Tyrus Berry (GMU)

Time: 10:45 am on Thursday, October 6th, 2022

Manifold Learning and Grid-Free Methods with the Spectral Exterior Calculus (SEC)

The geometric prior (the assumption that data lies on or near a manifold) is one important method of understanding how machine learning methods are sometimes able to side-step the curse-of-dimensionality. Algorithms that leverage the geometric prior will often be able to explicitly or implicitly reparametrize problems using the smaller intrinsic dimensionality of the hidden manifold. However, beyond explaining the success of these algorithms, the geometric prior also opens up new tools to address applications such as uncertainty quantification, forecasting, filtering, and control. The natural toolbox for many of these problems is the exterior calculus of differential forms, which transfers many of the operations and theorems of analysis to the nonlinear manifold context. Recently, there have been multiple techniques developed that build consistent discretizations of the exterior calculus, however most of these require a `grid' (meaning a simplicial complex). This sophisticated data structure can be difficult to construct even on known manifolds, and is extremely difficult to construct starting from noisy data sets. The Spectral Exterior Calculus (SEC) is an alternative construction which is formulated entirely in terms of harmonic analysis on the manifold, and crucially can be recovered without a simplicial complex. The SEC's grid-free approach opens up the exterior calculus toolbox to data-driven learning problems, as well as higher dimensional or evolving manifolds where simplicial complexes are difficult to construct or maintain. This talk will introduce the central ideas of the Spectral Exterior Calculus as well as the tools that have been developed so far and discuss its advantages and disadvantages as well as some open problems connected to harmonic analysis on manifolds.