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Spectral-NFP: Certified Low-Rank Curvature Majorization for Accelerating WMMSE

Jianhang Zhu, Tsung-Hui Chang, Kaiming Shen

arXiv:2609.27369v1cs.ITmath.IT

TL;DR

Large-array WMMSE and FP transmit updates require costly dense operations, while scaled-identity curvature bounds can discard important eigenvalue structure. Spectral-NFP retains selected dominant curvature eigenpairs and bounds the residual subspace, enabling rank-controlled updates and a projected-gradient interpretation. In the reported experiments, low retained ranks achieve near-WMMSE WSR with substantially shorter update times.

  • Problem

    WMMSE and FP transmit updates require costly dense operations for large arrays, while replacing curvature with a scaled identity can discard eigenvalue structure and yield a loose lower bound.

    Source: §I · p. 1, §IV · p. 5

  • Method

    Spectral-NFP retains selected dominant curvature eigenpairs and upper-bounds the remaining curvature with a scaled identity, with retained rank controlling the transition from NFP toward the exact update.

    Source: §I · p. 2, §VII · p. 13

  • Results

    Across tested antenna dimensions from 32 to 512, ranks 8 and 16 attain at least 99.354% and 99.921% of WMMSE final WSR, respectively; at Nt = 512, their update times are 5.84% and 6.83% of WMMSE-EVD time.

    Source: §VII · p. 13

  • Takeaways & Limitations

    The retained rank offers a practical quality-cost tradeoff, with higher user loads and transmit powers favoring more retained curvature directions.

    Source: §VII · p. 13

    Under the finite-dimensional Wishart analysis, the ordered eigenvalue integral generally lacks a simple closed form and becomes difficult to evaluate as Nt grows.

    Source: B. Finite-Dimensional Wishart… · p. 9

Abstract

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Weighted sum-rate maximization in multicell multiple-input multiple-output (MIMO) networks is commonly addressed by the weighted minimum mean-square error (WMMSE) algorithm or fractional programming (FP), both of which, after fixing their auxiliary variables, solve a power-constrained quadratic transmit problem that requires costly dense operations for large arrays. Replacing the underlying curvature matrix with a scaled identity can avoid the matrix inverse operation and thereby reduce complexity, but it discards the curvature eigenvalue structure and yields a loose lower bound. We propose Spectral-NFP, which retains selected dominant curvature eigenpairs and uses a scaled identity matrix to bound the curvature on the remaining subspace from above. The retained rank thus traces a continuous path from NFP to the exact WMMSE transmit update. With the surrogate curvature fixed, Spectral-NFP can be interpreted as Euclidean projected-gradient ascent after a linear coordinate transformation, admitting Nesterov-type acceleration. We derive a lower bound on the one-step transmit-objective gain of Spectral-NFP relative to WMMSE, expressed in terms of the curvature eigenvalues. Under an idealized Wishart model, we analyze this bound in both finite dimensions and the large-system limit, obtaining an asymptotic rank-selection rule. Experimental results show that retaining at most 20% of the transmit dimension, and often less than 10%, achieves more than 99% of the WMMSE WSR. In large-array settings, the measured update time is below 20% of that required by WMMSE.

I. INTRODUCTION

The introduction frames multicell MIMO WSR maximization as a nonconvex beamforming problem whose WMMSE and FP transmit updates become costly for large arrays. Spectral-NFP preserves selected curvature directions to trade update quality against computation, and the paper analyzes this tradeoff and evaluates it in multicell settings.

I. INTRODUCTION

  • Global optimization can solve certain formulations optimally, but its complexity limits practical use to small- or moderate-sized instances; scalable stationary-point methods are therefore used for large coordinated arrays.Convex or conic formulations do not extend directly to general multicell WSR maximization, where beamformers affect desired signals and interference.Source: §I · p. 1
  • WMMSE and FP reduce the fixed-auxiliary transmit step to a power-constrained quadratic problem, but direct updates require repeated full-dimensional shifted-curvature inversions during multiplier bisection.Full eigendecomposition avoids those repeated inversions but still decomposes the full-dimensional curvature matrix.Source: §I · pp. 1–2
  • Spectral-NFP retains dominant curvature eigenpairs and upper-bounds the residual subspace with a scaled identity, tightening NFP's lower bound while avoiding a full-dimensional inverse.Increasing retained rank moves the update from NFP toward the exact WMMSE/FP transmit update, controlling the accuracy–complexity tradeoff.Source: §I · pp. 1–2
  • The theory bounds Spectral-NFP's one-step gain ratio relative to WMMSE using curvature eigenvalues, with Wishart-based finite-dimensional guarantees and a large-system rank-selection rule.For fixed surrogate curvature, the method also admits a projected-gradient interpretation after a linear coordinate transformation, motivating Nesterov acceleration.Source: §I · p. 2

