Recent years have seen an enormous growth in the applications of topology to other disciplines from the biological sciences to materials science, and from dynamical systems to cosmology and engineering. Many of these are factored through “topological data analysis” (TDA), but not all, with notable exceptions among those from dynamics and from engineering. All of these applications are due to topology’s capacity to define precise invariants from imprecise data: topological invariants (like the winding number of a curve around a point) are usually discrete and have some stability properties (here, to arbitrary perturbations that don’t move points as much as their distance to that point) that make them attractive.
However, topological stability is quite different from ordinary statistical stability. A single outlier can completely change the apparent topology of a space. One of the ways of dealing with this, persistence, has had numerous applications within pure math in recent years (in differential geometry, group theory, and approximation theory, to name three). The study of topology of random processes, and how the randomness perturbs topology is thus arising as an important scientific issue with potentially very wide significance. This workshop will bring together workers who have been dealing with this in different settings and in different ways, which should lead to progress in both application domains, and in a longer run, on the fundamental problems.
This workshop will include a poster session. In order to propose a poster, you must first register for the workshop, and then submit a poster proposal using the form that will become available on this page after you register. The registration form should not be used to propose a poster. The deadline for proposing a poster is March 3, 2023.
The correspondence of max-flow to the absolute permeability of porous systems
Speaker: Vanessa Robins (The Australian National University)
The absolute permeability of porous media quantifies the resistance of the material to fluid flow through it. This is an important parameter for various technological applications including ground water hydrology, hydrocarbon recovery, and microfluidics. There are scaling relationships between the geometric structure of a porous domain and its absolute permeability within a given class of structure. However, there exists no universal relationship between permeability and structure. We investigate the max-flow min-cut theorem to provide new insights into the structure of porous domains that most influence absolute permeability. The max-flow min-cut theorem states that the maximum flow through any network is exactly the sum of the edge weights that define the minimum cut. We hypothesize that the min-cut can be related to network permeability. We demonstrate that flow in porous media can be generalized and modeled as described by the max-flow min-cut theorem, which provides an approach to measure the absolute permeability of 3-dimensional digital images of porous media. The max-flow of a network is found to correspond to its absolute permeability for over four orders of magnitude and identifies structural regions that result in significant energy dissipation. The findings are beneficial for the design of porous materials, as a subroutine for digital rock studies, the simplification of large network models, and further fundamental studies on the structure and flow properties of porous media.
10:00-10:30 CDT
Coffee Break
10:30-11:15 CDT
Large deviation principle for geometric and topological functionals and associated point processes
Speaker: Takashi Owada (Purdue University)
We prove a large deviation principle for the point process associated to k-element connected components in R^d with respect to the connectivity radii decaying to 0. The random points are generated from a homogeneous Poisson point process or the corresponding binomial point process, so that the connecting radius is of the sparse regime. The rate function for the obtained large deviation principle can be represented as relative entropy. As an application, we deduce large deviation principles for various functionals and point processes appearing in stochastic geometry and topology. As concrete examples of topological invariants, we consider persistent Betti numbers of geometric complexes and the number of Morse critical points of the min-type distance function. The large deviation principle for the volume of k-nearest neighbor balls is also discussed.
This is joint work with Christian Hirsch.
11:30-12:30 CDT
Lunch
12:30-13:15 CDT
Distortion, on the Average and in Expectation
Speaker: Herbert Edelbrunner (ISTA – Institute of Science and Technology Austria)
We generalize the concept of the Voronoi path of a line to more general shapes and compute the distortion constant, which describes how it changes volume on the average. Although initially asked for a Poisson point process, the distortion is a characteristic property of the space rather than the point process. In other words, the constant ratio of the perimeter of a circle and its pixelation—and the analogous ratios for spheres in three and higher dimensions—hold for all smoothly embedded shapes on average.
This is joint work with Anton Nikitenko.
13:35-2:10 CDT
Identifying Nonlinear Dynamics with High Confidence from Sparse Data
Speaker: Konstantin Mishaikow (Rutgers University)
We introduce a novel procedure that, given sparse data generated from a stationary deterministic nonlinear dynamical system, can characterize specific local and/or global dynamic behavior with rigorous probability guarantees. More precisely, the sparse data is used to construct a statistical surrogate model based on a Gaussian process (GP). The dynamics of the surrogate model is interrogated using combinatorial methods and characterized using algebraic topological invariants (Conley index). The GP predictive distribution provides a lower bound on the confidence that these topological invariants, and hence the characterized dynamics, apply to the unknown dynamical system (a sample path of the GP).
