Gaussian Processes

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Gaussian processes model unknown functions by using kernels to describe how values at different inputs relate, producing both predictions and uncertainty estimates. Research develops kernels for nonstationary, multi-output, deep, and structured data, alongside scalable inference methods such as inducing points, Vecchia approximations, Kronecker algebra, and matrix-free computation. These methods support applications including Bayesian optimization, spatial and time-series forecasting, robotics, scientific simulation, and uncertainty-aware classification. A central tension is how to scale inference and adapt kernels to complex data while retaining accurate, well-calibrated uncertainty and reliable decisions.

Latest papers

115 papers24 papers in the last four weeks, up 71% on the four weeks before. 0.1% of all new papers.

Apr 1, 2026

Inverse-Free Sparse Variational Gaussian Processes

Gaussian processes (GPs) offer appealing properties but are costly to train at scale. Sparse variational GP (SVGP) approximations reduce cost yet still rely on Cholesky decompositions of kernel matrices, ill-suited to low-precision, massively parallel hardware. While one can construct valid variational bounds that rely only on matrix multiplications (matmuls) via an auxiliary matrix parameter, optimising them with off-the-shelf first-order methods is challenging. We make the inverse-free approach practical by proposing a better-conditioned bound and deriving a matmul-only natural-gradient update for the auxiliary parameter, markedly improving stability and convergence. We further provide simple heuristics, such as step-size schedules and stopping criteria, that make the overall optimisation routine fit seamlessly into existing workflows. Across regression and classification benchmarks, we demonstrate that our method 1) serves as a drop-in replacement in SVGP-based models (e.g., deep GPs), 2) recovers similar performance to traditional methods, and 3) can be faster than baselines when well tuned.

Stefano Cortinovis, Laurence Aitchison, Stefanos Eleftheriadis +1

University of Oxford · University of Bristol

Sparse Gaussian Processes Gaussian Processes Variational Inference Gaussian Process Regression Natural Gradient Methods

Mar 24, 2026

Digital Twin-Assisted Measurement Design and Channel Statistics Prediction

Prediction of wireless channels and their statistics is a fundamental procedure for ensuring performance guarantees in wireless systems. Statistical radio maps powered by Gaussian processes (GPs) offer flexible, non-parametric frameworks, but their performance depends critically on the choice of mean and covariance functions. These are typically learned from dense measurements without exploiting environmental geometry. Digital twins (DTs) of wireless environments leverage computational power to incorporate geometric information; however, they require costly calibration to accurately capture material and propagation characteristics. This work introduces a hybrid channel prediction framework that leverages uncalibrated DTs derived from open-source maps to extract geometry-induced prior information for GP prediction. These structural priors are fused with a small number of channel measurements, enabling data-efficient prediction of channel statistics across the entire environment. By exploiting the uncertainty quantification inherent to GPs, the framework supports principled measurement selection by identifying informative probing locations under resource constraints. Through this integration of imperfect DTs with statistical learning, the proposed method reduces measurement overhead, improves prediction accuracy, and establishes a practical approach for resource-efficient wireless channel prediction.

Robin J. Williams, Mahmoud Saad Abouamer, Petar Popovski

Department of Electronic Systems, Aalborg University, Denmark

Radio Propagation Modeling Channel Estimation Digital Twins Gaussian Processes

Mar 22, 2026

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 Bayes Bayesian Inference Gaussian Processes Gaussian Process Regression

Mar 18, 2026

On Additive Gaussian Processes for Wind Farm Power Prediction

Population-based Structural Health Monitoring (PBSHM) aims to share information between similar machines or structures. This paper takes a population-level perspective, exploring the use of additive Gaussian processes to reveal variations in turbine-specific and farm-level power models over a collected wind farm dataset. The predictions illustrate patterns in wind farm power generation, which follow intuition and should enable more informed control and decision-making.

