Titles & Abstracts

Day 1 (Feb 14 JST)

Speaker: Ippei Obayashi

Title: Stable volumes for persistent homology

Abstract:

Persistent homology enables us to characterize the shape of data quantitatively using topological structures such as connected components, rings, and cavities. A persistence diagram is the output of persistent homology, and it has the information of the shape of the data. Each point on the diagram corresponds to a homological structure such as a ring or a cavity. To utilize the diagram, we want to identify the structure. Homological optimization techniques are used to identify such a structure. However, such techniques often suffer from instability to noise, and we sometimes miss the minimal building blocks.

In this talk, I propose a novel method, "stable volumes," to overcome the problem. I will show the mathematical formalization and algorithms of stable volumes. We will prove some good properties of stable volumes for (n-1)-th persistent homology. We will also present examples of the application of stable volume to synthetic and realistic data. The method is already implemented in HomCloud. The results are available at arxiv: https://arxiv.org/abs/2109.11711.



Speaker: Magnus Bakke Botnan

Title: Signed barcodes for multiparameter persistence

Abstract:

Moving from persistent homology in one parameter to multiparameter persistence comes at a significant increase in complexity. In particular, the notion of a barcode does not generalize straightforwardly. However, in this talk, I will show how it is possible to assign a unique barcode to a multiparameter persistence module if one is willing to take Z-linear combinations of intervals. The theoretical discussion will be complemented by numerical experiments. This is joint work with Steffen Oppermann and Steve Oudot.



Speaker: Omer Bobrowski

Title: Persistent cycle registration and bootstrap

Abstract:

The motivating question for this talk is the following: Suppose that we are given a persistence diagram computed from random data. Pick a feature (cycle) of interest in the diagram. Can we determine whether this feature represents a "statistically significant" phenomenon, or was it generated purely by chance? We propose a novel approach to answer this question using a bootstrap-like method. The key idea is that if we resample the data, we expect features that represent essential phenomena in the underlying system to appear frequently in the resamples. The main ingredient in this framework is a new "cycle-registration" method that allows us to match persistent cycles that appear in two different samples.


This is joint work with Yohai Reani (Technion).

Day 2 (Feb 15 JST)

Speaker: Erik Lundberg

Title: Homotopy types of random cubical complexes associated to Bernoulli site percolation
Abstract:

We present results on the homotopy types of a random cubical complexes arising from the Bernoulli site percolation model. We investigate the homotopy counting measure of the random cubical complex (this normalized counting measure counts connected components according to homotopy type). Looking within an increasingly large viewing window, we get a sequence of random probability measures that converges in probability to a deterministic probability measure (this limiting homotopy measure was introduced by Sarnak and Wigman in the context of nodal sets of random wave functions). The main goal of the talk is to present quantitative estimates showing how the limiting homotopy measure depends on the coloring probability p of the percolation model. Our results show a qualitative change in the homotopy measure as p crosses the so-called percolation threshold. Specializing to the case of two dimensions, we also present empirical results that raise further questions on the p-dependence of the limiting homotopy measure. This is joint work with K. Alex Dowling.



Speaker: Cheng Xin

Title: Decomposition and Stability of Multi-parameter Persistence Modules
Abstract:

Decomposition is an important problem to study multi-parameter persistence modules. Unlike 1-parameter cases, a multi-parameter persistence module might not have a decomposition over interval modules and the decomposition structure can be very complicated. The bottleneck distance based on the optimal matching between decompositions of two multi-parameter persistence modules in general does not enjoy the stability property with respect to the interleaving distance. Meanwhile, it is NP-hard to compute the interleaving distance between two multi-parameter persistence modules.

Here we would like to talk about some techniques of decomposition and approximation which can help us derive some kinds of stability-like properties for different kinds of multi-parameter modules as far as we can. We first propose an algorithm to compute a closest rectangle module to an interval module as an approximation deviated by a measurable interleaving distance. Then we can build a rectangle decomposable module for a given interval decomposable module as an approximation which enjoys a stability-like property. Based on that, we introduce a strategy to generalize the matching distance for multi-parameter persistence modules in general.

This talk is based on joint work with Tamal K. Dey.



