Forthcoming events in this series


Fri, 07 Feb 2025
15:00
L4

Decomposing Multiparameter Persistence Modules

Jan Jendrysiak
(TU Graz)

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Abstract

Dey and Xin (J. Appl.Comput.Top. 2022) describe an algorithm to decompose finitely presented multiparameter persistence modules using a matrix reduction algorithm. Their algorithm only works for modules whose generators and relations are distinctly graded. We extend their approach to work on all finitely presented modules and introduce several improvements that lead to significant speed-ups in practice.


Our algorithm is FPT with respect to the maximal number of relations with the same degree and with further optimisation we obtain an O(n3) algorithm for interval-decomposable modules. As a by-product to the proofs of correctness we develop a theory of parameter restriction for persistence modules. Our algorithm is implemented as a software library aida which is the first to enable the decomposition of large inputs.

This is joint work with Tamal Dey and Michael Kerber.

Fri, 24 Jan 2025
15:00
L4

Efficient computation of the persistent homology of Rips complexes

Katharine Turner
(Australian National University)

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Abstract

Given a point cloud in Euclidean space and a fixed length scale, we can create simplicial complexes (called Rips complexes) to represent that point cloud using the pairwise distances between the points. By tracking how the homology classes evolve as we increase that length scale, we summarise the topology and the geometry of the “shape” of the point cloud in what is called the persistent homology of its Rips filtration. A major obstacle to more widespread take up of persistent homology as a data analysis tool is the long computation time and, more importantly, the large memory requirements needed to store the filtrations of Rips complexes and compute its persistent homology. We bypass these issues by finding a “Reduced Rips Filtration” which has the same degree-1 persistent homology but with dramatically fewer simplices.

The talk is based off joint work is with Musashi Koyama, Facundo Memoli and Vanessa Robins.

Fri, 06 Dec 2024
15:00
L5

From single neurons to complex human networks using algebraic topology

Lida Kanari
(École Polytechnique Fédérale de Lausanne (EPFL))

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Abstract

Topological data analysis, and in particular persistent homology, has provided robust results for numerous applications, such as protein structure, cancer detection, and material science. In the field of neuroscience, the applications of TDA are abundant, ranging from the analysis of single cells to the analysis of neuronal networks. The topological representation of branching trees has been successfully used for a variety of classification and clustering problems of neurons and microglia, demonstrating a successful path of applications that go from the space of trees to the space of barcodes. In this talk, I will present some recent results on topological representation of brain cells, with a focus on neurons. I will also describe our solution for solving the inverse TDA problem on neurons: how can we efficiently go from persistence barcodes back to the space of neuronal trees and what can we learn in the process about these spaces. Finally, I will demonstrate how algebraic topology can be used to understand the links between single neurons and networks and start understanding the brain differences between species. The organizational principles that distinguish the human brain from other species have been a long-standing enigma in neuroscience. Human pyramidal cells form highly complex networks, demonstrated by the increased number and simplex dimension compared to mice. This is unexpected because human pyramidal cells are much sparser in the cortex. The number and size of neurons fail to account for this increased network complexity, suggesting that another morphological property is a key determinant of network connectivity. By comparing the topology of dendrites, I will show that human pyramidal cells have much higher perisomatic (basal and oblique) branching density. Therefore greater dendritic complexity, a defining attribute of human L2 and 3 neurons, may provide the human cortex with enhanced computational capacity and cognitive flexibility.

Fri, 15 Nov 2024
15:00
L5

On the Limitations of Fractal Dimension as a Measure of Generalization

Inés García-Redondo
(Imperial College)
Abstract
Bounding and predicting the generalization gap of overparameterized neural networks remains a central open problem in theoretical machine learning. There is a recent and growing body of literature that proposes the framework of fractals to model optimization trajectories of neural networks, motivating generalization bounds and measures based on the fractal dimension of the trajectory. Notably, the persistent homology dimension has been proposed to correlate with the generalization gap. In this talk, I will present an empirical evaluation of these persistent homology-based generalization measures, with an in-depth statistical analysis. This study reveals confounding effects in the observed correlation between generalization and topological measures due to the variation of hyperparameters. We also observe that fractal dimension fails to predict generalization of models trained from poor initializations; and reveal the intriguing manifestation of model-wise double descent in these topological generalization measures. This is joint work with Charlie B. Tan, Qiquan Wang, Michael M. Bronstein and Anthea Monod.
 
