Graph $p$-Laplacian eigenpairs as saddle points of a family of spectral energy functions
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915136406552576 |
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| author | Deidda, Piero Segala, Nicola Putti, Mario |
| author_facet | Deidda, Piero Segala, Nicola Putti, Mario |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_07056 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Graph $p$-Laplacian eigenpairs as saddle points of a family of spectral energy functions Deidda, Piero Segala, Nicola Putti, Mario Numerical Analysis 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. |
| title | Graph $p$-Laplacian eigenpairs as saddle points of a family of spectral energy functions |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2405.07056 |