Speaker: Chao Chen

Title: Topology-Driven Learning for Biomedical Imaging Informatics
Abstract:

Thanks to decades of technology development, we are now able to visualize in high quality complex biomedical structures such as neurons, vessels, trabeculae and breast tissues. We need innovative methods to fully exploit these structures, which encode important information about underlying biological mechanisms. In this talk, we explain how topology, i.e., connected components, handles, loops, and branches, can be seamlessly incorporated into different parts of a learning pipeline. Under the hood is a formulation of the topological computation as a differentiable operator, based on the theory of topological data analysis. This leads to a series of novel methods for segmentation, generation, and analysis of these topology-rich biomedical structures. We will also briefly mention how topological information can be used in graph neural networks and noise/attack-robust machine learning.

Day 3 (Feb 16 JST)

Speaker: Emerson G. Escolar

Title: Interval Decomposability/Approximation of Persistence Modules, and their Computation
Abstract:

In contrast to one-parameter persistent homology, there is no complete discrete invariant for multiparameter persistence, and the classification of all indecomposable modules is known to be a wild problem. One approach is to consider the subclass of interval-decomposable persistence modules, which are direct sums of interval representations. In this talk, we discuss several topics related to this approach. In particular, we introduce a new invariant for 2-parameter persistence modules called the compressed multiplicity generalizing the dimension vector and the rank invariant. In addition, we discuss an "interval-decomposable approximation" using Mobius inversion. Finally, we provide example computations using the GAP package pmgap (https://github.com/emerson-escolar/pmgap) under development.


This talk is based on joint works with Hideto Asashiba, Mickaël Buchet, Ken Nakashima, and Michio Yoshiwaki.



Speaker: Mathieu Carrière

Title: Statistical analysis of Mapper for stochastic and multivariate filters
Abstract:

Reeb spaces, as well as their discretized versions called Mappers, are common descriptors used in Topological Data Analysis, with plenty of applications in various fields of science, such as computational biology and data visualization, among others. The stability and quantification of the rate of convergence of the Mapper to the Reeb space has been studied a lot in recent works, focusing on the case where a scalar-valued filter is used for the computation of Mapper. On the other hand, much less is known in the multivariate case, when the codomain of the filter is Rp, and in the general case, when it is a general metric space (Z, dZ), instead of R. In this talk, I will introduce a slight modification of the usual Mapper construction and give risk bounds for estimating the Reeb space using this estimator. Our approach applies in particular to the setting where the filter function used to compute Mapper is also estimated from data, such as the eigenfunctions of PCA. I will also provide preliminary applications of this estimator in statistics and machine learning for different kinds of target filters.


This is joint work with Bertrand Michel.



Speaker: András Mészáros

Title: The local weak limit of $k$-dimensional hypertrees
Abstract:

Let $\mathcal{C}(n,k)$ be the set of $k$-dimensional simplicial complexes $C$ over a fixed set of $n$ vertices such that:


\begin{enumerate}[(1)]

\item $C$ has a complete $k-1$-skeleton;

\item $C$ has precisely ${{n-1}\choose {k}}$ $k$-faces;

\item the homology group $H_{k-1}(C)$ is finite.

\end{enumerate}


Consider the probability measure on $\mathcal{C}(n,k)$ where the probability of a simplicial complex $C$ is proportional to $|H_{k-1}(C)|^2$. For any fixed $k$, we determine the local weak limit of these random simplicial complexes as $n$ tends to infinity.


This local weak limit turns out to be the same as the local weak limit of the $1$-out $k$-complexes investigated by Linial and Peled.

Day 4 (Feb 17 JST)

Speaker: D. Yogeshwaran

Title: Poisson process approximation for critical points of random distance function

Abstract:

We shall consider extremal critical points (along with scaled critical radius) of a random distance function and show a Poisson process approximation result for the same. This is obtained as a consequence of a general Poisson process approximation result for stabilizing functionals of Poisson processes that arise in stochastic geometry. The bounds are derived for the Kantorovich-Rubinstein distance between a point process and an appropriate Poisson point process. This is a joint work with Omer Bobrowski (Technion) and Matthias Schulte (Hamburg Institute of Technology).


Speaker: Shu Kanazawa

Title: On the large deviation principle for persistence diagrams of random cubical filtration

Abstract:

A random cubical filtration is an increasing family of random cubical sets, which are the union of randomly generated higher-dimensional unit cubes with integer coordinates. The objective of this work is to investigate the asymptotic behavior of the (random) persistence diagrams of a random cubical filtration model as the window size tends to infinity. Recently, the strong law of large numbers for the persistence diagrams was proved by Hiraoka, Miyanaga, and Tsunoda, which states that the persistence diagram converges vaguely to a deterministic measure almost surely.

