Volume Ratio, Sparsity, and Minimaxity Under Unitarily Invariant Norms


This paper studies non-asymptotic minimax estimation of high-dimensional matrices and provides tight minimax rates for a large collection of loss functions in an exceedingly variety of problems via info-theoretic ways. Based mostly on the convex geometry of finite-dimensional Banach areas, we first develop a volume ratio approach for determining minimax estimation rates of unconstrained mean matrices under all unitarily invariant norm losses, which turn out to only rely on the norm of identity matrix. In addition, we tend to establish the minimax rates for estimating normal mean matrices with submatrix sparsity, where the sparsity constraint introduces a further term in the speed which, in distinction to the unconstrained case, is set by the smoothness (Lipschitz constant) of the norm. This technique is additionally applicable to the low-rank matrix completion downside and extends well beyond the additive noise model. In explicit, it yields tight rates in covariance matrix estimation and Poisson rate matrix estimation issues for all unitarily invariant norms.

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