A nonlinear PDE-based method for sparse deconvolution

TL;DR

This paper introduces a novel nonlinear PDE approach for sparse deconvolution that preserves the $ ext{l}_1$ norm, enhances sparsity asymptotically, and improves convergence speed and reconstruction quality in $ ext{l}_1$ minimization problems.

Contribution

It proposes a new PDE-based method that acts as a plug-in to existing algorithms, improving efficiency and accuracy in sparse deconvolution tasks.

Findings

PDE preserves the $ ext{l}_1$ norm while reducing residuals.

Solution becomes sparser asymptotically, confirmed numerically.

Numerical experiments show faster convergence and better reconstruction quality.

Abstract

In this paper, we introduce a new nonlinear evolution partial differential equation for sparse deconvolution problems. The proposed PDE has the form of continuity equation that arises in various research areas, e.g. fluid dynamics and optimal transportation, and thus has some interesting physical and geometric interpretations. The underlying optimization model that we consider is the standard $\ell_1$ minimization with linear equality constraints, i.e. $\min_u\{\|u\|_1 : Au=f\}$ with $A$ being an under-sampled convolution operator. We show that our PDE preserves the $\ell_1$ norm while lowering the residual $\|Au-f\|_2$. More importantly the solution of the PDE becomes sparser asymptotically, which is illustrated numerically. Therefore, it can be treated as a natural and helpful plug-in to some algorithms for $\ell_1$ minimization problems, e.g. Bregman iterative methods introduced for sparse reconstruction problems in [W. Yin, S. Osher, D. Goldfarb, and J. Darbon,SIAM J. Imaging Sci., 1 (2008), pp. 143-168]. Numerical experiments show great improvements in terms of both convergence speed and reconstruction quality.