Multi-Norm Beamforming Techniques

• Multi-norm beamforming is a technique that employs various norm formulations to model uncertainty and regularize beamformer outputs, enhancing robustness and sparsity.
• It encompasses methods such as robust adaptive beamforming, downlink 3D-MIMO, and speech enhancement, each leveraging different norm constraints for improved performance.
• Norm choice directly impacts SINR, power consumption, and computational efficiency, guiding optimization strategies across diverse signal processing applications.

Multi-norm beamforming denotes beamforming formulations in which several norms are used to model uncertainty, regularize the beamformer output, or both. In the cited literature, this includes robust adaptive beamforming for a general-rank signal model with matrix induced $\ell_{p,q}$-norm uncertainty, downlink 3D-MIMO beamforming with $\ell_1$-norm bounded CSI uncertainty and extensions to other vector and mixed norms, and speech enhancement beamforming that minimizes output power together with an $\ell_1$-norm penalty under a distortionless constraint (Huang et al., 2021, Liu et al., 2018, Qin et al., 24 Jul 2025). The common structure is that norm choice changes either the geometry of the admissible perturbations or the sparsity profile encouraged at the beamformer output.

1. Norms and uncertainty models

For $X\in\mathbb C^{M\times N}$, the induced matrix $\ell_{p,q}$-norm is defined by

$\|X\|_{p,q}=\max_{\|z\|_q=1}\|Xz\|_p.$

Equivalently,

$$\|X\|_{p,q}=\max\{\|X z\|_p:\;z\in\mathbb C^N,\;\|z\|_q=1\}.$$

In Huang and Vorobyov’s robust adaptive beamforming formulation, this norm is used to constrain matrix perturbations introduced into a factorization of the presumed desired-signal covariance. For the least-squares problem

$$\min_{\Delta:\;\|\Delta\|_{p,q}\le\eta}\;\|(X+\Delta)\,y-b\|_p,$$

the optimal value admits the closed form

$$\min_{\|\Delta\|_{p,q}\le\eta}\,\|(X+\Delta)y-b\|_p =\max\Bigl\{\|Xy-b\|_p-\eta\,\|y\|_q,\;0\Bigr\},$$

with equality obtained by a suitable rank-one construction of the worst-case $\Delta^\star$ when $\ell_1$0 (Huang et al., 2021).

In vector-uncertainty models for downlink beamforming, the same norm logic appears through channel-error sets. For user $\ell_1$1, the $\ell_1$2-bounded uncertainty set is

$\ell_1$3

The same framework also admits $\ell_1$4-norm uncertainty,

$\ell_1$5

mixed $\ell_1$6-norm uncertainty,

$\ell_1$7

and intersections of ellipsoids or hyper-rectangles. For a general $\ell_1$8-norm bound $\ell_1$9, the dual norm $\ell_1$0 satisfies $\ell_1$1, and the worst-case inner product obeys

$\ell_1$2

which is the basic device used to translate norm-bounded uncertainty into conic constraints (Liu et al., 2018).

2. General-rank robust adaptive beamforming

For a general-rank signal model, the presumed signal covariance is written as

$\ell_1$3

and the uncertainty is introduced through

$\ell_1$4

Applying the closed-form least-squares result yields the inner worst-case minimization

$\ell_1$5

The associated worst-case SINR maximization is

$\ell_1$6

which is reformulated, after dropping the zero-clamp since the optimum is nonnegative, as

$\ell_1$7

With $\ell_1$8, the final problem is

$\ell_1$9

This recasts worst-case SINR maximization as the maximization of the difference between an $X\in\mathbb C^{M\times N}$0-norm function and an $X\in\mathbb C^{M\times N}$1-norm function under a convex quadratic constraint (Huang et al., 2021).

The same paper studies a generalized RAB problem in which the $X\in\mathbb C^{M\times N}$2-term is replaced by an $X\in\mathbb C^{M\times N}$3-term and the matrix uncertainty is bounded by $X\in\mathbb C^{M\times N}$4. The inner result becomes

$X\in\mathbb C^{M\times N}$5

and the corresponding design is

$X\in\mathbb C^{M\times N}$6

The resulting family of beamformers depends explicitly on the choice of $X\in\mathbb C^{M\times N}$7 rather than only on the conventional Frobenius-norm case $X\in\mathbb C^{M\times N}$8 (Huang et al., 2021).

