Wireless Linear Computation Broadcast

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Wireless Linear Computation Broadcast

arXiv:2608.05692v1 [cs.IT] 6 Aug 2026

Shuo Tan, Graduate Student Member, IEEE, and Syed A. Jafar, Fellow, IEEE

or auxiliary measurements [13]. A conceptually analogous situation arises when receiver-side cached model components serve as side information, and the network delivers low-dimensional demand descriptors (e.g., via LoRA-style adaptation [14]). Existing progress on CBC, especially its linear specialization, Linear Computation Broadcast (LCBC), where the demands and side information are linear functions, has mainly focused on the information-theoretic capacity of LCBC for zero-error finite-field computation over an ideal (noiseless) broadcast channel (BC). The first CBC scheme in [9] is restricted to the 2-user case. The 3-user LCBC solution in [15] is derived from linear rank inequalities for three desired spaces and three corresponding side-information spaces. Beyond three users, a symmetric K-user LCBC construction is developed in [16] utilizing asymptotic interference alignment [17]. An achievable scheme for general K-user LCBC was proposed in [18] using linear-programming-based design principles inspired by the 3-user characterization in [15]. Consequently, translating information-theoretic LCBC schemes into practical wireless downlink designs remains challenging. LCBC schemes thus far are primarily developed over finite fields and under a zero-error criterion, whereas practical applications often require real-valued (over R) [19] and approximate computation under distortion metrics such as the mean-square error (MSE). Moreover, existing LCBC Index Terms—Linear computation broadcast, Gaussian MIMO formulations are largely physical-layer-agnostic: they model broadcast channel, noisy side information, weighted sum-MSE, the broadcast medium as a noiseless common information pipe, transceiver optimization. i.e., the trivial-channel setting in which every user receives the same broadcast information [9], [15], [16], [18]. They therefore I. I NTRODUCTION do not jointly account for MIMO antenna configurations, Artificial intelligence (AI) is reshaping communication channel noise, and transmit power constraints, which are systems toward learning-assisted and computation-centric data important considerations in wireless communications [6], dissemination [1]–[4]. The coupling of communication and [20], [21]. To the best of our knowledge, the corresponding computation has been predominantly explored in uplink information-theoretic LCBC problem with user-dependent nonmultiple-access channel (MAC) settings, where waveform trivial channels remains open. By explicitly accounting for superposition facilitates over-the-air function computation [5], a distinct MIMO channel at each user, WLCBC provides a [6] and model aggregation [7], [8]. significant stepping stone toward this broader setting. The downlink counterpart presents a distinct challenge: the Conventional MIMO downlink designs, such as weighted transmitter must simultaneously serve heterogeneous functional minimum mean-square error (WMMSE) [22], [23], optimize demands across users, each with user-specific side information. linear filters for independent data streams, treating unintended An information-theoretic abstraction for downlink function signals as interference. In our setting, however, users compute delivery is Computation Broadcast (CBC) [9], which studies the correlated linear functions of a shared source. Consequently, a dissemination of user-specific functions of a common source transmitted spatial direction can simultaneously benefit multiple in the presence of receiver-dependent side information. This users, with its utility governed by their heterogeneous side abstraction aligns with distributed edge deployments—e.g., information. autonomous driving [10], UAV swarms [11], and VR sysInformation-theoretic broadcast schemes using Slepian–Wolf, tems [12]—where devices extract demand-relevant transforms index, or lattice coding [24]–[27] also exploit receiver side of shared observations while leveraging locally cached context information. Yet, they inherently target the exact recovery of discrete messages, fundamentally differing from our objective The authors are with the Department of Electrical Engineering and Computer Science, University of California at Irvine, Irvine, CA 92697 USA (e-mail: of distortion-minimizing continuous estimation. shuot8@uci.edu; syed@uci.edu). Closer to our continuous-amplitude setting are Wyner–Ziv

Abstract—A linear computation broadcast (LCBC) problem comprises K users (receivers) and a transmitter. The users wish to compute various (vector) linear functions of a common dataset, and possess in advance heterogeneous side information corresponding to various other linear computations. The goal for the broadcast transmitter, who knows the dataset and all desired and side-information functions, is to satisfy all demands as efficiently as possible. Prior work has explored the informationtheoretic capacity of LCBC for zero-error finite-field computation over an ideal (noiseless) broadcast channel (BC). This paper develops a wireless LCBC (WLCBC) framework for the Gaussian MIMO BC with noisy receiver-side information under a transmit power constraint and arbitrary antenna configurations. In the WLCBC setting, a Gaussian source is linearly precoded for broadcast; each user requests a linear function of the source and forms an estimate by linearly combining its channel observation with its noisy side information. Assuming perfect channel state information, we cast the centralized joint linear transceiver design as a weighted sum-MSE minimization problem and propose an efficient alternating optimization algorithm. For a fixed precoder, the optimal decoders are the linear minimum mean-square error (LMMSE) estimators. For fixed decoders, the precoder update reduces to a convex quadratically constrained quadratic program which leads to a semi-closed-form solution parameterized by a single dual variable. The resulting algorithm guarantees a monotonic decrease in the objective and convergence of the weighted sum-MSE (WSMSE) objective sequence. Simulations demonstrate pronounced robustness gains over natural baselines obtained from prior works.

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and hybrid source–channel coding schemes [28]–[31]. While ing WSMSE objective sequence is monotonically nonthese works establish asymptotic limits for reconstruction under increasing and convergent; moreover, every accumulation side information, they do not translate to finite-dimensional point of the iterate sequence satisfies the KKT conditions linear transceiver designs. Specifically, they lack mechanisms of the joint WSMSE design problem. to jointly manage MIMO channels, noisy side observations, The remainder of this paper is organized as follows. and user-specific computation demands. Section II presents the system model. Section III develops To bridge this gap, we propose the wireless LCBC (WLCBC) the alternating optimization algorithm, with convergence and framework for Gaussian MIMO broadcast channels. We jointly optimality proofs given in Section IV. Section V presents optimize the linear precoder and decoders to minimize the numerical simulations, and Section VI concludes the paper. weighted sum-MSE (WSMSE) under a sum-power constraint. Notation: Bold lowercase and uppercase letters denote vectors Crucially, each user’s noisy side information induces a posterior and matrices, respectively. (·)⊤ denotes transpose, In denotes residual covariance Πi that carefully weights the source direc- the n×n identity matrix, and ∥·∥F denotes the Frobenius norm. tions based on their remaining uncertainty. Consequently, while N (0, Σ) denotes a real Gaussian distribution with mean 0 and the resulting block-coordinate updates retain familiar LMMSE covariance Σ. For a matrix A, rank(A), and tr(A) denote and QCQP structures, their core mathematical operators are its rank and trace, respectively. For matrices A1 , . . . , An the fundamentally shaped by the geometric interplay of user notation blkdiag(A1 , . . . , An ) represents their block-diagonal demands and side information. concatenation. The Kronecker product is denoted by ⊗. The main contributions are as follows: • Centralized linear Gaussian MIMO formulation with II. S YSTEM M ODEL noisy side information and user-dependent channels: For a K-user WLCBC system, this section specifies (i) the We formulate LCBC with noisy side information over Gaussian source and users’ linear computation demands, (ii) a K-user real Gaussian MIMO broadcast channel under side information available at each user, (iii) the T -use MIMO a transmit power constraint and perfect CSI, explicitly broadcast channel and its block-stacked representation, and (iv) accounting for a distinct MIMO channel at each user, as the linear transceiver architecture together with the WSMSE well as heterogeneous antenna configurations and userdesign objective. Table I summarizes the key notation and dependent noise levels. Each receiver has imperfect side dimensions used throughout the system model. information modeled as a correlated linear observation corrupted by additive Gaussian noise. Using the MSE TABLE I between each user’s demand and its estimate as the distorD IMENSION GUIDE FOR THE WLCBC SYSTEM MODEL . tion metric, we obtain a centralized WSMSE-minimization framework that jointly optimizes the transmit precoder Notation Definition and dimension and the per-user linear decoders, while accounting for T , M , Ni Number of channel uses, transmit antennas, noisy channels and side information. and receive antennas of user i. • WLCBC-aware LMMSE decoding via residual subr = T M , mi = T Ni Block-extended transmit and receive dimensions. space geometry: For a fixed precoder, the optimal ′ Demand and side-information dimensions. LMMSE decoder is intrinsically shaped by the poste- di , di n x∈R Common Gaussian source vector. rior residual covariance Πi . Geometrically, rather than Vi ∈ Rn×di ; Demand matrix and target linear function. enforcing a rigid orthogonal projection, Πi acts as a wi = Vi⊤ x ∈ Rdi ′ structure that weights the demanded LCBC directions by Vi′ ∈ Rn×di ; Side-information matrix and noisy observa′ ⊤ ′ ′ their remaining uncertainty—smoothly attenuating well- wi = Vi x + n′i ∈ Rdi tion. Centralized block precoder and transmit observed components while preserving unobserved ones. E ∈ R⊤n×r ; r s=E x∈R vector. Computationally, coupling this residual geometry with a Hi ∈ Rmi ×r ; Block-stacked MIMO channel and received thin-SVD reduction yields an efficient Schur-complement vector. yi = Hi s + ni ∈ Rmi ′ update, completely circumventing the high-dimensional zi ∈ Rmi +di Composite observation obtained by stacking yi and wi′ . inversion of the full composite covariance matrix. ′ Di ∈ R(mi +di )×di ; Linear decoder and target-function estimate. • Convex QCQP precoder update with a single dual vari⊤ b i = Di zi w able: For fixed decoders, we show that the precoder design reduces to a convex quadratically constrained quadratic program (QCQP). By analyzing the KKT conditions, we obtain a semi-closed-form solution parameterized by a A. Source and computation demands single optimal Lagrange multiplier. This reduces the highWe model the common source vector x ∈ Rn as an isotropic dimensional precoder optimization to a one-dimensional Gaussian random vector, x ∼ N (0, I ). Let K ≜ {1, . . . , K} n root-finding problem, which can be solved efficiently using denote the user set. Each user i ∈ K aims to compute a linear a safeguarded Newton–bisection search. function of x: • Alternating optimization with monotonic descent and wi = Vi⊤ x ∈ Rdi , (1) KKT characterization: Combining the decoder and precoder updates yields an alternating optimization proce- where Vi ∈ Rn×di is the demand matrix and di is the demand dure with parallel per-user decoder updates. The result- dimension.