Fri, 08 Nov 2024
15:00
L5

Topological Analysis of Bone Microstructure, Directed Persistent Homology and the Persistent Laplacian for Data Science

Ruben Sanchez-Garcia
(University of Southampton)

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Abstract

In this talk, I will give an overview of recent joint work on Topological Data Analysis (TDA). The first one is an application of TDA to quantify porosity in pathological bone tissue. The second is an extension of persistent homology to directed simplicial complexes. Lastly, we present an evaluation of the persistent Laplacian in machine learning tasks. This is joint work with Ysanne Pritchard, Aikta Sharma, Claire Clarkin, Helen Ogden, and Sumeet Mahajan; David Mendez; and Tom Davies and Zhengchao Wang, respectively.
 

Fri, 01 Nov 2024
15:00
L5

Generalized Multiple Subsampling for Persistent Homology

Yueqi Cao
(Imperial College London)
Abstract

Persistent homology is infeasible to compute when a dataset is very large. Inspired by the bootstrapping method, Chazal et al. (2014) proposed a multiple subsampling approach to approximate the persistence landscape of a massive dataset. In this talk, I will present an extension of the multiple subsampling method to a broader class of vectorizations of persistence diagrams and to persistence diagrams directly. First, I will review the statistical foundation of the multiple subsampling approach as applied to persistence landscapes in Chazal et al. (2014). Next, I will talk about how this analysis extends to a class of vectorized persistence diagrams called Hölder continuous vectorizations. Finally, I will address the challenges in applying this method to raw persistence diagrams for two measures of centrality: the mean persistence measure and the Fréchet mean of persistence diagrams. I will demonstrate these methods through simulation results and applications in estimating data shapes. 

Fri, 18 Oct 2024

15:00 - 16:00
L5

Extended Pareto grid: a tool to compute the matching distance in biparameter persistent homology

Francesca Tombari
((University of Oxford))
Abstract

Multiparameter persistence is an area of topological data analysis that synthesises the geometric information of a topological space via filtered homology. Given a topological space and a function on it, one can consider a filtration given by the sublevel sets of the space induced by the function and then take the homology of such filtration. In the case when the filtering function assumes values in the real plane, the homological features of the filtered object can be recovered through a "curved" grid on the plane called the extended Pareto grid of the function. In this talk, we explore how the computation of the biparameter matching distance between regular filtering functions on a regular manifold depends on the extended Pareto grid of these functions. 
 

Fri, 14 Jun 2024

15:00 - 16:00
L5

The bifiltration of a relation, extended Dowker duality and studying neural representations

Melvin Vaupel
(Norweign University of Science and Technology)
Abstract

To neural activity one may associate a space of correlations and a space of population vectors. These can provide complementary information. Assume the goal is to infer properties of a covariate space, represented by ochestrated activity of the recorded neurons. Then the correlation space is better suited if multiple neural modules are present, while the population vector space is preferable if neurons have non-convex receptive fields. In this talk I will explain how to coherently combine both pieces of information in a bifiltration using Dowker complexes and their total weights. The construction motivates an interesting extension of Dowker’s duality theorem to simplicial categories associated with two composable relations, I will explain the basic idea behind it’s proof.

Fri, 07 Jun 2024

15:00 - 16:00
L5

Morse Theory for Group Presentations and Applications

Ximena Fernandez
(Mathematical Institute, University of Oxford)
Abstract

Discrete Morse theory serves as a combinatorial tool for simplifying the structure of a given (regular) CW-complex up to homotopy equivalence, in terms of the critical cells of discrete Morse functions. In this talk, I will introduce a refinement of this theory that not only ensures homotopy equivalence with the simplified CW-complex but also guarantees a Whitehead simple homotopy equivalence. Furthermore, it offers an explicit description of the construction of the simplified Morse complex and provides bounds on the dimension of the complexes involved in the Whitehead deformation.
This refined approach establishes a suitable theoretical framework for addressing various problems in combinatorial group theory and topological data analysis. I will show applications of this technique to the Andrews-Curtis conjecture and computational methods for inferring the fundamental group of point clouds.

