# Characteristic Kernels and Infinitely Divisible Distributions

Yu Nishiyama, Kenji Fukumizu.

Year: 2016, Volume: 17, Issue: 180, Pages: 1−28

#### Abstract

We connect shift-invariant characteristic kernels to infinitely divisible distributions on $\mathbb{R}^{d}$. 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 twofold. First, we show, using the Levy--Khintchine formula, that any shift- invariant kernel given by a bounded, continuous, and symmetric probability density function (pdf) of an infinitely divisible distribution on $\mathbb{R}^d$ is characteristic. We mention some closure properties 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 $m_P(x)$, $x \in \mathcal{X}$, and (ii) kernel mean RKHS inner products ${\left\langle m_P, m_Q \right\rangle _{\mathcal{H}}}$, for probability measures $P, Q$. If $P, Q$, and kernel $k$ are Gaussians, then the computation of (i) and (ii) results in Gaussian pdfs that are tractable. We generalize this Gaussian combination to more general cases in the class of infinitely divisible distributions. We then introduce a *conjugate* kernel and a *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 $\alpha$-stable distributions and a rich class of generalized hyperbolic distributions, where the Laplace, Cauchy, and Student's $t$ distributions are included.