Research group
Topology
Fri, 21 Feb 2025
15:00
L4

Monodromy in bi-parameter persistence modules

Sara Scaramuccia
(University of Rome Tor Vergata)

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Abstract

Informally, monodromy captures the behavior of objects when one circles around a singularity. In persistent homology, non-trivial monodromy has been observed in the case of biparameter filtrations obtained by sublevel sets of a continuous function [1]. One might consider the fundamental group of an admissible open subspace of all lines defining linear one-parameter reductions of a bi-parameter filtration. Monodromy occurs when this fundamental group acts non-trivially on the persistence space, i.e. the collection of all the persistence diagrams obtained for each linear one-parameter reduction of the bi-parameter filtration. Here, under some tameness assumptions, we formalize the monodromy behavior in algebraic terms, that is in terms of the persistence module associated with a bi-parameter filtration. This allows to translate monodromy in terms of persistence module presentations as bigraded modules. We prove that non-trivial monodromy involves generators within the same summand in the direct sum decomposition of a persistence module. Hence, in particular interval-decomposable persistence modules have necessarily trivial monodromy group.

The work is under development and it is a joint collaboration with Octave Mortain from the École Normale Superieure, Paris.
 
[1] A. Cerri, M. Ethier, P. Frosini, A study of monodromy in the computation of multidimensional persistence, in: Proc. 17th IAPR Int. Conf. Discret. Geom. Comput. Imag., 2013: pp. 1–12.
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, 14 Feb 2025
15:00
L4

Distance-from-flat persistent homology transforms

Nina Otter
(Inria Saclay)

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Abstract
The persistent homology transform (PHT) was introduced in the field of Topological Data Analysis about 10 years ago, and has since been proven to be a very powerful descriptor of Euclidean shapes. The PHT consists of scanning a shape from all possible directions and then computing the persistent homology of sublevel set filtrations of the respective height functions; this results in a sufficient and continuous descriptor of Euclidean shapes. 
 
In this talk I will introduce a generalisation of the PHT in which we consider arbitrary parameter spaces and sublevel-set filtrations with respect to any function. In particular, we study transforms, defined on the Grassmannian AG(m,n) of affine subspaces of n-dimensional Euclidean space, which allow to scan a shape by probing it with all possible affine m-dimensional subspaces P, for fixed dimension m, and by then computing persistent homology of sublevel-set filtrations of the function encoding the distance from the flat P. We call such transforms "distance-from-flat PHTs". I will discuss how these transforms generalise known examples, how they are sufficient descriptors of shapes and finally present their computational advantages over the classical persistent homology transform introduced by Turner-Mukherjee-Boyer. 
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, 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.

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