Abstract mathematical concepts/tools appeared in machine learning research I am interested in knowing about abstract mathematical concepts, tools or methods that have come up in theoretical machine learning. By "abstract" I mean something that is not immediately related to that realm. For instance, a concept from mathematical optimization does not qualify since optimization is directly related to the training of deep networks. In contrast, to me Topological Data Analysis is a non-trivial example of applying algebraic topology to data analysis.
Here are few examples that I have encountered in the literature (all in the context of deep learning).


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*Betti numbers have been utilized to introduce a complexity measure
which could be used for comparing deep and shallow architectures:
https://www.elen.ucl.ac.be/Proceedings/esann/esannpdf/es2014-44.pdf

*A connection between Sharkovsky's Theorem and the expressive power of deep networks:
https://arxiv.org/pdf/1912.04378.pdf

*An application of Riemannian geometry:
https://arxiv.org/pdf/1606.05340.pdf

*Algebraic geometry naturally comes up in studying neural networks with polynomial activation functions. This paper discusses functional varieties associated with such networks: 
https://arxiv.org/abs/1905.12207
I find it useful to compile a list of such research works on ML that draw on pure math. 
 A: Probably, one the most striking  is the "UMAP" (Uniform manifold approximation and projection)  - a method of dimensional reduction in machine learning. The authors of the method use CATEGORY THEORY for its discovery. Well, there are certain discussions to what extent category theory is really required, see John Baez blog and references there in, still it is the author's original viewpoint how the method has been discovered. (The algorithm/implementation can be understood without category theory).
The method become quite quickly very popular - gaining 748 citations in two years according to google scholar. And found applications in many fields including bioinformatics (UMAP Nature), as well as capable to produce beautiful images MO355631.
It is similar to previously widely used method - tSNE (T-distributed stochastic neighbor embedding), however, quite often produces  better results with less computational efforts, thus beating the predecessor both in quality and speed. 
The documentations can be found here: UMAP docs.     