Simon M. Brealy, Lawrence A. Bull, Daniel S. Brennan +4

Dynamics Research Group, Department of Mechanical Engineering, The University of Sheffield, Mappin Street, Sheffield, S1 3JD, UK · Computational Statistics and Machine Learning Group, Department of Engineering, University of Cambridge, Cambridge, CB3 0FA, UK · Vattenfall R&D, Vattenfall AB, Älvkarleby, Sweden

Wind Power Forecasting Gaussian Processes Gaussian Process Regression

Mar 18, 2026

Real-Time Online Learning for Model Predictive Control using a Spatio-Temporal Gaussian Process Approximation

Learning-based model predictive control (MPC) can enhance control performance by correcting for model inaccuracies, enabling more precise state trajectory predictions than traditional MPC. A common approach is to model unknown residual dynamics as a Gaussian process (GP), which leverages data and also provides an estimate of the associated uncertainty. However, the high computational cost of online learning poses a major challenge for real-time GP-MPC applications. This work presents an efficient implementation of an approximate spatio-temporal GP model, offering online learning at constant computational complexity. It is optimized for GP-MPC, where it enables improved control performance by learning more accurate system dynamics online in real-time, even for time-varying systems. The performance of the proposed method is demonstrated by simulations and hardware experiments in the exemplary application of autonomous miniature racing.

Lars Bartels, Amon Lahr, Andrea Carron +1

Institute for Dynamic Systems and Control, ETH Zürich, CH-8092 Zürich, Switzerland.

Model Predictive Control Gaussian Processes Gaussian Process Regression Residual Dynamics Learning Robot Dynamics Learning

Mar 17, 2026

Simplex-to-Euclidean Bijection for Conjugate and Calibrated Multiclass Gaussian Process Classification

We propose a conjugate and calibrated Gaussian process (GP) model for multi-class classification by exploiting the geometry of the probability simplex. Our approach uses Aitchison geometry to map simplex-valued class probabilities to an unconstrained Euclidean representation, turning classification into a GP regression problem with fewer latent dimensions than standard multi-class GP classifiers. This yields conjugate inference and reliable predictive probabilities without relying on distributional approximations in the model construction. The method is compatible with standard sparse GP regression techniques, enabling scalable inference on larger datasets. Empirical results show well-calibrated and competitive performance across synthetic and real-world datasets.

Bernardo Williams, Harsha Vardhan Tetali, Arto Klami +1

Department of Computer Science, University of Helsinki, Finland

Gaussian Processes Multiclass Classification Probability Calibration

Mar 16, 2026

Kriging via variably scaled kernels

Classical Gaussian processes and Kriging models are commonly based on stationary kernels, whereby correlations between observations depend exclusively on the relative distance between scattered data. While this assumption ensures analytical tractability, it limits the ability of Gaussian processes to represent heterogeneous correlation structures. In this work, we investigate variably scaled kernels as an effective tool for constructing non-stationary Gaussian processes by explicitly modifying the correlation structure of the data. Through a scaling function, variably scaled kernels alter the correlations between data and enable the modeling of targets exhibiting abrupt changes or discontinuities. We analyse the resulting predictive uncertainty via the variably scaled kernel power function and clarify the relationship between variably scaled kernels-based constructions and classical non-stationary kernels. Numerical experiments demonstrate that variably scaled kernels-based Gaussian processes yield improved reconstruction accuracy and provide uncertainty estimates that reflect the underlying structure of the data

Gianluca Audone, Francesco Marchetti, Emma Perracchione +1

Faculty of Engineering and Physical Sciences, University of Southampton, United Kindom · Dipartimento di Matematica “Tullio Levi-Civita”, Universit`a di Padova, Italy · Dipartimento di Scienze Matematiche Giuseppe Luigi Lagrange, Politecnico di Torino, Italy +1

Gaussian Processes Gaussian Process Regression Kernel Methods Uncertainty Quantification

Mar 11, 2026

GGMPs: Generalized Gaussian Mixture Processes

Conditional density estimation is complicated by multimodality, heteroscedasticity, and strong non-Gaussianity. Gaussian processes (GPs) provide a principled nonparametric framework with calibrated uncertainty, but standard GP regression is limited by its unimodal Gaussian predictive form. We introduce the Generalized Gaussian Mixture Process (GGMP), a GP-based method for multimodal conditional density estimation in settings where each input may be associated with a complex output distribution rather than a single scalar response. GGMP combines local Gaussian mixture fitting, cross-input component alignment and per-component heteroscedastic GP training to produce a closed-form Gaussian mixture predictive density. The method is tractable, compatible with standard GP solvers and scalable methods, and avoids the exponentially large latent-assignment structure of naive multimodal GP formulations. Empirically, GGMPs improve distributional approximation on synthetic and real-world datasets with pronounced non-Gaussianity and multimodality.

