  In multiple-input multiple-output (MIMO) device-to-device (D2D) networks,
interference and rank-deficient channels are the critical bottlenecks for
achieving high degrees of freedom (DoFs). In this paper, we propose a
reconfigurable intelligent surface (RIS) assisted interference alignment
strategy to simultaneous mitigate the co-channel interference and cope with
rank-deficient channels, thereby improving the feasibility of interference
alignment conditions and in turn increasing the achievable DoFs. The key
enabler is a general low-rank optimization approach that maximizes the
achievable DoFs by jointly designing the phase-shift and transceiver matrices.
To address the unique challenges of the coupled optimization variables, we
develop a block-structured Riemannian pursuit method by solving fixed-rank and
unit modulus constrained least square subproblems along with rank increase.
Finally, to reduce the computational complexity and achieve good DoF
performance, we develop unified Riemannian conjugate gradient algorithms to
alternately optimize the fixed-rank transceiver matrix and the unit modulus
constrained phase shifter by exploiting the non-compact Stiefel manifold and
the complex circle manifold, respectively. Numerical results demonstrate the
effectiveness of deploying an RIS and the superiority of the proposed
block-structured Riemannian pursuit method in terms of the achievable DoFs and
the achievable sum rate.
