The geometry of nodal sets and outlier detection

JOURNAL OF NUMBER THEORY

卷:185

Article

Cheng, Xiuyuan¹  Mishne, Gal¹  Steinerberger, Stefan² 

[1] Yale Univ, Appl Math Program, New Haven, CT 06510 USA

[2] Yale Univ, Dept Math, New Haven, CT 06510 USA

关键词: Laplacian eigenfunctions;    Nodal sets;    Outlier detection;    Paley graphs;    Fractals;   

DOI  :  10.1016/j.jnt.2017.09.021

来源: Elsevier

【 摘 要 】

Let (M, g) be a compact manifold and let -Delta phi(k) = lambda(k)phi(k) be the sequence of Laplacian eigenfunctions. We present a curious new phenomenon which, so far, we only managed to understand in a few highly specialized cases: the family of functions f(N) : M -> R >= 0 f(N)(x)=Sigma(k <= N) 1/root lambda(k)vertical bar phi(k)(x)vertical bar/parallel to phi(k)parallel to L-infinity(M) and their extrema seem strangely suited for the detection of anomalous points on the manifold. It may be heuristically interpreted as the sum over distances to the nearest nodal line and potentially hints at a new phenomenon in spectral geometry. We give rigorous statements on the unit square [0,1](2) (where minima localize in Q(2)) and on Paley graphs (where f(N) recovers the geometry of quadratic residues of the underlying finite field F-p). Numerical examples show that the phenomenon seems to arise on fairly generic manifolds. (C) 2017 Elsevier Inc. All rights reserved.

【 授权许可】

Free