2019/06/28 by Philipp J. di Dio, di Dio, Philipp J.
Chemistry · Computer Science · Medicine · #14P99 #30E05 #35R30 #44A60 #65D32 #Algebraic Geometry (math.AG) #Bayesian Methods and Mixture Models #Drug Transport and Resistance Mechanisms #FOS: Mathematics #Functional Analysis (math.FA) #Molecular spectroscopy and chirality #Probability (math.PR) #Statistics Theory (math.ST)
paper · pdf · doi:10.48550/arxiv.1907.00790
openalex publication_date 2019/06/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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:ℝ[x1,…,xn]≤ 2d→ℝ which can be represented by a sum of \binomn+2dn - n⋅ \binomn+dn + \binomn2 Gaussians but not less. Hence, for any d∈ℕ and ε>0 we find an n∈ℕ such that L can be represented by a sum of (1-ε)⋅\binomn+2dn Gaussians but not less. An upper bound is \binomn+2dn-1.