Information-Theoretic Limits for the Matrix Tensor Product
This work provides foundational theoretical limits for a broad class of data science problems, including sparse PCA and network analysis, with potential impact across machine learning and statistics.
The paper tackles the high-dimensional inference problem of matrix tensor products, generalizing models like spiked matrices and stochastic block models, and derives exact analytical formulas for mutual information and minimum mean-squared error in the Bayes optimal setting, with non-asymptotic bounds and high-dimensional scaling results.
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 and the stochastic block model used in network analysis. The main results are single-letter formulas (i.e., analytical expressions that can be approximated numerically) for the mutual information and the minimum mean-squared error (MMSE) in the Bayes optimal setting where the distributions of all random quantities are known. We provide non-asymptotic bounds and show that our formulas describe exactly the leading order terms in the mutual information and MMSE in the high-dimensional regime where the number of rows $n$ and number of columns $d$ scale with $d = O(n^α)$ for some $α< 1/20$. 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 between the overlap and the free energy that is based on continuous-time I-MMSE relations.