Schedule for: 17w5108 - Topological Data Analysis: Developing Abstract Foundations

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

If TDA wants to move away from the study of persistence diagrams a posteriori and towards the construction of analytical models or predictions of persistence diagrams from calculus-based physics-type models, an analytical (rather than algorithmic) study of the persistence map must be undertaken. Beyond stability, remarkably few things are known about the persistence map. In this talk, I will provide a complete characterization of the image and the fiber of the persistence map for time series (functions from the interval to the real line) as detailed in a recent arXiv preprint (arxiv.org/abs/1706.06059). This talk will require very little background knowledge and will contain several open problems.

I have been thinking about this problem for a couple of years, first alone, then with a grad student (Ashleigh Thomas), and then also with a postdoc (Justin Curry, also at this BIRS workshop). As more points of view enter, the investigation becomes more nuanced. The ideas are still under development but are at the point where a group discussion would be valuable and potentially decisive. Beyond providing an opportunity to present background on new phenomena that arise with multiple parameters, and being fodder for substantial active discussion, what I have to say about it falls squarely under the heading of "foundations of other branches of mathematics that may not be used extensively in the field so far but could be relevant" -- the "other branches" being commutative algebra (graded rings), combinatorics (posets), and algebraic geometry (real algebraic sets). These provide instructions regarding not only theoretical considerations, but also algorithmic ones: how do you input a persistence module into a computer so as to carry out statistical inference with it? A lot of this is still open -- a lot remains to be done, especially when it comes to translating the theory into useful statistics. It bears mentioning that all of this is in service to an important question in evolutionary biology for which the goal is to analyze a specific dataset consisting of images of veins in fruit fly wings.

Meet in the Corbett Hall Lounge for a guided tour of The Banff Centre campus.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

In this talk I will build on the generalized notion of a persistence diagram introduced by A. Patel to multidimensional persistence. I will introduce several lattice-theoretic notions which reduce to known results in the one dimensional parameter case, but generalize to higher dimensional cases. With this interpretation, I will also give a possible definition for a family of distances between diagrams which include the classical bottleneck and wasserstein metrics currently used in applications and time permitting sketch the proof for bottleneck stability for this type of multidimensional persistence.

Many topological computation tasks, as well as stability results, assume that the input is a "clean" (finite) metric space or with very limited type of noise (e.g, with bounded Hausdorff-type distance). In this talk, I will describe three different ways to model noise in metric, and how to perform denoising so as to produce the more friendly form of Hausdorff-type noise. I will specifically focus on the case when the target metric is induced from a graph; however the observed graph is a (randomly) perturbed version of the true graph. I will also discuss some open problems at the end.

I'll discuss recent work on trying to adapt persistent homology methods to datasets that exhibit asymmetry. Natural candidates are the Rips and Cech filtrations. Whereas the Rips filtration can unambiguously be generalized directly, generalizing the Cech filtration gives rise to two different versions: the sink and the source filtrations. It turns out that the Rips filtration imposes a symmetrization on the data whereas the Cech filtrations do not, thus making them more suitable for the analysis of intrinsically asymmetric data. By generalizing a theorem of Dowker we can prove that the persistent homologies of these two Cech filtrations are isomorphic. We establish the stability of these constructions under a metric between networks that generalizes the Gromov-Hausdorff distance. I'll also describe some results ve that characterize the persistence diagrams of some likely "motifs" in real (e.g. biological) networks: cycle-networks, which are directed analogues of the standard (discrete) circle.

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Data science is a term that is commonly used to denote the systematic study of data that cannot be modeled using traditional techniques, either because of the size of the data or because of the ‘complexity’ of the data, e.g. when the data is heterogeneous, or unstructured. Data science is a young discipline, and the development of its foundations is in its infancy. Topological data analysis (TDA) is built on tools from algebraic topology, which gives a well-understood theoretical framework to study features of data. Issues inherent to the study of data make it necessary to have abstract principles that guide the development of the field. Therefore, understanding how TDA is placed within data science is a crucial aspect of building solid foundations for TDA. Conversely, one could ask which foundational questions fall within the purview of TDA, and whether techniques and ideas used in TDA could help in guiding the development of foundations for data science. Thus two main questions arise: 1) TDA is well suited to studying certain types of data; how can this sort of data be characterized more generally, and what issues are inherent to their analysis irrespective of the tools one uses? How are/should these issues be taken into account in current/future TDA practices?; and 2) What can TDA do for data science? Could techniques used in TDA (e.g. functoriality) be useful in tackling foundational issues in e.g. machine learning?

