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Keep Pace to Learn: Movable Antennas Restore the Backpropagated Gradient Under User Mobility
Xianzhong Li, Yuzhi Yang, Tianhao Guo
TL;DR
Terminal motion can rotate the physical return gradient away from a descent direction, motivating a mechanical approach that relates antenna displacement to the return’s spatial correlation. The paper derives speed and bias–jitter conditions for mean-direction preservation and reports that partial compensation can still support learning in a later positive correlation lobe.
Problem
Terminal movement between forward and return transmissions can rotate the returned gradient, so the paper studies how positioning affects its conditional mean direction.
Source: §I · p. 1Method
The paper relates actuator-limited residual displacement to spatial correlation and averages independent centered Gaussian positioning jitter under a scalar conditional-mean model.
Source: §I · p. 1, §VI · p. 6Results
At 30 km/h and κ = 0.7, a second-positive-lobe residual had Re ϕ = 0.463 and accuracy of 98.35%; at zero bias, finite Gaussian jitter leaves the conditional mean direction positive.
Source: B. Mechanical Interpretation · p. 5, §VI · p. 6Takeaways & Limitations
For mean-direction preservation, effective bias must be controlled; finite Gaussian jitter can weaken the scaling without reversing its direction.
Source: §VI · p. 6, A. Rail Error and Vector Error · p. 7The gradient-direction conclusion relies on scalar conditional regression given positioning error and on positioning-error independence from forward information.
Source: §VI · p. 7
Abstract
from arXiv · showhide
When a mobile terminal takes part in over-the-air training, its own motion rotates the gradient that travels back through it; a movable antenna that keeps pace with the terminal rotates it back---without any channel estimate. We give a speed law: under a scalar conditional-mean model of the physical return, landing the residual displacement inside the first positive lobe of the spatial correlation keeps the conditional mean gradient a descent direction, which, given sufficient travel, fixes a critical actuator-to-terminal speed ratio. We further prove a bias--jitter law: under the same scalar conditional-mean model and with zero effective bias, independent zero-mean Gaussian jitter of any finite amplitude scales the conditional mean gradient without reversing its direction, whereas a sub-wavelength systematic bias can reverse it. This result motivates prioritizing bias calibration to preserve gradient direction. Simulations on 3GPP CDL-A channels show that an antenna keeping pace with the terminal achieves near-CSI accuracy without digital channel correction, reaching 98.58 % compared with 98.59% for the ideal-CSI reference. The simulations also show that partial compensation can support learning when the residual displacement falls within a later positive correlation lobe, so a faster actuator does not necessarily yield higher accuracy. Separate positioning simulations show a sharp accuracy loss under systematic bias but little degradation under zero-mean Gaussian jitter of comparable spatial scale.
I. INTRODUCTION
Terminal motion can rotate a physically returned gradient away from a descent direction, prompting the paper to ask whether antenna counter-movement can restore it. The paper relates that mechanism to gradient direction through speed and positioning-error laws, while distinguishing its contribution from established counter-movement and physical backpropagation.
I. INTRODUCTION
- The paper asks whether a movable antenna can keep pace with terminal motion and restore a useful return gradient, then develops speed and bias–jitter laws for mean-direction preservation.The laws are stated under a scalar conditional-mean model; the introduction also describes CDL-A/C evaluation and operating boundaries.Source: §I · pp. 1–2
B. Counter-Movement and Position Reuse
- Counter-movement, movable-antenna response models –[4], and reciprocal MIMO backward computation are established; their relation to gradient direction is the stated contribution.Position selection for a static channel and opposite antenna motion are not claimed as new.Source: §I · p. 1, B. Counter-Movement and Position Reuse · p. 2
A. From Channel Shaping to Gradient Direction
- Unlike channel-shaping work that targets power and other channel properties, this work quantifies how restoring the prior position relates to the direction of a delayed gradient –[4].Maximizing return power alone cannot certify an acute angle with the derivative from the earlier forward pass.Source: A. From Channel Shaping to Gradient… · p. 2
C. Physical Returns and Imperfect Backward Paths
- Reciprocal MIMO backward computation requires the return to represent the forward adjoint, while mobility changes the operator performing differentiation rather than merely transporting a computed gradient.Acute alignment alone does not imply convergence under arbitrary steps, noise, or nonlinear parameterizations.Source: C. Physical Returns and Imperfect… · p. 2
D. Prototypes and the Remaining Experimental Gap
- A feedback-controlled movable-antenna prototype demonstrates controlled positioning, but not gradient restoration during a moving terminal’s exchange [14].That measurement would require return-time residual position, coherent RF phase, and physical/reference gradient-angle measurements.Source: D. Prototypes and the Remaining… · p. 2
III. SYSTEM MODEL
The system returns a loss-derived signal through a reciprocal channel, while terminal motion and antenna counter-movement determine the antenna’s world-position displacement. The model links that displacement to spatial correlation under a locally stationary field and specified endpoint conditions.
