282 research outputs found
Harmonic Exponential Families on Manifolds
In a range of fields including the geosciences, molecular biology, robotics
and computer vision, one encounters problems that involve random variables on
manifolds. Currently, there is a lack of flexible probabilistic models on
manifolds that are fast and easy to train. We define an extremely flexible
class of exponential family distributions on manifolds such as the torus,
sphere, and rotation groups, and show that for these distributions the gradient
of the log-likelihood can be computed efficiently using a non-commutative
generalization of the Fast Fourier Transform (FFT). We discuss applications to
Bayesian camera motion estimation (where harmonic exponential families serve as
conjugate priors), and modelling of the spatial distribution of earthquakes on
the surface of the earth. Our experimental results show that harmonic densities
yield a significantly higher likelihood than the best competing method, while
being orders of magnitude faster to train.Comment: fixed typ
Convolution of some slanted half-plane mappings with harmonic strip mappings
In this paper, we show that the convolution of generalized half-plane mapping and harmonic vertical strip mapping with dilatation eⁱᶱ zⁿ (n ∈ N, θ ∈ R) is convex in a particular direction and also solve the problem proposed by Z. Liu et al. [Convolutions of harmonic half-plane mappings with harmonic vertical strip mappings, Filomat, 31 (2017), no. 7, 1843–1856].Publisher's VersionPMID-123
Differentiability properties and characterization of H-convex functions
The present thesis deals with a number of geometric properties of convex functions in a non-Euclidean framework. This setting is represented by the so-called Sub-Riemannian space, also called a Carnot-Caratheodory (CC) space, that can be thought of as a space where the metric structure is a constrained geometry and one can move only along a prescribed set of directions depending on the point. We will study first order and second order reguarity of h-convex functions using h-subdifferentials. Moreover the distributional notion of h-convexity is considered. In this context we will prove that for all stratified groups an h-convex distribution is represented by an h-convex function. In the last chapter we address the study of convexity in general CC space. Here we prove a quantitative Lipschitz estimate for convex functions
Spectral gap properties for linear random walks and Pareto's asymptotics for affine stochastic recursions
Let be the Euclidean -dimensional space, (resp
) a probability measure on the linear (resp affine) group
(resp H= \Aff (V)) and assume that is the projection of on
. We study asymptotic properties of the iterated convolutions (resp if , i.e asymptotics of
the random walk on defined by (resp ), if the subsemigroup
(resp.\ ) generated by the support of
(resp ) is "large". We show spectral gap properties for the
convolution operator defined by on spaces of homogeneous functions of
degree on , which satisfy H{\"o}lder type conditions. As a
consequence of our analysis we get precise asymptotics for the potential kernel
, which imply its asymptotic
homogeneity. Under natural conditions the -space is a
-boundary; then we use the above results and radial Fourier Analysis
on to show that the unique -stationary measure
on is "homogeneous at infinity" with respect to dilations
(for t\textgreater{}0), with a tail measure depending
essentially of and . Our proofs are based on the simplicity of
the dominant Lyapunov exponent for certain products of Markov-dependent random
matrices, on the use of renewal theorems for "tame" Markov walks, and on the
dynamical properties of a conditional -boundary dual to
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