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In this paper we introduce a general framework for the study of limits of relational structures in general and graphs in particular, which is based on a combination of model theory and (functional) analysis. We show how the various…
Motivated by the studies of twisted generalized Reed-Solomon (TGRS) codes, we initiate the study of twisted elliptic curve codes (TECCs) in this paper. In particular, we study a class of TECCs with one twist. The parity-check matrices of…
We study quantum statistical inference tasks of hypothesis testing and their canonical variations, in order to review relations between their corresponding figures of merit---measures of statistical distance---and demonstrate the crucial…
Comparative analysis of scalar fields in scientific visualization often involves distance functions on topological abstractions. This paper focuses on the merge tree abstraction (representing the nesting of sub- or superlevel sets) and…
Tree-Sliced methods have recently emerged as an alternative to the traditional Sliced Wasserstein (SW) distance, replacing one-dimensional lines with tree-based metric spaces and incorporating a splitting mechanism for projecting measures.…
For $d\ge 2$ and an odd prime power $q$, consider the vector space $\mathbb{F}_q^d$ over the finite field $\mathbb{F}_q$, where the distance between two points $(x_1,\ldots,x_d)$ and $(y_1,\ldots,y_d)$ is defined as $\sum_{i=1}^d…
This paper brings together three distinct theories with the goal of quantifying shape textures with complex morphologies. Distance fields are central objects in shape representation, while topological data analysis uses algebraic topology…
We introduce the notion of multiscale covariance tensor fields (CTF) associated with Euclidean random variables as a gateway to the shape of their distributions. Multiscale CTFs quantify variation of the data about every point in the data…
Energy statistics are estimators of the energy distance that depend on the distances between observations. The idea behind energy statistics is to consider a statistical potential energy that would parallel Newton's gravitational potential…
In this paper, we address the complexity barrier inherent in Fourier-Motzkin elimination (FME) and cylindrical algebraic decomposition (CAD) when eliminating a block of (existential) quantifiers. To mitigate this, we propose exploiting…
In some scientific fields, a scaling is able to modify the topology of an observed object. Our goal in the present work is to introduce a new formalism adapted to the mathematical representation of this kind of phenomenon. To this end, we…
Modelling cosmic voids as spheres in Euclidean space, the notion of a de-Sitter configuration space is introduced. It is shown that a uniform distribution over this configuration space yields a power-law approximating the void size…
This paper introduces a set of numerical methods for Riemannian shape analysis of 3D surfaces within the setting of invariant (elastic) second-order Sobolev metrics. More specifically, we address the computation of geodesics and geodesic…
This paper reviews several Riemannian metrics and evolution equations in the context of diffeomorphic shape analysis. After a short review of of various approaches at building Riemannian spaces of shapes, with a special focus on the…
Building trees to represent or to fit distances is a critical component of phylogenetic analysis, metric embeddings, approximation algorithms, geometric graph neural nets, and the analysis of hierarchical data. Much of the previous…
Visual insights into a wide variety of statistical methods, for both didactic and data analytic purposes, can often be achieved through geometric diagrams and geometrically based statistical graphs. This paper extols and illustrates the…
In the elastic shape analysis approach to shape matching and object classification, plane curves are represented as points in an infinite-dimensional Riemannian manifold, wherein shape dissimilarity is measured by geodesic distance. A…
An ultrametric space or infinity-metric space is defined by a dissimilarity function that satisfies a strong triangle inequality in which every side of a triangle is not larger than the larger of the other two. We show that search in…
Distance measures play an important role in shape classification and data analysis problems. Topological distances based on Reeb graphs and persistence diagrams have been employed to obtain effective algorithms in shape matching and scalar…
Data describing the three-dimensional structure of physical networks is increasingly available, leading to a surge of interest in network science to explore the relationship between the shape and connectivity of physical networks. We…