Past Applied Topology Seminar

The Persistent Homology Transform, and the Euler Characteristic Transform are topological analogs of the Radon transform that can be used in statsistical shape analysis. In this talk I will consider an interesting variant called the Extended Persistent Homology Transform (XPHT) which replaces the normal persistent homology with extended persistent homology. We are particularly interested in the application of the XPHT to binary images. This paper outlines an algorithm for efficient calculation of the XPHT exploting relationships between the PHT of the boundary curves to the XPHT of the foreground.

Topological data analysis (TDA) is a field with tools to quantify the shape of data in a manner that is concise and robust using concepts from algebraic topology. Persistent homology, one of the most popular tools in TDA, has proven useful in applications to time series data, detecting shape that changes over time and quantifying features like periodicity. In this talk, I will present two applications using tools from TDA to study time series data: the first using zigzag persistence, a generalization of persistent homology, to study bifurcations in dynamical systems and the second, using the shape of weighted, directed networks to distinguish periodic and chaotic behavior in time series data.

Acyclic partial matchings on simplicial complexes play an important role in topological data analysis by facilitating efficient computation of (persistent) homology groups. Here we describe probabilistic properties of critical simplex counts for such matchings on clique complexes of Bernoulli random graphs. In order to accomplish this goal, we generalise the notion of a dissociated sum to a multivariate setting and prove an abstract multivariate central limit theorem using Stein's method. As a consequence of this general result, we are able to extract central limit theorems not only for critical simplex counts, but also for generalised U-statistics (and hence for clique counts in Bernoulli random graphs) as well as simplex counts in the link of a fixed simplex in a random clique complex.

The path signature is a characterization of paths that originated in Chen's iterated integral cochain model for path spaces and loop spaces. More recently, it has been used to form the foundations of rough paths in stochastic analysis, and provides an effective feature map for sequential data in machine learning. In this talk, we return to the topological foundations in Chen's construction to develop generalizations of the signature.

In this talk, I'll present inequalities bounding the number of critical cells in a filtered cell complex on the one hand, and the entries of the Betti tables of the multi-parameter persistence modules of such filtrations on the other hand. Using the Mayer-Vietoris spectral sequence we first obtain strong and weak Morse inequalities involving the above quantities, and then we improve the weak inequalities achieving a sharp lower bound for the number of critical cells. Furthermore, we prove a sharp upper bound for the minimal number of critical cells, expressed again in terms of the entries of Betti tables. This is joint work with Andrea Guidolin (KTH, Stockholm). The full paper is posted online as arxiv:2108.11427.

What do we use metrics on persistent modules for? Is it only to asure  stability of some constructions? 

In my talk I will describe why I care about such metrics, show how to construct a rich space of them and illustrate how  to use

them for analysis. 

We will consider modeling shapes and fields via topological and lifted-topological transforms. 

Specifically, we show how the Euler Characteristic Transform and the Lifted Euler Characteristic Transform can be used in practice for statistical analysis of shape and field data. The Lifted Euler Characteristic is an alternative to the. Euler calculus developed by Ghrist and Baryshnikov for real valued functions. We also state a moduli space of shapes for which we can provide a complexity metric for the shapes. We also provide a sheaf theoretic construction of shape space that does not require diffeomorphisms or correspondence. A direct result of this sheaf theoretic construction is that in three dimensions for meshes, 0-dimensional homology is enough to characterize the shape.

One of the prevailing paradigms in data analysis involves comparing groups of samples to statistically infer features that discriminate them. However, many modern applications do not fit well into this paradigm because samples cannot be naturally arranged into discrete groups. In such instances, graph techniques can be used to rank features according to their degree of consistency with an underlying metric structure without the need to cluster the samples. Here, we extend graph methods for feature selection to abstract simplicial complexes and present a general framework for clustering-independent analysis. Combinatorial Laplacian scores take into account the topology spanned by the data and reduce to the ordinary Laplacian score when restricted to graphs. We show the utility of this framework with several applications to the analysis of gene expression and multi-modal cancer data. Our results provide a unifying perspective on topological data analysis and manifold learning approaches to the analysis of point clouds.

In this talk I will present four case studies of sheaves and cosheaves in topological data analysis. The first two are examples of (co)sheaves in the small:

(1) level set persistence---and its efficacious computation via discrete Morse theory---and,

(2) decorated merge trees and Reeb graphs---enriched topological invariants that have enhanced classification power over traditional TDA methods. The second set of examples are focused on (co)sheaves in the large:

(3) understanding the space of merge trees as a stratified map to the space of barcodes and

(4) the development of a new "sheaf of sheaves" that organizes the persistent homology transform over different shapes.

