Geodesic Calculus on Latent Spaces

• URL: http://arxiv.org/abs/2510.09468v1
• Date: Fri, 10 Oct 2025 15:25:03 GMT
• Title: Geodesic Calculus on Latent Spaces
• Authors: Florine Hartwig, Josua Sassen, Juliane Braunsmann, Martin Rumpf, Benedikt Wirth,
• Abstract summary: We develop tools for a discrete Riemannian calculus approximating classical geometric operators.<n>We learn an approximate projection onto the latent manifold by minimizing a denoising objective.<n>We evaluate our approach on various autoencoders trained on synthetic and real data.
• Score: 4.023417156982924
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

Related papers

• Estimating Dataset Dimension via Singular Metrics under the Manifold Hypothesis: Application to Inverse Problems [0.6138671548064356]
  We propose a framework to deal with three key tasks: estimating the intrinsic dimension of the manifold, constructing appropriate local coordinates, and learning mappings between ambient and manifold spaces.<n>We focus on estimating the ID of datasets by analyzing the numerical rank of the VAE decoder pullback metric.<n>The estimated ID guides the construction of an atlas of local charts using a mixture of invertible VAEs, enabling accurate manifold parameterization and efficient inference.
  arXiv  Detail & Related papers  (2025-07-09T21:22:59Z)
• Riemannian generative decoder [11.074080383657453]
  We present a new method for learning representations based on manifold-valued latents.<n>Our method is compatible with existing architectures and yields interpretable latent spaces aligned with data geometry.<n>We validate our approach on three case studies -- a synthetic branching diffusion process, human migrations inferred from mitochondrial DNA, and cells undergoing a cell division cycle.
  arXiv  Detail & Related papers  (2025-06-23T21:06:13Z)
• Follow the Energy, Find the Path: Riemannian Metrics from Energy-Based Models [63.331590876872944]
  We propose a method for deriving Riemannian metrics directly from pretrained Energy-Based Models.<n>These metrics define spatially varying distances, enabling the computation of geodesics.<n>We show that EBM-derived metrics consistently outperform established baselines.
  arXiv  Detail & Related papers  (2025-05-23T12:18:08Z)
• Riemann$^2$: Learning Riemannian Submanifolds from Riemannian Data [12.424539896723603]
  Latent variable models are powerful tools for learning low-dimensional manifold from high-dimensional data.<n>This paper generalizes previous work and allows us to handle complex tasks in various domains, including robot motion synthesis and analysis of brain connectomes.
  arXiv  Detail & Related papers  (2025-03-07T16:08:53Z)
• Score-based Pullback Riemannian Geometry: Extracting the Data Manifold Geometry using Anisotropic Flows [10.649159213723106]
  We propose a framework for data-driven Riemannian geometry that is scalable in both geometry and learning.<n>We show that the proposed framework produces high-quality geodesics passing through the data support.<n>This is the first scalable framework for extracting the complete geometry of the data manifold.
  arXiv  Detail & Related papers  (2024-10-02T18:52:12Z)
• Improving embedding of graphs with missing data by soft manifolds [51.425411400683565]
  The reliability of graph embeddings depends on how much the geometry of the continuous space matches the graph structure. We introduce a new class of manifold, named soft manifold, that can solve this situation. Using soft manifold for graph embedding, we can provide continuous spaces to pursue any task in data analysis over complex datasets.
  arXiv  Detail & Related papers  (2023-11-29T12:48:33Z)
• The Fisher-Rao geometry of CES distributions [50.50897590847961]
  The Fisher-Rao information geometry allows for leveraging tools from differential geometry. We will present some practical uses of these geometric tools in the framework of elliptical distributions.
  arXiv  Detail & Related papers  (2023-10-02T09:23:32Z)
• Short and Straight: Geodesics on Differentiable Manifolds [6.85316573653194]
  In this work, we first analyse existing methods for computing length-minimising geodesics. Second, we propose a model-based parameterisation for distance fields and geodesic flows on continuous manifold. Third, we develop a curvature-based training mechanism, sampling and scaling points in regions of the manifold exhibiting larger values of the Ricci scalar.
  arXiv  Detail & Related papers  (2023-05-24T15:09:41Z)
• VTAE: Variational Transformer Autoencoder with Manifolds Learning [144.0546653941249]
  Deep generative models have demonstrated successful applications in learning non-linear data distributions through a number of latent variables. The nonlinearity of the generator implies that the latent space shows an unsatisfactory projection of the data space, which results in poor representation learning. We show that geodesics and accurate computation can substantially improve the performance of deep generative models.
  arXiv  Detail & Related papers  (2023-04-03T13:13:19Z)
• A singular Riemannian geometry approach to Deep Neural Networks I. Theoretical foundations [77.86290991564829]
  Deep Neural Networks are widely used for solving complex problems in several scientific areas, such as speech recognition, machine translation, image analysis. We study a particular sequence of maps between manifold, with the last manifold of the sequence equipped with a Riemannian metric. We investigate the theoretical properties of the maps of such sequence, eventually we focus on the case of maps between implementing neural networks of practical interest.
  arXiv  Detail & Related papers  (2021-12-17T11:43:30Z)
• Manifold Learning via Manifold Deflation [105.7418091051558]
  dimensionality reduction methods provide a valuable means to visualize and interpret high-dimensional data. Many popular methods can fail dramatically, even on simple two-dimensional Manifolds. This paper presents an embedding method for a novel, incremental tangent space estimator that incorporates global structure as coordinates. Empirically, we show our algorithm recovers novel and interesting embeddings on real-world and synthetic datasets.
  arXiv  Detail & Related papers  (2020-07-07T10:04:28Z)
• Disentangling by Subspace Diffusion [72.1895236605335]
  We show that fully unsupervised factorization of a data manifold is possible if the true metric of the manifold is known. Our work reduces the question of whether unsupervised metric learning is possible, providing a unifying insight into the geometric nature of representation learning.
  arXiv  Detail & Related papers  (2020-06-23T13:33:19Z)
• Computationally Tractable Riemannian Manifolds for Graph Embeddings [10.420394952839242]
  We show how to learn and optimize graph embeddings in certain curved Riemannian spaces. Our results serve as new evidence for the benefits of non-Euclidean embeddings in machine learning pipelines.
  arXiv  Detail & Related papers  (2020-02-20T10:55:47Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.