Mixtures Closest to a Given Measure: A Semidefinite Programming Approach

2025/09/26 by Srećko Đurašinović, Srećko Ðurašinović, Đurašinović, Srećko +5
Computer Science · Mathematics · #Cluster analysis #Convergence (economics) #Functional Equations Stability Results #Measure (data warehouse) #Mixing (physics) #Mixture model #Parametric statistics #Semidefinite programming #cs.LG #math.OC

paper · pdf · doi:10.48550/arxiv.2509.22879

published in arXiv (Cornell University) (Cornell University) · 23 pages, 2 algorithms, 1 table, 4 figures

openalex publication_date 2025/09/26 · openalex created_date 2025/10/19 · arxiv created 2026/06/23 · arxiv updated 2026/08/06 · openalex updated_date 2026/08/06

Abstract

Mixture models, such as Gaussian mixture models, are widely used in machine learning to represent complex data distributions. A key challenge, especially in high-dimensional settings, is to determine the mixture order and estimate the mixture parameters. We study the problem of approximating a target measure, available only through finitely many of its moments, by a mixture of distributions from a parametric family (e.g., Gaussian, exponential, Poisson), with approximation quality measured by the 2-Wasserstein or the total variation distance. Unlike many existing approaches, the parameter set is not assumed to be finite; it is modeled as a compact basic semi-algebraic set. We introduce a hierarchy of semidefinite relaxations with asymptotic convergence to the desired optimal value. In addition, when a certain rank condition is satisfied, the convergence is even finite and recovery of an optimal mixing measure is obtained. We also present an application to clustering, where our framework serves either as a stand-alone method or as a preprocessing step that yields both the number of clusters and strong initial parameter estimates, thereby accelerating convergence of standard (local) clustering algorithms.

Citations

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Related

• Clustering subgaussian mixtures by semidefinite programming
• Gaussian mixtures closest to a given measure via optimal transport
• Clustering Mixtures with Almost Optimal Separation in Polynomial Time
• Settling the Polynomial Learnability of Mixtures of Gaussians
• Semidefinite programming on population clustering: a global analysis