mlsanityData to 7 Oct 2026

Fast and Robust Shortest Paths on Manifolds Learned from Data

arXiv 1901.07229 ↗First posted 22 Jan 2019stat.ML cs.LGcited by 25

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Abstract (arXiv)

We propose a fast, simple and robust algorithm for computing shortest paths and distances on Riemannian manifolds learned from data. This amounts to solving a system of ordinary differential equations (ODEs) subject to boundary conditions. Here standard solvers perform poorly because they require well-behaved Jacobians of the ODE, and usually, manifolds learned from data imply unstable and ill-conditioned Jacobians. Instead, we propose a fixed-point iteration scheme for solving the ODE that avoids Jacobians. This enhances the stability of the solver, while reduces the computational cost. In experiments involving both Riemannian metric learning and deep generative models we demonstrate significant improvements in speed and stability over both general-purpose state-of-the-art solvers as well as over specialized solvers.

Key facts. Cited by 25 papers in the map (data through 7 Oct 2026). First posted on arXiv 22 Jan 2019.

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2021: 8; 2022: 2; 2023: 7; 2024: 4; 2025: 3; 2026: 1.

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  1. 2024Metric Flow Matching for Smooth Interpolations on the Data Manifoldcited by 74
  2. 2023Norm-guided latent space exploration for text-to-image generationcited by 43
  3. 2021Learning Riemannian Manifolds for Geodesic Motion Skillscited by 23
  4. 2021Pulling back information geometrycited by 23
  5. 2024Manifold Integrated Gradients: Riemannian Geometry for Feature Attributioncited by 21
  6. 2023Riemannian Laplace approximations for Bayesian neural networkscited by 14
  7. 2021A prior-based approximate latent Riemannian metriccited by 11
  8. 2021Exploring the Latent Space of Autoencoders with Interventional Assayscited by 10
  9. 2021On Linear Interpolation in the Latent Space of Deep Generative Modelscited by 8
  10. 2024Pulling back symmetric Riemannian geometry for data analysiscited by 8
  11. 2023RDA-INR: Riemannian Diffeomorphic Autoencoding via Implicit Neural Representationscited by 7
  12. 2025Follow the Energy, Find the Path: Riemannian Metrics from Energy-Based Modelscited by 7
13 more
  1. 2022Reactive Motion Generation on Learned Riemannian Manifoldscited by 6
  2. 2023Intrinsic Gaussian Process on Unknown Manifolds with Probabilistic Metricscited by 6
  3. 2024Decoder ensembling for learned latent geometriescited by 6
  4. 2021Bayesian Quadrature on Riemannian Data Manifoldscited by 5
  5. 2022Mario Plays on a Manifold: Generating Functional Content in Latent Space through Differential Geometrycited by 5
  6. 2021Learning Identity-Preserving Transformations on Data Manifoldscited by 4
  7. 2023An Intrinsic Approach to Scalar-Curvature Estimation for Point Cloudscited by 4
  8. 2025Geodesic Calculus on Implicitly Defined Latent Manifoldscited by 2
  9. 2021Linear-Time Probabilistic Solutions of Boundary Value Problemscited by 1
  10. 2023Geometric path augmentation for inference of sparsely observed stochastic nonlinear systemscited by 1
  11. 2023Geometric constraints improve inference of sparsely observed stochastic dynamicscited by 1
  12. 2025A Likely Geometry of Generative Modelscited by 0
  13. 2026Intrinsic-Hybrid Latent Diffusion Models for Generative Modeling on Unknown Manifoldscited by 0