Persistent homology was shown by Carlsson and Zomorodian to be homology of graded chain complexes with coefficients in the graded ring $\kk[t]$. As such, the behavior of persistence modules -- graded modules over $\kk[t]$ is an important part in the analysis and computation of persistent homology.
In this paper we present a number of facts about persistence modules; ranging from the well-known but under-utilized to the reconstruction of techniques to work in a purely algebraic approach to persistent homology. In particular, the results we present give concrete algorithms to compute the persistent homology of a simplicial complex with torsion in the chain complex.
We demonstrate that an equivalence of categories using $\varepsilon$-interleavings as a fundamental component exists between the model of persistence modules as graded modules over a polynomial ring and the model of persistence modules as modules over the total order of the real numbers.
One of the main reasons for topological persistence being useful in data analysis is that it is backed up by a stability (isometry) property: persistence diagrams of $1$-parameter persistence modules are stable in the sense that the bottleneck distance between two diagrams equals the interleaving distance between their generating modules. However, in multi-parameter setting this property breaks down in general. A simple special case of persistence modules called rectangle decomposable modules is known to admit a weaker stability property. Using this fact, we derive a stability-like property fo…
This short note establishes explicit and broadly applicable relationships between persistence-based distances computed locally and globally. In particular, we show that the bottleneck distance between two zigzag persistence modules restricted to an interval is always bounded above by the distance between the unrestricted versions. While this result is not surprising, it could have different practical implications. We give two related applications for metric graph distances, as well as an extension for the matching distance between multiparameter persistence modules.
The classical persistence algorithm computes the unique decomposition of a persistence module implicitly given by an input simplicial filtration. Based on matrix reduction, this algorithm is a cornerstone of the emergent area of topological data analysis. Its input is a simplicial filtration defined over the integers $\mathbb{Z}$ giving rise to a $1$-parameter persistence module. It has been recognized that multiparameter version of persistence modules given by simplicial filtrations over $d$-dimensional integer grids $\mathbb{Z}^d$ is equally or perhaps more important in data science applicat…
We study how to obtain partial matchings using the block function $\mathcal{M}_f$, induced by a morphism $f$ between persistence modules. $\mathcal{M}_f$ is defined algebraically and is linear with respect to direct sums of morphisms. We study some interesting properties of $\mathcal{M}_f$, and provide a way to obtain $\mathcal{M}_f$ using matrix operations.
In 2009, Chazal et al. introduced $ε$-interleavings of persistence modules. $ε$-interleavings induce a pseudometric $d_I$ on (isomorphism classes of) persistence modules, the interleaving distance. The definitions of $ε$-interleavings and $d_I$ generalize readily to multidimensional persistence modules. In this paper, we develop the theory of multidimensional interleavings, with a view towards applications to topological data analysis. We present four main results. First, we show that on 1-D persistence modules, $d_I$ is equal to the bottleneck distance $d_B$. This result, which first appeared…
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 stability theorem for persistent homology is a central result in topological data analysis. While the original formulation of the result concerns the persistence barcodes of $\mathbb{R}$-valued functions, the result was later cast in a more general algebraic form, in the language of \emph{persistence modules} and \emph{interleavings}. In this paper, we establish an analogue of this algebraic stability theorem for zigzag persistence modules. To do so, we functorially extend each zigzag persistence module to a two-dimensional persistence module, and establish an algebraic stability theorem f…
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…
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…
We give a self-contained treatment of the theory of persistence modules indexed over the real line. We give new proofs of the standard results. Persistence diagrams are constructed using measure theory. Linear algebra lemmas are simplified using a new notation for calculations on quiver representations. We show that the stringent finiteness conditions required by traditional methods are not necessary to prove the existence and stability of the persistence diagram. We introduce weaker hypotheses for taming persistence modules, which are met in practice and are strong enough for the theory still…
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…
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…
The matching distance is a pseudometric on multi-parameter persistence modules, defined in terms of the weighted bottleneck distance on the restriction of the modules to affine lines. It is known that this distance is stable in a reasonable sense, and can be efficiently approximated, which makes it a promising tool for practical applications. In this work, we show that in the 2-parameter setting, the matching distance can be computed exactly in polynomial time. Our approach subdivides the space of affine lines into regions, via a line arrangement. In each region, the matching distance restrict…
The goal of this work is to extend the standard persistent homology pipeline for exploratory data analysis to the 2-D persistence setting, in a practical, computationally efficient way. To this end, we introduce RIVET, a software tool for the visualization of 2-D persistence modules, and present mathematical foundations for this tool. RIVET provides an interactive visualization of the barcodes of 1-D affine slices of a 2-D persistence module $M$. It also computes and visualizes the dimension of each vector space in $M$ and the bigraded Betti numbers of $M$. At the heart of our computational ap…
