  The current paper presents a novel machinery for studying non-asymptotic
minimax estimation of high-dimensional matrices, which yields tight minimax
rates for a large collection of loss functions in a variety of problems.
  Based on the convex geometry of finite-dimensional Banach spaces, we first
develop a volume ratio approach for determining minimax estimation rates of
unconstrained normal mean matrices under all squared unitarily invariant norm
losses. In addition, we establish the minimax rates for estimating mean
matrices with submatrix sparsity, where the sparsity constraint introduces an
additional term in the rate whose dependence on the norm differs completely
from the rate of the unconstrained problem. Moreover, the approach is
applicable to the matrix completion problem under the low-rank constraint.
  The new method also extends beyond the normal mean model. In particular, it
yields tight rates in covariance matrix estimation and Poisson rate matrix
estimation problems for all unitarily invariant norms.
