Approximations to Gaussian processes based on inducing variables, combined with variational inference techniques, enable state-of-the-art sparse approaches to infer GPs at scale through mini batch-based learning. In this work, we address one limitation of sparse GPs, which is due to the challenge in dealing with a large number of inducing variables without imposing a special structure on the inducing inputs. In particular, we introduce a novel hierarchical prior, which imposes sparsity on the set of inducing variables. We treat our model variationally, and we experimentally show considerable computational gains compared to standard sparse GPs when sparsity on the inducing variables is realized considering the nearest inducing inputs of a random mini-batch of the data. We perform an extensive experimental validation that demonstrates the effectiveness of our approach compared to the state-of-the-art. Our approach enables the possibility to use sparse GPs using a large number of inducing points without incurring a prohibitive computational cost.
The class of Gaussian Process (GP) methods for Temporal Difference learning has shown promise for data-efficient model-free Reinforcement Learning. In this paper, we consider a recent variant of the GP-SARSA algorithm, called Sparse Pseudo-input Gaussian Process SARSA (SPGP-SARSA), and derive recursive formulas for its predictive moments. This extension promotes greater memory efficiency, since previous computations can be reused and, interestingly, it provides a technique for updating value estimates on a multiple timescales
Generalized additive models (GAMs) are a widely used class of models of interest to statisticians as they provide a flexible way to design interpretable models of data beyond linear models. We here propose a scalable and well-calibrated Bayesian treatment of GAMs using Gaussian processes (GPs) and leveraging recent advances in variational inference. We use sparse GPs to represent each component and exploit the additive structure of the model to efficiently represent a Gaussian a posteriori coupling between the components.
University of Sheffield（谢菲尔德大学）；University of Oxford（牛津大学）
This paper introduces a new sparse spatio-temporal structured Gaussian process regression framework for online and offline Bayesian inference. This is the first framework that gives a time-evolving representation of the interdependencies between the components of the sparse signal of interest. A hierarchical Gaussian process describes such structure and the interdependencies are represented via the covariance matrices of the prior distributions. The inference is based on the expectation propagation method and the theoretical derivation of the posterior distribution is provided in the paper. Th…
University of Stuttgart（斯图加特大学）；ETAS GmbH（ETAS有限公司）
Modeling sequential data has become more and more important in practice. Some applications are autonomous driving, virtual sensors and weather forecasting. To model such systems, so called recurrent models are frequently used. In this paper we introduce several new Deep recurrent Gaussian process (DRGP) models based on the Sparse Spectrum Gaussian process (SSGP) and the improved version, called variational Sparse Spectrum Gaussian process (VSSGP). We follow the recurrent structure given by an existing DRGP based on a specific variational sparse Nyström approximation, the recurrent Gaussian pro…
EURECOM（欧洲通信研究院）；University of California, Irvine（加州大学欧文分校）
Autoencoders and their variants are among the most widely used models in representation learning and generative modeling. However, autoencoder-based models usually assume that the learned representations are i.i.d. and fail to capture the correlations between the data samples. To address this issue, we propose a novel Sparse Gaussian Process Bayesian Autoencoder (SGPBAE) model in which we impose fully Bayesian sparse Gaussian Process priors on the latent space of a Bayesian Autoencoder. We perform posterior estimation for this model via stochastic gradient Hamiltonian Monte Carlo. We evaluate …
Aalto University（阿尔托大学）；RIKEN Center for AI Project（理化学研究所人工智能项目中心）
Sparse variational Gaussian process (SVGP) methods are a common choice for non-conjugate Gaussian process inference because of their computational benefits. In this paper, we improve their computational efficiency by using a dual parameterization where each data example is assigned dual parameters, similarly to site parameters used in expectation propagation. Our dual parameterization speeds-up inference using natural gradient descent, and provides a tighter evidence lower bound for hyperparameter learning. The approach has the same memory cost as the current SVGP methods, but it is faster and…
Sparse Gaussian Processes are a key component of high-throughput Bayesian Optimisation (BO) loops; however, we show that existing methods for allocating their inducing points severely hamper optimisation performance. By exploiting the quality-diversity decomposition of Determinantal Point Processes, we propose the first inducing point allocation strategy designed specifically for use in BO. Unlike existing methods which seek only to reduce global uncertainty in the objective function, our approach provides the local high-fidelity modelling of promising regions required for precise optimisation…
Gaussian processes are powerful models for probabilistic machine learning, but are limited in application by their $O(N^3)$ inference complexity. We propose a method for deriving parametric families of kernel functions with compact spatial support, which yield naturally sparse kernel matrices and enable fast Gaussian process inference via sparse linear algebra. These families generalize known compactly-supported kernel functions, such as the Wendland polynomials. The parameters of this family of kernels can be learned from data using maximum likelihood estimation. Alternatively, we can quickly…
