Gaussian Processes

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10 papers in the last four weeks, up 233% on the four weeks before. 0.1% of all new papers.

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May 4, 2026cs.LG

Inducing Permutation Invariant Priors in Bayesian Optimization for Carbon Capture and Storage Applications

Bayesian Optimization is an iterative method, tailored to optimizing expensive black box objective functions. Surrogate models like Gaussian Processes, which are the gold standard in Bayesian Optimization, can be inefficient for inputs with permutation symmetries, as the most common kernels employed are better suited for vector inputs rather than unordered sets of items. Motivated by this issue, we turn to permutation invariant Bayesian Optimization for well placement in Carbon Capture and Storage projects. The high fidelity black box simulator is instructed to operate wells under group control, giving rise to permutation symmetries within injector and producer groups that cannot be exploited with standard GP kernels. In this work, our main contribution is a novel Gaussian Process kernel (GP-Perm) that encodes permutation invariance by comparing sets through a stable divergence between their induced empirical representations, and can be combined with standard kernels for additional vector-valued inputs. As a learned invariant baseline, we also consider a Deep Kernel Learning model (DKL-DS) using the Deep Sets architecture to learn a permutation-invariant embedding. We evaluate the proposed methodology across 8 use cases, comprising seven synthetic benchmarks and one realistic CCS case study (Johansen formation)

Sofianos Panagiotis Fotias, Vassilis Gaganis

School of Mining and Metallurgical Engineering, National Technical University of Athens, Athens, 15773, Greece

Bayesian OptimizationGaussian Processes
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Apr 30, 2026eess.SP

Sequential Inference for Gaussian Processes: A Signal Processing Perspective

The proliferation of capable and efficient machine learning (ML) models marks one of the strongest methodological shifts in signal processing (SP) in its nearly 100-year history. ML models support the development of SP systems that represent complex, nonlinear relationships with high predictive accuracy. Adapting these models often requires sequential inference, which differs both theoretically and methodologically from the usual paradigm of ML, where data are often assumed independent and identically distributed. Gaussian processes (GPs) are a flexible yet principled framework for modeling random functions, and they have become increasingly relevant to SP as statistical and ML methods assume a more prominent role. We provide a self-contained, tutorial-style overview of GPs, with a particular focus on recent methodological advances in sequential, incremental, or streaming inference. We introduce these techniques from a signal-processing perspective while bridging them to recent advances in ML. Many of the developments we survey have direct applications to state-space modeling, sequential regression and forecasting, anomaly detection in time series, sequential Bayesian optimization, adaptive and active sensing, and sequential detection and decision-making. By organizing these advances from a signal-processing perspective, we intend to equip practitioners with practical tools and a coherent roadmap for deploying sequential GP models in real-world systems.

Daniel Waxman, Fernando Llorente, Petar M. Djurić

1Basis Research Institute · 3Computing and Data Sciences Directorate, Brookhaven National Laboratory · Department of Electrical and Computer Engineering, Stony Brook University

Gaussian ProcessesGaussian Process Regression
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Apr 30, 2026cs.LG

Predicting Covariate-Driven Spatial Deformation for Nonstationary Gaussian Processes

Nonstationary Gaussian processes (GPs) are essential for modeling complex, locally heterogeneous spatial data. A common modeling approach is the spatial deformation method that warps the domain to recover isotropy. However, this static method does not account for changes in spatial correlation induced by covariates, limiting its ability to predict nonstationary GPs under new covariate conditions. To enable predictive modeling of the deformation method, we propose to model the spatial deformation as a function of covariates. The spaces of diffeomorphic deformations and Euclidean covariate vectors are connected by characterizing deformations as generated by velocity fields living in a Lie algebra. To overcome the estimation instability caused by high-order interactions between multiple covariates in a general Lie algebra, we prove that those interactions can be truncated with a moderate physical assumption. Based on the theoretical results, a concise functional form of deformations driven by multiple covariates can be established, and an efficient estimation-inference algorithm is developed for out-of-sample nonstationary GP prediction with limited covariate-deformation sample pairs. The effectiveness and generalizability of the method are demonstrated on a simulation study and two case studies, in the fields of manufacturing and geostatistics, respectively.

