Time-adaptive infinite-dimensional Gaussian process regression on manifolds

MD Ruiz-Medina; AE Madrid; A Torres-Signes; JM Angulo

stat.MLMar 22, 2026

Time-adaptive infinite-dimensional Gaussian process regression on manifolds

Authors: MD Ruiz-Medina, AE Madrid, A Torres-Signes, JM Angulo

Abstract

This paper proposes a new formulation of functional Gaussian Process regression on manifolds, based on an Empirical Bayes approach, in the spatiotemporal random field context. We apply the machinery of tight Gaussian measures in separable Hilbert spaces, exploiting the invariance property of covariance kernels under the group of isometries of the manifold. The identification via characteristic function of these measures with the infinite product of one-dimensional Gaussian measures is then obtained, in terms of the eigenfunctions of the Laplace-Beltrami operator on the manifold. The involved time-varying angular spectrum constitutes the key tool for dimension reduction in the implementation of this regression approach, adopting a suitable truncation scheme depending on the functional sample size. The simulation study and synthetic data application illustrate the performance of the proposed functional regression predictor.

Figures & tables

Panel of Figure 1 — Figure 1 : Informative priors for hyperparameters characterizing spatiotemporal covariance function subfamilies 1 and 2. Regularity hyperparameter priors for $\gamma$ (top-left), $\nu$ (top-center) and $\varpi$ (top-right). Memory and noise hyperparameter priors for $\beta$ (bottom-left), $\alpha$ (bottom-center), and $\sigma(t)$ (bottom-right)

Panel of Figure 2 — Figure 2 : Subfamily 1 . Realization of conditional FGP spherical functional time series model (left), spherical functional posterior mean (center), and observation spherical functional time series model (right), $\mathbb{D}=\left\{1,21,41,61,81,101,121,141,161,181\right\}\subset\mathbb{T}$

Panel of Figure 3 — Figure 3 : Subfamily 1 . The time-varying Empirical Mean Quadratic Errors (EMQEs) are displayed for the following values of the functional sample size $T$ , number of spherical nodes $N$ , truncation parameter $TR$ , prior hyperparameter sample size $M$ , and number of repetitions $R$ : $T=50,$ $N=500,$ $TR=\log(T)\simeq 4,$ $M=100,$ $R=200$ (top-left); $T=300,$ $N=150,$ $TR=\log(T)\simeq 6,$ $M=50,$ $R=400$ (top-right); $T=300,$ $N=200,$ $TR=\log(T)\simeq 6,$ $M=50,$ $R=400$ (center-left); $T=500,$ $N=50,$ $TR=\log(T)\simeq 6,$ $M=100,$ $R=400$ (center-right); $T=300,$ $N=200,$ $TR=[T^{\varrho}]_{-},$ with $\varrho=1/2.45,$ $M=50,$ $R=400$ (bottom-left); $T=300,$ $N=250,$ $TR=[T^{\varrho}]_{-},$ with $\varrho=1/2.45,$ $M=50,$ $R=400$ (bottom-right)

Panel of Figure 4 — Figure 4 : Subfamily 1 . Theoretical (blue dotted line) and posterior (red dotted line) time linear correlation of the projected FGP model, at the eigenspaces specified by the logarithmic and power-function truncation schemes analyzed. The following values of the functional sample size $T$ , number of spherical nodes $N$ , truncation parameter $TR$ , prior hyperparameter sample size $M$ , and number of replicates $R$ are displayed: $T=50,N=500,TR=\log(T)\simeq 4,M=100,R=200$ (top-left); $T=50,N=50,TR=\log(T)\simeq 4,M=50,R=400$ (top-right); $T=50,N=50,TR=\log(T)\simeq 4,M=50,R=500$ (center-left); $T=50,N=500,TR=6,M=100,R=200$ (center-right); $T=50,N=50,TR=6,M=50,R=400$ (bottom-left); $T=500,N=50,TR=6,M=100,R=400$ (bottom-right)

Panel of Figure 5 — Figure 5 : Subfamily 1 . Empirical mean of the bias, $\mathbb{D}=\left\{1,64,127,190,253,316,379,442\right\}\subset\mathbb{T}$ (top), $\mathbb{D}=\left\{1,40,79,118,157,196,235,274\right\}\subset\mathbb{T}$ (bottom). The following values of the functional sample size $T$ , number of spherical nodes $N$ , truncation parameter $TR$ , prior hyperparameter sample size $M$ , and number of replicates $R$ are displayed: $T=500,N=50,TR=\log(T)\simeq 6,M=100,R=400$ (top-left); $T=500,N=150,TR=\log(T)\simeq 6,M=50,R=200$ (top-right); $T=300,N=150,TR=\log(T)\simeq 6,M=50,R=400$ (bottom-left); $T=300,N=200,TR=[T^{\varrho}]_{-}=10,\varrho=1/2.45,M=50,R=400$ (bottom-center); $T=300,N=250,TR=[T^{\varrho}]_{-}=10,\varrho=1/2.45,M=50,R=400$ (bottom-right)

