Geometry-Aware Resource Allocation for Network-Level ISAC Systems

Christos Masouros; Jingheng Zheng; Kaitao Meng; Lei Cao; Luting Kong; Weiwen Weng; Weiyan Chen; Wenzhi Li; Xiao-Yang Wang; Yang Liu

REVIEW 4 major objections 5 minor 39 references

Geometry-Aware Resource Allocation for Network-Level ISAC Systems

T0 review · 4 major / 5 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read For cooperative OFDMA sensing, the optimal time-frequency resource allocation distributes each transceiver's aperture along the gradient direction of the Cramér–Rao lower bound.

desk verdict The two-UE closed form is solid and VGPA is a sensible heuristic, but the multi-UE 'proportional-to-gradient' optimality claim is not proven and as stated contradicts the paper's own KKT solution. read the letter →

arxiv 2607.29060 v1 pith:FXMMG5Q7 submitted 2026-07-31 eess.SP

Xiao-Yang Wang , Luting Kong , Lei Cao , Yang Liu , Jingheng Zheng , Weiwen Weng , Weiyan Chen , Wenzhi Li

show 2 more authors

Kaitao Meng Christos Masouros

This is my paper · ORCID

classification eess.SP

keywords network-levelISACcooperativesensingresourceallocationCramér–RaolowerboundOFDMAgeometricwater-fillingfrequencyaperturelocalizationaccuracy

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 asks how a multi-antenna base station should divide time-frequency resources — which subcarriers and symbol slots — among many user devices that jointly illuminate a target for sensing. The central claim is that the optimal division is not fixed by channel quality alone: it is controlled by each transceiver's aperture, defined as the variance of its allocated subcarrier or symbol indices, and the best allocation follows the gradient of the sensing Cramér–Rao lower bound with respect to those apertures. For two transmitters the paper gives a closed-form solution; for many transmitters it proves, under a statistical model treating UE geometry as i.i.d. draws from a fixed joint distribution, that distributing apertures proportionally to this gradient minimizes the expected CRLB, a rule it names geometric water-filling. Because delay and Doppler estimation are mathematically isomorphic, the same rule covers both ranging and velocity sensing. The paper then converts the continuous optimal apertures into discrete OFDMA subcarrier sets with a linear-complexity Variance-Guided Partitioning Algorithm, and numerical results show near-lower-bound sensing performance while honoring each UE's communication-rate requirement.

What carries the argument

The central object is the aperture, the variance of the subcarrier indices (frequency aperture) or symbol indices (time aperture) assigned to a UE; it enters the Fisher information matrix and therefore sets the CRLB. The proof rests on three pieces: the two-transmitter closed form, which exposes the geometric water-filling intuition that resources are cross-leveraged between partners; the asymptotic multi-transmitter gradient, computed by modeling UE angles and distances as i.i.d. draws from a fixed joint PDF and applying the law of large numbers, which yields a gradient direction independent of the operating point (eq. (38)); and the variance-stride relationship (Prop. 4), which lets an alg

What would settle it

For a fixed small network (e.g., K=4 transceivers with a chosen, non-uniform coupled angle–distance distribution), numerically grid-search the continuous aperture split under a total-aperture constraint to locate the exact CRLB minimizer, and compare it with the allocation proportional to the gradient in eq. (38) computed from the assumed PDF. If the exact minimizer does not lie on that gradient direction, or if the expected-CRLB landscape has a minimum inconsistent with that direction, Proposition 3 fails. A lighter check: evaluate ∂CRLB/∂σ_k^2 at several feasible aperture points and test whe

Watch

Extended reading notes

Core claim

In cooperative OFDMA sensing, the frequency-domain aperture of each transmitter — the variance of its assigned subcarrier indices — is the resource that controls localization accuracy, and the whole allocation problem reduces to distributing this variance under a total-aperture budget. For K=2 the optimal apertures have a closed form that reveals cross-linked geometric gains: each node's allocation is scaled by the partner's angular quality. For K>2, using a law-of-large-numbers approximation with i.i.d. UE angles and distances, the paper derives the expected gradient of the CRLB with respect to the apertures, proves the CRLB decreases with every aperture unless the UE sits at angle pi (coll

Load-bearing premise

The proof for multi-transmitter networks assumes UE angles and distances are independent draws from a fixed probability distribution and that the network is large enough for the law of large numbers to replace random sums by expectations; if that does not hold at the actual network size, allocating apertures proportionally to the fixed gradient is not proven to minimize the CRLB.

