A Gaussian process (GP) model can agree with the data for two reasons: its assumptions are right, or the chosen inputs could never have shown that they are wrong. The distinction can be checked from the design before any responses are observed. Every model implies relations that its noiseless responses must satisfy at the chosen inputs, such as the middle value lies on the line through its two neighbours. For GPs built from finitely many features, these relations are exactly the null space of the kernel matrix. Gale duality gives them a geometric interpretation, in which each observation has a vector and the smallest groups of observations that can expose an error are the circuits. For other kernels the relations become soft: response patterns may be improbable under the prior rather than algebraically impossible. A standard test then combines two kinds of evidence. Structural evidence comes from a violated relation and grows without limit as the noise falls. Prior-based evidence only says that a departure is improbable under the prior. With all inputs at the two ends of an interval, for example, a GP can reject a straight line against a large curvature, but only because the implied intercept is improbable, never because curvature was seen. In simulations the predicted power matched the observed rejection rates. Choosing the next input by predicted power raised the power against a localised discrepancy from 0.48 to 0.72, against 0.51 when choosing by predictive variance, and a grid in two dimensions contained exact tests of additivity that a Latin hypercube lacked. The test itself is classical. The contribution is the prospective reading of that test: before observing the responses, the design already determines what kind of contradiction it can produce.
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 νj=E[(∂f/∂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 Matérn 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.
Jia Cai
Department of Statistics, George Mason University Fairfax, Virginia, USA
We present a theoretically grounded Gaussian process framework that leverages neural feature maps to construct expressive kernels. We show that the learned feature map can be interpreted as an optimal low-rank approximation to a Gram matrix derived from an implied RKHS, from which we establish consistency of the GP posterior. We further analyse the spectral properties of the induced kernels and introduce product feature-map kernels to address oversmoothing. This simple yet powerful approach enables fast, scalable, and accurate exact GP inference with minimal upfront work. The flexibility of kernel design supports seamless application to both regression and classification tasks across diverse data modalities, including tabular inputs and structured domains such as images. On benchmark datasets, this approach surpasses pre-existing methods in terms of accuracy and training and prediction efficiency.
Gaussian process (GP) predictive distributions are commonly used in Bayesian optimization (BO) to guide the selection of evaluation points for expensive objective functions. The choice of kernel and hyperparameters has a strong influence on the exploration--exploitation trade-off. For minimization, sampling criteria such as expected improvement (EI) depend on both the probability mass below the current best value and the shape of the predictive distribution in this region. This article studies goal-oriented calibration of GP predictive distributions below a low threshold t in the noiseless setting, for standard GP models with hyperparameters selected by maximum likelihood. We consider two complementary forms of calibration below t for inputs distributed according to a reference measure μ: occurrence calibration over the design space and thresholded μ-calibration on sublevel sets of the form {x∈X,f(x)≤t}. We propose tcGP, a post-hoc method that combines these two forms of calibration for GP predictive distributions below t. With fixed GP hyperparameters, the exact EI sampling criterion based on tcGP generates a sequence of evaluation points that is dense in the design space. Experiments on standard benchmarks show improved lower-tail calibration and BO performance relative to standard GP models and globally calibrated GP models.
Aurélien Pion, Emmanuel Vazquez
Transvalor S.A., Biot, France · Univ. Paris-Saclay, CNRS, CentraleSup´elec, L2S, Gif-sur-Yvette, France