On the Dual Geometry of Laplacian Eigenfunctions

Abstract

We discuss the geometry of Laplacian eigenfunctions −∆φ = λφ on compact manifolds (M, g) and combinatorial graphs G = (V, E). The 'dual' geometry of Laplacian eigenfunctions is well understood on T d (identified with Z d ) and R n (which is self-dual). The dual geometry is of tremendous role in various fields of pure and applied mathematics. The purpose of our paper is to point out a notion of similarity between eigenfunctions that allows to reconstruct that geometry. Our measure of 'similarity' α(φ λ , φµ) between eigenfunctions φ λ and φµ is given by a global average of local correlationswhere p(t, x, y) is the classical heat kernel and e −tλ + e −tµ = 1. 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.2010 Mathematics Subject Classification. 35J05, 35P05, 42C10, 65T60, 81Q50, 94A11.

Approximating pointwise products of Laplacian eigenfunctions

“…Already the simpler question of understanding L 2 −size of the product is highly nontrivial: a seminal result of Burq-Gérard-Tzetkov [8] states e µ e λ L 2 min(λ 1/4 , µ 1/4 ) e λ L 2 e µ L 2 on compact two-dimensional manifolds without boundary (this has been extended to higher dimensions [4,7]). A recent result of the third author [23] (see also [9]) shows that one would generically (i.e., on typical manifolds in the presence of quantum chaos) expect e i (x)e j (x) to be mainly supported at eigenfunctions having their eigenvalue close to max {λ i , λ j } and that deviation from this phenomenon, as in the case of Fourier series on T for example, requires eigenfunctions to be strongly correlated at the wavelength in a precise sense.…”

Natural Graph Wavelet Packet Dictionaries

“…However, one important drawback of the above method is that the construction of the spectral filters F j solely depends on the eigenvalue distribution (except some flexibility in choosing the filter pair (h, g), and the dilation parameters {s j }) and does not reflect how the eigenvectors behave. [5,20,31,40,[48][49][50][51].…”

“…Second, in the long term, this is a first method of systematically using the novel concept of eigenvector dual geometry [5,31,49]. This direction can set a path for future modification of spectral graph theory applications to incorporate dual geometry.…”

“…This includes: the Spectral Graph Wavelet Transform [16]; Diffusion Wavelets [7]; extensions to spectral graph convolutional networks [29]. However, these dictionaries do not fully address the relationships among eigenvectors [5,31,49], which should be utilized for graph dictionary construction; instead, they focus on the eigenvalue distributions to organize the corresponding eigenvectors (although there are some works, e.g., [43,44,55], which recognized the graph structures strongly influence the eigenvector behaviors). These relationships among eigenvectors can result from eigenvector localization in different clusters, differing scales in multi-dimensional data, etc.…”

“…These relationships among eigenvectors can result from eigenvector localization in different clusters, differing scales in multi-dimensional data, etc. These notions of similarity and difference between eigenvectors, while studied in the eigenfunction literature [5,31,49], have yet to be incorporated into building localized dictionaries on graphs.…”