This paper considers the unsupervised filtering problem for large-dimensional linear and Gaussian systems, a setup in which the optimal Kalman filter (KF) might not be usable due to the exorbitant computational cost and storage requirements. For this problem, we propose two efficient algorithms based on the variational Bayesian (VB) approach. The first algorithm is an extension of a recent VBKF algorithm to the unsupervised framework, whereas the second algorithm is an accelerated version of the first one, derived based on the adaptation of subspace optimization methods in Hilbert spaces into the space of probability density functions. Furthermore, both algorithms account for the sparsity in the state and observations through heavy-tailed Student-t priors. Results of numerical experiments conducted on a dynamical tomography problem to assess the performances of the proposed schemes are presented.