II. WEIGHTED SUM-RATE MAXIMIZATION PROBLEM

  • The paper formulates the task as maximizing weighted user rates over transmit beamformers subject to each base station's power budget.The multicell MIMO setting includes desired signals, intracell and intercell interference, and channel state information from each base station to each user.Source: §II · p. 3

III. WMMSE TRANSMIT SUBPROBLEM AND NFP

With auxiliary variables fixed, FP and WMMSE yield the same concave quadratic transmit problem. The exact per-base-station update enforces its power constraint through a multiplier.

III. WMMSE TRANSMIT SUBPROBLEM AND NFP

  • With auxiliary variables fixed, FP's transformed objective is separately concave in the beamformers and auxiliary variables, enabling iterative optimization of those blocks.The quadratic transform follows the Lagrangian dual transform in recasting the original objective.Source: A. FP and WMMSE Transmit Update · p. 4
  • The transmit-dependent objective terms form a concave quadratic problem, and separate base-station power constraints yield per-cell updates with a power multiplier selected by bisection.WMMSE gives the same auxiliary-variable and transmit updates as FP for the considered WSR problem.Source: A. FP and WMMSE Transmit Update · p. 4

B. NFP as Isotropic Curvature Majorization

NFP simplifies the transmit update by majorizing curvature with a scaled identity, but this can make its quadratic lower bound loose. Spectral-NFP retains selected leading eigenpairs and bounds only the residual subspace.

B. NFP as Isotropic Curvature Majorization

  • NFP replaces the curvature matrix with a scaled identity, making the constrained surrogate update a residual step followed by Euclidean projection but assigning the largest eigenvalue to every direction.This removes matrix inversion and bisection from the update, yet can yield a loose lower bound when only a few eigenvalues are near the maximum.Source: B. NFP as Isotropic Curvature… · pp. 4–5

IV. PROPOSED SPECTRAL-NFP

  • Spectral-NFP preserves selected dominant eigenpairs and uses a scaled-identity upper bound only on the residual subspace, tightening the lower bound relative to NFP.The retained rank controls the quality-cost tradeoff, while scalar bisection enforces the per-base-station power constraint without matrix inversion inside the search.Source: §IV · p. 5

A. Rank-r Spectral Majorizer

  • The rank-r majorizer preserves the leading r curvature eigenvalues and replaces the discarded eigenvalues with the largest discarded eigenvalue.At r = 0 it is the NFP scaled identity; at r = Nt − 1 it equals the exact curvature matrix.Source: A. Rank-r Spectral Majorizer · p. 5

B. Spectral-NFP Transmit Update

With auxiliary variables fixed, Spectral-NFP upper-bounds curvature using retained eigenpairs and a scaled identity on the residual subspace, producing a tighter lower-bound surrogate than NFP. Its power-constrained update uses retained-subspace projections and scalar bisection, with rank connecting the NFP and exact WMMSE updates.

B. Spectral-NFP Transmit Update

  • The curvature majorizer lies between the original curvature and its largest-eigenvalue identity bound, so the surrogate remains a lower bound and is tighter than NFP.The functions touch at the current beamformer.Source: B. Spectral-NFP Transmit Update · p. 5
  • The beamformer update uses retained-eigenspace projections, and at r = Nt−1 the surrogate becomes the exact FP/WMMSE transmit update.At r = 0, the surrogate is NFP.Source: B. Spectral-NFP Transmit Update · p. 6
  • The per-base-station power multiplier is zero when the unconstrained power is feasible; otherwise scalar bisection enforces the power budget.The scalar power function is continuous and nonincreasing for η ≥ 0.Source: B. Spectral-NFP Transmit Update · p. 6
  • The algorithm alternates updates of Z, Y, Γ, the curvature majorizer, and the beamformers until the objective converges.It initializes beamformers at feasible values under the power constraint.Source: B. Spectral-NFP Transmit Update · p. 6

C. Projected Gradient in Transformed Coordinates

With the auxiliary variables and spectral majorizers fixed, a linear coordinate transformation makes the Spectral-NFP update Euclidean projected-gradient ascent on the transformed transmit objective. In the original coordinates, it is a Kℓ,r-metric projected-gradient method for fq.