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Fig. 1. K-user WLCBC system model. The transmitter precodes x into s = E⊤ x; user i combines (yi , wi′ ) to estimate wi = Vi⊤ x.

With this representation, (2) becomes

B. MIMO broadcast channel Consider transmission from an M -antenna transmitter to an Ni -antenna user i over a block of T channel uses. To evaluate the distortion–communication tradeoff consistently across T , we assume a common underlying channel sequence (p) {Hi [t]}t≥1 . A T -use model employs the first T blocks, thereby strictly nesting the T -use system within the (T +1)-use system. 1) Per-use signal model: At channel use t ∈ {1, . . . , T }, the transmitted signal is s[t] ∈ RM , and the received signal at user i is (p) yi [t] = Hi [t] s[t] + ni [t],

2 ni [t] ∼ N (0, σc,i INi ), (2)

(p)

where Hi [t] ∈ RNi ×M denotes the per-use physical channel 2 matrix and σc,i is the channel-noise variance. C. Correlated side information User i is aided by correlated side information modeled as a noisy linear observation of the source: ⊤

′

wi′ = Vi′ x + n′i ∈ Rdi , n×d′i

2 n′i ∼ N (0, σs,i Id′i ).

(3)

yi = Hi s + ni ∈ Rmi ,

2 ni ∼ N (0, σc,i Imi ),
(p) (p) mi ×r Hi ≜ blkdiag Hi [1], . . . , Hi [T ] ∈ R .

(5)

[T ]

We write Hi when explicitly emphasizing the dependence on T , suppressing the superscript when T is fixed. (p) (p) 2) Block-fading special case: If Hi [t] ≡ Hi for all t, (p) then Hi = IT ⊗ Hi . D. Linear precoding and power constraint The transmitter adopts a linear precoding structure across the T channel uses via a precoding matrix E ∈ Rn×r : s = E⊤ x ∈ Rr .

(6)

Equivalently, E = [E[1], . . . , E[T ]] with E[t] ∈ Rn×M , so that s[t] = E[t]⊤ x. Let Pavg denote the average transmit-power budget per antenna–time coordinate. T  1   1 1 X  E ∥s[t]∥2 = E ∥s∥2 = ∥E∥2F ≤ Pavg . T M t=1 r r

(7)

Here, Vi′ ∈ R characterizes the side-information subspace, We normalize Pavg = 1, so the block power constraint is 2 while σs,i is the corresponding noise variance. 2 1) Block stacking and extended dimensions: Define the ∥E∥F ≤ r. All noise variables are mutually independent, and block-extended transmit dimension r ≜ T M and the block- independent of the source x. extended receive dimension of user i as mi ≜ T Ni . We stack E. Composite observation and linear decoding the per-use signals as     User i stacks its channel observation and side information s[1] yi [1]  ..    as   s ≜  .  ∈ Rr , yi ≜  ...  ∈ Rmi . (4) yi mi +d′i zi ≜ . (8) ′ ∈R s[T ] yi [T ] wi

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Combining (5), (6), and (3) gives the linear Gaussian model  

H i E⊤ 2 2 b i ∼ N 0, blkdiag σc,i zi = x+b ni , n Imi , σs,i Id′i . ⊤ Vi′ (9) Since (wi , zi ) are jointly Gaussian, the MMSE estimator coincides with the LMMSE estimator [32]. Accordingly, user ′ i employs a linear decoder Di ∈ R(mi +di )×di and forms b i = D⊤ w i zi .

Remark 1 (Canonical normalization). Without loss of generality, we use the canonical normalization x ∼ N (0, In ) and identity noise covariances. For any full-rank source/noise covariances, invertible whitening maps reduce the model to this form; the induced linear transformations are absorbed into (Vi , Vi′ , Hi , Di ) and into the corresponding power constraint. The entire framework is illustrated in Fig. 1.

(10) III. P ROPOSED A LTERNATING O PTIMIZATION

The corresponding MSE is   2 Ei (E, Di ) ≜ E ∥wi − D⊤ i zi ∥ .

(11)

Conformally partitioning the decoder according to (8) yields   ′ Di,y Di = , Di,y ∈ Rmi ×di , Di,w ∈ Rdi ×di . (12) Di,w Then the MSE exactly decomposes as 2

′ Ei (E, Di ) = Vi − EH⊤ i Di,y − Vi Di,w F 2 2 + σc,i ∥Di,y ∥2F + σs,i ∥Di,w ∥2F .

(13)

The first term represents the squared residual of the noiseless algebraic recovery condition, while the remaining terms capture the noise amplification induced by the decoder. At finite SNR, enforcing exact algebraic alignment may be severely suboptimal if it dictates ill-conditioned decoder matrices. Consequently, (13) reveals WLCBC as a physical-layer regularization of the classical LCBC problem, optimally balancing algebraic accuracy against noise robustness. F. Centralized Information Assumptions and Local Execution

We solve the WSMSE problem in (15) via alternating optimization (AO), i.e., exact block coordinate descent over the precoder E and the decoder set {Di }i∈K . Although (15) is not jointly convex, it is convex in each block separately: (i) for fixed E, each user decoder update is a strictly convex quadratic problem admitting a unique LMMSE solution; (ii) for fixed {Di }, the precoder update reduces to a convex QCQP with a KKT-characterized global minimizer. We first perform a onetime preprocessing that removes noise-only side-information dimensions (without loss of optimality), then present the decoder and precoder updates, and finally summarize the overall AO procedure in Algorithm 1. All proofs are deferred to Section IV. Throughout this section, we adopt the stacked representation in Section II: r = T M , mi = T Ni , and Hi ∈ Rmi ×r denotes the stacked channel matrix.