Minghao Gu, Weizhi Lin, Qiang Huang

Daniel J. Epstein Department of Industrial & Systems Engineering, Univeristy of Southern California · Charles W. Davidson College of Engineering, San José State University

Gaussian Processes
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Apr 21, 2026math.PR

Phase Transitions in the Fluctuations of Functionals of Random Neural Networks

We establish central and non-central limit theorems for sequences of functionals of the Gaussian output of an infinitely-wide random neural network on the d-dimensional sphere . We show that the asymptotic behaviour of these functionals as the depth of the network increases depends crucially on the fixed points of the covariance function, resulting in three distinct limiting regimes: convergence to the same functional of a limiting Gaussian field, convergence to a Gaussian distribution, convergence to a distribution in the Qth Wiener chaos. Our proofs exploit tools that are now classical (Hermite expansions, Diagram Formula, Stein-Malliavin techniques), but also ideas which have never been used in similar contexts: in particular, the asymptotic behaviour is determined by the fixed-point structure of the iterative operator associated with the covariance, whose nature and stability governs the different limiting regimes.

Simmaco Di Lillo, Leonardo Maini, Domenico Marinucci

RoMaDS - Department of Mathematics, University of Rome Tor Vergata, Rome, Italy

Gaussian Processes
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Apr 21, 2026math.OC

An Efficient Spatial Branch-and-Bound Algorithm for Global Optimization of Gaussian Process Posterior Mean Functions

We study the deterministic global optimization of trained Gaussian process posterior mean functions over hyperrectangular domains. Although the posterior mean function has a compact closed-form representation, its global optimization is challenging because it remains nonlinear and nonconvex. Existing exact deterministic approaches become increasingly difficult to scale as the number of training data points grows, leading to approximation-based methods that improve tractability by optimizing a modified (inexact) objective. In this work, we propose PALM-Mean, a piecewise-analytic lower-bounding framework embedded in reduced-space spatial branch-and-bound. At each node, kernel terms that are locally important are replaced by a sign-aware piecewise-linear relaxation in an appropriate scalar distance variable, while the remaining terms are bounded analytically in closed form. We show this hybrid approach yields a valid lower bound for the posterior mean, while limiting the size of the branch-and-bound subproblems. We establish validity of the node lower bounds and ε\varepsilon-global convergence of the resulting algorithm. Computational results on synthetic benchmarks and real-world application problems show that PALM-Mean improves scalability relative to representative general-purpose deterministic global solvers, particularly as the number of training data points increases.

Wei-Ting Tang, Akshay Kudva, Calvin Tsay +1

Department of Chemical and Biological Engineering, University of Wisconsin-Madison, 1415 Engineering Drive, Madison, 53706, Wisconsin, USA. · Department of Chemical and Biomolecular Engineering, The Ohio State University, 151 W. Woodruf Avenue, Columbus, 43210, Ohio, USA. · Department of Computing, Imperial College London, Exhibition Rd, South Kensington, SW7 2AZ, London, United Kingdom.

Bayesian OptimizationGlobal OptimizationGaussian Processes
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Apr 3, 2026cs.CV

GP-4DGS: Probabilistic 4D Gaussian Splatting from Monocular Video via Variational Gaussian Processes

We present GP-4DGS, a novel framework that integrates Gaussian Processes (GPs) into 4D Gaussian Splatting (4DGS) for principled probabilistic modeling of dynamic scenes. While existing 4DGS methods focus on deterministic reconstruction, they are inherently limited in capturing motion ambiguity and lack mechanisms to assess prediction reliability. By leveraging the kernel-based probabilistic nature of GPs, our approach introduces three key capabilities: (i) uncertainty quantification for motion predictions, (ii) motion estimation for unobserved or sparsely sampled regions, and (iii) temporal extrapolation beyond observed training frames. To scale GPs to the large number of Gaussian primitives in 4DGS, we design spatio-temporal kernels that capture the correlation structure of deformation fields and adopt variational Gaussian Processes with inducing points for tractable inference. Our experiments show that GP-4DGS enhances reconstruction quality while providing reliable uncertainty estimates that effectively identify regions of high motion ambiguity. By addressing these challenges, our work takes a meaningful step toward bridging probabilistic modeling and neural graphics.

