Moment ideals of local Dirac mixtures

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.Comment: 26 pages, 3 figure

Parameter recovery for moment problems on algebraic varieties

The thesis studies truncated moment problems and related reconstruction techniques. It transfers the main aspects of Prony's method from finitely-supported measures to the classes of signed or non-negative measures supported on algebraic varieties of any dimension. The Zariski closure of the support of these measures is shown to be determined by finitely many moments and can be computed from the kernel of moment matrices. Moreover, several reconstruction algorithms are developed which are based on the computation of generalized eigenvalues and allow to recover the components of mixtures of such measures