A Covariance-Surrogate Framework for Movable-Antenna Enabled Anti-Jamming with Unknown Jammers

Lebin Chen; Ming-Min Zhao; Min-Jian Zhao; Qingqing Wu; Rui Zhang

REVIEW 4 major objections 4 minor 1 cited by

A Covariance-Surrogate Framework for Movable-Antenna Enabled Anti-Jamming with Unknown Jammers

T0 review · 4 major / 4 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read This paper establishes that a movable-antenna receiver can suppress unknown jammers by optimizing antenna positions against a blockwise sample-covariance surrogate anchored at the current position, with a local error bound that makes trust-

desk verdict A genuinely useful surrogate framework with a real gap between its anchor-locality bound and the algorithm that is supposed to exploit it. read the letter →

arxiv 2512.20380 v2 pith:5T7FIVB3 submitted 2025-12-23 eess.SP cs.ITmath.IT

Lebin Chen , Ming-Min Zhao , Qingqing Wu , Min-Jian Zhao , Rui Zhang This is my paper

classification eess.SPcs.ITmath.IT

keywords movableantennaanti-jammingunknownjammersMVDRbeamformingsamplecovariancesurrogatetrust-regionoptimizationoutputSINRtwo-timescale

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper tries to show that a receiver with movable antennas can defend against jammers whose channels are completely unknown, by estimating the interference-plus-noise covariance from received snapshots at the current antenna positions and optimizing positions against that estimate. The central result is a local error bound: the surrogate objective tracks the true signal-to-interference-plus-noise ratio (SINR) objective within a tube whose width grows linearly with distance from the anchor and shrinks to zero as the snapshot budget grows. Because of that, an optimization method that deliberately keeps antenna moves inside a trust region can convert surrogate improvements into genuine SINR gains. The paper also proves that a seemingly more robust historical-average surrogate carries a permanent geometric bias and is worse near the anchor. If true, this gives a practical anti-jamming scheme that needs no jammer directions or powers, only snapshots and knowledge of the legitimate user's channel.

What carries the argument

The key object is the anchor-based sample-covariance surrogate: replace the unknown covariance R(x) in the MVDR SINR objective by R_hat(x_i), the sample covariance over a block of T snapshots taken at the current antenna position x_i. Its fidelity is controlled by two ingredients: Lipschitz continuity of the steering vectors and of the covariance with respect to antenna position, and a matrix concentration bound on the sample covariance, which together yield the local error bound E|g-g_hat| <= C_R(L_R||x-x_i|| + epsilon(T)). The second mechanism is the trust-region step-size control in PTRSO, which keeps iterates inside the region where the surrogate is reliable and projects candidates onto

What would settle it

Take the paper's simulation setting and add a deliberate mismatch to the desired-user channel used by the algorithm, for example a 5-degree bias in all desired-path AoAs while the true channel remains unperturbed. If the true output SINR of PTRSO falls back toward the fixed-ULA baseline under such a small mismatch, the known-h0 assumption is decisive; if the gain largely survives, the method is robust to h0 error.

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Extended reading notes

Core claim

The discovery is that the unknown, position-dependent covariance R(x) can be replaced by the blockwise sample covariance R_hat(x_i) measured at the current antenna-position anchor, and the resulting surrogate objective g_hat(x) = h0(x)^H R_hat(x_i)^{-1} h0(x) is provably close to the true minimum-variance distortionless-response (MVDR) objective g(x) = h0(x)^H R(x)^{-1} h0(x) in a neighborhood of the anchor. The paper proves E|g(x)-g_hat(x)| <= C_R (L_R ||x-x_i|| + epsilon(T)), where the snapshot error epsilon(T) vanishes as T grows, so the mismatch is controlled by keeping updates close to the anchor and using enough snapshots. Maximizing the surrogate within a trust region therefore also i

Load-bearing premise

The load-bearing premise is that the legitimate user's channel h0(x) = A(x)alpha is known to the receiver for every antenna position x; if the receiver's AoA or path-gain estimates for the desired user are wrong, both the distortionless MVDR constraint and the surrogate objective are corrupted, and the paper gives no sensitivity analysis or estimation protocol for this error.