Recently, the statistics of short bars in persistence diagrams started to draw a lot of attention (especially from material sciences) as a descriptor of functions or spaces. I will talk about reformulation of some known results in terms of the PH point processes. New results deal with the generic PH-dimension for Lipschitz or Hoelder functions on manifolds, and statistics of short bars (and their chiralities) for Brownian motions.

Multi-graded betti numbers, one of the only invariant we have to study multi-modules, are highly not stable with respect to the interleaving distance. However, they arise from the existence of free minimal resolutions and we propose to show how we can equip the derived category of persistent modules (the one in which resolutions « naturally » live) with a derived distance to make those resolutions stable. As a byproduct, we get the non-existence of a right (or left) exact functor from the category of modules to the one that decomposes in thin summands.

The lines or grids of vector spaces appearing in the study of persistence can be seen as special instances of representations of posets (= presheaves on posets). In the finite case, representations of posets (or arbitrary quivers with relations) have independently been studied in the context of finite dimensional algebras. In my talk I will review the setup and some classical results of representation theory of finite dimensional algebras which one might hope to apply in the study of persistence.

I'll present the persistent homotopy type distance d_HT to compare two real valued functions defi ned on possibly different but homotopy equivalent topological spaces. The underlying idea in the defi nition of d_HT is to measure the minimal shift that is necessary to apply to one of the two functions in order that the sublevel sets of the two functions become homotopically equivalent. This distance is interesting in connection with persistent homology. Indeed, dHT still provides an upper bound for the bottleneck distance between the persistence diagrams of the intervening functions. Moreover, because homotopy equivalences are weaker than homeomorphisms, this implies a lifting of the standard stability results provided by the supremum distance and by the natural pseudo-distance d_NP. From a different standpoint, d_HT extends the supremum distance and d_NP in two ways. First, appropriately restricting the category of objects to which d_HT applies, it can be made to coincide with the other two distances. Secondly, dHT has an interpretation in terms of interleavings that naturally places it in the family of distances used in persistence theory. This is joint work with Patrizio Frosini and Facundo Memoli.

We introduce and study homotopy interleavings between filtered topological spaces. These are homotopy-invariant analogues of interleavings, objects commonly used in topological data analysis to articulate stability and inference theorems. Intuitively, whereas a strict interleaving between filtered spaces X and Y certifies that X and Y are approximately isomorphic, a homotopy interleaving between X and Y certifies that X and Y are approximately weakly equivalent. The main results of this paper are that homotopy interleavings induce an extended pseudometric dHI on filtered spaces, and that this is the universal pseudometric satisfying natural stability and homotopy invariance axioms. To motivate these axioms, we also observe that dHI (or more generally, any pseudometric satisfying these two axioms and an additional “homology bounding” axiom) can be used to formulate lifts of fundamental TDA theorems from the algebraic (homological) level to the level of filtered spaces. Joint work with Andrew J. Blumberg (UT Austin).

I will interpret the persistence diagram of Cohen-Steiner, Edelsbrunner, and Harer as the Möbius inversion of the rank function. I will then show how to generalize the persistence diagram to the setting of constructible persistence modules valued in any small symmetric monoidal category.

The goal of this talk is to present some on-going work on proving stability for certain subcategories of multipersistence modules, specifically the ones that are decomposable into direct sums of free, rectangle, or interval modules. This is joint work with Magnus Botnan and Jérémy Cochoy.

In their paper "The theory of multidimensional persistence", Carlsson and Zomorodian write "Our study of multigraded objects shows that no complete discrete invariant exists for multidimensional persistence. We still desire a discriminating invariant that captures persistent information, that is, homology classes with large persistence." In this talk I will discuss how tools from commutative algebra give computable invariants able to capture homology classes with large persistence. Specifically, multigraded associated primes provide a stratification of the region where a multigraded module does not vanish, while multigraded Hilbert series and local cohomology give a measure of the size of components of the module supported on different strata. These invariants generalize in a suitable sense the invariant for the one-parameter case.

In this informal talk I will briefly describe three recent and current projects of mine outside of TDA proper but using TDA ideas or calling for more TDA interaction. The projects come from three different fields - symplectic geometry, theoretical machine learning and computational neuroscience - and the hope is to spur conversations with some of you regarding future directions.

