Symmetric Gauss-Seidel Technique Based Alternating Direction Methods of Multipliers for Transform Invariant Low-Rank Textures Problem

TL;DR

This paper introduces a symmetric Gauss-Seidel based ADMM method for solving the Transform Invariant Low-Rank Textures problem, providing guaranteed convergence and improved efficiency over traditional ADMM approaches.

Contribution

The paper develops a novel sGS-ADMM method with convergence guarantees for TILT, demonstrating its practical efficiency and reduced computational cost compared to standard ADMM.

Findings

sGS-ADMM guarantees convergence for TILT.

sGS-ADMM outperforms traditional ADMM in efficiency.

Only one variable update is needed in the sGS approach.

Abstract

Transform Invariant Low-Rank Textures, referred to as TILT, can accurately and robustly extract textural or geometric information in a 3D from user-specified windows in 2D in spite of significant corruptions and warping. It was discovered that the task can be characterized, both theoretically and numerically, by solving a sequence of matrix nuclear-norm and $\ell_1$-norm involved convex minimization problems. For solving this problem, the direct extension of Alternating Direction Method of Multipliers (ADMM) in an usual Gauss-Seidel manner often performs numerically well in practice but there is no theoretical guarantee on its convergence. In this paper, we resolve this dilemma by using the novel symmetric Gauss-Seidel (sGS) based ADMM developed by Li, Sun \& Toh (Math. Prog. 2016). The sGS-ADMM is guaranteed to converge and we shall demonstrate in this paper that it is also practically efficient than the directly extended ADMM. When the sGS technique is applied to this particular problem, we show that only one variable needs to be re-updated, and this updating hardly imposes any excessive computational cost. The sGS decomposition theorem of Li, Sun \& Toh (arXiv: 1703.06629) establishes the equivalent between sGS-ADMM and the classical ADMM with an additional semi-proximal term, so the convergence result is followed directly. Extensive experiments illustrate that the sGS-ADMM and its generalized variant have superior numerical efficiency over the directly extended ADMM.