A. Forward and Return Computations
- The receiver evaluates the loss derivative and returns a normalized signal; reciprocal propagation and conjugation yield the transmitter-side gradient through the local Jacobian-adjoint map.The movable antenna is at the terminal that receives the forward signal and transmits the return.Source: A. Forward and Return Computations · p. 2
B. World Position, Relative Motion, and Travel
- During an exchange, the terminal advances by vτ while the co-located antenna counter-moves relative to its body, subject to actuator speed and available opposite-direction travel.The speed bound is relative to the terminal; usable travel depends on the preceding exchange’s final rail position, and neglecting acceleration and settling is optimistic for hardware.Source: B. World Position, Relative Motion,… · p. 3
C. A Static Spatial Field During One Exchange
- The spatial correlation is modeled from a normalized nonnegative angular power measure; restoring the moving endpoint’s prior world position cancels its own path phases when the other endpoint and scattering environment stay fixed.The planar uniform angular-spectrum Bessel formula is distinct from spherical isotropy, and deterministic path equality does not establish scalar conditional regression at nonzero displacement.Source: C. A Static Spatial Field During One… · p. 3
D. Positioning Error and What Is Known
Positioning error is decomposed into a fixed bias and an independent centered random component, with Gaussian jitter used for the stated theorem. The section also specifies what the descent criterion requires and clarifies that avoiding a channel estimate does not remove positioning and motion-information needs.
D. Positioning Error and What Is Known
- Positioning error is modeled as a fixed bias plus an independent centered error, which is Gaussian with finite covariance for Theorem 2.The text distinguishes rail-aligned Gaussian error from isotropic vector error and states that the control still needs motion, timing, and position information even without a channel estimate.Source: D. Positioning Error and What Is Known · p. 3
IV. REVIEW OF THE DESCENT CRITERION
- Under the stated conditional-mean criterion, the negative conditional mean returned gradient is a strict first-order descent direction exactly when Re ρ > 0.For real-linear maps, a complex ρ requires complex linearity or a separate task-specific check; the theorem explicitly assumes the scalar conditional-mean model.Source: §IV · p. 3
A. A Direction Condition Is Not a Learning-Rate Rule
Positive mean alignment supports a sufficiently small stochastic step under finite conditional second moment, but does not guarantee an arbitrary learning rate. The speed law connects actuator limits and available travel to residual displacement, while the array discussion limits when a scalar correlation test applies.
A. A Direction Condition Is Not a Learning-Rate Rule
- Strict positive mean alignment permits a sufficiently small step when the conditional second moment is finite, but does not bound that step away from zero as correlation vanishes.The classical phase condition is not an arbitrary-stepsize convergence claim; Adam, clipping, and projections further distinguish a returned gradient from the executed update.Source: A. A Direction Condition Is Not a… · p. 4
B. What Changes for an Array
- For arrays, a trace correlation cannot generally replace the task-dependent operator; exact geometric restoration instead makes the return channel equal the forward channel path by path.Partial restoration lacks that universal identity, and broader CDL configurations may require vectorized channel prediction.Source: B. What Changes for an Array · p. 4
V. THEOREM 1: THE SPEED LAW
- Under the scalar conditional-mean model, landing the residual displacement where Re ϕm(rmin) > 0 suffices for strict mean descent, subject to the speed law’s available-travel condition.When travel does not bind, the condition becomes a critical actuator-to-terminal speed ratio; in the planar isotropic case the first real-part zero is tied to the first positive zero of J0.Source: §V · pp. 4–5
A. Boundary Cases and the Meaning of Critical Speed
Critical speed marks entry into the connected positive-real region containing the origin, not a global accuracy threshold. Boundary cases and later positive lobes qualify what the first-lobe guarantee means.
