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We develop an exact and scalable algorithm for one-dimensional Gaussianprocess regression with Mat\'ern correlations whose smoothness parameter $\nu$is a half-integer. The proposed algorithm only requires $\mathcal{O}(\nu^3 n)$operations and $\mathcal{O}(\nu n)$ storage. This leads to a linear-cost solversince $\nu$ is chosen to be fixed and usually very small in most applications.The proposed method can be applied to multi-dimensional problems if a full gridor a sparse grid design is used. The proposed method is based on a novel theoryfor Mat\'ern correlation functions. We find that a suitable rearrangement ofthese correlation functions can produce a compactly supported function, calleda "kernel packet". Using a set of kernel packets as basis functions leads to asparse representation of the covariance matrix that results in the proposedalgorithm. Simulation studies show that the proposed algorithm, whenapplicable, is significantly superior to the existing alternatives in both thecomputational time and predictive accuracy.

Kernel Packet: An Exact and Scalable Algorithm for Gaussian Process Regression with Mat\'ern Correlations