Related papers: Geometry of Similarity Comparisons
A linear transformation f(S) of configurational entropy with length scale dependent coefficients as a measure of spatial inhomogeneity is considered. When a final pattern is formed with periodically repeated initial arrangement of point…
This paper is a continuation of the papers [2,3,4,5,6]. In this paper the osculating spaces of arbitrary order of a manifold embedded in Euclidean space are considered. A better estimation of their dimensions as well as the description of…
The study of the topological structure of complex networks has fascinated researchers for several decades, and today we have a fairly good understanding of the types and reoccurring characteristics of many different complex networks.…
Fr\'echet regression extends classical regression methods to non-Euclidean metric spaces, enabling the analysis of data relationships on complex structures such as manifolds and graphs. This work establishes a rigorous theoretical analysis…
The similarity between objects is significant in a broad range of areas. While similarity can be measured using off-the-shelf distance functions, they may fail to capture the inherent meaning of similarity, which tends to depend on the…
Uniform continuity bounds on entropies are generally expressed in terms of a single distance measure between a pair of probability distributions or quantum states, typically, the total variation distance or trace distance. However, if an…
WWe define the notion of a random metric space and prove that with probability one such a space is isometricto the Urysohn universal metric space. The main technique is the study of universal and random distance matrices; we relate the…
This paper studies the properties of a new lower bound for the natural pseudo-distance. The natural pseudo-distance is a dissimilarity measure between shapes, where a shape is viewed as a topological space endowed with a real-valued…
Different types of two- and three-dimensional representations of a finite metric space are studied that focus on the accurate representation of the linear order among the distances rather than their actual values. Lower and upper bounds for…
A variety of statistics based on sample spacings has been studied in the literature for testing goodness-of-fit to parametric distributions. To test the goodness-of-fit to a nonparametric class of univariate shape-constrained densities,…
Various kinds of data are routinely represented as discrete probability distributions. Examples include text documents summarized by histograms of word occurrences and images represented as histograms of oriented gradients. Viewing a…
Starting from the equations of motion in a 1 + 1 static, diagonal, Lorentzian spacetime, such as the Schwarzschild radial line element, I find another metric, but with Euclidean signature, which produces the same geodesics x(t). This…
We investigate a relations of almost isometric embedding and almost isometry between metric spaces and prove that with respect to these relations: (1) There is a countable universal metric space. (2) There may exist fewer than continuum…
Two well studied invariants of a complex projective variety are the unit Euclidean distance degree and the generic Euclidean distance degree. These numbers give a measure of the algebraic complexity for "nearest" point problems of the…
Distance geometry explores the properties of distance spaces that can be exactly represented as the pairwise Euclidean distances between points in $\mathbb{R}^d$ ($d \geq 1$), or equivalently, distance spaces that can be isometrically…
This writeup describes ongoing work on designing and testing a certain family of correspondences between compact metric spaces that we call \emph{embedding-projection correspondences} (EPCs). Of particular interest are EPCs between spheres…
Following Ghomi and Tabachnikov we study topological obstructions to totally skew embeddings of a smooth manifold M in Euclidean spaces. This problem is naturally related to the question of estimating the geometric dimension of the stable…
We give new estimates for the extrinsic radius of compact hypersurfaces of the Euclidean space and the open hemisphere in terms of high order mean curvatures. Then we prove pinching results corresponding to theses estimates. We show that…
Persistence diagrams are objects that play a central role in topological data analysis. In the present article, we investigate the local and global geometric properties of spaces of persistence diagrams. In order to do this, we construct a…
Many graph processing algorithms require determination of shortest-path distances between arbitrary numbers of node pairs. Since computation of exact distances between all node-pairs of a large graph, e.g., 10M nodes and up, is…