相关论文: Persistence and Sheaves

This article introduces an algorithm to compute the persistent homology of a filtered complex with various coefficient fields in a single matrix reduction. The algorithm is output-sensitive in the total number of distinct persistent…

We investigate combinatorial dynamical systems on simplicial complexes considered as {\em finite topological spaces}. Such systems arise in a natural way from sampling dynamics and may be used to reconstruct some features of the dynamics…

A basic result in the theory of holomorphic functions of several complex variables is the following special case of the work of H. Cartan on the sheaf cohomology on Stein domains ([10], or see [14] or [16] for more modern treatments).

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…

We construct a spectral sequence associated to a stratified space, which computes the compactly supported cohomology groups of an open stratum in terms of the compactly supported cohomology groups of closed strata and the reduced cohomology…

Unravelling the block structure of a network is critical for studying macroscopic features and community-level dynamics. The weighted stochastic block model (WSBM), a variation of the traditional stochastic block model, is designed for…

(Makes a Gamma-acylic coherent resolution of a coherent sheaf on a projection scheme.)

Persistent homology (PH) is a rigorous mathematical theory that provides a robust descriptor of data in the form of persistence diagrams (PDs) which are 2D multisets of points. Their variable size makes them, however, difficult to combine…

We develop persistent homology in the setting of filtrations of (Cech) closure spaces. Examples of filtrations of closure spaces include metric spaces, weighted graphs, weighted directed graphs, and filtrations of topological spaces. We use…

Link prediction is an important learning task for graph-structured data. In this paper, we propose a novel topological approach to characterize interactions between two nodes. Our topological feature, based on the extended persistent…

We set up the theory for a distributed algorithm for computing persistent homology. For this purpose we develop linear algebra of persistence modules. We present bases of persistence modules, and give motivation as for the advantages of…

This paper associates a persistence module to a contact vector field $X$ on the ideal boundary of a Liouville manifold. The persistence module measures the dynamics of $X$ on the region $\Omega$ where $X$ is positively transverse to the…

Let $\omega$ be a differential $q$-form defining a foliation of codimension $q$ in a projective variety. In this article we study the singular locus of $\omega$ in various settings. We relate a certain type of singularities, which we name…

In this paper we present a new approach to computing homology (with field coefficients) and persistent homology. We use concepts from discrete Morse theory, to provide an algorithm which can be expressed solely in terms of simple graph…

Persistent homology is a central methodology in topological data analysis that has been successfully implemented in many fields and is becoming increasingly popular and relevant. The output of persistent homology is a persistence diagram --…

Persistent homology, a central tool of topological data analysis, provides invariants of data called barcodes (also known as persistence diagrams). A barcode is simply a multiset of real intervals. Recent work of Edelsbrunner, Jablonski,…

We propose an approach to learning with graph-structured data in the problem domain of graph classification. In particular, we present a novel type of readout operation to aggregate node features into a graph-level representation. To this…

We study some aspects of the relationship between A^1-homotopy theory and birational geometry. We study the so-called A^1-singular chain complex and zeroth A^1-homology sheaf of smooth algebraic varieties over a field k. We exhibit some…

We study the relation between augmentations and sheaves in the context of framed oriented links. In this set up, we find slightly more sheaves than augmentations. After removing the sporadic sheaves, we construct a bijective correspondence…

We introduce a novel set of observables associated to the rapidly developing field of persistent homology for the quantitative characterization of nuclear collisions and their evolution. Persistent homology allows for the identification of…