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 descriptions of the "life span" or "persistence" of a homology class. As a case study, we analyze the power filtration on unweighted graphs, and provide explicit bounds for the life spans of homology classes in persistence diagrams in all dimensions.
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…
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.
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.
We consider sequences of absolute and relative homology and cohomology groups that arise naturally for a filtered cell complex. We establish algebraic relationships between their persistence modules, and show that they contain equivalent information. We explain how one can use the existing algorithm for persistent homology to process any of the four modules, and relate it to a recently introduced persistent cohomology algorithm. We present experimental evidence for the practical efficiency of the latter algorithm.
TDA (topological data analysis) is a relatively new area of research related to importing classical ideas from topology into the realm of data analysis. Under the umbrella term TDA, there falls, in particular, the notion of persistent homology, which can be described in a nutshell, as the study of scale dependent homological invariants of datasets. In these notes, we provide a terse self contained description of the main ideas behind the construction of persistent homology as an invariant feature of datasets, and its stability to perturbations.
This paper proposes a stable volume and a stable volume variant, referred to as a stable sub-volume, for more reliable data analysis using persistent homology. In prior research, an optimal cycle and similar ideas have been proposed to identify the homological structure corresponding to each birth-death pair in a persistence diagram. While this is helpful for data analysis using persistent homology, the results are sensitive to noise. In this paper, stable volumes and stable sub-volumes are proposed to solve this problem. For a special case, we prove that a stable volume is the robust part of …
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…
By general case we mean methods able to process simplicial sets and chain complexes not of finite type. A filtration of the object to be studied is the heart of both subjects persistent homology and spectral sequences. In this paper we present the complete relation between them, both from theoretical and computational points of view. One of the main contributions of this paper is the observation that a slight modification of our previous programs computing spectral sequences is enough to compute also persistent homology. By inheritance from our spectral sequence programs, we obtain for free pe…
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.
We define persistent homology groups over any set of spaces which have inclusions defined so that the corresponding directed graph between the spaces is acyclic, as well as along any subgraph of this directed graph. This method simultaneously generalizes standard persistent homology, zigzag persistence and multidimensional persistence to arbitrary directed acyclic graphs, and it also allows the study of more general families of topological spaces or point-cloud data. We give an algorithm to compute the persistent homology groups simultaneously for all subgraphs which contain a single source an…
The literature in persistent homology often refers to a "structure theorem for finitely generated graded modules over a graded principal ideal domain". We clarify the nature of this structure theorem in this context.
In this paper we study how randomly generated knots occupy a volume of space using topological methods. To this end, we consider the evolution of the first homology of an immersed metric neighbourhood of a knot's embedding for growing radii. Specifically, we extract features from the persistent homology of the Vietoris-Rips complexes built from point clouds associated to knots. Statistical analysis of our data shows the existence of increasing correlations between geometric quantities associated to the embedding and persistent homology based features, as a function of the knots' lengths. We fu…
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.
Persistent homology is a multiscale method for analyzing the shape of sets and functions from point cloud data arising from an unknown distribution supported on those sets. When the size of the sample is large, direct computation of the persistent homology is prohibitive due to the combinatorial nature of the existing algorithms. We propose to compute the persistent homology of several subsamples of the data and then combine the resulting estimates. We study the risk of two estimators and we prove that the subsampling approach carries stable topological information while achieving a great redu…
Let $P$ be a finite poset. We will show that for any reasonable $P$-persistent object $X$ in the category of finite topological spaces, there is a $P-$ weighted graph, whose clique complex has the same $P$-persistent homology as $X$.
The subject of persistent homology has vitalized applications of algebraic topology to point cloud data and to application fields far outside the realm of pure mathematics. The area has seen several fundamentally important results that are rooted in choosing a particular algebraic foundational theory to describe persistent homology, and applying results from that theory to prove useful and important results. In this survey paper, we shall examine the various choices in use, and what they allow us to prove. We shall also discuss the inherent differences between the choices people use, and specu…
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…
The Čech and Rips constructions of persistent homology are stable with respect to perturbations of the input data. However, neither is robust to outliers, and both can be insensitive to topological structure of high-density regions of the data. A natural solution is to consider 2-parameter persistence. This paper studies the stability of 2-parameter persistent homology: We show that several related density-sensitive constructions of bifiltrations from data satisfy stability properties accommodating the addition and removal of outliers. Specifically, we consider the multicover bifiltration, She…
University of South Africa（南非大学）；African Institute for Mathematical Sciences, Research and Innovation Centre（非洲数学科学研究所研究与创新中心）；Post Urban Ventures；the University of Queensland（昆士兰大学）
We consider generic curves in R^2, i.e. generic C^1 functions f from S^1 to R^2. We analyze these curves through the persistent homology groups of a filtration induced on S^1 by f. In particular, we consider the question whether these persistent homology groups uniquely characterize f, at least up to re-parameterizations of S^1. We give a partially positive answer to this question. More precisely, we prove that f=goh, where h:S^1-> S^1 is a C^1-diffeomorphism, if and only if the persistent homology groups of sof and sog coincide, for every s belonging to the group Sigma_2 generated by reflecti…
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.
