  In this paper, we develop an approach to recursively estimate the quadratic
risk for matrix recovery problems regularized with spectral functions. Toward
this end, in the spirit of the SURE theory, a key step is to compute the (weak)
derivative and divergence of a solution with respect to the observations. As
such a solution is not available in closed form, but rather through a proximal
splitting algorithm, we propose to recursively compute the divergence from the
sequence of iterates. A second challenge that we unlocked is the computation of
the (weak) derivative of the proximity operator of a spectral function. To show
the potential applicability of our approach, we exemplify it on a matrix
completion problem to objectively and automatically select the regularization
parameter.
