  In this paper, we investigate the precoder design for user-centric network
(UCN) massive multiple-input multiple-output (mMIMO) downlink with matrix
manifold optimization. In UCN mMIMO systems, each user terminal (UT) is served
by a subset of base stations (BSs) instead of all the BSs, facilitating the
implementation of the system and lowering the dimension of the precoders to be
designed. By proving that the precoder set satisfying the per-BS power
constraints forms a Riemannian submanifold of a linear product manifold, we
transform the constrained precoder design problem in Euclidean space to an
unconstrained one on the Riemannian submanifold. Riemannian ingredients,
including orthogonal projection, Riemannian gradient, retraction and vector
transport, of the problem on the Riemannian submanifold are further derived,
with which the Riemannian conjugate gradient (RCG) design method is proposed
for solving the unconstrained problem. The proposed method avoids the inverses
of large dimensional matrices, which is beneficial in practice. The complexity
analyses show the high computational efficiency of RCG precoder design.
Simulation results demonstrate the numerical superiority of the proposed
precoder design and the high efficiency of the UCN mMIMO system.