We revisit the Gaussian process model with spherical harmonic features and study connections between the associated RKHS, its eigenstructure and deep models. Based on this, we introduce a new class of kernels which correspond to deep models of continuous depth. In our formulation, depth can be estimated as a kernel hyper-parameter by optimizing the evidence lower bound. Further, we introduce sparseness in the eigenbasis by variational learning of the spherical harmonic phases. This enables scaling to larger input dimensions than previously, while also allowing for learning of high frequency va…
We present an algorithm to identify sparse dependence structure in continuous and non-Gaussian probability distributions, given a corresponding set of data. The conditional independence structure of an arbitrary distribution can be represented as an undirected graph (or Markov random field), but most algorithms for learning this structure are restricted to the discrete or Gaussian cases. Our new approach allows for more realistic and accurate descriptions of the distribution in question, and in turn better estimates of its sparse Markov structure. Sparsity in the graph is of interest as it can…
Online anomaly detection of time-series data is an important and challenging task in machine learning. Gaussian processes (GPs) are powerful and flexible models for modeling time-series data. However, the high time complexity of GPs limits their applications in online anomaly detection. Attributed to some internal or external changes, concept drift usually occurs in time-series data, where the characteristics of data and meanings of abnormal behaviors alter over time. Online anomaly detection methods should have the ability to adapt to concept drift. Motivated by the above facts, this paper pr…
Sparse, knot-based Gaussian processes have enjoyed considerable success as scalable approximations to full Gaussian processes. Certain sparse models can be derived through specific variational approximations to the true posterior, and knots can be selected to minimize the Kullback-Leibler divergence between the approximate and true posterior. While this has been a successful approach, simultaneous optimization of knots can be slow due to the number of parameters being optimized. Furthermore, there have been few proposed methods for selecting the number of knots, and no experimental results exi…
The use of Gaussian process models is typically limited to datasets with a few tens of thousands of observations due to their complexity and memory footprint. The two most commonly used methods to overcome this limitation are 1) the variational sparse approximation which relies on inducing points and 2) the state-space equivalent formulation of Gaussian processes which can be seen as exploiting some sparsity in the precision matrix. We propose to take the best of both worlds: we show that the inducing point framework is still valid for state space models and that it can bring further computati…
Aalto University（阿尔托大学）；The Alan Turing Institute（艾伦·图灵研究所）
Approximate Bayesian inference methods that scale to very large datasets are crucial in leveraging probabilistic models for real-world time series. Sparse Markovian Gaussian processes combine the use of inducing variables with efficient Kalman filter-like recursions, resulting in algorithms whose computational and memory requirements scale linearly in the number of inducing points, whilst also enabling parallel parameter updates and stochastic optimisation. Under this paradigm, we derive a general site-based approach to approximate inference, whereby we approximate the non-Gaussian likelihood …
Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室）；University of California, Santa Cruz（加州大学圣克鲁兹分校）；Lawrence Livermore National Laboratory（劳伦斯利弗莫尔国家实验室）
Gaussian processes are a flexible Bayesian nonparametric modelling approach that has been widely applied but poses computational challenges. To address the poor scaling of exact inference methods, approximation methods based on sparse Gaussian processes (SGP) are attractive. An issue faced by SGP, especially in latent variable models, is the inefficient learning of the inducing inputs, which leads to poor model prediction. We propose a regularization approach by balancing the reconstruction performance of data and the approximation performance of the model itself. This regularization improves …
Multi-task learning models using Gaussian processes (GP) have been developed and successfully applied in various applications. The main difficulty with this approach is the computational cost of inference using the union of examples from all tasks. Therefore sparse solutions, that avoid using the entire data directly and instead use a set of informative "representatives" are desirable. The paper investigates this problem for the grouped mixed-effect GP model where each individual response is given by a fixed-effect, taken from one of a set of unknown groups, plus a random individual effect fun…
We present a method for Temporal Difference (TD) learning that addresses several challenges faced by robots learning to navigate in a marine environment. For improved data efficiency, our method reduces TD updates to Gaussian Process regression. To make predictions amenable to online settings, we introduce a sparse approximation with improved quality over current rejection-based sparse methods. We derive the predictive value function posterior and use the moments to obtain a new algorithm for model-free policy evaluation, SPGP-SARSA. With simple changes, we show SPGP-SARSA can be reduced to a …
