{"id":"84562857-ee1d-43b3-a40d-6fb1cfbab886","arxiv_id":"2411.13305","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The authors derive an asymptotic weighted sensing-plus-communication mutual information expression and a projected gradient ascent beamforming algorithm for large MIMO ISAC under statistical CSI.","lead":"This paper derives an approximate closed-form expression for the mutual information of a MIMO integrated sensing and communication system using only channel statistics, and uses it to optimize transmit beamforming. A smart generalist might care because it offers a tractable way to design large antenna arrays that simultaneously communicate and sense without perfect channel knowledge.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The closed-form MI in Prop. 1 rests on an unstated distributional/freeness assumption for S; E[SS†]=I_M alone does not justify Eqs. (65)-(66) or the transfer from [17], so the deterministic objective and PGA are not yet established as stated.","rationale":"The paper is a legitimate extension of the authors' earlier operator-valued free probability framework to a weighted sensing-plus-communication MI. The derivation of Prop. 2 from Prop. 1 is sketched in the appendix, and the numerical agreement at N_t=N_r=N_u=16 is visually good, which gives some independent support. My concern is narrower: the theorem as stated has an unstated hypothesis on S. The authors explicitly specify the distribution of P and Q_l but not of S, and only impose E[SS†]=I_M. Since the linearization trick and the subordination equations (65)-(66) depend on the second-order and asymptotic freeness structure of S, this gap is load-bearing. If the intended model is i.i.d. Gaussian S with variance 1/Ns, the main result is plausible and the paper's conditional verdict is appropriate; but as written, the conditions under which Prop. 1 holds are not established. This does not amount to an internal contradiction if one reads the numerical section as implicitly using i.i.d. S, but the theoretical claim in Prop. 1 should be tightened or the result should be conditioned on an explicit random-matrix model for S. I therefore see no reason to change the reader's CONDITIONAL verdict.","tokens_in":11626,"tokens_out":9254,"duration_ms":103771,"concrete_test":"Set M=Ns and take S=I_M, which satisfies the stated E[SS†]=I_M. Keep all other channel parameters as in Section IV, compute the sensing MI in Eq. (5) directly by Monte Carlo averaging over P and Q_l, and compare it with the theoretical value from Eq. (49) using Prop. 1/2. If the two disagree beyond the small-N fluctuations seen in Fig. 2, the stated assumptions are insufficient and Prop. 1 requires an explicit random-matrix model for S.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Proposition 1 (Eqs. (29)-(39)) is the foundation of the weighted asymptotic MI in Eq. (49) and of the PGA gradient in Eq. (56), but its hypotheses are not met by the assumptions stated for S. Section II-A only requires E[SS†]=I_M. This is a second-order condition; it does not 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), nor does it imply that S and S† are asymptotically free from the deterministic unitary matrices and from \\hat G. For example, a deterministic S=I_M (with M=Ns) satisfies E[SS†]=I_M, but for this S, ζ(D)=D, not a scalar multiple of the identity, so Eq. (65) fails and B1=\\hat G\\hat G† has a different limiting spectrum. The proof of Prop. 1 is only a one-sentence reference to [17, Prop. 2]; the freeness/subordination step for this particular linearized ISAC matrix is not demonstrated. The numerical agreement in Fig. 2 uses Gaussian S (if the customary i.i.d. model is intended), so it does not validate the proposition under the assumptions actually written. The closed-form objective and the PGA update are therefore justified only after adding an explicit i.i.d./Haar-type assumption on S and a proof (or precise citation) that the free subordination carries over.