A. Boundary Cases and the Meaning of Critical Speed
- When v = 0, no counter-movement is needed; if vτ < r0, every nonnegative actuator speed meets the threshold, but equality does not guarantee strict descent.At vτ = r0, a stationary actuator gives zero first-order mean alignment when the zero is finite; if no real-part zero exists, every finite nonnegative residual is in the first positive region.Source: A. Boundary Cases and the Meaning of… · p. 5
- The “critical” condition means entering the connected positive-real region containing the origin, not a unique global accuracy threshold or monotonic accuracy improvement.For isotropic scattering, later positive Bessel lobes can also yield mean descent, while an anisotropic discrete spectrum can recur.Source: A. Boundary Cases and the Meaning of… · p. 5
- The first-lobe condition r < r0 is a robust certificate, not a necessary condition for every successful learning outcome.Later positive regions can support mean descent, and a general angular spectrum may have recurring positive regions.Source: A. Boundary Cases and the Meaning of… · p. 5
B. Mechanical Interpretation
Mechanical compensation depends separately on actuator speed, travel, and the residual position left at return. CDL-A results show that later positive correlation lobes can support learning, while the bias–jitter analysis distinguishes centered Gaussian jitter from systematic bias under its scalar conditional-mean assumptions.
B. Mechanical Interpretation
- Speed and travel are separate constraints: increasing actuator speed cannot overcome insufficient travel, and extra travel cannot compensate for insufficient speed within the exchange.For variable velocity, the relevant command uses integrated displacement and actuator reachability rather than substituting endpoint velocity into the constant-speed law.Source: B. Mechanical Interpretation · p. 5
- Across 24 moving cells, Re ϕ(r) orders accuracy with Spearman ρs = 0.953; the first real-part zero is r = 0.3518λ.The first accuracy drop below 80% occurs at κ = 0.9 versus theoretical κ∗ = 0.910 at 30 km/h, and at κ = 0.8 versus κ∗ = 0.729 at 10 km/h; at 3 km/h every tested κ ≥ 0.5 stays above 80%.Source: B. Mechanical Interpretation · p. 5, Figure 2 · p. 6
- At 30 km/h and κ = 0.7, accuracy is 98.35% in the second positive lobe (Re ϕ = 0.463); κ = 0.8 falls in a negative lobe and performs worse than κ = 0.5 or 0.7.This finite-grid observation is not a design rule; at 10 km/h, κ = 0.8 falls below 80% despite positive Re ϕ = 0.375.Source: B. Mechanical Interpretation · p. 5
VI. THEOREM 2: THE BIAS–JITTER LAW
- At zero effective bias, centered Gaussian jitter of any finite amplitude scales the conditional mean gradient by a positive real factor without reversing its direction.Random positioning requires averaging the spatial correlation over the error law; substituting jitter standard deviation into J0 does not perform that average.Source: §VI · pp. 5–6
- With independent, identically distributed two-sided placement errors, Proposition 1 gives a nonnegative ensemble correlation at zero bias, strictly positive unless ψP(ku) vanishes for µ-almost every u.The proposition concerns ensemble correlation, not by itself the conditional mean gradient direction.Source: §VI · p. 7
- A descent-direction conclusion from the Gaussian jitter result requires the scalar conditional-mean model and error independence from forward information.The ensemble-correlation result alone does not establish the conditional mean gradient direction; the arbitrary-error correlation condition also depends on the angular spectrum, and uniform error can violate it.Source: §VI · pp. 6–7
A. Rail Error and Vector Error
The Gaussian-jitter attenuation law differs between rail and isotropic vector error models. At zero bias it remains positive for finite jitter, but can become arbitrarily weak as jitter grows.
A. Rail Error and Vector Error
- Rail and isotropic vector errors have different attenuation laws, so the one-dimensional projection factor cannot be used for Σ = σ^2I.Both models retain positivity at zero bias.Source: A. Rail Error and Vector Error · p. 7
- For uniform planar angular spectrum and rail jitter, the derivation uses a = π^2σ^2/λ^2 and the identity cos^2 θ = (1 + cos 2θ)/2.Source: A. Rail Error and Vector Error · p. 7
- At zero bias, Gaussian jitter preserves the conditional-mean direction through a positive attenuation factor, but the factor has no jitter-independent positive lower bound.In the planar rail model it tends to zero as σ grows without bound.Source: A. Rail Error and Vector Error · p. 7
B. Calibration Priority and a Direction Margin
Under the stated Gaussian scalar conditional-mean model, zero effective bias lets jitter attenuate without reversing direction, while systematic bias can reverse it. A separate centered-error example shows why zero mean alone is insufficient.
