39 research outputs found
Extensions of Positive Definite Functions: Applications and Their Harmonic Analysis
We study two classes of extension problems, and their interconnections: (i)
Extension of positive definite (p.d.) continuous functions defined on subsets
in locally compact groups ; (ii) In case of Lie groups, representations of
the associated Lie algebras by unbounded skew-Hermitian
operators acting in a reproducing kernel Hilbert space (RKHS)
.
Why extensions? In science, experimentalists frequently gather spectral data
in cases when the observed data is limited, for example limited by the
precision of instruments; or on account of a variety of other limiting external
factors. Given this fact of life, it is both an art and a science to still
produce solid conclusions from restricted or limited data. In a general sense,
our monograph deals with the mathematics of extending some such given partial
data-sets obtained from experiments. More specifically, we are concerned with
the problems of extending available partial information, obtained, for example,
from sampling. In our case, the limited information is a restriction, and the
extension in turn is the full positive definite function (in a dual variable);
so an extension if available will be an everywhere defined generating function
for the exact probability distribution which reflects the data; if it were
fully available. Such extensions of local information (in the form of positive
definite functions) will in turn furnish us with spectral information. In this
form, the problem becomes an operator extension problem, referring to operators
in a suitable reproducing kernel Hilbert spaces (RKHS). In our presentation we
have stressed hands-on-examples. Extensions are almost never unique, and so we
deal with both the question of existence, and if there are extensions, how they
relate back to the initial completion problem.Comment: 235 pages, 42 figures, 7 tables. arXiv admin note: substantial text
overlap with arXiv:1401.478
Identifiability of interaction kernels in mean-field equations of interacting particles
We study the identifiability of the interaction kernels in mean-field
equations for intreacting particle systems. The key is to identify function
spaces on which a probabilistic loss functional has a unique minimizer. We
prove that identifiability holds on any subspace of two reproducing kernel
Hilbert spaces (RKHS), whose reproducing kernels are intrinsic to the system
and are data-adaptive. Furthermore, identifiability holds on two ambient L2
spaces if and only if the integral operators associated with the reproducing
kernels are strictly positive. Thus, the inverse problem is ill-posed in
general. We also discuss the implications of identifiability in computational
practice
Unsupervised learning of observation functions in state-space models by nonparametric moment methods
We investigate the unsupervised learning of non-invertible observation
functions in nonlinear state-space models. Assuming abundant data of the
observation process along with the distribution of the state process, we
introduce a nonparametric generalized moment method to estimate the observation
function via constrained regression. The major challenge comes from the
non-invertibility of the observation function and the lack of data pairs
between the state and observation. We address the fundamental issue of
identifiability from quadratic loss functionals and show that the function
space of identifiability is the closure of a RKHS that is intrinsic to the
state process. Numerical results show that the first two moments and temporal
correlations, along with upper and lower bounds, can identify functions ranging
from piecewise polynomials to smooth functions, leading to convergent
estimators. The limitations of this method, such as non-identifiability due to
symmetry and stationarity, are also discussed
Universal Kernels
In this paper we investigate conditions on the features of a continuous kernel so that it may approximate an arbitrary continuous target function uniformly on any compact subset of the input space. A number of concrete examples are given of kernels with this universal approximating property