A. Preprocessing: Rank reduction of noisy side information For user i, let de′i ≜ rank(Vi′ ) and take the thin SVD ⊤

Vi′ = U′i Σ′i T′i , (16) We adopt a centralized-design and local-execution framework. The transmitter is assumed to know ′ e′ e′ 2 2 {Hi , Vi , Vi′ , ωi , σc,i , σs,i }i∈K , together with the transmit- where U′i ∈ Rn×di and T′i ∈ Rdi ×di have orthonormal ′ ′ e e power budget, but not the realizations of the receiver noise columns and Σ′i ∈ Rdi ×di is diagonal. ⊤ terms ni and n′i . Premultiplying (3) by T′i yields the reduced side observaUsing this information, the transmitter jointly computes the tion optimal precoder E and the user-specific decoders {Di }i∈K . ⊤ e i′ ≜ T′i wi′ = Σ′i U′⊤ e ′i , e ′i ∼ N (0, Ide′ ), w n The transmitter uses E for transmission, and receiver i forms i x+n i b i = (Di )⊤ zi from its local observation. Thus, during w ⊤ execution, receiver i requires only (zi , Di ). since T′i preserves whiteness of n′i . G. WSMSE Transceiver Design Objective Our goal is to jointly design the linear precoder E and the per-user decoding matrices {Di }i∈K to minimize a weighted sum of users’ MSEs under the transmit power constraint. Define the global weighted sum-MSE (WSMSE) objective as X F (E, {Di }) ≜ ωi Ei (E, Di ), (14) i∈K

where ωi > 0 denotes the positive weight for user i. The joint transceiver design problem is formulated as min

E, {Di }i∈K

s.t.

F (E, {Di })

(15a)

∥E∥2F ≤ r.

(15b)

Lemma 1 (Removal of noise-only side-information dimensions). The orthogonal complement of span(T′i ) contains pure noise independent of x. It thus does not affect (i) the optimal LMMSE decoder for any fixed E, nor (ii) the optimal value of (15). Hence, it is without loss of optimality to discard the e i′ )⊤ ]⊤ . noise-only dimensions and work with e zi ≜ [yi⊤ , (w Remark 2 (Notation reset after reduction). Define the reduced e ′ ≜ U′ Σ′ ∈ Rn×de′i so that w e i′ = side-information matrix V i i i ′ ⊤ ′ e e i . After discarding the noise-only dimensions, we (Vi ) x + n redefine d′i ← de′i ,

e i′ , wi′ ← w

zi ← e zi ,

e ′. Vi′ ← V i

We still retain (U′i , Σ′i ) since they will be repeatedly used in the efficient decoder update.

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B. Decoder update: Optimal decoder for fixed precoder ⊤

Fix any feasible E and define Ri ≜ Hi E . Under the stacked MIMO model, E ∈ Rn×r with r = T M can be partitioned as E = [E[1], . . . , E[T ]] with E[t] ∈ Rn×M . Then Ri can be formed blockwise without explicitly building the large stacked matrix Hi :   (p) Hi [1] E[1]⊤   ..  ∈ Rmi ×n . Ri =  (17) .   (p) Hi [T ] E[T ]⊤ Using the reduced side information (Remark 2), the stacked observation admits the linear Gaussian form   Ri zi = Ai x + n̂i , Ai ≜ , n̂i ∼ N (0, Imi +d′i ). (18) (Vi′ )⊤ Proposition 2 (LMMSE decoder and efficient evaluation). For fixed E, the unique minimizer of Ei (E, Di ) is
−1 D⋆i = Ai A⊤ Ai Vi . (19) i + Imi +d′i Moreover, partition D⋆i conformably with zi = [yi⊤ , (wi′ )⊤ ]⊤ as  ⋆  ′ Di,y ⋆ Di = , D⋆i,y ∈ Rmi ×di , D⋆i,w ∈ Rdi ×di , D⋆i,w and define Φ11 ≜ Id′i + (Σ′i )2 , Λi ≜ (Σ′i )2 Φ−1 11 ,

weighting fundamentally distinguishes WLCBC from conventional MIMO designs that treat user demands as independent data streams. Remark 4 (Matrix-free LMMSE implementation and solve– ′⊤ domain selection). The quantities Φ−1 11 , Λi , and Si ≜ Ui Vi depend only on the side-information model and can therefore be cached during preprocessing. ′ At each AO iteration, define Gi ≜ Ri U′i ∈ Rmi ×di . Then the coefficient matrix and right-hand side in (21) can be evaluated without explicitly forming the dense matrix Πi : ⊤ ⊤ Imi + Ri Πi R⊤ i = Imi + Ri Ri − Gi Λi Gi ,

(23)

Ri Πi Vi = Ri Vi − Gi Λi Si .

(24)

The resulting mi × mi coefficient matrix is symmetric positive definite (SPD) and can be factorized by Cholesky. After obtaining D⋆i,y , the side-information decoder block is recovered ′ ⊤ ⋆ as D⋆i,w = Φ−1 11 Σi (Si − Gi Di,y ), avoiding the explicit ⊤ ⋆ formation of Vi − Ri Di,y . When n < mi , the equivalent push-through identity −1 ⊤ −1 (Imi + Ri Πi R⊤ Ri Πi = Ri (Π−1 i ) i + R i Ri )

moves the main factorization to the source domain, where ⊤ Π−1 = In + U′i (Σ′i )2 U′i . Hence, an implementation may i strategically select the receive- or source-domain form according to the smaller effective system dimension. Detailed per-iteration operation counts are provided in Section III-E.

(20)

⊤ Πi ≜ In − U′i Λi U′i ≻ 0.

Then D⋆i can be computed without inverting an (mi + d′i ) × (mi + d′i ) matrix via
⊤ −1

D⋆i,y = Imi + Ri Πi Ri Ri Πi Vi ,   ′ ′⊤ ⋆ D⋆i,w = Φ−1 Vi − R ⊤ i Di,y . 11 Σi Ui

(21) (22)

The matrix Πi in (20) acts as the posterior source covariance given the side information, i.e., Πi = Cov(x | wi′ ). Consequently, the conditional covariances governing the channel observation are ′ Cov(yi | wi′ ) = Imi +Ri Πi R⊤ i , Cov(yi , wi | wi ) = Ri Πi Vi .

Thus, the channel-decoder block D⋆i,y in (21) is precisely the conditional LMMSE estimator of wi from yi given wi′ . Remark 3 (Residual covariance and LCBC geometry). The posterior covariance Πi embeds the LCBC side-information geometry into the transceiver update. Its decomposition,
−1 ′ ⊤ ⊤ Πi = Pi,⊥ + U′i Id′i + (Σ′i )2 Ui , Pi,⊥ ≜ In − U′i U′i , reveals that Πi acts as a residual covariance rather than a rigid orthogonal projection. It preserves full uncertainty along the unobserved subspace Pi,⊥ , while optimally attenuating each side-information direction according to its reliability. Consequently, the residual uncertainty of any demanded direction v, given by v⊤ Πi v, explicitly depends on its spatial alignment with the noisy side information. This structural

C. Precoder update: Optimal solution of a convex QCQP for fixed decoders With the side-information geometry abstracted into the posterior covariances {Πi }, the precoder update transcends conventional stream-wise power allocation. Instead, it steers MIMO resources toward demand directions based jointly on their residual uncertainty, channel alignment, and decoderinduced weighting. Fix an arbitrary decoder set {Di }i∈K . The corresponding precoder block of (15) is X min ωi Ei (E, Di ) s.t. ∥E∥2F ≤ r. (25) E∈Rn×r

i∈K

Partition each fixed decoder conformably with zi = [yi⊤ , (wi′ )⊤ ]⊤ as   ′ Di,y Di = , Di,y ∈ Rmi ×di , Di,w ∈ Rdi ×di . Di,w Define (with Vi′ denoting the reduced matrix after Remark 2) ′⊤ di ×n di ×r Ci ≜ Vi⊤ − D⊤ , Bi ≜ D⊤ . i,w Vi ∈ R i,y Hi ∈ R (26) The following lemma gives the quadratic reduction of the precoder subproblem; its proof is deferred to Section IV.

