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5932 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 61, NO. 23, DECEMBER 1, 2013

On the Pareto-Optimal Beam Structure and Designfor Multi-User MIMO Interference Channels

Juho Park, Member, IEEE, and Youngchul Sung, Senior Member, IEEE

Abstract—In this paper, the Pareto-optimal beam structurefor multi-user multiple-input multiple-output (MIMO) interfer-ence channels is investigated and a necessary condition for anyPareto-optimal transmit signal covariance matrix is presentedfor the -pair Gaussian MIMO interference channel. It is shownthat any Pareto-optimal transmit signal covariance matrix at atransmitter should have its column space contained in the unionof the signal spaces of the channel matrices from the transmitterto all receivers. Based on this necessary condition, an efficientparameterization for the beam search space is proposed. Theproposed parameterization is given by the product manifold ofa Stiefel manifold and a subset of a hyperplane and enables usto construct an efficient beam design algorithm by exploiting itsrich geometrical structure and existing tools for optimization onStiefel manifolds. Reduction in the beam search space dimensionand computational complexity by the proposed parameterizationand the proposed beam design approach is significant when thenumber of transmit antennas is much larger than the sum of thenumbers of receive antennas, as in upcoming cellular networksadopting massive MIMO technologies. Numerical results validatethe proposed parameterization and the proposed cooperativebeam design method based on the proposed parameterization forMIMO interference channels.

Index Terms—Beamforming, interference channels, mul-tiple-input multiple-output (MIMO), Pareto-optimality, Stiefelmanifolds.

I. INTRODUCTION

M ULTI-USER multiple antenna interference channelshave gained intensive interest from research commu-

nities in recent years because of the significance of properinterference control in current and future wireless networks.One of the break-through results in this area is interferencealignment by Cadambe and Jafar [2], which provides an effec-tive way to achieving maximum degrees-of-freedom (DoF) forMIMO interference channels. However, interference alignment

Manuscript received November 17, 2012; revised April 28, 2013, August 14,2013, and August 25, 2013; accepted September 02, 2013. Date of publicationSeptember 16, 2013; date of current version October 31, 2013. The associateeditor coordinating the review of this manuscript and approving it for publica-tion was Prof. Martin Haardt. This work was supported by the Basic ScienceResearch Program through the National Research Foundation of Korea (NRF)funded by the Ministry of Education, Science and Technology (2010-0021269).J. Park is with the ETRI, Daejeon, 305-700, South Korea (e-mail:

[email protected]).Y. Sung is with the Department of Electrical Engineering, KAIST, Daejeon,

305-701, South Korea (e-mail: [email protected]).This paper has supplementary downloadable multimedia material available

at http://ieeexplore.ieee.org provided by the authors. This includes Matlab filesfor generating the curves in the paper. The material is 44KB in size.Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TSP.2013.2281784

is only DoF optimal, i.e., it is optimal at high signal-to-noiseratio (SNR), whereas in typical cellular networks most receiversexperiencing severe interference are located at cell edges andhence operate in the low or intermediate SNR regime. Theproblem of finding the capacity region and capacity-achievingschemes for general multi-user MIMO interference channels isstill an open problem. To circumvent this difficulty, Jorswiecket al. simplified the multiple antenna interference channelproblem with the practical assumption of single user decodingand investigated the condition and structure of optimal beam-forming schemes [3]. One of the meaningful optimality criteriarelevant to the multi-dimensional achievable rate region ofthe multiple antenna interference channel (with single userdecoding) is given by Pareto-optimality, i.e., achievabilityof ratetuples on the Pareto-boundary of the achievable rateregion. Under this framework, Jorswieck et al. showed formultiple-input single-output (MISO) interference channels thatany Pareto-optimal beam vector at a transmitter is a normalizedconvex combination of the zero-forcing (ZF) beam vector andthe matched-filtering (MF) beam vector in the case of twousers and a linear combination of the channel vectors from thetransmitter to all receivers in the general case of an arbitrarynumber of users [3]. Their result and subsequent results byother researchers provide useful parameterizations for thePareto-optimal beam search space for efficient coordinatedbeam design in MISO interference channels [4]–[9]. However,not many results for the Pareto-optimal beam structure andcondition for MIMO interference channels are available, al-though there exist some results on Pareto-optimality in limitedcircumstances [10]–[13] and available beam design algorithmsspecifically targeting (weighted) sum rate maximization basedon a projected gradient method [14], convexification with locallinear approximation [15], and conversion to weighted min-imum mean squre error (WMMSE) minimization [16]–[18].In this paper, we provide a necessary condition for Pareto-

optimal beamformers for the general -pair Gaussian MIMOinterference channel, where the -th transmitter-receiver pairhas transmit antennas and receive antennas, and showthat any Pareto-optimal transmit signal covariance matrix at atransmitter should have its column space contained in the unionof the signal spaces of the channel matrices from the trans-mitter to all receivers. Based on this, we provide an efficientparameterization for the beam search space not losing Pareto-optimality whose dimension is independent of the number oftransmit antennas and is determined only by ,when the number of transmit antennas at each transmitter islarger than or equal to the sum of the numbers of receive an-tennas at all receivers. The proposed parameterization is givenby the product manifold of a Stiefel manifold and a subset of ahyperplane and enables us to construct an efficient cooperative

1053-587X © 2013 IEEE

PARK AND SUNG: ON THE PARETO-OPTIMAL BEAM STRUCTURE 5933

beam design algorithm by exploiting its rich geometrical struc-ture and existing tools for optimization on Stiefel manifolds. Re-duction in the beam search space dimension and computationalcomplexity by the proposed parameterization and the proposedbeam design algorithm is significant, when forall as in upcoming cellular systems adopting massive MIMOtechnologies [19], [20]. Furthermore, the proposed beam designalgorithm does not need to fix the number of data streams fortransmission beforehand. It finds an (locally) optimal numberof data streams for a given finite SNR and, in the case of mul-tiple data stream transmission, guarantees orthogonality amongthe obtained transmit beam vectors for multiple data streams foreach user. This is beneficial because the optimal number of datastreams is not known for a finite SNR in most cases. (Some partof the paper was presented in ICASSP 2013 [1].)Notations and Organization In this paper, we will make

use of standard notational conventions. Vectors and matricesare written in boldface with matrices in capitals. All vectors arecolumn vectors. For amatrix , , , , and

indicate the Hermitian transpose, 2-norm, trace, rank, anddeterminant of , respectively. For an integer ,denotes the matrix composed of the first columns of .or denotes the element in the -th row and the -th columnof . denotes the column space of and de-notes the orthogonal complement of . denotes theorthogonal projection of a vector onto a linear subspace .

represents the orthogonal projectiononto and . For matrices and ,means that is positive semi-definite. stands for theidentity matrix of size (the subscript is omitted when unnec-essary). or denotes the matrix composed ofvectors and denotes the diagonalmatrix with elements . means that iscircularly-symmetric complex Gaussian-distributed with meanvector and covariance matrix . , , and denote thesets of real numbers, non-negative real numbers, and complexnumbers, respectively. denotes the -dimensional Euclideanspace and denotes the vector space of all complex -tuples.

is the set of all matrices with complex elements.For a complex number , denotes the real part of .The remainder of this paper is organized as follows. The

system model is described in Section II. In Section III, anecessary condition and a parameterization for Pareto-optimaltransmit beamformers for MIMO interference channels areprovided. In Section IV, a beam design algorithm under theobtained parameterization is presented. Numerical results areprovided in Section V, followed by conclusions in Section VI.

