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We consider the problem of computing a Gaussian approximation to theposterior distribution of a parameter given a large number N of observationsand a Gaussian prior, when the dimension of the parameter d is also large. Toaddress this problem we build on a recently introduced recursive algorithm forvariational Gaussian approximation of the posterior, called recursivevariational Gaussian approximation (RVGA), which is a single pass algorithm,free of parameter tuning. In this paper, we consider the case where theparameter dimension d is high, and we propose a novel version of RVGA thatscales linearly in the dimension d (as well as in the number of observationsN), and which only requires linear storage capacity in d. This is afforded bythe use of a novel recursive expectation maximization (EM) algorithm appliedfor factor analysis introduced herein, to approximate at each step thecovariance matrix of the Gaussian distribution conveying the uncertainty in theparameter. The approach is successfully illustrated on the problems of highdimensional least-squares and logistic regression, and generalized to a largeclass of nonlinear models.

The limited-memory recursive variational Gaussian approximation (L-RVGA)