Convex Approaches to Model Wavelet Sparsity Patterns

TL;DR

This paper introduces convex optimization methods using group-sparsity penalties to model wavelet coefficient dependencies, enabling efficient and exact solutions for inverse problems like deconvolution and compressed sensing.

Contribution

It proposes novel convex modeling approaches that incorporate wavelet dependency structures, improving reconstruction quality over traditional greedy or iterative methods.

Findings

Significantly better performance in deconvolution and compressed sensing tasks.

Methods are as computationally efficient as standard lasso approaches.

Convex formulations enable exact and efficient solutions.

Abstract

Statistical dependencies among wavelet coefficients are commonly represented by graphical models such as hidden Markov trees(HMTs). However, in linear inverse problems such as deconvolution, tomography, and compressed sensing, the presence of a sensing or observation matrix produces a linear mixing of the simple Markovian dependency structure. This leads to reconstruction problems that are non-convex optimizations. Past work has dealt with this issue by resorting to greedy or suboptimal iterative reconstruction methods. In this paper, we propose new modeling approaches based on group-sparsity penalties that leads to convex optimizations that can be solved exactly and efficiently. We show that the methods we develop perform significantly better in deconvolution and compressed sensing applications, while being as computationally efficient as standard coefficient-wise approaches such as lasso.