Local weak convergence is a powerful framework for study of sparse graph limits and has been successfully applied in obtaining exact expectation asymptotics in probabilistic combinatorial optimization​, statistical physics and random graph theory. In particular, it can be used to show that sum of lifetime sum of $H_0$-persistent diagram on a mean field model (complete graph with i.i.d. weights) converges to $\zeta(3)$, where $\zeta$ is the Riemann-zeta function. Further, using this framework the minimum cost function on the complete bipartite graph with i.i.d. weights was shown to converge to $\zeta(2)$. In this talk, we shall look at some underlying ideas behind such results and wonder about the possibility of extensions to random topology. As is to be expected, when we move from random graphs to random complexes, there will be fewer answers and more questions.

We show that the one-dimensional homology of the nerve complex $N(\mathcal{U})$ of a path-connected cover $\mathcal{U}$ of a domain $X$ cannot be richer than that of the domain $X$ itself. Intuitively, this result means that $H_1$-homology classes can only be be killed under a natural map from $X$ to the nerve complex $N(\mathcal{U})$. Equipping $X$ with a pseudometric $d$, we further refine this result and characterize the classes of $H_1(X)$ that may survive in the nerve complex using the notion of \emph{size} of the covering elements in $\mathcal{U}$. These fundamental results about nerve complexes then lead to an analysis of the $H_1$-homology of Reeb spaces, mappers and multiscale mappers.

In this talk I want to try to address the question of whether the study of random topology is useful for TDA. I will do that by examining the case-study of random geometric complexes. The plan is to briefly review recent advances in the study of random geometric complexes, and discuss the potential contribution to statistical TDA.

We've scheduled a short (20 mins) guided tour of the piano repair workshop. If you're interested in joining, come to the entrance of the Maclab Bistro (bar) just around the corner on the patio at noon.

Geometry works best when dealing with precisely defined shapes, but struggles in the presence of noise. Persistent homology can deal with noise but, compared to geometry, is imprecise. Is there a meaningful way to combine the two?

Classical Morse theory connects the topology of a manifold with critical points of a Morse function. Stratified Morse theory considers the topology of a stratified space with critical points of a stratified Morse function. Discrete Morse theory is considered as a combinatorial adaptation of Classical Morse theory, and is applied to simplicial or cell complexes. It seems natural to ask whether there could be a combinatorial adaptation of Stratified Morse theory. This talk will serve as a conversation starter, where I would like to discuss a (possibly) discrete version of stratified Morse theory and its (potential) application in data analysis.

I will present a simple-minded view of interleaving: we want two diagrams to fit together inside a bigger diagram. I will show how this idea can be used to draw pictures that prove a number of persistence results. Surprisingly (to us!) carefully working out the theory underlying these ideas led us to discover that interleaving distance and Gromov-Hausdorff distance are both special cases of a richer theory. Joint work with Vin de Silva and Jonathan Scott.

I will report on work in progress, joint with U. Bauer, M. Botnan and S. Oppermann. One of the main motivations for this project was to try to determine the complexity of 2-parameter clustering. Although this eventually led to purely representation theoretic results, the focus of this talk is what one of these results tells us in the realm of clustering. To be more precise, we show that the category of 2-parameter persistence modules with epimorphisms either horizontally, vertically, or both, is nearly as bad as the full category of 2-parameter persistence modules.

A look at some research directions inspired by studying spiral defect chaos in Rayleigh-Benard convection via topological methods.

Motivated by the attempt to build on the top of reasonable data ( e.g. weighted networks) so-called BF-theories (a kind of topological quantum field theories), we will extend the usual approach of multidimensional persistent (co-)homology by considering diagrams of incidence algebras of finite posets. By using results of Gerstenhaber et al. we will show what is the link between diagram cohomology, Hochschild cohomology, and multidimensional persistence connecting seemingly uncorrelated research areas as topological data analysis and topological quantum field theory.

Multidimensional persistent homology is highly relevant in the analysis of noisy data, as it offers the ability to filter by two or more parameters simultaneously. In this talk I will summarize the RIVET software project, which aims to enable the use of two-dimensional persistence in real-world applications. I will highlight current work with students at St. Olaf College to apply RIVET real data, particularly data arising from Wikipedia, networks, and the natural sciences. I will demonstrate ways in which 2-D persistence can identify topological and geometric structure in complex, high-dimensional, multi-parameter data.

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.