2019/07/22 by Yohann de Castro, de Castro, Yohann, Sébastien Gadat +6
Computer Science · Mathematics · Medicine · #Bayesian Methods and Mixture Models #FOS: Computer and information sciences #FOS: Mathematics #Gaussian Processes and Bayesian Inference #Machine Learning (cs.LG) #Machine Learning (stat.ML) #Markov Chains and Monte Carlo Methods #Medical Imaging Techniques and Applications #Statistical Methods and Inference #Statistics Theory (math.ST)
paper · pdf · doi:10.48550/arxiv.1907.10592
openalex publication_date 2019/07/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.