Precoder Design for Massive MIMO Downlink with Matrix Manifold Optimization

Massive multiple-input multiple-output (MIMO) is one of the key techniques in the fifth generation (5G) wireless networks and will play an important role in future 6G systems with further increased antenna number scale [1, 2]. In massive MIMO systems, the base station (BS) equipped with a large number of antennas can serve a number of user terminals (UTs) on the same time-frequency resource, providing enormous potential capacity gains and high energy efficiency [3, 4]. However, serving many users simultaneously causes serious inter-user interference, which may reduce the throughput. To suppress the interference and increase the system throughput, precoders should be properly designed for massive MIMO downlink (DL) transmission. Specifically, a linear precoder is particularly attractive due to its low complexity [5, 6, 7].

There are several criteria for the DL precoder design, including minimum mean square error (MMSE), weighted sum-rate (WSR), quality of service (QoS), signal-to-interference-plus-noise ratio (SINR), signal-to-leakage-plus-noise ratio (SLNR), energy efficiency (EE), etc. [8, 9, 10, 11, 12, 13]. Among these criteria, WSR is of great practical significance and widely considered in massive MIMO systems, as it directly aims at increasing the system throughput, which is one of the main objectives of communication systems. In addition, different power constraints may be imposed on the designs. Generally, power constraints for massive MIMO systems can be classified into three main categories: total power constraint (TPC), per-user power constraint (PUPC) and per-antenna power constraint (PAPC). Designing precoders under different power constraints usually forms different problems, for which different approaches are proposed in the literature. To be specific, the precoder design problem under TPC is typically formulated as a nonconvex WSR-maximization problem, which is transformed into an MMSE problem iteratively solved by alternating optimization in [9]. Alternatively, [14] solves the WSR-maximization problem in the framework of the minorize-maximize (MM) method by finding a surrogate function of the objective function. When designing precoders under PUPC, the received SINR of each user is usually investigated with QoS constraints. With the received SINR, the sum-rate maximization problem is formulated and solved by the Lagrange dual function [15, 11]. Precoders under PAPC are deeply coupled with each other and thus challenging to design. In [16], precoders under PAPC are designed by minimizing the distance between the precoders under TPC and those under PAPC. The WSR-maximization precoder design under PAPC using the dual coordinate ascent method (DCAM) is proposed in [17], where the original nonconvex WSR-maximization problem is approximated as a sequence of convex MMSE-minimization problems subject to PAPC, and each convex problem is solved by the DCAM.

In general, the designs of WSR-maximization precoders under the power constraints mentioned above can be formulated as optimization problems with equality constraints. Recently, manifold optimization has been extensively studied and successfully applied to many domains [18, 19, 20, 21], showing a great advantage in dealing with smooth objective functions with challenging equality constraints. In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. By revealing the inherent geometric properties of the equality constraints, manifold optimization reformulates the constrained problems in Euclidean space as unconstrained ones on manifold. By defining the Riemannian ingredients associated with a Riemannian manifold, several Riemannian methods are presented for solving the unconstrained problems on manifold. In addition, manifold optimization usually shows promising algorithms resulting from the combination of insight from differential geometry, optimization, and numerical analysis. To be specific, by revealing that the precoders under TPC, PUPC or PAPC are on different Riemannian submanifolds, we can leverage this insight to transform the constrained problems into unconstrained ones on these submanifolds. Therefore, manifold optimization can provide a potential way for designing optimal WSR-maximization precoders under different power constraints in a unified framework.

In this paper, we focus on WSR-maximization precoder design for massive MIMO DL transmission and propose a matrix manifold framework applicable to TPC, PUPC and PAPC. We reveal the geometric properties of the precoders under different power constraints and prove that the precoder sets satisfying TPC, PUPC and PAPC form three different Riemannian submanifolds, respectively, transforming the constrained problems in Euclidean space into unconstrained ones on Riemannian submanifolds. To facilitate a better understanding, we analyze the precoder designs under TPC, PUPC and PAPC in detail. All the ingredients required during the optimizations on Riemannian submanifolds are derived for the three power constraints. Further, we present three Riemannian design methods using Riemannian steepest descent (RSD), Riemannian conjugate gradient (RCG) and Riemannian trust region (RTR), respectively. Without the need to invert the large dimensional matrix during the iterations, Riemannian methods can efficiently save computational costs, which is beneficial in practice. Complexity analysis shows that the method using RCG is computationally efficient. The numerical results confirm the advantages of the RCG method in convergence speed and WSR performance.

