Robust Beamforming Design for Secure Near-Field ISAC Systems

Feng Wang; Guojun Han; Vincent K. N. Lau; Xin Wang; Ziqiang Chen

REVIEW 3 major objections 5 minor 16 references

Robust Beamforming Design for Secure Near-Field ISAC Systems

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper proposes a robust beamforming design for near-field secure ISAC that maximizes the minimum sensing beampattern gain while enforcing worst-case SINR constraints under bounded CSI errors, and reports that it outperforms…

desk verdict A new application of S-Procedure and SROCR to robust near-field secure ISAC, but the SROCR stopping rule can declare convergence without a rank-one solution, so the headline numbers need verification. read the letter →

arxiv 2507.12881 v1 pith:AEGBDPY3 submitted 2025-07-17 cs.IT eess.SPmath.IT

Ziqiang CHen , Feng Wang , Guojun Han , Xin Wang , Vincent K. N. Lau This is my paper · ORCID

classification cs.ITeess.SPmath.IT

keywords near-fieldISACsecurebeamformingrobustS-ProcedureSROCRsensingbeampatterngainworst-caseSINRCSIuncertainty

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 letter addresses a practical gap: near-field ISAC systems that must simultaneously communicate with users, sense targets, and keep eavesdroppers from intercepting, while the base station only holds imperfect channel estimates. The paper formulates the design as maximizing the minimum sensing beampattern gain subject to worst-case SINR constraints for every user and every eavesdropper under bounded channel-error norms, together with a transmit-power budget. Because the resulting problem is non-convex with semi-infinite constraints, the authors use the S-Procedure to turn each uncertainty constraint into a linear matrix inequality, then apply a sequential rank-one constraint relaxation (SROCR) method to enforce the rank-one beamforming structure. They report that this SROCR-based design converges within a few iterations and achieves a higher worst-case sensing beampattern gain than semidefinite relaxation and information-only baselines while preserving security constraints. If correct, this gives a tractable robust beamforming recipe for near-field ISAC under imperfect CSI.

What carries the argument

The two load-bearing tools are the S-Procedure and the SROCR algorithm. The S-Procedure converts each semi-infinite constraint, an inequality that must hold for every channel error inside a norm ball, into an equivalent linear matrix inequality by introducing a nonnegative slack variable, provided the error set is convex with nonempty interior. SROCR then handles the rank-one constraints by replacing $\operatorname{rank}(W_k)=1$ with the equivalent condition $\lambda_{\max}(W_k)=\operatorname{tr}(W_k)$, relaxing it to $u^{(i-1)H}_{k,\max} W_k u^{(i-1)}_{k,\max} \geq v^{(i-1)}_k \operatorname{tr}(W_k)$ using the leading eigenvector from the previous iteration, and increasing $v$ toward one until a rank-one solution is reached. The near-field steering vector $\alpha(r,\theta)$ with spherical-wavefront phases carries the geometric model that distinguishes this problem from far-field beamforming.

What would settle it

Run the proposed design with a fixed scenario, such as $N=64$, $K=4$, $L=2$, $M=2$, and $P_0=30$ dBm, then draw many error realizations uniformly from the declared norm balls of radii $\epsilon_{C,k}$, $\epsilon_{E,l}$, and $\epsilon_{T,m}$; if any realization drives a CU SINR below $\bar{\gamma}_{C,k}$ or an eavesdropper SINR above $\bar{\gamma}_{E,l,k}$, the S-Procedure reformulation is not actually worst-case, and the robustness claim fails.

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Extended reading notes

Core claim

The central claim is that robust secure beamforming for near-field ISAC can be formulated as a rank-one constrained semidefinite program with linear matrix inequalities obtained from the S-Procedure, and that this program can be solved by iterating a convex relaxation that gradually forces each communication beamforming matrix $W_k$ to satisfy $\lambda_{\max}(W_k)=\operatorname{tr}(W_k)$, i.e., to have rank one. The paper shows numerically that this SROCR-based solution converges to a local Karush-Kuhn-Tucker point in a few iterations, and that for the tested configurations it yields a larger minimum sensing beampattern gain than the SDR baseline, which drops rank-one constraints and may fail to return a feasible beamformer, and larger than an information-only transmission without dedicated sensing signals. Robustness is demonstrated by degrading gracefully as the normalized CSI error bound grows, with the worst-case sensing gain eventually collapsing as the target-channel error grows.

