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Coherent but Discordant: Rotate, Don't Invert, to Backpropagate Through an Aging Channel

Xianzhong Li, Yuzhi Yang, Tianhao Guo

arXiv:2610.23137v1eess.SPcs.ITmath.IT

TL;DR

Physical backpropagation returns a receiver’s loss gradient through a reciprocal channel, but terminal motion can rotate that gradient and make coherence magnitude an inadequate guide. The paper analyzes scalar and array corrections for this rotation, including phase correction, polar correction, and displacement-based port re-indexing. In simulations, phase correction improves scalar training under high coherence, while array corrections recover performance near trace-correlation zeros.

  • Problem

    The paper asks whether a gradient returned through an aged reciprocal channel remains useful, since motion can rotate it even when channel magnitude changes little.

    Source: §I · p. 1, §II · p. 2

  • Method

    The paper models the conditional mean of the return channel and corrects scalar returns by correlation phase and array returns by a unitary polar factor or displacement-based port re-indexing.

    Source: §I · p. 2, E. Re-indexing Without Digital Feedback · p. 4, §VII · p. 7

  • Results

    51.7% accuracy uncorrected versus 92.5% phase-corrected for the four tested cells with |ρ| > 0.9 and |arg ρ| > 90°; near the array zero, polar correction improves on scalar phase correction by 0.276 CE.

    Source: §V · p. 5

  • Takeaways & Limitations

    In the tested settings, correcting the return’s rotation preserves useful gradient information that scalar correction misses near array correlation zeros.

    Source: E. Re-indexing Without Digital Feedback · p. 4, §V · p. 5, §VII · p. 7

    The rotation statement is exact under the scalar conditional-mean model with a complex-linear backward map; array claims describe the main term of an identity with a task-dependent residual, and polar correction is not a guarantee when that residual dominates.

    Source: §VI · p. 7

Abstract

from arXiv · show

A channel can be almost perfectly coherent and still teach a transmitter the wrong direction. In physical ("over-the-air") backpropagation, the receiver sends its loss gradient back through the reciprocal channel; when the terminal moves in between, mobility does not merely add noise to the returned gradient, it rotates it. Under a scalar conditional mean model the rotation is set by the phase of the temporal correlation, so coherence, which measures only its magnitude, is the wrong ruler: in simulation, channels with absolute value of rho greater than 0.9 drive uncorrected training to 51.7% accuracy, while phase correction restores 92.5%. Arrays add a twist. Near zeros of the trace correlation, scalar correction fails even with rho known exactly, yet most of the information is still there: noise has moved it onto neighbouring antennas. Re-indexing the ports by the displacement, with zero digital feedback, recovers the lost performance in simulation, and so does rotating the returned seed with the unitary polar factor of the conditional mean operator, never inverting it; aligning each antenna's phase with its own past makes things worse. An exact array identity explains all three. The lesson fits in one line: correct the rotation, not the stretch.

I. INTRODUCTION

The paper studies whether physical backpropagation remains useful when motion ages the reciprocal return channel, arguing that channel rotation—not coherence magnitude—determines gradient direction. It models the return through a conditional mean and develops scalar and array corrections for that rotation.

I. INTRODUCTION

  • Motion can leave channel magnitude nearly unchanged while rotating the returned gradient enough to point uphill, so coherence magnitude alone is the wrong measure.For arrays, motion can also move useful correlation from each antenna to its neighbour.Source: §I · p. 1
  • 96.9% accuracy with polar correction approaches 97.2% for CSI at matched RF use in the tested near-zero geometry task.The comparison uses 8000 versus 7998 RF uses; the reported bit counts are code ledgers, not a MAC-level design or resource-saving claim.Source: §I · p. 1
  • Under the scalar conditional-mean model, the return is counterrotated by the phase of temporal correlation, whose value is set by the projected angular spectrum.Removing mean Doppler alone leaves rotation from spectral asymmetry.Source: §I · p. 1
  • At trace-correlation zeros, scalar correction loses the main correlation term even though cross-port correlation survives; displacement-based port re-indexing recovers task performance with zero digital feedback.Aligning each antenna with itself makes performance worse.Source: §I · pp. 1–2
  • The unitary polar factor of the conditional-mean operator corrects array rotation without inverting the operator, keeping the main term positive at unit correction norm.Its covariance-weighted version is the unitary Procrustes solution maximizing the main term over corrections of norm at most one.Source: §I · p. 2

II. SYSTEM MODEL

  • The system sends the receiver’s loss-gradient seed back through reciprocity, and the transmitter applies its local backward map to form a parameter update.Motion separates the forward and return phases; the model uses a shared zero-mean Gaussian angular field and assumes finite angular grid, proper Gaussian coefficients, and independent transmit columns.Source: §II · p. 2
  • For arrays, the conditional mean of the return channel is A H_f, with A formed from lagged and zero-lag receive covariances; the trace ratio ρ summarizes this operator.For a single antenna, the temporal correlation is the characteristic function of projected motion at 2πd.Source: §II · p. 2

III. THE PHASE CRITERION AND ITS SCOPE

Under a scalar conditional-mean model, the expected returned gradient is the true gradient multiplied by the conjugate temporal correlation. Its phase sets the mean-descent condition, while array and optimizer behavior require additional qualifications.

