# On the Dual Geometry of Laplacian Eigenfunctions

arXiv: https://arxiv.org/abs/1804.09816 · HTML: https://mlsanity.com/paper/1804.09816/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 Apr 2018.

## Built on by (most cited first, 1 of 1)

- [LDLE: Low Distortion Local Eigenmaps](https://mlsanity.com/paper/2101.11055/) (2021)

## Abstract (arXiv)

We discuss the geometry of Laplacian eigenfunctions -Delta φ = λ φ on compact manifolds (M,g) and combinatorial graphs G=(V,E). The 'dual' geometry of Laplacian eigenfunctions is well understood on mathbbTd (identified with mathbbZd) and ℝn (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(φλ, φμ) between eigenfunctions φλ and φμ is given by a global average of local correlations  α(φλ, φμ)2 = | φλ φμ |L2-2intM left( intM p(t,x,y)( φλ(y) - φλ(x))( φμ(y) - φμ(x)) dy right)2 dx, where p(t,x,y) is the classical heat kernel and e-t λ + e-t μ = 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.