High Dimensional Multi-Level Covariance Estimation and Kriging
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With the advent of big data sets much of the computational science and engineering communities have been moving toward data-driven approaches to regression and classification. However, they present a significant challenge due to the increasing size, complexity and dimensionality of the problems. In this paper a multi-level kriging method that scales well with dimensions is developed. A multi-level basis is constructed that is adapted to a random projection tree (or kD-tree) partitioning of the observations and a sparse grid approximation. This approach identifies the high dimensional underlying phenomena from the noise in an accurate and numerically stable manner. Furthermore, numerically unstable covariance matrices are transformed into well conditioned multi-level matrices without compromising accuracy. A-posteriori error estimates are derived, such as the sub-exponential decay of the coefficients of the multi-level covariance matrix. The multi-level method is tested on numerically unstable problems of up to 50 dimensions. Accurate solutions with feasible computational cost are obtained.
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