Variational graph p-Laplacian eigendecomposition under p-orthogonality constraints

Alessandro Lanza (), Serena Morigi () and Giuseppe Recupero ()
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Alessandro Lanza: University of Bologna
Serena Morigi: University of Bologna
Giuseppe Recupero: University of Bologna

Computational Optimization and Applications, 2025, vol. 91, issue 2, No 15, 787-825

Abstract: Abstract The p-Laplacian is a non-linear generalization of the Laplace operator. In the graph context, its eigenfunctions are used for data clustering, spectral graph theory, dimensionality reduction and other problems, as non-linearity better captures the underlying geometry of the data. We formulate the graph p-Laplacian nonlinear eigenproblem as an optimization problem under p-orthogonality constraints. The problem of computing multiple eigenpairs of the graph p-Laplacian is then approached incrementally by minimizing the graph Rayleigh quotient under nonlinear constraints. A simple reformulation allows us to take advantage of linear constraints. We propose two different optimization algorithms to solve the variational problem. The first is a projected gradient descent on manifold, and the second is an Alternate Direction Method of Multipliers which leverages the scaling invariance of the graph Rayleigh quotient to solve a constrained minimization under p-orthogonality constraints. We demonstrate the effectiveness and accuracy of the proposed algorithms and compare them in terms of efficiency.

Keywords: Graph p-Laplacian; Nonlinear eigendecomposition; Manifold optimization; Variational method (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s10589-024-00631-2

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