# Hierarchical-Hyperplane Kernels for Actively Learning Gaussian Process Models of Nonstationary Systems

arXiv: https://arxiv.org/abs/2303.10022 · HTML: https://mlsanity.com/paper/2303.10022/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 37 references that are in the map.
- First posted on arXiv 17 Mar 2023.

## Builds on (most cited first, 14 of 14)

- [Deep Kernel Learning](https://mlsanity.com/paper/1511.02222/) (2015)
- [Structure Discovery in Nonparametric Regression through Compositional Kernel Search](https://mlsanity.com/paper/1302.4922/) (2013)
- [Input Warping for Bayesian Optimization of Non-stationary Functions](https://mlsanity.com/paper/1402.0929/) (2014)
- [Automatic Construction and Natural-Language Description of Nonparametric Regression Models](https://mlsanity.com/paper/1402.4304/) (2014)
- [Non-Stationary Spectral Kernels](https://mlsanity.com/paper/1705.08736/) (2017)
- [Active Learning for Deep Gaussian Process Surrogates](https://mlsanity.com/paper/2012.08015/) (2020)
- [Active Learning of Linear Embeddings for Gaussian Processes](https://mlsanity.com/paper/1310.6740/) (2013)
- [Non-Stationary Gaussian Process Regression with Hamiltonian Monte Carlo](https://mlsanity.com/paper/1508.04319/) (2015)
- [Bayesian Active Learning with Fully Bayesian Gaussian Processes](https://mlsanity.com/paper/2205.10186/) (2022)
- [Active Learning for Gaussian Process Considering Uncertainties with Application to Shape Control of Composite Fuselage](https://mlsanity.com/paper/2004.10931/) (2020)
- [Safe Active Learning for Multi-Output Gaussian Processes](https://mlsanity.com/paper/2203.14849/) (2022)
- [Partitioned Active Learning for Heterogeneous Systems](https://mlsanity.com/paper/2105.08547/) (2021)
- [Structural Kernel Search via Bayesian Optimization and Symbolical Optimal Transport](https://mlsanity.com/paper/2210.11836/) (2022)
- [Scalable Gaussian Processes for Characterizing Multidimensional Change Surfaces](https://mlsanity.com/paper/1511.04408/) (2015)

## Built on by (most cited first, 4 of 4)

- [Amortized Inference for Gaussian Process Hyperparameters of Structured Kernels](https://mlsanity.com/paper/2306.09819/) (2023)
- [Polynomial Chaos Expanded Gaussian Process](https://mlsanity.com/paper/2405.01052/) (2024)
- [Amortized Active Learning for Nonparametric Functions](https://mlsanity.com/paper/2407.17992/) (2024)
- [Active Learning for Multiple Change Point Detection in Non-stationary Time Series with Deep Gaussian Processes](https://mlsanity.com/paper/2505.20452/) (2025)

## Abstract (arXiv)

Learning precise surrogate models of complex computer simulations and physical machines often require long-lasting or expensive experiments. Furthermore, the modeled physical dependencies exhibit nonlinear and nonstationary behavior. Machine learning methods that are used to produce the surrogate model should therefore address these problems by providing a scheme to keep the number of queries small, e.g. by using active learning and be able to capture the nonlinear and nonstationary properties of the system. One way of modeling the nonstationarity is to induce input-partitioning, a principle that has proven to be advantageous in active learning for Gaussian processes. However, these methods either assume a known partitioning, need to introduce complex sampling schemes or rely on very simple geometries. In this work, we present a simple, yet powerful kernel family that incorporates a partitioning that: i) is learnable via gradient-based methods, ii) uses a geometry that is more flexible than previous ones, while still being applicable in the low data regime. Thus, it provides a good prior for active learning procedures. We empirically demonstrate excellent performance on various active learning tasks.