Convergence of Manifold Filter-Combine Networks
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arXiv
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| Main Authors: | , , , , , , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866910656927629312 |
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| author | Johnson, David R. Chew, Joyce Viswanath, Siddharth De Brouwer, Edward Needell, Deanna Krishnaswamy, Smita Perlmutter, Michael |
| author_facet | Johnson, David R. Chew, Joyce Viswanath, Siddharth De Brouwer, Edward Needell, Deanna Krishnaswamy, Smita Perlmutter, Michael |
| contents | In order to better understand manifold neural networks (MNNs), we introduce Manifold Filter-Combine Networks (MFCNs). The filter-combine framework parallels the popular aggregate-combine paradigm for graph neural networks (GNNs) and naturally suggests many interesting families of MNNs which can be interpreted as the manifold analog of various popular GNNs. We then propose a method for implementing MFCNs on high-dimensional point clouds that relies on approximating the manifold by a sparse graph. We prove that our method is consistent in the sense that it converges to a continuum limit as the number of data points tends to infinity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_14639 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Convergence of Manifold Filter-Combine Networks Johnson, David R. Chew, Joyce Viswanath, Siddharth De Brouwer, Edward Needell, Deanna Krishnaswamy, Smita Perlmutter, Michael Machine Learning Signal Processing In order to better understand manifold neural networks (MNNs), we introduce Manifold Filter-Combine Networks (MFCNs). The filter-combine framework parallels the popular aggregate-combine paradigm for graph neural networks (GNNs) and naturally suggests many interesting families of MNNs which can be interpreted as the manifold analog of various popular GNNs. We then propose a method for implementing MFCNs on high-dimensional point clouds that relies on approximating the manifold by a sparse graph. We prove that our method is consistent in the sense that it converges to a continuum limit as the number of data points tends to infinity. |
| title | Convergence of Manifold Filter-Combine Networks |
| topic | Machine Learning Signal Processing |
| url | https://arxiv.org/abs/2410.14639 |