Related papers: Topological Applications of p-Adic Divergence and …
Let $G$ be an infinite, edge- and vertex-weighted graph with certain reasonable restrictions. We construct the heat kernel of the associated Laplacian using an adaptation of the parametrix approach due to Minakshisundaram-Pleijel in the…
We consider the heat operator acting on differential forms on spaces with complete and incomplete edge metrics. In the latter case we study the heat operator of the Hodge Laplacian with algebraic boundary conditions at the edge singularity.…
The $p$-Laplacian for graphs, as well as the vertex Laplace operator and the hyperedge Laplace operator for the general setting of oriented hypergraphs, are generalized. In particular, both a vertex $p$-Laplacian and a hyperedge…
This overview article makes the case for how topological concepts can enrich research in machine learning. Using the Euler Characteristic Transform (ECT), a geometrical-topological invariant, as a running example, I present different use…
In a previous work of the authors, a result to algorithmically compute the topology types of the level curves of an algebraic surface, is given. From this result, here we derive applications based on level curves to determine some…
Several algebraic and topological properties of subgradient projection operators are investigated and various examples are provided. Connections with Moreau's proximity operator are also made and acceleration schemes for subgradient…
Graph Laplacians computed from weighted adjacency matrices are widely used to identify geometric structure in data, and clusters in particular; their spectral properties play a central role in a number of unsupervised and semi-supervised…
We consider the Hodge Laplacian on manifolds with incomplete edge singularities, with infinite dimensional von Neumann spaces and intricate elliptic boundary value theory. We single out a class of its algebraic self-adjoint extensions. Our…
Laplace operators perturbed by meromorphic potential on the Riemann and separated type Klein surfaces are constructed and their indices are calculated by two different ways. The topological expressions for the indices are obtained from the…
In this paper, we introduce heat kernel coupling (HKC) as a method of constructing multimodal spectral geometry on weighted graphs of different size without vertex-wise bijective correspondence. We show that Laplacian averaging can be…
This paper introduces gradient, adjoint, and $p$-Laplacian definitions for oriented hypergraphs as well as differential and averaging operators for unoriented hypergraphs. These definitions are used to define gradient flows in the form of…
We study the use of the Euler characteristic for multiparameter topological data analysis. Euler characteristic is a classical, well-understood topological invariant that has appeared in numerous applications, including in the context of…
The tempered fundamental group of a p-adic analytic variety classifies \'etale coverings that become topological coverings (for the Berkovich topology) after finite \'etale base change. I study what can be recovered from the tempered…
The fermionic topological charge of lattice gauge fields, given in terms of a spectral flow of the Hermitian Wilson--Dirac operator, or equivalently, as the index of Neuberger's lattice Dirac operator, is shown to have analogous properties…
Discrete versions of the Laplace and Dirac operators haven been studied in the context of combinatorial models of statistical mechanics and quantum field theory. In this paper we introduce several variations of the Laplace and Dirac…
We study a new paradigm of knowledge transfer that aims at encoding graph topological information into graph neural networks (GNNs) by distilling knowledge from a teacher GNN model trained on a complete graph to a student GNN model…
We give some remarks on twisted determinant line bundles and Chern-Simons topological invariants associated with real hyperbolic manifolds. Index of a twisted Dirac operator is derived. We discuss briefly application of obtained results in…
In this paper we study a Kahler structure on finite points. In particular, we study the edge Laplacian of a graph twisted by the Kahler structure introduced in this paper. We also discuss a metric aspect from a twisted holomorphic…
Graph disaggregation is a technique used to address the high cost of computation for power law graphs on parallel processors. The few high-degree vertices are broken into multiple small-degree vertices, in order to allow for more efficient…
We study the behavior of the heat kernel of the Hodge Laplacian on a contact manifold endowed with a family of Riemannian metrics that blow-up the directions transverse to the contact distribution. We apply this to analyze the behavior of…