The remainder of the paper is organized as follows. In Section II, we introduce the preliminaries in the matrix manifold optimization. In Section III, we first formulate the WSR-maximization precoder design problem in Euclidean space. Then, we transform the constrained problems in Euclidean space under TPC, PUPC and PAPC to the unconstrained ones on Riemannian submanifolds and derive Riemannian ingredients in the matrix manifold framework. To solve the unconstrained problems on the Riemannian submanifolds, Section IV provides three Riemannian design methods and their complexity analyses. Section V presents numerical results and discusses the performance of the proposed precoder designs. The conclusion is drawn in Section VI.

Notations: Boldface lowercase and uppercase letters represent the column vectors and matrices, respectively. We write conjugate transpose of matrix $\mathbf{A}$ as $\mathbf{A}^{H}$ while $\mathrm{tr}\left(\mathbf{A}\right)$ and $\det\left(\mathbf{A}\right)$ denote the matrix trace and determinant of $\mathbf{A}$, respectively. $\Re\left\{\mathbf{A}\right\}$ means the real part of $\mathbf{A}$. Let the mathematical expectation be $\mathbb{E}\left\{\cdot\right\}$. $\mathbf{I}_{M}$ denotes the $M\times M$ dimensional identity matrix, whose subscript may be omitted for brevity. $\mathbf{0}$ represents the vector or matrix whose elements are all zero. The Hadamard product of $\mathbf{A}$ and $\mathbf{B}$ is $\mathbf{A}\odot\mathbf{B}$. Let $\mathbf{e}_{j}$ denote the vector with the $j$-th element equals $1$ while the others equal $0$. $\mathrm{diag}\left(\mathbf{b}\right)$ represents the diagonal matrix with $\mathbf{b}$ along its main diagonal and $\mathrm{diag}\left(\mathbf{A}\right)$ denotes the column vector of the main diagonal of $\mathbf{A}$. Similarly, $\mathbf{D}=\mathrm{blkdiag}\left\{\mathbf{A}_{1},\cdots,\mathbf{A}_{K}\right\}$ denotes the block diagonal matrix with $\mathbf{A}_{1},\cdots,\mathbf{A}_{K}$ on the diagonal and $\left[\mathbf{D}\right]_{i}$ denotes the $i$-th matrix on the diagonal, i.e., $\mathbf{A}_{i}$. A mapping $F$ from manifold $\mathcal{N}$ to manifold $\mathcal{M}$ is $F:\mathcal{N}\rightarrow\mathcal{M}:\mathbf{X}\mapsto\mathbf{Y}$ denoted as $F(\mathbf{X})=\mathbf{Y}$. The differential of $F\left(\mathbf{X}\right)$ is represented as $\mathrm{D}F\left(\mathbf{X}\right)$ while $\mathrm{D}F\left(\mathbf{X}\right)\left[{\boldsymbol{\xi}}_{\mathbf{X}}\right]$ or $\mathrm{D}F\left[{\boldsymbol{\xi}}_{\mathbf{X}}\right]$ means the directional derivative of $F$ at $\mathbf{X}$ along the tangent vector ${\boldsymbol{\xi}}_{\mathbf{X}}$.

In this section, we introduce some basic notions in the matrix manifold optimization and list some common notations in Table I, with $\mathcal{M}$ representing a Riemannian manifold. For notational simplicity, the superscripts for distinguishing different manifolds can be neglected when no confusion will arise. To avoid overwhelmingly redundant preliminaries, we focus on a coordinate-free analysis and omit charts and differential structures. For a complete introduction to smooth manifolds, we refer to references [22, 23, 24].

