  We investigate the problem of recovering a structured sparse signal from a
linear observation model with an uncertain dynamic grid in the sensing matrix.
The state-of-the-art expectation maximization based compressed sensing (EM-CS)
methods, such as turbo compressed sensing (Turbo-CS) and turbo variational
Bayesian inference (Turbo-VBI), have a relatively slow convergence speed due to
the double-loop iterations between the E-step and M-step. Moreover, each inner
iteration in the E-step involves a high-dimensional matrix inverse in general,
which is unacceptable for problems with large signal dimensions or real-time
calculation requirements. Although there are some attempts to avoid the
high-dimensional matrix inverse by majorization minimization, the convergence
speed and accuracy are often sacrificed. To better address this problem, we
propose an alternating estimation framework based on a novel subspace
constrained VBI (SC-VBI) method, in which the high-dimensional matrix inverse
is replaced by a low-dimensional subspace constrained matrix inverse (with the
dimension equal to the sparsity level). We further prove the convergence of the
SC-VBI to a stationary solution of the Kullback-Leibler divergence minimization
problem. Simulations demonstrate that the proposed SC-VBI algorithm can achieve
a much better tradeoff between complexity per iteration, convergence speed, and
performance compared to the state-of-the-art algorithms.
