A comment on Stein's unbiased risk estimate for reduced rank estimators

In the framework of matrix valued observables with low rank means, Stein's unbiased risk estimate (SURE) can be useful for risk estimation and for tuning the amount of shrinkage towards low rank matrices. This was demonstrated by Cand\`es et al. (2013) for singular value soft thresholding, which is a Lipschitz continuous estimator. SURE provides an unbiased risk estimate for an estimator whenever the differentiability requirements for Stein's lemma are satisfied. Lipschitz continuity of the estimator is sufficient, but it is emphasized that differentiability Lebesgue almost everywhere isn't. The reduced rank estimator, which gives the best approximation of the observation with a fixed rank, is an example of a discontinuous estimator for which Stein's lemma actually applies. This was observed by Mukherjee et al. (2015), but the proof was incomplete. This brief note gives a sufficient condition for Stein's lemma to hold for estimators with discontinuities, which is then shown to be fulfilled for a class of spectral function estimators including the reduced rank estimator. Singular value hard thresholding does, however, not satisfy the condition, and Stein's lemma does not apply to this estimator.Comment: 11 pages, 1 figur

Risk estimation for matrix recovery with spectral regularization

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.Comment: This version is an update of our original paper presented at ICML'2012 workshop on Sparsity, Dictionaries and Projections in Machine Learning and Signal Processin

Resolving the Issues of Capon and APES Approach for Projecting Enhanced Spectral Estimation

There are various applications on signal processing that is highly dependent on preciseness and accuracy of the outcomes in spectrum of signals. Hence, from the past two decades the research community has recognized the benefits, significance, as well as associated problems in carrying out a model for spectral estimation. While in-depth investigation of the existing literatures shows that there are various attempts by the researchers to solve the issues associated with spectral estimations, where majority of teh research work is inclined towards addressing problems associated with Capon and APES techniques of spectral analysis. Therefore, this paper introduces a very simple technique towards resolving the issues of Capon and APES techniques. The outcome of the study was analyzed using correlational factor and power spectral density to find the proposed system offers better spectral estimations compared to existing system