# Extrinsic Bayesian Optimizations on Manifolds

arXiv: https://arxiv.org/abs/2212.13886 · HTML: https://mlsanity.com/paper/2212.13886/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 12 of its 38 references that are in the map.
- First posted on arXiv 21 Dec 2022.

## Builds on (most cited first, 12 of 12)

- [A Tutorial on Bayesian Optimization](https://mlsanity.com/paper/1807.02811/) (2018)
- [Mat'ern Gaussian processes on Riemannian manifolds](https://mlsanity.com/paper/2006.10160/) (2020)
- [Constrained Efficient Global Optimization of Expensive Black-box Functions](https://mlsanity.com/paper/2211.00162/) (2022)
- [Optimization Techniques on Riemannian Manifolds](https://mlsanity.com/paper/1407.5965/) (2014)
- [Optimization of Graph Neural Networks with Natural Gradient Descent](https://mlsanity.com/paper/2008.09624/) (2020)
- [Geometry-aware Bayesian Optimization in Robotics using Riemannian Mat'ern Kernels](https://mlsanity.com/paper/2111.01460/) (2021)
- [High-Dimensional Bayesian Optimization via Nested Riemannian Manifolds](https://mlsanity.com/paper/2010.10904/) (2020)
- [Extrinsic Gaussian processes for regression and classification on manifolds](https://mlsanity.com/paper/1706.08757/) (2017)
- [Intrinsic Gaussian processes on complex constrained domains](https://mlsanity.com/paper/1801.01061/) (2018)
- [Accelerated Algorithms for Convex and Non-Convex Optimization on Manifolds](https://mlsanity.com/paper/2010.08908/) (2020)
- [Learning Subspaces of Different Dimension](https://mlsanity.com/paper/1404.6841/) (2014)
- [Communication Efficient Parallel Algorithms for Optimization on Manifolds](https://mlsanity.com/paper/1810.11155/) (2018)

## Abstract (arXiv)

We propose an extrinsic Bayesian optimization (eBO) framework for general optimization problems on manifolds. Bayesian optimization algorithms build a surrogate of the objective function by employing Gaussian processes and quantify the uncertainty in that surrogate by deriving an acquisition function. This acquisition function represents the probability of improvement based on the kernel of the Gaussian process, which guides the search in the optimization process. The critical challenge for designing Bayesian optimization algorithms on manifolds lies in the difficulty of constructing valid covariance kernels for Gaussian processes on general manifolds. Our approach is to employ extrinsic Gaussian processes by first embedding the manifold onto some higher dimensional Euclidean space via equivariant embeddings and then constructing a valid covariance kernel on the image manifold after the embedding. This leads to efficient and scalable algorithms for optimization over complex manifolds. Simulation study and real data analysis are carried out to demonstrate the utilities of our eBO framework by applying the eBO to various optimization problems over manifolds such as the sphere, the Grassmannian, and the manifold of positive definite matrices.