相关论文: Field choice problem in persistent homology
For nearly three decades, spatial games have produced a wealth of insights to the study of behavior and its relation to population structure. However, as different rules and factors are added or altered, the dynamics of spatial models often…
Persistent homology probes topological properties from point clouds and functions. By looking at multiple scales simultaneously, one can record the births and deaths of topological features as the scale varies. In this paper we use a…
The intrinsic connection between lattice theory and topology is fairly well established, For instance, the collection of open subsets of a topological subspace always forms a distributive lattice. Persistent homology has been one of the…
We propose a functorial framework for persistent homology based on finite topological spaces and their associated posets. Starting from a finite metric space, we associate a filtration of finite topologies whose structure maps are…
Persistent homology has been widely used to discover hidden topological structures in data across various applications, including music data. To apply persistent homology, a distance or metric must be defined between points in a point cloud…
Persistent homology is a popular tool in Topological Data Analysis. It provides numerical characteristics of data sets which reflect global geometric properties. In order to be useful in practice, for example for feature generation in…
We prove an extension to the simplicial Nerve Lemma which establishes isomorphism of persistent homology groups, in the case where the covering spaces are filtered. While persistent homology is now widely used in topological data analysis,…
In persistent homology analysis, interval modules play a central role in describing the birth and death of topological features across a filtration. In this work, we extend this setting, and propose the use of bipath persistent homology,…
This technical report introduces a novel approach to efficient computation in homological algebra over fields, with particular emphasis on computing the persistent homology of a filtered topological cell complex. The algorithms here…
Persistent homology analysis, a recently developed computational method in algebraic topology, is applied to the study of the phase transitions undergone by the so-called XY-mean field model and by the phi^4 lattice model, respectively. For…
Persistent homology is a mathematical tool used for studying the shape of data by extracting its topological features. It has gained popularity in network science due to its applicability in various network mining problems, including…
Techniques from computational topology, in particular persistent homology, are becoming increasingly relevant for data analysis. Their stable metrics permit the use of many distance-based data analysis methods, such as multidimensional…
This article studies the robust version of persistent homology based on trimming methodology to capture the geometric feature through support of the data in presence of outliers. Precisely speaking, the proposed methodology works when the…
Cohomological ideas have recently been injected into persistent homology and have for example been used for accelerating the calculation of persistence diagrams by the software Ripser. The cup product operation which is available at…
We introduce several geometric notions, including the width of a homology class, to the theory of persistent homology. These ideas provide geometric interpretations of persistence diagrams. Indeed, we give quantitative and geometric…
Algorithms for persistent homology and zigzag persistent homology are well-studied for persistence modules where homomorphisms are induced by inclusion maps. In this paper, we propose a practical algorithm for computing persistence under…
In this paper, we study the persistent homology of the offset filtration of algebraic varieties. We prove the algebraicity of two quantities central to the computation of persistent homology. Moreover, we connect persistent homology and…
We discuss and review recent developments in the area of applied algebraic topology, such as persistent homology and barcodes. In particular, we discuss how these are related to understanding more about manifold learning from random point…
This paper is a survey of persistent homology, primarily as it is used in topological data analysis. It includes the theory of persistence modules, as well as stability theorems for persistence barcodes, generalized persistence,…
In this paper we introduce a statistic, the persistent homology transform (PHT), to model surfaces in $\mathbb{R}^3$ and shapes in $\mathbb{R}^2$. This statistic is a collection of persistence diagrams - multiscale topological summaries…