For a manifold $\mathcal{N}$, a smooth mapping $\gamma:\mathbb{R}\rightarrow\mathcal{N}:t\mapsto\gamma(t)$ is termed a curve in $\mathcal{N}$. Let $\mathbf{X}$ be a point on $\mathcal{N}$ and $\mathcal{F}_{\mathbf{X}}(\mathcal{N})$ denote the set of smooth real-valued functions defined on a neighborhood of $\mathbf{X}$. A tangent vector ${\boldsymbol{\xi}}_{\mathbf{X}}$ to a manifold $\mathcal{N}$ at a point $\mathbf{X}$ is a mapping from $\mathcal{F}_{\mathbf{X}}(\mathcal{N})$ to $\mathbb{R}$ such that there exists a curve $\gamma$ on $\mathcal{N}$ with $\gamma(0)=\mathbf{X}$, satisfying ${\boldsymbol{\xi}}_{\mathbf{X}}f=\left.\frac{\mathrm{d}(f(\gamma(t)))}{\mathrm{d}t}\right|_{t=0}$ for all $f\in\mathcal{F}_{\mathbf{X}}(\mathcal{N})$. Such a curve $\gamma$ is said to realize the tangent vector ${\boldsymbol{\xi}}_{\mathbf{X}}$. For a manifold $\mathcal{N}$, every point $\mathbf{X}$ on the manifold is attached to a unique and linear tangent space, denoted by $T_{\mathbf{X}}\mathcal{N}$. $T_{\mathbf{X}}\mathcal{N}$ is a set of all the tangent vectors to $\mathcal{N}$ at $\mathbf{X}$. The tangent bundle is the set of all tangent vectors to $\mathcal{N}$ denoted by $T\mathcal{N}$, which is also a manifold. A vector field ${\boldsymbol{\xi}}$ on a manifold $\mathcal{N}$ is a smooth mapping from $\mathcal{N}$ to $T\mathcal{N}$ that assigns to each point $\mathbf{X}\in\mathcal{N}$ a tangent vector ${\boldsymbol{\xi}}_{\mathbf{X}}\in T_{\mathbf{X}}\mathcal{N}$. Particularly, every vector space $\mathcal{E}$ forms a linear manifold naturally. Assuming $\mathcal{N}$ is a vector space $\mathcal{E}$, we have $T_{\mathbf{X}}\mathcal{N}=\mathcal{E}$.

Let $F:\mathcal{N}_{1}\rightarrow\mathcal{N}_{2}$ be a smooth mapping from manifold $\mathcal{N}_{1}$ to manifold $\mathcal{N}_{2}$. The differential of $F$ at $\mathbf{X}$ is a linear mapping from $T_{\mathbf{X}}\mathcal{N}_{1}$ to $T_{\mathbf{Y}}\mathcal{N}_{2}$, denoted by $\mathrm{D}F\left(\mathbf{X}\right)\left[\cdot\right]$. Denote $\mathbf{Y}=F\left(\mathbf{X}\right)$ as a point on $\mathcal{N}_{2}$. Similarly, ${\boldsymbol{\xi}}_{\mathbf{Y}}=\mathrm{D}F\left(\mathbf{X}\right)\left[{\boldsymbol{\xi}}_{\mathbf{X}}\right]$ is a tangent vector to $\mathcal{N}_{2}$ at $\mathbf{Y}$, i.e., an element in $T_{\mathbf{Y}}\mathcal{N}_{2}$. In particular, if $\mathcal{N}_{1}$ and $\mathcal{N}_{2}$ are linear manifolds, $\mathrm{D}F\left(\mathbf{X}\right)$ reduces to the classical directional derivative [25]

The rank of $F$ at $\mathbf{X}$ is the dimension of the range of $\mathrm{D}F\left(\mathbf{X}\right)\left[\boldsymbol{{\boldsymbol{\xi}}}_{\mathbf{X}}\right]$. Fig. 1 is a simple illustration for geometric understanding.