14:35-15:00 CDT
Coffee Break
15:00-15:45 CDT
Data Analysis in Metric Spaces Through Metric Observables
Speaker: Washington Mio (Florida State University)
We discuss ways of probing the shape of data in metric spaces using metric observables (1-Lipschitz functions). This leads to a dimension reduction, data visualization, and statistical analysis technique that we call Principal Observable Analysis. We also discuss the construction of robust data summaries associated with such metric observables in the form of filtered merge trees, Reeb graphs, and Leray-Reeb pre-cosheaves.
Tuesday, March 21, 2023
9:00-9:45 CDT
Exploring the Topology of Word Embeddings
Speaker: Bei Wang (University of Utah)
Transformer-based language models such as BERT and its variants have found widespread use in natural language processing. A common way of using these models is to fine-tune them to improve their performance on a specific task. However, it is currently unclear how the fine-tuning process affects the underlying structure of the word embeddings from these models. In this talk, I will discuss recent efforts in exploring the topology of these fine-tuned word embeddings.
10:00-10:30 CDT
Coffee Break
10:30-11:15 CDT
On Complexity of Computing Bottleneck and Lexicographic Optimal Cycles in a Homology Class
Speaker: Erin Wolf-Chambers (St. Louis University)
Homology features of spaces which appear in applications, for instance 3D meshes, are among the most important topological properties of these objects. Given a non-trivial cycle in a homology class, we consider the problem of computing a representative in that homology class which is optimal. We study two measures of optimality, namely, the lexicographic order of cycles (the lex-optimal cycle) and the bottleneck norm (a bottleneck-optimal cycle). We give a simple algorithm for computing the lex-optimal cycle for a 1-homology class in a closed orientable surface. In contrast to this, our main result is that, in the case of 3-manifolds of size n² in the Euclidean 3-space, the problem of finding a bottleneck optimal cycle cannot be solved more efficiently than solving a system of linear equations with an n × n sparse matrix. From this reduction, we deduce several hardness results. Most notably, we show that for 3-manifolds given as a subset of the 3-space of size n², persistent homology computations are at least as hard as rank computation (for sparse matrices) while ordinary homology computations can be done in O(n² log n) time. To the best of our knowledge, this is the first such distinction between these two computations. Moreover, it follows that the same disparity exists between the height persistent homology computation and general sub-level set persistent homology computation for simplicial complexes in 3-space.
11:30-12:30 CDT
Lunch
12:30-13:15 CDT
Topology and Local Geometry of the Eden Model
Speaker: Benjamin Schweinhart (George Mason University)
The Eden cell growth model is a simple discrete stochastic process which produces a “blob” in Rd: start with one cube in the regular grid, and at each time step add a neighboring cube uniformly at random. This process has been used as a model for the growth of aggregations, tumors, and bacterial colonies and the healing of wounds, among other natural processes. Here, we study the topology and local geometry of the resulting structure, establishing asymptotic bounds for Betti numbers. Our main result is that the Betti numbers at time t grow at a rate between t(d-1)/d and Pd(t), where Pd(t) is the size of the site perimeter. Assuming a widely believed conjecture, this establishes the rate of growth of the Betti numbers in every dimension. We also present the results of computational experiments on finer aspects of the geometry and topology, such as persistent homology and the distribution of shapes of holes. This research is joint work with Fedor Manin and Érika Roldán.
13:35-14:20 CDT
Homology of Gaussian random chain complexes
Speaker: Matthew Kahle (Ohio State University)
We introduce a model of random chain complex over the reals. The main question we are interested in is: what does the homology of such a random chain complex typically look like? We show that under various hypothesis, homology tries to be as small as possible. This is joint work in progress with Ayat Ababneh.
14:35-15:00 CDT
Coffee Break
15:00-15:45 CDT
Persistence Diagrams, Euler Characteristics, and Mobius Inversion
Speaker: Primoz Skraba (Queen Mary University of London and Jozef Stefan Institute, Slovenia)
There are many ways of defining persistence diagrams. In this talk I will discuss the definition based on the Mobius inversion function which was introduced by Amit Patel under the name Generalized Persistence Diagrams. I will cover how this approach has appeared implicitly and explicitly in various results on persistence as well as various implications of this approach and (very) new developments. In particular, I will cover a surprising connection between Euler characteristics and persistence diagrams. This is joint work with Amit Patel.
Wednesday, March 22, 2023
9:00-9:45 CDT
Universality in Random Persistence Diagrams
Speaker: Omer Bobrowski (Technion – Israel Institute of Technology)
10:00-10:30 CDT
Coffee Break
10:30-11:15 CDT
Random fields beyond the null: building models from critical points
Speaker: Jonathan Taylor (Stanford University)
Random field theory (RFT) has proven successful at understanding the null behavior of smooth random fields under the null / centered model. The main tool used in such analyses is the Kac-Rice formula that essentially enables description of a Palm theory for the process. Using the Kac-Rice framework, we consider inference in models derived from the critical points of the process itself.
