234 research outputs found
Characteristic Kernels and Infinitely Divisible Distributions
We connect shift-invariant characteristic kernels to infinitely divisible
distributions on . 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 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 , , and
(ii) kernel mean RKHS inner products , for probability measures . If , and
kernel 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 -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
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
Random integral mappings give isomorphisms between the
sub-semigroups of the classical and the free-infinite divisible
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
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
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
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