Mathieu Carriere

Statistics and Machine Learning in Topological Data Analysis with Applications to Biology

Topological Data Analysis (TDA) is an area of data science which aims at characterizing data sets with their topological features in various dimensions. Examples of such features include the connected components, the loops or the cavities that are present in the data, and which are encoded in the two main descriptors of TDA, the so-called persistence diagram and Mapper. Even though these descriptors have proved useful in many applications, it is not straightforward to include them in automated processes, which are common in statistics and machine learning, mainly because the space of these descriptors lacks a lot of required properties, such as a well-defined addition or barycenter.

In this talk, I will recall the basics of TDA and review the current solutions that have been proposed in the past few years to merge TDA descriptors (persistence diagram and Mapper) with statistics and Machine Learning. Then, I will introduce some questions that remain open in this topic, and that are active fields of research as of today, such that the question of persistence diagram differentiability for deep learning, or the statistical analysis of the Mapper in the multivariate case. In the process, I will also illustrate these problems by providing applications on biological data, such as immunofluorescence images for breast cancer pathology and single-cell RNA sequencing for understanding the spinal cord cellular diversity.

Simon Cho

Nerve functors and homology in applied topology

We show how a choice of quantale induces a nerve functor, with both the Vietoris-Rips complex and the magnitude nerve arising this way for different choices of monoidal structure on R>=0. This together with a choice of “localization” accounts for the difference between persistent and magnitude homology, cf. Nina Otter’s earlier work on this subject. Lastly, we mention some application-oriented observations naturally suggested by the perspective mentioned above.

Ellen Gasparovic

Comparing and Analyzing Metric Graphs via Topology

Metric graphs are effective tools for modeling complex structures that arise in many real-world applications, such as road networks, vascular systems, and galaxy filaments. This talk will focus on the rich topological summary information that one may extract from these meaningful objects. For the purposes of graph comparison, we will investigate the relationship between two persistent homology-based distances defined on the space of metric graphs. We will also discuss how to obtain the persistent homology profiles of the Vietoris-Rips complexes of a large class of metric graphs. This talk covers joint work with Maria Gommel, Emilie Purvine, Radmila Sazdanovic, Bei Wang, Yusu Wang, Lori Ziegelmeier, Henry Adams, and Michal Adamazsek.

Haibin Hang

Correspondence modules, persistence sheaves, and their diagrams

 In this talk, I will introduce correspondence modules (c-modules), which generalize persistence and zigzag modules. I also will introduce the persistence sheaf of sections of a c-module, which is used to analyze its structure and prove an interval decomposition theorem. Several applications in which c-modules arise naturally will be discussed.

Jakob Hansen

Laplacians of Cellular Sheaves and their Applications

Given a chain complex of inner product spaces, discrete Hodge theory constructs a graded Laplacian operator whose kernel consists of distinguished homology representatives. When applied to the cochain complex of a cellular sheaf, this construction produces generalizations of the Laplacian operators familiar from spectral graph theory.  These linear operators offer a way to connect discrete algebraic topological aspects of cellular sheaves with more continuous, geometric concepts, allowing us to take advantage of local-to-global structure in real-world systems. This talk will outline the construction of sheaf Laplacians and sketch avenues for their application to realistic engineering and scientific problems.

Woojin Kim

Generalized persistence diagrams for persistence modules over posets

When a category C satisfies certain conditions, we define the notion of rank invariant for arbitrary poset-indexed functors F : P → 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 : P → 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 poset P of F : P → C in defining Patel’s generalized persistence diagram of F. By specializing our idea to zigzag persistence modules, we also show that the zigzag barcode of a Reeb graph can be obtained in a purely set-theoretic setting without passing to the category of vector spaces. This is a joint work with Facundo Memoli.

Darrick Lee

 Path Signatures and Kernel Methods

Path signatures form the 0-cochains of Chen’s iterated integral cochain model of path spaces. By considering a collection of time series as a path, we can leverage the path signatures as a reparametrization-invariant feature set for time series. Specifically, we will discuss how signatures can be used in kernel methods for time series analysis.

Cassie Micucci

Determining the Structure of the Actin Cytoskeleton via Bayesian Topological Learning

This presentation explores the computation of posterior distributions from a new Bayesian framework for persistence diagrams. We explain our proposed Bayesian paradigm, which adopts a point process characterization of persistence diagrams. This framework provides the flexibility to estimate the posterior cardinality and intensity of persistence diagrams simultaneously. We present a closed form of the posterior intensity and cardinality using Gaussian mixtures and binomial distributions. Based on this form, we implement an effective Bayes factor classification algorithm on filament network data of plant cells. This work is joint with Vasileios Maroulas and Farzana Nasrin.

