Space Of Multivariate Normal Distributions Research Articles

Global Geometry of Bayesian Statistics.

In the previous work of the author, a non-trivial symmetry of the relative entropy in the information geometry of normal distributions was discovered. The same symmetry also appears in the symplectic/contact geometry of Hilbert modular cusps. Further, it was observed that a contact Hamiltonian flow presents a certain Bayesian inference on normal distributions. In this paper, we describe Bayesian statistics and the information geometry in the language of current geometry in order to spread our interest in statistics through general geometers and topologists. Then, we foliate the space of multivariate normal distributions by symplectic leaves to generalize the above result of the author. This foliation arises from the Cholesky decomposition of the covariance matrices.

DTI Segmentation and Fiber Tracking Using Metrics on Multivariate Normal Distributions

Existing clustering-based methods for segmentation and fiber tracking of diffusion tensor magnetic resonance images (DT-MRI) are based on a formulation of a similarity measure between diffusion tensors, or measures that combine translational and diffusion tensor distances in some ad hoc way. In this paper we propose to use the Fisher information-based geodesic distance on the space of multivariate normal distributions as an intrinsic distance metric. An efficient and numerically robust shooting method is developed for computing the minimum geodesic distance between two normal distributions, together with an efficient graph-clustering algorithm for segmentation. Extensive experimental results involving both synthetic data and real DT-MRI images demonstrate that in many cases our method leads to more accurate and intuitively plausible segmentation results vis-a-vis existing methods.

Statistics on the Manifold of Multivariate Normal Distributions: Theory and Application to Diffusion Tensor MRI Processing

This paper is dedicated to the statistical analysis of the space of multivariate normal distributions with an application to the processing of Diffusion Tensor Images (DTI). It relies on the differential geometrical properties of the underlying parameters space, endowed with a Riemannian metric, as well as on recent works that led to the generalization of the normal law on Riemannian manifolds. We review the geometrical properties of the space of multivariate normal distributions with zero mean vector and focus on an original characterization of the mean, covariance matrix and generalized normal law on that manifold. We extensively address the derivation of accurate and efficient numerical schemes to estimate these statistical parameters. A major application of the present work is related to the analysis and processing of DTI datasets and we show promising results on synthetic and real examples.

A Riemannian approach to anisotropic filtering of tensor fields

Tensors are nowadays an increasing research domain in different areas, especially in image processing, motivated for example by diffusion tensor magnetic resonance imaging (DT-MRI). Up to now, algorithms and tools developed to deal with tensors were founded on the assumption of a matrix vector space with the constraint of remaining symmetric positive definite matrices. On the contrary, our approach is grounded on the theoretically well-founded differential geometrical properties of the space of multivariate normal distributions, where it is possible to define an affine-invariant Riemannian metric and express statistics on the manifold of symmetric positive definite matrices. In this paper, we focus on the contribution of these tools to the anisotropic filtering and regularization of tensor fields. To validate our approach we present promising results on both synthetic and real DT-MRI data.

Multivariate Normal Distributions Parametrized as a Riemannian Symmetric Space

The construction of a distance function between probability distributions is of importance in mathematical statistics and its applications. The distance function based on the Fisher information metric has been studied by a number of statisticians, especially in the case of the multivariate normal distribution (Gaussian) on Rn. It turns out that, except in the case n=1, where the Fisher metric describes the hyperbolic plane, it is difficult to obtain an exact formula for the distance function (although this can be achieved for special families with fixed mean or fixed covariance). We propose to study a slightly different metric on the space of multivariate normal distributions on Rn. Our metric is based on the fundamental idea of parametrizing this space as the Riemannian symmetric space SL(n+1)/SO(n+1). Symmetric spaces are well understood in Riemannian geometry, allowing us to compute distance functions and other relevant geometric data.