When persistence diagrams are formalized as the Möbius inversion of the birth–death function, they naturally generalize to the multi-parameter setting and enjoy many of the key properties, such as stability, that we expect in applications. The direct definition in the 2-parameter setting, and the corresponding brute-force algorithm to compute them, requireΩ(n^4) operations, where n is the complexity of the input. But the size of the generalized persistence diagram, C, can be as low as linear (and as high as cubic). We elucidate a connection between the 2-parameter and the ordinary 1-parameter settings, which allows us to design an output-sensitive algorithm, whose running time is in O(n^2 m + Cm), where m is the number of simplices in the input.

This paper introduces Möbius homology, a homology theory for representations of finite posets into abelian categories. While the connection between poset topology and Möbius functions is classical, we establish a direct connection between poset topology and Möbius inversions. More precisely, the Möbius homology categorifies the Möbius inversion because its Euler characteristic is equal to the Möbius inversion of the dimension function of the representation. We also introduce a homological version of Rota’s Galois Connection Theorem which relates the Möbius homology over two posets connected by a Galois connection. Our main application is to persistent homology over general posets. We show that under one definition, the persistence diagram is an Euler characteristic over a poset of intervals and hence Möbius homology is a categorification of the persistence diagram. This provides a new invariant for persistent homology over general posets. Finally, we use our homological Rota’s Galois Connection Theorem to prove several results about the persistence diagram.

2022

SIAGA

Edit Distance and Persistence Diagrams over Lattices

Alexander McCleary, and Amit Patel

SIAM Journal on Applied Algebra and Geometry, 2022

We build a functorial pipeline for persistent homology. The input to this pipeline is a filtered simplicial complex indexed by any finite metric lattice, and the output is a persistence diagram defined as the Möbius inversion of its birth-death function. We adapt the Reeb graph edit distance to each of our categories and prove that both functors in our pipeline are 1-Lipschitz, making our pipeline stable. Our constructions generalize the classical persistence diagram, and in this setting, the bottleneck distance is strongly equivalent to the edit distance.

Poincaré Duality for Generalized Persistence Diagrams of (co)Filtrations

We dualize previous work on generalized persistence diagrams for filtrations to cofiltrations. When the underlying space is a manifold, we express this duality as a Poincaré duality between their generalized persistence diagrams. A heavy emphasis is placed on the recent discovery of functoriality of the generalized persistence diagram and its connection to Rota’s Galois Connection Theorem.

We employ the recent discovery of functoriality for persistent homology to recast the Persistent Homology Transform of a geometric complex as a cosheaf of combinatorial persistence diagrams. Detailed examples are provided. We hope the recasting of the PH transform as a cellular cosheaf will allow for the adoption of existing methods from category theory and (co)sheaf theory to the study of shapes.

In this paper, we give lower bounds for the homology of the fibers of a map to a manifold. Using new sheaf theoretic methods, we show that these lower bounds persist over whole open sets of the manifold and that they are stable under perturbations of the map. This generalizes certain ideas of persistent homology to higher dimensions.

In this paper we prove an equivalence theorem originally observed by Robert MacPherson. On one side of the equivalence is the category of cosheaves that are constructible with respect to a locally cone-like stratification. Our constructibility condition is new and only requires that certain inclusions of open sets are sent to isomorphisms. On the other side of the equivalence is the category of functors from the entrance path category, which has points for objects and certain homotopy classes of paths for morphisms. When our constructible cosheaves are valued in Set we prove an additional equivalence with the category of stratified coverings.

Abel

Canonical Stratifications Along Bisheaves

Amit Patel, and Vidit Nanda

In Proceedings of the Abel Symposium 2018: Topological Data Analysis, 2020

A theory of bisheaves has been recently introduced to measure the homological stability of fibers of maps to manifolds. A bisheaf over a topological space is a triple consisting of a sheaf, a cosheaf, and compatible maps from the stalks of the sheaf to the stalks of the cosheaf. In this note we describe how, given a bisheaf constructible (i.e., locally constant) with respect to a triangulation of its underlying space, one can explicitly determine the coarsest stratification of that space for which the bisheaf remains constructible.

AMS

Bottleneck stability for generalized persistence diagrams

Alex McCleary, and Amit Patel

Proceedings of the American Mathematical Society. American Mathematical Society, Mar 2020

In this paper, we extend bottleneck stability to the setting of one dimensional constructible persistence modules valued in any skeletally small abelian category.