This talk is based on the article: Fernandez, X. Morse theory for group presentations. Trans. Amer. Math. Soc. 377 (2024), 2495-2523.

Fri, 31 May 2024

15:00 - 16:00
L5

Topology for spatial data from oncology and neuroscience

Bernadette Stolz-Pretzer
(École Polytechnique Fédérale de Lausanne (EPFL))
Abstract

State-of-the art experimental data promises exquisite insight into the spatial heterogeneity in tissue samples. However, the high level of detail in such data is contrasted with a lack of methods that allow an analysis that fully exploits the available spatial information. Persistent Homology (PH) has been very successfully applied to many biological datasets, but it is typically limited to the analysis of single species data. In the first part of my talk, I will highlight two novel techniques in relational PH that we develop to encode spatial heterogeneity of multi species data. Our approaches are based on Dowker complexes and Witness complexes. We apply the methods to synthetic images generated by an agent-based model of tumour-immune cell interactions. We demonstrate that relational PH features can extract biological insight, including the dominant immune cell phenotype (an important predictor of patient prognosis) and the parameter regimes of a data-generating model. I will present an extension to our pipeline which combines graph neural networks (GNN) with local relational PH and significantly enhances the performance of the GNN on the synthetic data. In the second part of the talk, I will showcase a noise-robust extension of Reani and Bobrowski’s cycle registration algorithm  (2023) to reconstruct 3D brain atlases of Drosophila flies from a sequence of μ-CT images.

Fri, 24 May 2024

15:00 - 16:00
L5

Applying stratified homotopy theory in TDA

Lukas Waas
(Univeristy of Heidelberg)
Abstract

 

The natural occurrence of singular spaces in applications has led to recent investigations on performing topological data analysis (TDA) on singular data sets. However, unlike in the non-singular scenario, the homotopy type (and consequently homology) are rather course invariants of singular spaces, even in low dimension. This suggests the use of finer invariants of singular spaces for TDA, making use of stratified homotopy theory instead of classical homotopy theory.
After an introduction to stratified homotopy theory, I will describe the construction of a persistent stratified homotopy type obtained from a sample with two strata. This construction behaves much like its non-stratified counterpart (the Cech complex) and exhibits many properties (such as stability, and inference results) necessary for an application in TDA.
Since the persistent stratified homotopy type relies on an already stratified point-cloud, I will also discuss the question of stratification learning and present a convergence result which allows one to approximately recover the stratifications of a larger class of two-strata stratified spaces from sufficiently close non-stratified samples. In total, these results combine to a sampling theorem guaranteeing the (approximate) inference of (persistent) stratified homotopy types from non-stratified samples for many examples of stratified spaces arising from geometrical scenarios.

Fri, 17 May 2024

15:00 - 16:00
L5

Persistent Minimal Models in Rational Homotopy Theory

Kelly Spry Maggs
(École Polytechnique Fédérale de Lausanne (EPFL))
Abstract
One-parameter persistence and rational homotopy theory are two different ‘torsion-free’ algebraic models of space. Each enhances the cochain complex with additional algebraic structure— persistence equips cochain complexes with an action of a polynomial coefficient ring; rational homotopy theory equips cochains complexes with a graded-commutative product.
 
The persistent minimal model we introduce in this talk reconciles these two types of algebraic structures. Generalizing the classical case, we will describe how persistent minimal models are built by successively attaching the persistent rational homotopy groups into the persistent CDGA model. The attaching maps dualize to a new invariant called the persistent rational k-invariant.
 
