The Completion of Covariance Kernels

29.10.2021 11:15 – 12:15

RESEARCH CENTER FOR STATISTICS SEMINAR / ABSTRACT

We consider the problem of positive-semidefinite continuation: extending a partially specified covariance kernel from a subdomain Ω of a square domain I x I to a covariance kernel on the entire domain I x I. For a broad class of domains Ω called serrated domains, we will present a complete theory. Namely, we will demonstrate that a canonical completion always exists and can be explicitly constructed. We will characterise all possible completions as suitable perturbations of the canonical completion, and determine necessary and sufficient conditions for a unique completion to exist. We shall interpret the canonical completion via the graphical model structure it induces on the associated Gaussian process. Furthermore, we will show how the estimation of the canonical completion reduces to the solution of a system of linear statistical inverse problems in the space of Hilbert-Schmidt operators, and derive rates of convergence under standard source conditions. Time allowing, we will discuss extensions of our theory to more general forms of domains.

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