Load-bearing premise

Every channel error lies inside a known norm ball of a given radius, and the near-field channel is dominated by the line-of-sight path; if real errors fall outside that ball or the LoS model is wrong, the worst-case guarantees do not hold.

Editorial extensions

If this is right

For a fixed scenario, the proposed design satisfies every CU worst-case SINR constraint and every eavesdropper leakage cap under any error realization inside the declared norm balls, while maximizing the minimum target beampattern gain.
Compared with SDR, the SROCR method returns an actual rank-one beamforming matrix rather than a possibly high-rank relaxation, so the solution is directly implementable.
Jointly allocating power to dedicated sensing signals and communication beams improves worst-case sensing gain over information-only transmission, so the sensing covariance $R_0$ is not wasted.
The design degrades gracefully with increasing CSI error at users and eavesdroppers, up to normalized error $0.9$ in the simulations, but is most sensitive to target-channel error, where the beampattern gain collapses as that error grows.
The per-iteration complexity is polynomial in the array size $N$ and the numbers of users, targets, and eavesdroppers, so the approach remains feasible for typical system sizes.

Reading between the lines

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

Because the S-Procedure step requires only convex error sets with nonempty interior, the same LMI structure could be reused for other convex uncertainty sets, such as ellipsoidal or elementwise bounds, without re-deriving the algorithm; only the LMI entries would change.
The reported sensitivity to target-channel error suggests that in practice more estimation effort should go into target position accuracy than into user or eavesdropper channel accuracy.
If the bounded-error assumption were replaced by stochastic errors, the S-Procedure step would need to give way to chance-constrained or distributionally robust reformulations, but the SROCR outer loop over rank-one constraints would likely carry over unchanged.
The same SROCR machinery should transfer to related near-field secure ISAC objectives, such as maximizing secrecy rate or minimizing power subject to sensing and secrecy constraints, since only the objective and the LMI definitions would change.

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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

Short version: this paper takes a real problem — robust secure beamforming for near-field ISAC with multiple users, targets, and eavesdroppers under norm-bounded CSI errors — and applies standard tools (S-Procedure to LMI, SROCR for rank-one constraints) in a way that is mostly algebraically correct. The formulation looks new relative to the cited work, and the complexity analysis is a useful addition. If the algorithm is repaired, this is a reasonable letter contribution.

The soft spot that matters is in Algorithm 1. The update rule sets v_k = min(1, λmax(W_k)/tr(W_k) + δ_k) with δ_k initialized at 0.1 and kept positive until the SDP is infeasible. Stopping triggers when v_k = 1 for all k and t has converged. With δ positive, v crosses 1 as soon as the eigenvalue ratio exceeds 0.9. The returned W_k need not be rank-one. Remark 1 says v=1 implies rank one, which is only true after solving the next subproblem with that v. As written, the algorithm can terminate without producing any w_k, and the curves in Section IV may be computing a value that no actual beamforming vector achieves. This is not a fatal flaw in the overall approach — one extra SROCR iteration with v=1 would force rank-one — but as published, the performance claims need rechecking.

Other issues are minor. The numerics lack error bars or any indication of how many channel realizations were used; the SDR baseline is not fully specified; and the bounded-error model, while standard, is doing a lot of work. The paper acknowledges near-field CSI acquisition remains challenging, and Fig. 4(c) honestly shows the design is very sensitive to target channel error.

Bottom line: this deserves a serious referee, but I would not accept until the stopping criterion is fixed and the simulations rerun. The new formulation is worth engaging with.

