Lolo Jones – Mathematics and Decision

$\begin{quote} \begin{center} \textbf{Diffusion geometry for data analysis} \end{center} \medskip We introduce diffusion geometry as a new framework for geometric and topological data analysis. Diffusion geometry uses the theory of Markov diffusion operators to define an exterior calculus on certain probability spaces, which allows many objects from Riemannian geometry to be constructed in this very general setting. Given a finite sample of data, we can define estimators for all these constructions, including vector fields and differential forms, wedge product, exterior derivative, Lie bracket, Levi-Civita connection, Hodge Laplacian and Riemann curvature tensor. Compared with existing approaches like persistent homology and local principal component analysis, diffusion geometry is explicitly related to existing theory in geometry, is significantly more robust to noise, significantly faster to compute, provides a richer topological description with the cup product on cohomology, and is naturally vectorised for statistics and machine learning. \end{quote}$