ℋ‐matrix techniques for approximating large covariance matrices and estimating its parameters

ℋ‐matrix techniques for approximating large covariance matrices and estimating its parameters

​A. Litvinenko, M. Genton, Y. Sun, D. Keyes, ℋ‐matrix techniques for approximating large covariance matrices and estimating its parameters, arXiv:1611.09293, 2017 
A. Litvinenko, M. Genton, Y. Sun, D. Keyes
ℋ‐matrix techniques for approximating large covariance matrices and estimating its parameters
2017
In this work the task is to use the available measurements to estimate unknown hyper-parameters (variance, smoothness parameter and covariance length) of the covariance function. We do it by maximizing the joint log-likelihood function. This is a non-convex and non-linear problem. To overcome cubic complexity in linear algebra, we approximate the discretised covariance function in the hierarchical (ℋ-) matrix format. The ℋ-matrix format has a log-linear computational cost and storage O(knlogn), where rank k is a small integer. On each iteration step of the optimization procedure the covariance matrix itself, its determinant and its Cholesky decomposition are recomputed within ℋ-matrix format. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)​