Efficient Consensus Model based on Proximal Gradient Method applied to Convolutional Sparse Problems

TL;DR

This paper introduces an efficient consensus algorithm based on the proximal gradient method for convolutional sparse problems, demonstrating improved performance over ADMM in convolutional dictionary learning and applicability to anomaly detection.

Contribution

It develops a novel proximal gradient-based consensus algorithm for convolutional sparse problems, with thorough theoretical analysis and demonstrated advantages over ADMM.

Findings

Outperforms ADMM in convolutional dictionary learning tasks

Applicable to various convex optimization problems with shared global variables

Effective in anomaly detection applications

Abstract

Convolutional sparse representation (CSR), shift-invariant model for inverse problems, has gained much attention in the fields of signal/image processing, machine learning and computer vision. The most challenging problems in CSR implies the minimization of a composite function of the form $min_x \sum_i f_i(x) + g(x)$, where a direct and low-cost solution can be difficult to achieve. However, it has been reported that semi-distributed formulations such as ADMM consensus can provide important computational benefits. In the present work, we derive and detail a thorough theoretical analysis of an efficient consensus algorithm based on proximal gradient (PG) approach. The effectiveness of the proposed algorithm with respect to its ADMM counterpart is primarily assessed in the classic convolutional dictionary learning problem. Furthermore, our consensus method, which is generically structured, can be used to solve other optimization problems, where a sum of convex functions with a regularization term share a single global variable. As an example, the proposed algorithm is also applied to another particular convolutional problem for the anomaly detection task.