Behaviors of φ-Exponential Distributions in Wasserstein Geometry and an Evolution Equation

Wasserstein Riemannian Geometry of Positive-definite Matrices∗

The Wasserstein distance on multivariate non-degenerate Gaussian densities is a Riemannian distance. After reviewing the properties of the distance and the metric geodesic, we derive an explicit form of the Riemannian metrics on positive-definite matrices and compute its tensor form with respect to the trace scalar product. The tensor is a matrix, which is the solution of a Lyapunov equation. W...

Some New Characterization Results on Exponential and Related Distributions

Natural gradient via optimal transport I

We study a natural Wasserstein gradient flow on manifolds of probability distributions with discrete sample spaces. We derive the Riemannian structure for the probability simplex from the dynamical formulation of the Wasserstein distance on a weighted graph. We pull back the geometric structure to the parameter space of any given probability model, which allows us to define a natural gradient f...

Using MODIS data for nonlinear hazard analysis of the Middle East aerosols

Aerosols are among the most important of atmospheric pollutants observed like the microscopic particulate matter in the lower parts of the troposphere. The main purpose of this study is introducing a new method based on satellite images processing results and nonlinear analysis (fractal based) to investigate the origin and dynamical mechanism of aerosols distribution in North Africa and the Mid...

Nonuniform Exponential Unstability of Evolution Operators in Banach Spaces

In this paper we consider a nonuniform unstability concept for evolution operators in Banach spaces. The relationship between this concept and the Perron condition is studied. Generalizations to the nonuniform case of some results of Van Minh, Räbiger and Schnaubelt are obtained. The theory we present here is applicable for general time varying linear equations in Banach spaces. 1. Evolution op...