11:30-12:30 CDT
Lunch
12:30-13:15 CDT
Combining network analysis and persistent homology for classifying behavior of time series
Speaker: Liz Munch (Michigan State University)
Persistent homology, the flagship method of topological data analysis, can be used to provide a quantitative summary of the shape of data. One way to pass data to this method is to start with a finite, discrete metric space (whether or not it arises from a Euclidean embedding) and to study the resulting filtration of the Rips complex. In this talk, we will discuss several available methods for turning a time series into a discrete metric space, such as the Takens embedding, k-nearest neighbor network, and ordinal partition network. Combined with persistent homology and machine learning methods, we show how this can be used to classify behavior and investigate bifurcations from time series in both synthetic and experimental data.
13:35-14:20 CDT
Approximating Persistent Homology for Large Datasets
Speaker: Anthea Monod (Imperial College)
Persistent homology is an important methodology from topological data analysis which adapts theory from algebraic topology to data settings and has been successfully implemented in many applications; it produces a summary in the form of a persistence diagram, which captures the shape and size of the data. Despite its popularity, persistent homology is simply impossible to compute for very large datasets which prohibits its widespread use in many big data settings. What can we do if we would like a representative persistence diagram for a very large dataset whose persistent homology we cannot compute due to size restrictions? We adapt here the classical statistical method of bootstrapping, namely, drawing and studying smaller subsamples from the large dataset. We show that the mean of the persistence diagrams of subsamples is a valid approximation of the persistence diagram of the large dataset and derive its convergence rate to the true persistent homology of the large dataset. We demonstrate our approach on synthetic and real data; furthermore, we give an example of the utility of our approach in a shape clustering problem where we are able to obtain accurate results with only 2% subsampled from the original dataset.
14:30-15:30 CDT
Hybrid Poster Session
Thursday, March 23, 2023
9:00-9:45 CDT
Persistent Homology of Periodic Maps
Speaker: Teresa Heiss (Institute of Science and Technology Austria (IST Austria))
Persistent homology is well-defined and well studied for tame functions, for example various ones arising from finite point sets. However, periodic functions — for example used to study periodic point sets, like the atoms of a crystal — are not tame. We therefore extend the definition of persistent homology to periodic functions, which is a non-trivial endeavor. In contrast to related work, we quantify how the multiplicities of persistence pairs tend to infinity with increasing window size, in a way that is stable under perturbations and invariant under different finite representations of the infinite periodic function. This project is still ongoing research, but I’ll explain what we already know and what we don’t know yet.
10:00-10:30 CDT
Coffee Break
10:30-11:15 CDT
Curvature Sets Over Persistence Diagrams
Speaker: Facundo Mémoli (Ohio State University)
We study a family of invariants of compact metric spaces that combines the Curvature Sets defined by Gromov in the 1980s with Vietoris-Rips Persistent Homology. For given integers k ≥ 0 and n ≥ 1 we consider the dimension k Vietoris-Rips persistence diagrams of all subsets of a given metric space with cardinality at most n. We call these invariants persistence sets and call their measure theoretic counterparts persistence measures. We establish that (1) computing these invariants is often significantly more efficient than computing the usual Vietoris-Rips persistence diagrams, (2) these invariants have very good discriminating power and, in many cases, capture information that is imperceptible through standard Vietoris-Rips persistence diagrams, and (3) they enjoy stability properties. We precisely characterize some of them in the case of spheres and surfaces with constant curvature using a generalization of Ptolemy’s inequality. We also identify a rich family of metric graphs for which one of the simplest members of the family of persistence sets fully recovers their homotopy type by studying split-metric decompositions. Along the way we prove some useful properties of Vietoris-Rips persistence diagrams using Mayer-Vietoris sequences. These yield a geometric algorithm for computing the Vietoris-Rips persistence diagram of a space X with cardinality 2k + 2 with quadratic time complexity as opposed to the much higher cost incurred by the usual algebraic algorithms relying on matrix reduction. This is joint work with Mario Gomez.
