Related papers: On the Structure of Isometrically Embeddable Metri…
It is shown that a possibly irreversible $C^2$ Finsler metric on the torus, or on any other compact Euclidean space form, whose geodesics are straight lines is the sum of a flat metric and a closed $1$-form. This is used to prove that if…
The past few years have seen a revived interest in quantum geometrical characterizations of band structures due to the rapid development of topological insulators and semi-metals. Although the metric tensor has been connected to many…
In several works, Mendel and Naor have introduced and developed theory surrounding a nonlinear expansion constant similar to the spectral gap for sequences of graphs, in which one considers embeddings of a graph $G$ into a metric space $X$…
Over the past two decades the theory of the Weil-Petersson metric has been extended to general Teichm\"uller spaces of infinite type, including for example the universal Teichm\"uller space. In this paper we give a survey of the main…
Learning faithful graph representations as sets of vertex embeddings has become a fundamental intermediary step in a wide range of machine learning applications. The quality of the embeddings is usually determined by how well the geometry…
A framework (a straight-line embedding of a graph into a normed space allowing edges to cross) is globally rigid if any other framework with the same edge lengths with respect to the chosen norm is an isometric copy. We investigate global…
An ultrametric topology formalizes the notion of hierarchical structure. An ultrametric embedding, referred to here as ultrametricity, is implied by a hierarchical embedding. Such hierarchical structure can be global in the data set, or…
Constructing latent vector representation for nodes in a network through embedding models has shown its practicality in many graph analysis applications, such as node classification, clustering, and link prediction. However, despite the…
A new geometrical definition of naturally reductive Finsler manifold using geodeic graph is proposed, with a possible generalization. Based on a construction from a recent paper by the authors, Finsler metrics based on naturally reductive…
Let X be quasi-isometric to either the mapping class group equipped with the word metric, or to Teichmuller space equipped with either the Teichmuller metric or the Weil-Petersson metric. We introduce a unified approach to study the coarse…
Classic grammars and regular expressions can be used for a variety of purposes, including parsing, intent detection, and matching. However, the comparisons are performed at a structural level, with constituent elements (words or characters)…
Real-world graphs have inherently complex and diverse topological patterns, known as topological heterogeneity. Most existing works learn graph representation in a single constant curvature space that is insufficient to match the complex…
Representing graphs as sets of node embeddings in certain curved Riemannian manifolds has recently gained momentum in machine learning due to their desirable geometric inductive biases, e.g., hierarchical structures benefit from hyperbolic…
We show the existence of generalized clusters of a finite or even infinite number of sets, with minimal total perimeter and given total masses, in metric measure spaces homogeneous with respect to a group acting by measure preserving…
This paper establishes a foundational framework for geometric learning in weighted projective spaces $\mathbb{P}_{\mathbb{q}}$ by introducing a hierarchical clustering algorithm governed by Finsler geometry. We define a scaling-invariant…
Although our main interest here is developing an appropriate analog, for diffeological vector pseudo-bundles, of a Riemannian metric, a significant portion is dedicated to continued study of the gluing operation for pseudo-bundles…
In this paper we introduce a graph structure, called subspace sum graph $\mathcal{G}(\mathbb{V})$ on a finite dimensional vector space $\mathbb{V}$ where the vertex set is the collection of non-trivial proper subspaces of a vector space and…
Many procedures in science, engineering and medicine produce data in the form of geometric shapes. Mathematically, a shape can be modeled as an un-parameterized immersed sub-manifold, which is the notion of shape used here. Endowing shape…
In recent work, the authors developed a simple method of constructing topological spaces from certain well-behaved partially ordered sets -- those coming from sequences of relations between finite sets. This method associates a given poset…
We review some basic concepts related to convex real projective structures from the differential geometry point of view. We start by recalling a Riemannian metric which originates in the study of affine spheres using the Blaschke connection…