In recent years, topological and geometric data analysis (TGDA) has emerged as a new and promising field for processing, analyzing and understanding complex data. Indeed, geometry and topology form natural platforms for data analysis, with geometry describing the ''shape'' behind data; and topology characterizing / summarizing both the domain where data are sampled from, as well as functions and maps associated with them. In this talk, I will show how topological (and geometric ideas) can be used to analyze graph data, which occurs ubiquitously across science and engineering. Graphs could be geometric in nature, such as road networks in GIS, or relational and abstract. I will particularly focus on the reconstruction of hidden geometric graphs from noisy data, as well as graph matching and classification. I will discuss the motivating applications, algorithm development, and theoretical guarantees for these methods. Through these topics, I aim to illustrate the important role that topological and geometric ideas can play in data analysis.

 In this work we systematically introduce relations to topological data analysis (TDA) in the categories of sets, simplicial complexes and vector spaces to characterize and study the general dynamical behaviors in a consistent way. The proposed framework not only offers new insights to the classical TDA methodologies, but also motivates new approaches to interesting applications of TDA in dynamical metric spaces, dynamical coverings, etc. The associated algorithm which produces barcode invariants, and relations in more general categories will also be discussed.

In this work we present a thorough study of the theoretical properties and devise efficient algorithms for the persistent Laplacian, an extension of the standard combinatorial Laplacian to the setting of simplicial pairs: pairs of simplicial complexes related by an inclusion, which was recently introduced by Wang, Nguyen, and Wei. 

In analogy with the non-persistent case, we establish that the nullity of the q-th persistent Laplacian equals the q-th persistent Betti number of any given simplicial pair which provides an interesting connection between spectral graph theory and TDA. 

We further exhibit a novel relationship between the persistent Laplacian and the notion of Schur complement of a matrix. This relation permits us to uncover a link with the notion of effective resistance from network circuit theory and leads to a persistent version of the Cheeger inequality.

This relationship also leads to a novel and fundamentally different algorithm for computing the persistent Betti number for a pair of simplicial complexes which can be significantly more efficient than standard algorithms. 

Active matter systems, ranging from liquid crystals to populations of cells and animals, exhibit complex collective behavior characterized by pattern formation and dynamic phase transitions. However, quantitative analysis of these systems is challenging, especially for heterogeneous populations of varying sizes, and typically requires expertise in formulating problem-specific order parameters. I will describe an alternative approach, using a combination of topological data analysis and machine learning, to investigate emergent behaviors in self-organizing populations of interacting discrete agents.

Given a persistence diagram with n points, we give an algorithm that produces a sequence of n persistence diagrams converging in bottleneck distance to the input diagram, the ith of which has i distinct (weighted) points and is a 2-approximation to the closest persistence diagram with that many distinct points. For each approximation, we precompute the optimal matching between the ith and the (i+1)st. Perhaps surprisingly, the entire sequence of diagrams as well as the sequence of matchings can be represented in O(n) space. The main approach is to use a variation of the greedy permutation of the persistence diagram to give good Hausdorff approximations and assign weights to these subsets. We give a new algorithm to efficiently compute this permutation, despite the high implicit dimension of points in a persistence diagram due to the effect of the diagonal. The sketches are also structured to permit fast (linear time) approximations to the Hausdorff distance between diagrams -- a lower bound on the bottleneck distance. For approximating the bottleneck distance, sketches can also be used to compute a linear-size neighborhood graph directly, obviating the need for geometric data structures used in state-of-the-art methods for bottleneck computation.

Reducing a chain complex (whilst preserving its homotopy-type) using algebraic Morse theory ([1, 2, 3]) gives the same end-result as Gaussian elimination, but AMT does it only on certain rows/columns and with several pivots (in all matrices simultaneously). Crucially, instead of doing costly row/column operations on a sparse matrix, it computes traversals of a bipartite digraph. This significantly reduces the running time and memory load (smaller fill-in and coefficient growth of the matrices). However, computing with AMT requires the construction of a valid set of pivots (called a Morse matching).

In [4], we discover a family of Morse matchings on any chain complex of free modules of finite rank. We show that every acyclic matching is a subset of some member of our family, so all maximal Morse matchings are of this type.

Both the input and output of AMT are chain complexes, so the procedure can be used iteratively. When working over a field or a local PID, this process ends in a chain complex with zero matrices, which produces homology. However, even over more general rings, the process often reveals homology, or at least reduces the complex so much that other algorithms can finish the job. Moreover, it also returns homotopy equivalences to the reduced complexes, which reveal the generators of homology and the induced maps $H_{*}(\varphi)$. 

We design a new algorithm for reducing a chain complex and implement it in MATHEMATICA. We test that it outperforms other CASs. As a special case, given a sparse matrix over any field, the algorithm offers a new way of computing the rank and a sparse basis of the kernel (or null space), cokernel (or quotient space, or complementary subspace), image, preimage, sum and intersection subspace. It outperforms built-in algorithms in other CASs.