3. Sequential SOCP approximation and norm-pair selection

For any rational $X\in\mathbb C^{M\times N}$9, $\ell_{p,q}$0 has an SOC-representable epigraph, and $\ell_{p,q}$1 is handled through a lower-affine approximation at the current iterate $\ell_{p,q}$2: $\ell_{p,q}$3 Each SOCP subproblem is then

$\ell_{p,q}$4

which is fully SOC-representable for any rational $\ell_{p,q}$5. The iterative scheme initializes a feasible $\ell_{p,q}$6, solves the SOCP to obtain $\ell_{p,q}$7, increments $\ell_{p,q}$8, and stops when

$\ell_{p,q}$9

The sequence of objective values $\|X\|_{p,q}=\max_{\|z\|_q=1}\|Xz\|_p.$0 is nondecreasing, and under mild boundedness assumptions the iterates converge to a locally stationary point. The stated complexity per iteration is that solving an SOCP in $\|X\|_{p,q}=\max_{\|z\|_q=1}\|Xz\|_p.$1 variables with $\|X\|_{p,q}=\max_{\|z\|_q=1}\|Xz\|_p.$2 second-order cones of dimension up to $\|X\|_{p,q}=\max_{\|z\|_q=1}\|Xz\|_p.$3 costs on the order of $\|X\|_{p,q}=\max_{\|z\|_q=1}\|Xz\|_p.$4 or better using interior-point methods (Huang et al., 2021).

The generalized $\|X\|_{p,q}=\max_{\|z\|_q=1}\|Xz\|_p.$5-minus-$\|X\|_{p,q}=\max_{\|z\|_q=1}\|Xz\|_p.$6 formulation uses the same sequential-SOCP approach provided $\|X\|_{p,q}=\max_{\|z\|_q=1}\|Xz\|_p.$7, and $\|X\|_{p,q}=\max_{\|z\|_q=1}\|Xz\|_p.$8 if one wants to linearize $\|X\|_{p,q}=\max_{\|z\|_q=1}\|Xz\|_p.$9, are rational. The paper describes a practical norm-pair selection rule: one fixes a small finite candidate set of pairs $$\|X\|_{p,q}=\max\{\|X z\|_p:\;z\in\mathbb C^N,\;\|z\|_q=1\}.$$0, for example $$\|X\|_{p,q}=\max\{\|X z\|_p:\;z\in\mathbb C^N,\;\|z\|_q=1\}.$$1, runs the sequential-SOCP algorithm for each pair, computes the actual array-output SINR over held-out snapshots, and selects the $$\|X\|_{p,q}=\max\{\|X z\|_p:\;z\in\mathbb C^N,\;\|z\|_q=1\}.$$2 whose beamformer achieves the largest actual SINR. This suggests that norm choice is treated as a model-selection variable rather than as a fixed convention (Huang et al., 2021).

4. Downlink 3D-MIMO beamforming under vector and mixed norms

In downlink 3D-MIMO, robust beamforming is posed as minimization of total transmit power under worst-case SINR constraints over the uncertainty set $$\|X\|_{p,q}=\max\{\|X z\|_p:\;z\in\mathbb C^N,\;\|z\|_q=1\}.$$3. With beamformers $$\|X\|_{p,q}=\max\{\|X z\|_p:\;z\in\mathbb C^N,\;\|z\|_q=1\}.$$4,

$$\|X\|_{p,q}=\max\{\|X z\|_p:\;z\in\mathbb C^N,\;\|z\|_q=1\}.$$5

and the robust design is

$$\|X\|_{p,q}=\max\{\|X z\|_p:\;z\in\mathbb C^N,\;\|z\|_q=1\}.$$6

The reformulation introduces $$\|X\|_{p,q}=\max\{\|X z\|_p:\;z\in\mathbb C^N,\;\|z\|_q=1\}.$$7, auxiliary variables $$\|X\|_{p,q}=\max\{\|X z\|_p:\;z\in\mathbb C^N,\;\|z\|_q=1\}.$$8, and the constant $$\|X\|_{p,q}=\max\{\|X z\|_p:\;z\in\mathbb C^N,\;\|z\|_q=1\}.$$9, so that

$$\min_{\Delta:\;\|\Delta\|_{p,q}\le\eta}\;\|(X+\Delta)\,y-b\|_p,$$0

Using $$\min_{\Delta:\;\|\Delta\|_{p,q}\le\eta}\;\|(X+\Delta)\,y-b\|_p,$$1, one obtains

$$\min_{\Delta:\;\|\Delta\|_{p,q}\le\eta}\;\|(X+\Delta)\,y-b\|_p,$$2

With $$\min_{\Delta:\;\|\Delta\|_{p,q}\le\eta}\;\|(X+\Delta)\,y-b\|_p,$$3, row vectors $$\min_{\Delta:\;\|\Delta\|_{p,q}\le\eta}\;\|(X+\Delta)\,y-b\|_p,$$4, and $$\min_{\Delta:\;\|\Delta\|_{p,q}\le\eta}\;\|(X+\Delta)\,y-b\|_p,$$5, one further gets

$$\min_{\Delta:\;\|\Delta\|_{p,q}\le\eta}\;\|(X+\Delta)\,y-b\|_p,$$6

All constraints are linear or second-order-cone constraints, so the final problem is an SOCP solvable efficiently by a standard SOCP solver such as CVX (Liu et al., 2018).

