# Synchronous deautoconvolution algorithm for discrete-time positive signals via I-divergence approximation

## TL;DR

This paper introduces a new iterative algorithm for approximating nonnegative signals using autoconvolution, employing I-divergence as the criterion, with proven convergence properties and demonstrated effectiveness through numerical experiments.

## Contribution

It develops a novel synchronous deautoconvolution algorithm based on I-divergence minimization and relaxation techniques, with proven convergence to Kuhn-Tucker points.

## Key findings

- Algorithm converges to Kuhn-Tucker points.
- Numerical experiments validate the algorithm's effectiveness.
- The method effectively handles nonnegativity constraints.

## Abstract

We pose the problem of the optimal approximation of a given nonnegative signal $y_t$ with the scalar autoconvolution $(x*x)_t$ of a nonnegative signal $x_t$, where $x_t$ and $y_t$ are signals of equal length. The $\mathcal{I}$-divergence has been adopted as optimality criterion, being well suited to incorporate nonnegativity constraints. To find a minimizer we derive an iterative descent algorithm of the alternating minimization type. The algorithm is based on the lifting of the original problem to a larger space, a relaxation technique developed by Csisz\'ar and Tusn\'ady in [Statistics \& Decisions (S1) (1984), 205--237] which, in the present context, requires the solution of a hard partial minimization problem. We study the asymptotic behavior of the algorithm exploiting the optimality properties of the partial minimization problems and prove, among other results, that its limit points are Kuhn-Tucker points of the original minimization problem. Numerical experiments illustrate the results.