Convex Coding

TL;DR

This paper introduces a convex relaxation approach to sparse coding, enabling more reliable solutions and demonstrating its effectiveness in image denoising tasks.

Contribution

It proposes a novel convex formulation of sparse coding and a boosting algorithm to address local minima issues in traditional methods.

Findings

Convex relaxation improves sparse coding stability.

Boosting algorithm effectively identifies code elements.

Enhanced image denoising performance.

Abstract

Inspired by recent work on convex formulations of clustering (Lashkari & Golland, 2008; Nowozin & Bakir, 2008) we investigate a new formulation of the Sparse Coding Problem (Olshausen & Field, 1997). In sparse coding we attempt to simultaneously represent a sequence of data-vectors sparsely (i.e. sparse approximation (Tropp et al., 2006)) in terms of a 'code' defined by a set of basis elements, while also finding a code that enables such an approximation. As existing alternating optimization procedures for sparse coding are theoretically prone to severe local minima problems, we propose a convex relaxation of the sparse coding problem and derive a boosting-style algorithm, that (Nowozin & Bakir, 2008) serves as a convex 'master problem' which calls a (potentially non-convex) sub-problem to identify the next code element to add. Finally, we demonstrate the properties of our boosted coding algorithm on an image denoising task.