Characteristic Kernels and Infinitely Divisible Distributions

We connect shift-invariant characteristic kernels to infinitely divisible distributions on ℝd. Characteristic kernels play an important role in machine learning applications with their kernel means to distinguish any two probability measures. The contribution of this paper is two-fold. First, we show, using the L'evy-Khintchine formula, that any shift-invariant kernel given by a bounded, continuous and symmetric probability density function (pdf) of an infinitely divisible distribution on ℝd is characteristic. We also present some closure property of such characteristic kernels under addition, pointwise product, and convolution. Second, in developing various kernel mean algorithms, it is fundamental to compute the following values: (i) kernel mean values mP(x), x ∈ X, and (ii) kernel mean RKHS inner products leftlangle mP, mQ rightrangleH, for probability measures P, Q. If P, Q, and kernel k are Gaussians, then computation (i) and (ii) results in Gaussian pdfs that is tractable. We generalize this Gaussian combination to more general cases in the class of infinitely divisible distributions. We then introduce a it conjugate kernel and it convolution trick, so that the above (i) and (ii) have the same pdf form, expecting tractable computation at least in some cases. As specific instances, we explore α-stable distributions and a rich class of generalized hyperbolic distributions, where the Laplace, Cauchy and Student-t distributions are included.