Vardaan Tekriwal, Mark D. Risser, Hengrui Luo +1

Applied Math & Computational Research Division, Lawrence Berkeley National Laboratory, Berkeley, CA · Department of Electrical Engineering & Computer Sciences, University of California, Berkeley, CA · Engineering Mathematics & Statistics Division, University of California, Berkeley, CA +3

Conditional Distribution Estimation Gaussian Processes Gaussian Mixture Models

Mar 9, 2026

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

\widetilde{\mathcal{O}}(H\sqrt{γ_TT})

, where

H

is the horizon,

T

is the number of time steps and

γ_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 Control Regret Minimization in RL Reinforcement Learning Bayesian RL Gaussian Processes Posterior Sampling

Mar 7, 2026

A Distributed Gaussian Process Model for Multi-Robot Mapping

We propose DistGP: a multi-robot learning method for collaborative learning of a global function using only local experience and computation. We utilise a sparse Gaussian process (GP) model with a factorisation that mirrors the multi-robot structure of the task, and admits distributed training via Gaussian belief propagation (GBP). Our loopy model outperforms Tree-Structured GPs \cite{bui2014tree} and can be trained online and in settings with dynamic connectivity. We show that such distributed, asynchronous training can reach the same performance as a centralised, batch-trained model, albeit with slower convergence. Last, we compare to DiNNO \cite{yu2022dinno}, a distributed neural network (NN) optimiser, and find DistGP achieves superior accuracy, is more robust to sparse communication and is better able to learn continually.

Seth Nabarro, Mark van der Wilk, Andrew J. Davison

Dyson Robotics Lab, Imperial College London · Department of Computer Science, University of Oxford

Sparse Gaussian Processes Gaussian Belief Propagation Gaussian Processes Multi-Robot Systems Distributed Training Bayesian Inference

Mar 7, 2026

Structured Multitask Gaussian Processes for Probabilistic Full-Body Human Motion Prediction

Robots that share space with people need human motion predictions with reliable uncertainty, not just accurate means. This paper presents a structured multitask variational Gaussian process (GP) for probabilistic full-body human motion prediction. The model integrates one-shot prediction of the full horizon, multitask outputs that couple future time steps, a factorization into one sparse variational GP per joint-dimension pair of the rotation representation, and a continuous 6D rotation representation with forward kinematics that preserves bone lengths. Its predictive distribution is available in closed form: a Gaussian with full temporal covariance for each rotation coordinate, which downstream planners operating in rotation coordinates can use without sampling, density estimation, or forward kinematics. On Human3.6M, the model achieves lower kernel-density-estimated negative log-likelihood (KDE NLL) than Motron and DLow at every prediction step, 22-52 nats below Motron. Empirical coverage is conservative at the 50% and 80% levels and close to nominal at 95% over a 2 s horizon. The probabilistic model uses 0.24 M parameters, about seven times fewer than Motron, and a separately trained 0.35 M-parameter deterministic variant has a mean angle error 3-22% higher than Motron's. Ablations confirm the benefit of the 6D representation and show that, at a fixed training budget, factorized joint outputs outperform coupled right-arm alternatives in NLL and CRPS while using less memory. These results show that compact, structured GPs provide analytic, temporally structured uncertainty with conservative empirical coverage for robotic systems.

Jinger Chong, Xiaotong Zhang, Kamal Youcef-Toumi

Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA.

