Learning latent representations in high-dimensional state spaces using polynomial manifold constructions
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866913162801971200 |
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| author | Geelen, Rudy Balzano, Laura Willcox, Karen |
| author_facet | Geelen, Rudy Balzano, Laura Willcox, Karen |
| contents | We present a novel framework for learning cost-efficient latent representations in problems with high-dimensional state spaces through nonlinear dimension reduction. By enriching linear state approximations with low-order polynomial terms we account for key nonlinear interactions existing in the data thereby reducing the problem's intrinsic dimensionality. Two methods are introduced for learning the representation of such low-dimensional, polynomial manifolds for embedding the data. The manifold parametrization coefficients can be obtained by regression via either a proper orthogonal decomposition or an alternating minimization based approach. Our numerical results focus on the one-dimensional Korteweg-de Vries equation where accounting for nonlinear correlations in the data was found to lower the representation error by up to two orders of magnitude compared to linear dimension reduction techniques. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_13748 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Learning latent representations in high-dimensional state spaces using polynomial manifold constructions Geelen, Rudy Balzano, Laura Willcox, Karen Numerical Analysis We present a novel framework for learning cost-efficient latent representations in problems with high-dimensional state spaces through nonlinear dimension reduction. By enriching linear state approximations with low-order polynomial terms we account for key nonlinear interactions existing in the data thereby reducing the problem's intrinsic dimensionality. Two methods are introduced for learning the representation of such low-dimensional, polynomial manifolds for embedding the data. The manifold parametrization coefficients can be obtained by regression via either a proper orthogonal decomposition or an alternating minimization based approach. Our numerical results focus on the one-dimensional Korteweg-de Vries equation where accounting for nonlinear correlations in the data was found to lower the representation error by up to two orders of magnitude compared to linear dimension reduction techniques. |
| title | Learning latent representations in high-dimensional state spaces using polynomial manifold constructions |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2306.13748 |