The Geometry of Nodal Sets and Outlier Detection

06/05/2017
by   Xiuyuan Cheng, et al.
0

Let (M,g) be a compact manifold and let -Δϕ_k = λ_k ϕ_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) = ∑_k ≤ N1/√(λ_k)|ϕ_k(x)|/ϕ_k_L^∞(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 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.

READ FULL TEXT

page 2

page 4

page 5

research
04/25/2018

On the Dual Geometry of Laplacian Eigenfunctions

We discuss the geometry of Laplacian eigenfunctions -Δϕ = λϕ on compact ...
research
04/27/2023

On Manifold Learning in Plato's Cave: Remarks on Manifold Learning and Physical Phenomena

Many techniques in machine learning attempt explicitly or implicitly to ...
research
07/28/2021

Large sample spectral analysis of graph-based multi-manifold clustering

In this work we study statistical properties of graph-based algorithms f...
research
06/16/2008

Manifold Learning: The Price of Normalization

We analyze the performance of a class of manifold-learning algorithms th...
research
09/14/2021

A geometric perspective on functional outlier detection

We consider functional outlier detection from a geometric perspective, s...

Please sign up or login with your details

Forgot password? Click here to reset