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We develop and analyze a projected particle Langevin optimization method tolearn the distribution in the Sch\"{o}nberg integral representation of theradial basis functions from training samples. More specifically, wecharacterize a distributionally robust optimization method with respect to theWasserstein distance to optimize the distribution in the Sch\"{o}nberg integralrepresentation. To provide theoretical performance guarantees, we analyze thescaling limits of a projected particle online (stochastic) optimization methodin 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 areflected It\^{o} diffusion-drift process. Moreover, the drift is also afunction of the law of the underlying process. Using It\^{o} lemma forsemi-martingales and Grisanov's change of measure for the Wiener processes, wethen derive a Mckean-Vlasov type partial differential equation (PDE) with Robinboundary conditions that describes the evolution of the empirical measure ofthe projected Langevin particles in the mean-field regime. In addition, weestablish the existence and uniqueness of the steady-state solutions of thederived PDE in the weak sense. We apply our learning approach to train radialkernels in the kernel locally sensitive hash (LSH) functions, where thetraining data-set is generated via a $k$-mean clustering method on a smallsubset of data-base. We subsequently apply our kernel LSH with a trained kernelfor image retrieval task on MNIST data-set, and demonstrate the efficacy of ourkernel learning approach. We also apply our kernel learning approach inconjunction with the kernel support vector machines (SVMs) for classificationof benchmark data-sets.

A Mean-Field Theory for Learning the Sch\"{o}nberg Measure of Radial Basis Functions