In practice, the equality constraint $F:\mathcal{N}\rightarrow 0$ in many optimization problems forms a mapping between two linear manifolds. The solutions satisfying the equality constraint constitute a set $\mathcal{M}=\left\{\mathbf{X}\in\mathcal{N}\mid F(\mathbf{X})=0\right\}$. The set $\mathcal{M}$ may admit several manifold structures, while it admits at most one differential structure that makes it an embedded submanifold of $\mathcal{N}$. Whether $\mathcal{M}$ forms an embedded submanifold mainly depends on the properties of $F\left(\mathbf{X}\right)$ [22, Proposition 3.3.2]. We assume $\mathcal{M}$ is an embedded submanifold of $\mathcal{N}$ in the rest of this section to facilitate the introduction. Note that $\mathbf{X}\in\mathcal{M}$ is still a point on $\mathcal{N}$. A level set of a real-valued function $F$ is the set of values $\mathbf{X}$ for which $F\left({\mathbf{X}}\right)$ is equal to a given constant. In particular, when $\mathcal{M}$ is defined as a level set of a constant-rank function $F$, the tangent space of $\mathcal{M}$ at $\mathbf{X}$ is the kernel of the differential of $F$ and a subspace of the tangent space of $\mathcal{N}$:

By endowing tangent space with an inner product $g_{\mathbf{X}}^{\mathcal{N}}\left(\cdot\right)$, we can define the length of tangent vectors in $T_{\mathbf{X}}\mathcal{N}$. Note that the subscript and the superscript in $g_{\mathbf{X}}^{\mathcal{N}}\left(\cdot\right)$ is used to distinguish the metric of different points on different manifolds for clarity, which may be omitted if no confusion will arise. $g_{\mathbf{X}}^{\mathcal{N}}\left(\cdot\right)$ is called Riemannian metric if it varies smoothly and the manifold is called Riemannian manifold. Since $T_{\mathbf{X}}{\mathcal{M}}$ can be regarded as a subspace of $T_{\mathbf{X}}{\mathcal{N}}$, the Riemannian metric $g_{\mathbf{X}}^{\mathcal{N}}\left(\cdot\right)$ on $\mathcal{N}$ induces a Riemannian metric $g_{\mathbf{X}}^{\mathcal{M}}\left(\cdot\right)$ on $\mathcal{M}$ according to

Note that ${\boldsymbol{\xi}}_{\mathbf{X}}$ and ${\boldsymbol{\zeta}}_{\mathbf{X}}$ on the right hand side are viewed as elements in $T_{\mathbf{X}}\mathcal{N}$. Endowed with the Riemannian metric (3), the embedded submanifold $\mathcal{M}$ forms a Riemannian submanifold.

In practice, it is suggested to utilize the orthogonal projection to obtain the elements in $T_{\mathbf{X}}{\mathcal{M}}$. With the Riemannian metric $g_{\mathbf{X}}^{\mathcal{N}}\left(\cdot\right)$, the $T_{\mathbf{X}}{\mathcal{N}}$ can be divided into two orthogonal subspaces as

where the normal space $N_{\mathbf{X}}{\mathcal{M}}$ is the orthogonal complement of $T_{\mathbf{X}}{\mathcal{M}}$ defined as

Thus, any ${\boldsymbol{\xi}}_{\mathbf{X}}\in T_{\mathbf{X}}{\mathcal{N}}$ can be uniquely decomposed into the sum of an element in $T_{\mathbf{X}}{\mathcal{M}}$ and an element in $N_{\mathbf{X}}{\mathcal{M}}$

where ${\boldsymbol{\Pi}}_{\mathbf{B}}^{\mathbf{A}}\left(\cdot\right)$ denotes the orthogonal projection from $\mathbf{A}$ to $\mathbf{B}$.

For a smooth real-valued function $f$ on a Riemannian submanifold $\mathcal{M}$, the Riemannian gradient of $f$ at $\mathbf{X}$, denoted by $\mathrm{grad}f_{\mathcal{M}}(\mathbf{X})$, is defined as the unique element in $T_{\mathbf{X}}{\mathcal{M}}$ that satisfies

Denote $\mathrm{grad}f_{\mathcal{M}}$ as the vector field of the Riemannian gradient on $\mathcal{M}$, whose subscript can be omitted when no confusion will arise. Note that, since $\mathrm{grad}f_{\mathcal{M}}(\mathbf{X})\in T_{\mathbf{X}}{\mathcal{M}}$, we have $\mathrm{grad}f_{\mathcal{N}}(\mathbf{X})\in T_{\mathbf{X}}\mathcal{N}$ and it can then be decomposed as (6).

