We develop a framework for Gaussian processes regression constrained by boundary value problems. The framework may be applied to infer the solution of a well-posed boundary value problem with a known second-order differential operator and boundary conditions, but for which only scattered observations of the source term are available. Scattered observations of the solution may also be used in the regression. The framework combines co-kriging with the linear transformation of a Gaussian process together with the use of kernels given by spectral expansions in eigenfunctions of the boundary value problem. Thus, it benefits from a reduced-rank property of covariance matrices. We demonstrate that the resulting framework yields more accurate and stable solution inference as compared to physics-informed Gaussian process regression without boundary condition constraints.
This work is concerned with the convergence of Gaussian process regression. A particular focus is on hierarchical Gaussian process regression, where hyper-parameters appearing in the mean and covariance structure of the Gaussian process emulator are a-priori unknown, and are learnt from the data, along with the posterior mean and covariance. We work in the framework of empirical Bayes, where a point estimate of the hyper-parameters is computed, using the data, and then used within the standard Gaussian process prior to posterior update. We provide a convergence analysis that (i) holds for any …
In this work, we investigate Gaussian process regression used to recover a function based on noisy observations. We derive upper and lower error bounds for Gaussian process regression with possibly misspecified correlation functions. The optimal convergence rate can be attained even if the smoothness of the imposed correlation function exceeds that of the true correlation function and the sampling scheme is quasi-uniform. As byproducts, we also obtain convergence rates of kernel ridge regression with misspecified kernel function, where the underlying truth is a deterministic function. The conv…
We introduce constrained Gaussian process (CGP), a Gaussian process model for random functions that allows easy placement of mathematical constrains (e.g., non-negativity, monotonicity, etc) on its sample functions. CGP comes with closed-form probability density function (PDF), and has the attractive feature that its posterior distributions for regression and classification are again CGPs with closed-form expressions. Furthermore, we show that CGP inherents the optimal theoretical properties of the Gaussian process, e.g. rates of posterior contraction, due to the fact that CGP is an Gaussian p…
University of Oxford（牛津大学）；Imperial College London（帝国理工学院）；University of Twente（特文特大学）
We rigorously prove that deep Gaussian process priors can outperform Gaussian process priors if the target function has a compositional structure. To this end, we study information-theoretic lower bounds for posterior contraction rates for Gaussian process regression in a continuous regression model. We show that if the true function is a generalized additive function, then the posterior based on any mean-zero Gaussian process can only recover the truth at a rate that is strictly slower than the minimax rate by a factor that is polynomially suboptimal in the sample size $n$.
Wuhan University（武汉大学）；The Hong Kong University of Science and Technology（香港科技大学）
This paper studies the approximation capacity of ReLU neural networks with norm constraint on the weights. We prove upper and lower bounds on the approximation error of these networks for smooth function classes. The lower bound is derived through the Rademacher complexity of neural networks, which may be of independent interest. We apply these approximation bounds to analyze the convergences of regression using norm constrained neural networks and distribution estimation by GANs. In particular, we obtain convergence rates for over-parameterized neural networks. It is also shown that GANs can …
We study posterior rates of contraction in Gaussian process regression with unbounded covariate domain. Our argument relies on developing a Gaussian approximation to the posterior of the leading coefficients of a Karhunen--Loéve expansion of the Gaussian process. The salient feature of our result is deriving such an approximation in the $L^2$ Wasserstein distance and relating the speed of the approximation to the posterior contraction rate using a coupling argument. Specific illustrations are provided for the Gaussian or squared-exponential covariance kernel.
