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REVIEW 3 major objections 5 minor 18 references

Mutual Information-oriented ISAC Beamforming Design for Large Dimensional Antenna Array

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Statistical CSI is enough: closed-form MI objective for ISAC beamforming.

desk verdict A useful application of the authors' own free-probability framework to statistical-CSI ISAC beamforming, but the main theorem is imported without the assumptions that make it valid. read the letter →

arxiv 2411.13305 v2 pith:XZW3LEQV submitted 2024-11-20 eess.SP

Shanfeng Xu , Yanshuo Cheng , Siqiang Wang , Xinyi Wang , Zhong Zheng , Zesong Fei This is my paper · ORCID
classification eess.SP
keywords ISACbeamformingmutualinformationfreeprobabilitystatisticalCSIMIMOprojectedgradientascentlargedimensionalantennaarray
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 studies transmit beamforming for a MIMO integrated sensing and communication system in which the terminal knows only the statistics of the communication channel and the extended-target sensing channel, not the instantaneous realizations. The central claim is that, as the antenna counts grow large, the weighted mutual information $I(W)=\rho L N_r I_s+(1-\rho)N_u I_c$ converges almost surely to a deterministic function of the beamforming matrix, and that this function has a closed-form expression in terms of operator-valued Cauchy transforms. The authors then maximize this closed-form objective with a projected gradient ascent algorithm. If correct, ISAC beamforming can be designed from channel statistics alone using a cheap, deterministic objective, which is precisely the regime relevant to large-dimensional arrays in dynamic environments.

What carries the argument

The linearization trick embeds the non-free product $\hat{G}SS^\dagger\hat{G}^\dagger$ into a larger block matrix $B_L$, in which deterministic and random blocks become asymptotically free. The operator-valued Cauchy transform of $B_L$ is then characterized by the subordination formula and decomposed by matrix inversion into the fixed-point equations of Proposition 1. The parameterized one-sided correlation matrices $\eta_l$, $\tilde\eta_l$, $\zeta$, $\tilde\zeta$, $\tau$, and $\tilde\tau$, defined in Eqs. (57)--(66), encode all the statistical CSI that enters the closed form. The PGA algorithm carries the optimization by evaluating and differentiating this deterministic objective.

What would settle it

Run a Monte Carlo comparison with a non-Gaussian symbol matrix, such as QPSK, while retaining $\mathbf{E}[\mathbf{S}\mathbf{S}^\dagger]=\mathbf{I}_M$; if the empirical weighted MI fails to converge to the closed-form expression as $N_t$, $N_r$, and $N_u$ grow with fixed ratios, the freeness assumptions behind Proposition 1 are invalid.

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Extended reading notes

Core claim

The paper asserts that both the sensing mutual information and the communication mutual information admit deterministic asymptotic limits whose Shannon transforms can be written in closed form. For the sensing Gram matrix $B_1=\hat{G}SS^\dagger\hat{G}^\dagger$, the paper's Proposition 1 gives the Cauchy transform as $G_{B_1}(z)=\frac{1}{LN_r}\operatorname{Tr}\big[G_{\tilde C}(z)\big]$, where $G_{\tilde C}(z)$ solves the fixed-point system (30)--(39) built from parameterized one-sided correlation matrices of the channels and the symbol stream, and Proposition 2 gives the corresponding Shannon transform as Eq. (40). An analogous system, Eqs. (41)--(46), gives the communication MI. The paper then proposes to maximize the weighted asymptotic MI by projected gradient ascent, with the gradient given in closed form in Eq. (56), and numerical results show that the closed-form expressions match Monte Carlo simulations and that the algorithm improves the weighted MI while converging in about three iterations.

Load-bearing premise

The derivation assumes that the deterministic and random blocks of the linearized matrix are asymptotically free and that the channel fluctuation entries are Gaussian; in particular, the paper only assumes $\mathbf{E}[\mathbf{S}\mathbf{S}^\dagger]=\mathbf{I}_M$ for the symbol matrix, which by itself does not guarantee freeness or the stated $\zeta$ formulas.

