June 2021 SuperMix: Sparse regularization for mixtures
Y. De Castro, S. Gadat, C. Marteau, C. Maugis-Rabusseau
Author Affiliations +
Ann. Statist. 49(3): 1779-1809 (June 2021). DOI: 10.1214/20-AOS2022

Abstract

This paper investigates the statistical estimation of a discrete mixing measure μ0 involved in a kernel mixture model. Using some recent advances in 1-regularization over the space of measures, we introduce a “data fitting and regularization” convex program for estimating μ0 in a grid-less manner from a sample of mixture law, this method is referred to as Beurling-LASSO.

Our contribution is two-fold: we derive a lower bound on the bandwidth of our data fitting term depending only on the support of μ0 and its so-called “minimum separation” to ensure quantitative support localization error bounds; and under a so-called “nondegenerate source condition” we derive a nonasymptotic support stability property. This latter shows that for a sufficiently large sample size n, our estimator has exactly as many weighted Dirac masses as the target μ0, converging in amplitude and localization towards the true ones. Finally, we also introduce some tractable algorithms for solving this convex program based on “Sliding Frank–Wolfe” or “Conic Particle Gradient Descent”.

Statistical performances of this estimator are investigated designing a so-called “dual certificate”, which is appropriate to our setting. Some classical situations as, for example, mixtures of super-smooth distributions (see, e.g., Gaussian distributions) or ordinary-smooth distributions (see, e.g., Laplace distributions), are discussed at the end of the paper.

Funding Statement

S. Gadat acknowledges funding from the French National Research Agency (ANR) under the Investments for the Future program (Investissements d’Avenir, grant ANR-17-EURE-0010) and for the grant MaSDOL—19-CE23-0017-01

Citation

Download Citation

Y. De Castro. S. Gadat. C. Marteau. C. Maugis-Rabusseau. "SuperMix: Sparse regularization for mixtures." Ann. Statist. 49 (3) 1779 - 1809, June 2021. https://doi.org/10.1214/20-AOS2022

Information

Received: 1 July 2019; Revised: 1 June 2020; Published: June 2021
First available in Project Euclid: 9 August 2021

MathSciNet: MR4298881
zbMATH: 1475.62129
Digital Object Identifier: 10.1214/20-AOS2022

Subjects:
Primary: 62G05 , 90C25
Secondary: 49M29

Keywords: Beurling Lasso , dual certificate , kernel approach , mixture recovery , super-resolution

Rights: Copyright © 2021 Institute of Mathematical Statistics

Vol.49 • No. 3 • June 2021
Back to Top