Optimal Estimation of Schatten Norms of a rectangular Matrix
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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. 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.
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