We investigate two empirical Bayes methods and a hierarchical Bayes method for adapting the scale of a Gaussian process prior in a nonparametric regression model. We show that all methods lead to a posterior contraction rate that adapts to the smoothness of the true regression function. Furthermore, we show that the corresponding credible sets cover the true regression function whenever this function satisfies a certain extrapolation condition. This condition depends on the specific method, but is implied by a condition of self-similarity. The latter condition is shown to be satisfied with pro…
Bayesian posterior distributions arising in modern applications, including inverse problems in partial differential equation models in tomography and subsurface flow, are often computationally intractable due to the large computational cost of evaluating the data likelihood. To alleviate this problem, we consider using Gaussian process regression to build a surrogate model for the likelihood, resulting in an approximate posterior distribution that is amenable to computations in practice. This work serves as an introduction to Gaussian process regression, in particular in the context of buildin…
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 to a broader class of operators in the Gaussian case, via the Loève isometry. Several covariance-driven pathwise invariances are illustrated, including fields with symmetric paths, centred paths, harmonic paths, or sparse paths. The proposed approach delivers a number of promisi…
Gaussian processes provide a powerful probabilistic kernel learning framework, which allows learning high quality nonparametric regression models via methods such as Gaussian process regression. Nevertheless, the learning phase of Gaussian process regression requires massive computations which are not realistic for large datasets. In this paper, we present a Gauss-Legendre quadrature based approach for scaling up Gaussian process regression via a low rank approximation of the kernel matrix. We utilize the structure of the low rank approximation to achieve effective hyperparameter learning, tra…
Gaussian process regression is a popular Bayesian framework for surrogate modeling of expensive data sources. As part of a broader effort in scientific machine learning, many recent works have incorporated physical constraints or other a priori information within Gaussian process regression to supplement limited data and regularize the behavior of the model. We provide an overview and survey of several classes of Gaussian process constraints, including positivity or bound constraints, monotonicity and convexity constraints, differential equation constraints provided by linear PDEs, and boundar…
We consider a Bayesian approach to model selection in Gaussian linear regression, where the number of predictors might be much larger than the number of observations. From a frequentist view, the proposed procedure results in the penalized least squares estimation with a complexity penalty associated with a prior on the model size. We investigate the optimality properties of the resulting estimator. We establish the oracle inequality and specify conditions on the prior that imply its asymptotic minimaxity within a wide range of sparse and dense settings for "nearly-orthogonal" and "multicollin…
Approximation algorithms are widely used in many engineering problems. To obtain a data set for approximation a factorial design of experiments is often used. In such case the size of the data set can be very large. Therefore, one of the most popular algorithms for approximation - Gaussian Process regression - can be hardly applied due to its computational complexity. In this paper a new approach for Gaussian Process regression in case of factorial design of experiments is proposed. It allows to efficiently compute exact inference and handle large multidimensional data sets. The proposed algor…
The focus of this work is on spatial variable selection for scalar-on-image regression. We propose a new class of Bayesian nonparametric models, soft-thresholded Gaussian processes and develop the efficient posterior computation algorithms. Theoretically, soft-thresholded Gaussian processes provide large prior support for the spatially varying coefficients that enjoy piecewise smoothness, sparsity and continuity, characterizing the important features of imaging data. Also, under some mild regularity conditions, the soft-thresholded Gaussian process leads to the posterior consistency for both p…
An interpolation error is an integral of the squared error of a regression model over a domain of interest. We consider the interpolation error for the case of misspecified Gaussian process regression: used covariance function differs from the true one. We derive the interpolation error for an infinite grid design of experiments. In particular, we show that for Matern 1/2 covariance function poor estimation of parameters only slightly affects the quality of interpolation. Then we proceed to numerical experiments that consider the misspecification for the most common covariance functions includ…
Gaussian Processes (GP) are widely used for probabilistic modeling and inference for nonparametric regression. However, their computational complexity scales cubicly with the sample size rendering them unfeasible for large data sets. To speed up the computations various distributed methods were proposed in the literature. These methods have, however, limited theoretical underpinning. In our work we derive frequentist theoretical guarantees and limitations for a range of distributed methods for general GP priors in context of the nonparametric regression model, both for recovery and uncertainty…
Gaussian Process (GP) regression is a flexible non-parametric approach to approximate complex models. In many cases, these models correspond to processes with bounded physical properties. Standard GP regression typically results in a proxy model which is unbounded for all temporal or spacial points, and thus leaves the possibility of taking on infeasible values. We propose an approach to enforce the physical constraints in a probabilistic way under the GP regression framework. In addition, this new approach reduces the variance in the resulting GP model.
Singapore University of Technology and Design（新加坡科技设计大学）；University of Oxford（牛津大学）；National University of Singapore（新加坡国立大学）
Gaussian processes (GP) are a widely used model for regression problems in supervised machine learning. Implementation of GP regression typically requires $O(n^3)$ logic gates. We show that the quantum linear systems algorithm [Harrow et al., Phys. Rev. Lett. 103, 150502 (2009)] can be applied to Gaussian process regression (GPR), leading to an exponential reduction in computation time in some instances. We show that even in some cases not ideally suited to the quantum linear systems algorithm, a polynomial increase in efficiency still occurs.
