# Calibrated Derivative-Process Sensitivity for Gaussian-Process Variable Selection

arXiv: https://arxiv.org/abs/2609.33549 · HTML: https://mlsanity.com/paper/2609.33549/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 17 of its 49 references that are in the map.
- First posted on arXiv 27 Sep 2026.

## Builds on (most cited first, 17 of 17)

- [High-Dimensional Bayesian Optimization with Sparse Axis-Aligned Subspaces](https://mlsanity.com/paper/2103.00349/) (2021)
- [Panning for Gold: Model-X Knockoffs for High-dimensional Controlled Variable Selection](https://mlsanity.com/paper/1610.02351/) (2016)
- [Controlling the false discovery rate via knockoffs](https://mlsanity.com/paper/1404.5609/) (2014)
- [Sequential Learning of Active Subspaces](https://mlsanity.com/paper/1907.11572/) (2019)
- [Variable selection for Gaussian processes via sensitivity analysis of the posterior predictive distribution](https://mlsanity.com/paper/1712.08048/) (2017)
- [Global Sensitivity Analysis with Dependence Measures](https://mlsanity.com/paper/1311.2483/) (2013)
- [Randomized incomplete U-statistics in high dimensions](https://mlsanity.com/paper/1712.00771/) (2017)
- [Bayesian Approximate Kernel Regression with Variable Selection](https://mlsanity.com/paper/1508.01217/) (2015)
- [Variable Selection Consistency of Gaussian Process Regression](https://mlsanity.com/paper/1912.05738/) (2019)
- [Kernel Ridge Regression Inference](https://mlsanity.com/paper/2302.06578/) (2023)
- [Active Learning for Derivative-Based Global Sensitivity Analysis with Gaussian Processes](https://mlsanity.com/paper/2407.09739/) (2024)
- [Towards a Unified Framework for Uncertainty-aware Nonlinear Variable Selection with Theoretical Guarantees](https://mlsanity.com/paper/2204.07293/) (2022)
- [Projection predictive model selection for Gaussian processes](https://mlsanity.com/paper/1510.04813/) (2015)
- [Variable Prioritization in Nonlinear Black Box Methods: A Genetic Association Case Study](https://mlsanity.com/paper/1801.07318/) (2018)
- [Kernel Knockoffs Selection for Nonparametric Additive Models](https://mlsanity.com/paper/2105.11659/) (2021)
- [Estimation and Hypothesis Testing of Derivatives in Smoothing Spline ANOVA Models](https://mlsanity.com/paper/2308.13905/) (2023)
- [Gradient-based Active Learning with Gaussian Processes for Global Sensitivity Analysis](https://mlsanity.com/paper/2601.11790/) (2026)

## Abstract (arXiv)

Automatic relevance determination (ARD), the default tool for variable selection in Gaussian-process (GP) regression, ranks inputs by inverse lengthscales -- which measure how fast a function varies, not how much an input contributes to prediction -- and offers no calibrated rule for deciding which inputs to keep. The prediction-centred alternative, the derivative sensitivity nuj = mathbbE[(partial f/partial xj)2], is available in closed form from a fitted GP, but turning it into a selection rule is harder than it looks: at a null input the estimator is a degenerate quadratic form, so Wald and Bernstein-von Mises cutoffs are anti-conservative, and the natural residual bootstrap is mis-scaled. We show that a studentized multiplier bootstrap of the GP derivative process repairs both, prove its validity through an invariance principle for quadratic forms, and obtain asymptotic family-wise and false-discovery-rate control across inputs. Over 100 replications the rule controls FDR wherever inputs are truly null, while uncalibrated derivative rankings breach the target by up to 2x and a Bernstein-von Mises cutoff by 2.2x; at matched FDR it loses no power; it holds under a Mat'ern kernel and input correlation up to 0.99; on real data with planted and authentic null inputs it admits 5-12x fewer spurious inputs; it costs 5-18% of the GP fit; and a block-averaged variant retains validity at cost linear in n.