Sparsity-Enhanced Multilayered Non-Convex Regularization with Epigraphical Relaxation for Debiased Signal Recovery

TL;DR

This paper introduces a novel epigraphical relaxation approach to multilayered non-convex regularization, enabling more accurate and less biased high-dimensional signal recovery, especially for images and videos.

Contribution

It develops a new epigraphical relaxation technique to compute proximity operators for multilayered non-convex regularization, improving bias reduction and global optimality in signal recovery.

Findings

Reduced bias in image recovery results.

Enhanced accuracy in principal component analysis.

Demonstrated convergence to global optima.

Abstract

This paper proposes a precise signal recovery method with multilayered non-convex regularization, enhancing sparsity/low-rankness for high-dimensional signals including images and videos. In optimization-based signal recovery, multilayered convex regularization functions based on the L1 and nuclear-norms not only guarantee a global optimal solution but also offer more accurate estimation than single-layered ones, thanks to their faithful modeling of structured sparsity and low-rankness in high-dimensional signals. However, these functions are known to yield biased solutions (estimated with smaller amplitude values than the true ones). To address this issue, multilayered non-convex regularization functions have been considered, although they face their own challenges: 1) their closed-form proximity operators are unavailable, and 2) convergence may result in a local optimal solution. In this paper, we resolve the two issues with an approach based on epigraphical relaxation (ER). First, ER decomposes a multilayered non-convex regularization function into the outermost function and epigraph constraints for the inner functions, facilitating the computation of proximity operators. Second, the relaxed regularization functions by ER are integrated into a non-convexly regularized convex optimization model to estimate a global optimal solution with less bias. Numerical experiments demonstrate the bias reduction achieved by the proposed method in image recovery and principal component analysis.