III. THE PHASE CRITERION AND ITS SCOPE

  • E[ĝ | F] = ρ* g under the scalar conditional-mean assumptions, so the uncorrected update is a mean descent direction exactly when Re ρ > 0.The backward map must be complex-linear, and the channel innovation and reverse noise must have zero conditional mean.Source: §III · p. 2
  • For arrays, the conditional mean depends on the full matrix A, not only on trace correlation ρ; a negative eigenvalue of its Hermitian part permits ascent directions.At the stated broad-spectrum setting, d = 0.3 and Re ρ = 0.243, while the Hermitian part of A has eigenvalue −0.129.Source: §III · p. 2
  • In a deterministic scalar quadratic model, Re a > 0 is necessary but not sufficient for convergence, and the admissible step size vanishes as the phase approaches 90°.This is intuition, not a convergence guarantee for stochastic neural-network training.Source: §III · p. 2
  • With Adam, fixed scalar and phase-only corrections differ by a constant scale and yield nearly identical trajectories, but under shared-rate SGD that scale changes the effective step.The scale effect is large where |ρ̂| is small.Source: §III · p. 2
  • For real-linear backward maps, the descent boundary also depends on c; whether practical encoders produce a large arg c remains open.The reported real-linear test has |arg c| ≤ 0.11°.Source: §III · p. 2

A. An Exact Criterion for Arrays

The array criterion separates a covariance-weighted trace main term from a task-dependent residual. It gives an exact descent test and shows that task-independent correction requires exact inversion.

A. An Exact Criterion for Arrays

  • The criterion uses a task-dependent matrix T whose trace is the squared norm of the batch-averaged true gradient.The formulation assumes nonzero batch-averaged gradient, positive-definite C00, and complex-linear sample maps.Source: A. An Exact Criterion for Arrays · pp. 2–3
  • Theorem 1 decomposes the array alignment into a covariance-weighted trace main term and a residual bounded by ∥T_r∥_F ∥B_0∥_F.A sufficient condition for mean descent is Re ρ_B > α∥B_0∥_F, with α = ∥T_r∥_F/∥ḡ∥².Source: A. An Exact Criterion for Arrays · p. 3
  • For an effective return operator B, mean descent is exactly characterized by Re tr(TᴴB) > 0; the uncorrected return has trace main term equal to ρ*.The trace condition is a sufficient summary, not a necessary criterion, because the residual can dominate.Source: A. An Exact Criterion for Arrays · p. 3
  • A correction works for every task with positive real gradient trace if and only if B = βI with Re β > 0; for invertible A, this requires M = β(Aᴴ)⁻¹.Thus exact inversion is the task-independent correction, with correction norm |β|/σ_min(A).Source: A. An Exact Criterion for Arrays · p. 3

IV. CONSEQUENCES FOR MOBILE GEOMETRY

A local expansion of temporal correlation separates phase and magnitude effects through cumulants of projected motion. The phase–Doppler relation is used to interpret training behavior.

IV. CONSEQUENCES FOR MOBILE GEOMETRY

  • Locally in displacement, the expansion of log ρ uses the cumulants κ_n of projected motion, requiring ρ ≠ 0 and a continuous logarithm branch.The expansion holds for sufficiently small d.Source: §IV · p. 3
  • Odd cumulants set the local correlation phase, while even cumulants set its magnitude.Source: §IV · p. 3
  • The relation between correlation phase, mean Doppler, and the third spectral moment is used to interpret training behavior.The relation itself is classical in Doppler measurement.Source: §IV · p. 3

A. Coherent but Discordant

For narrow clusters, coherence can remain high while the temporal-correlation phase grows with displacement. The narrow-cluster approximation predicts the 90° crossing, but coherence magnitude alone does not determine mean descent.

