Optimal estimation of Schatten norms of a rectangular matrix
Solène Thépaut, Nicolas Verzelen
The Annals of Statistics · 2024 · Volume 52 · Issue 4
Abstract
We consider the twin problems of estimating the effective rank and the Schatten norms ‖A‖s of a rectangular p×q matrix A from noisy observations. When s is an even integer, we introduce a polynomial-time estimator of ‖A‖s that achieves the minimax rate (pq)1/4. Interestingly, this optimal rate does not depend on the underlying rank of the matrix A. When s is not an even integer, the optimal rate is much slower. A simple thresholding estimator of the singular values achieves the rate (q∧p)(pq)1/4, which turns out to be optimal up to a logarithmic multiplicative term. The tight minimax rate is achieved by a more involved polynomial approximation method. This allows us to build estimators for a class of effective rank indices. As a byproduct, we also characterize the minimax rate for estimating the sequence of singular values of a matrix.
Keywords
62C20, 62H12, Effective rank, matrix estimation, polynomial approximation, Random matrix, Schatten norm
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