# The multidimensional truncated Moment Problem: Shape and Gaussian Mixture Reconstruction from Derivatives of Moments

## TL;DR

This paper develops a theory for representing moment functionals using Gaussian mixtures and polytopes, establishing bounds on the number of Gaussians needed for such representations.

## Contribution

It introduces derivatives of moments to analyze Gaussian mixture representations and determines minimal Gaussian counts for representing certain moment functionals.

## Key findings

- Identifies exact Gaussian counts needed for specific moment functionals.
- Establishes bounds on the number of Gaussians required for representation.
- Provides theoretical limits for Gaussian mixture reconstruction from moments.

## Abstract

In this paper we introduce the theory of derivatives of moments and (moment) functionals to represent moment functionals by Gaussian mixtures, characteristic functions of polytopes, and simple functions of polytopes. We study, among other measures, Gaussian mixtures, their reconstruction from moments and especially the number of Gaussians needed to represent moment functionals. We find that there are moment functionals $L:\mathbb{R}[x_1,\dots,x_n]_{\leq 2d}\to\mathbb{R}$ which can be represented by a sum of $\binom{n+2d}{n} - n\cdot \binom{n+d}{n} + \binom{n}{2}$ Gaussians but not less. Hence, for any $d\in\mathbb{N}$ and $\varepsilon>0$ we find an $n\in\mathbb{N}$ such that $L$ can be represented by a sum of $(1-\varepsilon)\cdot\binom{n+2d}{n}$ Gaussians but not less. An upper bound is $\binom{n+2d}{n}-1$.