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Negative Results for Reciprocity-Based Over-the-Air Split Learning Under Channel Aging: Stability, Alignment, and Geometry

Tianhao Guo, Xianzhong Li, Hao Luo

arXiv:2610.37188v1eess.SPcs.ITmath.IT

TL;DR

Channel aging can rotate and attenuate the derivative returned in reciprocity-based over-the-air split learning. Across seven case studies, the results show that local physical relations alone do not establish training advantages; predictions need links to task support, noise injection, and update moments.

  • Problem

    Channel aging rotates and attenuates the returned derivative in reciprocity-based over-the-air split learning, motivating tests of proposed remedies.

    Source: §I · p. 1

  • Method

    Seven case studies test performance forecasts and mechanisms, using a task-subspace second-moment recursion for the quadratic proxy.

    Source: §X · p. 6

  • Results

    Across tested settings, results were mixed: truncation slightly favored the archived cross-entropy comparison, power-based selection missed its predicted advantage, nonsymmetric assignment succeeded, and jitter outperformed fixed bias.

    Source: §II · p. 2, §III · p. 3, §VII · p. 5

  • Takeaways & Limitations

    Training-advantage predictions should specify task support, noise injection, and relevant update moments rather than infer gains from local phase, norm preservation, or spatial alignment.

    Source: §X · p. 6

    Empirical conclusions are limited to the specified small synthetic binary classifiers, channel laws, correction rules, and finite training budgets; larger datasets, deeper networks, regression, and hardware deployments were not tested.

    Source: §IX · p. 6

Abstract

from arXiv · show

In reciprocity-based over-the-air split learning, channel aging rotates and attenuates the derivative signal returned to the transmitter. Seven case studies combine numerical forecasts, mechanism checks, and competing predictions. Near the correlation zero, an archived SGD comparison finds truncated inversion 0.000813 lower in cross-entropy than polar correction, with a paired 95% interval for the difference from -0.001853 to 0.000060, and a retained-mode quadratic proxy slightly favors truncation. Selecting the return port with the largest received power falls short of the predicted five-point alignment advantage over random selection at both tested spacings, gaining at most 0.97 points, while the local inverse relation between amplitude and phase gradient is supported. Seven specified interference-induced accuracy-collapse predictions are rejected, and independent zero-mean interference preserves the conditional mean under a fixed linear chain. Per-subcarrier correction and spherical-wavefront compensation do not deliver their predicted accuracy gains. Focusing leaves both methods near chance in a single-mode model and adds no observed gain in a model with a line-of-sight component plus scattering. A successful port assignment under nonsymmetric motion refutes an exact-symmetry requirement. The competing positioning-error prediction is supported. Gaussian rail jitter with a standard deviation of 0.383 wavelengths yields high terminal test accuracy, whereas a fixed bias of the same magnitude yields near-chance accuracy. These cases distinguish valid local physical relations from unsupported training forecasts through task support, noise injection, and update moments.

I. SETTING AND EVIDENCE PROTOCOL

In reciprocity-based over-the-air split learning, the reverse channel applies the adjoint to a returned loss derivative, but terminal motion can rotate and attenuate it. This paper records forecasts that did not hold, separating performance tests, mechanism counterexamples, and supported competing predictions.

I. SETTING AND EVIDENCE PROTOCOL

  • When a terminal moves between forward and reverse transmissions, channel aging rotates and attenuates the returned derivative.The receiver sends the loss derivative back through the reverse link, where time-division-duplex reciprocity lets the channel apply the required adjoint.Source: §I · p. 1
  • The comparisons test proposed remedies including avoiding matrix inversion, selecting the strongest port, suppressing interference, correcting each subcarrier, and exploiting array symmetry.The study also reports analytic counterexamples.Source: §I · p. 1
  • The paper records forecasts that did not hold, distinguishing rejected performance forecasts, mechanism counterexamples, and supported competing predictions.Companion studies report designs that survived in this setting: movable antennas, waiting out single-antenna dead zones, and statistical repair at a fixed array [1], [2], [3].Source: §I · p. 1
  • Training predictions were fixed before each run, and the reported comparisons use eight seed-level repetitions per configuration.Accuracy is terminal test accuracy, cross-entropy uses natural logarithms, and accuracy differences are percentage points; subcarriers, examples, and time steps are not additional independent training repetitions.Source: §I · p. 1

A. Tasks and training configurations

The experiments use synthetic binary classifiers and specified quadratic and channel-only tests to examine seven proposed forecasts. The evidence separates performance rejections, mechanism counterexamples, and supported competing predictions.

