between the subspace spanned by the principal singular vectors of Y and that spanned by the singular vectors of S. It turns out that the limiting distributions depend on the structure of the singular vectors of S and the distribution of X, and thus they are nonuniversal. Statistical applications of our results to singular vector and singular subspace inferences are also discussed.