Related papers: On the geometry of generalized Gaussian distributi…
A finite dimensional-system whose physics is governed by a Gaussian distribution can be regarded as a subsystem of an infinite dimensional-underlying system described by a uniform distribution on the (infinite dimensional) sphere. In turn,…
The space of Gaussian measures on a Euclidean space is geodesically convex in the $L^2$-Wasserstein space. This space is a finite dimensional manifold since Gaussian measures are parameterized by means and covariance matrices. By…
The Riemannian metric on the manifold of positive definite matrices is defined by a kernel function $\phi$ in the form $K_D^\phi(H,K)=\sum_{i,j}\phi(\lambda_i,\lambda_j)^{-1} Tr P_iHP_jK$ when $\sum_i\lambda_iP_i$ is the spectral…
On a Riemannian manifold with a positive lower bound on the Ricci tensor, the distance of isoperimetric sets from geodesic balls is quantitatively controlled in terms of the gap between the isoperimetric profile of the manifold and that of…
The Riemannian geometry is one of the main theoretical pieces in Modern Mathematics and Physics. The study of Riemann Geometry in the relevant literature is performed by using a well defined analytical path. Usually it starts from the…
This work studies the entropic regularization formulation of the 2-Wasserstein distance on an infinite-dimensional Hilbert space, in particular for the Gaussian setting. We first present the Minimum Mutual Information property, namely the…
We formulate the Riemannian calculus of the probability set embedded with $L^2$-Wasserstein metric. This is an initial work of transport information geometry. Our investigation starts with the probability simplex (probability manifold)…
Comparing probability distributions is a core challenge across the natural, social, and computational sciences. Existing methods, such as Maximum Mean Discrepancy (MMD), struggle in high-dimensional and non-compact domains. Here we…
The space of probability distributions on a given sample space possesses natural geometric properties. For example, in the case of a smooth parametric family of probability distributions on the real line, the parameter space has a…
We give a simple probabilistic description of a transition between two states which leads to a generalized escort distribution. When the parameter of the distribution varies, it defines a parametric curve that we call an escort-path. The…
Quantum information measures such as the entropy and the mutual information find applications in physics, e.g., as correlation measures. Generalizing such measures based on the R\'enyi entropies is expected to enhance their scope in…
Building on the recently introduced notion of quantum Ricci curvature and motivated by considerations in nonperturbative quantum gravity, we advocate a new, global observable for curved metric spaces, the curvature profile. It is obtained…
Estimating statistical properties is fundamental in statistics and computer science. In this paper, we propose a unified quantum algorithm framework for estimating properties of discrete probability distributions, with estimating R\'enyi…
We propose a one-parameter family \ $\mathbb{R}_q$ \ of deformations of the reals, which is motivated by the generalized additivity of the Tsallis entropy. We introduce a generalized multiplication which is distributive with respect to the…
This paper is concerned with the study of a circular random distribution called geodesic Normal distribution recently proposed for general manifolds. This distribution, parameterized by two real numbers associated to some specific location…
We study the statistical geometry of random chords on n-dimensional spheres by deriving explicit analytical expressions for the chord length distribution and its associated structural properties. A critical threshold emerges at dimension…
We introduce diffusion geometry as a new framework for geometric and topological data analysis. Diffusion geometry uses the Bakry-Emery $\Gamma$-calculus of Markov diffusion operators to define objects from Riemannian geometry on a wide…
Embedding complex objects as vectors in low dimensional spaces is a longstanding problem in machine learning. We propose in this work an extension of that approach, which consists in embedding objects as elliptical probability…
The so-called $q$-z-\textit{R\'enyi Relative Entropies} provide a huge two-parameter family of relative entropies which includes almost all well-known examples of quantum relative entropies for suitable values of the parameters. In this…
The family of q-Gaussian and q-exponential probability densities fit the statistical behavior of diverse complex self-similar non-equilibrium systems. These distributions, independently of the underlying dynamics, can rigorously be obtained…