Dictionary Learning with Convex Update (ROMD)

TL;DR

This paper introduces ROMD, a convex optimization-based dictionary learning algorithm that updates the entire dictionary simultaneously, offering improved convergence guarantees and accuracy, especially with high sparsity and limited data.

Contribution

ROMD recasts dictionary learning as a convex problem by using rank-one matrix decomposition, enabling whole-dictionary updates with convergence guarantees.

Findings

ROMD achieves higher recovery accuracy than benchmark algorithms.

ROMD converges faster due to convex optimization approach.

ROMD performs well with high sparsity and limited data.

Abstract

Dictionary learning aims to find a dictionary under which the training data can be sparsely represented, and it is usually achieved by iteratively applying two stages: sparse coding and dictionary update. Typical methods for dictionary update focuses on refining both dictionary atoms and their corresponding sparse coefficients by using the sparsity patterns obtained from sparse coding stage, and hence it is a non-convex bilinear inverse problem. In this paper, we propose a Rank-One Matrix Decomposition (ROMD) algorithm to recast this challenge into a convex problem by resolving these two variables into a set of rank-one matrices. Different from methods in the literature, ROMD updates the whole dictionary at a time using convex programming. The advantages hence include both convergence guarantees for dictionary update and faster convergence of the whole dictionary learning. The performance of ROMD is compared with other benchmark dictionary learning algorithms. The results show the improvement of ROMD in recovery accuracy, especially in the cases of high sparsity level and fewer observation data.