The Extrapolation of Correlation

We discuss the problem of positive-semidefinite extension: extending a partially specified covariance kernel from a subdomain Ω of a rectangular 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 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 graphical model structure it induces on theassociated 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 inverse problems associated with stochastic processes. Based on joint work with in collaboration with K.G. Waghmare (EPFL)

Victor M. Panaretos is Professor of Mathematical Statistics and Director of the Mathematics Institute at the EPFL. He received his PhD in 2007 from UC Berkeley, advised by David Brillinger. Upon graduation he was appointed as Assistant Professor at the EPFL, where he rose the ranks to Full Professor. He received the Erich Lehmann Award for an Outstanding PhD (UC Berkeley, 2007), an ERC Starting Grant Award (2011) and was named "One of 40 extraordinary scientists under 40" by the World Economic Forum (2014). He is an Elected Member of the ISI (2008) and a Fellow of the IMS (2019). He was the Bernoulli Society Forum Lecturer in the 2019 EMS, and will be a plenary speaker at the XVI Latin American Congress in Probability and Mathematical Statistic (CLAPEM). He is an Associate Editor for Biometrika, and the Journal of the American Statistical Association (Theory and Methods), and previously for the Annals of Statistics, Annals of Applied Statistics, and Electronic Journal of Statistics. He has served the discipline from various posts, most notably currently being President-Elect of the Bernoulli Society.