The Geometry of Nodal Sets and Outlier Detection

Abstract

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 $f_N:M \rightarrow \mathbb{R}_{\geq 0}$ $$ f_N(x) = \sum_{k \leq N}{ \frac{1}{\sqrt{\lambda_k}} \frac{|\phi_k(x)|}{\|\phi_k\|_{L^{\infty}(M)}}}$$ seems 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 $\mathbb{Q}^2$) and on Paley graphs (where $f_N$ recovers the geometry of quadratic residues of the underlying finite field $\mathbb{F}_p$). Numerical examples show that the phenomenon seems to arise on fairly generic manifolds.