Quantization and clustering on Riemannian manifolds with an application to air traffic analysis

A Simple Approximation Method for the Fisher-Rao Distance between Multivariate Normal Distributions

We present a simple method to approximate the Fisher-Rao distance between multivariate normal distributions based on discretizing curves joining normal distributions and approximating the Fisher-Rao distances between successive nearby normal distributions on the curves by the square roots of their Jeffreys divergences. We consider experimentally the linear interpolation curves in the ordinary, natural, and expectation parameterizations of the normal distributions, and compare these curves with a curve derived from the Calvo and Oller's isometric embedding of the Fisher-Rao d-variate normal manifold into the cone of (d+1)×(d+1) symmetric positive-definite matrices. We report on our experiments and assess the quality of our approximation technique by comparing the numerical approximations with both lower and upper bounds. Finally, we present several information-geometric properties of Calvo and Oller's isometric embedding.

The Geometry of the Generalized Gamma Manifold and an Application to Medical Imaging

The Fisher information metric provides a smooth family of probability measures with a Riemannian manifold structure, which is an object in information geometry. The information geometry of the gamma manifold associated with the family of gamma distributions has been well studied. However, only a few results are known for the generalized gamma family that adds an extra shape parameter. The present article gives some new results about the generalized gamma manifold. This paper also introduces an application in medical imaging that is the classification of Alzheimer’s disease population. In the medical field, over the past two decades, a growing number of quantitative image analysis techniques have been developed, including histogram analysis, which is widely used to quantify the diffuse pathological changes of some neurological diseases. This method presents several drawbacks. Indeed, all the information included in the histogram is not used and the histogram is an overly simplistic estimate of a probability distribution. Thus, in this study, we present how using information geometry and the generalized gamma manifold improved the performance of the classification of Alzheimer’s disease population.