A. Tasks and training configurations

  • Classifiers use synthetic three-coordinate binary inputs, a complex linear encoder, a componentwise tanh(|y|^2) receiver operation, and a real linear logistic head.Each configuration uses batch size 32, eight evaluation seeds, and 8192 test examples, per tone for wideband tests.Source: A. Tasks and training configurations · p. 1
  • Table I specifies classifier configurations, including encoder and receiver dimensions, per-tone training, noise entries, and CSI-arm use.Its caption states that interference and phase distortion are additional to return thermal noise.Source: Table I · p. 2
  • The quadratic test uses a two- or three-mode task, while channel-only measurements use four transmit components and eight candidate return ports.The quadratic update uses eight seeds; channel-only measurements use eight seeds and 8192 independent fields per seed, without labels, networks, or training updates.Source: A. Tasks and training configurations · p. 2
  • Across seven cases, port-selection, interference-collapse, wideband-gain, and focusing-gain forecasts cross their numerical rejection lines.Nonsymmetric port assignment counters a necessity claim; the local amplitude–phase relation and the competing bias-versus-jitter prediction receive support.Source: A. Tasks and training configurations · p. 2

II. UNITARY POLAR CORRECTION VERSUS TRUNCATED INVERSION

Near the correlation zero, the nearby archived SGD comparison slightly favors truncated inversion in cross-entropy, but its paired interval spans zero. The quadratic proxy favors truncation on retained modes while showing that polar correction can be unstable when all modes are active.

II. UNITARY POLAR CORRECTION VERSUS TRUNCATED INVERSION

  • The proposed polar advantage extrapolated from noise amplification along weak singular directions and the norm-preserving polar approximation [5, Sec. 8.3.1],.The archived forecast required polar CE ≤0.08, truncated-inverse CE ≥0.28, and a gap exceeding 0.20; the precise neural-network comparison remains untested.Source: §II · p. 2
  • −0.00081335 is the archived SGD cross-entropy difference (truncated inverse minus polar), with paired 95% bootstrap interval [−0.001853, 0.000060].The quadratic stationary squared-gradient difference on the two retained task modes is −4.03688 × 10−7 (TSVD minus polar); stationary quadratic energy cannot be converted into a CE forecast.Source: §II · p. 2, Table II · p. 3
  • With all three modes active, the second-moment operator’s lifted spectral radius exceeds one for polar correction with innovation, while TSVD leaves an undamped noisy direction.The recursion’s spectral radius, rather than norm preservation of the correction matrix, controls whether a finite second-moment steady state exists [7].Source: §II · p. 3
  • The fixed-frame quadratic model and nearby SGD archive require comparisons to specify task support, noise injection point, and stability domain.Hard truncation deletes the weak-mode inverse gain; when the task uses only retained modes, that deletion need not impose the proposed noise penalty.Source: §II · p. 3

III. PORT SELECTION BY RECEIVED POWER AND CROSS-EXCHANGE PHASE ALIGNMENT

The predicted alignment advantage from selecting the highest-power return port does not appear at either tested spacing, although the local amplitude–phase-gradient relation is supported. The results distinguish local brightness from persistent cross-exchange alignment.

IV. PORT SELECTION BY RECEIVED POWER AND CROSS-EXCHANGE PHASE ALIGNMENT

  • −0.647 pp at the finer spacing and +0.970 pp at the coarser spacing: max-power selection misses the predicted 5-pp alignment advantage over random selection.Both global comparisons cross the rejection line even though the local relation holds.Source: §III · p. 3
  • 0.08694 is the local bright-to-dark phase-gradient ratio, with interval [0.08320, 0.09042], supporting the predicted local relation.The local relation does not establish that the brightest return port preserves alignment across exchanges.Source: §III · p. 3
  • 89.169% accuracy in the training-channel replay coincides with only 63.641% positive channel alignment and mean cosine 0.29121.An unconditional channel cosine does not provide the conditional gradient alignment, second moments, and step sizes used in standard descent analyses.Source: §III · p. 3

IV. CONDITIONAL-MEAN INVARIANCE UNDER UNSUPPRESSED INTERFERENCE

The specified interference-related collapse forecasts are rejected in the tested settings. Under a fixed linear chain, independent zero-mean contamination preserves the conditional mean, while the tested ranking metrics do not establish a single explanation for historical ZF loss.