II. SYSTEM MODEL

In this paper, we consider a Gaussian interference channelwith transmitter-receiver pairs, where the -th pair hastransmit antennas and receive antennas. We assume that

and for all .Due to interference from the unwanted transmitters, the receivedsignal vector at receiver is given by

(1)

where denotes the channel matrix from transmitterto receiver ; is the transmit signal vector at trans-mitter generated from Gaussian distribution ; andis the additive Gaussian noise vector at receiver with dis-

tribution . Here, the transmit signal covariance matrixat transmitter is chosen among the feasible

set

(2)

where is the transmit power for transmitter , and the rankconstraint is imposed to guarantee that the number of transmitdata streams is at least one and is less than or equal to the pos-sible maximum for transmitter ,

. Note that any number of transmit data streams fromone to the maximum is feasible within the feasible set .Due to the assumption of for

all , the channel matrix is a fat matrix (i.e., thenumber of its columns is larger than or equal to that of its rows)and its singular value decomposition (SVD) is given by

(3)

where is a unitary matrix;is a diagonal matrix composed of the singular values of ;

is a submatrix composed of orthonormalcolumn vectors that span the column space of ; and

is a submatrix composed of or-thonormal column vectors that span the null space of .Thus, and . From here on, we shall

refer to and as the parallel and vertical spaces of(or simply with slight abuse of notation), respectively.

For the purpose of beam design in later sections, we assumethat the channel information is known to all the transmitters.Under the assumption that interference is treated as noise at

each receiver, for a given set of transmit signal covariance ma-trices and a given set of realized channel ma-trices , the rate of the -th transmitter-re-ceiver pair is given by

(4)

for . Then, for the given set of realized channelmatrices, the achievable rate region of the MIMO interferencechannel with interference treated as noise is defined as the unionof rate-tuples that can be achieved by all possible combinationsof transmit covariance matrices:

(5)

The strong Pareto boundary of the rate region is a subset ofthe outer boundary of consisting of rate-tuples for which the


rate of any one user cannot be increased without decreasing therate of at least one other user [3], [5].In the rest of this paper, we shall investigate the Pareto-op-

timal1 transmit beam structure for the -pair Gaussian MIMOinterference channel and develop an efficient beam design algo-rithm based on the obtained Pareto-optimal beam structure.

III. A NECESSARY CONDITION FOR PARETO-OPTIMALITYFOR TRANSMIT BEAMFORMING IN MIMO

INTERFERENCE CHANNELS

In this section, we provide a necessary condition for Pareto-optimal transmit covariance matrices for the -pair GaussianMIMO interference channel by generalizing the MISO resultin [3] to the MIMO case, which reveals the structure of Pareto-optimal transmit beamformers. The necessary condition is givenin the following theorem.Theorem 1: For the -pair Gaussian MIMO interference

channel in which the channel matricesare randomly realized from any non-degenerate2 continuous dis-tribution and interference is treated as noise at each receiver, anyPareto-optimal transmit signal covariance matrix at trans-mitter should satisfy

(6)

and

(7)

Proof: Proof is by contraction. That is, we first supposethat a set of transmit signal covariance matrices that does notsatisfy (6) and (7) is Pareto-optimal and this set of transmitsignal covariance matrices yields a certain ratetuple. Then, weshow that there exists another set of transmit signal covariancematrices that yields a strictly better ratetuple than the previousset of transmit signal covariance matrices assumed to yield aPareto-optimal ratetuple. This contradicts to the assumption.See Appendix A for details.Equation (6) in Theorem 1 states that the column space of

any Pareto-optimal transmit signal covariance matrix at trans-mitter should be contained in the union of the parallel spacesof the channels from transmitter to all receivers. Although arigorous proof of Theorem 1 is not trivial, what the theoremstates is simple; the transmit beamforming matrix should be de-signed so that the transmitted signal affects the received signalat the desired receiver and/or the received interference at theundesired receivers, and the power should not be wasted into adimension that does not have any impact on the signal or the in-terference. In the case that for all , the par-allel space is simply the 1-dimensional linear subspace spannedby the matched filtering vector. Thus, the result in Theorem 1can be regarded as a generalization of the result in the MISOinterference channel by Jorswieck et al. [3] to general -pairMIMO interference channels.

1Pareto-optimality in this paper means strong Pareto-optimality.2Here, by non-degeneracy we mean that and

are linearly independent for all if, almost surely.

A. The Symmetric 2-User Case

In this subsection, we consider the symmetric two-user caseand present another representation for Pareto-optimal transmitsignal covariance matrices in this case.Corollary 1: In the two-user case in which the number of

receive antennas is the same andfor , any Pareto-optimal transmit signal covariance

matrix at transmitter 1 should satisfy

(8)

and , where

.Proof: See Appendix B

As in the MISO case [3], the Pareto-optimal beam spaceis contained in the union of the self-parallel space ofand the ZF or null space of the channel

to the other user in the two-user symmetric MIMO case.

B. Parameterization for the Pareto-Optimal Beam Structurein MIMO Interference Channels

Theorem 1 provides a necessary condition for Pareto-optimaltransmit signal covariance matrices for the -pair GaussianMIMO interference channel with interference treated as noise.Based on Theorem 1, in this section, we develop a concreteparameterization for Pareto-optimal transmit signal covari-ance matrices for the -pair Gaussian MIMO interferencechannel for construction of an efficient beam design algorithmin the next section. Here, we mainly focus on the case of

, although the parameterization result here canbe applied to the case of .Since ,

any Pareto-optimal transmit signal covariance matrix attransmitter can be expressed as

(9)

where is a positive semi-defi-nite matrix with rank less than or equal to , and is a

matrix composed of orthonormal columnsthat span and obtained by the thin QRfactorization of :

(10)

where is a upper triangular matrix.(Note that is a matrix and ithas full column rank almost surely for randomly realized chan-nels from a non-degenerate distribution.3) Since is Hermi-tian, i.e., self-adjoint, by the spectral theorem, it has the spectraldecomposition given by

(11)

where is a matrix with orthonormal columns,i.e., and is adiagonal matrix with nonnegative elements, i.e., for all

3The full column rank assumption is not necessary. In fact, the complexity ofthe beam design problem is reduced when the matrix does not have full columnrank. This step will be explained in Algorithm I in Section IV.


