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Efficient Gaussian Process Regression for Large Data Sets
Anjishnu Banerjee, David Dunson, Surya Tokdar
TL;DR
Gaussian process regression becomes computationally burdensome at large n, with matrix operations typically requiring O(n^3) computation and numerical stability degrading as data grow. The paper replaces subset selection with random projections of all data points into a lower-dimensional subspace, providing theoretical justification and better approximation and conditioning in simulations.
Problem
Gaussian process computations typically require O(n^3) operations, while large datasets create storage, processing, and numerical-stability bottlenecks.
Method
The paper uses random projections and stochastic matrix approximations to construct lower-dimensional representations without discrete knot selection.
Results
Random projections provide better approximation accuracy and dramatically better condition numbers than two comparison methods, achieving target errors with lower ranks across matrix sizes and error levels.
Takeaways & Limitations
The approach offers a theoretically justified and practically implementable alternative to subset-based Gaussian process approximations using all data points in a lower-dimensional subspace.
Takeaways & Limitations
Parallel computing performance, multivariate extensions, and theoretical convergence rates remain unexplored or ongoing work.
Abstract
from arXiv · showhide
Gaussian processes (GPs) are widely used in nonparametric regression, classification and spatio-temporal modeling, motivated in part by a rich literature on theoretical properties. However, a well known drawback of GPs that limits their use is the expensive computation, typically O($n^3$) in performing the necessary matrix inversions with $n$ denoting the number of data points. In large data sets, data storage and processing also lead to computational bottlenecks and numerical stability of the estimates and predicted values degrades with $n$. To address these problems, a rich variety of methods have been proposed, with recent options including predictive processes in spatial data analysis and subset of regressors in machine learning. The underlying idea in these approaches is to use a subset of the data, leading to questions of sensitivity to the subset and limitations in estimating fine scale structure in regions that are not well covered by the subset. Motivated by the literature on compressive sensing, we propose an alternative random projection of all the data points onto a lower-dimensional subspace. We demonstrate the superiority of this approach from a theoretical perspective and through the use of simulated and real data examples. Some Keywords: Bayesian; Compressive Sensing; Dimension Reduction; Gaussian Processes; Random Projections; Subset Selection
1 Introduction
Gaussian processes provide a probabilistic framework for modeling and prediction, but their computational cost, numerical instability, and approximation challenges limit use with large datasets. The paper motivates random projections as an alternative to subset-based approximations.
- Gaussian processes support probabilistic modeling and prediction for spatial effects and supervised machine-learning tasks.
- Evaluating GP posteriors and predictive summaries typically requires O(n^3) computation.
- Increasing dimension can reduce numerical accuracy because matrix inversion becomes more unstable, especially for nearly rank-deficient covariance matrices.
- Reduced-rank approximations address these costs but raise concerns about basis selection, covariance-kernel restrictions, predictive-variance underestimation, and knot placement.
- The paper proposes random projections to bypass discrete knot selection, improve numerical stability, and generalize knot-based methods.
2 Random Projection Approximation Methodology
The methodology replaces conditioning on selected knots with conditioning on a lower-dimensional random projection of all observed function values. It connects this construction to variance correction, matrix approximations, truncated expansions, and optimal Hilbert-space projections.
- Predictive-process approximations condition the GP on values at a finite set of knots, producing a kriged surface and a reduced-rank covariance.
- Subset-of-regressors and predictive-process methods are equivalent through the covariance approximation Qf,f = Kf,∗(K∗,∗)^-1K∗,f.
- These knot-based approximations can underestimate predictive variance, motivating an independent correction term that restores the diagonal covariance contribution.
- Random projection approximation conditions on Φf(X) instead of knot values, where Φ is an m × n full-row-rank matrix with normalized rows.
- The modified random-projection covariance adds a diagonal correction, qRM(x,z) = qRP(x,z) + δ(x,z){k(x,z) − qRP(x,z)}.
- Random projections recover the full-rank process when m = n and include knot-based methods as special cases.
- The method is optimal as an orthogonal Hilbert-space projection and can be related to Nyström approximations, partial Cholesky decompositions, and truncated Karhunen–Loève expansions.
3 Matrix Approximations & Projection Construction
The paper constructs reduced-rank Gaussian-process approximations by combining random projections with Nyström matrix approximation. The method targets near-optimal projections while reducing computational cost and improving approximation accuracy and numerical stability relative to subset-based alternatives.
- Optimal projection: The best rank-m approximation uses the first m eigenvectors of the covariance matrix as the projection basis.This ideal projection is computationally burdensome because obtaining the spectral decomposition also requires O(n^3) operations.
