Learning to solve TV regularized problems with unrolled algorithms

TL;DR

This paper introduces learned unrolled algorithms for faster solutions to 1D TV regularized problems, overcoming limitations of traditional iterative methods by developing novel derivative computation techniques.

Contribution

It proposes and analyzes two new methods for differentiating through proximal operators in unrolled algorithms, improving convergence speed for TV regularized problems.

Findings

Unrolled algorithms outperform traditional iterative methods in convergence speed.

Two approaches for derivative computation through proximal operators are developed and compared.

Experiments validate the effectiveness of the proposed methods on synthetic and real data.

Abstract

Total Variation (TV) is a popular regularization strategy that promotes piece-wise constant signals by constraining the $\ell_1$-norm of the first order derivative of the estimated signal. The resulting optimization problem is usually solved using iterative algorithms such as proximal gradient descent, primal-dual algorithms or ADMM. However, such methods can require a very large number of iterations to converge to a suitable solution. In this paper, we accelerate such iterative algorithms by unfolding proximal gradient descent solvers in order to learn their parameters for 1D TV regularized problems. While this could be done using the synthesis formulation, we demonstrate that this leads to slower performances. The main difficulty in applying such methods in the analysis formulation lies in proposing a way to compute the derivatives through the proximal operator. As our main contribution, we develop and characterize two approaches to do so, describe their benefits and limitations, and discuss the regime where they can actually improve over iterative procedures. We validate those findings with experiments on synthetic and real data.