Variational Learning ISTA

TL;DR

This paper introduces VLISTA, a probabilistic framework that jointly learns dictionary distributions and sparse reconstructions under varying sensing matrices using a variational approach and an augmented LISTA architecture.

Contribution

It proposes a novel variational learning method for dictionary learning and sparse recovery that accounts for uncertainty and adapts to different sensing conditions.

Findings

VLISTA learns calibrated uncertainties in sparse recovery.

The model effectively adapts to different sensing matrices.

Experimental results demonstrate improved robustness and uncertainty estimation.

Abstract

Compressed sensing combines the power of convex optimization techniques with a sparsity-inducing prior on the signal space to solve an underdetermined system of equations. For many problems, the sparsifying dictionary is not directly given, nor its existence can be assumed. Besides, the sensing matrix can change across different scenarios. Addressing these issues requires solving a sparse representation learning problem, namely dictionary learning, taking into account the epistemic uncertainty of the learned dictionaries and, finally, jointly learning sparse representations and reconstructions under varying sensing matrix conditions. We address both concerns by proposing a variant of the LISTA architecture. First, we introduce Augmented Dictionary Learning ISTA (A-DLISTA), which incorporates an augmentation module to adapt parameters to the current measurement setup. Then, we propose to learn a distribution over dictionaries via a variational approach, dubbed Variational Learning ISTA (VLISTA). VLISTA exploits A-DLISTA as the likelihood model and approximates a posterior distribution over the dictionaries as part of an unfolded LISTA-based recovery algorithm. As a result, VLISTA provides a probabilistic way to jointly learn the dictionary distribution and the reconstruction algorithm with varying sensing matrices. We provide theoretical and experimental support for our architecture and show that our model learns calibrated uncertainties.