- Motivation: O(n^3) computation makes exact covariance decomposition and inversion impractical for large Gaussian-process data sets.Matrix inversion also becomes unstable when covariance matrices are nearly rank deficient, as commonly occurs for densely sampled smooth processes.
- Random projection construction: Random matrix products KΩ generate vectors in the covariance matrix’s range, enabling accurate range capture with m << n when K is severely rank deficient.The construction uses low-dimensional embeddings with Johnson–Lindenstrauss distortion guarantees.
- Random projection construction: The proposed algorithm applies a Johnson–Lindenstrauss embedding, a rank-m projection, and a Nyström approximation to construct a near-optimal covariance decomposition.The implementation decomposes the small matrix ΦKΦT and uses matrix multiplications that can be parallelized.
- Accuracy guarantees: The approximation error can be controlled for fixed rank or fixed target accuracy, with its covariance-matrix error governing the divergence between full and approximate marginal distributions.The method relates projection quality to Frobenius-norm approximation error and Kullback–Leibler divergence.
- Empirical comparison: Random projections achieve better approximation accuracy, dramatically better condition numbers, and lower ranks for target errors than the compared subset-selection methods.In simulations, CPU times were comparable and became lower for random projections when rank differences were substantial.
4 Parameter Estimation And Illustrations
The paper specifies Bayesian inference for random-projection Gaussian-process regression and illustrates the method on simulated and real data. Random projections improve predictive accuracy in both real-data examples, especially for the Sarcos Robot arm dataset.
- The squared exponential kernel uses θ1 and θ2 as range and inverse scale parameters, respectively.
- Bayesian inference assigns priors to θ1, θ2, and σ2 and uses conditionally conjugate distributions for Gibbs sampling.
- The random-projection model replaces the latent function with gRM and uses a bias-corrected covariance approximation.
- The implementation precomputes projection matrices over a grid of θ1 values and rescales inverses when only θ2 changes.
- 10,000 simulated observations across three smoothness levels are split into 9,000 fitting points and a validation set, with target errors 0.1 and 0.01.
- Random projections improve predictive accuracy on both real datasets, with a significant improvement for the Sarcos Robot arm data.The examples contain 4,000 training and 177 test cases for abalone, and 44,484 training and 4,449 test cases for Sarcos.
5 Concluding Remarks
The paper presents random projections within a broad reduced-rank framework and reports gains in performance and stability over existing approaches. It also identifies parallelization, multivariate extensions, and convergence rates as future work.
- The framework encompasses nearly the full range of existing reduced-rank Gaussian-process approximations.
- The authors seek better performance and stability by stochastically finding solutions within this broad approximation framework.
- The paper connects machine-learning and statistical Gaussian-process approximations with matrix-approximation perspectives.
- Random projections are studied mainly as an approximation scheme, while their potential as an alternative model remains for future investigation.
- Parallel computing performance was not explored, and parallelization, multivariate extensions, and convergence rates remain future work.
Proof of Theorem 1
The proof identifies the random-projection approximation with a generalized projection of the covariance matrix. This establishes its connection to reduced singular-value-decomposition approximations.
- The reduced SVD approximation produced by Algorithm 1 equals the random-projection approximation.
- The generalized projection matrix has range spanned by KΦT and nullity given by the orthogonal complement of the range of ΦT.
- The covariance approximation is written as Ktr = PK, where P is constructed from K, Φ, and their induced projection spaces.
Proof of Theorem 2
The proof bounds the divergence between the full and random-projection Gaussian-process distributions through covariance approximation error. The resulting bound has the same order as the Frobenius-norm covariance error but is not optimal.
- The proof compares the full posterior πfull with the random-projection posterior πRP as multivariate normal distributions.
- The Kullback–Leibler divergence is decomposed into separate terms involving covariance determinants and eigenvalues.
- The covariance-error assumption is expressed using a Frobenius-norm bound between the full and approximated covariance matrices.
- The noise variance lower-bounds the eigenvalues of the approximated covariance-plus-noise matrix, controlling the inverse-matrix terms.
- The resulting Kullback–Leibler divergence is of the same order as the covariance-estimation error in Frobenius norm, but the bound is not optimal.
Example of inversion with the Woodbury matrix identity
The Woodbury matrix identity rewrites the marginalized inference or prediction calculation so that the diagonal structure in the decomposition avoids direct matrix inversion.
- The decomposition expresses the covariance term through an orthogonal factorization with U^T U = I.
- The method uses the Woodbury matrix identity for marginalized inference or prediction involving the covariance-plus-noise inverse.The cited passage introduces the identity in this calculation, although the displayed expression is incomplete.
- Because D^-2 + σ^-2I is diagonal, its inverse is obtained by reciprocating diagonal entries, avoiding direct matrix inversion.