Mijeong Kim, Jungtaek Kim, Bohyung Han

1ECE and 2IPAI, Seoul National University, Korea · University of Wisconsin–Madison, USA

3D Gaussian SplattingDynamic Scene ReconstructionGaussian Processes
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Mar 22, 2026stat.ML

Time-adaptive infinite-dimensional Gaussian process regression on manifolds

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.

MD Ruiz-Medina, AE Madrid, A Torres-Signes +1

Empirical BayesBayesian InferenceGaussian ProcessesGaussian Process Regression
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Mar 9, 2026stat.ML

Posterior Sampling Reinforcement Learning with Gaussian Processes for Continuous Control: Sublinear Regret Bounds for Unbounded State Spaces

We analyze the Bayesian regret of the Gaussian process posterior sampling reinforcement learning (GP-PSRL) algorithm. Posterior sampling is a heuristic for decision-making under uncertainty that has been used to develop successful algorithms for a variety of continuous control problems. However, theoretical work on GP-PSRL is limited. All known regret bounds either have a sub-optimal growth rate, require strong smoothness assumptions, or fail to properly account for the fact that the set of possible system states is unbounded. Through a recursive application of the Borell-Tsirelson-Ibragimov-Sudakov inequality, we show that, with high probability, the states actually visited by the algorithm are contained within a ball of near-constant radius. We then use the chaining method to control the regret suffered by GP-PSRL under weak smoothness conditions. Our main result is a Bayesian regret bound of the order O~(HγTT)\widetilde{\mathcal{O}}(H\sqrt{γ_TT}), where HH is the horizon, TT is the number of time steps and γTγ_T is the expected information gain. With this result, we resolve the limitations with prior theoretical work on PSRL, and provide the theoretical foundation and tools for analyzing PSRL in complex settings.

Hamish Flynn, Joe Watson, Ingmar Posner +1

Carnegie Mellon University · University of Oxford · Technical University of Darmstadt +2

RL ControlRegret Minimization in RLReinforcement LearningBayesian RLGaussian ProcessesPosterior Sampling
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Feb 26, 2026stat.ML

Regular Fourier Features for Nonstationary Gaussian Processes

Simulating a Gaussian process requires sampling from a high-dimensional Gaussian distribution, which scales cubically with the number of sample locations. Spectral methods address this challenge by exploiting the Fourier representation and treating the spectral density as a probability distribution suitable for Monte Carlo approximation. Although this probabilistic interpretation is valid for stationary processes, it is overly restrictive for the nonstationary case, where spectral densities are generally not probability measures. To avoid this limitation, we propose regular Fourier features for harmonizable processes with one-dimensional inputs. Our method discretizes the spectral representation directly, preserving the correlation structure among spectral weights without requiring probability assumptions. Assuming finite spectral support, this yields an efficient low-rank approximation that is positive semi-definite by construction and consistent under mild regularity conditions. When the spectral density is unknown, the framework also extends to kernel learning from data, which we explore as a proof of concept. We demonstrate the approximation on locally stationary and harmonizable mixture kernels, the latter with a complex-valued spectral density. As a feasibility study, we then apply the kernel-learning extension to real and synthetic data, where it matches competitive baselines.