In this talk, we are interested in the decay rate of the probability that the persistence diagram is far from the deterministic limiting measure. We show large deviation principles for Betti numbers, persistent Betti numbers, and the histograms generated by counting the birth-death pairs falling in each fine rectangular region. The key tool for the proofs is a general large deviation principle for regular nearly additive processes, established by Seppäläinen and Yukich. Time permitting, we will also discuss the ongoing work on how to provide the large deviation principle for the persistence diagrams themselves.

This talk is based on joint work with Yasuaki Hiraoka, Jun Miyanaga, and Kenkichi Tsunoda.



Speaker: Bastian Rieck

Title: Topological Representation Learning: A Differentiable Perspective

Abstract:

Modern machine learning techniques, in particular those in representation learning, are based on the concept of having dynamic representations of a data set that can be updated continuously during training. This is in stark contrast to the ideas of topological data analysis (TDA), which are, for the most part, of a fundamentally discrete nature. Fortunately, this gap started to be bridged recently!

In this talk, I will discuss recent advances in TDA that enable the development of topology-based representation learning techniques. I will provide a brief theoretical introduction to this problem and showcase applications for learning representations of structured and unstructured data sets.

Day 5 (Feb 18 JST)

Speaker: Michael Lesnick

Title: ℓ^p-Metrics on Multiparameter Persistence Modules

Abstract:

Joint work with Håvard Bjerkevik


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 ∈ [1,∞], we in fact introduce two such generalizations d_I^p and d_M^p, such that d_I^∞ equals the interleaving distance and d_M^∞ equals the matching distance. Finally, we show that on 1- or 2-parameter persistence modules over prime fields, d_I^p is the universal (i.e., largest) metric satisfying a natural stability property; this result extends a stability theorem of Skraba and Turner for the p-Wasserstein distance on barcodes in the 1-parameter case, and is also a close analogue of a universality property for the interleaving distance given by the second author. We also show that d_M^p ≤ d_I^p for all p ∈ [1,∞], extending an observation of Landi in the p = ∞ case. We observe that the distances d_M^p can be efficiently approximated. In a forthcoming paper, we apply some of these results to study the stability of (2-parameter) multicover persistent homology.



Speaker: Benjamin Blanchette

Title: Homological approximations in persistence theory, joint work with Thomas Brüstle and Eric Hanson

Abstract:

We define a class of invariants, which we call homological invariants, for persistence modules over a finite poset. Informally, a homological invariant is one that respects some exact structure and takes values in the free abelian group generated by a finite set of indecomposable modules. We focus in particular on groups generated by "spread modules", which are sometimes called "interval modules" in the persistence theory literature. We show that both the dimension vector and rank invariant are equivalent to homological invariants taking values in groups generated by spread modules. We also show that that the free abelian group generated by the "single-source" spread modules gives rise to a new invariant which is finer than the rank invariant.

There is a preprint with our work available here: https://arxiv.org/abs/2112.07632



Speaker: Bei Wang

Title: Hypergraph Co-Optimal Transport: Metric and Categorical Properties

Abstract:

Hypergraphs capture multi-way relationships in data, and they have consequently seen a number of applications in higher-order network analysis, computer vision, geometry processing, and machine learning. We develop the theoretical foundations in studying the space of

hypergraphs using ingredients from optimal transport. By enriching a hypergraph with probability measures on its nodes and hyperedges, as well as relational information capturing local and global structure, we obtain a general and robust framework for studying the collection of all hypergraphs. First, we introduce a hypergraph distance based on the co-optimal transport framework of Redko et al. and study its theoretical properties. Second, we formalize common methods for transforming a hypergraph into a graph as maps from the space of hypergraphs to the space of graphs and study their functorial properties and Lipschitz bounds. Finally, we demonstrate the versatility of our Hypergraph Co-Optimal Transport (HyperCOT) framework through various examples. This is joint work with Samir Chowdhury, Tom Needham, Ethan Semrad, and Youjia Zhou. Preprint is

available at https://arxiv.org/abs/2112.03904.