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.

翻译：欧几里得几何的持久遗产奠定了经典机器学习的基础，数十年来，机器学习主要针对欧几里得空间中的数据而发展。然而，现代机器学习日益遇到本质上非欧几里得的、具有丰富结构的数据。这类数据可能展现出复杂的几何、拓扑和代数结构：从时空曲率的几何，到大脑中神经元之间拓扑复杂的相互作用，再到描述物理系统对称性的代数变换。从这类非欧几里得数据中提取知识需要更广阔的数学视角。呼应19世纪催生非欧几里得几何的革命，一个新兴的研究方向正在用非欧几里得结构重新定义现代机器学习。其目标在于：将经典方法推广到具有几何、拓扑和代数特性的非常规数据类型。在本综述中，我们为这一快速发展的领域提供了一个易于理解的入门指南，并提出了一种图形化分类法，将最新进展整合成一个直观的统一框架。随后，我们提炼出对当前挑战的见解，并强调了该领域未来发展的激动人心的机遇。