Beamforming methods rely on training data to estimate the covariance matrix of the interference pulse noise. Their convergence slows down if the signal of interest is present in the training data, thus requiring a large numbers of training snapshots to maintain good performance. In a distributed array, in which the array nodes are connected to a fusion center via a wireless link, the estimation of the covariance matrix would require the communication of large amounts of data, and thus would consume significant power. We propose an approach that enables good beamforming performance while requiring substantially fewer data to be transmitted to the fusion center. The main idea is based on the fact that when the number of signal and interference sources is much smaller than the number of array sensors, the training data matrix is low rank. Thus, based on matrix completion theory, under certain conditions, the training data matrix can be recovered from a subset of its elements, i.e., based on sub-Nyquist samples of the array sensors. Following the recovery of the training data matrix, and to cope with the errors introduced during the matrix completion process, we propose a robust optimization approach, which obtains the beamforming weight vector by optimizing the worst-case performance. Numerical results show that combination of matrix completion and robust optimization is very successful in suppressing interference and achieving a near-optimal beamforming performance with only partial training data.