# Optimization on Manifolds via Graph Gaussian Processes

arXiv: https://arxiv.org/abs/2210.10962 · HTML: https://mlsanity.com/paper/2210.10962/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 31 of its 68 references that are in the map.
- First posted on arXiv 20 Oct 2022.

## Builds on (most cited first, 30 of 31)

- [A Tutorial on Bayesian Optimization](https://mlsanity.com/paper/1807.02811/) (2018)
- [Bayesian Optimization in a Billion Dimensions via Random Embeddings](https://mlsanity.com/paper/1301.1942/) (2013)
- [On Information Gain and Regret Bounds in Gaussian Process Bandits](https://mlsanity.com/paper/2009.06966/) (2020)
- [Mat'ern Gaussian processes on Riemannian manifolds](https://mlsanity.com/paper/2006.10160/) (2020)
- [Adaptive and Safe Bayesian Optimization in High Dimensions via One-Dimensional Subspaces](https://mlsanity.com/paper/1902.03229/) (2019)
- [Error estimates for spectral convergence of the graph Laplacian on random geometric graphs towards the Laplace--Beltrami operator](https://mlsanity.com/paper/1801.10108/) (2018)
- [Mat'ern Gaussian Processes on Graphs](https://mlsanity.com/paper/2010.15538/) (2020)
- [Bayesian Optimization of Combinatorial Structures](https://mlsanity.com/paper/1806.08838/) (2018)
- [Misspecified Gaussian Process Bandit Optimization](https://mlsanity.com/paper/2111.05008/) (2021)
- [Geodesic Exponential Kernels: When Curvature and Linearity Conflict](https://mlsanity.com/paper/1411.0296/) (2014)
- [A Brief Introduction to Manifold Optimization](https://mlsanity.com/paper/1906.05450/) (2019)
- [Concentration Inequalities for Statistical Inference](https://mlsanity.com/paper/2011.02258/) (2020)
- [Geometry-aware Bayesian Optimization in Robotics using Riemannian Mat'ern Kernels](https://mlsanity.com/paper/2111.01460/) (2021)
- [Bayesian Optimization Meets Riemannian Manifolds in Robot Learning](https://mlsanity.com/paper/1910.04998/) (2019)
- [Bayesian Optimization over Hybrid Spaces](https://mlsanity.com/paper/2106.04682/) (2021)
- [Bayesian Optimization with Exponential Convergence](https://mlsanity.com/paper/1604.01348/) (2016)
- [Inverse Problems and Data Assimilation](https://mlsanity.com/paper/1810.06191/) (2018)
- [Optimization on flag manifolds](https://mlsanity.com/paper/1907.00949/) (2019)
- [Graph Based Gaussian Processes on Restricted Domains](https://mlsanity.com/paper/2010.07242/) (2020)
- [Gaussian Process Landmarking on Manifolds](https://mlsanity.com/paper/1802.03479/) (2018)
- [The rational SPDE approach for Gaussian random fields with general smoothness](https://mlsanity.com/paper/1711.04333/) (2017)
- [The SPDE Approach to Mat'ern Fields: Graph Representations](https://mlsanity.com/paper/2004.08000/) (2020)
- [Global optimization using random embeddings](https://mlsanity.com/paper/2107.12102/) (2021)
- [On the Consistency of Graph-based Bayesian Learning and the Scalability of Sampling Algorithms](https://mlsanity.com/paper/1710.07702/) (2017)
- [Constrained global optimization of functions with low effective dimensionality using multiple random embeddings](https://mlsanity.com/paper/2009.10446/) (2020)
- [Continuum Limit of Posteriors in Graph Bayesian Inverse Problems](https://mlsanity.com/paper/1706.07193/) (2017)
- [Unlabeled Data Help in Graph-Based Semi-Supervised Learning: A Bayesian Nonparametrics Perspective](https://mlsanity.com/paper/2008.11809/) (2020)
- [Manifold Free Riemannian Optimization](https://mlsanity.com/paper/2209.03269/) (2022)
- [Local Regularization of Noisy Point Clouds: Improved Global Geometric Estimates and Data Analysis](https://mlsanity.com/paper/1904.03335/) (2019)
- [Graph-based Prior and Forward Models for Inverse Problems on Manifolds with Boundaries](https://mlsanity.com/paper/2106.06787/) (2021)

## Built on by (most cited first, 6 of 6)

- [Enhancing Gaussian Process Surrogates for Optimization and Posterior Approximation via Random Exploration](https://mlsanity.com/paper/2401.17037/) (2024)
- [Efficient optimization of expensive black-box simulators via marginal means, with application to neutrino detector design](https://mlsanity.com/paper/2508.01834/) (2025)
- [Active Learning for Manifold Gaussian Process Regression](https://mlsanity.com/paper/2506.20928/) (2025)
- [Bayesian Optimization on Networks](https://mlsanity.com/paper/2510.27643/) (2025)
- [A Representation Optimization Dichotomy, Lie-Algebraic Policy Optimization](https://mlsanity.com/paper/2603.25525/) (2026)
- [Exploration strategies for Gaussian process bandit optimization with inference regret guarantees](https://mlsanity.com/paper/2610.16187/) (2026)

## Abstract (arXiv)

This paper integrates manifold learning techniques within a Gaussian process upper confidence bound algorithm to optimize an objective function on a manifold. Our approach is motivated by applications where a full representation of the manifold is not available and querying the objective is expensive. We rely on a point cloud of manifold samples to define a graph Gaussian process surrogate model for the objective. Query points are sequentially chosen using the posterior distribution of the surrogate model given all previous queries. We establish regret bounds in terms of the number of queries and the size of the point cloud. Several numerical examples complement the theory and illustrate the performance of our method.