Learning Hierarchical and Topographic Dictionaries with Structured Sparsity

TL;DR

This paper introduces a new class of convex regularization functions that promote structured sparsity, enabling the learning of hierarchical and topographic dictionaries for natural image patches, with efficient optimization methods.

Contribution

It proposes a flexible structured sparsity regularization framework with overlapping groups, and applies it to learn dictionaries with hierarchical and topographic structures.

Findings

Structured sparsity improves dictionary learning for image patches.

The proposed methods produce dictionaries similar to topographic ICA.

Efficient optimization algorithms are reviewed and applied.

Abstract

Recent work in signal processing and statistics have focused on defining new regularization functions, which not only induce sparsity of the solution, but also take into account the structure of the problem. We present in this paper a class of convex penalties introduced in the machine learning community, which take the form of a sum of l_2 and l_infinity-norms over groups of variables. They extend the classical group-sparsity regularization in the sense that the groups possibly overlap, allowing more flexibility in the group design. We review efficient optimization methods to deal with the corresponding inverse problems, and their application to the problem of learning dictionaries of natural image patches: On the one hand, dictionary learning has indeed proven effective for various signal processing tasks. On the other hand, structured sparsity provides a natural framework for modeling dependencies between dictionary elements. We thus consider a structured sparse regularization to learn dictionaries embedded in a particular structure, for instance a tree or a two-dimensional grid. In the latter case, the results we obtain are similar to the dictionaries produced by topographic independent component analysis.