234 research outputs found

    Characteristic Kernels and Infinitely Divisible Distributions

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    We connect shift-invariant characteristic kernels to infinitely divisible distributions on Rd\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 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\mathbb{R}^d 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)m_P(x), x∈Xx \in \mathcal{X}, and (ii) kernel mean RKHS inner products ⟨mP,mQ⟩H{\left\langle m_P, m_Q \right\rangle_{\mathcal{H}}}, for probability measures P,QP, Q. If P,QP, Q, and kernel kk 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\alpha-stable distributions and a rich class of generalized hyperbolic distributions, where the Laplace, Cauchy and Student-t distributions are included

    Autoregressive Kernels For Time Series

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    We propose in this work a new family of kernels for variable-length time series. Our work builds upon the vector autoregressive (VAR) model for multivariate stochastic processes: given a multivariate time series x, we consider the likelihood function p_{\theta}(x) of different parameters \theta in the VAR model as features to describe x. To compare two time series x and x', we form the product of their features p_{\theta}(x) p_{\theta}(x') which is integrated out w.r.t \theta using a matrix normal-inverse Wishart prior. Among other properties, this kernel can be easily computed when the dimension d of the time series is much larger than the lengths of the considered time series x and x'. It can also be generalized to time series taking values in arbitrary state spaces, as long as the state space itself is endowed with a kernel \kappa. In that case, the kernel between x and x' is a a function of the Gram matrices produced by \kappa on observations and subsequences of observations enumerated in x and x'. We describe a computationally efficient implementation of this generalization that uses low-rank matrix factorization techniques. These kernels are compared to other known kernels using a set of benchmark classification tasks carried out with support vector machines

    On a method of introducing free-infinitely divisible probability measures

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    Random integral mappings I(a,b]h,rI^{h,r}_{(a,b]} give isomorphisms between the sub-semigroups of the classical (ID,∗)(ID, \ast) and the free-infinite divisible (ID,⊞)(ID,\boxplus) probability measures. This allows us to introduce new examples of such measures and their corresponding characteristic functionals.Comment: 16 page

    Multidimensional limit theorems for homogeneous sums: a general transfer principle

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    The aim of the present paper is to establish the multidimensional counterpart of the \textit{fourth moment criterion} for homogeneous sums in independent leptokurtic and mesokurtic random variables (that is, having positive and zero fourth cumulant, respectively), recently established in \cite{NPPS} in both the classical and in the free setting. As a consequence, the transfer principle for the Central limit Theorem between Wiener and Wigner chaos can be extended to a multidimensional transfer principle between vectors of homogeneous sums in independent commutative random variables with zero third moment and with non-negative fourth cumulant, and homogeneous sums in freely independent non-commutative random variables with non-negative fourth cumulant

    On powers of Stieltjes moment sequences, II

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    We consider the set of Stieltjes moment sequences, for which every positive power is again a Stieltjes moment sequence, we and prove an integral representation of the logarithm of the moment sequence in analogy to the L\'evy-Khinchin representation. We use the result to construct product convolution semigroups with moments of all orders and to calculate their Mellin transforms. As an application we construct a positive generating function for the orthonormal Hermite polynomials.Comment: preprint, 21 page

    Infinitely divisible matrices

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