Editorial extensions

If this is right

  • Beamforming can be updated from second-order channel statistics alone, removing the need for instantaneous CSI at the ISAC terminal.
  • The weighting factor $\rho$ gives a direct knob for trading sensing MI against communication MI, as demonstrated in the Pareto-type trade-off figure.
  • The PGA algorithm converges within about three iterations in the simulations, which makes online reconfiguration of the beamforming matrix practical.
  • The closed-form deterministic objective matches Monte Carlo results, so it can replace expensive simulations for large-array ISAC design studies.

Reading between the lines

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

  • Editorial inference: the same linearization-plus-subordination recipe likely extends to other ISAC channel models, such as RIS-assisted or doubly-scattered propagation, since the proof outline already follows the pattern used for multi-RIS MIMO.
  • Editorial inference: if the data symbol matrix is drawn from a finite alphabet rather than a Gaussian model, the freeness assumptions behind the closed form may fail; a testable extension is to simulate non-Gaussian $S$ and measure the gap to the formula.
  • Editorial inference: the closed-form objective could enable water-filling-like power allocation across data streams and analytical sensitivity studies of MI with respect to antenna counts, beyond the fixed transmit-power projection used here.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper considers a large-dimensional MIMO ISAC system in which an ISAC UE transmits to a BS and senses an extended target, and it proposes to optimize the transmit beamforming matrix W by maximizing a weighted asymptotic mutual information (MI) comprising sensing and communication components. The main technical claim is that, using operator-valued free probability and a linearization trick, the weighted asymptotic MI admits the closed-form deterministic characterization in Propositions 1 and 2, Eqs. (29)-(49). Based on this characterization, the authors propose a projected gradient ascent (PGA) algorithm, Eqs. (51)-(56), and demonstrate numerically that the closed-form MI matches Monte Carlo simulations and that the PGA algorithm improves the weighted MI.

Significance. If the asymptotic characterization were fully established, the paper would offer a useful design tool: a deterministic, closed-form objective for ISAC beamforming under statistical CSI, avoiding the need for instantaneous channel estimates, together with a fast first-order algorithm. The Monte Carlo validation in Fig. 2 and the convergence demonstration in Fig. 3 are valuable and support the plausibility of the derived expressions in the specific simulation model. However, the central proposition is inherited from prior work via a one-sentence outline, and several hypothesis and dimension issues in the stated model must be resolved before the closed-form objective and the PGA gradient are justified as stated.

