Séance Séminaire

Colloquium de Mathématiques

The Extrapolation of Correlation

We discuss the problem of positive-semidefinite continuation: extending a partially specified positive semi-definite kernel from a subdomain Ω of a rectangular domain I x I to a positive semidefinite kernel on the entire domain I x I. For a broad class of domains Ω called serrated domains, we present a complete theory. Namely,  we demonstrate that a canonical completion always exists and can be explicitly constructed. We characterise all possible completions as suitable perturbations of the canonical completion, and determine necessary and sufficient conditions for a unique completion to exist. We interpret the canonical completion via the correlation structure it induces on the associated Gaussian process. Furthermore, we show how the determination of the canonical completion reduces to the solution of a system of linear inverse problems in the space of Hilbert-Schmidt operators, and derive rates of convergence when the kernel is to be empirically estimated. We conclude by providing extensions of our theory to more general forms of domains, and by demonstrating how our results can be used in statistical problems associated with stochastic processes. Based on joint work with in collaboration with K.G. Waghmare (EPFL).