Lemma 3 (Quadratic form of the precoder subproblem). Fix {Di }i∈K and define Ci , Bi by (26). Then the per-user MSE admits the decomposition Ei (E, Di ) = ∥Ci − Bi E⊤ ∥2F + ∥Di,y ∥2F + ∥Di,w ∥2F . (27)

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Consequently, after dropping the terms independent of E, the precoder subproblem (25) is equivalent to X min ωi ∥Ci − Bi E⊤ ∥2F . (28) 2 ∥E∥F ≤r

Let dtot ≜

µlo ← µN , else set µhi ← µN . Terminate when |ϕ(µ) − r|/r ≤ εµ for a prescribed tolerance εµ . Finally, compute E⋆ (µ⋆ ) by solving the SPD system (Q + µ⋆ Ir )X = P⊤ ,

i∈K

E⋆ (µ⋆ ) = X⊤ ,

P

i∈K di and define the weighted concatenations √  √ ⊤ n×dtot C≜ ω1 C⊤ , 1 , . . . , ωK CK ∈ R (29) √  √ ⊤ ⊤ r×dtot B≜ ω1 B1 , . . . , ωK BK ∈ R .

or equivalently via the
diagonal formula E⋆ (µ⋆ ) 1 1 ⊤ e diag P λ1 +µ⋆ , . . . , λr +µ⋆ U .

=

Then (28) can be written compactly as the convex QCQP

Algorithm 1 WLCBC-Aware Alternating Transceiver Design ′ 2 2 min (30) Input: {Hi , Vi , Vi , σc,i , σs,i }i∈K , weights {ωi }i∈K , power ∥E∥2F ≤r budget, tolerance ϵ, and maximum iteration count tmax . ⋆ ⋆ Proposition 4 (Optimal precoder structure). Given the fixed Output: E , {Di }i∈K . ⊤ r×r ⊤ 1: Preprocessing: For each i, compute thin SVD Vi′ = decoders {Di }, let Q ≜ BB ∈ R and P ≜ CB ∈ ⊤ n×r U′i Σ′i T′i and store (U′i , Σ′i ). R . An optimal solution to the convex QCQP (30) is 2: Initialize a feasible E(0) with ∥E(0) ∥2F ≤ r; set t ← 0.
−1 E⋆ (µ) = P Q + µIr , (31) 3: for i = 1 to K do (0) (0) 4: Compute Di,y and Di,w via (21)–(22) using E(0) . where µ ≥ 0 satisfies complementary slackness: either µ = 0 5: end for if ∥E⋆ (0)∥2F ≤ r, or µ > 0 such that ∥E⋆ (µ)∥2F = r. (If Q (0) 6: Evaluate F (0) ← F (E(0) , {Di }i∈K ). ⋆ † is singular and µ = 0, we take E (0) = PQ as a minimum7: repeat norm minimizer.) 8: t ← t + 1. Remark 5 (Implementation: safeguarded Newton for µ⋆ ). 9: Decoder update (parallel over i): The power constraint (7) is enforced via µ⋆ , which can be 10: for i = 1 to K do computed efficiently as follows. We avoid forming (Q+µIr )−1 11: Ri ← Hi (E(t−1) )⊤ . (t) (t) explicitly. If the unconstrained minimizer E⋆ (0) = PQ† 12: Form Φ11 and Λi via (20), and update Di,y , Di,w satisfies ∥E⋆ (0)∥2F ≤ r, then the constraint is inactive and via (21)–(22) using the matrix-free implementation µ⋆ = 0. Otherwise, µ⋆ > 0 is the unique root of ϕ(µ) = r, in Remark 4. where 13: end for 14: Precoder update: ϕ(µ) ≜ ∥E⋆ (µ)∥2F , E⋆ (µ) = P(Q + µIr )−1 . 15: Form C, B via (29); set P ← CB⊤ , Q ← BB⊤ . Since ϕ(µ) is continuous and strictly decreasing on µ > 0 16: Compute unconstrained solution E0 ← PQ† . (cf. Proposition 4 and its proof), we solve ϕ(µ) = r via a 17: if ∥E0 ∥2F ≤ r then safeguarded Newton (Newton–bisection) method. 18: µ⋆ ← 0; E(t) ← E0 . A numerically stable way to evaluate ϕ(µ) and ϕ′ (µ) is to 19: else diagonalize the positive semi-definite (PSD) matrix Q once per 20: Find µ⋆ > 0 such that ∥P(Q + µ⋆ Ir )−1 ∥2F = r via AO iteration: the safeguarded Newton method in Remark 5. 21: Solve (Q + µ⋆ Ir )X = P⊤ ; set E(t) ← X⊤ . ⊤ Q = U diag(λ1 , . . . , λr )U , λj ≥ 0, 22: end if (t) e ≜ PU = [p̃1 , . . . , p̃r ]. Then and define P 23: Evaluate F (t) ← F (E(t) , {Di }i∈K ). 24: until the stopping criterion (32) holds or t = tmax . r r X X ∥p̃j ∥22 ∥p̃j ∥22 ′ 25: Final synchronization: ϕ(µ) = , ϕ (µ) = −2 . (λj + µ)2 (λj + µ)3 26: Set E⋆ ← E(t) and recompute all {D⋆ j=1 j=1 i }i∈K via (21)–(22) using E⋆ . Newton’s update is 27: Construct the decoder in the original observation coordiϕ(µ) − r ϕ(µ) − r nates and provide D⋆i,dep to user i. = µ + Pr µN ← µ − 2 . ′ ∥p̃ ∥ j ϕ (µ) 2 j=1 (λj +µ)2 3 ∥C − EB∥2F .

To safeguard the iterations, maintain a bracket [µlo , µhi ] such that ϕ(µlo ) ≥ r ≥ ϕ(µhi ). A universal one-shot choice is µlo = 0 and ∥P∥F µhi = √ , r which guarantees ϕ(µhi ) ≤ r (see the proof of Proposition 4). Starting from µ(0) = (µlo + µhi )/2, iterate: (i) compute µN by Newton’s update; (ii) if µN ∈ / (µlo , µhi ) or is not finite, set µN ← (µlo + µhi )/2; (iii) update the bracket: if ϕ(µN ) > r set

D. Overall AO algorithm and stopping rule (t)

Define F (t) ≜ F (E(t) , {Di }). We terminate the AO loop when the relative improvement satisfies F (t) − F (t−1) ≤ ϵ, F (t−1)

(32)

or when a prescribed maximum iteration count is reached.

7

Then the objective sequence {F (t) }t≥0 is monotonically We count dense arithmetic operations and exclude the one- non-increasing and convergent. Moreover, the iterate seis bounded, and every accumulation point of time side-information SVD preprocessing. Iteration-invariant quence (t) (t) {(E , {D −1 ′ i }i∈K )}t≥0 is a coordinatewise minimum of (15). quantities (Φ11 , Λi , Si ) are assumed to be cached. Here, di Consequently, every accumulation point satisfies the KKT denotesPthe reduced informative rank after Remark 2, and K conditions of (15). d ≜ d. E. Per-Iteration Complexity and Scalability

tot

i=1

i

For user i, forming Ri = Hi E⊤ costs O(mi M n) operations due to the block-diagonal structure of Hi . Using the matrixfree receive-domain implementation in Remark 4, the decoder update requires  Cdec,i = O mi M n + m2i n + m3i + mi (n + mi )di  (33) + mi (n + mi + di )d′i .

IV. P ROOFS A. Proof of Lemma 1 Proof. Let T⊥ be any orthonormal basis of the orthogonal complement of span(T), so that [T, T⊥ ] is square orthonormal. ⊤ Premultiplying w′ = V′⊤ x + n′ by [T⊤ , T⊤ ⊥ ] yields  ′    ′ e e w n ΣU⊤ x = + ′ , ′ w⊥ n⊥ 0

