Mixtures Closest to a Given Measure: A Semidefinite Programming Approach

A new mathematical framework uses semidefinite programming to approximate an unknown probability distribution by a mixture of simpler parametric distributions, addressing a long-standing challenge in machine learning and statistics [1]. The approach, detailed in a preprint by Srećko Đurašinović of Nanyang Technological University, Jean B. Lasserre, and Victor Magron, tackles the problem of fitting a mixture model when only finitely many moments of the target measure are known [1]. Mixture models, such as Gaussian mixture models, are widely used to represent complex data distributions, but determining the correct number of components and their parameters remains difficult, particularly in high-dimensional settings [2]. The authors model the parameter set not as a finite collection of points, as is common in many existing methods, but as a compact basic semi-algebraic set [1]. They then construct a hierarchy of semidefinite relaxations that converges asymptotically to the optimal approximation, measured by either the 2-Wasserstein distance or the total variation distance [3]. When a specific rank condition holds, the convergence becomes finite, and the method recovers an exact optimal mixing measure [4]. The work has direct implications for clustering. Standard algorithms such as k-means partition data by minimizing within-cluster variances and are known to converge quickly to a local optimum via iterative refinement, a process similar to the expectation–maximization algorithm used for Gaussian mixtures [5]. The proposed semidefinite framework can serve as a preprocessing step that supplies both the number of clusters and initial parameter estimates, accelerating the convergence of these local methods [1]. The preprint was first submitted on 26 September 2025, with a file size of 205 KB, and a revised version followed on 23 June 2026, weighing 280 KB [1]. It was also submitted to the 2026 International Conference on Learning Representations (ICLR) under the primary area of optimization [3]. The authors note that the framework applies to parametric families including Gaussian, exponential, and Poisson distributions [2].