References

[1]  M. Jöllenbeck, Algebraic Discrete Morse Theory and Applications to Commutative Algebra, Thesis, (2005).

[2]  D.N. Kozlov, Discrete Morse theory for free chain complexes, C. R. Math. 340 (2005), no. 12, 867–872.

[3]  E. Sköldberg, Morse theory from an algebraic viewpoint, Trans. Amer. Math. Soc. 358 (2006), no. 1, 115–129.

[4]  L. Lampret, Chain complex reduction via fast digraph traversal, arXiv:1903.00783.

The well known node2vec algorithm has been used to explore network structures and represent the nodes of a graph in a vector space in a way that reflects the structure of the graph. Random walks in node2vec have been used to study the local structure through pairwise interactions. Our motivation for this project comes from a desire to understand higher-order relationships by a similar approach. To this end, we propose an extension of node2vec to a method for representing the k-simplices of a simplicial complex into Euclidean space. 

In this talk I outline a way to do this by performing random walks on simplicial complexes, which have a greater variety of adjacency relations to take into account than in the case of graphs. The walks on simplices are then used to obtain a representation of the simplices. We will show cases in which this method can uncover the roles of higher order simplices in a network and help understand structures in graphs that cannot be seen by using just the random walks on the nodes. 

Crystal Structure Prediction aims to reveal the properties that stable crystalline arrangements of a molecule have without stepping foot in a laboratory, consequently speeding up the discovery of new functional materials. Since it involves producing large datasets that themselves have little structure, an appropriate classification of crystals could add structure to these datasets and further streamline the process. We focus on geometric invariants, in particular introducing the density fingerprint of a crystal. After exploring its computations via Brillouin zones, we go on to show how it is invariant under isometries, stable under perturbations and complete at least for an open and dense space of crystal structures.

Ollivier Ricci curvature is a notion originated from Riemannian Geometry and suitable for applying on different settings from smooth manifolds to discrete structures such as (directed) hypergraphs. In the past few years, alongside Forman Ricci curvature, this curvature as an edge based measure, has become a popular and powerful tool for network analysis. This notion is defined based on optimal transport problem (Wasserstein distance) between sets of probability measures supported on data points and can nicely detect some important features such as clustering and sparsity in their structures. After introducing this notion for (directed) hypergraphs and mentioning some of its properties, as one of the main recent applications, I will present the result of implementation of this tool for the analysis of chemical reaction networks. 

Giancarlo Rota once wrote of matroids that "It is as if one were to
condense all trends of present day mathematics onto a single
structure, a feat that anyone would a priori deem impossible, were it
not for the fact that matroids do exist" (Indiscrete Thoughts, 1997).
This makes matroid theory a natural hub through which ideas flow from
one field of mathematics to the next. At the end of our three-day
workshop, participants will understand the most common objects and
constructions in matroid theory to the depth suitable for exploring
many of these interesting connections. We will also pick up some
highly practical matroid tools for working through problems in
persistent homology, (optimal) cycle representatives, and other
objects of interest in TDA.

Day 3, Lecture 1

Circuits in persistent homology

Day 3, Lecture 2

Exercise: write your own persistent homology algorithm!
 

Giancarlo Rota once wrote of matroids that "It is as if one were to
condense all trends of present day mathematics onto a single
structure, a feat that anyone would a priori deem impossible, were it
not for the fact that matroids do exist" (Indiscrete Thoughts, 1997).
This makes matroid theory a natural hub through which ideas flow from
one field of mathematics to the next. At the end of our three-day
workshop, participants will understand the most common objects and
constructions in matroid theory to the depth suitable for exploring
many of these interesting connections. We will also pick up some
highly practical matroid tools for working through problems in
persistent homology, (optimal) cycle representatives, and other
objects of interest in TDA.

Day 2, Lecture 1, 10-10.45am

Matroid representations, continued

Day 2, Lecture 2, 11-11.45am

Matroids in homological algebra

Giancarlo Rota once wrote of matroids that "It is as if one were to
condense all trends of present day mathematics onto a single
structure, a feat that anyone would a priori deem impossible, were it
not for the fact that matroids do exist" (Indiscrete Thoughts, 1997).
This makes matroid theory a natural hub through which ideas flow from
one field of mathematics to the next. At the end of our three-day
workshop, participants will understand the most common objects and
constructions in matroid theory to the depth suitable for exploring
many of these interesting connections. We will also pick up some
highly practical matroid tools for working through problems in
persistent homology, (optimal) cycle representatives, and other
objects of interest in TDA.

Condensed outline

Day 1, Lecture 1, 10-10.45am

Definitions There are many definitions of matroids. Here's how to organize them.
Examples We work with matroids every day. Here are a few you have seen.
Important properties What's so great about a matroid?

Day 1, Lecture 2, 11-11.45am

The essential operations: deletion, contraction, and dualization
Working with matroids: matrix representations