Thus, the additional per-iteration dependence on the reduced ′ ′ side-information rank d′i is strictly linear. If the source- where w⊥ = T⊤ ⊥ n is independent of x and contains no domain form in Remark 4 is used, the factorization-related information about x. Hence, any linear estimator can set its ′ terms O(m2i n + m3i ) are replaced by O(mi n2 + n3 ). The K coefficient on w⊥ to zero without affecting MSE, and the independent decoder updates can be executed in parallel. optimal value is unchanged. The precoder update consists of forming {Ci , Bi }K i=1 , accumulating P and Q, performing one spectral factorization B. Proof of Proposition 2 of Q, and running an Nµ -step safeguarded scalar dual search. Proof. Fix user i and omit the user index when clear. Under Aggregating these steps gives the reduced model, we have X K ′ 2 w = V⊤ x, z = Ax + n̂, (37) Cprec = O (ndi di + mi M di ) + nrdtot + r dtot i=1 (34)  where x ∼ N (0, In ), n̂ ∼ N (0, Im+d′ ), and x ⊥ n̂. 3 2 + r + nr + Nµ r . The optimal linear MMSE estimator satisfies the orthogonality principle E[(w − D⊤ z)z⊤ ] = 0, i.e., Here, the terms r3 + nr2 correspond to the eigendecomposition E[wz⊤ ] = D⊤ E[zz⊤ ]. (38) e = PU, while Nµ r accounts of Q and the multiplication P for the scalar Newton–bisection evaluations after the spectral We compute the cross-covariance and covariance matrices as quantities are cached. follows: Consequently, one full AO iteration has serial work and E[wz⊤ ] = E[V⊤ x(Ax + n̂)⊤ ] (39) parallel decoder-update depth = V⊤ E[xx⊤ ]A⊤ = V⊤ A⊤ , K X ser CAO = Cdec,i + Cprec , (35) and i=1 E[zz⊤ ] = AE[xx⊤ ]A⊤ + E[n̂n̂⊤ ] par (40) CAO = max Cdec,i + Cprec . (36) ′. = AA⊤ + I i∈K

m+d

For homogeneous antenna and demand dimensions (mi = Substituting these into (38) yields T N , r = T M , di = d, and d′i = d′ ), the dominant cubic dense V⊤ A⊤ = D⊤ (AA⊤ + Im+d′ ). (41) factorization burden under the receive-domain implementation is O(KT 3 N 3 + T 3 M 3 ). The first term arises from the K Since AA⊤ + Im+d′ ≻ 0, the solution is unique and equals receive-domain SPD factorizations of size T N , and the second D⋆ = (AA⊤ + Im+d′ )−1 AV, (42) from the eigendecomposition of the T M × T M matrix Q. Since the scalar dual search costs only O(Nµ T M ) once the which recovers (19). spectral quantities are cached, it is not the dominant operation. Partition the decoder D⋆ and matrix A conformably into Other matrix-multiplication terms retained in (33) and (34) two blocks:  ⋆   may dominate in regimes where n or dtot scales faster than Dy R the block antenna dimensions. D⋆ = , A = . (43) ΣU⊤ D⋆w Proposition 5 (Convergence and stationarity of AO). Let ⊤ ⋆ (t) {(E(t) , {Di }i∈K )}t≥0 be the sequence generated by Algo- Let Φ ≜ AA + Im+d′ . The normal equation ΦD = AV can be written in block form as rithm 1, and define      RV Φ00 Φ01 D⋆y (t) (t) (t) = , (44) F ≜ F (E , {Di }i∈K ). Φ10 Φ11 D⋆w ΣU⊤ V

8

where the block components are defined as ⊤

Φ00 = Im + RR , Φ10 = Φ⊤ 01 ,

Φ01 = RUΣ, Φ11 = Id′ + Σ2 .

(45)

Eliminating D⋆w via the Schur complement of Φ11 gives: −1 ⋆ ⊤ (Φ00 − Φ01 Φ−1 11 Φ10 )Dy = RV − Φ01 Φ11 ΣU V. (46)

Using Λ = Σ2 (Id′ + Σ2 )−1 and Π = I − UΛU⊤ , this simplifies to (Im + RΠR

⊤

)D⋆y = RΠV,

where strict negativity holds because P ̸= 0 and (Q + µIr )−3/2 ≻ 0. Therefore, ϕ(µ) is continuous and strictly decreasing on µ > 0, and satisfies ϕ(µ) → 0 as µ → ∞. By complementary slackness, if ∥E⋆ (0)∥2F ≤ r then µ⋆ = 0; otherwise there exists a unique µ⋆ > 0 such that ϕ(µ⋆ ) = r. Moreover, since Q ⪰ 0, we have ∥(Q + µIr )−1 ∥2 ≤ 1/µ for µ > 0, hence 2

ϕ(µ) = P(Q + µIr )−1 F 2

≤ ∥P∥2F · (Q + µIr )−1 2

(47)

which yields (21). Finally, the second block equation gives ⊤ ⊤ ⋆ D⋆w = Φ−1 11 ΣU (V − R Dy ),

(48)

which is (22). C. Proof of Lemma 3 ⊤ ′ b i = D⊤ Proof. The estimate is w i,y yi + Di,w wi . Substituting the signal model yields   ′⊤ ⊤ ⊤ b i = Vi⊤ − D⊤ wi − w V − D H E x i i,w i i,y (49) ⊤ ′ − D⊤ i,y ni − Di,w ni .

Using x ∼ N (0, In ), independent noises with identity covariances, and independence across x, ni , n′i , the cross terms vanish, and the MSE equals (27). Dropping the terms constant in E yields (28). D. Proof of Proposition 4 Proof. Consider (30): min∥E∥2F ≤r ∥C − EB∥2F . Using Q = BB⊤ and P = CB⊤ , the objective can be written as ∥C − EB∥2F = tr(EQE⊤ ) − 2 tr(P⊤ E) + tr(CC⊤ ), (50) where the last term is constant in E. The Lagrangian is L(E, µ) = tr(EQE⊤ ) − 2 tr(P⊤ E) + µ(∥E∥2F − r), µ ≥ 0. (51) Stationarity gives ∇E L = 2E(Q + µIr ) − 2P = 0,

(52)

E⋆ (µ) = P(Q + µIr )−1 .

(53)

hence This proves (31) for any µ > 0 since Q + µIr ≻ 0. For µ = 0, if Q is singular we may take E⋆ (0) = PQ† as a minimum-norm minimizer. It remains to characterize µ. If P = 0, then (53) yields E⋆ (µ) = 0 for all µ ≥ 0 and the constraint is inactive. Assume P ̸= 0 hereafter. For µ > 0, define
ϕ(µ) ≜ ∥E⋆ (µ)∥2F = tr P(Q + µIr )−2 P⊤ . (54) Since Q + µIr ≻ 0, ϕ(µ) is differentiable and ϕ′ (µ) = −2 tr P(Q + µIr )−3 P⊤ = −2 (Q + µIr )−3/2 P⊤ <0

2 F

(55)

(56)

∥P∥2F ≤ µ2 √ In particular, µhi = ∥P∥F / r ensures ϕ(µhi ) ≤ r and can be used to bracket the unique root in a safeguarded Newton search (Remark 5). E. Proof of Proposition 5 Proof. At iteration t, the decoder update minimizes the objective for fixed E(t−1) , hence (t)

(t−1)

F (E(t−1) , {Di }i∈K ) ≤ F (E(t−1) , {Di

}i∈K ).

(57)

Then, the precoder update minimizes the objective for fixed (t) {Di }i∈K under the constraint ∥E∥2F ≤ r, hence (t)

(t)

F (E(t) , {Di }i∈K ) ≤ F (E(t−1) , {Di }i∈K ).

(58)

Combining (57) and (58) yields F (t) ≤ F (t−1) for all t ≥ 1, so {F (t) }t≥0 is monotonically non-increasing. Since each user MSE is nonnegative, we have F (t) ≥ 0 for all t, and therefore {F (t) } converges. Next, define the precoder feasible set E ≜ {E ∈ Rn×r : ∥E∥2F ≤ r}, which is compact. Hence {E(t) }t≥0 ⊂ E is bounded. For each user i, the decoder update is given in closed form by Proposition 2:
−1 D⋆i (E) = Ai (E)Ai (E)⊤ + Imi +d′i Ai (E)Vi , where Ai (E) depends affinely on E. Because Ai (E)Ai (E)⊤ + Imi +d′i ⪰ Imi +d′i ≻ 0 for all E ∈ E, the matrix inversion is well-defined and continuous. Thus, D⋆i (E) is a continuous function of E. Therefore, because E is compact, the image (t) {D⋆i (E) : E ∈ E} is compact, implying that {Di }t≥0 is bounded for every i ∈ K. Hence the full iterate sequence (t) {(E(t) , {Di }i∈K )}t≥0 is bounded and admits at least one accumulation point. Algorithm 1 is an exact two-block coordinate-descent method for the continuously differentiable problem (15): for fixed E, the decoder block is minimized exactly by Proposition 2; for fixed {Di }i∈K , the precoder block is minimized exactly by Proposition 4. Specifically, the precoder minimizer is unique when the power constraint is active (µ⋆ > 0). When inactive (µ⋆ = 0), the minimum-norm selection PQ† yields a well-defined single-valued update, ensuring that the blockminimization mapping is closed as required by [33]. Therefore, by standard convergence results for exact two-block coordinatedescent methods [33], [34], every accumulation point of the iterate sequence is a coordinatewise minimum of (15).

