# The Geometry of Nodal Sets and Outlier Detection

## TL;DR

This paper introduces a new spectral geometric function based on eigenfunctions that appears effective for detecting anomalous points on manifolds, supported by rigorous results and numerical evidence.

## Contribution

It proposes a novel spectral function for outlier detection on manifolds and provides rigorous analysis in specific cases like the unit square and Paley graphs.

## Key findings

- The function localizes minima at rational points on the unit square.
- It recovers quadratic residue geometry on Paley graphs.
- Numerical examples suggest the phenomenon is widespread.

## 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.