Referee Report

3 major / 5 minor

Summary. The letter studies robust transmit beamforming for a near-field secure integrated sensing and communication (ISAC) system with multiple CUs, sensing targets, and eavesdroppers under norm-bounded CSI errors. The authors formulate a max-min optimization that maximizes the worst-case sensing beampattern gain subject to worst-case SINR lower bounds for CUs, SINR upper bounds for eavesdroppers, and a total power constraint. They apply the S-Procedure to convert the semi-infinite CSI-uncertainty constraints into LMIs, then use the sequential rank-one constraint relaxation (SROCR) algorithm to handle the rank-one beamforming constraints. Numerical results are presented to show convergence and performance gains over SDR, information-only, and far-field baselines.

Significance. If the proposed algorithm as stated produced valid rank-one beamformers and the numerical comparisons are reproducible, the work would be a useful extension of robust secure beamforming to near-field ISAC. The S-Procedure transformations in Section III-A are standard and appear algebraically correct, and the paper provides a complexity estimate and explicitly discusses limitations of near-field CSI acquisition. However, the rank-one termination flaw in Algorithm 1 and the under-specified numerical baselines currently prevent the central claims from being accepted as stated. The contribution is potentially fixable, but the validation path requires substantive changes to the algorithm and the experiments.

major comments (3)

[Sec. III-B, Algorithm 1, Remark 1] The stopping rule in Algorithm 1 (Step 6) can terminate with a non-rank-one W_k. In Step 4, v^(i)_k = min(1, lambda_max(W^(i)_k)/tr(W^(i)_k) + delta^(i)_k), and delta^(0)=0.1 is halved only when (P3) is infeasible. If (P3) remains feasible, delta stays 0.1, so v=1 is reached when lambda_max/tr >= 0.9, not only when rank(W_k)=1. Since the loop stops immediately after v=1, the returned W_k need not be rank-one, and Remark 1's assertion that 'when v^(i)_k = 1, we have rank(W_k)=1' is inconsistent with the update. Such a W_k cannot be factored as w_k w_k^H, so no valid beamforming vector is delivered and the objective t in Figs. 2-4 does not correspond to a feasible solution of (P1). The algorithm should either continue until lambda_max/tr(W_k) >= 1-epsilon, solve one final SDP with v=1, or verify rank-one status and apply a valid extraction procedure before reporting results.
[Sec. III-B, Remark 1] The claimed convergence to a KKT point of the rank-one constrained problem (P2) relies on Theorem 1 of [16], but the authors do not verify that the hypotheses of that theorem hold for Algorithm 1 as written. In particular, the update of delta^(i)_k does not drive delta to zero on the feasible path, and the criterion v=1 can be satisfied without the rank-one relaxation being tight. Without a delta-to-zero mechanism or an explicit check that the rank-one constraint is active, the convergence guarantee is not established.
[Sec. IV, Figs. 2-4] The numerical evaluation does not specify whether the plotted points are single random deployments or averages, no error bars or confidence intervals are given, and the SDR baseline is not fully defined (in particular, no rank-one recovery or Gaussian randomization step is described). The far-field and LoS/NLoS curves are also not accompanied by the exact channel model used. Since the abstract's claim of outperformance is empirical, these omissions prevent the reader from assessing the statistical significance and reproducibility of the comparison.

minor comments (5)

[Sec. V] The conclusion contains a typo: 'SROCR agorithm' should be 'SROCR algorithm'.
[Sec. IV, Fig. 4] The sentence introducing Fig. 4 repeats 'versus eta_C,k and eta_C,k'; the second quantity should presumably be eta_E,l.
[Sec. III-B, Remark 1] Remark 1's statement that 'the trace of W_k approaches one' is not implied by any normalization in (P3); the SROCR constraint compares lambda_max to tr(W_k) but does not fix tr(W_k). Please rephrase.
[Sec. II-A and Fig. 3(c)] Footnote 2 says the design can be extended to LoS/NLoS scenarios, but Fig. 3(c) already reports a LoS/NLoS curve; the NLoS channel model used in that figure should be given explicitly.
[Sec. I] The phrase 'near-field unlink/downlink ISAC performance' appears to contain a typo ('unlink' should be 'uplink').

