Persistence has proved to be a valuable tool to analyze real world data robustly. Several approaches to persistence have been attempted over time, some topological in flavor, based on the vector space-valued homology functor, other combinatorial, based on arbitrary set-valued functors. To unify the study of topological and combinatorial persistence in a common categorical framework, we give axioms for a generalized rank function on objects in a target category, so that functors to that category induce persistence functions. We port the interleaving and bottleneck distances to this novel framework and generalize classical equalities and inequalities. Unlike sets and vector spaces, in many categories the rank of an object does not identify it up to isomorphism: to preserve information about the structure of persistence modules, we define colorable ranks, persistence diagrams and prove the equality between multicolored bottleneck distance and interleaving distance in semisimple Abelian categories. To illustrate our framework in practice, we give examples of multicolored persistent homology on filtered topological spaces with a group action and labeled point cloud data.
In this paper we provide an explicit connection between level-sets persistence and derived sheaf theory over the real line. In particular we construct a functor from 2-parameter persistence modules to sheaves over $\mathbb{R}$, as well as a functor in the other direction. We also observe that the 2-parameter persistence modules arising from the level sets of Morse functions carry extra structure that we call a Mayer-Vietoris system. We prove classification, barcode decomposition, and stability theorems for these Mayer-Vietoris systems, and we show that the aforementioned functors establish a p…
In this paper we study multidimensional persistence modules [5,13] via what we call tame functors and noise systems. A noise system leads to a pseudo-metric topology on the category of tame functors. We show how this pseudo-metric can be used to identify persistent features of compact multidimensional persistence modules. To count such features we introduce the feature counting invariant and prove that assigning this invariant to compact tame functors is a 1-Lipschitz operation. For 1-dimensional persistence, we explain how, by choosing an appropriate noise system, the feature counting invaria…
We consider the question of defining interleaving metrics on generalized persistence modules over arbitrary preordered sets. Our constructions are functorial, which implies a form of stability for these metrics. We describe a large class of examples, inverse-image persistence modules, which occur whenever a topological space is mapped to a metric space. Several standard theories of persistence and their stability can be described in this framework. This includes the classical case of sublevelset persistent homology. We introduce a distinction between `soft' and `hard' stability theorems. While…
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, and Mrozek suggests an equivalent description of barcodes as functors R -> Mch, where R is the poset category of real numbers and Mch is the category whose objects are sets and whose morphisms are matchings (i.e., partial injective functions). Such functors form a category Mch^R whose morphisms are the natural transformations. Thus, this interpretation of barc…
We believe three ingredients are needed for further progress in persistence and its use: invariants not relying on decomposition theorems to go beyond 1-dimension, outcomes suitable for statistical analysis and a setup adopted for supervised and machine learning. Stable rank, a continuous invariant for multidimensional persistence, was introduced in W. Chacholski et al. - Multidimensional persistence and noise, 2017. In the current paper we continue this work by demonstrating how one builds an efficient computational pipeline around this invariant and uses it in inference in case of one parame…
By invoking the reflection functors introduced by Bernstein, Gelfand, and Ponomarev in 1973, in this paper we define a metric on the space of all zigzag modules of a given length, which we call the reflection distance. We show that the reflection distance between two given zigzag modules of the same length is an upper bound for the $\ell^1$-bottleneck distance between their respective persistence diagrams.
