Algorithms for Kullback-Leibler Approximation of Probability Measures in Infinite Dimensions

Understand

In this paper we study algorithms to find a Gaussian approximation to a target measure defined on a Hilbert space of functions; the target measure itself is defined via its density with respect to a reference Gaussian measure.

• We employ the Kullback-Leibler divergence as a distance and find the best Gaussian approximation by minimizing this distance.
• It then follows that the approximate Gaussian must be equivalent to the Gaussian reference measure, defining a natural function space setting for the underlying calculus of variations problem.
• We introduce a computational algorithm which is well-adapted to the required minimization, seeking to find the mean as a function, and parameterizing the covariance in two different ways: through low rank perturbations of the reference covariance; and through Schr\"odinger potential perturbations of the inverse reference covariance.