On the convergence of an efficient algorithm for Kullback-Leibler approximation of spectral densities

TL;DR

This paper presents a new efficient algorithm for approximating spectral densities using Kullback-Leibler divergence, with proven local convergence and numerical evidence suggesting potential global convergence.

Contribution

It introduces a variational approach and a nonlinear iterative algorithm for spectral density approximation, with a rigorous proof of local convergence.

Findings

Proven local convergence of the proposed algorithm.

Numerical evidence suggests possible global convergence.

The method effectively solves the Kullback-Leibler approximation problem.

Abstract

This paper deals with a method for the approximation of a spectral density function among the solutions of a generalized moment problem a` la Byrnes/Georgiou/Lindquist. The approximation is pursued with respect to the Kullback-Leibler pseudo-distance, which gives rise to a convex optimization problem. After developing the variational analysis, we discuss the properties of an efficient algorithm for the solution of the corresponding dual problem, based on the iteration of a nonlinear map in a bounded subset of the dual space. Our main result is the proof of local convergence of the latter, established as a consequence of the Central Manifold Theorem. Supported by numerical evidence, we conjecture that, in the mentioned bounded set, the convergence is actually global.