Panel of Figure 6 — Figure 6 : Subfamily 1 . Original FGP sample value (left), and posterior mean approximation (right), $\mathbb{D}=\left\{1,8,15,22,29,36,43,50\right\}\subset\mathbb{T},$ for $T=150,N=500,TR=4,M=100,R=200$

Panel of Figure 7 — Figure 7 : Subfamily 2 . Realization of conditional FGP spherical functional time series model (left), posterior spherical functional mean (center), and observation spherical functional time series model (right), $\mathbb{D}=\left\{1,11,21,31,41,51,61,71,81,91\right\}\subset\mathbb{T}$

Panel of Figure 8 — Figure 8 : Subfamily 2 . Time-varying EMQEs. $T=50,N=500,TR=\log(T)\simeq 4,M=100,R=400$ (left); $T=500,N=50,TR=\log(T)\simeq 6,M=100,R=400$ (right)

Panel of Figure 9 — Figure 9 : Subfamily 2 . FGP theoretical model (blue dotted line) and posterior time linear correlation reconstruction (red dotted line). $T=50,N=500,TR=\log(T)\simeq 4,M=100,R=400$ (left); $T=500,N=50,TR=\log(T)\simeq 6,M=100,R=400$ (right)

Panel of Figure 10 — Figure 10 : Subfamily 2 . Empirical mean of the bias over the $R$ replicates. $\mathbb{D}=\left\{1,39,77,115,153,191,229,267\right\}\subset\mathbb{T},$ and $T=500,N=50,TR=10,M=100,R=400$ (top-left); $T=500,N=150,TR=10,M=50,R=200$ (top-right); $T=300,N=150,TR=10,M=50,R=400$ (center-left); $T=300,N=200,TR=10,M=50,R=400$ (center-right); $T=300,N=250,TR=10,M=50,R=400$ (bottom-left); $T=50,N=500,TR=10,M=100,R=200$ (bottom-right)

Panel of Figure 11 — Figure 11 : Original values (left), and axial (zonal) isotropic approximation (right) of hourly dataset of ERA5 downward solar radiation flux (one day at each month during the period September-February, corresponding to autumn-winter)

Panel of Figure 12 — Figure 12 : Downward solar radiation flux, $\mathbb{D}=\left\{50,68,86,104,122,140,158,176\right\}\subset\mathbb{T}$ . Unrestricted (left) and restricted (right) model to autumn-winter, involving different declination and zenith angles

Panel of Figure 13 — Figure 13 : Temporal spherical functional regressor . Zonal isotropic approximation to atmospheric pressure at high cloud bottom, from inverted equation ( 28 ), $\mathbb{D}=\left\{60,76,92,108,124,140,156,172\right\}\subset\mathbb{T}$ (left). Unstructured spatiotemporal error term, $\mathbb{D}=\left\{1,21,41,61,81,101,121,141\right\}\subset\mathbb{T}$ (right)

Figure 14 : One realization of the axial (zonal) isotropic temporal spherical functional observation process, $\mathbb{D}=\left\{1,21,41,61,81,101,121,141\right\}\subset\mathbb{T}$

Panel of Figure 15 — Figure 15 : 5-fold cross-validation errors, $\mathbb{D}=\left\{84,97,110,123,136,149,162,175\right\}\subset\mathbb{T}$ , for $T=183,N=180,TR=8,M=40,R=100,$ EBFGP based nonparametric series posterior generalized least-squares estimator (top-left) and linear functional predictor, non-incorporating the posterior covariance operator structure (top-right). For $T=183,N=180,TR=8,M=60,R=200,$ EBFGP based nonparametric series posterior generalized least-squares estimator (bottom-left) and linear functional predictor, non-incorporating the posterior covariance operator structure (bottom-right)

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Scalable Gaussian process inference via neural feature maps

Anthony Stephenson

School of Mathematics University of Bristol

Gaussian Process Kernel Method Feature Learning Low-Rank Structure

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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.

Abstract

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