Editorial extensions

If this is right

In OFDMA network-level ISAC, sensing resources should be allocated to transceivers in proportion to their geometric sensitivity, not merely their channel SNR.
Transceivers whose angle to the target is collinear with the target-receiver axis should receive minimal aperture, since their contribution to the CRLB vanishes.
The same gradient-based allocation rule applies to velocity estimation as to ranging, because delay and Doppler estimation are mathematically isomorphic.
The Variance-Guided Partitioning Algorithm offers a scalable, roughly linear-complexity way to turn continuous optimal apertures into discrete subcarrier sets while satisfying per-UE communication-rate constraints.
According to the paper's numerical results, uniform aperture allocation can underperform even random allocation when transceiver geometry is asymmetric, because it wastes budget on geometrically blind nodes.

Reading between the lines

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

A natural adaptive extension would estimate the joint angle–distance distribution online from tracking data and recompute the aperture gradient, turning water-filling into a control policy for mobile targets or users.
The same variance–geometry coupling should generalize beyond apertures to other sensing resource dimensions such as power or antenna selection, since the FIM depends only on weighted sums of steering-vector outer products.
For finite small networks, an iterative scheme that re-evaluates the exact CRLB gradient at the current allocation point would likely beat the fixed asymptotic direction; comparing the two would measure how much the law-of-large-numbers assumption actually costs.
When the i.i.d. assumption fails — as in clustered or street-canyon deployments — the fixed-gradient theorem is not strictly applicable, and the paper's own street-canyon results suggest that a geometry-only (angle-based) allocation already captures most of the gain, pointing to a robustness hierarchy worth formalizing.

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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

Quick take: this paper deserves a serious referee, but not for the theorem it headlines. What is actually new is the two-UE aperture allocation and the geometry-aware framing over prior work [21]. The two-UE derivation is clean Lagrange multiplier work, and the simulation confirms the closed form. The time-frequency isomorphism is a useful organizing idea, and VGPA is a reasonable low-complexity heuristic that handles communication-rate constraints in the experiments.

The soft spot is exactly where the reader put it. Proposition 3 is not established. The asymptotic passage replaces E[A/B] by E[A]/E[B] in (26), then treats the gradient direction as fixed. Even granting the i.i.d. model, a fixed gradient direction does not imply proportional allocation. The paper's own two-UE case demonstrates this: the optimum satisfies the KKT condition Γ_k/σ_k^4 ∝ N_k, giving σ_1^2/σ_2^2 = sqrt(Γ1 N2/(Γ2 N1)), not σ^2 proportional to ∇CRLB. A genuinely fixed linear gradient on a simplex would push to a corner, not to an interior proportional point. So the 'geometric water-filling principle' is currently a heuristic, not a theorem. The K>2 simulations compare against the paper's own 'optimal' allocation rather than an independent lower bound or exhaustive search, so they do not close that gap.

What I would not do is discard the paper. The two-UE result is correct and useful; the algorithm design is practical; the street-canyon and Gaussian-mixture experiments are the right kind of robustness checks. The multi-UE section can be reframed as an asymptotic heuristic, with Proposition 3 replaced by a claim like 'under the LLN approximation, a reasonable strategy is to assign apertures according to sensitivity, and VGPA achieves that target.' That version is publishable after revision. The paper should also disclose the expectation-ratio approximation and stop calling the heuristic a proof. Missing code and data are minor here; this is an algorithm paper for TWC, not a benchmark submission.

Bottom line: accept for review. A good referee should push for the multi-UE rewrite, but a desk reject would discard a sound two-UE contribution and a practical heuristic that the simulations show works.