IV. CONDITIONAL-MEAN INVARIANCE UNDER UNSUPPRESSED INTERFERENCE

  • 0.0382% for white noise and 0.8835% for multiuser interference are the relative changes in the fixed-chain complex gradient projection coefficient, below the 3% support target.The complete-vector GSNR rank correlation is 0.3466, below its 0.80 line; the scalar proxy is 0.8556, showing the definitions are not interchangeable.Source: §IV · pp. 3–4
  • ZF loses 8.0261 pp under SGD and 7.2891 pp under Adam versus their respective single-user controls, but these observations leave the historical loss without a single established explanation.The loss ratio is 1.1011, below the frozen rejection line of 2 for optimizer amplification.Source: §IV · p. 4
  • For a fixed linear correction chain, independent zero-mean contamination preserves the conditional mean and can increase variance without creating a systematic reverse direction.Interference power alone is therefore insufficient to predict collapse.Source: §IV · p. 4
  • For the frozen linear, independent-message, and AP proxies, all seven specified collapse predictions fail.The tests cover unsuppressed corrected returns, a best single access point, projected returns with self-interference, and centralized correction with AP phase jitter.Source: §IV · pp. 3–4

V. PER-SUBCARRIER CORRECTION IN WIDEBAND AGING CHANNELS

Per-subcarrier correction was predicted to outperform a shared statistical polar correction, but none of the three tests met its predicted gain. The results show that instantaneous frequency selectivity need not produce useful dispersion in the statistical forward–return operator.

V. PER-SUBCARRIER CORRECTION IN WIDEBAND AGING CHANNELS

  • The tests predicted that global correction would fail and local correction would gain at least 10, 6, and 4 pp in D1–D3, respectively.D1 used independent taps with common spatial statistics; D2 used distinct tap angles; D3 also made array and motion phases frequency dependent at fractional bandwidth 0.20.Source: §V · p. 4
  • All three relative local-minus-global accuracy advantages crossed their rejection lines (Table IV); in D3, the gain was 0.008202 pp, with paired interval [−0.005722, 0.020981] pp.D3's CE advantage was −0.000014820, with an interval spanning zero.Source: §V · p. 4, Table IV · p. 4
  • In D1, same-tone delay phases cancel inside each covariance, yielding common forward–return operators across tones.In D2, invertible V gives A_k = VDV^−1 for every tone, so neither protocol creates the proposed statistical dispersion.Source: §V · p. 4
  • D3 produced real operator dispersion, with maximum pairwise Frobenius distance 0.8375, but no material learning advantage for local correction at the prescribed endpoint.The paper recommends checking forward–return operator dispersion before paying for separate calibration.Source: §V · p. 4

VI. SPHERICAL-WAVEFRONT FOCUSING AND TEMPORAL DECORRELATION

The paper tested whether spherical-wavefront focusing restores learning under temporal decorrelation. It found no predicted near-field advantage in either a single-mode model or a model with a line-of-sight component and scattering.

VI. SPHERICAL-WAVEFRONT FOCUSING AND TEMPORAL DECORRELATION

  • E1 predicted at least 95% near-field accuracy and a 15 pp gain, while E2 predicted at least 96% accuracy and a 20 pp gain.E1 is a single-mode model; E2 adds scattering to a LOS-plus-scattering model with Rician factor 3.Source: §VI · p. 5
  • The near-field accuracy advantage was −0.2029 pp in E1 and −0.0092 pp in E2, triggering both rejection rules.Both corrections performed poorly in E1, while both performed well in E2.Source: §VI · p. 5, Table V · p. 5
  • In E1, spatial alignment cannot increase the magnitude of the random scalar temporal gain's correlation, which was approximately −0.000848 at the tested displacement.Spherical and common radial phase compensation already overlapped by 0.992546 in power.Source: §VI · p. 5
  • In E2, the coherent LOS component retains a cross-time outer product when scattering decorrelates, so both corrections retain useful LOS information.Its high near-field accuracy therefore does not show that curvature recovered lost scattering coherence.Source: §VI · p. 5
  • The two models separate spatial phase compensation from temporal predictability.The focusing forecast was motivated by coherent aperture gain in near-field beamforming.Source: §VI · p. 5

VII. TRACE-MAXIMIZING PORT ASSIGNMENT UNDER NONSYMMETRIC MOTION

A covariance-weighted port assignment achieved high accuracy under nonsymmetric motion, refuting exact motion symmetry as a requirement for successful permutation. Further nonsymmetric directions also yielded high assignment-selected accuracies.