The same work makes the multi-norm generalization explicit. For a general $$\min_{\Delta:\;\|\Delta\|_{p,q}\le\eta}\;\|(X+\Delta)\,y-b\|_p,$$7-norm uncertainty bound $$\min_{\Delta:\;\|\Delta\|_{p,q}\le\eta}\;\|(X+\Delta)\,y-b\|_p,$$8, the “numerator” constraint becomes

$$\min_{\Delta:\;\|\Delta\|_{p,q}\le\eta}\;\|(X+\Delta)\,y-b\|_p,$$9

with the corresponding dual norm $$\min_{\|\Delta\|_{p,q}\le\eta}\,\|(X+\Delta)y-b\|_p =\max\Bigl\{\|Xy-b\|_p-\eta\,\|y\|_q,\;0\Bigr\},$$0. The paper also notes that $$\min_{\|\Delta\|_{p,q}\le\eta}\,\|(X+\Delta)y-b\|_p =\max\Bigl\{\|Xy-b\|_p-\eta\,\|y\|_q,\;0\Bigr\},$$1 can be represented via linear constraints and a positive slack, $$\min_{\|\Delta\|_{p,q}\le\eta}\,\|(X+\Delta)y-b\|_p =\max\Bigl\{\|Xy-b\|_p-\eta\,\|y\|_q,\;0\Bigr\},$$2 is polyhedral and can be recast with auxiliary variables $$\min_{\|\Delta\|_{p,q}\le\eta}\,\|(X+\Delta)y-b\|_p =\max\Bigl\{\|Xy-b\|_p-\eta\,\|y\|_q,\;0\Bigr\},$$3, and mixed $$\min_{\|\Delta\|_{p,q}\le\eta}\,\|(X+\Delta)y-b\|_p =\max\Bigl\{\|Xy-b\|_p-\eta\,\|y\|_q,\;0\Bigr\},$$4 constraints lead to conic combinations of rotated-SOC and SOC constraints. The final result is that, for any desired $$\min_{\|\Delta\|_{p,q}\le\eta}\,\|(X+\Delta)y-b\|_p =\max\Bigl\{\|Xy-b\|_p-\eta\,\|y\|_q,\;0\Bigr\},$$5-norm or mixed-norm uncertainty, each worst-case term translates into either a linear, an SOC, or a rotated-SOC constraint, so the robust beamforming problem remains in the class of conic programs solvable in polynomial time (Liu et al., 2018).

5. Speech enhancement with power-plus-sparsity beamforming

In Qin et al., the beamforming stage follows a dual-path MCLP dereverberation step. At each frame-frequency bin, the early estimate is

$$\min_{\|\Delta\|_{p,q}\le\eta}\,\|(X+\Delta)y-b\|_p =\max\Bigl\{\|Xy-b\|_p-\eta\,\|y\|_q,\;0\Bigr\},$$6

The beamformer seeks $$\min_{\|\Delta\|_{p,q}\le\eta}\,\|(X+\Delta)y-b\|_p =\max\Bigl\{\|Xy-b\|_p-\eta\,\|y\|_q,\;0\Bigr\},$$7 that minimizes output power together with an $$\min_{\|\Delta\|_{p,q}\le\eta}\,\|(X+\Delta)y-b\|_p =\max\Bigl\{\|Xy-b\|_p-\eta\,\|y\|_q,\;0\Bigr\},$$8-norm penalty while preserving the target direction $$\min_{\|\Delta\|_{p,q}\le\eta}\,\|(X+\Delta)y-b\|_p =\max\Bigl\{\|Xy-b\|_p-\eta\,\|y\|_q,\;0\Bigr\},$$9: $\Delta^\star$0 Here $\Delta^\star$1 is the output power term, $\Delta^\star$2 is the sample-wise $\Delta^\star$3 norm over the complex output, $\Delta^\star$4 weights the sparsity penalty, and $\Delta^\star$5 is the steering vector of the desired source. The paper states that speech STFT frames are sparse in magnitude, and that the additional $\Delta^\star$6 penalty encourages the beamformer output to concentrate energy in a few TF bins, thereby further suppressing diffuse noise and small residual reverberation (Qin et al., 24 Jul 2025).