. Thus, any Pareto-optimal transmit signal covariance matrixat transmitter is expressed as

(12)

which is a spectral decomposition of since. Note here that is known to

the transmitter under the assumption of known channel infor-mation and fixed for a given set of realized channel matrices

. Note also that (12) incorporates thecondition (6) of Theorem 1 only. In the case of ,we have the full transmission power condition (7) additionally.Applying this full power constraint to (12), we have

(13)

where for all and (a) holds by the fact that. Thus, any Pareto-optimal transmit

signal covariance matrix can be parameterized by the two ma-trices and with constraints and ,respectively. Especially, ’s satisfying form a

special subset of called the Stiefel manifold.

Definition 1 (Stiefel Manifold [21]) : The (compact) Stiefelmanifold (or ) is the set of all complex ma-trices with orthonormal columns, i.e.,

(14)

Note that is a vector space over with the normalmatrix addition and the scalar multiplication as vector additionand scalar multiplication. The Stiefel manifold is a sub-manifold of the vector space [21]. Now, we present ourparameterization result for Pareto-optimal beamforming in the-pair MIMO interference channel when in the

following theorem.Theorem 2: Any Pareto-optimal transmit signal covariance

matrix at transmitter for the -pair Gaussian MIMO inter-ference channel with is parameterized4 by theproduct manifold :

(15)

where is the Stiefel manifold of orthonormal

-frames in and is a subset in the first quadrantof a hyperplane in the Euclidean space defined by

(16)

Proof: Combining Theorem 1 and (9), (11), (12) and (13),we have the result.Note that is an embedded manifold within the original

high dimensional space . The main advantage of the param-

4The parameterization is based on a necessary condition. Hence, not all pointsin correspond to Pareto-optimal points, but the parameterization capturesall Pareto-optimal points.

eterization in Theorem 2 is that the dimension of the param-eter space (or beam search space) not losing Pareto-optimalitydoes not depend on the number of transmit antennas when

and the proposed parameterization significantlyreduces the dimension of the beam search space as compared tothe original search space , when . Thus, theproposed parameterization can be useful for upcoming cellulardownlink cooperative transmission with massive MIMO tech-nologies [19], [20] in which large-scale transmit antenna arraysare adopted at basestations while each mobile terminal still hasa limited number of receive antennas. The exact dimension ofthe parameter space for transmitter is given by

(17)

This is because the dimension of is given by [21,p.27, ] and the dimension of in is given by . In ad-dition to the independence of the parameter space dimension onthe number of transmit antennas, the parameterization in The-orem 2 enables us to exploit the rich geometrical structure ofStiefel manifolds and hyperplanes for optimal search for beamdesign. This will become clear shortly in the next section.Now, consider the case that . In this case, The-

orem 1 is not so helpful, but a parameterization similar to thatin Theorem 2 can be obtained by directly applying spectral de-composition to with rank less than or equal to

. The spectral decomposition of in this case isgiven by

(18)

where is a matrix with orthonormal columns, i.e.,and is a positive semi-definite

diagonal matrix. Thus, the parameter space is given by, where is a subset of a half space of ,

defined as.

IV. THE PROPOSED BEAM DESIGN ALGORITHM

In this section, we provide an efficient beam design algorithmunder the parameterization in (15) containing all Pareto-op-timal beamformers by exploiting the geometric structure of theparameter space. Here, we consider a centralized beam designapproach under the assumption that all channel information isavailable. For example, in cellular systems, all channel infor-mation from cooperating basestations can be delivered to thebasestation combiner (BSC), and the BSC can compute beam-forming matrices for all the basestations under its control andinform the computed beamforming matrices to the basestationsunder its control. When fast communication between the BSCand the basestations is available, such a method can be used inpractice.

A. The Overall Algorithm Structure: A Utility Function-BasedApproach

Our approach to beam design is based on the utility functionbased method in [9], [22], [23]. In this approach, we define autility function :

(19)


The utility function is chosen to represent the desired systemperformance metric. We assume that is a bounded smoothfunction of . Examples of useful utility functionssatisfying the above condition include [23]

(20)

In addition to the mapping (19), due to Theorem 2, we have thefollowing mapping:

(21)

which is determined by the rate formula (4) andin (12). Here, we only need to consideras our beam search space owing to Theorem 2.

The composition of the two mappings is given by

(22)

Note that this mapping is the desired mapping from the beamsearch space containing all Pareto-optimal beams to the set ofutility values and that is a smooth function on the product man-ifold by the smoothness assumption on andthe smoothness of the rates as functions of .Then, the utility-maximizing beam design problem is formu-lated as

(23)

where is given by (15). Although simultaneous optimiza-tion of to maximize the utility functionis difficult, the optimization (23) can efficiently be solved byan alternating optimization technique. That is, we fix all other

except and update the unfixed pa-rameters in order that the utility function is maximized.After this update, the next is picked for update. Thisprocedure continues until it converges. The proposed overall al-gorithm is described below.

Algorithm 1: The Proposed Beam Design Algorithm—TheOverall Structure

Requirements:

• Channel information

• Maximum available transmit power

• Utility function

• Stopping tolerance

Preprocessing:

• Obtain by QR factorization ofas in (10) for all .

• In the above QR factorization step, the rank of isrevealed. Based on the revealed rank5 , set the numberof rows of as and set the number of its columns

as . In this step, the proper Stiefel manifold for isidentified and it is .

Iteration:

Initialization:

•

•Initialize and properly6 for all

while

for

(24)

end for

end while

Postprocessing

• Check the rank of to determine the number ofdata streams for transmitter .

• Construct a beamformer matrix for transmitter as

(25)where .

• At transmitter , generate zero-mean unit-power i.i.d.data streams and construct the data vector withthe generated data streams. Then, construct the signalvector and transmit through antennas. Then,the signal vector has the desired signal covariancematrix . Typically, i.i.d. data streams are fromindependent channel encoders.

