# Elementary topology and machine learning

The machine learning bubble may be overinflated, but it is not about to burst. Interdisciplinary research in this field is grounded in sound theory and has numerous empirical breakthroughs to show for. As it finds more and more applications and concentrates public research funding, many of us are still wondering: how can mathematics contribute?
Case study of an interaction between elementary topology and machine learning’s binary classification problem. Following classical theorems, we obtain a topologically accurate solution.

Correction: It should be $\sup_{i,j}| E_{i,j}^n - (k+1)^2 \int_{R_{i,j}^n} f | = o(1/n)$ and $k=k(n)$ grow correspondingly slower with $n$.

PDF note here.

# Drawings

I use these grids to construct smooth approximations to bivariate angular probability densities (of interest in protein bioinformatics) and to specify semiparametric priors on bivariate angular density spaces. The grid on the Torus illustrates the proper angular wrapping behavior of planar grid. I drew them in Mathematica.

Code and variants below.

# The basics

Amino acids are small molecules of the form

where $R$ is a side chain called the $R$-group. There are 20 different amino acids found in proteins, each characterized by its $R$-group.

Peptides and proteins are chains of amino acids. Proteins are long such chains, whereas peptides and polypeptides are shorter ones. The amino acids are linked together by peptide bonds: