In 2009, Chazal et al. introduced -interleavings of persistence modules.interleavings induce a pseudometric d I on (isomorphism classes of) persistence modules, the interleaving distance. The definitions of -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 in an earlier preprint of this paper, has since appeared in several other places, and is now known as the isometry theorem.Second, we present a characterization of the -interleaving relation on multidimensional persistence modules. This expresses transparently the sense in which two -interleaved modules are algebraically similar.Third, using this characterization, we show that when we define our persistence modules over a prime field, d I satisfies a universality property. This universality result is the central result of the paper. It says that d I satisfies a stability property generalizing one which d B is known to satisfy, and that in addition, if d is any other pseudometric on multidimensional persistence modules satisfying the same stability property, then d ≤ d I . We also show that a variant of this universality result holds for d B , over arbitrary fields.Finally, we show that d I restricts to a metric on isomorphism classes of finitely presented multidimensional persistence modules. * mlesnick@ima.umn.edu.

We define a simple, explicit map sending a morphism f : M → 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 coker f .As an immediate corollary, we obtain a new proof of the algebraic stability theorem for persistence barcodes [5,9], a fundamental result in the theory of persistent homology. In contrast to previous proofs, ours shows explicitly how a δ-interleaving morphism between two persistence modules induces a δ-matching between the barcodes of the two modules. Our main result also specializes to a structure theorem for submodules and quotients of persistence modules.

Characterization of transient intermediate or transition states is crucial for the description of biomolecular folding pathways, which is, however, difficult in both experiments and computer simulations. Such transient states are typically of low population in simulation samples. Even for simple systems such as RNA hairpins, recently there are mounting debates over the existence of multiple intermediate states. In this paper, we develop a computational approach to explore the relatively low populated transition or intermediate states in biomolecular folding pathways, based on a topological data analysis tool, MAPPER, with simulation data from large-scale distributed computing. The method is inspired by the classical Morse theory in mathematics which characterizes the topology of high-dimensional shapes via some functional level sets. In this paper we exploit a conditional density filter which enables us to focus on the structures on pathways, followed by clustering analysis on its level sets, which helps separate low populated intermediates from high populated folded/unfolded structures. A successful application of this method is given on a motivating example, a RNA hairpin with GCAA tetraloop, where we are able to provide structural evidence from computer simulations on the multiple intermediate states and exhibit different pictures about unfolding and refolding pathways. The method is effective in dealing with high degree of heterogeneity in distribution, capturing structural features in multiple pathways, and being less sensitive to the distance metric than nonlinear dimensionality reduction or geometric embedding methods. The methodology described in this paper admits various implementations or extensions to incorporate more information and adapt to different settings, which thus provides a systematic tool to explore the low-density intermediate states in complex biomolecular folding systems.

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 R-valued functions, the result was later cast in a more general algebraic form, in the language of persistence modules and 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 for these extensions. One part of our argument yields a stability result for free two-dimensional persistence modules. As an application of our main theorem, we strengthen a result of Bauer et al. on the stability of the persistent homology of Reeb graphs. Our main result also yields an alternative proof of the stability theorem for level set persistent homology of Carlsson et al. IntroductionPersistence Modules. Let Vec denote the category of vector spaces over some fixed field k, and let vec denote the subcategory of finite dimensional vector spaces. We define a persistence module to be a functor M : P → Vec, for P a poset. We will often refer to such M as a P-indexed module. If M takes values in vec, we say M is pointwise finite dimensional (p.f.d.). The P-indexed persistence modules form a category Vec P whose morphisms are the natural transformations.Persistence modules are the basic algebraic objects of study in the theory of persistent homology. The theory begins with the study of 1-D persistence modules, i.e. functors R → Vec or Z → Vec, where R and Z are taken to have the usual total orders. The structure theorem for 1-D persistence modules [25,43] tells us that the isomorphism type of a p.f.d. 1-D persistence module M is completely described by a collection B(M ) of intervals in R, called the barcode of M ; B(M ) specifies the decomposition of M into indecomposable summands. Persistent Homology. In topological data analysis, one often studies a data set by associating to the data a persistence module. To do so, we first associate to our data a filtration, i.e., a functor F : R → Top such that the map F a → F b is an inclusion whenever a ≤ b. For example, if our data is an R-valued function γ : T → R, for T a topological space, we may take F to be the sublevel set filtration S ↑ (γ), defined by, this indeed gives a filtration. If our data set is instead a point cloud, we often consider a Vietoris-Rips orČech filtration; see e.g. [10] for details. * Letting H i : Top → Vec denote the i th singular homology functor with coefficients in k, we obtain a (typically p.f.d.) persistence module H i F for any i ≥ 0. The barcodes B(H i F) serve as concise descriptors of the coarse-scale, global, non-linear geometric structure of the data set. These descriptors have been applied to many problems in science and engineering, e.g., to natural scene statistics, evolutionary biology, periodicity detection in gene expression data, sensor networks, and cluster...

Motivated by applications to topological data analysis, we give an efficient algorithm for computing a (minimal) presentation of a bigraded K[x, y]-module M , where K is a field. The algorithm takes as input a short chain complex of free modulesmemory, where |F i | denotes the size of a basis of F i . We observe that, given the presentation computed by our algorithm, the bigraded Betti numbers of M are readily computed. We also introduce a different but related algorithm, based on Koszul homology, which computes the bigraded Betti numbers without computing a presentation, with these same complexity bounds. These algorithms have been implemented in RIVET, a software tool for the visualization and analysis of two-parameter persistent homology. In experiments on topological data analysis problems, our approach outperforms the standard computational commutative algebra packages Singular and Macaulay2 by a wide margin.