VII. TRACE-MAXIMIZING PORT ASSIGNMENT UNDER NONSYMMETRIC MOTION

  • At displacement 0.5λ and direction 30°, a complete seven-port permutation reached 99.5392% accuracy, with a 2.1875% negative-gradient fraction in the return-noise-free diagnostic.The frozen prediction was accuracy below 65% for every nonsymmetric permutation and a negative-gradient fraction of at least 50%.Source: §VII · p. 5
  • Assignment-selected accuracies ranged from 99.5422% to 99.6841% on further nonsymmetric directions.The permutation maximizes the covariance-weighted score Re tr(C00AHΠ).Source: §VII · p. 5
  • The successful nonsymmetric assignments reject exact motion symmetry as a requirement for port permutation.Exact symmetry was a working hypothesis in this study.Source: §VII · p. 5

VIII. ZERO-MEAN POSITIONING JITTER VERSUS FIXED DISPLACEMENT BIAS

The tested competing prediction distinguished centered Gaussian positioning jitter from a fixed displacement bias. At the same 0.383λ error parameter, jitter yielded high accuracy while fixed bias yielded near-chance accuracy.

VIII. ZERO-MEAN POSITIONING JITTER VERSUS FIXED DISPLACEMENT BIAS

  • The tested prediction was that centered Gaussian jitter remains learnable while a comparable fixed bias does not.The protocols tested this competing prediction, not whether jitter moves a nonzero-displacement correlation root.Source: §VIII · p. 5
  • At 0.383λ, G1 accuracy was 50.3235% for fixed bias and 98.4222% for independent jitter; G2 accuracy was 53.2944% for bias and 98.4909% for rigid jitter.The two archives used independent seed sets.Source: §VIII · p. 5
  • The archives use one-dimensional centered Gaussian positioning error in an isotropic field; the separate isotropic-vector identity preserves existing real-correlation zeros and signs.Its Gaussian characteristic-function multiplier is positive.Source: §VIII · pp. 5–6
  • For centered errors, averaging differs from evaluating J0 at a fixed bias: the positive expectation decays toward zero as jitter grows but does not ensure every exchange is positively correlated.At σ = 0.383λ, G2 records a 29.6163% probability of a negative scalar J0(2πe).Source: §VIII · p. 5

IX. SCOPE

The conclusions are bounded to specified small synthetic classifiers, channel laws, correction rules, and finite budgets; code and archived data are available on reasonable request.

IX. SCOPE

  • The empirical conclusions cover only the specified small synthetic binary classifiers, channel laws, correction rules, and finite training budgets.Larger datasets, deeper networks, regression tasks, and hardware deployments lie outside the tested domain.Source: §IX · p. 6
  • Several evaluated settings remain bounded by their stated models, including prescribed linear processing, frequency-domain channels, and single-mode or LOS-plus-white-scattering near-field laws.The wideband experiments exclude intrasymbol Doppler and intercarrier interference.Source: §IX · p. 6
  • Code and archived data are available from the corresponding author on reasonable request.Source: §IX · p. 6

X. CONCLUSION

The conclusion links physical-layer design rules to the training update distribution across seven cases. It argues that training advantages require explicit task support, noise injection, and update-moment connections.

X. CONCLUSION

  • A useful design prediction must link physical-layer rules to task support, noise injection, and the relevant training-update moments.The seven cases distinguish supported local physical relations from training forecasts that lack that link.Source: §X · p. 6
  • The quadratic proxy makes this link through a task-subspace second-moment recursion.Source: §X · p. 6
  • The remaining comparisons test concrete performance forecasts under the stated tasks and budgets, with predictions fixed before evaluation and retained alongside counterexamples.Source: §X · p. 6

APPENDIX A OPERATORS AND FROZEN EXPERIMENTAL

This appendix documents the frozen experimental configurations, signal conventions, and evaluation procedures underlying the paper’s tests, including channel models, calibration and correction pipelines, and comparison designs. It also specifies the G1/G2 displacement-error setup and conditional Gaussian field generation, distinguishing independent port errors, shared errors, fixed bias, and centered jitter.

