  We study the sparse recovery problem with an underdetermined linear system
characterized by a Kronecker-structured dictionary and a Kronecker-supported
sparse vector. We cast this problem into the sparse Bayesian learning (SBL)
framework and rely on the expectation-maximization method for a solution. To
this end, we model the Kronecker-structured support with a hierarchical
Gaussian prior distribution parameterized by a Kronecker-structured
hyperparameter, leading to a non-convex optimization problem. The optimization
problem is solved using the alternating minimization (AM) method and a singular
value decomposition (SVD)-based method, resulting in two algorithms. Further,
we analytically guarantee that the AM-based method converges to the stationary
point of the SBL cost function. The SVD-based method, though it adopts
approximations, is empirically shown to be more efficient and accurate. We then
apply our algorithm to estimate the uplink wireless channel in an intelligent
reflecting surface-aided MIMO system and extend the AM-based algorithm to
address block sparsity in the channel. We also study the SBL cost to show that
the minima of the cost function are achieved at sparse solutions and that
incorporating the Kronecker structure reduces the number of local minima of the
SBL cost function. Our numerical results demonstrate the effectiveness of our
algorithms compared to the state-of-the-art.
