Hypergraph $p$-Laplacian equations for data interpolation and semi-supervised learning
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910904813092864 |
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| author | Shi, Kehan Burger, Martin |
| author_facet | Shi, Kehan Burger, Martin |
| contents | Hypergraph learning with $p$-Laplacian regularization has attracted a lot of attention due to its flexibility in modeling higher-order relationships in data. This paper focuses on its fast numerical implementation, which is challenging due to the non-differentiability of the objective function and the non-uniqueness of the minimizer. We derive a hypergraph $p$-Laplacian equation from the subdifferential of the $p$-Laplacian regularization. A simplified equation that is mathematically well-posed and computationally efficient is proposed as an alternative. Numerical experiments verify that the simplified $p$-Laplacian equation suppresses spiky solutions in data interpolation and improves classification accuracy in semi-supervised learning. The remarkably low computational cost enables further applications. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_12601 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Hypergraph $p$-Laplacian equations for data interpolation and semi-supervised learning Shi, Kehan Burger, Martin Numerical Analysis Machine Learning 35R02, 65D05 Hypergraph learning with $p$-Laplacian regularization has attracted a lot of attention due to its flexibility in modeling higher-order relationships in data. This paper focuses on its fast numerical implementation, which is challenging due to the non-differentiability of the objective function and the non-uniqueness of the minimizer. We derive a hypergraph $p$-Laplacian equation from the subdifferential of the $p$-Laplacian regularization. A simplified equation that is mathematically well-posed and computationally efficient is proposed as an alternative. Numerical experiments verify that the simplified $p$-Laplacian equation suppresses spiky solutions in data interpolation and improves classification accuracy in semi-supervised learning. The remarkably low computational cost enables further applications. |
| title | Hypergraph $p$-Laplacian equations for data interpolation and semi-supervised learning |
| topic | Numerical Analysis Machine Learning 35R02, 65D05 |
| url | https://arxiv.org/abs/2411.12601 |