Beyond Euclid - An Illustrated Guide to Modern Machine Learning - Geometry, Topology, and Algebraic Structures

Aug 3, 2026

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Euclidean geometry has long shaped classical machine learning, which historically assumes data living in Euclidean space. Modern learning increasingly deals with structured, non-Euclidean data carrying geometric, topological, and algebraic organization, from spacetime curvature to neuron interactions and symmetry-preserving transformations in physical systems. Extracting insight from these settings requires a broader mathematical toolkit. This review offers an accessible entry point to the fast-growing field of non-Euclidean machine learning and proposes a graphical taxonomy unifying recent advances, then identifies current challenges and future opportunities.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages36

Reading Time1 h 31 min

AccessFree to download

Introduction

Mathematical origins: beyond Euclid

Non-Euclidean revolution in machine learning

Core taxonomy and organization

Background, data structure, and model ontology

Software ecosystem and application domains

Mathematical properties

Topological properties

Geometric properties

Algebraic properties

Beyond Euclid: An Illustrated Guide to Modern Machine Learning with Geometric, Topological, and Algebraic Structures Sophia Sanborn∗1 Johan Mathe∗2 Mathilde Papillon∗1 Domas Buracas3 Hansen J Lillemark3 Christian Shewmake3 ,5 Abby Bertics 1 Xavier Pennec4 Nina Miolane 1 ,2 ,3 1University of California, Santa Barbara 2Atmo, Inc. 3New Theory AI 4Universit Cte d’Azur & Inria 5University of California, Berkeley arXiv :2407 .09468v 1 [ cs .LG] 12 Jul 2024 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. Index Terms—Geometric Deep Learning, Geometry, Topology, Algebra, Machine Learning I. INTRODUCTION For nearly two millennia, Euclid’s Elements of Geometry formed the backbone of our understanding of space and shape. This ‘Euclidean’ view of geometry—characterized by flat planes and straight lines—remained unquestioned until the 19th century. Only then, did mathematicians venture “beyond” to develop the principles of non-Euclidean geometry on curved spaces. Their pioneering work revealed that there is no singular, true geometry. Instead, Euclidean geometry is but one in a mathematical universe of geometries, each of which can be used to illuminate different structures in nature—from the mechanics of celestial bodies embracing the curvature of spacetime to the topologically and algebraically complex electri- cal patterns of neurons in natural and artificial neural networks. This non-Euclidean revolution was part of a greater trend towards generalization and abstraction in 19th and 20th century mathematics. In addition to expanding the realm of geometry, mathematicians proceeded to define more abstract notions of space, freed from rigid geometric concepts like distances and angles. This gave rise to the field of topology, which examines the properties of a space that are preserved under continuous transformations such as stretching and bending. By abstracting away from the rigidity of geometric structures, topology emphasizes more general spatial properties such as continuity and connectedness. Indeed, two structures that look very different from a geometric perspective may be considered topologically equivalent. The famous of example of this is a donut and coffee mug, which are topologically equivalent since one can be continuously deformed into the other. This notion of abstract equivalence was supported by the simultaneous development of the field of abstract algebra, which examines the symmetries of an object—the transformations that leave its fundamental structure unchanged. These mathematical ideas quickly found applications in the natural sciences, and revolutionized how we model the world. A similar revolution is now unfolding in machine learning. In the last two decades, a burgeoning body of research has expanded the horizons

FAQs

Why does modern machine learning require going beyond Euclidean geometry?

Many real-world datasets have intrinsic structure that is not naturally Euclidean, exhibiting geometric, topological, or algebraic organization. Modeling and learning from such data benefits from generalizing classical methods to non-Euclidean settings.

What mathematical concepts does the review emphasize for non-Euclidean learning?

It highlights geometry, topology, and abstract algebra as complementary perspectives. Geometry supports measurements like distance and angle, topology captures continuity and connectedness, and algebra characterizes symmetries through structure-preserving transformations.

How is the field organized in this review?

The authors build a coherent taxonomy based on both the structure of the data and the structure of the machine learning model. The review introduces mathematical background, analyzes data structure, presents an ontology for machine learning and deep learning, surveys open-source libraries, and discusses key application domains.

Aug 3, 2026

Upload by Oliver