We will first discuss some work with Mauro Maggioni and Raanan Schul that "explains" the robustness of Laplace eigenfunctions as effective local coordinate systems for knowledge building. The result is also flexible in the sense that the proof breaks down into rather simple elements, which can easily be interpreted in a wide number of settings. While this may be very satisfying from a theoretical point of view, we will point out several numerical challenges that remain. We also discuss some theoretical challenges. One of these is to unify spectral methods with (multiscale) SVD (aka principle components analysis). A numerically appealing fact for this last problem is that multiscale SVD has essentially the same running time as a single SVD pass, and is known to encode a large amount of geometric information for data sets embedded in Euclidean space.
Audio (MP3 File, Podcast Ready)
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