  We consider recovery of low-rank matrices from noisy data by shrinkage of
singular values, in which a single, univariate nonlinearity is applied to each
of the empirical singular values. We adopt an asymptotic framework, in which
the matrix size is much larger than the rank of the signal matrix to be
recovered, and the signal-to-noise ratio of the low-rank piece stays constant.
For a variety of loss functions, including Mean Square Error (MSE - square
Frobenius norm), the nuclear norm loss and the operator norm loss, we show that
in this framework there is a well-defined asymptotic loss that we evaluate
precisely in each case. In fact, each of the loss functions we study admits a
unique admissible shrinkage nonlinearity dominating all other nonlinearities.
We provide a general method for evaluating these optimal nonlinearities, and
demonstrate our framework by working out simple, explicit formulas for the
optimal nonlinearities in the Frobenius, nuclear and operator norm cases. For
example, for a square low-rank n-by-n matrix observed in white noise with level
$\sigma$, the optimal nonlinearity for MSE loss simply shrinks each data
singular value $y$ to $\sqrt{y^2-4n\sigma^2 }$ (or to 0 if
$y<2\sqrt{n}\sigma$). This optimal nonlinearity guarantees an asymptotic MSE of
$2nr\sigma^2$, which compares favorably with optimally tuned hard thresholding
and optimally tuned soft thresholding, providing guarantees of $3nr\sigma^2$
and $6nr\sigma^2$, respectively. Our general method also allows one to evaluate
optimal shrinkers numerically to arbitrary precision. As an example, we compute
optimal shrinkers for the Schatten-p norm loss, for any p>0.
