SuperMix: Sparse Regularization for Mixtures - Archive ouverte HAL Access content directly
Journal Articles Annals of Statistics Year : 2021

SuperMix: Sparse Regularization for Mixtures

Yohann de Castro
Sébastien Gadat
  • Function : Author
  • PersonId : 857773


This paper investigates the statistical estimation of a discrete mixing measure µ0 involved in a kernel mixture model. Using some recent advances in l1-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 twofold: 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 "non-degenerate source condition" we derive a non-asymptotic 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 e.g. mixtures of super-smooth distributions (e.g. Gaussian distributions) or ordinary-smooth distributions (e.g. Laplace distributions), are discussed at the end of the paper.
Fichier principal
Vignette du fichier
Preprint-Gadatetal-v2.pdf (911.97 Ko) Télécharger le fichier
Origin : Files produced by the author(s)

Dates and versions

hal-02190117 , version 1 (22-07-2019)
hal-02190117 , version 2 (18-06-2020)



Yohann de Castro, Sébastien Gadat, Clément Marteau, Cathy Maugis. SuperMix: Sparse Regularization for Mixtures. Annals of Statistics, 2021, 49 (3), pp.1779 - 1809. ⟨10.1214/20-AOS2022⟩. ⟨hal-02190117v2⟩
239 View
232 Download



Gmail Facebook Twitter LinkedIn More