A Geometrical Study of Matching Pursuit Parametrization

TL;DR

This paper analyzes how discretizing dictionary parameters affects Matching Pursuit decompositions, using differential geometry to prove convergence properties and improve atom selection through geometric optimization.

Contribution

It introduces a geometric framework for understanding discretized dictionaries in Matching Pursuit and demonstrates how optimization enhances convergence.

Findings

Discrete dictionaries with sufficient density match continuous MP convergence.

Gradient ascent optimization improves atom selection, reducing weakness factor.

Numerical experiments confirm theoretical predictions on signals and images.

Abstract

This paper studies the effect of discretizing the parametrization of a dictionary used for Matching Pursuit decompositions of signals. Our approach relies on viewing the continuously parametrized dictionary as an embedded manifold in the signal space on which the tools of differential (Riemannian) geometry can be applied. The main contribution of this paper is twofold. First, we prove that if a discrete dictionary reaches a minimal density criterion, then the corresponding discrete MP (dMP) is equivalent in terms of convergence to a weakened hypothetical continuous MP. Interestingly, the corresponding weakness factor depends on a density measure of the discrete dictionary. Second, we show that the insertion of a simple geometric gradient ascent optimization on the atom dMP selection maintains the previous comparison but with a weakness factor at least two times closer to unity than without optimization. Finally, we present numerical experiments confirming our theoretical predictions for decomposition of signals and images on regular discretizations of dictionary parametrizations.