Human Motion Prediction Gaussian Processes Uncertainty Quantification

Mar 1, 2026

Probabilistic Learning and Generation in Deep Sequence Models

Despite exceptional predictive performance of Deep sequence models (DSMs), the main concern of their deployment centers around the lack of uncertainty awareness. In contrast, probabilistic models quantify the uncertainty associated with unobserved variables with rules of probability. Notably, Bayesian methods leverage Bayes' rule to express our belief of unobserved variables in a principled way. Since exact Bayesian inference is computationally infeasible at scale, approximate inference is required in practice. Two major bottlenecks of Bayesian methods, especially when applied in deep neural networks, are prior specification and approximation quality. In Chapter 3 & 4, we investigate how the architectures of DSMs themselves can be informative for the design of priors or approximations in probabilistic models. We first develop an approximate Bayesian inference method tailored to the Transformer based on the similarity between attention and sparse Gaussian process. Next, we exploit the long-range memory preservation capability of HiPPOs (High-order Polynomial Projection Operators) to construct an interdomain inducing point for Gaussian process, which successfully memorizes the history in online learning. In addition to the progress of DSMs in predictive tasks, sequential generative models consisting of a sequence of latent variables are popularized in the domain of deep generative models. Inspired by the explicit self-supervised signals for these latent variables in diffusion models, in Chapter 5, we explore the possibility of improving other generative models with self-supervision for their sequential latent states, and investigate desired probabilistic structures over them. Overall, this thesis leverages inductive biases in DSMs to design probabilistic inference or structure, which bridges the gap between DSMs and probabilistic models, leading to mutually reinforced improvement.

Wenlong Chen

Imperial College London Department of Computing

Bayesian Inference Sequence Modeling Bayesian Neural Networks Latent Variable Models Gaussian Processes

Feb 27, 2026

Scalable Gaussian process modeling of parametrized spatio-temporal fields

We introduce a scalable Gaussian process (GP) framework with deep product kernels for data-driven learning of parametrized spatio-temporal fields over fixed or parameter-dependent domains. The proposed framework learns a continuous representation, enabling predictions at arbitrary spatio-temporal coordinates, independent of the training data resolution. We leverage Kronecker matrix algebra to formulate a computationally efficient training procedure with complexity that scales nearly linearly with the total number of spatio-temporal grid points. A key feature of our approach is the efficient computation of the posterior variance at essentially the same computational cost as the posterior mean (exactly for Cartesian grids and via rigorous bounds for unstructured grids), thereby enabling scalable uncertainty quantification. Numerical studies on a range of benchmark problems demonstrate that the proposed method achieves accuracy competitive with operator learning methods such as Fourier neural operators and deep operator networks. On the one-dimensional unsteady Burgers' equation, our method surpasses the accuracy of projection-based reduced-order models. These results establish the proposed framework as an effective tool for data-driven surrogate modeling, particularly when uncertainty estimates are required for downstream tasks.

Srinath Dama, Prasanth B. Nair

University of Toronto Institute for Aerospace Studies, 4925 Dufferin Street, Toronto, M3H 5T6, Ontario, Canada

Gaussian Processes Gaussian Process Regression PDE Surrogate Modeling Uncertainty Quantification

Feb 26, 2026

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 Embeddings Gaussian Processes Low-Rank Approximation Kernel Methods Spectral Methods

Feb 26, 2026

From Shallow Bayesian Neural Networks to Gaussian Processes: General Convergence, Identifiability and Scalable Inference

In this work, we study scaling limits of shallow Bayesian neural networks (BNNs) via their connection to Gaussian processes (GPs), with an emphasis on statistical modeling, identifiability, and scalable inference. We first establish a general convergence result from BNNs to GPs by relaxing assumptions used in prior formulations, and we compare alternative parameterizations of the limiting GP model. Building on this theory, we propose a new covariance function defined as a convex mixture of components induced by four widely used activation functions, and we characterize key properties including positive definiteness and both strict and practical identifiability under different input designs. For computation, we develop a scalable maximum a posterior (MAP) training and prediction procedure using a Nyström approximation, and we show how the Nyström rank and anchor selection control the cost-accuracy trade-off. Experiments on controlled simulations and real-world tabular datasets demonstrate stable hyperparameter estimates and competitive predictive performance at realistic computational cost.

