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Topological data analysis is becoming a popular way to study high dimensional feature spaces without any contextual clues or assumptions. This paper concerns itself with one popular topological feature, which is the number of…
The class of statistical manifolds with divisible cubic forms arises from affine differential geometry. We examine the geodesic connectedness of affine connections on this class of statistical manifolds. In information geometry, the…
A statistical manifold is a pseudo-Riemannian manifold endowed with a Codazzi structure. This structure plays an important role in Information Geometry and its related fields, e.g., a statistical model admits this structure with the…
In this paper, we propose a novel space-time geometric representation of human landmark configurations and derive tools for comparison and classification. We model the temporal evolution of landmarks as parametrized trajectories on the…
Imaging systems are represented as linear operators, and their singular value spectra describe the structure recoverable at the operator level. Building on an operator-based information-theoretic framework, this paper introduces a minimal…
Algebraic topology methods have recently played an important role for statistical analysis with complicated geometric structured data such as shapes, linked twist maps, and material data. Among them, \textit{persistent homology} is a…
This survey contains a selection of topics unified by the concept of positive semi-definiteness (of matrices or kernels), reflecting natural constraints imposed on discrete data (graphs or networks) or continuous objects (probability or…
Geodesics become an essential element of the geometry of a semi-Riemannian manifold. In fact, their differences and similarities with the (positive definite) Riemannian case, constitute the first step to understand semi-Riemannian Geometry.…
Modern formulation of Finsler geometry of a manifold M utilizes the equivalence between this geometry and the Riemannian geometry of VTM, the vertical bundle over the tangent bundle of M, treating TM as the base space. We argue that this…
We first introduce a class of divergence measures between power spectral density matrices. These are derived by comparing the suitability of different models in the context of optimal prediction. Distances between "infinitesimally close"…
We study a Riemannian metric on the cone of symmetric positive-definite matrices obtained from the Hessian of the power potential function $(1-\det(X)^\beta)/\beta$. We give explicit expressions for the geodesics and distance function,…
Wasserstein distance, especially among symmetric positive-definite matrices, has broad and deep influences on development of artificial intelligence (AI) and other branches of computer science. A natural idea is to describe the geometry of…
Representations in the form of Symmetric Positive Definite (SPD) matrices have been popularized in a variety of visual learning applications due to their demonstrated ability to capture rich second-order statistics of visual data. There…
Symmetric Positive Definite (SPD) matrices have been widely used in medical data analysis and a number of different Riemannian met-rics were proposed to compute with them. However, there are very few methodological principles guiding the…
Riemannian neural networks, which extend deep learning techniques to Riemannian spaces, have gained significant attention in machine learning. To better classify the manifold-valued features, researchers have started extending Euclidean…
A general model for geometric structures on differentiable manifolds is obtained by deforming infinitesimal symmetries. Specifically, this model consists of a Lie algebroid, equipped with an affine connection compatible with the Lie…
Geometric graphs are a special kind of graph with geometric features, which are vital to model many scientific problems. Unlike generic graphs, geometric graphs often exhibit physical symmetries of translations, rotations, and reflections,…
We study the Hilbert geometry induced by the Siegel disk domain, an open bounded convex set of complex square matrices of operator norm strictly less than one. This Hilbert geometry yields a generalization of the Klein disk model of…
The differential-geometric structure of the manifold of smooth shapes is applied to the theory of shape optimization problems. In particular, a Riemannian shape gradient with respect to the first Sobolev metric and the Steklov-Poincar\'{e}…
Planar homography, with eight degrees of freedom (DOFs), is fundamental in numerous computer vision tasks. While the positional offsets of four corners are widely adopted (especially in neural network predictions), this parameterization…