{"id":"6548001b-ec7e-4a6e-9581-7ea8fc64dd39","arxiv_id":"2507.00942","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Capacity-achieving SK-type feedback schemes are given for dirty paper channels with state estimation at the receiver, including a new optimal rate-distortion region for the multiple-access channel.","lead":"This paper constructs Schalkwijk-Kailath style feedback codes for dirty paper channels in which the receiver must both decode the message and estimate the interfering state. The authors prove their schemes achieve the optimal rate-distortion trade-offs, and they determine a new optimal region for the two-user multiple-access version.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Converse of Theorem 1 uses a false Jensen step in Appendix A: the average of (1-b_i^2)(1-rho_i^2) need not be bounded by gamma(1-rho^2), so the outer bound is unproved.","rationale":"The paper's central claim is Theorem 1, the optimal rate-distortion region of the DP-MAC with SE-R and feedback. The reader's declared weakest assumption is the unproved transplant of the Ozarow/Rosenzweig SK scheme to the equivalent channel with state lambda S. That is a real gap, but it is an omission of a standard-looking argument. My independent check of Appendix A finds a more concrete problem: the exact Jensen step used to upper-bound the individual rates is false. The counterexample is not exotic; it uses valid Gaussian covariance matrices satisfying the paper's own notation, and it shows the averaged term can exceed gamma(1-rho^2) by a substantial margin. Because the outer bound is half of the optimality claim, the theorem is not proved by the manuscript. I do not claim the theorem is false; the union over rho may still cover the true region, and the erroneous step may be replaceable by a sharper converse. But as written, the proof has an invalid inequality, so the central claim is unverified. This is why I recommend UNVERDICTED rather than ACCEPT or REJECT, and it is why I only partially agree with the reader's choice of weakest assumption: the achievability delegation is concerning, but the converse Jensen step is the place where the argument demonstrably breaks under a concrete computation.","tokens_in":12282,"tokens_out":28225,"duration_ms":306691,"concrete_test":"Recompute the inequality leading to (A16) with the covariance sequence above: if (1/n) sum_i (1-b_i^2)(1-rho_i^2) > gamma(1-rho^2), the Jensen step is false and the converse proof needs repair. To decide whether Theorem 1 itself is false, solve the corrected per-letter outer bound and check whether any point (R1,R2,D) satisfying the per-letter bounds (A11)-(A14) lies outside the union of R(rho) in (21); if such a point exists, the theorem must be revised, and if not, the theorem may survive with a different converse argument.","verdict_should_be":"UNVERDICTED","load_bearing_attack":"Appendix A derives the individual rate bounds in (A16) by passing from the per-letter bound R1 <= (1/n) sum_i 0.5 log(1 + P1(1-b_i^2)(1-rho_i^2)/sigma^2) to R1 <= 0.5 log(1 + gamma P1 (1-rho^2)/sigma^2), where gamma = (1/n) sum_i (1-b_i^2) and rho = (1/n) sum_i rho_i. This implication is false without extra hypotheses. A valid covariance sequence violating it is obtained with P1=P2=Q=1, sigma^2=1, c_i=0 for all i; for half the indices take b_i=0.99, rho_i=1, a_i=sqrt(1-b_i^2) (the matrix [[1,a_i,b_i],[a_i,1,0],[b_i,0,1]] is positive semidefinite), and for the other half take b_i=0, rho_i=0, a_i=0. Then gamma=(0.0199+1)/2 ~ 0.510, rho=0.5, while (1/n) sum_i (1-b_i^2)(1-rho_i^2) = 0.5, which exceeds gamma(1-rho^2) ~ 0.382. Hence the step used to obtain (A16) fails, and the converse part of Theorem 1 is not established as written. The achievability half is additionally delegated to [10], [11] after (29), but the converse gap is explicit and concretely checkable.