Gracielle Antunes de Araújo, Flávio B. Gonçalves

Universidade do Estado de Minas Gerais (UEMG), FaEnge – Faculdade de Engenharia. · Universidade Federal de Minas Gerais (UFMG), ICEx – Instituto de Ciências Exatas.

Nyström Method Bayesian Neural Networks Shallow Neural Networks Gaussian Processes Gaussian Process Regression

Feb 20, 2026

Quasi-Periodic Gaussian Process Predictive Iterative Learning Control

Repetitive motion tasks are common in robotics, but performance can degrade over time due to environmental changes and robot wear and tear. Iterative learning control (ILC) improves performance by using information from previous iterations to compensate for expected errors in future iterations. This work incorporates the use of Quasi-Periodic Gaussian Processes (QPGPs) into a predictive ILC framework to model and forecast disturbances and drift across iterations. Using a recent structural equation formulation of QPGPs, the proposed approach enables efficient inference with complexity

\mathcal{O}(p^3)

instead of

\mathcal{O}(i^2p^3)

, where

p

denotes the number of points within an iteration and

i

represents the total number of iterations, specially for larger

i

. This formulation also enables parameter estimation without loss of information, making continual GP learning computationally feasible within the control loop. By predicting next-iteration error profiles rather than relying only on past errors, the controller achieves faster convergence and maintains this under time-varying disturbances. We benchmark the method against both standard ILC and conventional Gaussian Process (GP)-based predictive ILC on three tasks, autonomous vehicle trajectory tracking, a three-link robotic manipulator, and a real-world Stretch robot experiment. Across all cases, the proposed approach converges faster and remains robust under injected and natural disturbances while reducing computational cost. This highlights its practicality across a range of repetitive dynamical systems.

Unnati Nigam, Radhendushka Srivastava, Faezeh Marzbanrad +1

IITB-Monash Research Academy, IIT Bombay, Mumbai, India · Department of Mathematics, IIT Bombay, Mumbai, India · Department of Electrical and Computer Systems Engineering, Monash University, Clayton, Melbourne, Australia

Iterative Learning Control Robotic Control Gaussian Processes Adaptive Control Gaussian Process Regression Robot Trajectory Tracking

Feb 16, 2026

Distributed Quantum Gaussian Processes for Multi-Agent Systems

Gaussian Processes (GPs) are a powerful tool for probabilistic modeling, but their performance is often constrained in complex, large-scale real-world domains due to the limited expressivity of classical kernels. Quantum computing offers the potential to overcome this limitation by embedding data into exponentially large Hilbert spaces, capturing complex correlations that remain inaccessible to classical computing approaches. In this paper, we propose a Distributed Quantum Gaussian Process (DQGP) method in a multi-agent setting to enhance modeling capabilities and scalability. To address the challenging non-Euclidean optimization problem, we develop a Distributed consensus Riemannian Alternating Direction Method of Multipliers (DR-ADMM) algorithm that aggregates local agent models into a global model. We evaluate the efficacy of our method through numerical experiments conducted on a quantum simulator in classical hardware. We use real-world, non-stationary elevation datasets of NASA's Shuttle Radar Topography Mission and synthetic datasets generated by Quantum Gaussian Processes. Beyond modeling advantages, our framework highlights potential computational speedups that quantum hardware may provide, particularly in Gaussian processes and distributed optimization.

Meet Gandhi, George P. Kontoudis

Colorado School of Mines, Golden, USA

Gaussian Processes Quantum Machine Learning Multi-Agent Systems Distributed Optimization

Feb 16, 2026

Frequentist Regret Analysis of Gaussian Process Thompson Sampling via Fractional Posteriors

We study Gaussian Process Thompson Sampling (GP-TS) for sequential decision-making over compact, continuous action spaces and provide a frequentist regret analysis based on fractional Gaussian process posteriors, without relying on domain discretization as in prior work. We show that the variance inflation commonly assumed in existing analyses of GP-TS can be interpreted as Thompson Sampling with respect to a fractional posterior with tempering parameter