There are several interesting features about the proposed beamdesign algorithm.• First, it is not necessary to predetermine the number of

data streams for the algorithm. Although there exist someasymptotic results on optimal DoF at high SNR [2], the optimalnumber of independent data streams for transmission is notknown for finite SNR in most cases except the known factthat the maximum number of data streams for transmitter is. Our parameterization for the beam search space includes

all possible numbers of data streams less than or equal to .Thus, if the algorithm converges at least to a local optimum,the algorithm will find the (at least locally) optimal number ofdata streams for given SNR automatically. When the maximumnumber of data streams is not optimal, the algorithm wouldreturn not in the interior of but on theboundary such as a corner or an edge of . Furthermore,in case of multiple data stream transmission for transmitter, perfect orthogonality among the transmit beam vectors formultiple streams of transmitter is guaranteed and power isoptimally allocated to these multiple orthogonal streams since

has orthonormal columns and these


columns are properly scaled by .In previous beam design algorithms for MIMO interferencechannels such as [24], the number of beam vectors for each useris fixed beforehand with an orthonormal condition on multiplebeam vectors of each user [24].• Any transmit signal covariance matrix can be imple-

mented by a beamforming matrix as in (25).• Due to the non-convexity of utility functions with respect to

(w.r.t.) (note the rate formula (4)), the con-vergence of the proposed algorithm to the global optimum is notguaranteed, but the proposed algorithm converges to a locallyoptimal point by the monotone convergence theorem [25, p. 45]if the step (24) returns a non-decreasing value compared to thecurrent one at each iteration. This is because the utility functionis upper bounded and the proposed algorithm yields a monoton-ically non-decreasing sequence of utility function values underthe assumption of proper operation of the step (24). Further-more, in this case the proposed algorithm is stable since it mono-tonically converges.Thus, an efficient and successful implementation of the step

(24) is critical to the overall beam design algorithm. Such an im-plementation is possible and available because of the geometryof our parameterization . The problem (24) involves opti-mization on a Stiefel manifold, which is well established [21],[26]. In the next subsections, we briefly introduce some basicfacts about Stiefel manifolds and then present our algorithm im-plementing (24) based on the steepest descent method or theNewton method on Stiefel manifolds of Edelman et al. [26].

B. Preliminaries: Riemannian Geometry on Stiefel Manifolds

Since geometry of hyperplanes or half spaces is simple, wehere provide some basic facts about the Stiefel manifoldthat are necessary to understand the subalgorithm implementingthe step (24). For a detailed explanation of the Stiefel manifoldand its geometry, please refer to [21], [26]. For general Rie-mannian geometry, please refer to [27], [28].Tangent spaces:[26] The tangent space at a pointis given by

(26)

where . That is, a tangent vector at isa matrix s.t. is skew-Hermitian.The canonical metric:[26] For two tangent vectors andin , the canonical metric is defined as

(27)

Geodesics: A geodesic on a manifold is a curve on the man-ifold whose velocity vector field is constant along the curvew.r.t. a given affine connection [28]. A geodesic formula for theStiefel manifold w.r.t. the Levi-Civita connection is givenby the following theorem by Edelman et al. :Theorem 3 (Edelman et al. [26]): Let be a point in

and be a tangent vector in . Then, the geodesic on the

5If is not of full column rank, the problem size simply reduces.6See Appendix C for details.

Stiefel manifold emanating from in the direction is givenby the curve

(28)

where

(29)

is the thin QR decomposition of with beingand being , and and are matrices

given by the following matrix exponential

(30)

where .Gradient: [26] For a smooth function on the Stiefel mani-

fold, i.e., , the gradient of at w.r.t. the canon-ical metric is defined as the tangent vector gradsatisfying for all tangent vec-tors at , where is the matrix composed of partialderivatives of w.r.t. the elements of , i.e., .The gradient of at is given by

(31)

Hessian: [21] For a general Riemannian manifold , theHessian operator of a smooth function at a pointis defined as a linear operator: with

for all , where is the Levi-Civita connection on . Just as in the Euclidean case, a smoothfunction on admits Taylor expansion. Let , whereis a retraction.7 Then, in a neighborhood of , we have

(32)

Thus, the stationary point of the RHS of (32) satisfies theNewton equation:

(33)

The Hessian operator can be computed for complex Stiefel man-ifolds as well as real Stiefel manifolds. For detail, please referto [26] and [29]. A simple way to handle the complex case is toconvert the complex matrix to a real matrix with doubled size,to express the cost function in terms of the converted real matrixas in [30], and to use the Hessian operator for the real case in[26].

C. The Subalgorithm: Steepest Descent or Newton Method onthe Product Manifold

Notice that the cost function in (24) is a smooth mappingfrom to when

are fixed. By exploiting the product structure of the parameterspace, the optimization problem (24) can be solved by an alter-nating technique again in a similar way to the technique in [31].That is, first we fix and update by the steepest descent orNewton method on the Stiefel manifold [21], [26].

7A retraction is a smooth mapping from to with andis an identity map, where is the differential of . The exponential

map is an example of retraction. [21]


Next, we fix and update by the steepest descent or Newtonmethod on . We continue this iteration until we have satis-factory convergence. The subalgorithm implementing the step(24) is given below.

Algorithm 2: The Subalgorithm for (24)

Requirements:

• Cost function . Set .

• Step sizes and

• Stopping tolerance

Initialization:

•

•

while

step

Fix . Given the current ,

1. Compute the movement direction vector.

* For the steepest descent method, in (31).

* For the Newton method, compute as in [26], [29].

2. Move from to , whereis the exponential map at . That is,

move from in direction to in (28) along thegeodesic given by Theorem 3. Then, .

step

Fix . Given the current ,

1. Compute the movement direction vector .

* For the steepest descent method, compute the gradientvector of at , and .

* For the Newton method, compute the Hessian matrixof at , and

2. Obtain the projection of to thetangent space .

3. Move from to the direction on . That is,

(34)

end while

•

The step 2 in the step is to maximize the utility withthe constraint that the points still stay in the Stiefel manifold

. Note that for the step, is a conventional

multi-variable scalar function, i.e., it is .Thus, the ordinary gradient vector and the ordinary Hessianmatrix for a function defined on a Euclidean space are valid.Furthermore, the step is simple since a hyperplane is flat andthus its geometry is induced by projection from its embedding

Euclidean space. In (34), is still onthe hyperplane containing but it may be outside(i.e., not in the first quadrant). Projection back to can bedone by simple scaling of after checking thecoordinate values of . That is, if thereexists a negative value at some coordinate, isscaled down and then added to so that the value at thatcoordinate becomes zero.An attracting aspect of the steepest descent method on the

Stiefel manifolds is that its local convergence with a proper stepsize selection rule such as the Armijo rule [32] is established forcontinuously differentiable cost functions [21, Theorem 4.3.1].That is, the steepest descent method with a proper step size se-lection rule converges to a stationary point reachable from aninitialization point. Thus, Algorithm 2 with the steepest descentmethod with a proper step size selection rule returns an im-proved point than the current point since the current point is usedas the initialization point. Due to this monotone property of thesubalgorithm, the overall algorithm, Algorithm 1, converges toa locally optimal point at least. Furthermore, the complexity ofthe subalgorithm is not prohibitive. Formulas for andcan be precomputed and stored for typical utility functions. Thematrix exponential in (30) involves a matrix with small size

. There exist even simpler alternative ways togenerating a curve with a given tangent vector other than thegeodesic [21], [33]. The subalgorithm presented here is onlyone example among many possible implementations for opti-mization on Stiefel manifolds and a variety of different methodsare available to compromise complexity and performance [21],[29].In [14], an algorithm based on a projected gradient method

[34] was proposed for the problem of sum rate maximizationin MIMO interference channels by applying gradient and pro-jection operations directly to the signal covariance matrix .However, the method does not exploit any necessary conditionfor Pareto-optimality for and the resulting dimension reduc-tion. In the proposed algorithm here, gradient descent for a gen-eral utility function is performed in a more efficient way on amanifold with reduced dimensions, especially when the numberof transmit antenna is large.

D. A Design Example: Weighted Sum Rate Maximization

The proposed beam design method based on the proposed pa-rameterization can be applied to any differentiable utility func-tions. In this subsection, we provide a specific example for theproposed beam design method. Here, we consider the coopera-tive beam design for commonly considered weighted sum ratemaximization based on the steepest descent on the product man-ifold . The weighted sum rate maximizationproblem is formulated as

(35)

where ,

, and isthe set of weighting factors. To compute the gradient of the ob-jective function with respect to and , we manipulate the


rate formula of receiver as follows. First, consider the case of.

(36)

Thus, the (Wirtinger) derivative of w.r.t. for isgiven by [35]

(37)

For the gradient of w.r.t. , we only need to consider thediagonal elements of since the off-diag-onal elements are fixed to zero. In (36), define .In the case of , from (36), the gradient of w.r.t. for

is obtained as

(38)

where is the -th column of and the second equalityis from [35]. Therefore,

(39)

Next, consider the derivative of with respect to ( ).In this case, we have

(40)

From (40), the derivatives of w.r.t. and are respec-tively given by (41) and (42) at the bottom of the page, for

. Note that the most complicated computation for ob-taining the derivatives and is inversion of

matrices. Finally, the derivatives of the overall costfunction w.r.t. and in the case of weighted sum rate max-imization are given respectively by

(43)

and

(44)

(41)

(42)


Fig. 1. CCDF of the number of iterations required for the outer loop algorithm,Algorithm 1, to converge : (a) , ,(b) , , and (c) ,

( for all ).

for . With the obtained derivatives, Algorithm 1with the subalgorithm, Algorithm 2, can be run. For an arbitrarydifferentiable utility function , the derivative ofthe utility function w.r.t. and can easily be obtained byusing ((37) and (41)) and ((39) and (42))and the chain rule of differentiation.

V. NUMERICAL RESULTS

In this section, we provide some numerical results to vali-date our beam design method based on the parameterization

for the beam search space for MIMOinterference channels. To validate the proposed method, weconsidered the sum rate maximization problem discussed inSection IV-D and used the overall algorithm, Algorithm 1,equipped with the steepest descent subalgorithm, Algorithm2, with the Armijo step size selection rule [32]. Throughoutsimulations, we set for simplicity;each element of the channel matrix was generated i.i.d.according to ; and the additive noise variance wasone. (Hence, (or simply ) is SNR.)First, we examined the convergence of the overall al-

gorithm equipped with the steepest descent subalgorithmwith the Armijo step size rule. Fig. 1 shows the comple-mentary cumulative distribution function (CCDF) of thenumber of iterations for Algorithm 1 to converge in threecases: Case (a) , , Case(b) , , and Case (c) ,

, where dB for allin each case. The CCDF was obtained based on 500 randomchannel realizations in each case. Convergence of the innerloop (Algorithm 2) and the outer loop (Algorithm 1) was de-clared when the rate difference was below and (i.e.,

and ), respectively. At transmitter , initialand for the outer loop were chosen to correspond

to the simple ZF beamformer. It is seen in Fig. 1 that whenand , Algrithm 1 converges

in most cases within 5 iterations, whereas for the other twocases, roughly 10 iterations are required for the algorithm to

TABLE ITHE AVERAGE NUMBER OF ITERATIONS FOR ALGORITHM 2 TO CONVERGE( ): (A) , , (B) ,

, (C) ,

converge for most channel realizations. The average numbersof iterations for Algorithm 1 to converge were 2.24, 3.15, and3.00 in cases (a), (b), and (c), respectively.Next, we examined the convergence behavior of the inner

loop, Algorithm 2, based on the steepest descent with the Armijostep size rule. Fig. 2 shows the CCDF of the required number ofiterations for Algorithm 2 to converge for the three cases shownin Fig. 1. Since the average number of iterations for the outerloop to converge is roughly three, we plotted three CCDF curvesin each sub-figure corresponding to the first three calls of Algo-rithm 2 by the outer loop. It is seen that as the iteration numberof the outer loop increases, the required number of iterations forthe inner loop to converge becomesmuch smaller. Table I showsthe average number of iterations requried for the inner loop toconverge.Fig. 3 shows the learning curves of the proposed algorithm

for three different channel realizations in the case of ,with dB for all

corresponding to case (c) in Figs. 1 and 2. Fig. 3(a) showsthe learning curve of the outer loop of Algorithm 1. As we ex-pect from the CCDF curve in Fig. 1, the outer loop convergeswithin a few iterations. Figs. 3(b) and 3(c) respectively showthe learning curves of the inner loop of Algorithm 2 when it iscalled in the first and second iteration of the outer loop of Al-gorithm 1. It is seen that the inner loop, Algorithm 2, yields amonotone increasing sequence, as expected, since we used theArmijo rule in the steepest gradient method, which guaranteesthe monotone increasing behavior.With convergence of the proposed algorithm confirmed, we

examined the weighted sum rate performance of the proposedbeam design algorithm. Fig. 4 shows the rate-tuples of severalbeam design methods for a random channel realization in thecase of , for two different powersettings dB and dB for all . We consid-ered the single-user eigen-beamforming, ZF beamforming (i.e.,channel inversion), and regularized ZF (i.e., regularized channelinversion) in addition to the proposed beam design method. Wesolved the weighted sum rate maximization problem with theproposed beam design algorithm with varying weights so thatwe can obtain rate-tuples at different locations of the rate re-gion. As expected, there is a considerable gap between simplebeamforming schemes and the proposed scheme in the low SNRcase of dB for all , and the ZF and regularized ZFbeamformers catch up the proposed design scheme in the highSNR case.Fig. 5 shows the sum rate performance of the proposed

method w.r.t. transmit power for three different system pa-rameter settings: (a) , ,(b) , , and (c) ,

. For each setting, one set ofchannel matrices was randomly realized. The table in Fig. 5


Fig. 2. CCDF of the number of iterations required for the inner loop, Algorithm 2, to converge ( ): (a) , , (b) ,, and (c) , .