Carnegie-Mellon University（卡内基梅隆大学）；MIT-IBM Watson AI Lab（麻省理工学院-IBM沃森人工智能实验室）；IBM Research（IBM研究院）
We introduce a new scalable approximation for Gaussian processes with provable guarantees which hold simultaneously over its entire parameter space. Our approximation is obtained from an improved sample complexity analysis for sparse spectrum Gaussian processes (SSGPs). In particular, our analysis shows that under a certain data disentangling condition, an SSGP's prediction and model evidence (for training) can well-approximate those of a full GP with low sample complexity. We also develop a new auto-encoding algorithm that finds a latent space to disentangle latent input coordinates into well…
We face network data from various sources, such as protein interactions and online social networks. A critical problem is to model network interactions and identify latent groups of network nodes. This problem is challenging due to many reasons. For example, the network nodes are interdependent instead of independent of each other, and the data are known to be very noisy (e.g., missing edges). To address these challenges, we propose a new relational model for network data, Sparse Matrix-variate Gaussian process Blockmodel (SMGB). Our model generalizes popular bilinear generative models and cap…
Knot-based, sparse Gaussian processes have enjoyed considerable success as scalable approximations to full Gaussian processes. Problems can occur, however, when knot selection is done by optimizing the marginal likelihood. For example, the marginal likelihood surface is highly multimodal, which can cause suboptimal knot placement where some knots serve practically no function. This is especially a problem when many more knots are used than are necessary, resulting in extra computational cost for little to no gains in accuracy. We propose a one-at-a-time knot selection algorithm to select both …
We introduce a new class of inter-domain variational Gaussian processes (GP) where data is mapped onto the unit hypersphere in order to use spherical harmonic representations. Our inference scheme is comparable to variational Fourier features, but it does not suffer from the curse of dimensionality, and leads to diagonal covariance matrices between inducing variables. This enables a speed-up in inference, because it bypasses the need to invert large covariance matrices. Our experiments show that our model is able to fit a regression model for a dataset with 6 million entries two orders of magn…
Sparse Gaussian processes and various extensions thereof are enabled through inducing points, that simultaneously bottleneck the predictive capacity and act as the main contributor towards model complexity. However, the number of inducing points is generally not associated with uncertainty which prevents us from applying the apparatus of Bayesian reasoning for identifying an appropriate trade-off. In this work we place a point process prior on the inducing points and approximate the associated posterior through stochastic variational inference. By letting the prior encourage a moderate number …
Shenzhen Institutes of Advanced Technology, Chinese Academy of Sciences（中国科学院深圳先进技术研究院）
Gaussian process is one of the most popular non-parametric Bayesian methodologies for modeling the regression problem. It is completely determined by its mean and covariance functions. And its linear property makes it relatively straightforward to solve the prediction problem. Although Gaussian process has been successfully applied in many fields, it is still not enough to deal with physical systems that satisfy inequality constraints. This issue has been addressed by the so-called constrained Gaussian process in recent years. In this paper, we extend the core ideas of constrained Gaussian pro…
University of Edinburgh（爱丁堡大学）；Pharmatics Ltd（Pharmatics有限公司）
One of the fundamental tasks of science is to find explainable relationships between observed phenomena. One approach to this task that has received attention in recent years is based on probabilistic graphical modelling with sparsity constraints on model structures. In this paper, we describe two new approaches to Bayesian inference of sparse structures of Gaussian graphical models (GGMs). One is based on a simple modification of the cutting-edge block Gibbs sampler for sparse GGMs, which results in significant computational gains in high dimensions. The other method is based on a specific co…
Gaussian processes (GPs) provide a probabilistic nonparametric representation of functions in regression, classification, and other problems. Unfortunately, exact learning with GPs is intractable for large datasets. A variety of approximate GP methods have been proposed that essentially map the large dataset into a small set of basis points. Among them, two state-of-the-art methods are sparse pseudo-input Gaussian process (SPGP) (Snelson and Ghahramani, 2006) and variablesigma GP (VSGP) Walder et al. (2008), which generalizes SPGP and allows each basis point to have its own length scale. Howev…
We propose a parallelizable sparse inverse formulation Gaussian process (SpInGP) for temporal models. It uses a sparse precision GP formulation and sparse matrix routines to speed up the computations. Due to the state-space formulation used in the algorithm, the time complexity of the basic SpInGP is linear, and because all the computations are parallelizable, the parallel form of the algorithm is sublinear in the number of data points. We provide example algorithms to implement the sparse matrix routines and experimentally test the method using both simulated and real data.
