Schedule

María José Jiménez: Geometric complexes for spatial biology

Abstract: Recent technical progress has enabled the generation of multiplex histological images that are converted into point clouds with spatial localization of different cell types. These point clouds can be analyzed using various geometric constructions to model the 2D data from multiple cell types. Among them, Dowker complexes, witness complexes [1], and the novel chromatic alpha complexes [2] are used to represent spatial relations among different data types. Specifically, chromatic alpha complexes capture spatial proximity relations for point cloud data consisting of points of varying types or "colors." This method allows for encoding information about the spatial distribution of points for each color, as well as the interplay between these subsets, facilitating the study of cell pattern co-localization and even multi-localization. Persistent homology computations from these complexes may yield various vectorizations [3] that can be employed as input for machine learning algorithms to deduce properties of the spatial distribution of the different cell types. This is joint work with colleagues at Oxford’s Centre for TDA.

Aina Ferrà Marcús: Importance attribution in neural networks through reconstruction of univariate functions

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Abstract: Since deep learning models are often regarded as "blackboxes" it is important to develop interpretable pipelines in order to work in a medical environment. Understanding the decision that a neural network is taking is crucial to build up trust for practitioners and spread the use of the models. We propose a method to analyze time series data with a neural network using a matrix of persistence landscapes obtained with topological data analysis. Using a gating layer, we are able to identify the most relevant landscape levels for a classification task, thus working as an attribution system. We demonstrate the usefulness of the method in the case of arrhythmia classification. Furthermore, we discuss general conditions to reconstruct a univariate function from its set of directional persistence diagrams.

José Manuel Ros Rodrigo: Towards persistent homology as a tool to analyze deep neural networks

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Abstract: This presentation will focus on two thesis projects from the Mathematics and Computer Science degrees: Persistent Homology as a Tool to Analyze Neural Networks and Computational Topology Software for the Structural Analysis of Neural Networks Underlying Deep Learning Models. Both projects explore the application of persistent homology as a mathematical tool for analyzing neural networks, building upon S. Watanabe's pioneering article [1]. In addition to introducing algorithms for the systematic calculation of persistence diagrams associated with a neural network, we provide an interpretation of these diagrams. These works represent a promising first step in understanding the structural richness of deep learning models through mathematical tools.

Rubén Ballester: Correlation topology in neural networks: Implications for generalization

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Abstract: Understanding how neural networks generalize to unseen data is crucial for designing robust and reliable models. Recent studies have highlighted a link between the persistent homology of different aspects of neural networks, such as their weights or activations, and their ability to generalize. In this talk, we will focus on exploring this connection, particularly examining how persistence diagrams derived from network activations in response to datasets equipped with a correlation dissimilarity correlate with the generalization gap, a quantitative measure of how well a network generalizes. Additionally, we will demonstrate how to leverage correlation-based persistence diagrams for formulating regularization terms that enhance the overall accuracy of neural networks during training, outperforming traditional regularization approaches on our experiments. 

Andrea Guidolin: Computing relative Betti diagrams of persistence modules over finite posets using Koszul complexes

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Abstract: A recent trend in TDA is to study persistence modules over a finite poset via relative homological invariants. Relative homological algebra extends constructions of standard homological algebra by redefining the notion of projective module, which depends on the choice of a family of “basic” modules. In this talk, we study relative resolutions, which are a way of approximating a given persistence module by a sequence of persistence modules that are projective relative to the chosen family. Under certain conditions, the multiplicities of the basic persistence modules in a relative resolution are unique, and define invariants called relative Betti diagrams. Using Koszul complexes, relative Betti diagrams can be computed in a simple and local way, avoiding the computation of the entire relative resolution. The talk is based on a joint work with Wojciech Chachólski, Isaac Ren, Martina Scolamiero, and Francesca Tombari.

Álvaro Torras-Casas: Planar alpha complex persistence via the Mayer-Vietoris spectral sequence 

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Abstract: The alpha filtration is a well-known combinatorial structure used to deduce geometric and topological properties of point samples in low dimensions. In particular, it is used to compute persistent homology, which encodes information such as clusters or cycles by means of a barcode summary. When a dataset is large, both the computation of the alpha filtration and persistent homology become slow and memory expensive. In this talk, I will present recent work on the distribution of such calculation by computing the Mayer-Vietoris spectral sequence. Our approach considers points in two dimensions which are partitioned by a grid dividing the ambient space. Such partition is used to deduce a cover of the alpha filtration and to determine the communication between processes. I will also explain properties of the spectral sequence and, in particular, a condition that ensures its collapse on the second page. Finally, I will describe some of the details of our algorithm, the related software  as well as experiments. This is joint work with Freya Jensen from Heidelberg University.

Rubén Sánchez-García: The persistent Laplacian for data science

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Abstract: Persistent homology is arguably the most successful technique in Topological Data Analysis. It combines homology, a topological feature of a data set, with persistence, which tracks the evolution of homology over different scales. The persistent Laplacian is a recent theoretical development that combines persistence with the combinatorial Laplacian, a higher-order extension of the well-known graph Laplacian. Crucially, the Laplacian encodes both the homology of a data set and some additional geometric information not captured by homology. In this talk, I will summarise our investigation into the use of the persistent Laplacian on real data for downstream machine learning classification and regression tasks. We extend the persistent Laplacian to cubical complexes so it can be used on images and then evaluate its performance on the MNIST and MoleculeNet datasets, showing that it consistently outperforms persistent homology across tasks. Our results encourage the use of persistent Laplacian based features in learning tasks. This is joint work with Tom Davies and Zhengchao Wan.

Eduardo Sáenz de Cabezón: A combinatorial approach to (multi-)persistent homology of simplicial complexes

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Abstract: Given an abstract simplicial complex, we construct combinatorial filtrations and multi-filtrations of it. These can be read in terms of monomial ideals, which, in turn, can be seen as multi-graded modules in a multi-variate polynomial ring. We then study homological invariants of these modules and relate them to the persistent homology of the original complex and the associated combinatorial filtrations. This is joint work with F. Mohammadi and H. Wynn.