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