F. Cagliari, B. Di Fabio, M. Ferri
A recent result on size functions is extended to higher homology modules: the persistent homology based on a multidimensional measuring function is reduced to a 1-dimensional one. This leads to a stable distance for multidimensional persistent homology. Some reflections on i-essentiality of homological critical values conclude the paper.
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Multidimensional persistence studies topological features of shapes by analyzing the lower level sets of vector-valued functions. The rank invariant completely determines the multidimensional analogue of persistent homology groups. We prove that multidimensional rank invariants are stable with respect to function perturbations. More precisely, we construct a distance between rank invariants such that small changes of the function imply only small changes of the rank invariant. This result can be obtained by assuming the function to be just continuous. Multidimensional stability opens the way t…
Fast computation of persistent homology representatives with involuted persistent homology
针对持久上同调计算无法直接提取同调代表元的问题，提出一种利用上同调信息加速直接持久同调计算的新方法，在Rips复形上能显著快于标准方法地提取代表元。
We investigate to what extent persistent homology benefits from the properties of the usual homology theory.
We show that the interleaving distance between the persistent singular homology and the persistent Čech homology of a homologically locally connected filtration consisting of paracompact Hausdorff spaces is 0.
In this paper, we study further properties and applications of weighted homology and persistent homology. We introduce the Mayer-Vietoris sequence and generalized Bockstein spectral sequence for weighted homology. For applications, we show an algorithm to construct a filtration of weighted simplicial complexes from a weighted network. We also prove a theorem that allows us to calculate the mod $p^2$ weighted persistent homology given some information on the mod $p$ weighted persistent homology.
In this paper we develop the theory of weighted persistent homology. In 1990, Robert J. Dawson was the first to study in depth the homology of weighted simplicial complexes. We generalize the definitions of weighted simplicial complex and the homology of weighted simplicial complex to allow weights in an integral domain $R$. Then we study the resulting weighted persistent homology. We show that weighted persistent homology can tell apart filtrations that ordinary persistent homology does not distinguish. For example, if there is a point considered as special, weighted persistent homology can t…
Relative persistent homology
针对大尺度点云相对持续同调计算效率低的问题，提出相对Delaunay-Čech复形方法，可通过点云嵌入高维的Delaunay复形构建，实现低维欧氏空间中相对持续同调的高效计算。
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, the usual Nerve Lemma does not provide isomorphism of persistent homology groups. Our argument involves some homological algebra: the key point being that although the maps produced in the standard proof of the Nerve Lemma do not commute as maps of chain complexes, the maps they induce on homology do.
An axiomatic treatment of Persistent Homology
本研究构建了适用于任意阶数的持续同调公理化框架，证明了经典同调Eilenberg-Steenrod公理的持续版本及其约化版本的存在性与唯一性，为持续同调提供了公理化基础。
A directed persistent homology theory for dissimilarity functions
本文提出一种针对有向单纯复形、可检测奇数维持有向环的持续同调理论，建立其与经典持续同调的联系并证明稳定性。
Decomposition of Persistent Homology and Spectral Sequences
研究域上过滤链复形的持续同调与谱序列关系，提出基于持续同调分解的方法，证明在一般假设下两者捕获相同信息。
The Euler calculus -- an integral calculus based on Euler characteristic as a valuation on constructible functions -- is shown to be an incisive tool for answering questions about injectivity and invertibility of recent transforms based on persistent homology for shape characterization.
Hypergraph is the most general model for complex networks involving group interactions. Taking the ideas of path homology from Alexander Grigor'yan, Yong Lin, Yuri Muranov and Shing-Tung Yau [18-22], Stephane Bressan, Jingyan Li and the authors of this article introduced embedded homology of hypergraphs [6] in 2019, which has leaded to successful applications in protein-ligand binding network [24, 25] in 2021. A fundamental question arising from practical applications is about the stability of the persistent embedded homology of hypergraphs. In this paper, we prove the stability of the persist…
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, vectorization of persistence barcodes, as well as some applications.
