Distribution (differential geometry)

About: Distribution (differential geometry) is a research topic. Over the lifetime, 911 publications have been published within this topic receiving 10149 citations.

Integrated and separable EGR distribution manifold

Abstract: An exhaust gas recirculation (EGR) manifold received within an intake manifold. The EGR manifold includes a plurality of holes for delivery of recirculated exhaust gas within the flowpath of the intake manifold. Both linear and round shapes are contemplated for the EGR manifold.

An Exact Decomposition Theorem and a Unified View of Some Related Distributions for a Class of Exponential Transformation Models on Symmetric Cones

Abstract: A class of exponential transformation models is defined on symmetric cones $\Omega$ with the group of automorphisms on $\Omega$ as the acting group. We show that these models are reproductive and the exponent of their joint distribution for a given sample of size $q$ can be split into $q$ independent components, introducing one sample point at a time. The automorphism group can be factorized into the group of positive dilation and another group. Accordingly, the symmetric cone $\Omega$ can be seen as the direct product of $\mathbb{R}^+$ and a unit orbit, and every $x$ in $\Omega$ can be identified by its orbital decomposition. We derive the distributions of the independent components of the exponent, of the "length" of $x$, the "direction" of $x$, the conditional distribution of the direction given the length and other distributions for a given sample. The Wishart distribution and the hyperboloid distribution are two special cases in the class we define. We also give a unified view of several distributions which are usually treated separately.

On the Support of Minimizers of Causal Variational Principles

Abstract: A class of causal variational principles on a compact manifold is introduced and analyzed both numerically and analytically. It is proved under general assumptions that the support of a minimizing measure is either completely timelike, or it is singular in the sense that its interior is empty. In the examples of the circle, the sphere and certain flag manifolds, the general results are supplemented by a more detailed and explicit analysis of the minimizers. On the sphere, we get a connection to packing problems and the Tammes distribution. Moreover, the minimal action is estimated from above and below.

Statistics on the Stiefel manifold: Theory and applications

Abstract: A Stiefel manifold of the compact type is often encountered in many fields of engineering including, signal and image processing, machine learning, numerical optimization and others. The Stiefel manifold is a Riemannian homogeneous space but not a symmetric space. In previous work, researchers have defined probability distributions on symmetric spaces and performed statistical analysis of data residing in these spaces. In this paper, we present original work involving definition of Gaussian distributions on a homogeneous space and show that the maximum-likelihood estimate of the location parameter of a Gaussian distribution on the homogeneous space yields the Frechet mean (FM) of the samples drawn from this distribution. Further, we present an algorithm to sample from the Gaussian distribution on the Stiefel manifold and recursively compute the FM of these samples. We also prove the weak consistency of this recursive FM estimator. Several synthetic and real data experiments are then presented, demonstrating the superior computational performance of this estimator over the gradient descent based nonrecursive counter part as well as the stochastic gradient descent based method prevalent in literature.