Sydney Dynamics Group: Wormell -- Spectral convergence of diffusion maps

Dear all, 

This week, Friday June 26, Caroline Wormell (USyd) will give a talk at 4pm (Sydney time)
via Zoom.  

Zoom link: https://uni-sydney.zoom.us/j/97886080891 

Meeting ID: 978 8608 0891 

Title: Spectral convergence of diffusion maps 

Abstract: 
Diffusion maps is a manifold learning algorithm widely used for dimensionality
reduction.  Using a sample from a distribution, it approximates the eigenvalues and
eigenfunctions of associated Laplace-Beltrami operators.  Theoretical bounds on the
approximation error are however generally much weaker than the rates that are seen in
practice.  We present new approaches to improve the error bounds in the model case where
the distribution is supported on a hypertorus.  For the data sampling (variance)
component of the error we make spatially localised compact embedding estimates on
certain Hardy spaces; we study the deterministic (bias) component as a perturbation of
the Laplace-Beltrami operator’s associated PDE, and apply relevant spectral stability
results.  These techniques enable long-standing pointwise error bounds to be matched for
both the spectral data and the norm convergence of the operator discretisation.  

We also introduce an alternative normalisation for diffusion maps based on Sinkhorn
weights.  This normalisation approximates a Langevin diffusion on the sample and yields
a symmetric operator approximation.  We prove that it has better convergence compared
with the standard normalisation on flat domains, and present a highly efficient
algorithm to compute the Sinkhorn weights.  


Past talks can be found on the YouTube channel:
https://www.youtube.com/channel/UCZqgDJ21wbdzMbeIdealpUg/ 

I hope to see you all online.  

Lachlan