Sparse Regularization for Mixture Problems

This paper investigates the statistical estimation of a discrete mixing measure $\mu^0$ involved in a kernel mixture model. Using some recent advances in $\ell_1$-regularization over the space of measures, we introduce a "data fitting + regularization" convex program for estimating $\mu^0$ in a grid-less manner, 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 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 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. The statistical performances of this estimator are investigated designing a so-called "dual certificate", which will be appropriate to our setting. Some classical situations, as e.g., Gaussian or ordinary smooth mixtures (e.g., Laplace distributions), are discussed at the end of the paper. We stress in particular that our method is completely adaptive w.r.t. the number of components involved in the mixture.