Study on estimating distances between covariance operators and Gaussian processes.
problem Estimating distances between covariance operators and Gaussian processes.
method Riemannian distances, concentration results for Hilbert space-valued random variables, RKHS covariance and cross-covariance operators.
result Both distances converge in the Hilbert-Schmidt norm and can be consistently and efficiently estimated.
Extends Gaussian process theory to Banach spaces.
problem Extending Gaussian process theory to Banach spaces.
method Investigates the connection between Gaussian processes and Gaussian random elements in reproducing kernel Banach spaces.
result Characterizes positive definite functions that arise from covariance operators in Banach space setting.
This study approximates distances between Gaussian processes and covariance operators using RKHS.
problem Approximating distances between Gaussian processes and covariance operators from finite samples.
method Using reproducing kernel Hilbert space (RKHS) covariance and cross-covariance operators, the study shows how to consistently and efficiently estimate Sinkhorn divergence from finite samples.
result Convergence rates are dimension-independent and of the same order as Hilbert-Schmidt distance.
Develops a new method for functional regression that works with non-Gaussian data.
problem Limited models for regression in function spaces with Gaussian process priors.
method Introduces Neural Operator Flows (OpFlow) for non-Gaussian function spaces.
result OpFlow enables robust and accurate uncertainty quantification for functional regression.
This work extends Gaussian process priors to neural operators for function space mappings.
problem Improving uncertainty quantification in deep neural networks.
method Extending Gaussian process priors to neural operators with conditions for convergence and computation of covariance functions.
result Arbitrary-depth neural operators with Gaussian kernels converge to function-valued GPs, enabling posterior computation in regression scenarios.
NOGaP uses neural operators and GPs to solve PDEs with uncertainty quantification.
problem Lack of uncertainty measures in neural operator solutions for PDEs.
method NOGaP combines neural operators with Gaussian Processes to provide probabilistic solutions.
result NOGaP offers improved prediction accuracy and uncertainty quantification.
Proposes a new model for traffic flow on directed graphs.
problem Modeling advection on directed graphs for traffic flow.
method Reformulates graph advection operator as finite difference scheme; proposes DGAMGP model.
result Effective modeling of traffic flow and uncertainty as an advective process.
Physics-informed GP regression solves eigenvalue problems by identifying non-trivial eigenspaces.
problem Solving eigenvalue problems of linear operators with trivial solutions.
method Constructing a transfer function-type indicator using physics-informed Gaussian Process posterior.
result The posterior covariance is non-trivial only for eigenvalues of the operator, indicating non-trivial eigenspaces.
A new framework for recycling Gaussian process approximations.
problem Efficiently combining multiple Gaussian process approximations.
method Construct variational ensembles using a dictionary of fitted Gaussian processes.
result Framework allows for various tasks and scalability.
Bayesian layer improves image segmentation and out-of-distribution detection.
problem Outlier detection in image segmentation.
method Parameter-efficient hierarchical convolutional Gaussian Processes in Wasserstein-2 space.
result Uncertainty estimates improve out-of-distribution detection.
Scalable Gaussian Process Operator tackles high-dimensional PDEs.
problem Scaling Gaussian Process Operators to high-dimensional, data-intensive regimes.
method Nearest-neighbor-based local kernel approximations, sparse kernel approximation, structured Kronecker factorizations, operator-aware kernel structures, task-informed mean functions.
result Consistently achieves high accuracy across varying discretization scales.
Paper introduces a Gaussian Process for operator learning in computational mechanics.
problem Efficient and accurate solutions for large datasets with reliable uncertainty quantification.
method Gaussian Process (GP) embedded in a neural operator framework with stochastic dual descent (SDD) algorithm.
result Improves GP resolution independence and scalability for high-dimensional and non-linear systems.
LUNO linearizes neural operators to quantify their predictive uncertainty.
problem Quantifying the predictive error of neural operators for high-stakes simulations.
method Model linearization to push weight-space uncertainty forward to predictions.
result LUNO provides a practical and theoretically sound way to apply Bayesian methods to neural operators.
Accelerates GPR with localized kernels for faster performance.
problem Speeding up Gaussian process regression.
method Localization kernels applied at each data point to down-weight distant points, leading to a sparsified Gram matrix.
result Significant speedups with competitive performance compared to other methods.
Generalizes randomized SVD for better matrix approximations using Gaussian vectors.
problem Computing accurate rank-k approximations of matrices with limited data.
method Extends randomized SVD to multivariate Gaussian vectors, incorporating prior knowledge and using Gaussian processes.
result Demonstrates improved accuracy in approximating matrices and Hilbert-Schmidt operators.
