  The truncated singular value decomposition (SVD) of the measurement matrix is
the optimal solution to the_representation_ problem of how to best approximate
a noisy measurement matrix using a low-rank matrix. Here, we consider the
(unobservable)_denoising_ problem of how to best approximate a low-rank signal
matrix buried in noise by optimal (re)weighting of the singular vectors of the
measurement matrix. We exploit recent results from random matrix theory to
exactly characterize the large matrix limit of the optimal weighting
coefficients and show that they can be computed directly from data for a large
class of noise models that includes the i.i.d. Gaussian noise case.
  Our analysis brings into sharp focus the shrinkage-and-thresholding form of
the optimal weights, the non-convex nature of the associated shrinkage function
(on the singular values) and explains why matrix regularization via singular
value thresholding with convex penalty functions (such as the nuclear norm)
will always be suboptimal. We validate our theoretical predictions with
numerical simulations, develop an implementable algorithm (OptShrink) that
realizes the predicted performance gains and show how our methods can be used
to improve estimation in the setting where the measured matrix has missing
entries.