B. Calibration Priority and a Direction Margin
- With zero effective bias, independent Gaussian rail jitter changes the mean-gradient magnitude without reversing its sign; systematic bias can reverse it.The effective offset is beff = rmin c_m + b, so uncompensated kinematic residual contributes to the bias.Source: B. Calibration Priority and a… · p. 7
- The sufficient bound k∥beff∥ < π/2 keeps every projected bias phase within the positive-cosine region, separating the bias sign margin from jitter attenuation.It is a worst-spectrum bound, not the isotropic first-zero radius.Source: B. Calibration Priority and a… · p. 7
C. Why Zero Mean Alone Is Insufficient
- A centered equal-probability rail error can still produce negative mean correlation in a negative Bessel lobe, so zero mean alone does not guarantee direction preservation.This is an analytic counterexample, not simulated evidence; Theorem 2 concerns Gaussian scalar conditional means.Source: C. Why Zero Mean Alone Is Insufficient · p. 7
VII. 3GPP CDL EVALUATION AND POSITIONING EVIDENCE
The evaluation reports restoration without digital channel correction and examines how CDL-A accuracy varies with actuator speed ratio. Its results are bounded by the tested configurations and channel settings.
A. Evidence Organization and Experimental Scope
- The studies use fixed configurations and eight paired environmental seeds; retained bootstrap intervals describe only finite tested configurations, not independent replication across frequencies or examples.G53 uses CDL-A/C and G60 CDL-A at 3 and 30 km/h with a 40 ms delay; G60 retains ray angles and phases.Source: A. Evidence Organization and… · p. 8
B. Restoration Without Digital Channel Correction
- In G31, the movable-antenna S0 return reaches 98.59% mean accuracy, versus 51.47% for fixed-antenna S0 and 98.74% for CSI.The zero residual is built into this frozen shared-angular-spectrum model; it is not hardware evidence or a proof for arbitrary endpoint motion.Source: B. Restoration Without Digital Channel… · p. 8
C. Keeping Pace in the CDL-A Speed Study
- At 30 km/h, κ = 0.8 reaches 46.379%, below κ = 0.5 at 54.750%; the paired difference is −8.371 percentage points, with a 95% interval of [−14.124, −3.046].The sampled CDL-A diagnostics show real correlation and mean gradient cosine of approximately −0.332 and −0.388 at the high-speed κ = 0.8 point; these do not constitute an exact scalar reduction.Source: C. Keeping Pace in the CDL-A Speed Study · p. 8
- At κ = 1, G60 accuracy is 98.579% at both terminal speeds, with paired mean CE excess over CSI of 0.001143.Figure 3 presents the full speed grid.Source: C. Keeping Pace in the CDL-A Speed Study · p. 8, Figure 3 · p. 9
- Two tested speeds and four ratios cannot locate a critical curve, and the boundary applies to the tested linear spatial correction and time-separable model.The paper identifies waiting as the demonstrated escape within that correction family, not the only possible remedy with CSI or other feedback.Source: C. Keeping Pace in the CDL-A Speed Study · pp. 8–9
D. Fixed Bias and Gaussian Jitter Separate Sharply
The positioning-error results distinguish fixed bias from Gaussian jitter: large jitter retains high accuracy, while bias can sharply reduce it. Baseline comparisons also report polar correction and waiting results, and a separate time-separable setting where delayed returns outperform spatial correction.
