OPTIMAL ESTIMATION OF SCHATTEN NORMS OF A RECTANGULAR MATRIX

• 成果类型: Article
• 署名作者: Thepaut, Solene; Verzelen, Nicolas
• 署名单位: Safran S.A.; INRAE; Universite de Montpellier
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/24-AOS2374
• 发表日期: 2024
• 页码: 1334-1359
• 关键词: minimax estimation; functionals

摘要

We consider the twin problems of estimating the effective rank and the Schatten norms HAHs of a rectangular p x q matrix A from noisy observations. When s is an even integer, we introduce a polynomial-time estimator of HAHs 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 boolean AND 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.