Editorial extensions

If this is right

Anti-jamming antenna placement becomes feasible with zero jammer-side prior: only T snapshots per block and the known desired-user channel are needed.
The snapshot budget and the step size jointly control performance: more snapshots per block tighten the surrogate error, so true output SINR rises with T and saturates, while moves kept near the anchor prevent surrogate gains from evaporating.
Using all past covariance estimates is counterproductive near a moving anchor; the current-anchor sample covariance is the correct local model for trust-region updates.
PTRSO reaches a stationary point of the surrogate with O(N_r^2) per-iteration cost, making it more scalable than Newton-type methods with explicit Hessian factorizations.
In simulations, the optimized array places deeper nulls at the jammer angles than a fixed uniform linear array, giving higher true output SINR across user SNR, snapshot number, jammer count, and array size.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Because the error bound depends on the covariance Lipschitz constant and the snapshot error, the trust-region radius could be set adaptively from estimated values of these quantities instead of using the fixed heuristic update factors.
The most exposed point is the known-desired-channel assumption: in a real deployment the receiver would need a separate tracking loop for the legitimate user's AoAs and path gains, and the method's practical robustness to imperfect h0 remains untested.
The geometric-bias result suggests a broader design principle: whenever a position-dependent covariance is replaced by a sample estimate, the estimate must be re-centered at the current geometry; averaging over geometries is not a free variance-reduction step.
A natural extension is to schedule the anchor-update rate and block length T based on the measured surrogate residual, which would let the two-timescale framework track slowly moving jammers rather than only static interference.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

The real contribution here is the covariance-surrogate idea: replace the unknown position-dependent interference-plus-noise covariance with a blockwise sample covariance at an anchor, and prove a local tube bound on the mismatch between the surrogate and true MVDR objective (Eq. 20). The derivation is self-contained: Lipschitz continuity of steering/covariance plus a concentration inequality, and the bound has the right dependence on snapshot budget and anchor displacement. The historical-average bias theorem is new and interesting, though weaker than the text sometimes suggests. The numerical section is honest—true SINR is computed with the true covariance, not assumed equal to the surrogate—and PTRSO consistently beats the baselines by 3–5 dB across the swept parameters.

The main problem is that PTRSO does not enforce the condition Eq. (20) needs. The bound is in terms of distance to the anchor x_i; the trust region controls distance from the current iterate x_k, and the acceptance ratio measures how well the quadratic model matches the surrogate, not how close the iterate stays to x_i. Nothing stops a block of accepted TR steps from drifting far away from the anchor, so the claim that TR maintains the locality required by Eq. (20) is not actually supported. The convergence guarantee cited from [35] is for the surrogate problem, not for stationarity near the anchor, so the abstract's "near the anchor" is an overstatement. This needs a fix—e.g., one TR step per covariance block, an explicit anchor-distance constraint, or a proved drift bound—before the central transfer argument is solid.

Other soft spots are lesser. The receiver is assumed to know h0(x) exactly for all x, including AoAs and gains; that is common in MA papers but deserves a sensitivity check. The bound assumes Rhat(x_i) is nonsingular; no diagonal loading is discussed for T

Still, the paper is a serious piece of work. The formulation is relevant, the theoretical core is mostly sound, and the numerics back the practical claim. I would send it to peer review, but with a referee specifically asked to check the anchor-locality gap. If that gap is closed or honestly bounded, the paper is publishable.