9

Finally, let (Ē, {D̄i }i∈K ) be any accumulation point. Since it is coordinatewise minimal, each decoder block satisfies the unconstrained optimality condition ∇Di F = 0, while Ē satisfies the KKT conditions of the precoder subproblem minE∈E F (E, {D̄i }i∈K ) (i.e., stationarity of the Lagrangian with complementary slackness for the constraint ∥E∥2F ≤ r). Together, these are precisely the KKT conditions of the joint problem (15).

B. Experiment 1: WLCBC versus SJ-LCBC baseline (two-user) We specialize in K = 2 and benchmark WLCBC against SJLCBC. The purpose of this experiment is to compare robustness against a canonical LCBC reference in a noisy setting; it is not intended as a comparison against all possible noisy-broadcast baselines. In this experiment, we use an identity-channel SISO AWGN broadcast link over T = r channel uses, i.e., M = Ni = 1 and Hi = Ir for all i, so that mi = r and

V. N UMERICAL R ESULTS 2 2 This section evaluates the proposed WLCBC framework, also ni ∼ N (0, σch Ir ), n′i ∼ N (0, σside,i Id′i ). (61) referred to as the over-the-air (OTA) setting, through Monte Carlo simulations under linear-Gaussian models. We consider In the SNR sweeps below, we set γc = γs and denote the three experiments over two channel settings. Experiment 1 common SNR by γ. Let c ∈ {Base, OTA} denote the scheme index; the precoder uses an identity-channel AWGN broadcast link in a two-user ′ n×r is E and the user-i decoder is Di,c ∈ R(r+di )×di , so c ∈R comparison with the Sun–Jafar LCBC (SJ-LCBC) [9], including   an examination of the respective roles of precoder and decoder that y ⊤ optimization. Experiments 2 and 3 use i.i.d. real Gaussian b i,c = Di,c i′ . w wi MIMO broadcast channels. Experiment 2 studies the channeluse-versus-distortion tradeoff across different antenna configura- Both schemes are compared using Etotal in (60) under the tions, including heterogeneous per-user SNRs, the convergence same constraint ∥Ec ∥2F ≤ r. behavior of AO, and sensitivity to random initialization, while Baseline (SJ-LCBC). SJ-LCBC is an algebraic alignmentExperiment 3 compares WLCBC with a matched direct-output based construction designed for noiseless side information MIMO-BC baseline to examine the impact of its encoder- (and noiseless channels) [9]. In each trial, after generating subspace restriction. A key observation in Experiment 2 is (Vi , Vi′ ), we fix the broadcast dimension r for both schemes that, at sufficiently high SNRs, a sharp drop in MSE from according to the noiseless subspace-decomposition prescription the interference-limited regime to the noise-limited regime as in [9] (which yields r = 1 for the toy instance below and the number of channel uses increases may indicate, for each r = 4 for the heterogeneous instance below), construct the fixed antenna configuration, the minimum number of channel corresponding SJ precoder/decoder, and scale the precoder to uses required by the considered linear WLCBC design to satisfy ∥EBase ∥2F ≤ r. The resulting SJ scheme is then applied attain a prescribed distortion level. This provides an empirical directly to the noisy observations. connection to the classical LCBC. Proposed WLCBC (OTA). We use over-the-air (OTA) to refer to the WLCBC setting, which jointly optimizes the A. Common Monte Carlo setup and performance metric precoder and decoders by alternating minimization of the MSE For each user i ∈ K ≜ {1, . . . , K}, the demand matrix objective under channel noise and noisy side information. ′ Vi ∈ Rn×di and the side-information matrix Vi′ ∈ Rn×di (a) Toy instance: We first consider (K, n, r, di , d′i ) = are generated as independent random orthonormal bases, i.e., (2, 2, 1, 1, 1), which appears in [9] as a canonical two-user Vi⊤ Vi = Idi and Vi′⊤ Vi′ = Id′i , corresponding to isotropically example. Let vi , vi′ ∈ R2 be the columns of Vi , Vi′ , define random di - and d′i -dimensional subspaces of Rn . G ≜ [v1′ v2′ ], and let e1 = [1, 0]⊤ , e2 = [0, 1]⊤ . The Whenever channel noise and side-information noise are (unnormalized) SJ precoder is isotropic Gaussian, we parameterize their variances by the Eraw = [v1′ 0]G−1 v2 + [0 v2′ ]G−1 v1 , (62) channel and side-information SNRs (in dB), denoted by γc √ and γs , respectively, via which we normalize as E = r E /∥E ∥ . For user Base

2 σch = 10−γc /10 ,

2 σside,i = 10−γs /10 .

(59)

Unless otherwise stated, we use uniform user weights ωi = 1/|K|. The AO-based WLCBC optimization terminates when the iteration count reaches tmax = 250 or when the relative convergence tolerance falls below ϵ = 10−10 . For the precoder subproblem, the Lagrange multiplier µ⋆ is computed via the safeguarded Newton method described in Remark 5. We report 2 the MSE for user i as Ei (E, Di ) ≜ E ∥wi − D⊤ i zi ∥ . The corresponding weighted sum-MSE is X Etotal = ωi Ei (E, Di ). (60) i∈K

This quantity is reported in Figs. 2, 3, 4, 6, and 7, whereas Fig. 5 reports the individual user MSEs {Ei }i∈K .

raw

raw F

i ∈ {1, 2} (with j = 3 − i), the baseline decoder is a two-tap linear combiner of the received symbol and the one-dimensional side-information observation:  ⊤ −1  ei G (vi − vj ) Di,Base = ∈ R2×1 . (63) ∥EBase ∥F Fig. 2 plots Etotal versus the common SNR γ. As γ increases, the gap shrinks, and the two schemes nearly coincide at γ = 20 dB. This behavior is consistent with the fact that SJ-LCBC is tailored to the noiseless/alignment regime, whereas WLCBC is substantially more robust in the noisy regime and approaches the same high-SNR limit as noise vanishes. (b) Heterogeneous instance, precoder/decoder roles: We next consider a higher-dimensional heterogeneous configuration n = 7, r = 4, (d1 , d′1 ) = (4, 2), and (d2 , d′2 ) = (3, 2). To

10

Fig. 2. Two-user toy instance (n, r) = (2, 1) with (d1 , d′1 ) = (1, 1) and (d2 , d′2 ) = (1, 1): weighted sum-MSE Etotal versus γ = SNRch,side (common channel and side-information SNR). Here M = N1 = N2 = 1, T = r = 1.

Fig. 3. Heterogeneous two-user instance (n, r) = (7, 4) with (d1 , d′1 ) = (4, 2) and (d2 , d′2 ) = (3, 2): ablation comparing (i) SJ-LCBC with the Sun– Jafar precoder and decoder, (ii) the fixed Sun–Jafar precoder with OTAoptimized decoders, and (iii) full WLCBC with AO precoder and decoder, versus γ = SNRch,side (common channel and side-information SNR). Here M = N1 = N2 = 1, T = r = 4.

