  We consider the problem of computing a Gaussian approximation to the
posterior distribution of a parameter given a large number N of observations
and a Gaussian prior, when the dimension of the parameter d is also large. To
address this problem we build on a recently introduced recursive algorithm for
variational Gaussian approximation of the posterior, called recursive
variational Gaussian approximation (RVGA), which is a single pass algorithm,
free of parameter tuning. In this paper, we consider the case where the
parameter dimension d is high, and we propose a novel version of RVGA that
scales linearly in the dimension d (as well as in the number of observations
N), and which only requires linear storage capacity in d. This is afforded by
the use of a novel recursive expectation maximization (EM) algorithm applied
for factor analysis introduced herein, to approximate at each step the
covariance matrix of the Gaussian distribution conveying the uncertainty in the
parameter. The approach is successfully illustrated on the problems of high
dimensional least-squares and logistic regression, and generalized to a large
class of nonlinear models.
