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This research advances the geometric understanding of autoencoder latent spaces by introducing a discrete Riemannian calculus that computes geodesic paths and exponential maps directly on implicit manifolds. The proposed framework offers robustness against representation inaccuracies while remaining independent of specific autoencoder architectures, enabling flexible analysis through various Riemannian geometries. By evaluating these tools across synthetic and real datasets, the study provides a rigorous mathematical foundation for interpreting low-dimensional data representations without relying solely on classical Euclidean assumptions.
Geodesic Calculus on Implicitly Defined Latent Manifolds
arXiv:2510.09468v3 Announce Type: replace Abstract: 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.
Abstract: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.
Machine Learning (cs.LG)
arXiv:2510.09468 [cs.LG]
(or arXiv:2510.09468v3 [cs.LG] for this version)
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