Past Applied Topology Seminar

Training deep learning models involves searching for a good model over the space of possible architectures and their parameters. Discovering models that exhibit robust generalization to unseen data and tasks is of paramount for accurate and reliable machine learning. Generalization, a hallmark of model efficacy, is conventionally gauged by a model's performance on data beyond its training set. Yet, the reliance on vast training datasets raises a pivotal question: how can deep learning models transcend the notorious hurdle of 'memorization' to generalize effectively? Is it feasible to assess and guarantee the generalization prowess of deep neural networks in advance of empirical testing, and notably, without any recourse to test data? This inquiry is not merely theoretical; it underpins the practical utility of deep learning across myriad applications. In this talk, I will show that scrutinizing the training dynamics of neural networks through the lens of topology, specifically using 'persistent-homology dimension', leads to novel bounds on the generalization gap and can help demystifying the inner workings of neural networks. Our work bridges deep learning with the abstract realms of topology and learning theory, while relating to information theory through compression.

In this talk I will describe a new model for time-varying graphs (TVGs) based on persistent topology and cosheaves. In its simplest form, this model presents TVGs as matrices with entries in the semi-ring of subsets of time; applying the classic Kleene star construction yields novel summary statistics for space networks (such as STARLINK) called "lifetime curves." In its more complex form, this model leads to a natural featurization and discrimination of certain Earth-Moon-Mars communication scenarios using zig-zag persistent homology. Finally, and if time allows, I will describe recent work with David Spivak and NASA, which provides a complete description of delay tolerant networking (DTN) in terms of an enriched double category.

Say we want to view the function-Rips multifiltration as an estimator. Then, what is the target? And what kind of consistency, bias, or convergence rate, should we expect? In this talk I will present on-going joint work with Ethan André (Ecole Normale Supérieure) that aims at laying the algebro-topological ground to start answering these questions.

Pseudo metrics between persistence modules can be defined starting from Noise Systems [1].  Such metrics are used to compare the modules directly or to extract stable vectorisations. While the stability property directly follows from the axioms of Noise Systems, finding algorithms or closed formulas to compute the distances or associated vectorizations  is often a difficult problem, especially in the multi-parameter setting. In this seminar I will show how extra properties of Noise Systems can be used to define algorithms. In particular I will describe how to compute stable vectorisations with respect to Wasserstein distances [2]. Lastly I will discuss ongoing work (with D. Lundin and R. Corbet) for the computation of a geometric distance (the Volume Noise distance) and associated invariants on interval modules.

[1] M. Scolamiero, W. Chachólski, A. Lundman, R. Ramanujam, S. Oberg. Multidimensional Persistence and Noise, (2016) Foundations of Computational Mathematics, Vol 17, Issue 6, pages 1367-1406. doi:10.1007/s10208-016-9323-y.

[2] J. Agerberg, A. Guidolin, I. Ren and M. Scolamiero. Algebraic Wasserstein distances and stable homological invariants of data. (2023) arXiv: 2301.06484.

I will discuss various aspects of multi-parameter persistence related to representation theory and decompositions into indecomposable summands, based on joint work with Magnus Botnan, Steffen Oppermann, Johan Steen, Luis Scoccola, and Benedikt Fluhr.

A classification of indecomposables is infeasible; the category of two-parameter persistence modules has wild representation type. We show [1] that this is still the case if the structure maps in one parameter direction are epimorphisms, a property that is commonly satisfied by degree 0 persistent homology and related to filtered hierarchical clustering. Furthermore, we show [2] that indecomposable persistence modules are dense in the interleaving distance, and that being nearly-indecomposable is a generic property of persistence modules. On the other hand, the two-parameter persistence modules arising from interleaved sets (relative interleaved set cohomology) have a very well-behaved structure [3] that is encoded as a complete invariant in the extended persistence diagram. This perspective reveals some important but largely overlooked insights about persistent homology; in particular, it highlights a strong reason for working at the level of chain complexes, in a derived category [4].

[1] Ulrich Bauer, Magnus B. Botnan, Steffen Oppermann, and Johan Steen, Cotorsion torsion triples and the representation theory of filtered hierarchical clustering, Adv. Math. 369 (2020), 107171, 51. MR4091895

[2] Ulrich Bauer and Luis Scoccola, Generic multi-parameter persistence modules are nearly indecomposable, 2022.

[3] Ulrich Bauer, Magnus Bakke Botnan, and Benedikt Fluhr, Structure and interleavings of relative interlevel set cohomology, 2022.