Referee Report

4 major / 4 minor

Summary. The paper studies an uplink movable-antenna (MA) receiver that must suppress unknown jammers while jointly optimizing antenna positions and an MVDR beamformer. Since the true interference-plus-noise covariance R(x) is unknown and position-dependent, the authors replace it with a blockwise sample covariance R̂(x_i) at an antenna-position anchor x_i, yielding the surrogate objective ĝ(x)=h0(x)^H R̂(x_i)^{-1} h0(x). The main theoretical result is a local bound, Eq. (20), stating that E|g(x)−ĝ(x)| ≤ C_R(L_R‖x−x_i‖+ε(T)), so the surrogate approximates the true SINR objective near the anchor when the snapshot budget T is large. The paper also proves a geometric-bias result (Theorem 1) showing that a historical-average covariance surrogate is locally worse than the anchor-based surrogate. A projected trust-region algorithm, PTRSO, is proposed for the position optimization, and numerical results show higher output SINR than the tested baselines across SNR, T, number of antennas, and number of jammers.

Significance. If the locality-transfer argument is made rigorous, the paper offers a genuinely useful nonparametric route to MA position optimization against noncooperative jammers, avoiding jammer DoA or power priors. The theoretical scaffolding—Lipschitz covariance bounds, matrix concentration, and the historical-average bias theorem—is self-contained and goes beyond a purely algorithmic contribution. The numerical study is also careful in reporting both surrogate and true SINR, and the consistent gap between them is a meaningful check on the claims. However, the central link between Eq. (20) and the implemented algorithm is currently not established, and several technical gaps (sample-covariance nonsingularity, proof of the convergence guarantee, sensitivity to the assumed-known desired channel) need to be addressed before the paper's main claims are supportable.

major comments (4)

[§4-A, Eq. (31), Algorithm 1] The paper's central transfer argument is not connected to Algorithm 1. Eq. (20) bounds E|g(x)−ĝ(x)| by C_R(L_R‖x−x_i‖+ε(T)), so it justifies optimizing ĝ only when x stays near the anchor x_i. But the TR subproblem (31) is centered at the current iterate x_k with constraint ‖x_p‖≤Δ_k, and the acceptance ratio ρ_k in (36) compares q_k with ĝ, not ĝ with g. Algorithm 1 has no constraint, penalty, or stopping rule enforcing ‖x_k−x_i‖ small. Since Δ_k regulates the per-step displacement from x_k, not the cumulative displacement from x_i, accepted steps can accumulate and make δ(x_k) large, voiding the tube argument. Section IV-B's statement that the TR 'confines each update within a neighborhood where ... the mismatch between the surrogate ĝ(x) and true objective g(x) is small' conflates the neighborhood of x_k with that of x_i. Please either modify the algorithm to control anchor drift (e.g
[Eqs. (13), (18), (20), Algorithm 1] The proposed surrogate and all theoretical bounds require R̂(x_i)^{-1}, but the blockwise sample covariance (13) is singular when T<N_r and can be ill-conditioned even when T is only slightly larger than N_r. Lemma 3 gives a bound on E‖R̂−R‖ but does not imply a uniform lower bound on λ_min(R̂); the constant C_R=2k√L‖α‖/σ_n^4 in Eq. (20) appears to rely on such an inverse-norm bound. No diagonal loading, ridge regularization, or pseudo-inverse strategy is discussed. This is not a minor edge case: Fig. 8 sweeps T, and the paper claims robustness over snapshot budgets. Please introduce a regularized covariance estimator and derive the corresponding modification of Eqs. (18) and (20), or state the range of T for which the current bound and algorithm are valid.
[§4-A, abstract and contribution list] The abstract and the introduction assert that PTRSO 'is guaranteed to converge to a stationary point near the anchor,' but Section IV-A only says the procedure is 'guaranteed to converge to the set of stationary solutions of problem (14) [35]' without stating or proving a theorem. Problem (14) is nonconvex, the algorithm uses projection, an approximately solved TR subproblem via Steihaug-CG, and an acceptance rule based on ρ_k; none of these features is automatically covered by a generic TR convergence citation for the projected variant. Please state the precise convergence theorem with the required assumptions (e.g., bounded Hessians, gradient continuity, sufficient decrease after projection) and provide a proof or a counterexample-free argument. As written, the convergence guarantee is an unsupported assertion.
[§2-A, Eq. (2)] The paper assumes the desired-user channel h0(x)=A(x)α is known to the receiver for every antenna position x, including all path AoAs and complex gains. Both the MVDR constraint |w^H h0(x)|=1 and the surrogate objective ĝ(x) depend on exact h0. The claim of robustness to 'unknown jammers' is therefore conditional on a very strong legitimate-side assumption; in practice h0 must be estimated, and errors in AoAs or gains would distort the beamformer and the position objective. No estimation protocol or sensitivity analysis is provided. Please add a numerical robustness study with imperfect h0 or clearly scope the claim to the known-desired-channel regime.