When second-order derivative optimization algorithms, such as Newton method and trust-region method, are preferred, affine connection is indispensable [22]. Let $\mathfrak{X}(\mathcal{N})$ denote the set of smooth vector fields on $\mathcal{N}$, the affine connection on $\mathcal{N}$ is defined as a mapping $\nabla^{\mathcal{N}}:\mathfrak{X}(\mathcal{N})\times\mathfrak{X}(\mathcal{N})\rightarrow\mathfrak{X}(\mathcal{N}):(\boldsymbol{\eta},\boldsymbol{{\boldsymbol{\xi}}})\mapsto\nabla_{\boldsymbol{\eta}}^{\mathcal{N}}\boldsymbol{{\boldsymbol{\xi}}}$. When Riemannian manifold $\mathcal{M}$ is an embedded submanifold of a vector space, $\nabla_{\boldsymbol{\eta}_{\mathbf{X}}}^{\mathcal{M}}{\boldsymbol{\xi}}$ is called Riemannian connection and given by [22, Proposition 5.3.1]

Given a Riemannian connection $\nabla^{\mathcal{M}}$ defined on $\mathcal{M}$, the Riemannian Hessian of a real valued function $f$ at point $\mathbf{X}$ on $\mathcal{M}$ is the linear mapping $\operatorname{Hess}f(\mathbf{X})\left[\cdot\right]$ of $T_{\mathbf{X}}\mathcal{M}$ into itself defined as

The notion of moving along the tangent vector while remaining on the manifold is generalized by retraction. The retraction $R_{\mathbf{X}}^{\mathcal{N}}\left(\cdot\right)$, a smooth mapping from $T_{\mathbf{X}}\mathcal{N}$ to $\mathcal{N}$ [22, Definition 4.1.1], builds a bridge between the tangent space $T_{\mathbf{X}}\mathcal{N}$ and the manifold $\mathcal{N}$. Practically, the Riemannian exponential mapping forms a retraction but is computationally expensive to implement. For a Riemannian submanifold $\mathcal{M}$, the geometric structure of the manifold is utilized to define the retraction $R_{\mathbf{X}}^{\mathcal{M}}\left(\cdot\right)$ through the projection for affordable and efficient access.

Sometimes, we need to move the tangent vector from the current tangent space to another, which is not always straightforward as the tangent spaces are different in a nonlinear manifold. To address this difficulty, vector transport denoted by $\mathcal{T}_{\boldsymbol{\eta}_{\mathbf{X}}}^{\mathcal{N}}\left({\boldsymbol{\xi}}_{\mathbf{X}}\right)\in T_{R_{\mathbf{X}}\left(\boldsymbol{\eta}_{\mathbf{X}}\right)}\mathcal{N}$ is introduced, which specifies how to transport a tangent vector ${\boldsymbol{\xi}}_{\mathbf{X}}$ from a point $\mathbf{X}$ on $\mathcal{N}$ to another point $R_{\mathbf{X}}\left(\boldsymbol{\eta}_{\mathbf{X}}\right)$. Like $\mathrm{Hess}f(\mathbf{X})\left[\cdot\right]$, $\mathcal{T}_{\boldsymbol{\eta}_{\mathbf{X}}}^{\mathcal{N}}\left(\cdot\right)$ is an operator rather than a matrix. For the Riemannian submanifold, $\mathcal{T}_{\boldsymbol{\eta}_{\mathbf{X}}}^{\mathcal{M}}\left({\boldsymbol{\xi}}_{\mathbf{X}}\right)$ can be achieved by orthogonal projection [22]:

For geometric understanding, Fig. 2 is a simple illustration.

In this section, we first present the WSR-maximization precoder design problems under TPC, PUPC and PAPC, respectively, for massive MIMO DL in Euclidean space. Then, we show that the precoder set satisfying TPC forms a sphere and the precoder set satisfying PUPC or PAPC forms an oblique manifold. Further, we prove that the precoder sets form three different Riemannian submanifolds, from which we transform the constrained problems in Euclidean space to unconstrained ones on these Riemannian submanifolds. Sequentially, we derive Riemannian ingredients, including orthogonal projection, Riemannian gradient, Riemannian Hessian, retraction and vector transport, which are needed for precoder design in matrix manifold framework.