pinpoint where the gain comes from, we compare three designs in Fig. 3: (i) SJ-LCBC (precoder and decoder): the original 2 SJ precoder and decoder applied to the noisy setting; (ii) SJ the same block power constraint ∥E∥F ≤ r = T M , which precoder with OTA-optimized decoders: the SJ precoder is corresponds to unit average power per transmit dimension kept fixed, while the decoders are re-optimized using the OTA (antenna–time slot). Under the above channel normalization, decoder update; (iii) WLCBC (AO precoder and decoder): both the nominal received signal power per receive antenna satisfies i  precoder and decoders are optimized by the proposed AO 1 h (p) 1  2 E ∥s[t]∥22 ≤ 1, E ∥H [t]s[t]∥ 2 = i procedure. Ni M Fig. 3 shows that noise-aware decoder optimization provides substantial robustness gains in the low-SNR regime, but is not so increasing M does not artificially increase SNR; it only sufficient by itself. The remaining performance improvement, increases the available spatial DoF in the stacked model. variance particularly at moderate and high SNRs, is primarily attributable Accordingly, we parameterize the channel noise 2 −γc /10 directly by the target receive SNR (in dB) as σ = 10 , ch to optimizing the precoder. At high SNR (γ ≳ 30 dB), the two so that γ represents the nominal per-antenna receive SNR. c curves that share the SJ precoder nearly overlap, indicating that We fix a K = 4-user instance with n = 24 and (di , d′i ) = decoder mismatch becomes negligible as noise vanishes; the remaining gap to WLCBC (roughly 8 dB for Etotal ≈ 10−2 ) (4, 12) for all i ∈ K, and sweep T ∈ {1, . . . , 10}. The sidesubspaces are generated as in Section V-A. The reflects the benefit of a noise-aware precoder that avoids ill- information ′ choice d = 12 is made relative to the source dimension i conditioned alignment and mitigates noise enhancement. n = 24. Since Vi and Vi′ are independently generated random orthonormal bases, d′i = n/2 provides substantial but C. Experiment 2: Channel-use-versus-distortion transition in incomplete side information: on average, half of the demand Gaussian MIMO BC energy is aligned with the side-information subspace, while the We evaluate WLCBC over i.i.d. real Gaussian MIMO corresponding observations remain noisy. In general, increasing broadcast channels and investigate how the channel-use-versus- d′i reduces the posterior residual uncertainty over more source distortion tradeoff connects to the communication-efficiency directions and therefore tends to lower the MSE and may shift viewpoint of LCBC. the waterfall to a smaller T . 1) Gaussian MIMO setup and dimensionality scaling: For Throughout this experiment, the channel and side(p) each user i and channel use t, the entries of Hi [t] ∈ RNi ×M information SNRs are matched for each user. For Fig. 4, we set are drawn i.i.d. as γc = γs and sweep their common value over {20, 40, 80} dB.   Three antenna configurations are compared in Fig. 4: (i) SISO 1 (p) [Hi [t]]a,b ∼ N 0, , (M = 1, Ni = 1), (ii) MISO (M = 2, Ni = 1), and (iii) M heterogeneous MIMO (M = 4, Ni ∈ {2, 2, 4, 4}). independently across (i, t). The 1/M variance normalization Because the per-channel-use/per-antenna power constraint is yields unit average gain per receive antenna, i.e., kept fixed, increasing T increases both the number of available h i (p) (p) observations and the total energy available over the block. E Hi [t]Hi [t]⊤ = INi . 2) Results: Interference-limited to noise-limited transition: As in (4), transmitting over T channel uses induces the Fig. 4 exhibits a consistent two-regime behavior, most visible stacked dimensions r = T M and mi = T Ni . We impose at high SNR. For small T , the distortion is relatively insensitive

11

Fig. 4. K = 4-user MIMO Gaussian WLCBC, n = 24 and (di , d′i ) = (4, 12): total MSE Etotal versus channel uses T for SNRch ∈ {20, 40, 80} dB. The antenna configurations are shown above each subplot. Recall r = T M and mi = T Ni in (4), and ∥E∥2F ≤ r in (7).

to SNR: even large reductions in noise yield only marginal MSE improvement. This is the interference-limited regime. The stacked channel does not yet provide enough effective spatiotemporal DoF to reliably resolve all users’ demand components, so the residual error is dominated by interference-limited geometry rather than noise. Once T exceeds a configurationdependent threshold T ⋆ , the distortion undergoes a sharp “waterfall” and drops by multiple orders of magnitude, after which the curves become strongly SNR-dependent, consistent with a noise-limited regime. The observed waterfall can be interpreted through a simple dimension count. The stacked transmit and receive dimensions are r = T M and mi = T Ni , so the channel-provided source subspace available to user i has dimension at most min{n, r, mi }. In the present random-subspace setting, (di , d′i ) = (4, 12) and di + d′i < n, so the demand generically contributes Fig. 5. Per-user MSE versus the number of channel uses T for K = 4, n = 24, four dimensions outside the side-information subspace. This (di , d′i ) = (4, 12), M = 2, and Ni = 1. Users 1–4 have SNRch,i = yields the necessary per-user condition min{T M, T Ni } ≥ 4, SNRside,i ∈ {10, 20, 40, 80} dB, respectively. corresponding to T ≥ 4 for SISO and MISO and T ≥ 2 for the bottleneck users in heterogeneous MIMO. These counts are consistent with the observed MISO and MIMO waterfalls, the channel use bottleneck rather than by noise. After this while the later SISO transition is consistent with additional bottleneck is relieved, the two curves separate sharply according multiuser coupling through the common transmit subspace. to their respective noise levels. Thus, SNR has limited leverage Also, for a target distortion δ, the first simulated T at which before sufficient transmission dimensions become available, the optimized MSE falls below δ provides an empirical estimate but becomes decisive afterward. The 10- and 20-dB curves of the required number of channel uses under the proposed instead decrease more smoothly because noise masks the sharp separation between the interference- and noise-limited regimes. WLCBC design. 3) Per-user distortion under heterogeneous SNRs: We revisit Since all users share one precoder, these curves reflect a coupled its MISO configuration with K = 4, n = 24, (di , d′i ) = (4, 12), multiuser allocation rather than four independent point-to-point M = 2, Ni = 1, and T ∈ {1, . . . , 10}. Users 1–4 have links. 4) AO convergence across SNRs: We next examine the SNRch,i = SNRside,i ∈ {10, 20, 40, 80} dB, respectively, and are jointly optimized with uniform weights and a common iteration-wise behavior of AO for a fixed heterogeneous-MIMO precoder. Fig. 5 reveals a regime-dependent role of SNR. For realization with K = 4, n = 24, (di , d′i ) = (4, 12), M = 4, T ≤ 4, the 40- and 80-dB curves remain close despite their (N1 , N2 , N3 , N4 ) = (2, 2, 4, 4), and T = 2. The common 40-dB SNR gap, and both undergo a waterfall near T = 5. channel and side-information SNR is 20, 40, or 80 dB. Their nearly common transition point suggests that, at high Because the pointwise minimum of non-increasing sequences SNR, the onset of the waterfall is governed primarily by is also non-increasing, the envelopes in Fig. 6 inherit the same

12

D. Experiment 3: WLCBC versus a matched direct-output MIMO-BC baseline We next compare WLCBC with a matched same-task directoutput MIMO-BC baseline. Define ⊤ ⊤ ⊤ Vtot ≜ [V1 , . . . , VK ], wtot ≜ [w1⊤ , . . . , wK ] = Vtot x.

Similar to traditional MIMO-BC schemes, the baseline considers only each user’s requested output streams and then transmits their linear mixtures: ⊤ sdir = Pwtot = PVtot x,

Edir = Vtot P⊤ .

(64)

Consequently, its feasible broadcast directions satisfy col(Edir ) ⊆ span(Vtot ) ⊆ | {z } Fig. 6. AO convergence (pointwise lower envelopes of the unclipped Etotal ) for (K, n, M, T ) = (4, 24, 4, 2), (di , d′i ) = (4, 12), (Ni )4i=1 = (2, 2, 4, 4), and common channel/side-information SNRs of 20, 40, and 80 dB.

TABLE II B EST MSE AMONG R RANDOM STARTS ON ONE FIXED MIMO REALIZATION . SNRch,side R

20 dB

40 dB

80 dB

1 5 20 50 100

1.6267 × 10−2 1.6267 × 10−2 1.6081 × 10−2 1.6074 × 10−2 1.6074 × 10−2

2.4701 × 10−4 2.2014 × 10−4 2.2014 × 10−4 2.0642 × 10−4 2.0642 × 10−4

1.3741 × 10−7 6.4918 × 10−8 5.4543 × 10−8 5.4543 × 10−8 5.4543 × 10−8

direct-output baseline can use

Rn |{z}

.