Referee Report

4 major / 5 minor

Summary. The paper proposes a two-stage framework for allocating time-frequency resources among distributed OFDMA transceivers in a network-level ISAC system, with the goal of minimizing the CRLB of target position estimation. For the two-transmitter case, Proposition 1 gives a closed-form optimal allocation of frequency apertures (variance of subcarrier indices) under a total variance budget; this part is a standard convex optimization result. For the multi-transmitter case, the paper uses an asymptotic law-of-large-numbers argument to claim that the gradient of the expected CRLB has a fixed direction, and then states Proposition 3, asserting that allocating aperture proportions along this gradient minimizes the expected CRLB. The authors call this 'geometric water-filling.' They then formulate the discrete subcarrier assignment as an NP-hard integer partitioning problem and propose the Variance-Guided Partitioning Algorithm (VGPA), a two-stage heuristic combining variance-guided initialization with Metropolis-based local search. Numerical simulations compare VGPA against random and uniform allocation in two-UE, multi-UE, Gaussian-mixture, and street-canyon scenarios.

Significance. If the multi-transmitter optimality claim were correct, the geometric water-filling principle would be a genuinely useful design insight for network-level ISAC, and the paper would make a strong theoretical contribution. The two-UE closed-form result is clean and appears correct given the stated CRLB model; it is a worthwhile contribution. The proposed VGPA is low-complexity and the numerical comparisons suggest that it can outperform geometry-agnostic baselines. However, the central multi-transmitter theorem is not established: the asymptotic derivation uses mathematically invalid expectation approximations, and the inference from a fixed gradient direction to proportional allocation is a non sequitur. As a result, the paper's headline claim of optimal resource allocation is not supported. The numerical evaluation of VGPA is also circular in that the algorithm is constructed to enforce the very ratios that Proposition 3 claims to be optimal. The practical heuristic may still be useful, but the paper as submitted does not provide a sound theoretical foundation for it.

major comments (4)

[§III-C.2, Eq. (38), Proposition 3] The inference from a fixed gradient direction to proportional allocation is invalid. If ∇E[CRLB] = c·h with h constant, then the objective is, to first order, linear in the aperture vector. Under a linear budget constraint Σ N_k σ²_k ≤ E_total, the constrained minimizer of a linear function is an extreme point (put the entire budget into the most sensitive coordinate, subject to bounds), not the proportional interior point σ² ∝ h. The two-UE solution in Appendix A, Eq. (46), satisfies σ²_1/σ²_2 = sqrt(Γ₁N₂/(Γ₂N₁)), which is not proportional to the gradient of CRLB. Thus Proposition 3 is not merely unproved; it is false as stated. No proof or appendix is provided for it.
[§III-C.1, Eq. (26)] Equation (26) replaces E[(A−B)/(ρ C)] with (E[A]−E[B])/(ρ E[C]), where A = ∂g₁/∂v_k · g₂, B = g₁·∂g₂/∂v_k, and C = g₂². This is mathematically invalid unless strong higher-order moment and correlation conditions are established, which the paper does not do. Chebyshev's LLN does not justify replacing the expectation of a ratio by the ratio of expectations. Every subsequent expression, including the non-positive derivative in Proposition 2 and the fixed gradient in Eq. (38), inherits this error. Eq. (37) is therefore not the derivative of the true expected CRLB.
[§III-C.1, Eqs. (27)-(31)] The factorization leading to Eqs. (27) and (28) is not justified. For example, E[f(k,k′)(f(k,k′)−f(k′,k)) v_k v_k′] is rewritten as E[f(k,k′)(f(k,k′)−f(k′,k))] E[v_k v_k′], which requires independence between the angular terms and the distance-dependent terms v_k, v_k′; the paper only assumes a joint PDF p(θ,d), not independence. Moreover, θ_{k1} is sometimes treated as a deterministic index (e.g., Eqs. (29)-(33)) and sometimes as random (as in the stated expectation over geometry). The quantity E[∂CRLB/∂σ²_{k1}] is not well-defined under the current notation.
[§V, Figs. 6-9] The numerical evaluation does not test the optimality claim. VGPA is explicitly constructed to match the GWF ratios: Algorithm 1 uses the variance ratios α from the geometric water-filling strategy and Eq. (40) projects the partition onto those ratios. Comparing VGPA against random and uniform allocation therefore only shows that this particular heuristic beats those baselines; it does not validate Proposition 3. The abstract's claim that the scheme 'approaches the theoretical performance lower bound' is supported only in the two-UE simulations (Fig. 5); no comparable multi-UE lower-bound comparison is presented.