α\in (0,1)

. We derive a kernel-agnostic regret bound expressed in terms of the information gain parameter

γ_t

and the posterior contraction rate

ε_t

, and identify conditions on the Gaussian process prior under which

ε_t

can be controlled. As special cases of our general bound, we recover regret of order

\tilde{\mathcal{O}}(T^{\frac{1}{2}})

for the squared exponential kernel,

\tilde{\mathcal{O}}(T^{\frac{2ν+3d}{2(2ν+d)}} )

for the Matérn-

ν

kernel, and a bound of order

\tilde{\mathcal{O}}(T^{\frac{2ν+3d}{2(2ν+d)}})

for the rational quadratic kernel. Overall, our analysis provides a unified and discretization-free regret framework for GP-TS that applies broadly across kernel classes.

Somjit Roy, Prateek Jaiswal, Anirban Bhattacharya +2

Department of Statistics, Texas A&M University, College Station, TX 77843, USA · Daniels School of Business, Purdue University, West Lafayette, IN 47907, USA · Department of Statistics, University of Wisconsin-Madison, Madison, WI 53706, USA

Gaussian Process Bandits Thompson Sampling Bayesian Optimization Gaussian Processes Posterior Sampling Multi-Armed Bandits

Nov 22, 2025

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 Models Gaussian Processes Multi-Output Gaussian Processes Clinical Prediction Suicide Risk Modeling Clinical Risk Prediction

Nov 20, 2025

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 Inference Gaussian Processes

Oct 24, 2025

Deep Jump Gaussian Processes for Surrogate Modeling of High-Dimensional Piecewise Continuous Functions

We introduce Deep Jump Gaussian Processes (DJGP), a novel method for surrogate modeling of a piecewise continuous function on a high-dimensional domain. DJGP addresses the limitations of conventional Jump Gaussian Processes (JGP) in high-dimensional input spaces by integrating region-specific, locally linear projections with JGP modeling. These projections employ region-dependent matrices to capture local low-dimensional subspace structures, making them well suited to the inherently localized modeling behavior of JGPs, a variant of local Gaussian processes. To control model complexity, we place a Gaussian Process prior on the projection matrices, allowing them to evolve smoothly across the input space. The projected inputs are then modeled with a JGP to capture piecewise continuous relationships with the response. This yields a distinctive two-layer deep learning of GP/JGP. We further develop a scalable variational inference algorithm to jointly learn the projection matrices and JGP hyperparameters. Rigorous theoretical analysis and extensive empirical studies are provided to justify the proposed approach. In particular, we derive an oracle error bound for DJGP and decompose it into four distinct sources of error, which are then linked to practical implications. Experiments on synthetic and benchmark datasets demonstrate that DJGP achieves superior predictive accuracy and more reliable uncertainty quantification compared with existing methods.

Yang Xu, Chiwoo Park

Industrial & Systems Engineering, University of Washington, Seattle, WA, USA

Deep Gaussian Processes Surrogate Modeling Gaussian Processes Gaussian Process Regression Uncertainty Quantification

Sep 23, 2025

MAGIC: Multi-task Gaussian process for joint imputation and classification in healthcare time series

Time series analysis has emerged as an important tool for improving patient diagnosis and management in healthcare applications. However, these applications commonly face two critical challenges: time misalignment and data sparsity. Traditional approaches address these issues through a two-step process of imputation followed by prediction. We propose MAGIC (Multi-tAsk Gaussian Process for Imputation and Classification), a novel unified framework that simultaneously performs class-informed missing value imputation and label prediction within a hierarchical multi-task Gaussian process coupled with functional logistic regression. To handle intractable likelihood components, MAGIC employs Taylor expansion approximations with bounded error analysis, and parameter estimation is performed using EM algorithm with block coordinate optimization supported by convergence analysis. We validate MAGIC through two healthcare applications: prediction of post-traumatic headache improvement following mild traumatic brain injury and prediction of in-hospital mortality within 48 hours after ICU admission. In both applications, MAGIC achieves superior predictive accuracy compared to existing methods. The ability to generate real-time and accurate predictions with limited samples facilitates early clinical assessment and treatment planning, enabling healthcare providers to make more informed treatment decisions.