Zero-inflated datasets, which have an excess of zero outputs, are commonly encountered in problems such as climate or rare event modelling. Conventional machine learning approaches tend to overestimate the non-zeros leading to poor performance. We propose a novel model family of zero-inflated Gaussian processes (ZiGP) for such zero-inflated datasets, produced by sparse kernels through learning a latent probit Gaussian process that can zero out kernel rows and columns whenever the signal is absent. The ZiGPs are particularly useful for making the powerful Gaussian process networks more interpre…
We propose a similarity measure for sparsely sampled time course data in the form of a log-likelihood ratio of Gaussian processes (GP). The proposed GP similarity is similar to a Bayes factor and provides enhanced robustness to noise in sparse time series, such as those found in various biological settings, e.g., gene transcriptomics. We show that the GP measure is equivalent to the Euclidean distance when the noise variance in the GP is negligible compared to the noise variance of the signal. Our numerical experiments on both synthetic and real data show improved performance of the GP similar…
University of Cambridge（剑桥大学）；ETH Zürich（苏黎世联邦理工学院）；Warwick University（华威大学）
Large, multi-dimensional spatio-temporal datasets are omnipresent in modern science and engineering. An effective framework for handling such data are Gaussian process deep generative models (GP-DGMs), which employ GP priors over the latent variables of DGMs. Existing approaches for performing inference in GP-DGMs do not support sparse GP approximations based on inducing points, which are essential for the computational efficiency of GPs, nor do they handle missing data -- a natural occurrence in many spatio-temporal datasets -- in a principled manner. We address these shortcomings with the de…
Gaussian processes (GPs) provide a nonparametric representation of functions. However, classical GP inference suffers from high computational cost and it is difficult to design nonstationary GP priors in practice. In this paper, we propose a sparse Gaussian process model, EigenGP, based on the Karhunen-Loeve (KL) expansion of a GP prior. We use the Nystrom approximation to obtain data dependent eigenfunctions and select these eigenfunctions by evidence maximization. This selection reduces the number of eigenfunctions in our model and provides a nonstationary covariance function. To handle nonl…
Good sparse approximations are essential for practical inference in Gaussian Processes as the computational cost of exact methods is prohibitive for large datasets. The Fully Independent Training Conditional (FITC) and the Variational Free Energy (VFE) approximations are two recent popular methods. Despite superficial similarities, these approximations have surprisingly different theoretical properties and behave differently in practice. We thoroughly investigate the two methods for regression both analytically and through illustrative examples, and draw conclusions to guide practical applicat…
Standard sparse pseudo-input approximations to the Gaussian process (GP) cannot handle complex functions well. Sparse spectrum alternatives attempt to answer this but are known to over-fit. We suggest the use of variational inference for the sparse spectrum approximation to avoid both issues. We model the covariance function with a finite Fourier series approximation and treat it as a random variable. The random covariance function has a posterior, on which a variational distribution is placed. The variational distribution transforms the random covariance function to fit the data. We study the…
Creating low dimensional representations of a high dimensional data set is an important component in many machine learning applications. How to cluster data using their low dimensional embedded space is still a challenging problem in machine learning. In this article, we focus on proposing a joint formulation for both clustering and dimensionality reduction. When a probabilistic model is desired, one possible solution is to use the mixture models in which both cluster indicator and low dimensional space are learned. Our algorithm is based on a mixture of sparse Gaussian processes, which is cal…
Modeling sequential data has become more and more important in practice. Some applications are autonomous driving, virtual sensors and weather forecasting. To model such systems so called recurrent models are used. In this article we introduce two new Deep Recurrent Gaussian Process (DRGP) models based on the Sparse Spectrum Gaussian Process (SSGP) and the improved variational version called Variational Sparse Spectrum Gaussian Process (VSSGP). We follow the recurrent structure given by an existing DRGP based on a specific sparse Nyström approximation. Therefore, we also variationally integrat…
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.