Classical persistent homology is a powerful mathematical tool for shape comparison. Unfortunately, it is not tailored to study the action of transformation groups that are different from the group Homeo(X) of all self-homeomorphisms of a topological space X. This fact restricts its use in applications. In order to obtain better lower bounds for the natural pseudo-distance d_G associated with a subgroup G of Homeo(X), we need to adapt persistent homology and consider G-invariant persistent homology. Roughly speaking, the main idea consists in defining persistent homology by means of a set of ch…
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 machine learning, it needs to be effectively computable. Classical homology is a computable topological invariant because of the Mayer-Vietoris exact and spectral sequences associated to coverings of a space. We state and prove versions of the Mayer-Vietoris theorem for persistent homology under mild and commonplace assumptions. This is done through the use of a …
We introduce a fractal dimension for a metric space defined in terms of the persistent homology of extremal subsets of that space. We exhibit hypotheses under which this dimension is comparable to the upper box dimension; in particular, the dimensions coincide for subsets of $\mathbb{R}^2$ whose upper box dimension exceeds $1.5.$ These results are related to extremal questions about the number of persistent homology intervals of a set of $n$ points in a metric space.
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.
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…
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…
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.
Graz University of Technology（格拉茨工业大学）；University of Modena and Reggio Emilia（摩德纳-雷焦艾米利亚大学）；University of Utah（犹他大学）
Topological data analysis and its main method, persistent homology, provide a toolkit for computing topological information of high-dimensional and noisy data sets. Kernels for one-parameter persistent homology have been established to connect persistent homology with machine learning techniques. We contribute a kernel construction for multi-parameter persistence by integrating a one-parameter kernel weighted along straight lines. We prove that our kernel is stable and efficiently computable, which establishes a theoretical connection between topological data analysis and machine learning for …
This paper shows a mathematical formalization, algorithms and computation software of volume optimal cycles, which are useful to understand geometric features shown in a persistence diagram. Volume optimal cycles give us concrete and optimal homologous structures, such as rings or cavities, on a given data. The key idea is the optimality on $(q + 1)$-chain complex for a $q$th homology generator. This optimality formalization is suitable for persistent homology. We can solve the optimization problem using linear programming. For an alpha filtration on $\mathbb{R}^n$, volume optimal cycles on an…
Over the past two decades, topological data analysis has emerged as a field of applied mathematics with new applications and algorithmic developments appearing rapidly. Two fundamental computations in this field are persistent homology and zigzag homology. In this paper, we show how these computations in the most general case reduce to finding a canonical form of a matrix associated with a type A quiver representation, which in turn can be computed using factorizations of associated matrices. We show how to use arbitrary induced maps on homology for computation, providing a framework that goes…
In this paper we study the relationship between a very classical algebraic object associated to a filtration of spaces, namely a spectral sequence introduced by Leray in the 1940's, and a more recently invented object that has found many applications -- namely, its persistent homology groups. We show the existence of a long exact sequence of groups linking these two objects and using it derive formulas expressing the dimensions of each individual groups of one object in terms of the dimensions of the groups in the other object. The main tool used to mediate between these objects is the notion …
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.
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.
EPFL（洛桑联邦理工学院）；IST Austria (Institute of Science and Technology Austria)（奥地利科学技术研究所）；Mathematical Sciences Institute, The Australian National University（澳大利亚国立大学数学科学研究所）；University of New England（新英格兰大学）；Research School of Physics, The Australian National University（澳大利亚国立大学物理研究学院）
We derive the relationship between the persistent homology barcodes of two dual filtered CW complexes. Applied to greyscale digital images, we obtain an algorithm to convert barcodes between the two different (dual) topological models of pixel connectivity.
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.
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.
Institute of Mathematics for Industry, Kyushu University（九州大学产业数学研究所）；School of Interdisciplinary Mathematical Sciences, Meiji University（明治大学跨学科数理科学学院）
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.
Persistent magnitude homology on finite metric space
Chongqing University of Technology（重庆理工大学）；University of North Carolina at Charlotte（北卡罗来纳大学夏洛特分校）；Yanqi Lake Beijing Institute of Mathematical Sciences and Applications（北京国际数学研究中心）
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.
Computation of degree-1 persistent homology on larger point-clouds using the Reduced Vietoris-Rips filtration
Australian National University（澳大利亚国立大学）；Centre for Systems Biology Dresden（德累斯顿系统生物学中心）；Max Planck Institute of Cell Biology and Genetics（马克斯·普朗克细胞生物学与遗传学研究所）；Max Planck institute for the Physics of Complex Systems（马克斯·普朗克复杂系统物理研究所）