A. Coherent but Discordant

  • |rho| stays near one while the phase grows linearly for a narrow cluster.This occurs when κ2 ≈ 0 and κ1 ≈ cos(θ̄ − θm).Source: A. Coherent but Discordant · p. 3
  • At θ̄ − θm = 35°, d90 is 0.305; exact first crossings are 0.313 for κ = 20 and 0.308 for κ = 50.The approximation requires cos(θ̄ − θm) to be bounded away from zero and fails for near-perpendicular motion.Source: A. Coherent but Discordant · p. 3
  • Coherence measures |rho|, whereas mean descent depends on arg rho.Source: A. Coherent but Discordant · p. 3

B. When Mean-Doppler Derotation Suffices

Mean-Doppler derotation can leave a residual phase, so scalar phase correction may still be needed. At a trace-correlation zero, array cross-port correlation can remain substantial and is inaccessible to scalar correction.

B. When Mean-Doppler Derotation Suffices

  • For a broad spectrum at d = 0.3, Re ρ̃ = 0.498 while the corresponding de-meaned predictor has an indefinite Hermitian part.Thus, positive scalar-model evidence after derotation does not ensure descent for arrays.Source: B. When Mean-Doppler Derotation Suffices · p. 3
  • Derotating by mean Doppler leaves a residual phase led by the third cumulant; for isotropic scattering, rho changes sign despite κ1 = κ3 = 0.For isotropic scattering, derotation cannot correct the resulting 180° rotation; a pilot-estimated phase correction is a direct remedy.Source: B. When Mean-Doppler Derotation Suffices · p. 3

C. Zeros of the Trace Correlation

  • At the first isotropic trace-correlation zero, the whitened cross-covariance retains singular values 0.999, 0.936, and 0.009.Motion brings array elements close to former neighboring positions, so scalar corrections cannot exploit the surviving cross-port structure.Source: C. Zeros of the Trace Correlation · p. 3

D. Polar (Unitary Procrustes) Correction

Polar correction removes rotation while retaining stretch, producing a positive main term without inversion. It maximizes a specified trace objective, and its weighted version is non-inferior to the unweighted version in the tested near-zero setting.

D. Polar (Unitary Procrustes) Correction

  • At the isotropic first zero, polar correction raises ρB from approximately 0 uncorrected to 0.576, with lower bound 0.445.It removes cancellation in the main term, but does not guarantee descent for every task.Source: D. Polar (Unitary Procrustes) Correction · pp. 3–4
  • The polar factor removes rotation while leaving the stretch, and its correction norm is one rather than an inverse that can amplify noise.The conditional innovation-power bound is the same as for an uncorrected return; actual power depends on alignment with Q and can exceed it.Source: D. Polar (Unitary Procrustes) Correction · p. 3
  • Among corrections with norm at most one, the covariance-weighted choice maximizes the main term, while the polar correction maximizes the unweighted trace.Source: D. Polar (Unitary Procrustes) Correction · p. 4
  • The weighted and unweighted solutions are non-inferior within a CE margin of 0.02 in the tested near-zero setting.They differ only through C00.Source: D. Polar (Unitary Procrustes) Correction · p. 4

E. Re-indexing Without Digital Feedback

Displacement-based port re-indexing recovers neighboring cross-port correlations without digital feedback, but the tested rule is limited to positive axial motion and its performance depends on normalization and noise. Across tested displacements, the shift rule and broad-spectrum shift-plus-phase variant show strong results, with qualifications at tie points.

E. Re-indexing Without Digital Feedback

  • For isotropic scattering at d ≈ 0.383 with Δ = 0.5, diagonal correlation is zero while adjacent-port correlation is approximately 0.869.The displacement-based re-indexing pairs a return port with the former position of a neighboring forward port.Source: E. Re-indexing Without Digital Feedback · p. 4
  • The tested mapping discards the last k input components and silences the first k output ports, without wraparound or power redistribution.The tested rule covers positive axial motion with k < Nr; negative motion and all-ports-unmatched cases are untested.Source: E. Re-indexing Without Digital Feedback · p. 4
  • The shift sends Skδ/L with fixed L, no pilots, correction bits, or per-sample norm field, but still uses an analog return.Silent ports do not reduce RF slots; every example still has a return observation.Source: E. Re-indexing Without Digital Feedback · p. 4
  • At all 20 non-tie points, Sk⋆ has the lowest eight-seed mean CE among S0, S1, and S2, with maximum mean regret zero within the 0.02 criterion.This is not per-seed optimality or a universal geometric theorem.Source: E. Re-indexing Without Digital Feedback · p. 4
  • At tie points, only 5/8 seeds change sign in each endpoint pair, and the negative-endpoint intervals include zero, so the data do not establish a population sign change.The reported tie-point mean regrets are descriptive.Source: E. Re-indexing Without Digital Feedback · p. 4
  • CE(shift+phase)−CE(polar) = 0.000999, with 95% interval [0.000174, 0.001930], supporting non-inferiority at margin 0.02 while retaining a small detectable deficit.Both use 8000 RF; shift+phase uses 12 304 control bits versus polar’s 74 016.Source: E. Re-indexing Without Digital Feedback · p. 4

V. NUMERICAL RESULTS

Across the tested geometries, channel coherence alone did not predict whether an aged physical return remained useful: phase correction recovered performance when highly coherent channels rotated the gradient. Near array correlation zeros, spatial correction and port re-indexing recovered task performance, while residual effects and limits remained.

