Optimal Estimation of Schatten Norms of a rectangular Matrix

TL;DR

This paper develops minimax-optimal estimators for Schatten norms and effective rank of rectangular matrices from noisy data, revealing different rates depending on whether the Schatten norm order is even or not.

Contribution

It introduces polynomial-time estimators that achieve the minimax rates for Schatten norms, especially for even integers, and characterizes the rates for non-integer cases and singular value sequences.

Findings

Optimal rate for even s is (pq)^{1/4} independent of rank.

Thresholding estimator achieves near-optimal rate for non-integer s.

Polynomial approximation method achieves tight minimax rates for Schatten norms.

Abstract

We consider the twin problems of estimating the effective rank and the Schatten norms $\|{\bf A}\|_{s}$ of a rectangular $p\times q$ matrix ${\bf A}$ from noisy observations. When $s$ is an even integer, we introduce a polynomial-time estimator of $\|{\bf A}\|_s$ that achieves the minimax rate $(pq)^{1/4}$. Interestingly, this optimal rate does not depend on the underlying rank of the matrix. When $s$ is not an even integer, the optimal rate is much slower. A simple thresholding estimator of the singular values achieves the rate $(q\wedge p)(pq)^{1/4}$, 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.