A new model for curves on manifolds using rolling operations.
problem Modeling curves on manifolds without explicit parametrization.
method Using rolling operations to construct Gaussian processes on manifolds.
result Conditions for the rolling of mean to equal Fréchet mean and estimators of parameters.
Bayesian Gaussian Processes layer detects out-of-distribution data in medical imaging.
problem Detecting out-of-distribution data in medical imaging tasks.
method Parameter-efficient hierarchical convolutional Gaussian Processes in Wasserstein-2 space.
result Uncertainty estimates enable superior out-of-distribution detection compared to previous methods.
The paper introduces a method for learning nonparametric Volterra kernels using Gaussian processes.
problem Learning nonparametric nonlinear operators from data.
method NVKM model using Volterra series and Gaussian processes for unobserved and observed input functions.
result The NVKM model can perform both single and multiple output regression and system identification.
We introduce the concept of numerical Gaussian processes, which we define as Gaussian processes with covariance functions resulting from temporal discretization of time-dependent partial differential equations. Numerical Gaussian processes, by construction, are designed to deal with cases where: (1) all we observe are …
Quantum-assisted Gaussian process speeds up data regression.
problem High computational complexity of Gaussian process regression for large datasets.
method Quantum-assisted sparse Gaussian process regression using random Fourier features.
result Achieves polynomial-order computational speedup compared to classical methods.
We study pathwise invariances of centred random fields that can be controlled through the covariance. A result involving composition operators is obtained in second-order settings, and we show that various path properties including additivity boil down to invariances of the covariance kernel. These results are extended…
This work introduces the concept of parametric Gaussian processes (PGPs), which is built upon the seemingly self-contradictory idea of making Gaussian processes parametric. Parametric Gaussian processes, by construction, are designed to operate in "big data" regimes where one is interested in quantifying the uncertaint…
This work leverages recent advances in probabilistic machine learning to discover conservation laws expressed by parametric linear equations. Such equations involve, but are not limited to, ordinary and partial differential, integro-differential, and fractional order operators. Here, Gaussian process priors are modifie…
Gaussian processes for dynamical systems with Koopman equivariance.
problem Forecasting and learning representations of nonlinear dynamical systems.
method Koopman-equivariant Gaussian processes with linear time-invariant responses and trajectory-based equivariance.
result Enhanced forecasting performance compared to kernel-based methods.
A latent force model is a Gaussian process with a covariance function inspired by a differential operator. Such covariance function is obtained by performing convolution integrals between Green's functions associated to the differential operators, and covariance functions associated to latent functions. In the classica…
GP CC-OPF solves uncertain power grid optimization with Gaussian Process.
problem Uncertainty in power grid operations due to high renewables integration.
method Data-driven Gaussian Process regression for solving non-convex CC-OPF problem.
result Effective economic dispatch optimization in uncertain power grids.
This paper provides an algorithm for simulating improper (or noncircular) complex-valued stationary Gaussian processes. The technique utilizes recently developed methods for multivariate Gaussian processes from the circulant embedding literature. The method can be performed in O(nlog2n) operations, where…
Banded matrices can be used as precision matrices in several models including linear state-space models, some Gaussian processes, and Gaussian Markov random fields. The aim of the paper is to make modern inference methods (such as variational inference or gradient-based sampling) available for Gaussian models with band…
New Gaussian processes for Riemannian manifolds enable uncertainty quantification.
problem Modeling functions on Riemannian manifolds with uncertainty.
method Generalized Matérn Gaussian processes on compact manifolds via spectral theory.
result Efficient training of Riemannian Matérn Gaussian processes using scalable techniques.
The paper develops divergences for Gaussian processes and RKHS settings.
problem Estimating divergences in infinite-dimensional spaces.
method Formulations of Alpha Log-Det divergences, continuity in norm, laws of large numbers, consistent estimation from finite samples.
result Infinite-dimensional divergences can be estimated from finite-dimensional versions with dimension-independent sample complexities.
LVM-GP solves PDEs with uncertainty using latent variables and Gaussian processes.
problem Uncertainty quantification in PDE solutions with noisy data.
method Combines latent variable model and Gaussian process for uncertainty-aware prediction.
result Efficiently captures functional dependencies and robust uncertainty quantification.
Develops intrinsic Gaussian process regression for manifold-valued data.
problem Lack of intrinsic Gaussian process methods for manifold-valued response variables.
method Proposes an intrinsic covariance structure and a novel intrinsic Gaussian process regression model.
result Establishes asymptotic properties and shows posterior consistency.
