Graph $p$-Laplacian eigenpairs as saddle points of a family of spectral energy functions

TL;DR

This paper reformulates the graph p-Laplacian eigenpair problem as a constrained eigenvalue problem, introduces spectral energy functions whose saddle points correspond to eigenpairs, and develops gradient-based methods for computation.

Contribution

It provides a novel constrained formulation, links eigenpairs to saddle points of energy functions, and proposes new numerical algorithms for p-Laplacian eigenpair computation.

Findings

Establishes a correspondence between p-Laplacian eigenpairs and saddle points of energy functions.

Proves the uniqueness of the first eigenpair via a specific energy function.

Develops gradient-based algorithms for computing eigenpairs for p in (2, ∞).

Abstract

We address the problem of computing the graph $p$-Laplacian eigenpairs for $p\in (2,\infty)$. We propose a reformulation of the graph $p$-Laplacian eigenvalue problem in terms of a constrained weighted Laplacian eigenvalue problem and discuss theoretical and computational advantages. We provide a correspondence between $p$-Laplacian eigenpairs and linear eigenpair of a constrained generalized weighted Laplacian eigenvalue problem. As a result, we can assign an index to any $p$-Laplacian eigenpair that matches the Morse index of the $p$-Rayleigh quotient evaluated at the eigenfunction. In the second part of the paper we introduce a class of spectral energy functions that depend on edge and node weights. We prove that differentiable saddle points of the $k$-th energy function correspond to $p$-Laplacian eigenpairs having index equal to $k$. Moreover, the first energy function is proved to possess a unique saddle point which corresponds to the unique first $p$-Laplacian eigenpair. Finally we develop novel gradient-based numerical methods suited to compute $p$-Laplacian eigenpairs for any $p\in(2,\infty)$ and present some experiments.