Moment ideals of local Dirac mixtures

TL;DR

This paper investigates the algebraic structure of moment ideals from local Dirac measures and their mixtures, providing generators, methods for parameter estimation, and applications in signal processing and statistics.

Contribution

It introduces generators for moment ideals of first order local Diracs and demonstrates how to derive the Pareto distribution's moment ideal, connecting algebraic methods with practical applications.

Findings

Derived generators for first order local Dirac moment ideals

Showed how to obtain Pareto distribution's moment ideal from basic cases

Applied elimination theory and Prony's method for mixture parameter estimation

Abstract

In this paper we study ideals arising from moments of local Dirac measures and their mixtures. We provide generators for the case of first order local Diracs and explain how to obtain the moment ideal of the Pareto distribution from them. We then use elimination theory and Prony's method for parameter estimation of finite mixtures. Our results are showcased with applications in signal processing and statistics. We highlight the natural connections to algebraic statistics, combinatorics and applications in analysis throughout the paper.