Metrics assess uncertainty structure and distribution for regression models.
problem Quantifying uncertainty in high-dimensional and nonlinear regression tasks.
method Two bounded comparison metrics for uncertainty structure and distribution.
result DNNs and DNOs provide encouraging uncertainty metric values in high dimensions.
Quantitative modeling of post-transcriptional regulation process is a challenging problem in systems biology. A mechanical model of the regulatory process needs to be able to describe the available spatio-temporal protein concentration and mRNA expression data and recover the continuous spatio-temporal fields. Rigorous…
NOVI improves deep Gaussian process inference with neural generators and regularized Stein discrepancy.
problem Intractable exact inference in deep Gaussian processes.
method NOVI uses a neural generator to approximate the posterior distribution and minimizes Regularized Stein Discrepancy.
result NOVI achieves 93.56% classification accuracy on CIFAR10, outperforming state-of-the-art methods.
Efficiently trains deep Gaussian processes with sparse approximations.
problem High computational complexity in training and inference for DGP models.
method Tensor Markov Gaussian Processes (TMGP) and hierarchical expansion to create DTMGP model.
result DTMGP model achieves superior computational efficiency compared to existing DGP models.
Improved outlier detection in hierarchical Gaussian Processes using Wasserstein-2 kernels.
problem Outlier detection limitations in stacked Gaussian Processes.
method Proposed a hybrid kernel combining Euclidean and Wasserstein-2 distances, emphasizing variance in Wasserstein-2 computations.
result Improved performance and enhanced out-of-distribution detection on various datasets.
New method uses Gaussian processes for solving linear PDEs with boundary conditions.
problem Solving linear PDEs with boundary conditions.
method Boundary Ehrenpreis--Palamodov Gaussian Processes (B-EPGPs).
result Significant accuracy and resource improvements over existing methods.
The paper considers the problem of global optimization in the setup of stochastic process bandits. We introduce an UCB algorithm which builds a cascade of discretization trees based on generic chaining in order to render possible his operability over a continuous domain. The theoretical framework applies to functions u…
A novel multi-resolution Gaussian process model for efficient time traversal.
problem Inference for long sequences with fast and slow transitions is difficult.
method A novel Gaussian process state-space architecture composed of multiple components, each trained on a different resolution.
result The combined model allows efficient inference for arbitrarily long sequences with complex dynamics.
New method for spatiotemporal data regression using Gaussian processes.
problem Regression in spatiotemporal random fields.
method Empirical Bayes approach, tight Gaussian measures, truncation scheme.
result Effective dimension reduction through time-varying angular spectra.
This work develops discrete Gaussian models for vector-valued data on triangular meshes.
problem Discrete representation of continuous vector-valued environmental data.
method Develops discrete intrinsic Gaussian processes for vector-valued data on triangular meshes using discrete differential operators.
result Models can capture harmonic flows, incorporate boundary conditions, and model non-stationary data.
This dissertation advances scalable Gaussian processes using iterative methods and pathwise conditioning.
problem The classical Gaussian process formulation is not scalable for large datasets and modern hardware.
method Combining iterative methods and pathwise conditioning to improve scalability.
result Significantly reduced memory requirements and facilitated application to larger datasets.
SGPA calibrates transformer uncertainty for safety-critical tasks.
problem Uncertainty estimation in transformer models for safety-critical domains.
method Bayesian inference in transformer's output space using sparse Gaussian processes.
result SGPA-based Transformers improve in-distribution calibration and out-of-distribution robustness.
Development systems for deep learning (DL), such as Theano, Torch, TensorFlow, or MXNet, are easy-to-use tools for creating complex neural network models. Since gradient computations are automatically baked in, and execution is mapped to high performance hardware, these models can be trained end-to-end on large amounts…
Scalable Gaussian processes with latent Kronecker structure for large datasets.
problem Limited scalability of Gaussian processes for large datasets.
method Leveraging latent Kronecker structure, projecting kernel matrix onto latent Kronecker product, using iterative linear system solvers and pathwise conditioning.
result Outperforms state-of-the-art sparse and variational GPs on real-world datasets with up to five million examples.
The paper integrates multiple Gaussian process predictions using Monte Carlo sampling.
problem Accurate prediction of variables using multiple models.
method Log-linear pooling of Gaussian process predictions, combined with Monte Carlo sampling.
result The log-linear pooling method improves prediction accuracy compared to linear pooling.
We consider a modification of the covariance function in Gaussian processes to correctly account for known linear constraints. By modelling the target function as a transformation of an underlying function, the constraints are explicitly incorporated in the model such that they are guaranteed to be fulfilled by any sam…