This is joint work with Samuel Lavenir and Kathryn Hess: https://arxiv.org/abs/2312.08326


 

Fri, 03 May 2024

15:00 - 16:00
L5

Local systems for periodic data

Adam Onus
(Queen Mary University of London)
Abstract

 

Periodic point clouds naturally arise when modelling large homogenous structures like crystals. They are naturally attributed with a map to a d-dimensional torus given by the quotient of translational symmetries, however there are many surprisingly subtle problems one encounters when studying their (persistent) homology. It turns out that bisheaves are a useful tool to study periodic data sets, as they unify several different approaches to study such spaces. The theory of bisheaves and persistent local systems was recently introduced by MacPherson and Patel as a method to study data with an attributed map to a manifold through the fibres of this map. The theory allows one to study the data locally, while also naturally being able to appeal to local systems of (co)sheaves to study the global behaviour of this data. It is particularly useful, as it permits a persistence theory which generalises the notion of persistent homology. In this talk I will present recent work on the theory and implementation of bisheaves and local systems to study 1-periodic simplicial complexes. Finally, I will outline current work on generalising this theory to study more general periodic systems for d-periodic simplicial complexes for d>1. 

Fri, 26 Apr 2024

15:00 - 16:00
L5

Lagrangian Hofer metric and barcodes

Patricia Dietzsch
(ETH Zurich)
Further Information

Patricia is a Postdoc in Mathematics at ETH Zürich, having recently graduated under the supervision of Prof. Paul Biran.

Patricia is working in the field of symplectic topology. Some key words in her current research project are: Dehn twist, Seidel triangle, real Lefschetz fibrations and Fukaya categories. Besides this, she is a big fan of Hofer's metric, expecially of the Lagrangian Hofer metric and the many interesting open questions related to it. 

Abstract

 

This talk discusses an application of Persistence Homology in the field of Symplectic Topology. A major tool in Symplectic Topology are Floer homology groups. These are algebraic invariants that can be associated to pairs of Lagrangian submanifolds. A richer algebraic invariant can be obtained using 
filtered Lagrangian Floer theory. This gives rise to a persistence module and a barcode. Its bar lengths are invariants for the pair of Lagrangians. 
 
We explain how these numbers can be used to estimate the Lagrangian Hofer distance between the two Lagrangians: It is a well-known stability result  that the bar lengths are lower bounds of the distance. We show how to get an upper bound of the distance in terms of the bar lengths in the special case of equators in a cylinder.
Fri, 08 Mar 2024

15:00 - 16:00
L6

Topological Perspectives to Characterizing Generalization in Deep Neural Networks

Tolga Birdal
((Imperial College)
Further Information

 

Dr. Tolga Birdal is an Assistant Professor in the Department of Computing at Imperial College London, with prior experience as a Senior Postdoctoral Research Fellow at Stanford University in Prof. Leonidas Guibas's Geometric Computing Group. Tolga has defended his master's and Ph.D. theses at the Computer Vision Group under Chair for Computer Aided Medical Procedures, Technical University of Munich led by Prof. Nassir Navab. He was also a Doktorand at Siemens AG under supervision of Dr. Slobodan Ilic working on “Geometric Methods for 3D Reconstruction from Large Point Clouds”. His research interests center on geometric machine learning and 3D computer vision, with a theoretical focus on exploring the boundaries of geometric computing, non-Euclidean inference, and the foundations of deep learning. Dr. Birdal has published extensively in leading academic journals and conference proceedings, including NeurIPS, CVPR, ICLR, ICCV, ECCV, T-PAMI, and IJCV. Aside from his academic life, Tolga has co-founded multiple companies including Befunky, a widely used web-based image editing platform.

Abstract

 

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.

 

Fri, 01 Mar 2024

15:00 - 16:00
L6

Applied Topology TBC

Zoe Cooperband
(University of Pennsylvania)
Further Information

Dr  Zoe Copperband is a member of the Penn Engineering GRASP Laboratory. Her recent preprint, Towards Homological Methods in Graphic Statics, can be found here.

Fri, 16 Feb 2024

15:00 - 16:00
L5

Morse Theory for Tubular Neighborhoods

Antoine Commaret
(INRIA Sophia-Antipolis)
Abstract
Given a set $X$ inside a Riemaniann manifold $M$ and a smooth function $f : X -> \mathbb{R}$, Morse Theory studies the evolution of the topology of the closed sublevel sets filtration $X_c = X \cap f^{-1}(-\infty, c]$ when $c \in \mathbb{R}$ varies using properties on $f$ and $X$ when the function is sufficiently generic. Such functions are called Morse Functions . In that case, the sets $X_c$ have the homotopy type of a CW-complex with cells added at every critical point. In particular, the persistent homology diagram associated to the sublevel sets filtration of a Morse Function is easily understood. 
 
