One idea in the exciting new area of Diffusion Geometry is to use certain eigenfunctions as new local coordinates on a data set (e.g.
a collection of documents). These coordinates are surprisingly robust under perturbation of the underlying sets and have been empirically
observed to provide local coordinates on rather large patches. In this talk we discuss the mechanism that explains this robustness: it turns out to be a "hidden" (i.e. simple, though previously unobserved) feature of the Riemann Mapping Theorem for simply connected planar domains that is quite general. This simple feature also works on manifolds of arbitrary dimension. The idea is to use standard estimates for the Heat Kernel to pick eigenfunctions providing local coordinates on a large
Ball, and the diameter of that ball is optimal up to universal constants. (In more technical language, which will be explained in
English, the main result is an analogue of the Distortion Theorems for conformal mappings.)
The lecture will not assume any familiarity with the proof of the Riemann mapping or related estimates. Instead we will look at simple
estimates for the Heat Kernel. Here we are allowed to choose between the Dirichlet Heat Kernel (related to absorbing random walk) or the Neumann version (related to random walk that reflects off the boundary). It is the second case that occurs in Diffusion Geometry. By reexamining the statement of the Riemann mapping theorem and using standard facts from study of the Laplace operator we will be led to an algorithm for picking out d eigenfunctions that provide "robust" local coordinates in a large
neighborhood of a given point in a smooth manifold of dimension d. The model to think of is the d dimensional unit cube or torus, where the sine or Fourier eigenfunctions easily give rise to robust local coordinate systems. (Notice that on the circle the two eigenfunctions
sine and cosine provide local coordinate systems in different patches. Neither one by itself gives a global coordinate patch.) We also discuss the setting of Diffusion Geometry on a finite set where there are again "heat" eigenfunctions.
Audio (MP3 File, Podcast Ready)
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