2021/11/26 by Thépaut, Solène, Verzelen, Nicolas
#FOS: Mathematics #Statistics Theory (math.ST)
paper · doi:10.48550/arxiv.2111.13551
We consider the twin problems of estimating the effective rank and the Schatten norms ‖\bf A‖s of a rectangular p× 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.