Fig. 3. Learning curve of the proposed algorithm for , and for : (a) learning curve of the outerloop, (b) learning curve of the inner loop in the first call by the outer loop, and (c) learning curve of the inner loop in the second call by the outer loop (curves stopwhen convergence reached for ).

summarizes the corresponding obtained rank of the converged, , by the proposed beam design method. Note

that in the low SNR regime indeed the proposed beam designmethod does not yield a beamformer with the maximum avail-able number of data streams, , for all . Futhermore, it tellswhich transmitter should not use the available (single-user)maximum number of data streams for sum rate maximization.It is expected that at low SNR the optimal strategy does not usethe maximum number of data streams since all power can beallocated in the best direction, as in the single-user MIMO case.Due to the separate parameterization for and in the beamsearch space , the proposed method can clearly identifythe optimal rank of the beamforming matrix by checking thediagonal elements of .Finally, we compared the proposed beam design method

with other beam design methods available for sum rate max-imization since other methods exist in the case of sum ratemaximization in MIMO interference channels [16], [18],[36]. Fig. 6 shows the sum rate performance of the proposedalgorithm and other beamformer design methods providedin [16], [18], [36] averaged over 50 channel realizations forthree different cases: (a) , ,(b) , , and (c) ,

. The performance of simpleeigen-beamforming, ZF (i.e., channel inversion) beaforming,and regularized ZF (i.e., regularized channel inversion)

[37] is also included. The initialization point for the proposedalgorithm and the other methods was chosen as the eigen-beam-forming when is under 5 dB and the ZF beamformingotherwise. It is seen in Fig. 6 that the performance of the pro-posed method and that of the other methods [16], [18], [36] arealmost identical and the advanced algorithms perform equallywell in the sum rate maximization case. This validates the pro-posed design approach based on the proposed parameterization.Furthermore, we considered the MISO case in which there existseveral algorithms that yield globally optimal beamformingvectors, e.g., [23], and compared the sum rate performanceof the proposed algorithm to that of the algorithm in [23] inthe MISO case. Fig. 7 shows the corresponding result. It isseen that the performance of the proposed algorithm is almostidentical to that of the globally-optimal algorithm in [23].

VI. CONCLUSION

We have considered the Pareto-optimal beam structure formulti-user MIMO interference channels and have provided anecessary condition for any Pareto-optimal transmit signal co-variance matrix for the -pair Gaussian MIMO interferencechannel. We have shown that any Pareto-optimal transmit signalcovariance matrix at a transmitter should have its column space


Fig. 4. Rate pairs of several beam design methods for ,: (a) dB and

(b) .

contained in the union of the signal spaces of the channel ma-trices from the transmitter to all receivers. Based on this neces-sary condition, we have proposed an efficient parameterizationfor the beam search space, given by the product manifold of aStiefel manifold and a subset of a hyperplane. Based on the pro-posed parameterization, we have developed an efficient beamdesign algorithm for maximizing any differentiable utility func-tion by exploiting the geometrical structure of the beam searchspace and existing tools for optimization on Stiefel manifolds.Numerical results validate the proposed beam design approachbased on the proposed parameterization for the beam searchspace.

APPENDIX APROOF OF THEOREM 1

Proof: Proof is by contradiction. First, we considerthe case that . Suppose that the matrix

has rank .8

Then, there exists an orthonormal basis thatspans , i.e.,

. Now, suppose that a set of covariance ma-trices is strongly Pareto-optimal and that

8When , the condition (6) is trivially satisfied.

Fig. 5. Performance of the proposed method: (a) weighted sum rate with re-spect to transmit power and (b) the obtained number of data streams ( in

in the table denotes the obtained rank of for transmitter-receiverpair —(1) , , (2) ,

, and (3) , ).

at transmitter . Then, we canexpress as

where , , and . Here,implies that for some

. Let be such an index and let

(45)

with . Then, andis positive semi-definite.9 Thus, is a valid transmit signal

covariance matrix. Now consider the rate-tuple that is achievedby . Let the interference covariance ma-trix at receiver be denoted by

(46)

Then, with the new set of transmit signal covariance matrices,the rate of the -th transmitter-receiver pair is given by

9The positive semi-definiteness of can be shown as in [38]. First,by

the definition of and . For any vector orthogonal to ,we have by thepositive semi-definiteness of . Since any vector in is contained in

. The claim follows.


Fig. 6. Sum rate performance of the proposed algorithm and other methodsin [16], [18], [36] averaged over 50 channel realizations: (a) ,

, (b) , , and (c) ,.

(47)

Fig. 7. Sum rate performance (averaged over 50 channel realizations) of theproposed algorithm and the globally-optimal beam design algorithm in [23] ina MISO case: , (A ZF initialization is usedwhen SNR and the MF initialization is used otherwise).

where step (a) holds because andhence . Similarly, the rate of the -th transmitter-receiver pair ( ) is given by

(48)

where step holds again becauseand hence . Therefore, the rate-tuple doesnot change by replacing with

.Now, construct another transmit signal covariance matrix

as

(49)

where satisfies while for all . Suchexists almost surely in under the assump-

tion of non-degenerate random channel realization, because theevent has measure zero.10 Here, is

10The dimension of is at most which is strictly less

than by the assumption . Under the non-degenerate channeldistribution assumption, the probability that a randomly realized subspace

of is contained in another randomly realized subspace

of with dimension strictly less than is zero.


chosen so that (this is possible since

. See (45).) and

(50)Thus, is a valid transmit signal covariance matrix. Now con-sider the rate-tuple that is achieved by .Here, we define

(51)

Then, the rate of the -th transmitter-receiver pair receiver () is given by

(52)

where step holds by the construction of and step (d) holdsby (48). On the other hand, the rate of the -th transmitter-re-ceiver pair with is given by

(53)

where step holds by , stepholds by Lemma 1, and step (g) holds by (47). This con-

tradicts our assumption that the set oftransmit signal covariance matrices is Pareto-optimal. There-fore, we have

Next, suppose that but. Then, by the same argument as before, there

almost surely exists such that and forall , when . Let

(54)

where is chosen to be so that

. Then, the rate of the -th transmitter-receiver pair( ) does not change by the same argument as in (52) and therate of the -th transmitter-receiver pair strictly increases by thesame argument as in (53). Thus, in the case of ,each transmitter should use full power for Pareto optimality.Now, consider the case of . In this

case, for randomly realizedchannel matrices under the non-degenerate channel dis-tribution assumption and (6) is trivially true. Finally,

by the defini-tion of . (See (3).)

Lemma 1: Under the same conditions as in Theorem 1, wehave

(55)

where such that and for all

, and .