2018

JACT

Generalized persistence diagrams

Amit Patel

Journal of Applied and Computational Topology, Jun 2018

We generalize the persistence diagram of Cohen-Steiner, Edelsbrunner, and Harer to the setting of constructible persistence modules valued in a symmetric monoidal category. We call this the \emphtype \mathcalA persistence diagram of a persistence module. If the category is also abelian, then we define a second \emphtype \mathcalB persistence diagram. In addition, we show that both diagrams are stable to all sufficiently small perturbations of the module. The type \mathcalB persistence diagram carries less information than the type \mathcalA persistence diagram, but it enjoys a stronger stability theorem.

The Reeb graph is a construction which originated in Morse theory to study a real-valued function defined on a topological space. More recently, it has been used in various applications to study noisy data which creates a desire to define a measure of similarity between these structures. Here, we exploit the fact that the category of Reeb graphs is equivalent to the category of a particular class of cosheaf. Using this equivalency, we can define an ‘interleaving’ distance between Reeb graphs which is stable under the perturbation of a function. Along the way, we obtain a natural construction for smoothing a Reeb graph to reduce its topological complexity. The smoothed Reeb graph can be constructed in polynomial time.

2013

HHA

Homology and robustness of level and interlevel sets

Paul Bendich, Herbert Edelsbrunner, Dmitriy Morozov, and 1 more author

Given a continuous function f : X \to \mathcalR on a topological space, we consider the preimages of intervals and their homology groups and show how to read the ranks of these groups from the extended persistence diagram of f. In addition, we quantify the robustness of the homology classes under perturbations of f using well groups, and we show how to read the ranks of these groups from the same extended persistence diagram. The special case X = \mathcalR^3 has ramifications in the fields of medical imaging and scientific visualization.

Using topological degree theory, we present a fast algorithm for computing the well diagram, a quantitative property, of a vector field on Euclidean space.

2011

Top Viz

The Stability of the Apparent Contour of an Orientable 2-Manifold

Herbert Edelsbrunner, Dmitriy Morozov, and Amit Patel

The (apparent) contour of a smooth mapping from a 2-manifold to the plane, f :M \to \mathcalR^2, is the set of critical values, that is, the image of the points at which the gradients of the two component functions are linearly dependent. Assuming M is compact and orientable and measuring difference with the erosion distance, we prove that the contour is stable.

FoCM

Quantifying Transversality by Measuring the Robustness of Intersections

Herbert Edelsbrunner, Dmitriy Morozov, and Amit Patel

Foundations of Computational Mathematics, Jun 2011

By definition, transverse intersections are stable under infinitesimal perturbations. Using persistent homology, we extend this notion to a measure. Given a space of perturbations, we assign to each homology class of the intersection its robustness, the magnitude of a perturbation in this space necessary to kill it, and then we prove that the robustness is stable. Among the applications of this result is a stable notion of robustness for fixed points of continuous mappings and a statement of stability for contours of smooth mappings.

2010

ESA

The Robustness of Level Sets

Paul Bendich, Herbert Edelsbrunner, Dmitriy Morozov, and 1 more author

We define the robustness of a level set homology class of a function }f: {\backslashmathbb X} \backslashto {\backslashmathbb R}}as the magnitude of a perturbation necessary to kill the class. Casting this notion into a group theoretic framework, we compute the robustness for each class, using a connection to extended persistent homology. The special case }{\backslashmathbb X} = {\backslashmathbb R}^3}has ramifications in medical imaging and scientific visualization.

MFCS

Persistent Homology under Non-uniform Error

Paul Bendich, Herbert Edelsbrunner, Michael Kerber, and 1 more author

In Mathematical Foundations of Computer Science 2010, Jun 2010

Using ideas from persistent homology, the robustness of a level set of a real-valued function is defined in terms of the magnitude of the perturbation necessary to kill the classes. Prior work has shown that the homology and robustness information can be read off the extended persistence diagram of the function. This paper extends these results to a non-uniform error model in which perturbations vary in their magnitude across the domain.

Generalizing the concept of a Reeb graph, the Reeb space of a multivariate continuous mapping identifies points of the domain that belong to a common component of the preimage of a point in the range. We study the local and global structure of this space for generic, piecewise linear mappings on a combinatorial manifold.