minor comments (5)

[Eq. (12)] The definition of E_total is badly typeset: the expression 'NX n=1 ... NX m=1' is garbled, making the summation range and the variance definition unclear. Please correct the notation.
[§III-B, after Eq. (13)] The text says 'we first optimize the objective with respect to σ²_1 and σ²_1'; the second variable should be σ²_2.
[§IV-A] Typo: 'the minimized communication rate of each UE shuld be satisfied' should be 'should'.
[§V, paragraph before Fig. 7] The text refers to 'Fig. 6(a)' and 'Fig. 6(b)' when discussing the results shown in Fig. 7. Please renumber the cross-references.
[Fig. 4] The figure annotation appears to give α* = 0.632 while the text reports α = 0.607. Please reconcile the values.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity; the core two-UE derivation is self-contained, while the multi-UE optimality gap is a rigor issue, not a circular reduction.

full rationale

The two-transmitter result is derived rather than assumed: the FIM in (3) is co-cited to [11] and external [33], eq. (11) expands CRLB=Tr(F_N^{-1}), and Appendix A solves the Lagrange problem (42)-(48) to obtain the closed forms (15)-(16). No parameters are fitted and the constrained minimizer is obtained from stationarity, so this part is self-contained. The multi-transmitter chain is where the paper is weak, but the weakness is not circularity. Eq. (26) approximates E[A/B] by E[A]/E[B], and the move from a 'fixed' gradient direction (38) to Proposition 3 (proportional allocation minimizes the expected CRLB) is asserted rather than derived from KKT/boundary conditions; in fact a fixed gradient direction alone does not imply that the constrained minimizer is the proportional interior point. That is a mathematical non-sequitur / omitted proof, not an output equivalent to its input by construction. Same-group prior work [11],[21] supplies the FIM/aperture-variance model, but the FIM is independently cited to [33] and the variance-budget identity is elementary; there is no uniqueness theorem or unverified ansatz chain that forces the result by self-citation. The only mild tautology is in the evaluation: VGPA's objective (39)/(41) explicitly penalizes deviation from the target ratios alpha, so Fig. 7(a) showing that the achieved ratios track alpha mainly confirms that the optimizer enforces its own constraint, not that Proposition 3 is globally true. This weakens the validation but is not a circular step in the derivation. Overall: no significant circularity; score 2 reflects the same-group model citations and the validation tautology, not a reduction of the central claim to its inputs.

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

The central derivation rests on the CRLB model from [11,21,33] (with author overlap), on the time-frequency isomorphism, and on an i.i.d./LLN asymptotic model for K>2. No free parameters are fit to data; the aperture ratios α are outputs of the optimization, not fitted constants. The VGPA validation is partly self-consistent because the algorithm is explicitly designed to realize the α it is compared against.

assumptions (6)

domain assumption The FIM in (3) is F_N = ρ Σ_k d_{1,k}^{-β} d_2^{-β} σ_k^2 [a_k², a_k b_k; a_k b_k, b_k²], i.e., sensing information scales linearly with the variance of allocated subcarrier indices.
Cited to [11], [21], [33]; author overlap with [11] and [21]. If this model is invalid, the optimality results do not transfer to real OFDMA ISAC receivers.
domain assumption Delay and Doppler estimation are mathematically isomorphic, so time-aperture optimization is structurally identical to frequency-aperture optimization.
Invoked in the Remark after the aperture definitions (Section II) to claim time-domain results without a separate derivation.
domain assumption For K>2, UE angles/distances are i.i.d. draws from a fixed joint PDF p(θ,d), and sums can be replaced by expectations via Chebyshev's law of large numbers.
Used in Section III-C eqs. (27)-(31) to derive the asymptotic CRLB gradient and the fixed gradient direction in (38).
ad hoc to paper Angles are uniformly distributed over [0,2π) and independent of distances in the closed-form gradient example.
Example in Section III-C; the paper claims the GWF strategy applies beyond this independence assumption, but the explicit optimality statement is derived under it.
standard math Metropolis refinement with a sufficiently slow cooling schedule converges to the global optimum with probability one.
Invoked in Section IV-C citing [38]; the paper does not specify the cooling schedule or validate convergence for this particular objective.
domain assumption The variance budget constraint N₁σ₁²+N₂σ₂²≤E_total and the total-variance decomposition (51)-(52) hold with subset means aligned to the global mean.
Used in problem (13) and Appendix C; assumes between-subset variance vanishes, which is a design choice imposed later by the mean-consistency constraint.