Wrocław University of Technology（弗罗茨瓦夫理工大学）；Opole University of Technology（奥波莱理工大学）
In this paper, we present the Gaussian process regression as the predictive model for Quality-of-Service (QoS) attributes in Web service systems. The goal is to predict performance of the execution system expressed as QoS attributes given existing execution system, service repository, and inputs, e.g., streams of requests. In order to evaluate the performance of Gaussian process regression the simulation environment was developed. Two quality indexes were used, namely, Mean Absolute Error and Mean Squared Error. The results obtained within the experiment show that the Gaussian process performe…
University of Sheffield（谢菲尔德大学）；Department of Computer Science（计算机科学系）；Sheffield, UK（英国谢菲尔德）
This paper introduces a method to approximate Gaussian process regression by representing the problem as a stochastic differential equation and using variational inference to approximate solutions. The approximations are compared with full GP regression and generated paths are demonstrated to be indistinguishable from GP samples. We show that the approach extends easily to non-linear dynamics and discuss extensions to which the approach can be easily applied.
Gaussian Process (GP) regression models typically assume that residuals are Gaussian and have the same variance for all observations. However, applications with input-dependent noise (heteroscedastic residuals) frequently arise in practice, as do applications in which the residuals do not have a Gaussian distribution. In this paper, we propose a GP Regression model with a latent variable that serves as an additional unobserved covariate for the regression. This model (which we call GPLC) allows for heteroscedasticity since it allows the function to have a changing partial derivative with respe…
University of Southern California（南加州大学）；Max-Planck-Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）
Locally weighted regression was created as a nonparametric learning method that is computationally efficient, can learn from very large amounts of data and add data incrementally. An interesting feature of locally weighted regression is that it can work with spatially varying length scales, a beneficial property, for instance, in control problems. However, it does not provide a generative model for function values and requires training and test data to be generated identically, independently. Gaussian (process) regression, on the other hand, provides a fully generative model without significan…
Gaussian processes offer a flexible kernel method for regression. While Gaussian processes have many useful theoretical properties and have proven practically useful, they suffer from poor scaling in the number of observations. In particular, the cubic time complexity of updating standard Gaussian process models make them generally unsuitable for application to streaming data. We propose an algorithm for sequentially partitioning the input space and fitting a localized Gaussian process to each disjoint region. The algorithm is shown to have superior time and space complexity to existing method…
University of Bristol（布里斯托大学）；University of Bath（巴斯大学）
We introduce Latent Gaussian Process Regression which is a latent variable extension allowing modelling of non-stationary multi-modal processes using GPs. The approach is built on extending the input space of a regression problem with a latent variable that is used to modulate the covariance function over the training data. We show how our approach can be used to model multi-modal and non-stationary processes. We exemplify the approach on a set of synthetic data and provide results on real data from motion capture and geostatistics.
Direct quantile regression involves estimating a given quantile of a response variable as a function of input variables. We present a new framework for direct quantile regression where a Gaussian process model is learned, minimising the expected tilted loss function. The integration required in learning is not analytically tractable so to speed up the learning we employ the Expectation Propagation algorithm. We describe how this work relates to other quantile regression methods and apply the method on both synthetic and real data sets. The method is shown to be competitive with state of the ar…
Caltech（加州理工学院）；University of Southern California（南加州大学）
Low-rank tensor regression, a new model class that learns high-order correlation from data, has recently received considerable attention. At the same time, Gaussian processes (GP) are well-studied machine learning models for structure learning. In this paper, we demonstrate interesting connections between the two, especially for multi-way data analysis. We show that low-rank tensor regression is essentially learning a multi-linear kernel in Gaussian processes, and the low-rank assumption translates to the constrained Bayesian inference problem. We prove the oracle inequality and derive the ave…
University of Cambridge（剑桥大学）；Imperial College London（帝国理工学院）
Recent work in scalable approximate Gaussian process regression has discussed a bias-variance-computation trade-off when estimating the log marginal likelihood. We suggest a method that adaptively selects the amount of computation to use when estimating the log marginal likelihood so that the bias of the objective function is guaranteed to be small. While simple in principle, our current implementation of the method is not competitive computationally with existing approximations.