The optimization is handled by ADMM. Auxiliary variables $\Delta^\star$7 are introduced as scalar copies of the beamformer output,

$\Delta^\star$8

with dual variables $\Delta^\star$9 for these equalities and $\ell_1$00 for the distortionless constraint. The augmented Lagrangian is

$\ell_1$01

The $\ell_1$02-update is a small quadratic program with one linear constraint. Defining

$\ell_1$03

and

$\ell_1$04

one obtains

$\ell_1$05

The $\ell_1$06-update is complex soft-thresholding,

$\ell_1$07

and the dual updates are

$\ell_1$08

$\ell_1$09

Since the objective is convex in $\ell_1$10 and the constraints are affine, the paper states that ADMM converges to the global optimum under standard assumptions. In the complete pipeline, Stage 1 is dual-path MCLP, which removes late reverberation by minimizing $\ell_1$11 of the dereverberated multichannel signals via PALM, and Stage 2 is multi-norm beamforming, which removes spatial noise and further sharpens sparsity by minimizing output power plus $\ell_1$12 under a distortionless constraint via ADMM (Qin et al., 24 Jul 2025).

6. Reported performance, norm trade-offs, and scope

The reported numerical results emphasize that norm choice affects both beamformer quality and computational profile. In the induced $\ell_1$13-norm RAB experiments, for $\ell_1$14, $\ell_1$15, and an Intel Xeon E5-1620 v3 @3.5 GHz, the $\ell_1$16-design with $\ell_1$17 requires $\ell_1$18 s per trial, $\ell_1$19 requires $\ell_1$20 s, and $\ell_1$21 requires $\ell_1$22 s. Under $\ell_1$23 dB and SNR from $\ell_1$24 dB to $\ell_1$25 dB, the $\ell_1$26 beamformer outperforms $\ell_1$27 by $\ell_1$28–$\ell_1$29 dB SINR gain across SNR; the generalized design $\ell_1$30 yields a further $\ell_1$31 dB gain over $\ell_1$32; and the CPU–SINR trade-off reported is that $\ell_1$33 is fastest with second-best SINR, whereas $\ell_1$34 is slightly slower but gives the best SINR (Huang et al., 2021).

In the 3D-MIMO study, the $\ell_1$35-bounded uncertainty model is reported to consume less beamforming power than the conventional spherical uncertainty under the same SINR thresholds. At $\ell_1$36 and $\ell_1$37 dB, the $\ell_1$38-robust design needs approximately $\ell_1$39 units of power, whereas spherical $\ell_1$40-robust needs approximately $\ell_1$41, with the perfect-CSI baseline flat at approximately $\ell_1$42. For $\ell_1$43 and SINR target increasing from $\ell_1$44 to $\ell_1$45 dB, the $\ell_1$46-robust power grows to approximately $\ell_1$47 at $\ell_1$48 dB, the $\ell_1$49-robust design to approximately $\ell_1$50, and the perfect-CSI baseline to approximately $\ell_1$51 (Liu et al., 2018).

In the speech-enhancement setting, the multi-norm beamformer is reported to consistently outperform both the cascade WPE+MVDR and the unified WPD beamformer, which omits the extra $\ell_1$52 term, in PESQ and SI-SNR across a wide range of reverberation times $\ell_1$53 and SNRs, particularly in high reverberation scenarios (Qin et al., 24 Jul 2025).

Setting	Norm mechanism	Reported outcome
General-rank RAB	$\ell_1$54 and generalized $\ell_1$55 objective under matrix induced norm uncertainty	Actual array-output SINR and CPU-time vary with $\ell_1$56
Downlink 3D-MIMO	$\ell_1$57-bounded CSI uncertainty and extensions to other vector and mixed norms	Lower transmit power than spherical $\ell_1$58-uncertainty for the same worst-case SINR
Speech enhancement	Output power plus $\ell_1$59-norm penalty under a distortionless constraint	Better PESQ and SI-SNR than WPE+MVDR and WPD

A common simplification is to equate multi-norm beamforming solely with replacing an $\ell_1$60 or Frobenius uncertainty bound by $\ell_1$61. The cited works show a broader technical scope: multiple norms may appear in an induced matrix error model, in vector or mixed uncertainty sets, or directly in the beamformer objective as a joint power-and-sparsity criterion. The solver class also depends on the formulation rather than on the phrase “multi-norm” itself: sequential SOCP approximation is used for the nonconvex $\ell_1$62 RAB family, direct SOCP reformulation is used for the 3D-MIMO robust design, and ADMM is used for the speech-enhancement beamformer. This suggests that “multi-norm beamforming” is best understood as a family of norm-parameterized beamforming designs rather than as a single optimization template.