Mixtures Closest To A Given Measure: A Semidefinite Programming Approach

Srecko Durasinovic, Jean B. Lasserre, Victor Magron
Proceedings of the 43rd International Conference on Machine Learning, PMLR 306:27281-27303, 2026.

Abstract

Mixture models, such as Gaussian mixture models (GMMs), 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 ($\operatorname{W_2}$) or the total variation ($\operatorname{TV}$) 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

Cite this Paper

BibTeX
@InProceedings{pmlr-v306-durasinovic26a, title = {Mixtures Closest To A Given Measure: A Semidefinite Programming Approach}, author = {Durasinovic, Srecko and Lasserre, Jean B. and Magron, Victor}, booktitle = {Proceedings of the 43rd International Conference on Machine Learning}, pages = {27281--27303}, year = {2026}, editor = {Zhang, Tong and Dudik, Miroslav and Jaggi, Martin and Agarwal, Alekh and Li, Sharon and Schuurmans, Dale and Zhu, Jerry and Berkenkamp, Felix and Dong, Hanze and Bietti, Alberto}, volume = {306}, series = {Proceedings of Machine Learning Research}, month = {06--11 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v306/main/assets/durasinovic26a/durasinovic26a.pdf}, url = {https://proceedings.mlr.press/v306/durasinovic26a.html}, abstract = {Mixture models, such as Gaussian mixture models (GMMs), 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 ($\operatorname{W_2}$) or the total variation ($\operatorname{TV}$) 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} }
Endnote
%0 Conference Paper %T Mixtures Closest To A Given Measure: A Semidefinite Programming Approach %A Srecko Durasinovic %A Jean B. Lasserre %A Victor Magron %B Proceedings of the 43rd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2026 %E Tong Zhang %E Miroslav Dudik %E Martin Jaggi %E Alekh Agarwal %E Sharon Li %E Dale Schuurmans %E Jerry Zhu %E Felix Berkenkamp %E Hanze Dong %E Alberto Bietti %F pmlr-v306-durasinovic26a %I PMLR %P 27281--27303 %U https://proceedings.mlr.press/v306/durasinovic26a.html %V 306 %X Mixture models, such as Gaussian mixture models (GMMs), 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 ($\operatorname{W_2}$) or the total variation ($\operatorname{TV}$) 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
APA
Durasinovic, S., Lasserre, J.B. & Magron, V.. (2026). Mixtures Closest To A Given Measure: A Semidefinite Programming Approach. Proceedings of the 43rd International Conference on Machine Learning, in Proceedings of Machine Learning Research 306:27281-27303 Available from https://proceedings.mlr.press/v306/durasinovic26a.html.

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