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Curvature is a fundamental geometric characteristic of smooth spaces. In recent years different notions of curvature have been developed for combinatorial discrete objects such as graphs. However, the connections between such discrete…

Graphs on surfaces is an active topic of pure mathematics belonging to graph theory. It has also been applied to physics and relates discrete and continuous mathematics. In this paper we present a formal mathematical description of the…

In recent years, discrete spaces such as graphs attract much attention as models for physical spacetime or as models for testing the spirit of non-commutative geometry. In this work, we construct the differential algebras for graphs by…

In this paper, we introduce a definition of discrete conformality for triangulated surfaces with flat cone metrics and describe an algorithm for solving the problem of prescribing curvature, that is to deform the metric discrete conformally…

Mapping complex input data into suitable lower dimensional manifolds is a common procedure in machine learning. This step is beneficial mainly for two reasons: (1) it reduces the data dimensionality and (2) it provides a new data…

Recent research on graph embedding has achieved success in various applications. Most graph embedding methods preserve the proximity in a graph into a manifold in an embedding space. We argue an important but neglected problem about this…

A generic method for combinatorial constructions of intrinsic geometrical spaces is presented. It is based on the well known inverse sequences of finite graphs that determine (in the limit) topological spaces. If a pattern of the…

Weighted Gaussian Curvature is an important measurement for images. However, its conventional computation scheme has low performance, low accuracy and requires that the input image must be second order differentiable. To tackle these three…

Given a planar graph derived from a spherical, euclidean or hyperbolic tessellation, one can define a discrete curvature by combinatorial properties, which after embedding the graph in a compact 2d-manifold, becomes the Gaussian curvature.

This paper presents a mathematical framework for analyzing machine learning models through the geometry of their induced partitions. By representing partitions as Riemannian simplicial complexes, we capture not only adjacency relationships…

To definite and compute differential invariants, like curvatures, for triangular meshes (or polyhedral surfaces) is a key problem in CAGD and the computer vision. The Gaussian curvature and the mean curvature are determined by the…

Metric graphs are structures obtained by associating edges in a standard graph with segments of the real line and gluing these segments at the vertices of the graph. The resulting structure has a natural metric that allows for the study of…

We address the "inverse problem" for discrete geometry, which consists in determining whether, given a discrete structure of a type that does not in general imply geometrical information or even a topology, one can associate with it a…

We investigate whether there is a relationship between the discrete Bakry-\'{E}mery curvature of a graph and the smooth curvature of an ambient surface into which the graph is embedded geodesically. As we used weighted graphs as test…

We derive pointwise curvature estimates for graphical mean curvature flows in higher codimensions. To the best of our knowledge, this is the first such estimates without assuming smallness of first derivatives of the defining map. An…

Graph embedding approaches attempt to project graphs into geometric entities, i.e, manifolds. The idea is that the geometric properties of the projected manifolds are helpful in the inference of graph properties. However, if the choice of…

Directed graphs have asymmetric connections, yet the current graph clustering methodologies cannot identify the potentially global structure of these asymmetries. We give a spectral algorithm called di-sim that builds on a dual measure of…

Geometrical properties of spacetime are difficult to study in nonperturbative approaches to quantum gravity like Causal Dynamical Triangulations (CDT), where one uses simplicial manifolds to define the gravitational path integral, instead…

Motivated by the search for geometric observables in nonperturbative quantum gravity, we define a notion of coarse-grained Ricci curvature. It is based on a particular way of extracting the local Ricci curvature of a smooth Riemannian…

We introduce a novel formulation for geometry on discrete points. It is based on a universal differential calculus, which gives a geometric description of a discrete set by the algebra of functions. We expand this mathematical framework so…