The London TDA seminar is a research seminar gathering researchers and practitioners in Topological Data Analysis based in and around London. It takes place four times a year in the School of Mathematical Sciences building at Queen Mary University of London, in Mile End.
Schedule 23.05.2023
The seminar will take place in MB-503 at 11am.
11am: Katharine Turner (Australian National University)
Title: The Extended Persistent Homology Transform for Manifolds with Boundary
Abstract: The Persistent Homology Transform (PHT) is a topological transform which can be use to quantify the difference between subsets of Euclidean space. To each unit vector the transform assigns the persistence module of the height function over that shape with respect to that direction. The PHT is injective on piecewise-linear subsets of Euclidean space, and it has been demonstrably useful in diverse applications. One shortcoming is that shapes with different essential homology (i.e., Betti numbers) have an infinite distance between them. The theory of extended persistence for Morse functions on a manifold was developed by Cohen-Steiner, Edelsbrunner and Harer in 2009 to quantify the support of the essential homology classes. By using extended persistence modules of height functions over a shape, we obtain the extended persistent homology transform (XPHT) which provides a finite distance between shapes even when they have different Betti numbers. I will discuss how the XPHT of a manifold with boundary can be deduced from the XPHT of the boundary which allows for efficient calculation. James Morgan has implemented the required algorithms for 2-dimensional binary images as a forthcoming R-package. Work is also with Vanessa Robins.
11.30am: Anthony Baptista (QMUL)
Title: Zoo guide to Network embedding
Abstract: Networks are intrinsically combinatorial objects (i.e., interconnected nodes, where certain pairs of nodes are connected by links), with no a priori ambient space, nor node geometric information such as `coordinates'. Network embedding (a.k.a representation learning) is the process of assigning such a latent space (a.k.a embedding space) to a network. This is typically done by mapping the nodes to a geometric space, such as a Euclidean space, while preserving some properties of the nodes, links, and/or network. Overall, network embedding methods are used for learning a low-dimensional vector representation from a high-dimensional network. The relationships between nodes in the network are represented by their distance in the low-dimensional embedding space. Then, the low-dimension vector representation can be used for visualisation, and in a wide variety of downstream analyses, from network inference or link prediction to node classification or community detection. Over the past few years, there has been a significant surge in the number of embedding methods, making it challenging to navigate this fast-evolving field. I will present an overview of network embedding methods and introduce a new taxonomy that captures the latest developments in the field. Additionally, I will present a groundbreaking embedding technique that is able to project in the same latent space heterogeneous information. Finally, I will conclude by highlighting open questions and challenges that require further investigation in the field.
12pm: Ximena Fernandez (Durham)
Title: Intrinsic Persistent Homology via Density-based Metric Learning
Abstract: In this talk, I will explain a density-based method to address the problem of estimating topological features from noisy data in high dimensional Euclidean spaces under the manifold assumption. The key to our approach is to consider a sample metric known as Fermat distance to robustly infer the homology the space of data points. I will show that such sample metric space GH-converges almost surely to the manifold itself endowed with an intrinsic (Riemannian) metric that accounts for both the geometry of the manifold and the density that produces the sample. This fact, joint with the stability properties of persistent homology, implies the convergence of the associated persistence diagrams, which present advantageous properties. I will show that they are robust to the presence of (geometric) outliers in the input data and less sensitive to the particular embedding of the underlying manifold in the ambient space. Finally, I will exhibit a concrete application of these ideas to time series analysis, with examples in both real and simulated data. This is a joint work with E. Borghini, G. Mindlin and P. Groisman. 'Intrinsic persistent homology via density-based metric learning'. Journal of Machine Learning Research, 24 (2023) no. 75, 1-42.
2pm: Adam Brown (Oxford)
Title: A microlocal perspective of multiparameter persistent homology
Abstract: How might derived sheaf theory, microsupport, and stratifications contribute to our understanding of multi-parameter persistence? I aim to provide a high-level introduction to each subject, relying only on linear algebra and basic combinatorics, assuming no prior knowledge of derived categories. Through examples, I will hint at a geometrically motivated generalization of ‘barcodes’, and describe the remaining challenges to realize the proposed theory. This talk is based partly on joint work with Ondrej Draganov (IST Austria).
2.30pm: Adam Onus (QMUL)
Title: Sheafy Approaches to Periodic Cell Complexes
Abstract: Spatially periodic topological spaces arise out of necessity when simulating or modelling large homogenous structures on comparably small finite domains without introducing irregular boundary effects. One typically studies such a structure as being embedded in a flat d-dimensional torus, which equivalently represents a single unit cell of translations on Rd with periodic boundary conditions. However, topological information is lost in this way and the ambient toroidal topology introduces fake "toroidal" cycles which do not lift to meaningful homological information in the infinite cover. Previously I have shown various methods which provide a heuristic for recovering this information for periodic cell complexes, but here I will motivate a promising new sheaf theoretic approach based on work by MacPherson & Patel which appears to make much of this study more rigorous.
If you are interested in being added to the mailing list for this seminar, please contact Nina Otter at n dot last-name @ qmul.ac.uk.