[4] Ulrich Bauer and Benedikt Fluhr, Relative interlevel set cohomology categorifies extended persistence diagrams, 2022.

High-dimensional data is often assumed to be distributed near a smooth manifold. But should we really believe that? In this talk I will introduce HADES, an algorithm that quickly detects singularities where the data distribution fails to be a manifold.

By using hypothesis testing, rather than persistent homology, HADES achieves great speed and a strong statistical foundation. We also have a precise mathematical theorem for correctness, proven using optimal transport theory and differential geometry. In computational experiments, HADES recovers singularities in synthetic data, road networks, molecular conformation space, and images.

Paper link: https://arxiv.org/abs/2311.04171
Github link: https://github.com/uzulim/hades
 

Topological data analysis (TDA) is an emerging field of research, which considers the application of topology to data analysis. Recently, these methods have been successfully applied to research problems in the field of geographical information science (GIS). This includes the problems of Point of Interest (PoI), street network and weather analysis. In this talk I will describe how TDA can be used to provide solutions to these problems plus how these solutions compare to those traditionally used by GIS practitioners. I will also describe some of the challenges of performing interdisciplinary research when applying TDA methods to different types of data.

The preferential attachment model is a natural and popular random graph model for a growing network that contains very well-connected ``hubs''. Despite intense interest in the higher-order connectivity of these networks, their Betti numbers at higher dimensions have been largely unexplored.

In this talk, after a brief survey on random topology, we study the clique complexes of preferential attachment graphs, and we prove the asymptotics of the expected Betti numbers. If time allows, we will briefly discuss their homotopy connectedness as well. This is joint work with Gennady Samorodnitsky, Christina Lee Yu and Rongyi He, and it is based on the preprint https://arxiv.org/abs/2305.11259

In this talk, I will present joint work with Omer Bobrowski:  a series of statements regarding the behaviour of persistence diagrams arising from random point-clouds. I will present evidence that, viewed in the right way, persistence values obey a universal probability law, that depends on neither the underlying space nor the original distribution of the point-cloud.  I will present two versions of this universality: “weak” and “strong” along with progress which has been made in proving the statements.  Finally, I will also discuss some applications of this phenomena based on detecting structure in data.

In topological data analysis, persistence barcodes record the
persistence of homological generators in a one-parameter filtration
built on the data at hand. In contrast, computing the pointwise Euler
characteristic (EC) of the filtration merely records the alternating sum
of the dimensions of each homology vector space.

In this talk, we will show that despite losing the classical
"signal/noise" dichotomy, EC tools are powerful descriptors, especially
when combined with new integral transforms mixing EC techniques with
Lebesgue integration. Our motivation is fourfold: their applicability to
multi-parameter filtrations and time-varying data, their remarkable
performance in supervised and unsupervised tasks at a low computational
cost, their satisfactory properties as integral transforms (e.g.,
regularity and invertibility properties) and the expectation results on
the EC in random settings. Along the way, we will give an insight into
the information these descriptors record.

This talk is based on the work [https://arxiv.org/abs/2111.07829] and
the joint work with Olympio Hacquard [https://arxiv.org/abs/2303.14040].

Various constructions relevant to practical problems such as clustering and graph classification give rise to multiparameter persistence modules (MPPM), that is, linear representations of non-totally ordered sets. Much of the mathematical interest in multiparameter persistence comes from the fact that there exists no tractable classification of MPPM up to isomorphism, meaning that there is a lot of room for devising invariants of MPPM that strike a good balance between discriminating power and complexity of their computation. However, there is no consensus on what type of information we want these invariants to provide us with, and, in particular, there seems to be no good notion of “global” or “high persistence” features of MPPM.

With the goal of substantiating these claims, as well as making them more precise, I will start with an overview of some of the known invariants of MPPM, including joint works with Bauer and Oudot. I will then describe recent work of Bjerkevik, which contains relevant open questions and which will help us make sense of the notion of global feature in multiparameter persistence.

Deep convolutional neural networks such as GoogLeNet and ResNet have become ubiquitous in image classification tasks, whereas
transformer-based language models such as BERT and its variants have found widespread use in natural language processing. In this talk, I
will discuss recent efforts in exploring the topology of artificial neuron activations in deep learning, from images to word embeddings.
First, I will discuss the topology of convolutional neural network activations, which provides semantic insight into how these models
organize hierarchical class knowledge at each layer. Second, I will discuss the topology of word embeddings from transformer-based models.
I will explore the topological changes of word embeddings during the fine-tuning process of various models and discover model confusions in
the embedding spaces. If time permits, I will discuss on-going work in studying the topology of neural activations under adversarial attacks.
 