An exact computation of the persistent Betti numbers of a submanifold $X$ of a Euclidean space is possible only in a theoretical setting. In practical situations, only a finite sample of $X$ is available. We show that, under suitable density conditions, it is possible to estimate the multidimensional persistent Betti numbers of $X$ from the ones of a union of balls centered on the sample points; this even yields the exact value in restricted areas of the domain. Using these inequalities we improve a previous lower bound for the natural pseudodistance to assess dissimilarity between the shapes …
We demonstrate that an excision property holds for persistent homology groups. This property holds for a large class of filtrations, and in fact we show that given any filtration on a larger space, we can extend it to a filtration of two subspaces which guarantees that the excision property holds for the triple. This method also applies to the Mayer-Vietoris sequence in persistent homology introduced by DiFabio and Landi in 2011, extending their results to a much larger class of filtrations.
Persistent Mayer homology and persistent Mayer Laplacian
本文将Mayer同调与Mayer拉普拉斯算子引入持久化框架，提出持久Mayer同调和持久Mayer拉普拉斯算子，并通过理论分析与计算实验展示其在拓扑数据分析中的潜力。
Persistent magnitude homology on finite metric space
本文提出持久幅度同调，用于捕捉度量空间的多尺度几何和拓扑特征，并建立了其稳定性结果。
Differential Calculus on Hypergraphs and Mayer-Vietoris Sequences for the Constrained Persistent Homology
本文利用Kouzul复形研究超图离散微分演算，定义受约束（上）同调并证明其Mayer-Vietoris序列的函子性，进而提出受约束持续（上）同调作为计算大型网络高维持续同调的工具。
Computation of degree-1 persistent homology on larger point-clouds using the Reduced Vietoris-Rips filtration
提出一种算法，利用简化 Vietoris-Rips 过滤高效计算低维欧氏空间中更大点云的 1 次持续同调，降低了计算复杂度。
Persistence by Parts: Multiscale Feature Detection via Distributed Persistent Homology
本文提出一种基于广义Mayer–Vietoris原理和胞腔余层、谱序列的分布式持久同调方法，可按密度等特征局部化计算，用于点云多尺度分析，检测标准方法遗漏的不同尺度特征。
Persistent hypergraph homology and its applications
本文提出持续超图模型及新同调 $\hat{H}$ 和高效算法，用于计算持续条形码与 $\hat{H}$ 条形码，并通过分析狒狒、马拉维农村居民、会议、工作场所和高中等群体的面对面交互数据验证其有效性。
While standard persistent homology has been successful in extracting information from metric datasets, its applicability to more general data, e.g. directed networks, is hindered by its natural insensitivity to asymmetry. We study a construction of homology of digraphs due to Grigor'yan, Lin, Muranov and Yau, and extend this construction to the persistent framework. The result, which we call persistent path homology, can provide information about the digraph structure of a directed network at varying resolutions. Moreover, this method encodes a rich level of detail about the asymmetric structu…
Persistent Homology with Selective Rips complexes detects geodesic circles
本文提出利用持久同调和新型选择性Rips复形检测测地空间中的测地圆与瓶颈环路，并基于局部绕数概念证明其能终止1维同调类或产生2维同调类。
The aim of this article is to describe a new perspective on functoriality of persistent homology and explain its intrinsic symmetry that is often overlooked. A data set for us is a finite collection of functions, called measurements, with a finite domain. Such a data set might contain internal symmetries which are effectively captured by the action of a set of the domain endomorphisms. Different choices of the set of endomorphisms encode different symmetries of the data set. We describe various category structures on such enriched data sets and prove some of their properties such as decomposit…
Persistent hyperdigraph homology and persistent hyperdigraph Laplacians
本文提出超有向图同调与拓扑超有向图拉普拉斯算子，并通过过滤引入其持久化版本，以捕获有向结构化数据的多尺度拓扑持久性与同伦形状演化，为拓扑数据分析提供了新工具。
This paper introduces persistent homology, which is a powerful tool to characterize the shape of data using the mathematical concept of topology. We explain the fundamental idea of persistent homology from scratch using some examples. We also review some applications of persistent homology to materials researches and software for persistent homology data analysis. HomCloud, one of persistent homology software, is especially featured in this paper.
