Beyond Euclid: An Illustrated Guide to Modern Machine Learning with Geometric, Topological, and Algebraic Structures

TL;DR

This paper reviews the emerging field of machine learning that extends beyond Euclidean spaces, incorporating geometric, topological, and algebraic structures to handle complex, non-Euclidean data types.

Contribution

It provides an accessible overview and a unifying graphical taxonomy of recent advances in non-Euclidean machine learning, highlighting challenges and future opportunities.

Findings

Unified framework for non-Euclidean ML methods

Identification of key challenges in the field

Opportunities for integrating geometric, topological, and algebraic approaches

Abstract

The enduring legacy of Euclidean geometry underpins classical machine learning, which, for decades, has been primarily developed for data lying in Euclidean space. Yet, modern machine learning increasingly encounters richly structured data that is inherently nonEuclidean. This data can exhibit intricate geometric, topological and algebraic structure: from the geometry of the curvature of space-time, to topologically complex interactions between neurons in the brain, to the algebraic transformations describing symmetries of physical systems. Extracting knowledge from such non-Euclidean data necessitates a broader mathematical perspective. Echoing the 19th-century revolutions that gave rise to non-Euclidean geometry, an emerging line of research is redefining modern machine learning with non-Euclidean structures. Its goal: generalizing classical methods to unconventional data types with geometry, topology, and algebra. In this review, we provide an accessible gateway to this fast-growing field and propose a graphical taxonomy that integrates recent advances into an intuitive unified framework. We subsequently extract insights into current challenges and highlight exciting opportunities for future development in this field.