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# Magnitude Workshop at ETH Zürich

## April 24–26, HG F 26.1

# Schedule

## Wednesday, April 24

Time |
Speaker |
Title |
Abstract |
---|---|---|---|

9:00-10:00 | Maria Morariu | Introduction to the Magnitude of Finite Metric Spaces | The magnitude of a finite metric space is a numerical invariant, initially introduced in reproductive biology as a measure for the effective number of species The mathematical definition we will see today was introduced by Tom Leinster in 2010. In this talk, I will define the magnitude of finite metric spaces and show some examples and ways to compute it. I will also introduce two constructions and analyze how the magnitude behaves with respect to these constructions. In the end, I will define the magnitude function and discuss some of its properties. |

10:00-11:00 | Nadja Häusermann | Magnitude Homology | After a brief introduction to the magnitude of graphs, we will define the magnitude homology of graphs. The magnitude homology categorifies the magnitude and can be used to deduce properties of the magnitude, like the inclusion-exclusion principle. We will discuss several properties of the magnitude homology, including a Mayer-Vietoris type theorem. We will look at examples of magnitude homology of graphs; some examples we can do directly and for some we use our code written in SageMath. |

11:00-12:00 | Eudes Robert | On the connection between magnitude and persistence | The aim of this talk is to present a result from Govc and Hepworth establishing a relationship between persistence and magnitude. Magnitude is an invariant of metric spaces originally described by Leinster. Further work from Willerton, Hepworth and Otter showed it is strictly refined by algebraic objects - magnitude homology, and its cousin blurred magnitude homology. We will define blurred magnitude homology and show it is an instance of a construction from topological data analysis called a persistence module. Persistence modules are structured by a classification theorem; we will show that this so-called barcode decomposition of blurred magnitude homology is tightly related to the original invariant. |

12:00-14:00 | Lunch |
||

14:00 | Coffee |
||

Afternoon | Free for discussions |

## Thursday, April 25

Time |
Speaker |
Title |
Abstract |
---|---|---|---|

9:00-10:00 | Nick Kunz | Magnitude Dimension of Metric Spaces | Magnitude is an invariant of a metric space that, in the case of finite spaces, can be thought of as the effective number of points. Observing how magnitude changes when the space is viewed at different scales provides valuable insights. This idea is encapsulated in the magnitude dimension, which is the instantaneous growth rate of magnitude as the metric is scaled. In many cases, for example, for compact subspaces of R^{n}, the magnitude dimension aligns with the box-counting dimension, offering an additional method for computing fractal dimensions. Through a sampling process, we can effectively estimate the magnitude dimension of infinite metric spaces. This serves as a tool for estimating the intrinsic dimension of datasets, which is particularly useful for analyzing complex datasets. |

10:00-11:00 | Miguel O’Malley | Magnitude, Alpha magnitude, dimension and clustering | Magnitude, an isometric invariant of metric spaces, is known to bear rich connections to other desirable invariants, such as dimension, volume, and curvature. Connections between magnitude and persistent homology, a method to observe topological features in datasets, are well studied and fruitful. We leverage one such connection, persistent magnitude, to introduce alpha magnitude, a new invariant which bears many of the same properties of magnitude. We show in particular a strong connection to the Minkowski dimensions of compact subspaces of R^{n} and conjecture the connection exists in general. We further investigate some early methods involving both invariants in the clustering of point clouds and discuss preliminary results, associated topics, desirable qualities, and limitations thereof. |

11:00-12:00 | Katharina Limbeck | Metric Space Magnitude in Machine Learning | The magnitude of a metric space is a recently-established invariant capable of summarising the diversity and geometry of data across multiple scales. Describing the effective size of a space, magnitude is closely linked to entropy and even encodes the intrinsic dimensionality of a dataset. Nevertheless, despite its strong geometric properties, magnitude has only rarely been used for machine learning. Bridging this gap between theory and practice, this talk will give an overview of all existing applications of magnitude to machine learning. To demonstrate the expressivity of magnitude as a multi-scale summary of data, we then introduce our two most recent works. As a first application to deep learning, we quantify the learning process of neural networks using the magnitude dimension. Further, in the context of representation learning, we introduce novel multi-scale measures for evaluating the diversity of latent representations based on magnitude functions. Finally, we introduce magnipy, a package for computing magnitude in Python, and discuss how best to compute magnitude in practice. As such, this talk will not only explain how magnitude has been used so far, but also address its usability and potential for future machine learning research. |

12:00-14:00 | Lunch |
||

14:00 | Coffee |
||

Afternoon | Free for discussions |

## Friday, April 26

Time |
Speaker |
Title |
Abstract |
---|---|---|---|

9:00-10:00 | Julius von Rohrscheidt | TARDIS: Topological Algorithm for Robust DIscovery of Singularities | The so-called manifold hypothesis is a fundamental assumption in modern machine learning techniques like e.g. non-linear dimensionality reduction. This hypothesis assumes given data to lie on or close to a manifold, giving rise to methods which are built on the underpinnings of topology and geometry. However, recent discoveries suggest that most real-world datasets do not exhibit a manifold structure, and that reasonable performance of techniques which are based on this assumption cannot be guaranteed. We therefore suggest a method to test if the given data in fact satisfies the manifold hypothesis, by introducing a score that measures the euclidicity of a data point locally. This score enjoys strong theoretical guarantees, and turns out to be a useful tool for the detection of topological anomalies. |

10:00-11:00 | Ernst Röell | Topological VAEs for the Generation of Shapes | The Euler Characteristic Transform (ECT) is powerful topological invariant for the analysis of shapes and graphs. Recently, the ECT has been made differentiable for application in machine learning pipelines as a layer. The method, called the Differential Euler Characteristic Transform (DECT), is fast to compute and performs on par with more complex topological methods. Given this fast, direct and native integration we focus on the application of topology to generative models and hope to better understand if such a powerful statistics can be viewed as a natural latent space. We show that point clouds can be reconstructed based on their ECT and that it leads to good results with a simple architecture. Moreover, using the ECT as a natural loss term during training we further show that it enhances the quality of the reconstruction when combined with more traditional loss terms. |

11:00-12:00 | Jeremy Wayland | Mapping the Multiverse of Latent Representations | Echoing recent calls to counter reliability and robustness concerns in machine learning via multiverse analysis, we present Presto, a principled framework for mapping the multiverse of machine-learning models that rely on latent representations. Although such models enjoy widespread adoption, the variability in their embeddings remains poorly understood, resulting in unnecessary complexity and untrustworthy representations. Our framework uses persistent homology to characterize the latent spaces arising from different combinations of diverse machine-learning methods, (hyper)parameter configurations, and datasets, allowing us to measure their pairwise (dis)similarity and statistically reason about their distributions. As we demonstrate both theoretically and empirically, our pipeline preserves desirable properties of collections of latent representations, and it can be leveraged to perform sensitivity analysis, detect anomalous embeddings, or efficiently and effectively navigate hyperparameter search spaces. |

12:00-14:00 | Lunch |
||

14:00 | Coffee |
||

Afternoon | Free for discussions |