Osman Okutan

Existence and approximation of Reeb graphs

Although Reeb graphs are useful in approximating geodesic spaces, it is not known if they are really metric graphs in many cases. In this talk we address this problem by modifying the definition slightly. We then discuss how to approximate Reeb graphs.

Luis Scoccola

Quotient interleaving distances

The interleaving distance on persistent topological spaces is not homotopy invariant, but can be made so in a universal way by quotienting it by the equivalence relation given by weak equivalence. The distance thus obtained is the homotopy interleaving distance of Blumberg and Lesnick. I will explain how other distances, such as the Gromov-Hausdorff distance, can be recovered by quotienting an interleaving distance by an equivalence relation, and how to prove stability results and metric properties of these kinds of distances.

Don Sheehy

Conformal Change of Metrics in Data Analysis

I will survey a variety of algorithmic settings in which one would like to compute (or at least use) distances between points that are induced by local scaling of space.  I will track this idea through topics in mesh generation, surface reconstruction, and robotics.  Then, I will discuss some recent results on exact computation of such distances for the special case where the local scaling of space is proportional to the distance to the input.  This gives the first example of an exact computation for a so-called density-based distance.

Elchanan Solomon

Topological Transforms via Kernel Embeddings

Topological transforms are parametrized families of topological invariants, which, by analogy with transforms in signal processing, are much more discriminative than single measurements. The first two topological transforms to be defined were the Persistent Homology Transform and Euler Characteristic Transform, both of which apply to shapes embedded in Euclidean space. In order to define topological transforms for intrinsic metric objects, one needs a canonical way of choosing informative height functions for the purposes of computing sublevel set persistence diagrams or Euler curves. To that end, we define an integral operator, which we call the distance kernel operator, whose eigenfunctions and eigenvalues encode the large-scale geometry of our metric objects. We use these eigenvalues and eigenfunctions to construct height functions and compute topological invariants, and show that the resulting topological transform enjoys desirable stability and injectivity properties. This is joint work with Steve Oudot and Clément Maria.

Anastasios Stefanou

Algebraic and Metric Structure Theorems for Reeb Graphs and Formigrams.

In this talk I will present (i) my previous work on Reeb graphs, and also (ii) my recent work with F. Mémoli and W. Kim on formigrams.

Reeb graphs: Inspired by the interval-decomposition of persistence modules and the extended Newick format of phylogenetic networks, we show that, inside the larger category of partially ordered Reeb graphs, every Reeb graph with n leaves and first Betti number, s, is equal to a coproduct of at most 2^s trees with n leaves. An implication of this result, is that the isomorphism problem for Reeb graphs is fixed parameter tractable when the parameter is the first Betti number. In particular, we propose partially ordered Reeb graphs as a natural framework for modeling time consistent phylogenetic networks and define a certain Hausdorff distance as metric for these networks.

Formigrams: This notion is a natural generalization of dendrograms and has recently been proposed as a signature for studying the evolution of clusters in dynamic datasets across different time scales. Although its formulation is set-theoretic, the notion of formigram is deeply related to certain algebraic-topological methods used in topological data analysis, such as Reeb graphs and zigzag persistence modules. In this work, we define a poset structure on formigrams and we show that every formigram over X has a canonical decomposition into a join of simpler formigrams, which is analogous to the decomposition of persistence modules into direct sums of interval modules. Furthermore, we show that the interleaving distance between formigrams decomposes into a product metric of the interleaving distance between certain pre-cosheaves. This is analogous to the celebrated interleaving-bottleneck isometry theorem for persistence modules.

Chris Tralie

Topology-Guided Analysis And Synthesis of (Quasi)Periodic Phenomena in Multimedia Data

A large variety of multimedia data inference problems require analysis of repeated structures.  In medical video analysis, for instance, there is interest in analyzing stereotypical repetitive motor motions in videos of autism spectrum disorder patients, and in analyzing voice pathologies from high speed videos of vocal folds.  In this work, we show how the “shape” of these videos can be used to discover important properties.  We present a unified sliding window framework in which periodic patterns show up as loops and quasiperiodic patterns show up as flat tori.  Surprisingly, we also show that some periodic processes with harmonic structures lie on loops which bound twisted spaces such as the Moebius strip, and quantifying these structures is applicable both to rhythm hierarchy analysis in audio and detection of “biphonation” due to mucous in vibrating vocal folds.  In addition to detecting and quantifying periodicity, we can parameterize (possibly unordered) periodic data with circular coordinates, which we use both to analyze tempo/pulse rate and to create slow motion seamless templates of periodic time series and videos.  We end by imagining other patterns that could occur in multimedia data by reverse engineering Takens’ theorem.  This allows us to synthesize time series whose sliding window embeddings lie on arbitrary topological manifolds, and we showcase our technique by generating projective plane and Klein bottle time series.