C. Projected Gradient in Transformed Coordinates

  • When each Kℓ,r is positive definite, a bijective linear transformation converts the power constraints and transmit objective into transformed coordinates for projected-gradient ascent.The transformation is defined with the auxiliary variables and spectral majorizers held fixed.Source: C. Projected Gradient in Transformed… · p. 6
  • The projected-gradient update in transformed coordinates is equivalent to maximizing the Spectral-NFP surrogate and, in the original coordinates, is a Kℓ,r-metric projected-gradient method for fq.The transformed projection is Euclidean and applies jointly to beamformers subject to each cell power constraint.Source: C. Projected Gradient in Transformed… · pp. 6–7

D. Nesterov Acceleration

Spectral-NFP uses Nesterov-style extrapolation to form updates at extrapolated beamformers, without first projecting those beamformers onto the power-feasible set. Its leading-order cost is lower than WMMSE when the retained rank is much smaller than the transmit dimension.

D. Nesterov Acceleration

  • The Nesterov variant evaluates auxiliary variables and curvature matrices at extrapolated beamformers and applies the same power-constrained update without first projecting the extrapolated beamformers.The extrapolation coefficient follows Nesterov’s strategy.Source: D. Nesterov Acceleration · p. 7

1) Spectral-NFP Complexity:

2) WMMSE Complexity and Comparison:

  • For fixed user and stream counts, WMMSE has leading-order complexity O(Nt³), while WMMSE-EVD avoids matrix inversion inside the bisection search but retains the same cubic order.The exact update’s direct implementation inverts a shifted curvature matrix at each bisection step.Source: 2) WMMSE Complexity and Comparison · p. 7

E. Computational Complexity

V. THEORETICAL ANALYSIS

The theoretical analysis compares Spectral-NFP’s surrogate-certified one-step gain with the exact WMMSE transmit-update gain. It establishes rank-monotone gain guarantees and relates the guaranteed ratio to the residual curvature spectrum.

A. Certified Gain Relative to WMMSE

  • The rank-r surrogate gain certifies actual fixed-auxiliary improvement, and its ratio to the WMMSE gain lower-bounds the fraction of WMMSE improvement achieved when GW > 0.The comparison uses the same feasible current point and auxiliary variables.Source: A. Certified Gain Relative to WMMSE · pp. 7–8
  • The certified gain is nondecreasing with retained rank, and at r = Nt−1 the Spectral-NFP surrogate gain equals the WMMSE gain.The rank ordering applies for the same feasible current point, auxiliary variables, and exact rank-r spectral majorizers.Source: A. Certified Gain Relative to WMMSE · p. 8
  • The gain-ratio lower bound rises toward one as rank increases and discarded eigenvalues cluster near the smallest eigenvalue; a small λℓ,r+1 alone is insufficient.The residual spread must be small relative to the minimum regularized WMMSE curvature.Source: A. Certified Gain Relative to WMMSE · p. 8

B. Finite-Dimensional Wishart Certificate

The finite-dimensional certificate combines a deterministic gain bound with the ordered-eigenvalue density of an idealized Wishart curvature model. It gives a lower bound on the probability of attaining a target gain ratio, but its integral becomes difficult to evaluate as the transmit dimension grows.

B. Finite-Dimensional Wishart Certificate

  • The analysis models the curvature as a Gram matrix formed from effective-channel blocks whose entries are independent CN(0, 1), yielding a complex Wishart spectrum.The ordered eigenvalues of that spectrum provide the probability model used by the certificate.Source: B. Finite-Dimensional Wishart… · p. 9
  • The finite-dimensional certificate integrates the ordered-eigenvalue density over a region defined by a sufficient spectral condition for the target gain ratio.The condition removes the nonnegative, sample-dependent multiplier, so the resulting probability is a lower bound on attaining the target ratio.Source: B. Finite-Dimensional Wishart… · p. 9
  • For tolerated failure probability δ, the certificate selects the smallest retained rank r whose probability bound is at least 1 − δ.Because the events are nested in r, the criterion can be applied across ranks; the ordered Nt-dimensional integral becomes difficult to evaluate as Nt grows.Source: B. Finite-Dimensional Wishart… · p. 9

C. Asymptotic Rank-Selection Rule

The large-system analysis uses the limiting eigenvalue distribution and an extreme-eigenvalue result to relate retained rank fraction to the gain ratio. Under its Wishart-model conditions, this relation yields a rank-selection criterion, illustrated by a 90% asymptotic certificate at roughly 20% retained rank.

