Title: Observable dictionary learning for high-dimensional statistical inference,
Abstract: This paper introduces a method for efficiently inferring a high-dimensional
distributed quantity from a few observations. The quantity of interest (QoI) is
approximated in a basis (dictionary) learned from a training set. The
coefficients associated with the approximation of the QoI in the basis are
determined by minimizing the misfit with the observations. To obtain a
probabilistic estimate of the quantity of interest, a Bayesian approach is
employed. The QoI is treated as a random field endowed with a hierarchical
prior distribution so that closed-form expressions can be obtained for the
posterior distribution. The main contribution of the present work lies in the
derivation of \emph{a representation basis consistent with the observation
chain} used to infer the associated coefficients. The resulting dictionary is
then tailored to be both observable by the sensors and accurate in
approximating the posterior mean. An algorithm for deriving such an observable
dictionary is presented. The method is illustrated with the estimation of the
velocity field of an open cavity flow from a handful of wall-mounted point
sensors. Comparison with standard estimation approaches relying on Principal
Component Analysis and K-SVD dictionaries is provided and illustrates the
superior performance of the present approach.