When the data are sparse, optimization of hyperparameters of the kernel in Gaussian process regression by the commonly used maximum likelihood estimation (MLE) criterion often leads to overfitting. We show that choosing hyperparameters (in this case, kernel length parameter and regularization parameter) based on a criterion of the completeness of the basis in the corresponding linear regression problem is superior to MLE. We show that this is facilitated by the use of high-dimensional model representation (HDMR) whereby a low-order HDMR representation can provide reliable reference functions a…
University of Toronto（多伦多大学）；TTI-Chicago（芝加哥丰田技术研究所）；Adobe Research（Adobe研究院）
We propose an efficient optimization algorithm for selecting a subset of training data to induce sparsity for Gaussian process regression. The algorithm estimates an inducing set and the hyperparameters using a single objective, either the marginal likelihood or a variational free energy. The space and time complexity are linear in training set size, and the algorithm can be applied to large regression problems on discrete or continuous domains. Empirical evaluation shows state-of-art performance in discrete cases and competitive results in the continuous case.
In this paper, for efficient data collection with limited bandwidth, data-aided sensing is applied to Gaussian process regression that is used to learn data sets collected from sensors in Internet-of-Things systems. We focus on the interpolation of sensors' measurements from a small number of measurements uploaded by a fraction of sensors using Gaussian process regression with data-aided sensing. Thanks to active sensor selection, it is shown that Gaussian process regression with data-aided sensing can provide a good estimate of a complete data set compared to that with random selection. With …
When constructing a Bayesian Machine Learning model, we might be faced with multiple different prior distributions and thus are required to properly consider them in a sensible manner in our model. While this situation is reasonably well explored for classical Bayesian Statistics, it appears useful to develop a corresponding method for complex Machine Learning problems. Given their underlying Bayesian framework and their widespread popularity, Gaussian Processes are a good candidate to tackle this task. We therefore extend the idea of Mixture models for Gaussian Process regression in order to …
We develop a novel framework to accelerate Gaussian process regression (GPR). In particular, we consider localization kernels at each data point to down-weigh the contributions from other data points that are far away, and we derive the GPR model stemming from the application of such localization operation. Through a set of experiments, we demonstrate the competitive performance of the proposed approach compared to full GPR, other localized models, and deep Gaussian processes. Crucially, these performances are obtained with considerable speedups compared to standard global GPR due to the spars…
This paper deals with the speed of convergence of the learning curve in a Gaussian process regression framework. The learning curve describes the average generalization error of the Gaussian process used for the regression. More specifically, it is defined in this paper as the integral of the mean squared error over the input parameter space with respect to the probability measure of the input parameters. The main result is the proof of a theorem giving the mean squared error in function of the number of observations for a large class of kernels and for any dimension when the number of observa…
We introduce a new regression framework, Gaussian process regression networks (GPRN), which combines the structural properties of Bayesian neural networks with the non-parametric flexibility of Gaussian processes. This model accommodates input dependent signal and noise correlations between multiple response variables, input dependent length-scales and amplitudes, and heavy-tailed predictive distributions. We derive both efficient Markov chain Monte Carlo and variational Bayes inference procedures for this model. We apply GPRN as a multiple output regression and multivariate volatility model, …
Gaussian process (GP) regression is a powerful interpolation technique due to its flexibility in capturing non-linearity. In this paper, we provide a general framework for understanding the frequentist coverage of point-wise and simultaneous Bayesian credible sets in GP regression. As an intermediate result, we develop a Bernstein von-Mises type result under supremum norm in random design GP regression. Identifying both the mean and covariance function of the posterior distribution of the Gaussian process as regularized $M$-estimators, we show that the sampling distribution of the posterior me…
Bayesian nonparametric regression under a rescaled Gaussian process prior offers smoothness-adaptive function estimation with near minimax-optimal error rates. Hierarchical extensions of this approach, equipped with stochastic variable selection, are known to also adapt to the unknown intrinsic dimension of a sparse true regression function. But it remains unclear if such extensions offer variable selection consistency, i.e., if the true subset of important variables could be consistently learned from the data. It is shown here that variable consistency may indeed be achieved with such models …
