Beyond Euclid: An Illustrated Guide to Modern Machine Learning with Geometric, Topological, and Algebraic Structure, by Sanborn et al.

The two images below are from the article titled – Beyond Euclid: An Illustrated Guide to Modern Machine Learning with Geometric, Topological, and Algebraic Structure, by Sanborn et al. (https://arxiv.org/abs/2407.09468v1)

The paper talk about how “the mathematics of topology, geometry and algebra provide a conceptual framework to categorize the nature of data found in machine learning.” The two figures, present a “graphical taxonomy, to categorize the structures of data.” The article the authors discuss two types of data that we generally encounter: “either data as coordinates in space—for example the coordinate of the position of an object in a 2D space; or data as signals over a space—for example, an image is a 3D (RGB) signal defined over a 2D space. In each case, the space can either be a Euclidean space or it can be equipped with topological, geometric and algebraic structures.”

The paper goes on to then “review a large and disparate body of literature of non-Euclidean generalizations of algorithms classically defined for data residing in Euclidean spaces.” The algorithms presented assume that certain topological, algebraic, or geometric structures of the data / problem are known. However, it does not go into discussion on methods where such structures are not known. For example, methods that fall into the category of topological data analysis, metric learning, or group learning are not covered.