Arsalan Jawaid, Abdullah Karatas, Jörg Seewig

Institute of Measurement and Sensor Technology, University of Kaiserslautern-Landau, Kaiserslautern, Germany · Independent Researcher, Mannheim, Germany

Fourier Feature EmbeddingsGaussian ProcessesLow-Rank ApproximationKernel MethodsSpectral Methods
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Nov 22, 2025stat.ML

Improving Forecasts of Suicide Attempts for Patients with Little Data

Ecological Momentary Assessment (EMA) studies provide real-time data on suicidal thoughts and behaviors, but forecasting suicide attempts remains challenging: attempts are rare, and the pathways patients take to them are heterogeneous. Here, we investigate a cohort of patients from an EMA study with recorded suicide-related events. We show that a single model fit to all patients forecasts poorly, while idiographic (per-patient) models show improvement but overfit for those with little data. Based on this result, one may hypothesize that patients should be partitioned into subgroups---this way, similar patients' data can be pooled together to improve forecasts. However, we show that grouping patients at random already improves forecasts, with performance increasing monotonically with the number of groups. Moreover, we show that grouping patients by demographics yields worse forecasts than random groupings. From these results, we hypothesize that patient similarity is continuous, rather than discrete, and must be inferred from the data. This motivated us to use Latent Variable Multiple Output Gaussian Processes (LVMOGPs), adapted to our data. Preliminary results show that, even without careful kernel design, LVMOGPs already match the strongest baseline models on most metrics, and their latent spaces yield a similarity between patients that we can inspect directly. Because the cohort is conditioned on the outcome and the splits are not temporal, we read these results as evidence that idiographic structure exists and can be recovered, not as deployable forecasting performance---an area for future work.

Genesis Hang, Annie Chen, Hope Neveux +2

Wellesley College · Harvard University

Latent Variable ModelsGaussian ProcessesClinical PredictionSuicide Risk ModelingClinical Risk Prediction
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Nov 20, 2025cs.LG

Warm-Starting Iterative Gaussian Processes for Faster Sequential Inference

Efficient Gaussian process (GP) inference is critical for sequential decision-making tasks such as active learning, online prediction, and Bayesian optimization. Iterative approaches of approximating the GP posterior using solvers like conjugate gradients, stochastic gradient descent, or alternating projections avoid cubic costs, but often require many iterations to converge, limiting their efficacy when the posterior is updated frequently with new data. To address this, we introduce three warm-start strategies that exploit solutions of smaller linear systems to substantially speed-up convergence when updating the posterior with new data. Our methods are supported by theoretical analysis showing reduced initialization error in reproducing kernel Hilbert space (RKHS) distance, and by empirical results on regression benchmarks and Bayesian optimization tasks. Across solvers, warm-starting achieves speed-ups of up to 19x when solving to tolerance, and produces more accurate posterior estimates under fixed compute budgets, directly improving optimization performance. These results establish warm-starting as a simple, effective, and broadly applicable tool for scaling Gaussian processes in sequential settings.

Alan Yufei Dong, Jihao Andreas Lin, José Miguel Hernández-Lobato

Department of Engineering, University of Cambridge

Efficient InferenceGaussian Processes
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Jun 20, 2025stat.ML

Gaussian Processes and Reproducing Kernel Hilbert Spaces: Connections and Equivalences

This monograph studies the relations between two approaches using positive definite kernels: probabilistic methods using Gaussian processes, and non-probabilistic methods using reproducing kernel Hilbert spaces (RKHS). They are widely studied and used in machine learning, statistics, and numerical analysis. We study connections and equivalences for fundamental topics such as regression, interpolation, numerical integration, distributional discrepancies, and statistical dependence, as well as sample path properties of Gaussian processes. A unifying perspective for these equivalences is established, based on the equivalence between the Gaussian Hilbert space and the RKHS. The monograph serves as a basis to bridge many other methods based on Gaussian processes and reproducing kernels, which are developed in parallel by the two research communities.

Motonobu Kanagawa, Philipp Hennig, Dino Sejdinovic +1

EURECOM · University of Tübingen, Tübingen AI Center Max Planck Institute for Intelligent Systems · Adelaide University Australian Institute for Machine Learning +1

Gaussian ProcessesReproducing Kernel Hilbert SpacesKernel MethodsGaussian Process Regression
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