SUPERMIX: SPARSE REGULARIZATION FOR MIXTURES

Yohann de Castro

doi:10.1214/20-AOS2022

SUPERMIX: SPARSE REGULARIZATION FOR MIXTURES

2021-06-01

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AI

Cathy Maugis-Rabusseau

DOI：10.1214/20-AOS2022

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Abstract

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This paper investigates the statistical estimation of a discrete mixing measure mu(0) involved in a kernel mixture model. Using some recent advances in l(1)-regularization over the space of measures, we introduce a data fitting and regularization convex program for estimating mu(0) 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 mu(0) and its socalled 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 mu(0), 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 socalled 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.

Keywords:

Beurling Lasso

mixture recovery

dual certificate

kernel approach

super-resolution

SUPERMIX: SPARSE REGULARIZATION FOR MIXTURES

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