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Cite this review

Pith. "Pith review of Geometry-Aware Resource Allocation for Network-Level ISAC Systems." pith.science (2026). https://pith.science/paper/FXMMG5Q7

@misc{pith2026260729060,
  author       = {Pith},
  title        = {Pith review of: Geometry-Aware Resource Allocation for Network-Level ISAC Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FXMMG5Q7}},
  note         = {Machine review of arXiv:2607.29060}
}

read the original abstract

Network-level integrated sensing and communication (ISAC) is recognized as a transformative technology for next-generation mobile radio systems. By enabling collaboration among multiple transceivers, network-level ISAC can significantly enhance both communication and sensing performance through spatial diversity. However, existing resource allocation strategies typically overlook the impact of spatial geometry, where identical time-frequency resources contribute differently to sensing accuracy depending on the transceiver's location. This leaves the fundamental coupling between spatial topology and resource efficacy unclear, rendering optimal resource allocation a critical challenge for unlocking the full potential of network-level ISAC.To address this challenge, this paper investigates the optimal distribution of time-frequency resources across spatially distributed transceivers through a theoretically grounded two-stage framework. First, we analytically derive the optimal time and frequency aperture distributions for sensing, defined as the variances of the allocated symbol and subcarrier indices, respectively, under both two-transmitter and multi-transmitter scenarios. By exploiting the mathematical isomorphism between delay and Doppler estimation, we prove that the optimal resource allocation strategy follows the gradient direction of the Cramer-Rao Lower Bound (CRLB) with respect to the apertures. Second, to bridge the gap between theoretical aperture values and practical OFDMA constraints, such as the minimized communication rate of each user equipment (UE), we formulate the resource allocation as a combinatorial integer partitioning problem. To tackle the NP-hard nature of the formulated problem, a low-complexity Variance-Guided Partitioning Algorithm (VGPA) is proposed to jointly optimize the subcarrier and symbol patterns for communication and sensing.

Figures

Figures reproduced from arXiv: 2607.29060 by the authors.

**Figure 1.** System model. required communication rate, we formulate the subcarrier/symbol allocation as a variance-constrained integer partitioning problem. Since this problem is NP-hard, we develop a low-complexity Variance-Guided Partitioning Algorithm (VGPA). Numerical results demonstrate that the VGPA effectively approaches the theoretical performance lower bound in two-UE scenarios and significantly improve sensing perfo… view at source ↗

**Figure 2.** Frequency aperture computation for UE 1. The colorful boxes [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** Network-level distributed ISAC set-up. 0 0.2 0.4 0.6 0.8 1 Aperture Allocation Ratio , (UE1 share) 101 102 CRB (m 2 ) , * Sweep Curve = 0.632 Theoretical Lower Bound Derived Optimal Point The derived optimal point achieves the lowest CRLB 0.2 0.4 0.6 0.8 Aperture Allocation Ratio , (UE1 share) [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: The CRLB corresponding to different aperture allocation scheme in [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]

**Figure 6.** Figure 6: CDF of CRLB with different tranceivers distribution. [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]

**Figure 7.** Figure 7: Verification of resource allocation in five-UE scenarios under com [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]

**Figure 9.** Figure 9: Performance in an asymmetric urban street canyon scenario. (a) [PITH_FULL_IMAGE:figures/full_fig_p011_9.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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