We consider the minimum variance distortionless response (MVDR) beamforming problems where the array covariance matrix is rank deficient. The conventional approach handles such rank-deficiencies via diagonal loading on the covariance matrix. In this setting, we show that the array weights for optimal signal estimation can admit a sparse representation on the array manifold. To exploit this structure, we propose a convex regularizer in a grid-free fashion, which requires semidefinite programming. We then provide numerical evidence showing that the new formulation can significantly outperform diagonal loading when the regularization parameters are correctly tuned.