Sparse pseudo-point approximations for Gaussian process (GP) models provide a suite of methods that support deployment of GPs in the large data regime and enable analytic intractabilities to be sidestepped. However, the field lacks a principled method to handle streaming data in which both the posterior distribution over function values and the hyperparameter estimates are updated in an online fashion. The small number of existing approaches either use suboptimal hand-crafted heuristics for hyperparameter learning, or suffer from catastrophic forgetting or slow updating when new data arrive. T…
This paper describes and illustrates functionality of the spNNGP R package. The package provides a suite of spatial regression models for Gaussian and non-Gaussian point-referenced outcomes that are spatially indexed. The package implements several Markov chain Monte Carlo (MCMC) and MCMC-free Nearest Neighbor Gaussian Process (NNGP) models for inference about large spatial data. Non-Gaussian outcomes are modeled using a NNGP Polya-Gamma latent variable. OpenMP parallelization options are provided to take advantage of multiprocessor systems. Package features are illustrated using simulated and…
We present a planning framework for minimising the deterministic worst-case error in sparse Gaussian process (GP) regression. We first derive a universal worst-case error bound for sparse GP regression with bounded noise using interpolation theory on reproducing kernel Hilbert spaces (RKHSs). By exploiting the conditional independence (CI) assumption central to sparse GP regression, we show that the worst-case error minimisation can be achieved by solving a posterior entropy minimisation problem. In turn, the posterior entropy minimisation problem is solved using a Gaussian belief space planni…
University of Oxford（牛津大学）；EURECOM（欧洲通信研究院）；University of Adelaide（阿德莱德大学）
We investigate the connections between sparse approximation methods for making kernel methods and Gaussian processes (GPs) scalable to large-scale data, focusing on the Nyström method and the Sparse Variational Gaussian Processes (SVGP). While sparse approximation methods for GPs and kernel methods share some algebraic similarities, the literature lacks a deep understanding of how and why they are related. This may pose an obstacle to the communications between the GP and kernel communities, making it difficult to transfer results from one side to the other. Our motivation is to remove this ob…
Methods for inference and simulation of linearly constrained Gaussian Markov Random Fields (GMRF) are computationally prohibitive when the number of constraints is large. In some cases, such as for intrinsic GMRFs, they may even be unfeasible. We propose a new class of methods to overcome these challenges in the common case of sparse constraints, where one has a large number of constraints and each only involves a few elements. Our methods rely on a basis transformation into blocks of constrained versus non-constrained subspaces, and we show that the methods greatly outperform existing alterna…
This note is devoted to the comparison between two Nearest-neighbor Gaussian processes (NNGP) based models: the response NNGP model and the latent NNGP model. We exhibit that the comparison based on the Kullback-Leibler divergence (KL-D) from the NNGP based models to their parent GP based model can result in reverse conclusions in different parameter spaces. And we suggest a heuristic explanation on the phenomenon that the latent NNGP model tends to outperform the response NNGP model in approximating their parent GP based model.
We consider the problem of clustering data points in high dimensions, i.e. when the number of data points may be much smaller than the number of dimensions. Specifically, we consider a Gaussian mixture model (GMM) with non-spherical Gaussian components, where the clusters are distinguished by only a few relevant dimensions. The method we propose is a combination of a recent approach for learning parameters of a Gaussian mixture model and sparse linear discriminant analysis (LDA). In addition to cluster assignments, the method returns an estimate of the set of features relevant for clustering. …
A fast matching pursuit method using a Bayesian approach is introduced for sparse signal recovery. This method, referred to as nGpFBMP, performs Bayesian estimates of sparse signals even when the signal prior is non-Gaussian or unknown. It is agnostic on signal statistics and utilizes a priori statistics of additive noise and the sparsity rate of the signal, which are shown to be easily estimated from data if not available. nGpFBMP utilizes a greedy approach and order-recursive updates of its metrics to find the most dominant sparse supports to determine the approximate minimum mean square err…
We consider alternate formulations of recently proposed hierarchical Nearest Neighbor Gaussian Process (NNGP) models (Datta et al., 2016a) for improved convergence, faster computing time, and more robust and reproducible Bayesian inference. Algorithms are defined that improve CPU memory management and exploit existing high-performance numerical linear algebra libraries. Computational and inferential benefits are assessed for alternate NNGP specifications using simulated datasets and remotely sensed light detection and ranging (LiDAR) data collected over the US Forest Service Tanana Inventory U…