C. Asymptotic Rank-Selection Rule

  • The rank-selection analysis takes Nt and the factor count M to infinity at a fixed ratio under an i.i.d.-Gaussian factor model.The empirical distribution of normalized curvature eigenvalues converges almost surely to the Marchenko–Pastur distribution.Source: C. Asymptotic Rank-Selection Rule · p. 9
  • The Marchenko–Pastur quantile argument describes interior eigenvalues but does not determine the smallest eigenvalue, so a lower-edge limit is also needed.The quantile step applies when r/Nt stays away from both spectral endpoints.Source: C. Asymptotic Rank-Selection Rule · pp. 9–10
  • Under Proposition 4's conditions, the asymptotic gain-ratio bound yields a rank condition for achieving a prescribed target loss.The certificate uses the gain bound together with the limiting spectrum and rank scaling.Source: C. Asymptotic Rank-Selection Rule · p. 10
  • At c = 0.001 and target loss ϵ = 0.1, r/Nt = 0.2 gives an asymptotic expression of approximately 0.90046, while the 90% certificate requires at least 0.19627.In this example, c = 0.001 corresponds to M = 1000Nt, a strongly concentrated Wishart spectrum.Source: C. Asymptotic Rank-Selection Rule · p. 10

VI. NUMERICAL RESULTS

The numerical-results section compares Spectral-NFP with NFP and WMMSE across rate, runtime, scaling, SNR robustness, and the theoretical gain bound.

VI. NUMERICAL RESULTS

  • The evaluation compares Spectral-NFP with NFP and WMMSE in WSR and running time, as well as antenna and user-load scaling and SNR robustness.It also examines the theoretical gain bound.Source: §VI · p. 10
  • WMMSE-EVD is included as a WMMSE transmit-update implementation using a full EVD followed by scalar bisection in the eigenbasis.This implementation avoids repeated matrix inversions.Source: §VI · p. 10
  • The section assesses Spectral-NFP's performance against the theoretical gain bound in addition to empirical comparisons.Source: §VI · p. 10

A. Experimental Setup and Channel Scenarios

The experiments use a wrappedaround seven-cell network with distance-dependent path loss and independent small-scale phases. They average WSR across channel realizations and use common initialization and iteration settings across algorithms.

A. Experimental Setup and Channel Scenarios

  • The main scenario is a seven-cell wrappedaround hexagonal network with three sectors per cell and 800 m between adjacent base stations.Users are placed in their serving cells at least 300 m from their serving base station.Source: A. Experimental Setup and Channel… · p. 10
  • Each user has four receive antennas, and the main convergence comparison uses Nt = 256 with six users per cell.Scaling sweeps vary Nt from 32 to 512, user load from 3 to 24 per cell, and SNR from 70 to 120 dB.Source: A. Experimental Setup and Channel… · p. 10
  • Each setting averages WSR over 100 channel realizations, with algorithms initialized from the same RZF beamformers and run for 40 outer iterations.Unless otherwise stated, each user has two streams, weights are equal, each base station has power budget 100, and noise variance is 10−8.Source: A. Experimental Setup and Channel… · p. 10

B. Main Geometry-Based Multicell Results

On geometry-based multicell channels, low-rank Spectral-NFP approaches WMMSE performance across antenna, user-load, and SNR sweeps while reducing update time in large arrays. The paper also validates a curvature-spectrum lower bound on one-step surrogate gain and gives a rank-selection rule under an idealized Wishart model.

B. Main Geometry-Based Multicell Results

C. Scaling With Antenna Dimension and User Load

D. Robustness Across Operating SNR

E. Certified-Gain Validation

  • At r = 24, the mean and empirical 5th-percentile surrogate-gain ratios were 0.9953 and 0.9860, and the observed ratios satisfied the theoretical lower bound at all evaluated points.The corresponding mean lower bound was 0.9788.Source: E. Certified-Gain Validation · p. 12, Figure 6 · p. 12

VII. CONCLUSION

  • Retained rank controls the transition from scaled-identity NFP at r = 0 to the exact WMMSE/FP transmit update at r = Nt − 1, without a full curvature decomposition.The update retains leading curvature eigenpairs and upper-bounds the remaining eigenvalues.Source: §VII · pp. 12–13
  • The deterministic gain-ratio bound increases with retained rank; under an idealized Wishart curvature model, the analysis gives finite-dimensional probability guarantees and a large-system rank-selection rule.With fixed surrogate curvature, Spectral-NFP is equivalent to Euclidean projected-gradient ascent after a linear coordinate transformation, motivating a Nesterov-like extension.Source: §VII · p. 13
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