The notion of generalized rank invariant in the context of multiparameter persistence has become an important ingredient for defining interesting homological structures such as generalized persistence diagrams. Naturally, computing these rank invariants efficiently is a prelude to computing any of these derived structures efficiently. We show that the generalized rank over a finite interval $I$ of a $\mathbb{Z}^2$-indexed persistence module $M$ is equal to the generalized rank of the zigzag module that is induced on a certain path in $I$ tracing mostly its boundary. Hence, we can compute the g…
By analogy with methods of Spivak, there is a realization functor which takes a persistence diagram $Y$ in simplicial sets to an extended pseudo-metric space (or ep-metric space) $Re(Y)$. The functor $Re$ has a right adjoint, called the singular functor, which takes an ep-metric space $Z$ to a persistence diagram $S(Z)$. We give an explicit description of $Re(Y)$, and show that it depends only on the $1$-skeleton $sk_{1}Y$ of $Y$. If $X$ is a totally ordered ep-metric space, then there is an isomorphism $Re(V_{\ast}(X)) \cong X$, between the realization of the Vietoris-Rips diagram $V_{\ast}(X…
Although there is no doubt that multi-parameter persistent homology is a useful tool to analyse multi-variate data, efficient ways to compute these modules are still lacking in the available topological data analysis toolboxes. Other issues such as interpretation and visualization of the output remain difficult to solve. Software visualizing multi-parameter persistence diagrams is currently only available for 2-dimensional persistence modules. One of the simplest invariants for a multi-parameter persistence module is its rank invariant, defined as the function that counts the number of linearl…
We first introduce the notion of meta-rank for a 2-parameter persistence module, an invariant that captures the information behind images of morphisms between 1D slices of the module. We then define the meta-diagram of a 2-parameter persistence module to be the Möbius inversion of the meta-rank, resulting in a function that takes values from signed 1-parameter persistence modules. We show that the meta-rank and meta-diagram contain information equivalent to the rank invariant and the signed barcode. This equivalence leads to computational benefits, as we introduce an algorithm for computing th…
We develop a novel hierarchy for zero-dimensional persistence pairs, i.e., connected components, which is capable of capturing more fine-grained spatial relations between persistence pairs. Our work is motivated by a lack of spatial relationships between features in persistence diagrams, leading to a limited expressive power. We build upon a recently-introduced hierarchy of pairs in persistence diagrams that augments the pairing stored in persistence diagrams with information about which components merge. Our proposed hierarchy captures differences in branching structure. Moreover, we show how…
A theory of modules over posets is developed to define computationally feasible, topologically interpretable data structures, in terms of birth and death of homology classes, for persistent homology with multiple real parameters. To replace the noetherian hypothesis in the general setting of modules over posets, a "finitely encoded" condition is defined combinatorially and developed algebraically. It captures topological tameness of persistent homology. Poset-modules satisfying it can be specified by "fringe presentations" that reflect birth-and-death descriptions of persistence. A syzygy theo…
When a category $\mathcal{C}$ satisfies certain conditions, we define the notion of rank invariant for arbitrary poset-indexed functors $F:\mathbf{P} \rightarrow \mathcal{C}$ from a category theory perspective. This generalizes the standard notion of rank invariant as well as Patel's recent extension. Specifically, the barcode of any interval decomposable persistence modules $F:\mathbf{P} \rightarrow \mathbf{vec}$ of finite dimensional vector spaces can be extracted from the rank invariant by the principle of inclusion-exclusion. Generalizing this idea allows freedom of choosing the indexing p…
Invariants of persistence modules defined by order-embeddings
Départment de Mathématiques, Université de Sherbrooke（数学系，舍布鲁克大学）；Bishop’s University（主教大学）；Department of Mathematics, North Carolina State University（数学系，北卡罗来纳州立大学）
We study the decomposition of zero-dimensional persistence modules, viewed as functors valued in the category of vector spaces factorizing through sets. Instead of working directly at the level of vector spaces, we take a step back and first study the decomposition problem at the level of sets. This approach allows us to define the combinatorial notion of rooted subsets. In the case of a filtered metric space $M$, rooted subsets relate the clustering behavior of the points of $M$ with the decomposition of the associated persistence module. In particular, we can identify intervals in such a dec…
