OptShrink: An algorithm for improved low-rank signal matrix denoising by optimal, data-driven singular value shrinkage
This addresses a fundamental denoising problem in signal processing and data analysis, offering a theoretically optimal solution that outperforms existing methods, though it is incremental in refining low-rank matrix estimation.
The paper tackles the problem of denoising low-rank signal matrices from noisy measurements by deriving optimal singular value shrinkage weights from random matrix theory, showing that convex regularization methods like nuclear norm are suboptimal, and validating gains with simulations and an algorithm (OptShrink) that improves estimation even with missing data.
The truncated singular value decomposition (SVD) of the measurement matrix is the optimal solution to the_representation_ problem of how to best approximate a noisy measurement matrix using a low-rank matrix. Here, we consider the (unobservable)_denoising_ problem of how to best approximate a low-rank signal matrix buried in noise by optimal (re)weighting of the singular vectors of the measurement matrix. We exploit recent results from random matrix theory to exactly characterize the large matrix limit of the optimal weighting coefficients and show that they can be computed directly from data for a large class of noise models that includes the i.i.d. Gaussian noise case. Our analysis brings into sharp focus the shrinkage-and-thresholding form of the optimal weights, the non-convex nature of the associated shrinkage function (on the singular values) and explains why matrix regularization via singular value thresholding with convex penalty functions (such as the nuclear norm) will always be suboptimal. We validate our theoretical predictions with numerical simulations, develop an implementable algorithm (OptShrink) that realizes the predicted performance gains and show how our methods can be used to improve estimation in the setting where the measured matrix has missing entries.