In this talk, I will present the notion of projected barcodes and projected distances for multi-parameter persistence modules. Projected barcodes are defined as derived pushforward of persistence modules onto R. Projected distances come in two flavors: the integral sheaf metrics (ISM) and the sliced convolution distances (SCD). I will explain how the fibered barcode is a particular instance of projected barcodes and how the ISM and the SCD provide lower bounds for the convolution distance. 

Furthermore, in the case where the persistence module considered is the sublevel-sets persistence modules of a function f : X -> R^n, we will explain how, under mild conditions, the projected barcode of this module by a linear map u : R^n \to R is the collection of sublevel-sets barcodes of the composition uf . In particular, it can be computed using software dedicated to one-parameter persistence modules. This is joint work with Nicolas Berkouk.

Dimensionality reduction is the machine learning problem of taking a data set whose elements are described with potentially many features (e.g., the pixels in an image), and computing representations which are as economical as possible (i.e., with few coordinates). In this talk, I will present a framework to leverage the topological structure of data (measured via persistent cohomology) and construct low dimensional coordinates in classifying spaces consistent with the underlying data topology.

I will survey on the cluster structure of random geometric graphs in a regime that is less discussed in the literature. The statistics of interest include the number of k-components, the number of components, the number of vertices in the giant component, and the connectivity threshold. We show LLN and normal/Poisson approximation by Stein's method. Based on recent joint works with Mathew Penrose (Bath).

TBC

Multi-parameter persistence is a main research topic in topological data analysis. Major questions involve the computation and the structural properties
of persistence modules. In this context, I will sketch two very recent results:

(1) We define a natural bifiltration called the localized union-of-balls bifiltration that contains filtrations studied in the context of local persistent homology as slices. This bifiltration is not k-critical for any finite k. Still, we show that a representation of it (involving algebraic curves of low degree) can be computed exactly and efficiently. This is joint work with Matthias Soels (TU Graz).

(2) Every persistence modules permits a unique decomposition into indecomposable summands. Intervals are the simplest type of summands, but more complicated indecomposables can appear, and usually do appear in examples. We prove that for homology-dimension 0 and density-Rips bifiltration, at least a quarter of the indecomposables are intervals in expectation for a rather general class of point samples. Moreover, these intervals can be ``peeled off'' the module efficiently. This is joint work with Angel Alonso (TU Graz).

One-dimensional persistent homology encodes geometric information of data by means of a barcode decomposition. Often, one needs to relate the persistence barcodes of two datasets which are intrinsically linked, e.g. consider a sample from a large point cloud. Such connections are encoded through persistence morphisms; as in linear algebra, a (one-dimensional) persistence morphism is fully understood by fixing a basis in the domain and codomain and computing the associated matrix. However, in the literature and existing software, the focus is often restricted to interval decompositions of images, kernels and cokernels. This is the case of the Bauer-Lesnick matching, which is computed using the intervals from the image. Unfortunately, this matching has substantial differences from the structure of the persistence morphism in very simple examples. In this talk I will present an induced block function that is well-behaved in such examples. This block function is computed using the associated matrix of a persistence morphism and is additive with respect to taking direct sums of persistence morphisms. This is joint work with M. Soriano-Trigueros and R. Gonzalez-Díaz from Universidad de Sevilla.

Mapper and Ball Mapper are Topological Data Analysis tools used for exploring high dimensional point clouds and visualizing scalar–valued functions on those point clouds. Inspired by open questions in knot theory, new features are added to Ball Mapper that enable encoding of the structure, internal relations and symmetries of the point cloud. Moreover, the strengths of Mapper and Ball Mapper constructions are combined to create a tool for comparing high dimensional data descriptors of a single dataset. This new hybrid algorithm, Mapper on Ball Mapper, is applicable to high dimensional lens functions. As a proof of concept we include applications to knot and game  theory, as well as material science and cancer research. 

Using persistent homology to guide optimization has emerged as a novel application of topological data analysis. Existing methods treat persistence calculation as a black box and backpropagate gradients only onto the simplices involved in particular pairs. We show how the cycles and chains used in the persistence calculation can be used to prescribe gradients to larger subsets of the domain. In particular, we show that in a special case, which serves as a building block for general losses, the problem can be solved exactly in linear time. We present empirical experiments that show the practical benefits of our algorithm: the number of steps required for the optimization is reduced by an order of magnitude. (Joint work with Arnur Nigmetov.)