Proof: Since andare positive definite matrices and

we have [39, p.471, ]

(56)

where be the -th largest eigenvalue ofand be the -th largest eigenvalue of. Next, consider the difference of the traces of

the two matrices:

(57)

where the last inequality holds by the construction of satis-fying . By (56), (57) and the fact that the trace ofa matrix is the sum of its eigenvalues, there exists at least oneeigenvalue that is strictly larger than . Therefore, we have

APPENDIX BPROOF OF COROLLARY 1

Proof: The proof is given by showing the equivalence ofthe two subspaces:

(58)

Any vector in of the right-hand side (RHS) of(58) can be expressed as

(59)


for some , whereas any vector inof the left-hand side (LHS)

of (58) can be expressed as

(60)

for some . Equation (60) can be rewritten as

(61)

Furthermore, is invertible almost surely.11

Thus, there exists an isomorphism between andgiven by

(62)

(63)

to satisfy

(64)

Thus, the two subspaces are equivalent, i.e.,. Since

by Theorem 1, the claimfollows.

APPENDIX CINITIALIZATION OF ALGORITHM 1

Initial and should satisfy the orthonormality andpower constraints, i.e., and ,respectively. Since the performance of the proposed algorithmdepends on the initial point, one can choose some smart ini-tial point such as the zero-forcing solution or the eigen-beam-forming (matched filtering) solution (composed of the right sin-gular vectors corresponding to the non-zero singular values ofthe desired channel ), based on the operating SNR. Onewould expect that the ZF beamforming is nearly optimal at highSNR and the eigen-beamforming is nearly optimal at low SNR.Of course, one can choose the initial point randomly under thetwo constraints. Here we briefly explain how to set and

to generate such well-known initial points. As in Section II,we assume that all the channel matrices have full rank and

for all . First, note that the transmitsignal covariance matrix at transmitter is expressed as

(65)

where is defined in (10).

11 and are the parallel spaces of and , respectively. Theevent that is non-invertible requires that is con-tained in a strict subspace of with dimension less than determined by

. Such an event has measure zero for randomly realized channel matricesunder the non-degenerate channel distribution assumption.

1) MF initialization: From the SVD of the desired channel, (3), and from (65), the MF initialization can be made by

setting as

(66)

Note that in (66) satisfies the orthonormality constraint,i.e., .This is because is a projectionmatrix ontoand thus, .The initial power loading matrix

can be set by for allor by the solution of the single-user water-filling

algorithm for in (66).2) ZF initialization (ZF is feasible only when

): A zero-forcing initialization can be made similarly.First, we obtain the thin QR decomposition as follows:

(67)Since , we have

(68)

where . Here, sinceboth and have orthonormal columns (

). From the property of the QR factor-ization, we have

(69)

where . Thus, by choosing

(70)

we have

(71)

Here, denotes the submatrix of composed ofthe columns from the -th column to the -th column. Theinitial power loading matrix can be set as in the previous case.3) Random beam: Provided , generate as a random

matrix with orthonormal columns. One easy way to obtainingsuch a random matrix is by the SVD of a random ma-trix with appropriate size. can be generated randomly bychoosing an interior point of a half-space given by.


REFERENCES[1] J. Park and Y. Sung, “An efficient parameterization for Pareto-optimal

beamformers for -user MIMO interference channels,” in Proc.ICASSP, Vancouver, Canada, May 2013.

[2] V. R. Cadambe and S. A. Jafar, “Interference alignment and degrees offreedom of the -user interference channel,” IEEE Trans. Inf. Theory,vol. 54, pp. 3425–3441, Aug. 2008.

[3] E. Jorswieck, E. Larsson, and D. Danev, “Complete characterization ofthe Pareto boundary for the MISO interference channel,” IEEE Trans.Signal Process., vol. 56, pp. 5292–5296, Oct. 2008.

[4] E. Björnson, R. Zakhour, D. Gesbert, and B. Ottersten, “Cooperatemulticell precoding: Rate region characterization and distributedstrategies with instantaneous and statistical CSI,” IEEE Trans. SignalProcess., vol. 58, pp. 4298–4310, Aug. 2010.

[5] R. Zhang and S. Cui, “Cooperative interference management withMISO beamforming,” IEEE Trans. Signal Process., vol. 58, pp.5450–5458, Oct. 2010.

[6] R. Zakhour and D. Gesbert, “Distributed multicell-MISO precodingusing the layered virtual SINR framework,” IEEE Trans. WirelessCommun., vol. 9, pp. 2444–2448, Aug. 2010.

[7] R. Mochaourab and E. Jorswieck, “Optimal beamforming in interfer-ence networks with perfect local channel information,” IEEE Trans.Signal Process., vol. 59, pp. 1128–1141, Mar. 2011.

[8] X. Shang, B. Chen, and H. V. Poor, “Multiuser MISO interferencechannels with single-user detection: Optimality of beamforming andthe achievable rate region,” IEEE Trans. Inf. Theory, vol. 57, pp.4255–4273, Jul. 2011.

[9] J. Park, G. Lee, Y. Sung, and M. Yukawa, “Coordinated beamformingwith relaxed zero forcing: The sequential orthogonal projection com-bining method and rate control,” IEEE Trans. Signal Process., vol. 61,pp. 3100–3112, Jun. 2013.

[10] E. Björnson, M. Bengtsson, and B. Ottersten, “Pareto characterizationof the multicell MIMO performance region with simple receivers,”IEEE Trans. Signal Process., vol. 60, pp. 4464–4469, Aug. 2012.

[11] Z. Chen, S. A. Vorobyov, C.-X. Wang, and J. Thompson, “Paretoregion characterization for rate control in MIMO interference systemsand Nash bargaining,” IEEE Trans. Autom. Control, vol. 57, pp.3203–3208, Dec. 2012.

[12] P. Cao, E. Jorswieck, and S. Shi, “Pareto boundary of the rate regionfor single-stream MIMO interference channels: Linear transceiver de-sign,” IEEE. Trans. Signal Process., 2013.

[13] R. Mochaourab, P. Cao, and E. Jorswieck, “Alternating rate profileoptimization in single stream MIMO interference channels,” in Proc.ICASSP, Vancouver, Canada, May 2013, pp. 2104–2108.

[14] S. Ye and R. S. Blum, “Optimized signaling for MIMO interferencesystems with feedback,” IEEE Trans. Signal Process., vol. 51, pp.2839–2848, Nov. 2003.

[15] S.-J. Kim and G. B. giannakis, “Optimal resource allocation for MIMOad hoc cognitive radio networks,” IEEE Trans. Inf. Theory, vol. 57, pp.3117–3131, May 2011.

[16] Q. Shi, M. Razaviyayn, Z.-Q. Luo, and C. He, “An iteratively weightedMMSE approach to distributed sum-utility maximization for a MIMOinterfering broadcast channel,” IEEE Trans. Signal Process., vol. 59,pp. 4331–4340, Sep. 2011.

