The inverse problem of positive autoconvolution

TL;DR

This paper introduces a new iterative algorithm for optimally approximating a nonnegative signal by the autoconvolution of another nonnegative signal, using I-divergence as the criterion, with proven convergence and numerical validation.

Contribution

It develops an alternating minimization algorithm for the inverse autoconvolution problem with nonnegativity constraints, including convergence analysis and practical performance assessment.

Findings

The algorithm converges to Kuhn-Tucker points.

Numerical experiments show fast convergence.

Existence of an optimal approximation is established.

Abstract

We pose the problem of approximating optimally a given nonnegative signal with the scalar autoconvolution of a nonnegative signal. The I-divergence is chosen as the optimality criterion being well suited to incorporate nonnegativity constraints. After proving the existence of an optimal approximation we derive an iterative descent algorithm of the alternating minimization type to find a minimizer. The algorithm is based on the lifting technique developed by Csisz\'ar and Tusn\'adi and exploits the optimality properties of the related minimization problems in the larger space. We study the asymptotic behavior of the iterative algorithm and prove, among other results, that its limit points are Kuhn-Tucker points of the original minimization problem. Numerical experiments confirm the asymptotic results and exhibit the fast convergence of the proposed algorithm.