Risk estimation for matrix recovery with spectral regularization
This work addresses parameter tuning challenges in matrix recovery for practitioners, though it is incremental as it builds on existing SURE theory and proximal methods.
The paper tackles the problem of estimating quadratic risk for matrix recovery with spectral regularization, enabling automatic selection of regularization parameters, as demonstrated on a matrix completion task.
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.