[17] T. Bogale and L. Vandendorpe, “Weighted sum rate optimization fordownlink multiuser MIMO coordinated base station systems: Central-ized and distributed algorithms,” IEEE Trans. Signal Process., vol. 60,pp. 1876–1889, Apr. 2012.

[18] J. Kaleva, A. Tolli, andM. Juntti, “Weighted sum rate maximization forinterfering broadcast channel via successive convex approximation,” inProc. Globecom, Anaheim, CA, USA, Dec. 2012.

[19] T. L. Marzetta, “Noncooperative celular wireless with unlimitednumber of base station antennas,” IEEE Trans. Wireless Commun.,vol. 9, pp. 3590–3600, Nov. 2010.

[20] F. Rusek, D. Persson, B. K. Lau, E. G. Larsson, T. L. Marzetta, O.Edfor, and F. Tufvesson, “Scaling up MIMO: Opportunities and chal-lenges with very large arrays,” IEEE Signal Process. Mag., vol. 30, pp.40–60, Jan. 2012.

[21] R. A. Absil, R. Mahony, and R. Sepulchre, Optimization Algorithms onMatrix Manifolds. Princeton, NJ, USA: Princeton Univ. Press, 2007.

[22] E. Jorswieck and E. Larsson, “The MISO interference channel from agame-theoretic perspective: A combination of selfishness and altruismachieves Pareto optimality,” in Proc. ICASSP 2008, Las Vegas, NV,USA, Apr. 2008, pp. 2805–2808.

[23] W. Utschick and J. Brehmer, “Monotonic optimization framework forcoordinated beamforming in multicell networks,” IEEE Trans. SignalProcess., vol. 60, pp. 1899–1909, Apr. 2012.

[24] K. Gomadam,V. R. Cadambe, and S. A. Jafar, “A distributed numericalapproach to interference alignment and applications to wireless inter-ference networks,” IEEE Trans. Inf. Theory, vol. 57, pp. 3309–3322,Jun. 2011.

[25] W. Wade, An Introduction to Analysis. Upper Saddle River, NJ,USA: Prentice-Hall, 2004.

[26] A. Edelman, T. A. Arias, and S. T. Smith, “The geometry of algorithmswith orthogonality constraints,” SIAM J. Matrix Anal. Appl., vol. 20,no. 2, pp. 303–353, 1998.

[27] W. M. Boothby, An Introduction to Differential Manifolds and Rie-mannian Geometry, 2nd Ed ed. New York, NY, USA: Academic,2002.

[28] M. P. Do Carmo, Riemannian Geometry. Boston, MA, USA:Birkhauser, 1992.

[29] J. H. Manton, “Optimization algorithms exploiting unitary con-straints,” IEEE Trans. Signal Process., vol. 50, pp. 635–650, Mar.2002.

[30] I. E. Telatar, “Capacity of multi-antenna Gaussian channels,” Eur.Trans. Telecommun., vol. 10, pp. 585–596, Nov.–Dec. 1999.

[31] J. Peng and D. Paul, “A geometric approach to maximum likelihoodestimation of the functional principal components from sparse longi-tudinal data,” ArXiv pre-print [Online]. Available: http://arxiv.org/abs/0710.5343

[32] L. Armijo, “Minimization of functions having Lipschitz continuousfirst partial derivatives,” Pacific J. Math., vol. 16, no. 1, pp. 1–3, 1966.

[33] Z. Wen and W. Yin, “A feasible method for optimization with orthog-onality constraints,” Rice Univ., Houston, TX, USA, 2010.

[34] A. A. Goldstein, “Convex programming in Hilbert space,” Bull. Amer.Math. Soc., vol. 70, pp. 709–710, 1964.

[35] S. Kay, Fundamentals of Statistical Signal Processing: EstimationTheory. Englewood Cliffs, NJ, USA: Prentice-Hall, 1993.

[36] A. Liu, Y. Liu, H. Xiang, andW. Luo, “Polite water-filling for weightedsum-rate maximization in MIMO B-MAC networks under multiplelinear constraints,” IEEE Trans. Signal Process., vol. 60, pp. 834–847,Feb. 2012.

[37] C. B. Peel, B. M. Hochwald, and A. L. Swindlehurst, “A vector-per-turbation technique for near-capacity multiantenna multiuser commu-nication-part I: channel inversion and regularization,” IEEE Trans.Commun., vol. 53, pp. 195–202, Jan. 2005.

[38] J. Lindblom, E. Larsson, and E. Jorswieck, “Parametrization of theMISO IFC rate region: The case of partial channel state information,”IEEE Trans. Wireless Commun., vol. 9, pp. 500–504, Feb. 2010.

[39] R. A. Horn and C. R. Johnson, Matrix Analysis. Cambridge, U.K.:Cambridge Univ. Press, 1985.

Juho Park (S’06–M’13) received the B.S., M.S.,and Ph.D. degrees from the Korea Advanced Insti-tute of Science and Technology (KAIST), Daejeon,all in electrical engineering in 2006, 2008, and 2013,respectively.He is currently working at the B4G research team

in the Electronics and Telecommunications ResearchInstitute (ETRI), Daejeon. His research interests in-clude signal processing for communications and nextgeneration communication systems.

Youngchul Sung (S’92–M’93–SM’09) received theB.S. and M.S. degrees from Seoul National Univer-sity, Seoul, Korea, in electronics engineering in 1993and 1995, respectively. After working at LG Elec-tronics, Ltd., Seoul, from 1995 to 2000, he joined thePh.D. program and received the Ph.D. degree in elec-trical and computer engineering from Cornell Uni-versity, Ithaca, NY, in 2005.From 2005 until 2007, he was a Senior Engineer in

the Corporate R&D Center of Qualcomm, Inc., SanDiego, CA, and participated in design of WCDMA

basestation modem. Since 2007, he has been on the faculty of the Departmentof Electrical Engineering, Korea Advanced Institute of Science and Technology(KAIST), Daejeon. His research interests include wireless communications, sta-tistical signal processing, and asymptotic statistics with applications to next-generation wireless communications and related areas.Dr. Sung is an Associate Editor of the IEEE SIGNAL PROCESSING LETTERS,

a member of Signal and Information Processing Theory and Methods (SIPTM)Technical Committee of Asia-Pacific Signal and Information Processing Asso-ciation (APSIPA), Vice-Chair of IEEE ComSoc Asia-Pacific Board MCC, andTechnical Program Committee member of conferences, including Globecom2009–2013, ICC 2011/2014, APSIPA 2009–2012, DCOSS 2010/2013,MILCOM 2010, WiOpt 2009 and its sponsorship chair, and SAM 2008.Recently, he served as a Guest Editor of the IEEE JOURNAL ON SELECTEDAREAS IN COMMUNICATIONS Special Issue on “Theories and Methods forAdvanced Wireless Relays.”

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