  This paper investigates the statistical estimation of a discrete mixing
measure $\mu$0 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 $\mu$0 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 $\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 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
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.