major comments (3)
  1. [Section II-A; Appendix A, Eqs. (65)-(66); Proposition 1] The signal model only assumes E[SS†] = I_M. This second-order condition does not imply the one-sided correlation identities ζ(D) = (1/Ns)Tr(D)I_Ns and ~ζ(~D) = (1/Ns)Tr(~D)I_M used in Eqs. (65)-(66), nor does it guarantee the asymptotic freeness of S and S† from the deterministic unitary matrices and from the channel random components in the linearized matrix. For example, S = I_M with M = Ns satisfies the stated assumption, but then ζ(D) = D rather than a scalar multiple of the identity, so Eq. (65) fails and B1 = \hat G\hat G† has a different limiting spectrum. The proof of Proposition 1 is only a one-sentence reference to [17, Prop. 2]; the paper should either prove the subordination step for this particular linearized ISAC matrix or state precisely which hypotheses of [17] are satisfied, including an explicit distributional assumption on S such as i.i.d. Gaussian or Haar-distributed entries. As written, the deterministic objective (49) and the PGA gradient (56) are not justified under the assumptions stated in Section II-A.
  2. [Eqs. (5), (24), (30)-(39), (56)] The displayed dimensions in the linearization are inconsistent unless Nt = M, which is not assumed. In Eq. (5), \hat G = [(G1W)†, (G2W)†, ..., (GLW)†]† has size LM×Nr, so \hat GSS†\hat G† is LM×LM, whereas the determinant is taken over I_{LNr}; the required object is [G1W; ...; GLW] of size LNr×M, which is the size used in the block matrix BL in Eq. (24). Once this is corrected, the one-sided correlation functions in Eqs. (57)-(58) are defined for \tilde G_l, but the random block entering BL is \tilde G_lW. For non-unitary W the correct correlations should involve W†η_l(...)W and \tilde η_l(W...W†), and these W factors need to appear in Ψ(z), ~Ψ(z), and the gradient. As printed, Ψ(z) in Eq. (33) has dimension Nt×Nt while Π in Eq. (31) must be M×M unless Nt = M, and Eq. (36) appears to contain G†...G† rather than G†...G. This is load-bearing because the W-dependence in the variance profile is exactly what the PGA algorithm optimizes; the closed-form objective and gradient need to be rederived with consistent dimensions and explicit W dependence.
  3. [Proposition 2, Eq. (83)] The proof of Proposition 2 establishes only the derivative identity dVB1/dz = -1/z - GB1(-z). To conclude that the proposed expression is the Shannon transform, a boundary condition must also be verified, for example by showing that both sides have the same asymptotic limit as |z| → ∞ or, equivalently, as σ² → 0 or σ² → ∞. As written, the integration constant is not checked. This is a local gap in the proof of Proposition 2 and should be fixed by a short asymptotic argument.
minor comments (5)
  1. [Section IV, Fig. 2, Fig. 3, Fig. 4, Fig. 5] The text refers to 'Fig. 3' when discussing the accuracy of the unoptimized MI curves, but the corresponding caption is Fig. 2; later, the text refers to 'Fig. 5' for both the optimized MI versus SNR and the sensing-communication trade-off plot. Please renumber the figure citations to match the captions.
  2. [Section II-A and Section III] The symbol M is used both for the number of data streams and for the variance-profile matrix of the communication channel in Eqs. (63)-(64). This notational collision makes the dimension arguments in Proposition 1 difficult to follow; please use distinct symbols for the data-stream count and the variance-profile matrices.
  3. [Section III-B, Eq. (56)] The gradient display is garbled: there is an empty Eq. (55), and Eq. (56) mixes W†G~Ψ^{-1}GW with E_{i,j}†G†~Ψ^{-1}GW in a way that is hard to parse. Please rewrite the gradient derivation with explicitly defined intermediate quantities and consistent dimensions.
  4. [Section IV, Fig. 2] The claimed 'excellent' match between theoretical and Monte Carlo results is not quantified; adding mean squared error or confidence bands for the simulated curves would make the validation more convincing.
  5. [Introduction, first paragraph] The sentence beginning 'In the case of large dimensional antenna array, perfect CSI becomes challenging' is repeated awkwardly in the same paragraph and should be edited for clarity.

Circularity Check

1 steps flagged · score 4.0 of 10

No fitted or definitional circularity, but the central Cauchy transform in Prop. 1 is inherited from the authors' own [17] via a one-sentence proof, and the stated E[SS†]=I_M does not justify the S correlation identities used in that theorem.

  1. self citation load bearing [Section III-A, Proposition 1 and its proof (Eqs. (29)-(39)); text immediately before Proposition 1 and the proof paragraph.]
    "It can be observed that BL shares a similar structure with the matrix L proposed in [17, Prop. 2], which inspires us to use the method in [17]. ... Proof. The proof of this Proposition 1 is similar to the method presented in [17, Prop. 2]... First, we prove that the deterministic and random components of the linearized matrix are free. Then, by applying the subordination formula, we derive the equation for the operator-valued Cauchy transform. Finally, using the matrix inversion formula, we decompose the operator-valued Cauchy transform, thereby obtaining the above expressions."

    The deterministic Cauchy transform G_B1(z) in Proposition 1 is the load-bearing step: it is the input to the Shannon transform V_B1(z) in Proposition 2, which defines the weighted asymptotic MI in Eq. (49), which in turn supplies the PGA gradient in Eq. (56). The proposition is not actually proved in this paper; its proof is deferred to [17, Prop. 2], whose authors (Z. Zheng, S. Wang, Z. Fei, J. Yuan) overlap with the present author list (S. Wang, Z. Zheng, Z. Fei). The one-sentence outline does not demonstrate that freeness and subordination hold for this ISAC-specific linearized matrix B_L when S is assumed only to satisfy E[SS†]=I_M.