minor comments (4)

[Eq. (10)] The dimensions of l are inconsistent: U is (N_r+1)×N_r, so l should be (N_r+1)×1, not N_r×1 as written.
[Fig. 8] The x-axis starts at T=0, but a sample covariance with zero snapshots is undefined. The plot should start at T=1 or T=N_r, or the caption should clarify that T=0 is included only as a nominal point.
[Abstract and §4-B] The phrase 'stationary point near the anchor' is misleading: the convergence claimed in §4-A is to a stationary point of problem (14), and no result shows that this point is near the anchor x_i. The wording should be aligned with what is actually proved.
[Eq. (36)] The denominator pred_k in the ratio test should be explicitly guaranteed positive (or the case pred_k=0 handled) after projection, since d_k=0 or a degenerate subproblem solution could make the ratio undefined.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the surrogate-fidelity bound, historical-average bias theorem, and numerical SINR claims are independently derived or externally checked.

full rationale

I traced the derivation chain from the surrogate definition (14) to the fidelity bound (20) and the PTRSO algorithm. The bound is built from Lemmas 1-3 (Lipschitz steering vectors, Lipschitz covariance, sub-Gaussian covariance concentration) and the matrix inverse perturbation identity; it is not an input restated as a conclusion. The surrogate objective is defined by replacing R(x) with R-hat(x_i), and the bound quantifies the resulting error, so the relation is derived rather than definitional. Theorem 1's historical-average bias is proved from Lemmas 4-5 and an almost-sure non-degeneracy argument, not assumed. PTRSO's convergence guarantee is explicitly for the surrogate problem (14), and the numerical results are measured against the true ground-truth SINR computed with R(x), so the reported gains are not forced by fitting a parameter to the simulated output. Self-citations such as [20] support the background field-response/h0-known modeling assumption, but they are not invoked as a load-bearing uniqueness result and do not by themselves imply the theorems; the h0-known assumption is a scope limitation, not a circular step. A possible technical gap exists between the bound (20)'s dependence on distance to the anchor and the trust-region's control of only per-iteration step size, but that is a correctness/robustness concern rather than a case where a claimed prediction reduces to its inputs by construction.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

The central claims rest on standard mathematical tools (concentration inequalities, TR convergence) plus domain assumptions: known desired-user channel, stationary WSS jammers, slow antenna motion, nonsingular sample covariance, and continuous fading powers. No parameters were fitted to data to produce the reported SINR gains; the simulation parameters (e.g., σ_i²=10σ_s², T=100) are test-scenario inputs. The algorithm's TR thresholds are hand-set but standard. The main unstated risks are the nonsingularity/quality of R̂ for finite T and the strength of the known-h0 assumption.

free parameters (1)

TR hyperparameters η1, η2, γ1, γ2, Δ0, Δmax = η1=0.25, η2=0.75, γ1=0.25, γ2=2; Δ0, Δmax default
Hand-set algorithmic constants; they affect convergence behavior but are not fitted to the data and are not part of the theoretical bound. They are standard values for trust-region methods.

