Singular vector and singular subspace distribution for the matrix denoising model
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In this paper, we study the matrix denosing 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. 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 non-universal.
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