Title:
Dual Geometry of Laplacian Eigenfunctions with Applications to Graph Wavelets, Cuts, and Visualization
Abstract:
We discuss the geometry of Laplacian eigenfunctions on compact manifolds and combinatorial graphs.
The 'dual' geometry of Laplacian eigenfunctions is well understood on the torus and euclidean space,
and is of tremendous importance in various fields of pure and applied mathematics.
In this talk, we derive a measure of 'similarity' between eigenfunctions given by a global average of local correlations,
and show its relationship to pointwise products. This notion recovers all classical notions of duality
but is equally applicable to other (rough) geometries and graphs;
many numerical examples in different continuous and discrete settings illustrate the result.
This talk will also focus on the applications of discovering such a dual geometry,
namely in constructing anisotropic graph wavelet packets and anisotropic graph cuts.
This is joint work with Stefan Steinerberger, Haotian Li, and Naoki Saito.
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