Information-Theoretic Limits for the Matrix Tensor Product

05/22/2020

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by   Galen Reeves, et al.

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This paper studies a high-dimensional inference problem involving the matrix tensor product of random matrices. This problem generalizes a number of contemporary data science problems including the spiked matrix models used in sparse principal component analysis and covariance estimation. It is shown that the information-theoretic limits can be described succinctly by formulas involving low-dimensional quantities. On the technical side, this paper introduces some new techniques for the analysis of high-dimensional matrix-valued signals. Specific contributions include a novel extension of the adaptive interpolation method that uses order-preserving positive semidefinite interpolation paths and a variance inequality based on continuous-time I-MMSE relations.