Robust GP kernel / input metric

This example shows how robust covariance estimators from robustcov can be used as input-space metrics for Gaussian-process kernels.

robustcov does not implement Gaussian-process regression, kernel ridge regression, likelihoods, posterior inference, Bayesian optimization, or training loops. The GP library owns those pieces. robustcov only supplies robust input-space covariance geometry.

Result at a glance

The contaminated design points inflate the ordinary empirical input covariance. A GP kernel based on that non-robust geometry can oversmooth along an important input direction. The robust input metric is less distorted.

What the data represent

The synthetic training set has two input features. The response depends mostly on the second feature. A small number of contaminated rows are placed far away in that same direction with unrelated responses.

Why this estimator

FastMCD is used because this is a low-dimensional contaminated-design example with separable leverage points. The resulting robust precision matrix is passed into a scikit-learn-compatible Mahalanobis RBF kernel.

Reproduce the result

python examples/gp_robust_input_metric.py

Output from the run

robust GP kernel input-metric example
gp_empirical_input_covariance_rmse=0.3566
gp_robust_input_covariance_rmse=0.1813
empirical_covariance_diag= [0.6417 1.8476]
robust_covariance_diag=    [0.5296 0.0273]
saved diagnostics to results/use_cases/gp_robust_input_metric

Figures and diagnostics

How to read the result

Compare the empirical-kernel GP curve with the robust-kernel GP curve. The GP model and training machinery are the same; only the input-space covariance geometry changes.

What this does not prove

This is not a robust GP likelihood and it does not make the model robust to outliers in y. Output-side robustness belongs to the GP library through likelihoods, noise models, or inference choices.