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 uncertainty associated with noisy data. The proposed methodology circumvents the well-established need for stochastic variational inference, a scalable algorithm for approximating posterior distributions. The effectiveness of the proposed approach is demonstrated using an illustrative exam…
We propose a practical and scalable Gaussian process model for large-scale nonlinear probabilistic regression. Our mixture-of-experts model is conceptually simple and hierarchically recombines computations for an overall approximation of a full Gaussian process. Closed-form and distributed computations allow for efficient and massive parallelisation while keeping the memory consumption small. Given sufficient computing resources, our model can handle arbitrarily large data sets, without explicit sparse approximations. We provide strong experimental evidence that our model can be applied to lar…
Monge-Kantorovich distances, otherwise known as Wasserstein distances, have received a growing attention in statistics and machine learning as a powerful discrepancy measure for probability distributions. In this paper, we focus on forecasting a Gaussian process indexed by probability distributions. For this, we provide a family of positive definite kernels built using transportation based distances. We provide a probabilistic understanding of these kernels and characterize the corresponding stochastic processes. We prove that the Gaussian processes indexed by distributions corresponding to th…
Intrinsic Gaussian Process Regression Modeling for Manifold-valued Response Variable
International Institute of Finance, School of Management, University of Science and Technology of China（中国科学技术大学管理学院国际金融学院）；Department of Statistics and Data Science, College of Science, Southern University of Science and Technology（南方科技大学理学院统计与数据科学系）
The aim of this article is to present a novel parallelization method for temporal Gaussian process (GP) regression problems. The method allows for solving GP regression problems in logarithmic O(log N) time, where N is the number of time steps. Our approach uses the state-space representation of GPs which in its original form allows for linear O(N) time GP regression by leveraging the Kalman filtering and smoothing methods. By using a recently proposed parallelization method for Bayesian filters and smoothers, we are able to reduce the linear computational complexity of the temporal GP regress…
In this paper, we present a new statistical approach to the problem of incorporating experimental observations into a mathematical model described by linear partial differential equations (PDEs) to improve the prediction of the state of a physical system. We augment the linear PDE with a functional that accounts for the uncertainty in the mathematical model and is modeled as a {\em Gaussian process}. This gives rise to a stochastic PDE which is characterized by the Gaussian functional. We develop a {\em functional Gaussian process regression} method to determine the posterior mean and covarian…
We characterize the power-law asymptotics of learning curves for Gaussian process regression (GPR) under the assumption that the eigenspectrum of the prior and the eigenexpansion coefficients of the target function follow a power law. Under similar assumptions, we leverage the equivalence between GPR and kernel ridge regression (KRR) to show the generalization error of KRR. Infinitely wide neural networks can be related to GPR with respect to the neural network GP kernel and the neural tangent kernel, which in several cases is known to have a power-law spectrum. Hence our methods can be applie…
Quantification and minimization of uncertainty is an important task in the design of electromagnetic devices, which comes with high computational effort. We propose a hybrid approach combining the reliability and accuracy of a Monte Carlo analysis with the efficiency of a surrogate model based on Gaussian Process Regression. We present two optimization approaches. An adaptive Newton-MC to reduce the impact of uncertainty and a genetic multi-objective approach to optimize performance and robustness at the same time. For a dielectrical waveguide, used as a benchmark problem, the proposed methods…
Chennai Mathematical Institute（金奈数学研究所）；Indian Statistical Institute（印度统计研究所）
Gaussian Processes are widely used for regression tasks. A known limitation in the application of Gaussian Processes to regression tasks is that the computation of the solution requires performing a matrix inversion. The solution also requires the storage of a large matrix in memory. These factors restrict the application of Gaussian Process regression to small and moderate size data sets. We present an algorithm that combines estimates from models developed using subsets of the data obtained in a manner similar to the bootstrap. The sample size is a critical parameter for this algorithm. Guid…
A primary goal of computer experiments is to reconstruct the function given by the computer code via scattered evaluations. Traditional isotropic Gaussian process models suffer from the curse of dimensionality, when the input dimension is relatively high given limited data points. Gaussian process models with additive correlation functions are scalable to dimensionality, but they are more restrictive as they only work for additive functions. In this work, we consider a projection pursuit model, in which the nonparametric part is driven by an additive Gaussian process regression. We choose the …
