82 research outputs found
A Primer on Reproducing Kernel Hilbert Spaces
Reproducing kernel Hilbert spaces are elucidated without assuming prior
familiarity with Hilbert spaces. Compared with extant pedagogic material,
greater care is placed on motivating the definition of reproducing kernel
Hilbert spaces and explaining when and why these spaces are efficacious. The
novel viewpoint is that reproducing kernel Hilbert space theory studies
extrinsic geometry, associating with each geometric configuration a canonical
overdetermined coordinate system. This coordinate system varies continuously
with changing geometric configurations, making it well-suited for studying
problems whose solutions also vary continuously with changing geometry. This
primer can also serve as an introduction to infinite-dimensional linear algebra
because reproducing kernel Hilbert spaces have more properties in common with
Euclidean spaces than do more general Hilbert spaces.Comment: Revised version submitted to Foundations and Trends in Signal
Processin
Riemannian Gaussian distributions on the space of positive-definite quaternion matrices
Recently, Riemannian Gaussian distributions were defined on spaces of
positive-definite real and complex matrices. The present paper extends this
definition to the space of positive-definite quaternion matrices. In order to
do so, it develops the Riemannian geometry of the space of positive-definite
quaternion matrices, which is shown to be a Riemannian symmetric space of
non-positive curvature. The paper gives original formulae for the Riemannian
metric of this space, its geodesics, and distance function. Then, it develops
the theory of Riemannian Gaussian distributions, including the exact expression
of their probability density, their sampling algorithm and statistical
inference.Comment: 8 pages, submitted to GSI 201
Isotropic Multiple Scattering Processes on Hyperspheres
This paper presents several results about isotropic random walks and multiple
scattering processes on hyperspheres . It allows one to
derive the Fourier expansions on of these processes. A
result of unimodality for the multiconvolution of symmetrical probability
density functions (pdf) on is also introduced. Such
processes are then studied in the case where the scattering distribution is von
Mises Fisher (vMF). Asymptotic distributions for the multiconvolution of vMFs
on are obtained. Both Fourier expansion and asymptotic
approximation allows us to compute estimation bounds for the parameters of
Compound Cox Processes (CCP) on .Comment: 16 pages, 4 figure
Density estimation on the rotation group using diffusive wavelets
This paper considers the problem of estimating probability density functions
on the rotation group . Two distinct approaches are proposed, one based
on characteristic functions and the other on wavelets using the heat kernel.
Expressions are derived for their Mean Integrated Squared Errors. The
performance of the estimators is studied numerically and compared with the
performance of an existing technique using the De La Vall\'ee Poussin kernel
estimator. The heat-kernel wavelet approach appears to offer the best
convergence, with faster convergence to the optimal bound and guaranteed
positivity of the estimated probability density function
Asymptotic regime for impropriety tests of complex random vectors
Impropriety testing for complex-valued vector has been considered lately due
to potential applications ranging from digital communications to complex media
imaging. This paper provides new results for such tests in the asymptotic
regime, i.e. when the vector dimension and sample size grow commensurately to
infinity. The studied tests are based on invariant statistics named impropriety
coefficients. Limiting distributions for these statistics are derived, together
with those of the Generalized Likelihood Ratio Test (GLRT) and Roy's test, in
the Gaussian case. This characterization in the asymptotic regime allows also
to identify a phase transition in Roy's test with potential application in
detection of complex-valued low-rank subspace corrupted by proper noise in
large datasets. Simulations illustrate the accuracy of the proposed asymptotic
approximations.Comment: 11 pages, 8 figures, submitted to IEEE TS
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