Dual Geometry of Laplacian Eigenfunctions and Anisotropic Graph Wavelets

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. The purpose of this talk is to point out a notion of similarity between eigenfunctions that allows to reconstruct that geometry. Our measure of `similarity' between eigenfunctions is given by a global average of local correlations, and 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.