# Group kernels for Gaussian process metamodels with categorical inputs

arXiv: https://arxiv.org/abs/1802.02368 · HTML: https://mlsanity.com/paper/1802.02368/

## Key facts

- Cited by 23 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Feb 2018.

## Built on by (most cited first, 23 of 23)

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- [Scalable mixed-domain Gaussian process modeling and model reduction for longitudinal data](https://mlsanity.com/paper/2111.02019/) (2021)
- [Multi-objective robust optimization using adaptive surrogate models for problems with mixed continuous-categorical parameters](https://mlsanity.com/paper/2203.01996/) (2022)
- [A Unified Gaussian Process for Branching and Nested Hyperparameter Optimization](https://mlsanity.com/paper/2402.04885/) (2024)
- [Frequency-aware Surrogate Modeling With SMT Kernels For Advanced Data Forecasting](https://mlsanity.com/paper/2507.09694/) (2025)
- [A reproducible comparative study of categorical kernels for Gaussian process regression, with new clustering-based nested kernels](https://mlsanity.com/paper/2510.01840/) (2025)
- [MF-SCBO : Multi-fidelity Scalable Constrained Bayesian Optimization](https://mlsanity.com/paper/2609.29941/) (2026)
- [Hierarchical Gaussian Processes for Survey Weighting: Local Equivalent Weights in Structured Populations](https://mlsanity.com/paper/2610.20321/) (2026)

## Abstract (arXiv)

Gaussian processes (GP) are widely used as a metamodel for emulating time-consuming computer codes. We focus on problems involving categorical inputs, with a potentially large number L of levels (typically several tens), partitioned in G << L groups of various sizes. Parsimonious covariance functions, or kernels, can then be defined by block covariance matrices T with constant covariances between pairs of blocks and within blocks. We study the positive definiteness of such matrices to encourage their practical use. The hierarchical group/level structure, equivalent to a nested Bayesian linear model, provides a parameterization of valid block matrices T. The same model can then be used when the assumption within blocks is relaxed, giving a flexible parametric family of valid covariance matrices with constant covariances between pairs of blocks. The positive definiteness of T is equivalent to the positive definiteness of a smaller matrix of size G, obtained by averaging each block. The model is applied to a problem in nuclear waste analysis, where one of the categorical inputs is atomic number, which has more than 90 levels.