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Abstract:
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Low-rank matrices are pervasive throughout a broad range of fields, including statistics, machine learning, signal processing, and applied mathematics. In this paper, we propose a novel and user-friendly Euclidean representation of general low-rank matrices. Correspondingly, we establish a collection of technical and theoretical tools for locally analyzing the intrinsic perturbation of low-rank matrices. Namely, the referential matrix and the perturbed matrix both live on the same manifold of low-rank matrices. In particular, our analysis shows that, locally around the referential matrix, the sine-theta distance between subspaces is equivalent to the Euclidean distance between two appropriately selected orthonormal basis without orthogonal alignment. These tools are applicable to a broad range of statistical problems. Specific applications considered in detail include Bayesian sparse principal component analysis with non-intrinsic loss, efficient estimation in stochastic block models where the block probability matrix may be degenerate, and least-squares estimation in biclustering problems. Both Euclidean representation and the technical tools may be of independent interest.
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