aiqbench analysis
This arXiv paper uses a conditional invertible neural network, the "flow chart," to learn normal coordinates on Riemannian manifolds, so that exponential and logarithmic maps become fast Euclidean operations. The authors report a 60-70x median speedup over numerical baselines at comparable error, with results across sixteen geometries. Those speedups are benchmark results, not validated in production, but the approach matters for anyone running repeated geodesic computations in geometric ML.
Learning Normal Coordinates on Riemannian Manifolds with Flow Charts
arXiv:2610.28302v1 Announce Type: new Abstract: Riemannian manifolds are powerful models for problems with non-Euclidean structure. Recent work in geometric machine learning has focused on training deep neural networks to learn Riemannian metrics from high-dimensional data. However, many geometric algorithms require repeated evaluations of the exponential and logarithmic maps, whose high computational cost restricts the utility of learned Riemannian manifolds in practice. We propose to address this cost by training a conditional invertible neural network, called a flow chart, to approximate normal coordinates around points on a given Riemannian manifold. With this approach, the exponential and logarithmic maps can be approximated by fast Euclidean operations in the model's learned coordinate space. Our main contribution is an efficient loss function for training flow charts that is nonnegative and, under suitable domain assumptions, is zero precisely when the chart is normal. Across sixteen synthetic and data-derived geometries, flow charts are competitive with a specialized neural geodesic interpolator for logarithmic map estimation, while also being able to estimate the exponential map. When pretrained, flow charts provide a median speedup of 60-70x over numerical baselines at comparable logarithmic map estimation error, and their learned coordinates improve the accuracy of a numerical geodesic solver.
Abstract:Riemannian manifolds are powerful models for problems with non-Euclidean structure. Recent work in geometric machine learning has focused on training deep neural networks to learn Riemannian metrics from high-dimensional data. However, many geometric algorithms require repeated evaluations of the exponential and logarithmic maps, whose high computational cost restricts the utility of learned Riemannian manifolds in practice. We propose to address this cost by training a conditional invertible neural network, called a flow chart, to approximate normal coordinates around points on a given Riemannian manifold. With this approach, the exponential and logarithmic maps can be approximated by fast Euclidean operations in the model's learned coordinate space. Our main contribution is an efficient loss function for training flow charts that is nonnegative and, under suitable domain assumptions, is zero precisely when the chart is normal. Across sixteen synthetic and data-derived geometries, flow charts are competitive with a specialized neural geodesic interpolator for logarithmic map estimation, while also being able to estimate the exponential map. When pretrained, flow charts provide a median speedup of 60-70x over numerical baselines at comparable logarithmic map estimation error, and their learned coordinates improve the accuracy of a numerical geodesic solver.
Machine Learning (cs.LG); Differential Geometry (math.DG)
arXiv:2610.28302 [cs.LG]
(or arXiv:2610.28302v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2610.28302 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.
Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.
Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.
Full text under the publisher's open licence · 290 words
Related coverage
A Decentralized LiDAR-SLAM System with Certifiably Optimal Pose Graph Optimization
arXiv · Jun 3, 2026
Riemannian Optimization-based Whole-body Obstacle Avoidance for Mobile Continuum Robots in Cluttered Environments
arXiv · Nov 5, 2026
Geometry-Aware Uncertainty Quantification via Conformal Prediction on Manifolds
arXiv · Jun 9, 2026
Aerodynamic Promptness of Tilting-Propeller Multirotors
arXiv · Oct 14, 2026