full rationale

No fitted-input or definitional circularity is present: no parameter is calibrated to data, and the closed-form MI expressions are not equal to the model by construction. The Monte Carlo comparison in Fig. 2 is an external numerical benchmark and provides independent evidence for the algebra of Proposition 2 when S is drawn as Gaussian. The main caveats are (i) Proposition 1 is inherited from the authors' prior work [17] with only a proof sketch, and (ii) the stated assumption E[SS†]=I_M in Section II-A does not by itself imply the one-sided correlation identities ζ(D)=Ns^{-1}Tr(D)I_Ns and ~ζ(~D)=Ns^{-1}Tr(~D)I_M in Eqs. (65)-(66); a deterministic S=I_M satisfies the former but not the latter, so the variance-profile/freeness conditions are not fully specified. These are correctness and omitted-assumption risks rather than circular reductions by construction. Because the central theorem is load-bearing and is transferred from a same-group citation rather than independently re-derived, the circularity score is 4 rather than 0-2, but the numerical validation prevents a higher score.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The derivation leans on the Weichselberger channel model, perfect knowledge of channel statistics, and external random matrix theorems. There are no fitted free parameters and no new entities; the paper's contribution is the combination of these ingredients for the ISAC MI objective.

assumptions (4)
  • domain assumption Weichselberger MIMO channel model for both the sensing and communication channels
    Equations (3)-(4) assume the random part of each channel is a Gaussian matrix with variance profile M or N_l modulated by fixed unitary matrices; this is a modeling assumption, not derived.
  • domain assumption Perfect statistical CSI at the UE, including known unitaries, variance profiles, and noise variances
    The optimization problems (P2)-(P3) and the expressions in Propositions 1-2 require these statistics as inputs; the paper does not discuss estimation error or how they are obtained.
  • standard math Asymptotic freeness and convergence of empirical spectral distributions from the cited random matrix theory literature
    Equations (9)-(10), (19), and the linearization step rely on theorems from [12], [15], and [16], which are cited but not proved here.
  • ad hoc to paper Sufficient distributional and independence structure on S for freeness
    Only E[SS†]=I_M is stated in Section II-A, but the free-probability calculations require a stronger isotropic or independence structure, such as random matrix entries with matching higher moments. This assumption is not explicitly flagged.

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Pith. "Pith review of Mutual Information-oriented ISAC Beamforming Design for Large Dimensional Antenna Array." pith.science (2026). https://pith.science/paper/XZW3LEQV

@misc{pith2026241113305,
  author       = {Pith},
  title        = {Pith review of: Mutual Information-oriented ISAC Beamforming Design for Large Dimensional Antenna Array},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XZW3LEQV}},
  note         = {Machine review of arXiv:2411.13305}
}
read the original abstract

Existing integrated sensing and communication (ISAC) beamforming design were mostly designed under perfect instantaneous channel state information (CSI), limiting their use in practical dynamic environments. In this paper, we study the beamforming design for multiple-input multiple-output (MIMO) ISAC systems based on statistical CSI, with the weighted mutual information (MI) comprising sensing and communication perspectives adopted as the performance metric. In particular, the operator-valued free probability theory is utilized to derive the closed-form expression for the weighted MI under statistical CSI. Subsequently, an efficient projected gradient ascent (PGA) algorithm is proposed to optimize the transmit beamforming matrix with the aim of maximizing the weighted MI.Numerical results validate that the derived closed-form expression matches well with the Monte Carlo simulation results and the proposed optimization algorithm is able to improve the weighted MI significantly. We also illustrate the trade-off between sensing and communication MI.

Figures

Figures reproduced from arXiv: 2411.13305 by the authors.

Figure 1
Figure 1. Considered MIMO ISAC system. The received signal at BS can be expressed as Yc = HcWS + NC , (1) where Hc is the channel between UE and BS, NC ∈ C Nu×Ns is the additive white Gaussian noise (AWGN) at the receiving antennas of BS and NC ∼ CN (0, σ2 c INs ). The received echoes at the UE, Ys ∈ C Nr , can be expressed as Ys = X L l=1 GlWS + NS, (2) where Gl ∈ C Nr×Nt is the round-trip channel matrix between the l−th sca… view at source ↗
Figure 2
Figure 2. Communication MI and sensing MI versus SNR. [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Weighted MI versus the number of iterations, where [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Weighted MI versus SNR with different schemes, where [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]
Figure 5
Figure 5. Figure 5: Sensing MI versus Communication MI. with different numbers of antennas for the considered system in [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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