Topology meets Machine Learning: An Introduction using the Euler Characteristic Transform

Machine learning is shaping up to be the transformative technology of our times: Many of us have played with (and marveled at) models like ChatGPT, new breakthroughs in applications like healthcare research are announced on an almost daily basis, and new avenues for integrating these tools into scientific research are opening up, with some mathematicians already using large language models as proof assistants. This article aims to lift the veil and dispel some myths about machine learning; along the way, it will also show how machine learning itself can benefit from mathematical concepts. Indeed, from the outside, machine learning might look like a homogeneous entity, but in fact, the field is fractured and highly diverse. While the main thrust of the field arises from the undeniable engineering advances, with bigger and better models, there is also a strong community of applied mathematicians. Next to the classical drivers of machine-learning architectures, i.e., linear algebra and statistics, topology recently started to provide novel insights into the foundations of machine learning: Point-set topology, harnessing concepts like neighborhoods, can be used to extend existing algorithms from graphs to cell complexes [4]. Algebraic topology, making use of effective invariants like homology, improves the results of models for volume reconstruction [13]. Finally, differential topology, providing tools to study smooth properties of data, results in efficient methods for analyzing embedded (simplicial) complexes [6]. These (and many more) methods have now found a home in the nascent field of topological deep learning [8]. Before diving into concrete examples, let us first take a step back and discuss machine learning as such.