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*][1] 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).

 1. 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
 2. A connection between Sharkovsky's Theorem and the expressive power of deep networks:
    https://arxiv.org/pdf/1912.04378.pdf
 3. An application of Riemannian geometry:  
 https://arxiv.org/pdf/1606.05340.pdf
 4. 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. 
 

 
 
 
 
    

   

  [1]: https://en.wikipedia.org/wiki/Topological_data_analysis