- Precoder Design for Large MIMO Downlink with Matrix Manifold Optimization(arXiv)
Creator : Rui Sun, Chen Wang, An-An Lu, Xiqi Gao, Xiang-Gen Xia
Summary : We examine the weighted sum-rate (WSR) maximization linear precoder design for enormous multiple-input multiple-output (MIMO) downlink. We contemplate a single-cell system with a number of customers and suggest a unified matrix manifold optimization framework relevant to whole energy constraint (TPC), per-user energy constraint (PUPC) and per-antenna energy constraint (PAPC). We show that the precoders beneath TPC, PUPC and PAPC are on distinct Riemannian submanifolds, and remodel the constrained issues in Euclidean house to unconstrained ones on manifolds. In accordance with this, we derive Riemannian substances, together with orthogonal projection, Riemannian gradient, Riemannian Hessian, retraction and vector transport, that are wanted for precoder design within the matrix manifold framework. Then, Riemannian design strategies utilizing Riemannian steepest descent, Riemannian conjugate gradient and Riemannian belief area are supplied to design the WSR-maximization precoders beneath TPC, PUPC or PAPC. Riemannian strategies don’t contain the inverses of the massive dimensional matrices through the iterations, lowering the computational complexities of the algorithms. Complexity analyses and efficiency simulations reveal the benefits of the proposed precoder design
2. Vector Transport Free Riemannian LBFGS for Optimization on Symmetric Optimistic Particular Matrix Manifolds(arXiv)
Creator : Reza Godaz, Benyamin Ghojogh, Reshad Hosseini, Reza Monsefi, Fakhri Karray, Mark Crowley
Summary : This work concentrates on optimization on Riemannian manifolds. The Restricted-memory Broyden-Fletcher-Goldfarb-Shanno (LBFGS) algorithm is a generally used quasi-Newton methodology for numerical optimization in Euclidean areas. Riemannian LBFGS (RLBFGS) is an extension of this methodology to Riemannian manifolds. RLBFGS entails computationally costly vector transports in addition to unfolding recursions utilizing adjoint vector transports. On this article, we suggest two mappings within the tangent house utilizing the inverse second root and Cholesky decomposition. These mappings make each vector transport and adjoint vector transport id and due to this fact isometric. Id vector transport makes RLBFGS much less computationally costly and its isometry can also be very helpful in convergence evaluation of RLBFGS. Furthermore, beneath the proposed mappings, the Riemannian metric reduces to Euclidean inside product, which is way much less computationally costly. We deal with the Symmetric Optimistic Particular (SPD) manifolds that are helpful in varied fields reminiscent of knowledge science and statistics. This work opens a analysis alternative for extension of the proposed mappings to different well-known manifolds
