Geodesic Calculus on Implicitly Defined Latent Manifolds
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866914290623053824 |
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| author | Hartwig, Florine Sassen, Josua Braunsmann, Juliane Rumpf, Martin Wirth, Benedikt |
| author_facet | Hartwig, Florine Sassen, Josua Braunsmann, Juliane Rumpf, Martin Wirth, Benedikt |
| contents | Latent manifolds of autoencoders provide low-dimensional representations of data, which can be studied from a geometric perspective. We propose to describe these latent manifolds as implicit submanifolds of some ambient latent space. Based on this, we develop tools for a discrete Riemannian calculus approximating classical geometric operators. These tools are robust against inaccuracies of the implicit representation often occurring in practical examples. To obtain a suitable implicit representation, we propose to learn an approximate projection onto the latent manifold by minimizing a denoising objective. This approach is independent of the underlying autoencoder and supports the use of different Riemannian geometries on the latent manifolds. The framework in particular enables the computation of geodesic paths connecting given end points and shooting geodesics via the Riemannian exponential maps on latent manifolds. We evaluate our approach on various autoencoders trained on synthetic and real data. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_09468 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Geodesic Calculus on Implicitly Defined Latent Manifolds Hartwig, Florine Sassen, Josua Braunsmann, Juliane Rumpf, Martin Wirth, Benedikt Machine Learning 68T07, 53Z50 (Primary) 53B12 (Secondary) I.2.6 Latent manifolds of autoencoders provide low-dimensional representations of data, which can be studied from a geometric perspective. We propose to describe these latent manifolds as implicit submanifolds of some ambient latent space. Based on this, we develop tools for a discrete Riemannian calculus approximating classical geometric operators. These tools are robust against inaccuracies of the implicit representation often occurring in practical examples. To obtain a suitable implicit representation, we propose to learn an approximate projection onto the latent manifold by minimizing a denoising objective. This approach is independent of the underlying autoencoder and supports the use of different Riemannian geometries on the latent manifolds. The framework in particular enables the computation of geodesic paths connecting given end points and shooting geodesics via the Riemannian exponential maps on latent manifolds. We evaluate our approach on various autoencoders trained on synthetic and real data. |
| title | Geodesic Calculus on Implicitly Defined Latent Manifolds |
| topic | Machine Learning 68T07, 53Z50 (Primary) 53B12 (Secondary) I.2.6 |
| url | https://arxiv.org/abs/2510.09468 |