assumptions (6)

domain assumption h0(x)=A(x)α is known to the receiver for all feasible x
Stated in Sec. II-A before Eq. (2); the MVDR constraint and the surrogate objective require exact desired-channel knowledge. No sensitivity analysis is provided.
domain assumption Jamming and noise processes are WSS, zero-mean, and mutually independent with fixed powers σ_i², σ_n²
Sec. II-A; the blockwise sample covariance is only meaningful for stationary interference over the block duration.
domain assumption Antenna positions are fixed for T snapshots; the two-timescale block model
Sec. II-B and Fig. 2; this makes the per-block sample covariance a valid estimate at the current anchor and justifies the blockwise optimization.
domain assumption R̂(x_i) is nonsingular and close to R(x_i); no diagonal loading is used
Needed for the inversion in (14) and for Lemma 3's concentration bound; when T<Nr, the sample covariance is singular, which the paper does not address.
domain assumption Power vector c has an absolutely continuous distribution w.r.t. Lebesgue measure
Used in Appendix D to show B(x*)≠0 almost surely in Theorem 1; this is a modeling assumption on fading/path-power statistics.
standard math Vershynin's matrix concentration (Theorem in [37]) and Nocedal-Wright TR convergence theory [35] apply verbatim
Lemmas 3 and the PTRSO convergence claim rely on these external results; their applicability to the projection-after-step variant is not fully verified.

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Cite this review

Pith. "Pith review of A Covariance-Surrogate Framework for Movable-Antenna Enabled Anti-Jamming with Unknown Jammers." pith.science (2026). https://pith.science/paper/5T7FIVB3

@misc{pith2026251220380,
  author       = {Pith},
  title        = {Pith review of: A Covariance-Surrogate Framework for Movable-Antenna Enabled Anti-Jamming with Unknown Jammers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5T7FIVB3}},
  note         = {Machine review of arXiv:2512.20380}
}

read the original abstract

In this paper, we investigate a movable antenna (MA) enabled anti-jamming optimization problem, where a legitimate uplink system is exposed to multiple jammers with unknown jamming channels. To enhance the anti-jamming capability of the considered system, an MA array is deployed at the receiver, and the antenna positions and the minimum-variance distortionless-response (MVDR) receive beamformer are jointly optimized to maximize the output signal-to-interference-plus-noise ratio (SINR). The main challenge arises from the fact that the interference covariance matrix is unknown and nonlinearly dependent on the antenna positions. To overcome these issues, we propose a surrogate objective by replacing the unknown covariance with the sample covariance evaluated at the current antenna position anchor. Under a two-timescale framework, the surrogate objective is updated once per block (contains multiple snapshots) at the current anchor position, while the MVDR beamformer is adapted on a per-snapshot basis. We establish a local bound on the discrepancy between the surrogate and the true objective by leveraging matrix concentration inequalities, and further prove that a natural historical-averaging surrogate suffers from a non-vanishing geometric bias. Building on these insights, we develop a low-complexity projected trust-region (TR) surrogate optimization (PTRSO) algorithm that maintains the locality of each iteration via TR constraints and enforces feasibility through projection, which is guaranteed to converge to a stationary point near the anchor. Numerical results verify the effectiveness and robustness of the proposed PTRSO algorithm, which consistently achieves higher output SINR than existing baselines.

Figures

Figures reproduced from arXiv: 2512.20380 by the authors.

**Figure 2.** Illustration of the proposed two-timescale framewo [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Toy example of surrogate and true function geometry. [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: Toy example of one step iteration geometry. [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]

**Figure 5.** Figure 5: Output SINR (true and surrogate objective) versus us [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]

**Figure 6.** Figure 6: Effectiveness versus user SNR. Nr (number of antennas) 2 4 6 8 10 12 14 16 Output SINR (dB) -15 -10 -5 0 5 10 PTRSO (proposed) PGD PNM ULA [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]

**Figure 7.** Figure 7: Output SINR versus number of receive antennas. [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]

**Figure 10.** Figure 10: Comparison of beampatterns with different antenna [PITH_FULL_IMAGE:figures/full_fig_p010_10.png]

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