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":11899,"tokens_out":14521,"duration_ms":159323,"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":[{"comment":"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.","section":"Section II-A; Appendix A, Eqs. (65)-(66); Proposition 1"},{"comment":"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.","section":"Eqs. (5), (24), (30)-(39), (56)"},{"comment":"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.","section":"Proposition 2, Eq. (83)"}],"minor_comments":[{"comment":"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.","section":"Section IV, Fig. 2, Fig. 3, Fig. 4, Fig. 5"},{"comment":"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.","section":"Section II-A and Section III"},{"comment":"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.","section":"Section III-B, Eq. (56)"},{"comment":"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.","section":"Section IV, Fig. 2"},{"comment":"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.","section":"Introduction, first paragraph"}],"recommendation":"major_revision","confidential_remarks":"The central free-probability result is taken from the authors' own prior paper [17] with a one-sentence proof outline. Given the overlapping authorship, the transfer of [17, Prop. 2] to the present ISAC linearized matrix needs to be made checkable by the reader; a precise theorem statement with all hypotheses would resolve this. The topic and framework are within the scope of the journal, and the numerical evidence is promising, but the hypothesis and dimension issues in the main derivation must be addressed before the contribution is fully established."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: the paper does something genuinely useful—it takes a weighted sensing-plus-communication MI for MIMO ISAC under statistical CSI, derives a deterministic asymptotic expression via operator-valued free probability, and optimizes beamforming with projected gradient ascent. The problem is well motivated for 6G, and the numerical agreement with Monte Carlo is convincing at the antenna counts shown. If the math stands, this is a practical contribution to ISAC beamforming under imperfect CSI.\n\nThe novelty is moderate. The weighted-MI objective for ISAC under statistical CSI is new, as is applying the linearization trick to this system model. But the engine—operator-valued free probability, subordination, one-sided correlation matrices—comes directly from the authors' earlier [17] on multi-RIS MIMO. That is not a sin, but it means this is an application, not new theory.\n\nThe soft spots are real. The load-bearing Proposition 1 is not proved here; the proof is a one-sentence outline referencing [17, Prop. 2]. That could be acceptable if the reference were exact, but the linearized matrix in this paper (Eq. 24) is not identical to the one in [17], so the transfer needs at least a careful verification. More importantly, the assumptions on S are under-specified. Section II-A gives only E[SS†]=I_M. As the stress-test note points out, that second-order condition does not justify the one-sided correlation ζ(D)=Ns^{-1}Tr(D)I in Eq. (65), nor does it guarantee the freeness needed for the subordination step. A deterministic S=I_M satisfies the written assumption and violates Eq. (65). The fix is easy—state that S has i.i.d. entries with zero mean and variance 1/Ns, or Haar-distributed—but as written the theorem is not established. The numerical validation uses Gaussian S, presumably, so it does not test the written assumptions.\n\nMinor points: no error bars on the Monte Carlo curves, though the agreement looks good, and no code. PGA converging within three iterations is nice and plausible.\n\nBottom line: the central idea holds up if you add the missing distributional assumption on S and provide a real transfer of [17, Prop. 2] to this linearized matrix. That is a revision, not a rewrite. I would send this to peer review—it deserves referee time—and the authors can fix the gaps. I would not cite it in its current form, but I would read the revised version.","headline":"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.","tokens_in":12460,"tokens_out":2524,"would_cite":false,"duration_ms":26320,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Statistical CSI is enough: closed-form MI objective for ISAC beamforming.","keywords":["ISAC","beamforming","mutual information","free probability","statistical CSI","MIMO","projected gradient ascent","large dimensional antenna array"],"falsifier":"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.","tokens_in":11403,"feed_emoji":"📡","tokens_out":5825,"duration_ms":61906,"temperature":0.7,"pith_summary":"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.","feed_headline":"Closed-form MI unlocks ISAC beamforming from statistics alone","feed_subtitle":"Statistical CSI replaces perfect CSI, and a few gradient steps improve weighted sensing-plus-communication MI.