An Exposition on the Algebra and Computation of Persistent Homology
该研究以Peter Bubenik等学者提出的、将持续模视为偏序集范畴到数域向量空间范畴函子的现代表征为视角，阐释了经典持续同调矩阵约简算法背后的代数原理。
In recent years, A. Grigor'yan, Y. Lin, Y. Muranov and S.T. Yau [6, 7, 8, 9] constructed a path homology theory for digraphs. Later, S. Chowdhury and F. Memoli [3] studied the persistent path homology for directed networks. In this paper, we generalize the path homology theory for digraphs and construct a weighted path homology for weighted digraphs. We study the persistent weighted path homology for weighted digraphs and detect the effects of the weights on the persistent weighted path homology. We prove a persistent version of a Kunneth-type formula for joins of weighted digraphs.
Persistent homology is a branch of computational algebraic topology that studies shapes and extracts features over multiple scales. In this paper, we present an unsupervised approach that uses persistent homology to study divergent behavior in agricultural point cloud data. More specifically, we build persistence diagrams from multidimensional point clouds, and use those diagrams as the basis to compare and contrast different subgroups of the population. We apply the framework to study the cold hardiness behavior of 5 leading grape cultivars, with real data from over 20 growing seasons. Our re…
Persistent homology has recently emerged as a powerful technique in topological data analysis for analyzing the emergence and disappearance of topological features throughout a filtered space, shown via persistence diagrams. Additionally, (co)sheaves have proven to be powerful instruments in tracking locally defined data across global systems, resulting in innovative applications to network science. In this paper, we combine the topological results of persistent homology and the quantitative data tracking capabilities of cosheaf theory to develop novel techniques in network data flow analysis.…
Harmonic Persistent Homology
本文针对有限单纯复形过滤引入调和持久同调空间，将具体循环子空间关联至条形码各条，证明其在函数微扰下的稳定性，并揭示其与“本质单纯形”概念的联系及在最大化相对本质内容上的最优性。
We study the categorical framework for the computation of persistent homology, without reliance on a particular computational algorithm. The computation of persistent homology is commonly summarized as a matrix theorem, which we call the Matrix Structural Theorem. Any of the various algorithms for computing persistent homology constitutes a constructive proof of the Matrix Structural Theorem. We showthat the Matrix Structural Theorem is equivalent to the Krull-Schmidt property of the category of filtered chain complexes. We separately establish the Krull-Schmidt property by abstract categorica…
A U-match Algorithm for Persistent Relative Homology
本文提出了一种基于两步矩阵约简的U匹配算法，用于高效计算持久相对同调，提升拓扑数据分析在高维噪声数据中的应用能力。
Geometric Obstructions in Finsler Spaces and Torsion-Free Persistent Homology
本文研究Finsler空间中拓扑数据分析与持续同调的关联，通过整数矩阵分解寻找素数p以提取自由部分的孔结构。
We introduce weighted versions of the classical Čech and Vietoris-Rips complexes. We show that a version of the Vietoris-Rips Lemma holds for these weighted complexes and that they enjoy appropriate stability properties. We also give some preliminary applications of these weighted complexes.
Consider the space of continuous functions on a geometric tree $X$ whose persistent homology gives rise to a finite generic barcode $D$. We show that there are exactly as many path connected components in this space as there are merge trees whose barcode is $D$. We find that each component is homotopy equivalent to a configuration space on $X$ with specialized constraints encoded by the merge tree. For barcodes $D$ with either one or two intervals, our method also allows us to compute the homotopy type of this space of functions.