11:30-12:30 CDT
Lunch
12:30-13:15 CDT
Probabilistic Generative Frameworks for Sampling 3D Complex Shapes and Images
Speaker: Lorin Crawford (Microsoft Research)
The recent curation of large-scale databases with 3D surface scans of shapes has motivated the development of computational tools that better detect global patterns in morphological variation. Recent studies have focused on developing methods for the task of sub-image selection which aims at identifying physical features that best describe the variation between classes of 3D objects. A large piece in assessing the utility of these approaches is to demonstrate their performance on both simulated and real datasets. However, when creating a model for shape statistics, real data can be difficult to access and the sample sizes within these data are often small due to expensive collection procedures. Meanwhile, the landscape of current shape simulation methods has been mostly limited to approaches that use black-box inference—making it difficult to systematically assess the power and calibration of sub-image models. In this talk, we present a new statistical framework for simulating realistic 2D and 3D shapes based on probability distributions which can be learned from real data. We demonstrate this framework in two applications within computational biology: (1) cellular imagining of neutrophils and (2) mandibular molars from four different suborders of primates.
13:35-14:20 CDT
Some topological properties of subgaussian fields of Riemannian manifolds
Speaker: Daniel Perez (École normale supérieure – ENS)
We discuss some topological properties of the persistent homology of subgaussian fields on compact Riemannian manifolds. For this class of processes, we are able to infer many desirable properties regarding the distribution of the persistence diagrams of the process, as well as some of its representations.
We adopt the point of view of persistence measures and give out some results regarding the convergence of the empirical mean diagram towards the mean of the distribution (as defined by duality by the action of the persistence measures on measurable sets of $mathbb{R}^2$) and discuss the future perspectives of this subject.
14:30-15:15 CDT
Clustering of discrete measures via mean measure quantization: application to unsupervised vectorization of persistence diagrams
Speaker: Fred Chazal (INRIA)
Robust topological information commonly comes in the form of a set of persistence diagrams that can be seen as discrete measures and are uneasy to use in generic machine learning frameworks.
In this talk we will introduce a fast, learnt, unsupervised vectorization method for measures in Euclidean spaces and use it for clustering of distributions of discrete measures. The algorithm is simple and efficiently discriminates important space regions where meaningful differences to the ‘’mean’’ measure arise. Applied to persistence diagrams, we will show that it is proven to be able to separate clusters of persistence diagrams. We will illustrate the strength and robustness of our approach on a few synthetic and real data sets.
Friday, March 24, 2023
9:00-9:45 CDT
Using Zigzag Persistence for Bifurcation Detection
Speaker: Sarah Tymochko (University of California, Los Angeles (UCLA))
Bifurcations in a dynamical system are drastic behavioral changes, thus being able to detect the parameter values for which these bifurcations occur is essential to understanding the system overall. We develop a one-step method to study and detect bifurcations using zigzag persistent homology. While standard persistent homology has been used in this setting, it usually requires analyzing a collection of persistence diagrams, which in turn drives up the computational cost. Using zigzag persistence, we can capture topological changes in the state space of the dynamical system in only one persistence diagram.
10:00-10:30 CDT
Coffee Break
10:30-11:15 CDT
Erdos-Renyi perturbed random geometric graphs
Speaker: Yusu Wang (University of California, San Diego (UCSD))
Graphs and network data are ubiquitous across a wide spectrum of scientific and application domains. Often in practice, an input graph can be considered as an observed snapshot of a (potentially continuous) hidden domain or process. Subsequent analysis and inferences are then performed on this observed graph. In this talk, we will describe a natural network model: we assume that there is a true grpah wihch is a certain proximity graph for points sampled from a nice domain. Then the observed graph is an Erdos-Renyi (ER) type perturbed version of that true graph, where random edges can be inserted and deleted. We will describe two main results obtained: the first concerns the recovery of shortest path metric of the true graph from this ER-perturbed graph. The second concerns the behavior of the so-called clique number of the graph. This is joint work with M. Kahle, S. Parthasarathy, S. Sivakoff, and M. Tian.
11:30-12:15 CDT
Multi-Scale Analysis of Random Functions on Graphs
Speaker: Alex Strang (University of Chicago)
The global structure of edge flows, alternating functions on graph edges, and their simplicial extensions can be characterized via projection onto principled subspaces associated with graph operators. Classically, the edge-incidence matrix acts as a discrete gradient whose associated subspaces are widely used to decompose an edge flow into a conservative and rotational component. Simplicial homology extends this idea to complexes via chains of operators that map between simplices of neighboring dimension. The expected structure of randomly sampled functions is determined by the covariance of the sampled function, and the projector onto the associated subspace. We analyze the relation between expected structure, covariance, and graph topology, by expanding both the covariance and the projector onto powers of a weighted adjacency matrix defined by the operator and its adjoint. We show that, in this basis, the expected structure is easily interpreted using the coefficients of the associated power series, and that truncating the series can produce accurate approximations even when the graph topology is not known exactly. We then study the spectral characteristics of graph families that enable approximation with a small number of terms.
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