This paper addresses two questions: (a) can we identify a sensible class of 2-parameter persistence modules on which the rank invariant is complete? (b) can we determine efficiently whether a given 2-parameter persistence module belongs to this class? We provide positive answers to both questions, and our class of interest is that of rectangle-decomposable modules. Our contributions include: on the one hand, a proof that the rank invariant is complete on rectangle-decomposable modules, together with an inclusion-exclusion formula for counting the multiplicities of the summands; on the other ha…
Motivated both by theoretical and practical considerations in topological data analysis, we generalize the $p$-Wasserstein distance on barcodes to multiparameter persistence modules. For each $p\in [1,\infty]$, we in fact introduce two such generalizations $d_{\mathcal I}^p$ and $d_{\mathcal M}^p$, such that $d_{\mathcal I}^\infty$ equals the interleaving distance and $d_{\mathcal M}^\infty$ equals the matching distance. We show that on 1- or 2-parameter persistence modules over prime fields, $d_{\mathcal I}^p$ is the universal (i.e., largest) metric satisfying a natural stability property; th…
The algebraic stability theorem for $\mathbb{R}$-persistence modules is a fundamental result in topological data analysis. We present a stability theorem for $n$-dimensional rectangle decomposable persistence modules up to a constant $(2n-1)$ that is a generalization of the algebraic stability theorem, and also has connections to the complexity of calculating the interleaving distance. The proof given reduces to a new proof of the algebraic stability theorem with $n=1$. We give an example to show that the bound cannot be improved for $n=2$. We apply the same technique to prove stability result…
Computation of the interleaving distance between persistence modules is a central task in topological data analysis. For $1$-parameter persistence modules, thanks to the isometry theorem, this can be done by computing the bottleneck distance with known efficient algorithms. The question is open for most $n$-parameter persistence modules, $n>1$, because of the well recognized complications of the indecomposables. Here, we consider a reasonably complicated class called {\em $n$-parameter interval decomposable} modules whose indecomposables may have a description of non-constant complexity. We pr…
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 define a simple, explicit map sending a morphism $f:M \rightarrow N$ of pointwise finite dimensional persistence modules to a matching between the barcodes of $M$ and $N$. Our main result is that, in a precise sense, the quality of this matching is tightly controlled by the lengths of the longest intervals in the barcodes of $\ker f$ and $\mathop{\mathrm{coker}} f$. As an immediate corollary, we obtain a new proof of the algebraic stability of persistence, a fundamental result in the theory of persistent homology. In contrast to previous proofs, ours shows explicitly how a $δ$-interleaving …
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…
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…
This work concerns the theoretical foundations of persistence-based topological data analysis. We develop theory of topological inference in the multidimensional persistence setting, and directly at the (topological) level of filtrations rather than only at the (algebraic) level of persistent homology modules. Our main mathematical objects of study are interleavings. These are tools for quantifying the similarity between two multidimensional filtrations or persistence modules. They were introduced for 1-D filtrations and persistence modules by Chazal, Cohen-Steiner, Glisse, Guibas, and Oudot. …
The foundational character of certain algebraic structures as Boolean algebras and Heyting algebras is rooted in their potential to model classical and constructive logic, respectively. In this paper we discuss the contributions of algebraic logic to the study of persistence based on a new operation on the ordered structure of the input diagram of vector spaces and linear maps given by a filtration. Within the context of persistence theory, we give an analysis of the underlying algebra, derive universal properties and discuss new applications. We highlight the definition of the implication ope…
We present a row reduction algorithm to compute the barcode decomposition of persistence modules. This algorithm dualises the standard persistence one and clarifies the symmetry between clear and compress optimisations.
We show that computing the interleaving distance between two multi-graded persistence modules is NP-hard. More precisely, we show that deciding whether two modules are $1$-interleaved is NP-complete, already for bigraded, interval decomposable modules. Our proof is based on previous work showing that a constrained matrix invertibility problem can be reduced to the interleaving distance computation of a special type of persistence modules. We show that this matrix invertibility problem is NP-complete. We also give a slight improvement of the above reduction, showing that also the approximation …
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 most prominent areas of research in computational topology in the past 20 years. In this paper we will introduce an alternative interpretation of persistence based on the study of the order structure of its correspondent lattice. Its algorithmic construction leads to two operations on homology groups which describe a diagram of spaces as a complete Heyting alge…
Multidimensional persistence modules do not admit a concise representation analogous to that provided by persistence diagrams for real-valued functions. However, there is no obstruction for multidimensional persistent Betti numbers to admit one. Therefore, it is reasonable to look for a generalization of persistence diagrams concerning those properties that are related only to persistent Betti numbers. In this paper, the persistence space of a vector-valued continuous function is introduced to generalize the concept of persistence diagram in this sense. The main result is its stability under f…
A stability theorem for bigraded persistence barcodes
Department of Mathematics, Rider University, Lawrenceville, NJ 08648, USA；Mathematical Institute of the Serbian Academy of Sciences；Department of Mathematics；Mechanics, Moscow State University, Russia；Institute for Information Transmission Problems, Russian Academy of Sciences, Moscow, Russia；National Research University Higher School of Economics, Moscow, Russia；Department of Mathematics Education, Pusan National University, Busan, Republic of Korea；Statistics, University of Regina, Regina, Saskatchewan, Canada