Rigidity of terminal simplices in persistent homology
本文研究持续同调中终端单形在滤值函数扰动下的刚性，给出了一个依赖相邻维度条形码的充分条件，保证终端单形在 $ε$-扰动下保持不变。
Distributed Persistent Homology for 2D Alpha Complexes
本文提出面向二维 alpha 复形的分布式持续同调算法，通过网格覆盖与 Mayer-Vietoris 谱序列实现并行计算，并给出 C++/Open MPI 实现和实验验证。
Persistent Homology: A Pedagogical Introduction with Biological Applications
本文面向初学者系统介绍持续同调的核心概念与实现方法，并以3-1超螺旋DNA结构分析为例，推动其在生物及数据分析中的广泛应用。
Topological data analysis combines machine learning with methods from algebraic topology. Persistent homology, a method to characterize topological features occurring in data at multiple scales is of particular interest. A major obstacle to the wide-spread use of persistent homology is its computational complexity. In order to be able to calculate persistent homology of large datasets, a number of approximations can be applied in order to reduce its complexity. We propose algorithms for calculation of approximate sparse nerves for classes of Dowker dissimilarities including all finite Dowker d…
Graph-based representations of point-cloud data are widely used in data science and machine learning, including epsilon-graphs that contain edges between pairs of data points that are nearer than epsilon and kNN-graphs that connect each point to its k-nearest neighbors. Recently, topological data analysis has emerged as a family of mathematical and computational techniques to investigate topological features of data using simplicial complexes. These are a higher-order generalization of graphs and many techniques such as Vietoris-Rips (VR) filtrations are also parameterized by a distance epsilo…
Path representations in multiparameter persistent homology
针对单参数拓扑分析捕捉差异有限的问题，研究将多参数持续同调中的匹配距离直线推广为单调路径滤过，拓展了多参数拓扑分析的路径表示形式，并提供对应软件实现。
Persistent Topology studies topological features of shapes by analyzing the lower level sets of suitable functions, called filtering functions, and encoding the arising information in a parameterized version of the Betti numbers, i.e. the ranks of persistent homology groups. Initially introduced by considering real-valued filtering functions, Persistent Topology has been subsequently generalized to a multidimensional setting, i.e. to the case of $\R^n$-valued filtering functions, leading to studying the ranks of multidimensional homology groups. In particular, a multidimensional matching dista…
The extension of persistent homology to multi-parameter setups is an algorithmic challenge. Since most computation tasks scale badly with the size of the input complex, an important pre-processing step consists of simplifying the input while maintaining the homological information. We present an algorithm that drastically reduces the size of an input. Our approach is an extension of the chunk algorithm for persistent homology (Bauer et al., Topological Methods in Data Analysis and Visualization III, 2014). We show that our construction produces the smallest multi-filtered chain complex among a…
The distilled Vietoris Rips filtration for persistent homology and a new memory efficient algorithm
针对Vietoris Rips持久同调计算内存占用高的问题，提出蒸馏Vietoris Rips滤过构造，证明其与标准滤过持久同调同构，相关算法高度可并行、内存高效，可适配任意度量空间的点云。
Cofiltrations of spanning trees in multiparameter persistent homology
本文提出利用生成树显式构造多参数持续同调群的生成元，引入持续生成树条件并证明其余过滤的存在性，同时推广生成复形以处理高维情形。
The objective of this study is to detect and quantify the periodic behavior of the signals using topological methods. We propose to use delay-coordinate embeddings as a tool to measure the periodicity of signals. Moreover, we use persistent homology for analyzing the structure of point clouds of delay-coordinate embeddings. A method for finding the appropriate value of delay is proposed based on the autocorrelation function of the signals. We apply this topological approach to wheeze signals by introducing a model based on their harmonic characteristics. Wheeze detection is performed using the…
Rank invariants are a parametrized version of Betti numbers of a space multi-filtered by a continuous vector-valued function. In this note we give a sufficient condition for their finiteness. This condition is sharp for spaces embeddable in R^n.