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the proof pattern for Proposition 1: the paper states that the proof is similar to the method in [17, Prop. 2] for the linearized ISAC matrix.","marker":"[17]"},{"why":"Provides the operator-valued subordination theory and the linearization trick used to build the block matrix $B_L$ and derive the operator-valued Cauchy transform.","marker":"[15]"},{"why":"Gives the almost-sure convergence of the per-antenna MI to deterministic limits $I_s$ and $I_c$, which is the basis for the weighted asymptotic MI objective.","marker":"[12]"},{"why":"Justifies that the resolvent of $B_1$ converges almost surely to the Cauchy transform, connecting the empirical spectral measure to the deterministic formula.","marker":"[16]"},{"why":"Provides the log-det expression for the sensing MI in Eq. (5), the starting point for the sensing part of the weighted objective.","marker":"[11]"},{"why":"Supplies the Shannon-transform framework used to convert the Cauchy transforms $G_{B_1}(z)$ and $G_{B_2}(z)$ into the asymptotic MI expressions.","marker":"[13]"},{"why":"Stated as [18, Lemma 1], it gives the trace identities used in the proof of Proposition 2 to simplify the derivative of the Shannon transform.","marker":"[18]"},{"why":"Defines the Weichselberger MIMO channel model with joint correlation at both link ends, which is the channel model used for both sensing and communication links.","marker":"[10]"}],"fun_headline_variants":["Closed-form MI for ISAC beamforming with only statistical CSI","Free probability unlocks closed-form MI for large-scale ISAC beamforming","Weighted MI for ISAC from statistical CSI via closed-form gradient","Closed-form sensing and comm MI for massive MIMO ISAC","ISAC beamforming optimized via closed-form MI from statistical CSI"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Closed-form MI for ISAC beamforming with only statistical CSI","Free probability unlocks closed-form MI for large-scale ISAC beamforming","Weighted MI for ISAC from statistical CSI via closed-form gradient","Closed-form sensing and comm MI for massive MIMO ISAC","ISAC beamforming optimized via closed-form MI from statistical CSI"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000702,"raw_usage":{"total_tokens":3146,"prompt_tokens":904,"completion_tokens":2242,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":520,"completion_tokens_details":{"reasoning_tokens":2154}},"tokens_in":520,"tokens_out":2242,"duration_ms":17260,"temperature":1.0,"reasoning_tokens":2154,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T16:35:06.778245+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"On the mutu al information of multi-RIS assisted MIMO: From operator-v alued free probability aspect,","cited_arxiv_id":null,"evidence_quote":"Supplies the proof pattern for Proposition 1: the paper states that the proof is similar to the method in [17, Prop. 2] for the linearized ISAC matrix."},{"cited_title":"Analytic subo rdination theory of operator-valued free additive convolu tion and the solution of a general random matrix problem,","cited_arxiv_id":null,"evidence_quote":"Provides the operator-valued subordination theory and the linearization trick used to build the block matrix $B_L$ and derive the operator-valued Cauchy transform."},{"cited_title":"”Asymptotic analysis of double-scattering channels.” 201 1 Conference Record of the Forty Fifth Asilomar Conference on Signals, Systems a nd Computers (ASILOMAR)","cited_arxiv_id":null,"evidence_quote":"Gives the almost-sure convergence of the per-antenna MI to deterministic limits $I_s$ and $I_c$, which is the basis for the weighted asymptotic MI objective."},{"cited_title":"W., ”Strong convergence of the empiric al distribution of eigenvalues of large dimensional random matrices,” Journal of Multivariate Analysis, vol","cited_arxiv_id":null,"evidence_quote":"Justifies that the resolvent of $B_1$ converges almost surely to the Cauchy transform, connecting the empirical spectral measure to the deterministic formula."},{"cited_title":"Join t beamforming for RIS-assisted integrated sensing and comm unication systems,","cited_arxiv_id":null,"evidence_quote":"Provides the log-det expression for the sensing MI in Eq. (5), the starting point for the sensing part of the weighted objective."},{"cited_title":"Random matrix theory and wire less communications,","cited_arxiv_id":null,"evidence_quote":"Supplies the Shannon-transform framework used to convert the Cauchy transforms $G_{B_1}(z)$ and $G_{B_2}(z)$ into the asymptotic MI expressions."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Stated as [18, Lemma 1], it gives the trace identities used in the proof of Proposition 2 to simplify the derivative of the Shannon transform."},{"cited_title":"A stochastic MIMO channel model with joint correlation of b oth link ends,","cited_arxiv_id":null,"evidence_quote":"Defines the Weichselberger MIMO channel model with joint correlation at both link ends, which is the channel model used for both sensing and communication links."}],"review_version":1}