WLCBC can use

Thus, under the same transmit-dimension and power constraints, the direct-output baseline’s precoder is confined to the aggregate demand span span(Vtot ), whereas WLCBC can select precoding directions from the full source space Rn . We also equip the baseline with the same side-information-aware conditional LMMSE decoder as WLCBC, so that the comparison focuses on this encoder-subspace restriction rather than on decoder capability. We use the same setting as in Experiment 2, except that γc = γs = 40 dB. Since M = 2, the available transmit dimension is r = T M = 2T . The aggregate demand rank is q ≜ rank(Vtot ) = 16. For each realization, the two schemes use identical demand and side-information subspaces, channels, noise powers, and transmit-power constraints. As

descent property. The initial decrease is rapid: by iteration 10, the objective is reduced by factors of approximately 7, 34, and 66 at 20, 40, and 80 dB, respectively. The late-stage behavior, however, depends strongly on SNR. From iteration 50 to 250, the 20- and 40-dB curves decrease by only about 12% and 25%, whereas the 80-dB curve falls by nearly three additional orders of magnitude. This contrast is consistent with the lower-SNR cases becoming noise dominated relatively early, while the much lower noise floor at 80 dB keeps small residual errors visible to subsequent AO updates. The terminal values, 1.66×10−2 , 3.70×10−4 , and 4.62×10−8 , also track the corresponding noise-variance scales 10−2 , 10−4 , Fig. 7. Total MSE versus the number of channel uses T in Experiment 3, and 10−8 . Hence, the longer high-SNR tail reflects a lower with K = 4, n = 24, (di , d′i ) = (4, 12), M = 2, Ni = 1, γc = 40 dB, and γs = 40 dB for the SI-enabled schemes. The matched same-task observable error floor rather than weaker descent. We further examine the sensitivity of AO to random initialization using the same dimensions as in Fig. 6. For each R ∈ {1, 5, 20, 50, 100}, we report the minimum MSE among the R starts of the same random-start sequences. As shown in Table II, the best objective values stabilize after a moderate number of starts, with negligible improvement from R = 50 to R = 100. Thus, AO exhibits limited sensitivity to the random-start budget for this realization.

direct-output MIMO-BC baseline restricts col(E) to span(Vtot ), whereas WLCBC optimizes E over the full source space Rn . “SI” denotes receiver side information.

shown in Fig. 7, without side information, WLCBC and the direct-output baseline perform nearly identically. In this case, any precoding directions outside span(Vtot ) are uncorrelated with the requested outputs and merely waste transmit power. Thus, the optimal WLCBC precoder naturally lies within the aggregate demand span.

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The presence of side information changes the geometry of open several compelling avenues for future research. The useful transmitted directions. For user i, the relevant conditional real-valued formulation admits a direct extension to proper cross-covariance is Cov (wi , s | wi′ ) = Vi⊤ Πi E, where Πi = complex Gaussian baseband models by replacing transposes Cov(x | wi′ ) is the posterior residual covariance. Since Πi with Hermitian transposes and using the corresponding complex does not necessarily preserve span(Vtot ), a direction e ∈ covariance matrices. Robust transceiver designs under imperfect span(Vtot )⊥ may satisfy Vi⊤ e = 0 but Vi⊤ Πi e ̸= 0. Such a or statistical CSI remain an important direction for future direction is irrelevant to the demanded outputs in the marginal work. Furthermore, extending the current linear transceiver source geometry, but can become informative after conditioning framework to accommodate nonlinear computational demands on user i’s noisy side information. It therefore acts as a side- represents a promising frontier, particularly for functional information-complementary innovation direction. Although the delivery in distributed AI and machine learning applications. direct-output baseline exploits side information through its Finally, investigating hybrid digital–analog schemes could yield LMMSE receivers and the associated encoder optimization, additional robustness and hardware efficiency. its transmitted directions remain restricted to span(Vtot ). In contrast, WLCBC can additionally allocate transmit power to ACKNOWLEDGMENT such conditional innovation directions within the full source The authors acknowledge the use of ChatGPT (OpenAI) space. to improve the readability of this manuscript, to assist in At T = 4, incorporating side information reduces the MSE −1 developing simulation codes, and to support the formatting of the direct-output baseline from approximately 1.82 × 10 −3 of mathematical expressions. The authors have thoroughly to 5.51 × 10 , confirming that the baseline itself benefits subreviewed and edited the text, independently verified all anstantially from this additional information. However, WLCBC alytical derivations and simulation results, and assume full achieves a further order-of-magnitude improvement, dropping −4 responsibility for the overall content and integrity of the final the MSE to 4.17 × 10 —corresponding to an impressive publication. 92.4% reduction over the side-information-aided baseline. Conversely, at T = 8, the available transmit dimension fully accommodates the aggregate demand rank (r = q = 16). With R EFERENCES the baseline’s dimensional bottleneck largely eliminated, the [1] K. B. Letaief, W. Chen, Y. Shi, J. Zhang, and Y.-J. A. Zhang, “The performance gap between the schemes naturally narrows. roadmap to 6G: AI empowered wireless networks,” IEEE CommunicaVI. C ONCLUSION AND F UTURE W ORK This paper studied WLCBC over real Gaussian MIMO broadcast channels with noisy receiver-side information under a centralized linear transceiver design formulation. We formulated a WSMSE optimization problem under a unified stackedchannel power constraint and developed an efficient alternatingoptimization framework. The resulting AO procedure yields a monotonically non-increasing WSMSE objective sequence, and every accumulation point satisfies the KKT conditions of the joint design problem. For a fixed precoder, we established that the optimal LMMSE decoders admit an efficient implementation via a thin-SVD side-information reduction and a Schur-complement evaluation. For fixed decoders, the precoder update is cast as a convex QCQP, for which we derived a semi-closed-form solution parameterized by a single optimal multiplier. Numerical experiments demonstrated substantial robustness gains over classical two-user LCBC baselines in noisy regimes and revealed a clear channel-use-versusdistortion tradeoff across various antenna configurations. The results further illustrated the effects of heterogeneous per-user SNRs and the convergence behavior of AO. The matchedbaseline comparison also showed that, in the presence of side information, WLCBC can outperform a direct-output MIMOBC design by exploiting side-information-complementary directions in the full source space. In particular, the observed high-SNR distortion waterfall provides an empirical indication of a spatio-temporal transmission-dimension bottleneck in the considered finite-dimensional linear WLCBC design. While the proposed framework is developed for real-valued Gaussian channels under perfect CSI, its underlying principles

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Shuo Tan (Graduate Student Member, IEEE) received his B.E. degree in Computer Science (Excellence Plan) from Chongqing University, Chongqing, China, in 2025. He is currently pursuing the M.S./Ph.D. degrees with the Department of Electrical Engineering and Computer Science (EECS) at the University of California, Irvine (UCI), Irvine, CA, USA. He is a recipient of the UCI EECS Department Fellowship. His research interests include information theory and its applications.

Syed A. Jafar (Fellow, IEEE) received his B. Tech. from IIT Delhi, India, in 1997, M.S. from Caltech, USA, in 1999, and Ph.D. from Stanford, USA, in 2003, all in Electrical Engineering. His industry experience includes positions at Lucent Bell Labs and Qualcomm. He is a Chancellor’s Professor in the Department of Electrical Engineering and Computer Science and the Henry Samueli Endowed Chair in Engineering at the University of California, Irvine (UCI), Irvine, CA, USA. His research interests include multiuser information theory, wireless communications and network coding. Dr. Jafar is a recipient of the New York Academy of Sciences Blavatnik National Laureate in Physical Sciences and Engineering, the NSF CAREER Award, the ONR Young Investigator Award, the UCI Academic Senate Distinguished Mid-Career Faculty Award for Research, the School of Engineering Mid-Career Excellence in Research Award and the School of Engineering Maseeh Outstanding Research Award. His co-authored papers have received the IEEE Information Theory Society Paper Award, IEEE Communication Society and Information Theory Society Joint Paper Award, IEEE Communications Society Best Tutorial Paper Award, IEEE Communications Society Heinrich Hertz Award, IEEE Signal Processing Society Young Author Best Paper Award, IEEE Information Theory Society Jack Wolf ISIT Best Student Paper Award, and three IEEE GLOBECOM Best Paper Awards. Dr. Jafar received the UC Irvine EECS Professor of the Year award six times from the Engineering Students Council, a School of Engineering Teaching Excellence Award in 2012, and a Senior Career Innovation in Teaching Award in 2018. He was a University of Canterbury Erskine Fellow, an IEEE Communications Society Distinguished Lecturer, an IEEE Information Theory Society Distinguished Lecturer and a Thomson Reuters/Clarivate Analytics Highly Cited Researcher. He served as Associate Editor for IEEE TRANSACTIONS ON COMMUNICATIONS 2004-2009, for IEEE COMMUNICATIONS LETTERS 2008-2009 and for IEEE TRANSACTIONS ON INFORMATION THEORY 2009-2012.

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