  Estimating the direction of arrival (DOA) of sources is an important problem
in aerospace and vehicular communication, localization and radar. In this
paper, we consider a challenging multi-source DOA estimation task, where the
receiving antenna array is composed of non-coherent sub-arrays, i.e.,
sub-arrays that observe different unknown phase shifts at every snapshot (e.g.,
due to waiving the demanding synchronization of local oscillators across the
entire array). We formulate this problem as the reconstruction of joint sparse
and low-rank matrices, and solve the problem's convex relaxation. To scale the
optimization complexity with the number of snapshots better than
general-purpose solvers, we design an optimization scheme, based on integrating
the alternating direction method of multipliers and the accelerated proximal
gradient techniques, that exploits the structure of the problem. While the DOAs
can be estimated from the solution of the aforementioned convex problem, we
further show how an improvement is obtained if, instead, one estimates from
this solution only the sub-arrays' phase shifts. This is done using another,
computationally-light, convex relaxation that is practically tight. Using the
estimated phase shifts, "phase-corrected" observations are created and a final
plain ("coherent") DOA estimation method can be applied. Numerical experiments
show the performance advantages of the proposed strategies over existing
methods.