APPENDIX A OPERATORS AND FROZEN EXPERIMENTAL

  • Methods share underlying random draws within seeds, separate calibration, training, and test draws, and specify learning budgets, noise levels, and evaluation seeds.Classifier intervals use paired seed resampling; all reported intervals are two-sided 95% intervals.Source: Appendix A · p. 6

A. Signal conventions, learning, and calibration

  • Matrix-channel returns use an adjoint baseband convention, while single-receiver tests apply a receive-side map after reception; training uses locally computed receiver gradients and final-model evaluation.Corrections are estimated from pilot moments rather than fitted to training outcomes; quantization, norm recovery, and CSI references vary by experiment.Source: A. Signal conventions, learning, and… · pp. 6–7

B. Test A: correction, task support, and distinct calibrations

  • Test A uses distinct quadratic-proxy and classifier SGD setups, differing in task support, channel sampling, innovation covariance, correction cutoffs, and pilot-based estimation.The classifier uses the full random channel and estimates both corrections from the same 64 pilot pairs; the proxy uses exact matrices without pilots or quantization.Source: B. Test A: correction, task support,… · p. 7

C. Test B: field, selectors, and local statistic

  • Test B compares max-power, random, fixed, and geometric return-port selection in a channel-only field study, with a separate max-power training replay and a local phase-gradient statistic.The channel-only study uses spacings 0.2 and 0.5; the replay uses spacing 1/7, reconstructs 7680 training channels per seed, and applies no phase or inverse correction.Source: C. Test B: field, selectors, and local… · p. 7

D. Tests C1–C7: propagation and calibrated receive chains

  • Tests C1–C7 specify single-path multiuser and self-interference cases, multi-AP and phase-varying returns, and calibrated receive maps with pilot-noise subtraction.The setups include projected correction for two interferers and an explicitly test-oracle single-AP selection; C7 quantizes pilot and data-return vectors without supplying instantaneous AP phase.Source: D. Tests C1–C7: propagation and… · pp. 7–8

E. Tests C8–C10: GSNR and processing order

  • Tests C8–C10 define calibrated multiuser chains and compare naive, ZF, loaded-combiner, single-user, and CSI processing, alongside GSNR and gradient-alignment diagnostics.C9 fixes total data SIR at 0 dB and compares optimizers’ single-minus-ZF accuracy differences; C10 measures gradient cosine across interference angles and methods.Source: E. Tests C8–C10: GSNR and processing… · pp. 8–9

F. Tests D1–D3: wideband channels and averaging order

  • Tests D1–D3 define three wideband channel models and compare global correction fitted from averaged raw moments with independently fitted per-tone correction.The high-bandwidth D3 comparison fixes η = 0.20; neither method averages polar factors, and terminal tone metrics are averaged before seed-level inference.Source: F. Tests D1–D3: wideband channels and… · p. 9

G. Tests E1/E2: spherical geometry and constrained fits

  • Tests E1/E2 compare near-field diagonal and far-field correction fits under spherical geometry, with E2’s far-field arm constrained to a rank-one complex line.Both use the same 64 pilot pairs; E1 uses a full-space phase operator, whereas E2 projects after quantization and normalizes the scalar coefficient.Source: G. Tests E1/E2: spherical geometry and… · p. 9

I. Tests G1/G2: sample the error before the field

  • With a four-element half-wavelength ULA, G1 uses independent port errors and G2 compares shared versus independent errors; fixed bias sets r_i=0.383 and jitter sets b=0.The terminal displacement is 0.50, ideally compensated mechanically by −0.50; errors are redrawn independently each exchange.Source: §I · p. 10
  • Residuals are sampled before the conditional Gaussian field draw; replacing them with mean covariance changes the mixture distribution, and zero residual gives B=F.Bias/jitter arms use M=I, return scale L=16, and no pilots or phase correction; the reported table uses σ=0.383, with a supported G1 endpoint at σ=0.45. Only roundoff-scale negative eigenvalues of Q are clipped; an eigenvalue below −10^-9 fails validation.Source: §I · p. 10

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