# Latent Map Gaussian Processes for Mixed Variable Metamodeling

arXiv: https://arxiv.org/abs/2102.03935 · HTML: https://mlsanity.com/paper/2102.03935/

## Key facts

- Cited by 23 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 80 references that are in the map.
- First posted on arXiv 7 Feb 2021.

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

- [XGBoost: A Scalable Tree Boosting System](https://mlsanity.com/paper/1603.02754/) (2016)
- [Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning](https://mlsanity.com/paper/1506.02142/) (2015)
- [Machine Learning of Linear Differential Equations using Gaussian Processes](https://mlsanity.com/paper/1701.02440/) (2017)
- [Practical Multi-fidelity Bayesian Optimization for Hyperparameter Tuning](https://mlsanity.com/paper/1903.04703/) (2019)
- [Bayesian Optimization for Materials Design with Mixed Quantitative and Qualitative Variables](https://mlsanity.com/paper/1910.01688/) (2019)
- [The reparameterization trick for acquisition functions](https://mlsanity.com/paper/1712.00424/) (2017)
- [Local Gaussian process approximation for large computer experiments](https://mlsanity.com/paper/1303.0383/) (2013)
- [A Latent Variable Approach to Gaussian Process Modeling with Qualitative and Quantitative Factors](https://mlsanity.com/paper/1806.07504/) (2018)
- [Sequential Learning of Active Subspaces](https://mlsanity.com/paper/1907.11572/) (2019)
- [Data-Driven Topology Optimization with Multiclass Microstructures using Latent Variable Gaussian Process](https://mlsanity.com/paper/2006.15273/) (2020)
- [Probabilistic learning on manifolds constrained by nonlinear partial differential equations for small datasets](https://mlsanity.com/paper/2010.14324/) (2020)
- [Parametric Gaussian Process Regression for Big Data](https://mlsanity.com/paper/1704.03144/) (2017)

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

- [Multi-Fidelity Cost-Aware Bayesian Optimization](https://mlsanity.com/paper/2211.02732/) (2022)
- [A mixed-categorical correlation kernel for Gaussian process](https://mlsanity.com/paper/2211.08262/) (2022)
- [Data Fusion with Latent Map Gaussian Processes](https://mlsanity.com/paper/2112.02206/) (2021)
- [Multi-fidelity wavelet neural operator with application to uncertainty quantification](https://mlsanity.com/paper/2208.05606/) (2022)
- [Multi-fidelity Bayesian Optimization: A Review](https://mlsanity.com/paper/2311.13050/) (2023)
- [High-dimensional mixed-categorical Gaussian processes with application to multidisciplinary design optimization for a green aircraft](https://mlsanity.com/paper/2311.06130/) (2023)
- [A Gaussian Process Framework for Solving Forward and Inverse Problems Involving Nonlinear Partial Differential Equations](https://mlsanity.com/paper/2401.03492/) (2024)
- [Probabilistic Neural Data Fusion for Learning from an Arbitrary Number of Multi-fidelity Data Sets](https://mlsanity.com/paper/2301.13271/) (2023)
- [Practical multi-fidelity machine learning: fusion of deterministic and Bayesian models](https://mlsanity.com/paper/2407.15110/) (2024)
- [Unveiling Processing--Property Relationships in Laser Powder Bed Fusion: The Synergy of Machine Learning and High-throughput Experiments](https://mlsanity.com/paper/2409.00248/) (2024)
- [Sampling from Gaussian Processes: A Tutorial and Applications in Global Sensitivity Analysis and Optimization](https://mlsanity.com/paper/2507.14746/) (2025)
- [Constrained multi-fidelity Bayesian optimization with automatic stop condition](https://mlsanity.com/paper/2503.01126/) (2025)
- [Localized Physics-informed Gaussian Processes with Curriculum Training for Topology Optimization](https://mlsanity.com/paper/2503.15561/) (2025)
- [On the Effects of Heterogeneous Errors on Multi-fidelity Bayesian Optimization](https://mlsanity.com/paper/2309.02771/) (2023)
- [Distributional encoding for Gaussian process regression with qualitative inputs](https://mlsanity.com/paper/2506.04813/) (2025)
- [Unsupervised Anomaly Detection via Nonlinear Manifold Learning](https://mlsanity.com/paper/2306.09441/) (2023)
- [Multi-fidelity Design of Porous Microstructures for Thermofluidic Applications](https://mlsanity.com/paper/2310.18505/) (2023)
- [Weighted Euclidean Distance Matrices over Mixed Continuous and Categorical Inputs for Gaussian Process Models](https://mlsanity.com/paper/2503.02630/) (2025)
- [A preliminary data fusion study to assess the feasibility of Foundation Process-Property Models in Laser Powder Bed Fusion](https://mlsanity.com/paper/2503.16667/) (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)
- [Capabilities of Auto-encoders and Principal Component Analysis of the Reduction of Microstructural Images; Application on the Acceleration of Phase-Field Simulations](https://mlsanity.com/paper/2605.04229/) (2026)
- [On the Uncertainty Quantification Ability of Tabular Foundation Models](https://mlsanity.com/paper/2606.01427/) (2026)
- [On the Brittleness of Maximum Likelihood Estimation for Gaussian Process Hyperparameter Optimization](https://mlsanity.com/paper/2608.13793/) (2026)

## Abstract (arXiv)

Gaussian processes (GPs) are ubiquitously used in sciences and engineering as metamodels. Standard GPs, however, can only handle numerical or quantitative variables. In this paper, we introduce latent map Gaussian processes (LMGPs) that inherit the attractive properties of GPs and are also applicable to mixed data which have both quantitative and qualitative inputs. The core idea behind LMGPs is to learn a continuous, low-dimensional latent space or manifold which encodes all qualitative inputs. To learn this manifold, we first assign a unique prior vector representation to each combination of qualitative inputs. We then use a low-rank linear map to project these priors on a manifold that characterizes the posterior representations. As the posteriors are quantitative, they can be directly used in any standard correlation function such as the Gaussian or Matern. Hence, the optimal map and the corresponding manifold, along with other hyperparameters of the correlation function, can be systematically learned via maximum likelihood estimation. Through a wide range of analytic and real-world examples, we demonstrate the advantages of LMGPs over state-of-the-art methods in terms of accuracy and versatility. In particular, we show that LMGPs can handle variable-length inputs, have an explainable neural network interpretation, and provide insights into how qualitative inputs affect the response or interact with each other. We also employ LMGPs in Bayesian optimization and illustrate that they can discover optimal compound compositions more efficiently than conventional methods that convert compositions to qualitative variables via manual featurization.