Function reconstruction as a classical moment problem: A maximum entropy approach

TL;DR

This paper investigates reconstructing non-negative functions from finite moments using a maximum entropy approach, analyzing convergence and accuracy across diverse function types with various features.

Contribution

It introduces an iterative entropy optimization scheme for function reconstruction from moments and evaluates its effectiveness on complex functions with singularities and oscillations.

Findings

Convergence improves with more moments and iterations.

Maximum entropy solutions closely match exact functions in various scenarios.

The method effectively captures features like singularities and oscillations.

Abstract

We present a systematic study of the reconstruction of a non-negative function via maximum entropy approach utilizing the information contained in a finite number of moments of the function. For testing the efficacy of the approach, we reconstruct a set of functions using an iterative entropy optimization scheme, and study the convergence profile as the number of moments is increased. We consider a wide variety of functions that include a distribution with a sharp discontinuity, a rapidly oscillatory function, a distribution with singularities, and finally a distribution with several spikes and fine structure. The last example is important in the context of the determination of the natural density of the logistic map. The convergence of the method is studied by comparing the moments of the approximated functions with the exact ones. Furthermore, by varying the number of moments and iterations, we examine to what extent the features of the functions, such as the divergence behavior at singular points within the interval, is reproduced. The proximity of the reconstructed maximum entropy solution to the exact solution is examined via Kullback-Leibler divergence and variation measures for different number of moments.