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As datasets used in scientific applications become more complex, studying thegeometry and topology of data has become an increasingly prevalent part of thedata analysis process. This can be seen for example with the growing interestin topological tools such as persistent homology. However, on the one hand,topological tools are inherently limited to providing only coarse informationabout the underlying space of the data. On the other hand, more geometricapproaches rely predominately on the manifold hypothesis, which asserts thatthe underlying space is a smooth manifold. This assumption fails for manyphysical models where the underlying space contains singularities.  In this paper we develop a machine learning pipeline that captures fine-graingeometric information without having to rely on any smoothness assumptions. Ourapproach involves working within the scope of algebraic geometry and algebraicvarieties instead of differential geometry and smooth manifolds. In the settingof the variety hypothesis, the learning problem becomes to find the underlyingvariety using sample data. We cast this learning problem into a Maximum APosteriori optimization problem which we solve in terms of an eigenvaluecomputation. Having found the underlying variety, we explore the use ofGr\"obner bases and numerical methods to reveal information about its geometry.In particular, we propose a heuristic for numerically detecting points lyingnear the singular locus of the underlying variety.

Algebraic Machine Learning with an Application to Chemistry