  We develop and analyze a projected particle Langevin optimization method to
learn the distribution in the Sch\"{o}nberg integral representation of the
radial basis functions from training samples. More specifically, we
characterize a distributionally robust optimization method with respect to the
Wasserstein distance to optimize the distribution in the Sch\"{o}nberg integral
representation. To provide theoretical performance guarantees, we analyze the
scaling limits of a projected particle online (stochastic) optimization method
in the mean-field regime. In particular, we prove that in the scaling limits,
the empirical measure of the Langevin particles converges to the law of a
reflected It\^{o} diffusion-drift process. Moreover, the drift is also a
function of the law of the underlying process. Using It\^{o} lemma for
semi-martingales and Grisanov's change of measure for the Wiener processes, we
then derive a Mckean-Vlasov type partial differential equation (PDE) with Robin
boundary conditions that describes the evolution of the empirical measure of
the projected Langevin particles in the mean-field regime. In addition, we
establish the existence and uniqueness of the steady-state solutions of the
derived PDE in the weak sense. We apply our learning approach to train radial
kernels in the kernel locally sensitive hash (LSH) functions, where the
training data-set is generated via a $k$-mean clustering method on a small
subset of data-base. We subsequently apply our kernel LSH with a trained kernel
for image retrieval task on MNIST data-set, and demonstrate the efficacy of our
kernel learning approach. We also apply our kernel learning approach in
conjunction with the kernel support vector machines (SVMs) for classification
of benchmark data-sets.
