- Compressed sensing is a new emerging field dealing with the reconstruction of a sparse or, more precisely, a compressed representation of a signal from a relatively small number of observations, typically less than the signal dimension. In our previous work we have shown how the Kalman filter can be naturally applied for obtaining an approximate Bayesian solution for the compressed sensing problem. The resulting algorithm, which was termed CSKF, relies on a pseudo-measurement technique for enforcing the sparseness constraint. Our approach raises two concerns which are addressed in this paper. The first one refers to the validity of our approximation technique. In this regard, we provide a rigorous treatment of the CSKF algorithm which is concluded with an upper bound on the discrepancy between the exact (in the Bayesian sense) and the approximate solutions. The second concern refers to the computational overhead associated with the CSKF in large scale settings. This problem is alleviated here using an efficient measurement update scheme based on Krylov subspace method.