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 theoretical operations. We use iterated Morse decomposition, which allows us to sidetrack many problems related to the standard discrete Morse theory. In particular, this approach is provably correct in any dimension.
The present lack of a stable method to compare persistent homology groups with torsion is a relevant problem in current research about Persistent Homology and its applications in Pattern Recognition. In this paper we introduce a pseudo-distance d_T that represents a possible solution to this problem. Indeed, d_T is a pseudo-distance between multidimensional persistent homology groups with coefficients in an Abelian group, hence possibly having torsion. Our main theorem proves the stability of the new pseudo-distance with respect to the change of the filtering function, expressed both with resp…
We propose that the recently defined persistent homology dimensions are a practical tool for fractal dimension estimation of point samples. We implement an algorithm to estimate the persistent homology dimension, and compare its performance to classical methods to compute the correlation and box-counting dimensions in examples of self-similar fractals, chaotic attractors, and an empirical dataset. The performance of the $0$-dimensional persistent homology dimension is comparable to that of the correlation dimension, and better than box-counting.
We introduce Cubical Ripser for computing persistent homology of image and volume data (more precisely, weighted cubical complexes). To our best knowledge, Cubical Ripser is currently the fastest and the most memory-efficient program for computing persistent homology of weighted cubical complexes. We demonstrate our software with an example of image analysis in which persistent homology and convolutional neural networks are successfully combined. Our open-source implementation is available online.
The classical Künneth formula in algebraic topology describes the homology of a product space in terms of that of its factors. In this paper, we prove Künneth-type theorems for the persistent homology of the categorical and tensor product of filtered spaces. That is, we describe the persistent homology of these product filtrations in terms of that of the filtered components. In addition to comparing the two products, we present two applications in the setting of Vietoris-Rips complexes: one towards more efficient algorithms for product metric spaces with the maximum metric, and the other recov…
In this paper we extend the notion of the Euler characteristic to persistent homology and give the relationship between the Euler integral of a function and the Euler characteristic of the function's persistent homology. We then proceed to compute the expected Euler integral of a Gaussian random field using the Gaussian kinematic formula and obtain a simple closed form expression. This results in the first explicitly computable mean of a quantitative descriptor for the persistent homology of a Gaussian random field.
This paper aims to introduce a filtration analysis of sampled maps based on persistent homology, providing a new method for reconstructing the underlying maps. The key idea is to extend the definition of homology induced maps of correspondences using the framework of quiver representations. Our definition of homology induced maps is given by most persistent direct summands of representations, and the direct summands uniquely determine a persistent homology. We provide stability theorems of this process and show that the output persistent homology of the sampled map is the same as that of the u…
Given a metric space $X$ and a subspace $A\subset X$, we prove $A$ can generate various algebraic elements in persistent homology of $X$. We call such elements (algebraic) footprints of $A$. Our results imply that footprints typically appear in dimensions above the dimension of $A$. Higher-dimensional persistent homology thus encodes lower-dimensional geometric features of $X$. We pay special attention to a specific type of geodesics in a geodesic surface $X$ called geodesic circles. We explain how they may generate non-trivial odd-dimensional and two-dimensional footprints. In particular, we …
Persistent homology is a topological feature used in a variety of applications such as generating features for data analysis and penalizing optimization problems. We develop an approach to accelerate persistent homology computations performed on many similar filtered topological spaces which is based on updating associated matrix factorizations. Our approach improves the update scheme of Cohen-Steiner, Edelsbrunner, and Morozov for permutations by additionally handling addition and deletion of cells in a filtered topological space and by processing changes in a single batch. We show that the c…
We describe an approach to bounded-memory computation of persistent homology and betti barcodes, in which a computational state is maintained with updates introducing new edges to the underlying neighbourhood graph and percolating the resulting changes into the simplex stream feeding the persistence algorithm. We further discuss the memory consumption and resulting speed and complexity behaviours of the resulting algorithm.
In this paper we describe a model based on persistent homology that describes interactions between mathematicians in terms of collaborations. Some ideas from classical data analysis are used.
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