Related papers: Sheaves as a Framework for Understanding and Inter…

This document develops general concepts useful for extracting knowledge embedded in large graphs or datasets that have pair-wise relationships, such as cause-effect-type relations. Almost no underlying assumptions are made, other than that…

Many complicated network problems can be easily understood on small networks. Difficulties arise when small networks are combined into larger ones. Fortunately, the mathematical theory of sheaves was constructed to address just this kind of…

There is an interplay between models, specified by variables and equations, and their connections to one another. This dichotomy should be reflected in the abstract as well. Without referring to the models directly -- only that a model…

We introduce a new formal model -- based on the mathematical construct of sheaves -- for representing contradictory information in textual sources. This model has the advantage of letting us (a) identify the causes of the inconsistency; (b)…

This paper provides an overview of the applications of sheaf theory in deep learning, data science, and computer science in general. The primary text of this work serves as a friendly introduction to applied and computational sheaf theory…

A sensor integration framework should be sufficiently general to accurately represent all information sources, and also be able to summarize information in a faithful way that emphasizes important, actionable information. Few approaches…

This chapter explores dynamical structural equation models (DSEMs) and their nonlinear generalizations into sheaves of dynamical systems. It demonstrates these two disciplines on part of the food web in the Bering Sea. The translation from…

Sampling theory has traditionally drawn tools from functional and complex analysis. Past successes, such as the Shannon-Nyquist theorem and recent advances in frame theory, have relied heavily on the application of geometry and analysis.…

Integration of heterogeneous sensors is a challenging problem across a range of applications. Prominent among these are multi-target tracking, where one must combine observations from different sensor types in a meaningful way to track…

Knowledge graph embedding involves learning representations of entities -- the vertices of the graph -- and relations -- the edges of the graph -- such that the resulting representations encode the known factual information represented by…

A Sheaf Neural Network (SNN) is a type of Graph Neural Network (GNN) that operates on a sheaf, an object that equips a graph with vector spaces over its nodes and edges and linear maps between these spaces. SNNs have been shown to have…

We demonstrate that large language models can produce reasonable numerical ratings of the logical consistency of claims. We also outline a mathematical approach based on sheaf theory for lifting such ratings to hypertexts such as laws,…

Many structured systems admit locally consistent descriptions that nevertheless fail to globalize when constrained by an ambient reference or feasibility condition. Diagnosing such failures is naturally an evaluative problem: given a fixed…

Motivated by efforts to incorporate sheaves into networking, we seek to reinterpret pathfinding algorithms in terms of cellular sheaves, using Dijkstra's algorithm as an example. We construct sheaves on a graph with distinguished source and…

In this article, the theory of sheaves is studied from a categorical point of view. This perspective vastly generalizes the usual theory of sheaves of sets to a more abstract setting which allows us to investigate the theory of sheaves with…

We introduce and develop the theory of metric sheaves. A metric sheaf $\A$ is defined on a topological space $X$ such that each fiber is a metric model. We describe the construction of the generic model as the quotient space of the sheaf…

The performance of machine learning models relies heavily on the quality of input data, yet real-world applications often face significant data-related challenges. A common issue arises when curating training data or deploying models: two…

We propose a model for network community detection using topological data analysis, a branch of modern data science that leverages theory from algebraic topology to statistical analysis and machine learning. Specifically, we use cellular…

This book provides an inviting tour through sheaf theory, from the perspective of applied category theory and pitched at a less specialized audience than is typical with introductions to sheaves. The book makes it as easy as possible for…

Sheaf Neural Networks (SNNs) naturally extend Graph Neural Networks (GNNs) by endowing a cellular sheaf over the graph, equipping nodes and edges with vector spaces and defining linear mappings between them. While the attached geometric…