D. Fixed Bias and Gaussian Jitter Separate Sharply
- 8.438% and 12.708% of batches have negative true/estimated gradient inner products at the two large jitter levels, versus 47.917% at the fixed-bias point.The mean direction can survive Gaussian averaging even though individual updates sometimes reverse.Source: D. Fixed Bias and Gaussian Jitter… · p. 9
E. CDL-A/C Baselines: Digital Correction and Waiting
- 94.994% and 89.474% are waiting accuracies in CDL-A, while 98.724% and 97.301% are the CDL-C values.Polar correction has better mean CE in each cell, although some seeds favor waiting in the CDL-C high-speed comparison; the study does not measure end-to-end latency.Source: E. CDL-A/C Baselines: Digital… · p. 9
D. Fixed Bias and Gaussian Jitter Separate Sharply
- 50.323% accuracy with a 0.383λ fixed bias contrasts with 98.422% under independent Gaussian jitter at the same standard deviation.At σ = 0.45λ, independent jitter yields 98.380% accuracy; the fixed bias is slightly beyond the first isotropic root.Source: D. Fixed Bias and Gaussian Jitter… · p. 9, Figure 4 · p. 10
- 98.491% and 98.419% accuracy under common rigid Gaussian jitter at σ = 0.383λ and 0.45λ, respectively, contrasts with 53.294% for the fixed-bias control.The largest tested mean-accuracy difference between rigid and independent jitter is 0.093 percentage points, which does not establish equivalence of channel distributions or learning tasks.Source: D. Fixed Bias and Gaussian Jitter… · p. 9, Figure 4 · p. 10
E. CDL-A/C Baselines: Digital Correction and Waiting
- 98.504% and 98.429% are the CDL-A polar-correction accuracies at low and high speeds; CDL-C yields 99.029% and 98.463%.Figure 5 reports a 40 ms natural return delay; fixed-antenna S0 accuracies are near chance, and the individual profile-speed cells should not be conflated.Source: E. CDL-A/C Baselines: Digital… · p. 9, Figure 5 · p. 10
- G53 lacks an MA arm and G60 lacks a CDL-C MA arm, so the reported baselines do not provide a matched comparison across all methods and profiles.Figure 5's standard-channel baseline evidence is distinct from G60's MA speed result.Source: E. CDL-A/C Baselines: Digital… · pp. 9–10, Figure 5 · p. 10
F. The Separable-Channel Boundary
- 61.049% polar accuracy contrasts with 98.418% and 98.396% for negative-lobe and second-positive-lobe waiting in the tested time-separable TDL setting.The stated boundary is limited to the tested linear spatial correction and time-separable model; waiting is the demonstrated remedy within that correction family, not the only possible remedy.Source: F. The Separable-Channel Boundary · p. 10
A. What the Two Laws Establish
The two laws depend on stated channel and motion assumptions, and practical performance also depends on sensing, travel, and finite-budget effects. The conclusion and open problems emphasize boundaries to generalization and measurements needed to evaluate real systems.
A. What the Two Laws Establish
- The speed law combines reachability with scalar conditional-mean and local-map assumptions, while the Gaussian identity is broader than its learning conclusion.Neither law establishes a universal accuracy threshold, and smaller residuals need not improve finite-budget accuracy.Source: A. What the Two Laws Establish · pp. 10–11
B. Motion Sensing and the Cost of Returning Home
- Motion sensing, command latency, and synchronization affect residual displacement; finite-rail compensation also requires reset scheduling that costs time, energy, and learning information.Nameplate actuator speed alone cannot certify an exchange because attainable travel may be reduced by acceleration, braking, vibration, and RF settling.Source: B. Motion Sensing and the Cost of… · p. 11
C. Static Geometry Is Not a Static Radio
- Returning the antenna cancels its translation only in a static scene, and the stated speed result assumes one-dimensional, co-located motion.Moving scatterers, endpoint motion, oscillator drift, rotation, and platform-induced pattern changes remain outside that cancellation; multidimensional and near-field settings need further treatment.Source: C. Static Geometry Is Not a Static Radio · p. 11
D. Finite-Budget Learning and Array Effects
- A positive mean direction can be too weak for useful learning, and slowly varying or state-dependent errors need not satisfy the Gaussian-jitter conclusion.The simulations also use small tasks and differing update or feedback budgets, so their comparisons do not isolate mechanical-versus-digital causality or establish hardware savings.Source: D. Finite-Budget Learning and Array… · p. 11
IX. OPEN PROBLEMS
- Open problems include distinguishing correlation-zero, gradient-inner-product-zero, and finite-budget accuracy thresholds, and testing reachability and directional margin under multiuser deadlines and rail constraints.The proposed prototype measurements pair physical returns with reference gradients and separately assess position, channel, direction, and learning restoration.Source: §IX · p. 11
X. CONCLUSION
- Within the stated channel model and reachable travel, the conclusion presents movable antennas as preserving learning direction without a channel estimate, subject to finite-travel and measurement boundaries.The speed and bias–jitter laws are conditional-mean results; zero-mean independent Gaussian jitter can shrink the mean gradient without reversing it when effective bias is zero.Source: §X · pp. 11–12