In this talk, we will give a broad overview of the classical Morse Theory, i.e when $X$ is itself a manifold, before discussing how this regularity assumption can be relaxed. When $M$ is a Euclidean space, we will describe how to define a notion of Morse Functions, first on sets with positive reach (a result from Joseph Fu, 1988), and then for any tubular neighborhood of a set at a regular value of its distance function, i.e when $X = \{ x \in M, d_Y(x) \leq \varepsilon \}$ where $Y \subset M$ is a compact set and $\varepsilon > 0$ is a regular value of $d_Y$ the distance to $Y$ function.
 
 
If needed, here are three references :
 
Morse Theory , John Milnor, 1963
 
Curvature Measures and Generalized Morse Theory, Joseph Fu, 1988
Morse Theory for Tubular Neighborhoods, Antoine Commaret, 2024, Arxiv preprint https://arxiv.org/abs/2401.04034
Fri, 02 Feb 2024

15:00 - 16:00
L5

Algebraic and Geometric Models for Space Communications

Prof. Justin Curry
(University at Albany)
Further Information

Justin Curry is a tenured Associate Professor in the Department of Mathematics and Statistics at the University at Albany SUNY.

His research is primarily in the development of theoretical foundations for Topological Data Analysis via sheaf theory and category theory.

Abstract

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.

Fri, 26 Jan 2024

15:00 - 16:00
L5

Expanding statistics in phylogenetic tree space

Gillian Grindstaff
(Mathematical Institute)
Abstract
For a fixed set of n leaves, the moduli space of weighted phylogenetic trees is a fan in the n-pointed metric cone. As introduced in 2001 by Billera, Holmes, and Vogtmann, the BHV space of phylogenetic trees endows this moduli space with a piecewise Euclidean, CAT(0), geodesic metric. This has be used to define a growing number of statistics on point clouds of phylogenetic trees, including those obtained from different data sets, different gene sequence alignments, or different inference methods. However, the combinatorial complexity of BHV space, which can be most easily represented as a highly singular cube complex, impedes traditional optimization and Euclidean statistics: the number of cubes grows exponentially in the number of leaves. Accordingly, many important geometric objects in this space are also difficult to compute, as they are similarly large and combinatorially complex. In this talk, I’ll discuss specialized regions of tree space and their subspace embeddings, including affine hyperplanes, partial leaf sets, and balls of fixed radius in BHV tree space. Characterizing and computing these spaces can allow us to extend geometric statistics to areas such as supertree contruction, compatibility testing, and phylosymbiosis.


 

Fri, 19 Jan 2024

15:00 - 16:00
L4

The Function-Rips Multifiltration as an Estimator

Steve Oudot
(INRIA - Ecole Normale Supérieure)
Abstract

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.

Fri, 01 Dec 2023

15:00 - 16:00
L5

Computing algebraic distances and associated invariants for persistence

Martina Scolamiero
(KTH Stockholm)
Further Information

Martina Scolamiero is an Assistant Professor in Mathametics with specialization in Geometry and Mathematical Statistics in Artificial Intelligence.

Her research is in Applied and Computational Topology, mainly working on defining topological invariants which are suitable for data analysis, understanding their statistical properties and their applicability in Machine Learning. Martina is also interested in applications of topological methods to Neuroscience and Psychiatry.

Abstract

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.

Fri, 24 Nov 2023

15:00 - 16:00
L5

Indecomposables in multiparameter persistence

Ulrich Bauer
(TU Munich)
Further Information

Ulrich Bauer is an associate professor (W3) in the department of mathematics at the Technical University of Munich (TUM), leading the Applied & Computational Topology group. His research revolves around application-motivated concepts and computational methods in topology and geometry, popularized by application areas such as topological data analysis. Some of his key research areas are persistent homology, discrete Morse theory, and geometric complexes.

Abstract

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.

 

Tue, 21 Nov 2023
11:00
L1

Singularity Detection from a Data "Manifold"

Uzu Lim
(Mathematical Institute)

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Abstract

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