An important problem in the field of Topological Data Analysis is defining topological summaries which can be combined with traditional data analytic tools. In recent work Bubenik introduced the persistence landscape, a stable representation of persistence diagrams amenable to statistical analysis and machine learning tools. In this paper we generalise the persistence landscape to multiparameter persistence modules providing a stable representation of the rank invariant. We show that multiparameter landscapes are stable with respect to the interleaving distance and persistence weighted Wassers…
We introduce a refinement of the persistence diagram, the graded persistence diagram. It is the Mobius inversion of the graded rank function, which is obtained from the rank function using the unary numeral system. Both persistence diagrams and graded persistence diagrams are integer-valued functions on the Cartesian plane. Whereas the persistence diagram takes non-negative values, the graded persistence diagram takes values of 0, 1, or -1. The sum of the graded persistence diagrams is the persistence diagram. We show that the positive and negative points in the k-th graded persistence diagram…
A distance between maps via interleavings of relative Sullivan algebras
Shinshu University（信州大学）；Nippon Institute of Technology（日本工业大学）；Graduate School of Medicine, Science and Technology, Shinshu University（信州大学医学部理工学研究科）；Graduate School of Mathematics, Nagoya University（名古屋大学大学院数学研究科）；Faculty of Education, Kochi University（高知大学教育学部）
Shizuoka University（静冈大学）；Institute for Advanced Study, Kyoto University（京都大学高等研究院）；Osaka Central Advanced Mathematical Institute（大阪中央先进数学研究所）；École Polytechnique, Institut Polytechnique de Paris（巴黎理工学院巴黎综合理工学院）；Graduate School of Science, Kyoto University（京都大学理学研究科）
The concept of topological persistence, introduced recently in computational topology, finds applications in studying a map in relation to the topology of its domain. Since its introduction, it has been extended and generalized in various directions. However, no attempt has been made so far to extend the concept of topological persistence to a generalization of `maps' such as cocycles which are discrete analogs of closed differential forms, a well known concept in differential geometry. We define a notion of topological persistence for 1-cocycles in this paper and show how to compute its relev…
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…
The persistence diagram of Cohen-Steiner, Edelsbrunner, and Harer was recently generalized by Patel to the case of constructible persistence modules with values in a symmetric monoidal category with images. Patel also introduced a distance for persistence diagrams, the erosion distance. Motivated by this work, we extend the erosion distance to a distance of rank invariants of generalized persistence modules by using the generalization of the interleaving distance of Bubenik, de Silva, and Scott as a guideline. This extension of the erosion distance also gives, as a special case, a distance for…
Identifying and representing clusters in time-varying network data is of particular importance when studying collective behaviors emerging in nature, in mobile device networks or in social networks. Based on combinatorial, categorical, and persistence theoretic viewpoints, we establish a stable functorial pipeline for the summarization of the evolution of clusters in a time-varying network. We first construct a complete summary of the evolution of clusters in a given time-varying network over a set of entities $X$ of which takes the form of a formigram. This formigram can be understood as a ce…
Characterizing the dynamics of time-evolving data within the framework of topological data analysis (TDA) has been attracting increasingly more attention. Popular instances of time-evolving data include flocking/swarming behaviors in animals and social networks in the human sphere. A natural mathematical model for such collective behaviors is a dynamic point cloud, or more generally a dynamic metric space (DMS). In this paper we extend the Rips filtration stability result for (static) metric spaces to the setting of DMSs. We do this by devising a certain three-parameter "spatiotemporal" filtra…
An ideal invariant for multiparameter persistence would be discriminative, computable and stable. In this work we analyse the discriminative power of a stable, computable invariant of multiparameter persistence modules: the fibered bar code. The fibered bar code is equivalent to the rank invariant and encodes the bar codes of the 1-parameter submodules of a multiparameter module. This invariant is well known to be globally incomplete. However in this work we show that the fibered bar code is locally complete for finitely presented modules by showing a local equivalence of metrics between the i…
Suppose F is a special Gamma-space equipped with a natural transformation to the infinite symmetric power functor. Segal's infinite loop space machine associates with F a spectrum, denoted kF, equipped with a map to the integral Eilenberg-Mac Lane spectrum. In our previous work we constructed a filtration of kF by a sequence of spectra, which we called the "stable rank filtration of F". In this paper we give a new construction of the stable rank filtration. The new construction is combinatorial in nature and avoids the process of stabilization. In particular, we construct a sequence of special…