V. NUMERICAL RESULTS

  • Experiments used a three-element ULA, two transmit ports, and 128 angular components, with eight independent seeds except for the 16-seed direction check.The two tasks were a three-bit agreement geometry task and an eight-message task; methods and displacements shared paired seeds.Source: §V · p. 4
  • Across 16 geometries, cos(arg ρ) and the cosine of the conditional-mean return with the true task-averaged gradient agreed in sign, with maximum deviation 0.0057.The agreement was empirical for this task, not an implication of the stated proposition.Source: Figure 1 · p. 5, §V · p. 5
  • With estimated A, polar correction improved on scalar phase correction near the zero by 0.276 CE ([0.253, 0.305], 8/8) and was non-inferior to delaying the return.It passed the 0.02 CE non-inferiority margin in all 16 prespecified comparison cells.Source: §V · p. 5
  • With |ρ| > 0.9 and |arg ρ| > 90°, accuracy was 51.7% uncorrected, 92.5% phase-corrected, and 93.0% with CSI.The phase-correction CE gain was 0.454 (95% CI [0.433, 0.479]).Source: §V · p. 5, Figure 2 · p. 6
  • At d = 0.5 in the broad spectrum, accuracy was 61.7% with derotation versus 96.5% with phase correction.Across displacements, phase correction's CE advantage was 0.109 ([0.102, 0.114]); in the message task at d = 0.55, BLER was 5.17% with derotation versus 2.08% with phase correction.Source: §V · p. 5, Table II · p. 5, Figure 3 · p. 6
  • At d = 0.38, the uncorrected main term was 0.009∥¯g∥2 and 3 of 16 seeds ascended, whereas with polar correction it was 0.571∥¯g∥2 and all 16 descended.Across randomly drawn states not visited during training, residuals overturned positive main terms in 3.2% of uncorrected and 7.7% of polar-corrected pairs; training-state frequency was not measured.Source: §V · p. 6
  • Near the trace-correlation zero, port re-indexing improved on scalar phase correction by 0.257 CE ([0.219, 0.302]) and reached 96.8–96.9% accuracy, versus 96.9% for polar correction and 97.2% for CSI.Re-indexing used no pilots or digital feedback; per-antenna phase correction averaged 52.6–64.2% accuracy across the near-zero displacements.Source: §V · p. 6, Figure 4 · p. 7
  • Under SGD, where |ρ| ≤ 0.15, scalar correction was worse than phase correction by 0.267 CE ([0.224, 0.306]), with accuracy of 60.5% versus 86.7%.Correction factors reached 174 before projection; a magnitude floor's effect under SGD was not tested.Source: §V · p. 7

VI. DISCUSSION AND LIMITATIONS

The reported results are limited to the tested channel model, two small tasks, and optimizer settings. The proposed corrections also have specific boundaries: re-indexing requires a uniform linear array, known displacement, and leaves unmatched edge ports silent; polar correction is not a universal remedy.

VI. DISCUSSION AND LIMITATIONS

  • Results are simulations of two small tasks under a zero-mean Gaussian angular field with Kronecker transmit statistics and no line of sight.Samples are independent realizations at fixed displacement, and only Adam and plain SGD were tested.Source: §VI · p. 7
  • Port re-indexing assumes a uniform linear array moving along its axis with known displacement, and leaves edge ports without partners silent.For other geometries or motion directions, re-indexing does not apply directly.Source: §VI · p. 7
  • Polar correction removes cancellation in the main term but is not a guarantee, because the task-dependent residual can dominate in some parameter states.How often training visits those states is open; CSI-based training is stronger in aggregate in the tested settings.Source: §VI · p. 7

VII. CONCLUSION

Channel aging rotates returned gradients, while array motion can move useful correlation to neighbouring antennas. The conclusion is to correct that rotation rather than the channel's stretch.

VII. CONCLUSION

  • For scalar returns, correct the rotation with the phase of ρ; for arrays, use the unitary polar factor or re-index ports to their former positions.Port re-indexing applies to a uniform array moving along its axis.Source: §VII · p. 7
  • Near trace-correlation zeros, scalar correction fails because motion moves useful correlation from each antenna to its neighbour.The cross-port correlation retains most of the information.Source: §VII · p. 7
  • A single antenna can only trade time for information, whereas an array can also trade space.Source: §VII · p. 7

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