Dohyun Ku, Catherine D. Chong, Visar Berisha +2

H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology, GA, USA · Department of Neurology, Mayo Clinic Arizona, AZ, USA · College of Health Solutions and School of Electrical Computer and Energy Engineering, Arizona State University, AZ, USA

Time Series Imputation Time Series Classification Gaussian Processes Clinical Outcome Prediction Multi-Task Learning

Jun 20, 2025

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 Processes Reproducing Kernel Hilbert Spaces Kernel Methods Gaussian Process Regression

Feb 17, 2025

Spectral structure learning for clinical time series

We develop and evaluate a structure learning algorithm for clinical time series. Clinical time series are multivariate time series observed in multiple patients and irregularly sampled, challenging existing structure learning algorithms. We assume that our times series are realizations of StructGP, a k-dimensional multi-output or multi-task stationary Gaussian process (GP), with independent patients sharing the same covariance function. StructGP encodes ordered conditional relations between time series, represented in a directed acyclic graph. We implement an adapted NOTEARS algorithm, which based on a differentiable definition of acyclicity, recovers the graph by solving a series of continuous optimization problems. Simulation results show that up to mean degree 3 and 20 tasks, we reach a median recall of 0.93% [IQR, 0.86, 0.97] while keeping a median precision of 0.71% [0.57-0.84], for recovering directed edges. We further show that the regularization path is key to identifying the graph. With StructGP, we proposed a model of time series dependencies, that flexibly adapt to different time series regularity, while enabling us to learn these dependencies from observations.

Ivan Lerner, Anita Burgun, Francis Bach

Université Paris Cité, Inria, Inserm, HeKA, F-75015 Paris, France · Service d’informatique biomédicale, AP-HP, Hôpital Georges Pompidou, F-75015 Paris, France · SIERRA, Inria Paris, F-75015 Paris, France

Multi-Output Gaussian Processes Time-Series Causal Discovery Causal Structure Learning Irregular Time-Series Modeling Gaussian Processes

Jun 26, 2024

Second Maximum of a Gaussian Random Field and Exact (t-)Spacing test

In this article, we introduce the novel concept of the second maximum of a Gaussian random field defined on a Riemannian manifold: Once the field reaches its global maximum at a point

t_1

, the second maximum is the largest value of the field away from

t_1

, after the value and the gradient of the field at

t_1

have been removed. This statistic serves as a powerful tool for characterizing the distribution of the global maximum. By employing an ad-hoc Kac-Rice formula, we derive the explicit distribution of the maximum, conditioned on the second maximum and the independent part of the Riemannian Hessian (the part of the Hessian at

t_1

that is independent of the value and the gradient of the field at

t_1

). This approach yields an exact test based on the spacing between these maxima, which we refer to as the spacing test. We investigate the applicability of this test for detecting sparse alternatives in Gaussian symmetric tensors, continuous sparse deconvolution, and two-layered neural networks with smooth rectifiers. Theoretical results are supported by numerical experiments that illustrate the calibration and power of the proposed procedures. More generally, we provide a framework for applying the spacing test to continuous sparse regression for any Gaussian random field on a Riemannian manifold. Furthermore, when the variance-covariance function is known only up to a scaling factor, we derive an exact Studentized version of the procedure, coined the t-spacing test. We show that this test is perfectly calibrated under the null hypothesis and exhibits high power for detecting sparse alternatives. As a corollary, we derive an exact, non-asymptotic test for the spiked tensor model without requiring prior knowledge of the noise level.

Jean-Marc Azaïs, Federico Dalmao, Yohann De Castro

Institut de Mathématiques de Toulouse Université Paul Sabatier, France · DMEL, CENUR Litoral Norte Universidad de la República, Salto, Uruguay · Institut Camille Jordan, CNRS UMR 5208 École Centrale Lyon, France +1

Statistical Hypothesis Testing Sparse Recovery Gaussian Processes