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper solves a well-defined robust beamforming optimization and compares against independent baselines.

full rationale

The derivation chain is self-contained as an optimization design study. The paper defines a target objective (maximize the minimum worst-case sensing beampattern gain) under explicit constraints and solves the resulting non-convex problem with standard transformations (S-Procedure) and a standard algorithm (SROCR). No parameter is fitted to the reported performance metric, and the metric plotted in the numerical section is exactly the objective function of problem (P1), which is the normal and non-circular mode of evaluation for a design problem. The S-Procedure equivalence is justified by an external theorem ([15], Luo-Sturm-Zhang), and the SROCR convergence claim cites external work ([16], Cao-Thompson-Poor); no load-bearing claim rests on a self-citation. The comparisons against the SDR and information-only baselines are against independently defined alternatives. The only flagged concern, namely that the SROCR stopping rule may terminate before the returned W_k is strictly rank-one, is a correctness or numerical-recovery issue rather than a circularity, because the algorithm's output is not being used as evidence for the assumptions that produced it. Accordingly, no circular step is present.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces no fitted constants. It relies on standard optimization theorems and domain assumptions about the near-field channel and error model. No new physical entities are postulated.

assumptions (5)

standard math S-Procedure equivalence (Theorem 3.3 in [15])
Used to convert semi-infinite constraints (7) and (9) into LMIs (8) and (10). The theorem requires a non-empty interior of the uncertainty set, which holds for norm balls with epsilon > 0.
standard math SROCR convergence to a local KKT point (Theorem 1 in [16])
Algorithm 1's convergence claim is imported from [16]; the paper does not verify all theorem assumptions beyond stating the problem.
domain assumption LoS-dominant near-field channel model
Eq. (1) models channels as LoS steering vectors with distance-dependent pathloss; NLoS components are neglected for mmWave, which may not hold in dense scattering environments.
domain assumption Bounded CSI error model with known norm bounds epsilon
Section II.A assumes norm of Delta h <= epsilon with known epsilon; the robust constraints are only valid under this worst-case bounded-error model.
domain assumption Eavesdropper locations are already acquired via sensing
Footnote 1 states that eavesdroppers' locations have been obtained by target sensing methods, allowing their channels to be modeled; if locations are unknown, the secrecy constraints cannot be formulated.

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Cite this review

Pith. "Pith review of Robust Beamforming Design for Secure Near-Field ISAC Systems." pith.science (2026). https://pith.science/paper/AEGBDPY3

@misc{pith2026250712881,
  author       = {Pith},
  title        = {Pith review of: Robust Beamforming Design for Secure Near-Field ISAC Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AEGBDPY3}},
  note         = {Machine review of arXiv:2507.12881}
}

read the original abstract

This letter investigates the robust beamforming design for a near-field secure integrated sensing and communication (ISAC) system with multiple communication users (CUs) and targets, as well as multiple eavesdroppers. Taking into account the channel uncertainty constraints, we maximize the minimum sensing beampattern gain for targets, subject to the minimum signal-to-interference-plus-noise ratio (SINR) constraint for each CU and the maximum SINR constraint for each eavesdropper, as well as the ISAC transmit power constraint. The formulated design problem is non-convex. As a low-complexity suboptimal solution, we first apply the S-Procedure to convert semi-infinite channel uncertainty constraints into linear matrix inequalities (LMIs) and then use the state-of-the-art sequential rank-one constraint relaxation (SROCR) method to address the rank-one constraints. The numerical results show that the proposed ISAC beamforming design scheme outperforms the existing semidefinite relaxation (SDR) and other baseline schemes, and it significantly enhances security and robustness for near-field ISAC systems.

Figures

Figures reproduced from arXiv: 2507.12881 by the authors.

**Figure 2.** (a) The convergence performance of Algorithm 1; (b) T [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** The achieved minimum sensing beampattern gain; (a) T [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: The achieved minimum sensing beampattern gain versu [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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Reviewed August 6, 2026 · model on record in the stance chip above.

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