  Estimating a low rank matrix from its linear measurements is a problem of
central importance in contemporary statistical analysis. The choice of tuning
parameters for estimators remains an important challenge from a theoretical and
practical perspective. To this end, Stein's Unbiased Risk Estimate (SURE)
framework provides a well-grounded statistical framework for degrees of freedom
estimation. In this paper, we use the SURE framework to obtain degrees of
freedom estimates for a general class of spectral regularized matrix
estimators, generalizing beyond the class of estimators that have been studied
thus far. To this end, we use a result due to Shapiro (2002) pertaining to the
differentiability of symmetric matrix valued functions, developed in the
context of semidefinite optimization algorithms. We rigorously verify the
applicability of Stein's lemma towards the derivation of degrees of freedom
estimates; and also present new techniques based on Gaussian convolution to
estimate the degrees of freedom of a class of spectral estimators to which
Stein's lemma is not directly applicable.
