# Geometry-aware Bayesian Optimization in Robotics using Riemannian Mat'ern Kernels

arXiv: https://arxiv.org/abs/2111.01460 · HTML: https://mlsanity.com/paper/2111.01460/

## Key facts

- Cited by 24 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 56 references that are in the map.
- First posted on arXiv 2 Nov 2021.

## Builds on (most cited first, 20 of 20)

- [Poincar'e Embeddings for Learning Hierarchical Representations](https://mlsanity.com/paper/1705.08039/) (2017)
- [GPyTorch: Blackbox Matrix-Matrix Gaussian Process Inference with GPU Acceleration](https://mlsanity.com/paper/1809.11165/) (2018)
- [Learning Continuous Hierarchies in the Lorentz Model of Hyperbolic Geometry](https://mlsanity.com/paper/1806.03417/) (2018)
- [Efficiently Sampling Functions from Gaussian Process Posteriors](https://mlsanity.com/paper/2002.09309/) (2020)
- [Pymanopt: A Python Toolbox for Optimization on Manifolds using Automatic Differentiation](https://mlsanity.com/paper/1603.03236/) (2016)
- [Mat'ern Gaussian processes on Riemannian manifolds](https://mlsanity.com/paper/2006.10160/) (2020)
- [Mat'ern Gaussian Processes on Graphs](https://mlsanity.com/paper/2010.15538/) (2020)
- [From Trees to Continuous Embeddings and Back: Hyperbolic Hierarchical Clustering](https://mlsanity.com/paper/2010.00402/) (2020)
- [Pathwise Conditioning of Gaussian Processes](https://mlsanity.com/paper/2011.04026/) (2020)
- [Geodesic Exponential Kernels: When Curvature and Linearity Conflict](https://mlsanity.com/paper/1411.0296/) (2014)
- [Automatic LQR Tuning Based on Gaussian Process Global Optimization](https://mlsanity.com/paper/1605.01950/) (2016)
- [BoTorch: A Framework for Efficient Monte-Carlo Bayesian Optimization](https://mlsanity.com/paper/1910.06403/) (2019)
- [Bayesian Optimization Meets Riemannian Manifolds in Robot Learning](https://mlsanity.com/paper/1910.04998/) (2019)
- [Bayesian Optimization Using Domain Knowledge on the ATRIAS Biped](https://mlsanity.com/paper/1709.06047/) (2017)
- [High-Dimensional Bayesian Optimization via Nested Riemannian Manifolds](https://mlsanity.com/paper/2010.10904/) (2020)
- [Deep Kernels for Optimizing Locomotion Controllers](https://mlsanity.com/paper/1707.09062/) (2017)
- [Reproducing kernel Hilbert spaces on manifolds: Sobolev and Diffusion spaces](https://mlsanity.com/paper/1905.10913/) (2019)
- [Cautious Bayesian Optimization for Efficient and Scalable Policy Search](https://mlsanity.com/paper/2011.09445/) (2020)
- [On the Design of LQR Kernels for Efficient Controller Learning](https://mlsanity.com/paper/1709.07089/) (2017)
- [Interactive Trajectory Adaptation through Force-guided Bayesian Optimization](https://mlsanity.com/paper/1908.07263/) (2019)

## Built on by (most cited first, 24 of 24)

- [Recent Advances in Bayesian Optimization](https://mlsanity.com/paper/2206.03301/) (2022)
- [Stationary Kernels and Gaussian Processes on Lie Groups and their Homogeneous Spaces I: the compact case](https://mlsanity.com/paper/2208.14960/) (2022)
- [A Survey of Geometric Optimization for Deep Learning: From Euclidean Space to Riemannian Manifold](https://mlsanity.com/paper/2302.08210/) (2023)
- [Learning Riemannian Stable Dynamical Systems via Diffeomorphisms](https://mlsanity.com/paper/2211.03169/) (2022)
- [Stationary Kernels and Gaussian Processes on Lie Groups and their Homogeneous Spaces II: non-compact symmetric spaces](https://mlsanity.com/paper/2301.13088/) (2023)
- [MANTRA: The Manifold Triangulations Assemblage](https://mlsanity.com/paper/2410.02392/) (2024)
- [The GeometricKernels Package: Heat and Mat'ern Kernels for Geometric Learning on Manifolds, Meshes, and Graphs](https://mlsanity.com/paper/2407.08086/) (2024)
- [Bringing motion taxonomies to continuous domains via GPLVM on hyperbolic manifolds](https://mlsanity.com/paper/2210.01672/) (2022)
- [Alignment of Density Maps in Wasserstein Distance](https://mlsanity.com/paper/2305.12310/) (2023)
- [Isotropic Gaussian Processes on Finite Spaces of Graphs](https://mlsanity.com/paper/2211.01689/) (2022)
- [Optimization on Manifolds via Graph Gaussian Processes](https://mlsanity.com/paper/2210.10962/) (2022)
- [Posterior Contraction Rates for Mat'ern Gaussian Processes on Riemannian Manifolds](https://mlsanity.com/paper/2309.10918/) (2023)
- [Unraveling the Single Tangent Space Fallacy: An Analysis and Clarification for Applying Riemannian Geometry in Robot Learning](https://mlsanity.com/paper/2310.07902/) (2023)
- [Sample-efficient Bayesian Optimisation Using Known Invariances](https://mlsanity.com/paper/2410.16972/) (2024)
- [Residual Deep Gaussian Processes on Manifolds](https://mlsanity.com/paper/2411.00161/) (2024)
- [High-Dimensional Bayesian Optimization via Random Projection of Manifold Subspaces](https://mlsanity.com/paper/2412.16554/) (2024)
- [Visualizing Riemannian data with Rie-SNE](https://mlsanity.com/paper/2203.09253/) (2022)
- [On Probabilistic Pullback Metrics for Latent Hyperbolic Manifolds](https://mlsanity.com/paper/2410.20850/) (2024)
- [On power sum kernels on symmetric groups](https://mlsanity.com/paper/2211.05650/) (2022)
- [Extrinsic Bayesian Optimizations on Manifolds](https://mlsanity.com/paper/2212.13886/) (2022)
- [Non-parametric regression for robot learning on manifolds](https://mlsanity.com/paper/2310.19561/) (2023)
- [Efficient Sliced Wasserstein Distance Computation via Adaptive Bayesian Optimization](https://mlsanity.com/paper/2509.17405/) (2025)
- [Taxonomy-aware Dynamic Motion Generation on Hyperbolic Manifolds](https://mlsanity.com/paper/2509.21281/) (2025)
- [Information Theoretic Bayesian Optimization over the Probability Simplex](https://mlsanity.com/paper/2603.09793/) (2026)

## Abstract (arXiv)

Bayesian optimization is a data-efficient technique which can be used for control parameter tuning, parametric policy adaptation, and structure design in robotics. Many of these problems require optimization of functions defined on non-Euclidean domains like spheres, rotation groups, or spaces of positive-definite matrices. To do so, one must place a Gaussian process prior, or equivalently define a kernel, on the space of interest. Effective kernels typically reflect the geometry of the spaces they are defined on, but designing them is generally non-trivial. Recent work on the Riemannian Mat'ern kernels, based on stochastic partial differential equations and spectral theory of the Laplace-Beltrami operator, offers promising avenues towards constructing such geometry-aware kernels. In this paper, we study techniques for implementing these kernels on manifolds of interest in robotics, demonstrate their performance on a set of artificial benchmark functions, and illustrate geometry-aware Bayesian optimization for a variety of robotic applications, covering orientation control, manipulability optimization, and motion planning, while showing its improved performance.