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript studies Gaussian dirty-paper channels with receiver state estimation (SE-R) and noiseless feedback. In Section II it proposes an SK-type scheme that splits the transmitted signal into a state-carrying part and a message-carrying part and claims to achieve the known optimal region C^fb_dp in (4). In Section III it extends the construction to a two-user dirty-paper multiple-access channel (DP-MAC) with SE-R and feedback, stating in Theorem 1 that the optimal region is the union over 0≤ρ≤1 of R(ρ) in (21); achievability is by an SK-type scheme following [10] and [11], and the converse is in Appendix A. Section IV discusses a noisy state observation version and notes that a capacity-achieving SK-type scheme there remains open.","tokens_in":12532,"tokens_out":14857,"duration_ms":157162,"significance":"If the claims were established, the paper would provide two useful results: an explicit low-complexity feedback scheme for the single-user DPC-SE-R and a characterization showing that feedback enlarges the rate-distortion region of the DP-MAC-SE-R. The paper is not circular: the single-user result is checked against the known region (4), and the MAC region is compared with the no-feedback region of [9]. The authors also correctly refrain from claiming optimality in the noisy-observation case. However, the current proof has a false Jensen step in the converse of Theorem 1 and several incorrect or ambiguous displayed equations in the achievability part, so the main optimality claim is not yet supported.","major_comments":[{"comment":"The Jensen step used to obtain the individual rate bounds is false. From (A12), the proof gives R1 ≤ (1/n) Σ_i 0.5 log(1 + P1(1-b_i^2)(1-ρ_i^2)/σ^2). The text then defines γ = (1/n)Σ_i(1-b_i^2), ρ = (1/n)Σ_iρ_i, and concludes R1 ≤ 0.5 log(1 + γP1(1-ρ^2)/σ^2). This implication is not valid. For example, with P1=σ^2=1, take half the indices with b_i=0.99, ρ_i=1, a_i=√(1-b_i^2), and half with b_i=0, ρ_i=0, a_i=0; the covariance matrices are positive semidefinite, γ≈0.510, ρ=0.5, but (1/n)Σ_i(1-b_i^2)(1-ρ_i^2)=0.5, which exceeds γ(1-ρ^2)≈0.382. Since this step is load-bearing for the converse of Theorem 1, the outer bound is not established as written.","section":"Appendix A, Eq. (A16)"},{"comment":"The displayed fixed-point equation mixes ρ and ρ*. As written, σ^2(γP1+βP2+2√(γP1βP2)ρ+σ^2) = (βP2(1-ρ*^2)+σ^2)(γP1(1-ρ*^2)+σ^2) contains two variables and does not define the claimed unique positive root ρ*. The left-hand side should presumably contain ρ* in place of ρ; otherwise the definition of R(ρ*) and the achievability analysis are ambiguous.","section":"Section III-C1, Eq. (29)"},{"comment":"The achievability half of Theorem 1 is delegated to [10] and [11]. The sentence after (29) asserts that R(ρ) for 0≤ρ<ρ* is achieved by combining random coding with the SK-type scheme, and that the Ozarow/Rosenzweig construction applies directly to the equivalent channel Y=G1+G2+λS+η. This is the core of the achievability claim, and the equivalent channel has a common state λS in addition to the MAC structure. The paper should either provide the missing derivation or identify the precise theorem in [10] or [11] and verify that its hypotheses cover this channel.","section":"Section III-C1, after Eq. (29)"},{"comment":"The variance recursion in (14) is inconsistent with the claimed SK rate. With ε_t = ε_{t-1} - µ_t(Y_t - ωS_t) and Y_t - ωS_t = √(γP/α_{t-1})ε_{t-1} + η_t, the MMSE coefficient is µ_t = √(γP α_{t-1})/(γP+σ^2) and the variance update is α_t = α_{t-1} σ^2/(γP+σ^2). Equation (14) instead gives α_t = α_{t-1} γP/(γP+σ^2), which would lead to a per-use rate 1/2 log(1+σ^2/(γP)) rather than the claimed 1/2 log(1+γP/σ^2); the formula for α_n in the Decoding paragraph inherits this error. The index in 'µ_i' also appears to be 'µ_{t-1}'.","section":"Section II-B, Eq. (14)"},{"comment":"Equation (15) as printed is not a valid equality: θhat_t = θhat_{t-1} - µ_tY_t is rewritten as θhat_{t-1} - ωµ_tS_t - µ_t(Y_{t-1}-ωS_{t-1}), mixing Y_t and Y_{t-1}. To obtain (16), the update should be θhat_t = θhat_{t-1} - µ_t(Y_t - ωS_t). The displayed form should be corrected.","section":"Section II-B, Eq. (15)"}],"minor_comments":[{"comment":"There are typos in the text ('trad-off' appears twice), and the derivation of the noisy-observation region (40) is omitted; please add a short derivation or a precise reference for this formula.","section":"Section IV"},{"comment":"The indices in equations (11)-(14) are used inconsistently (t, t-1, and i in µ_i). Standardizing the index notation would help reproducibility.","section":"Section II-B, Eqs. (11)-(14)"}],"recommendation":"major_revision","confidential_remarks":"The main obstacle is the false Jensen step in Appendix A: the converse of Theorem 1 is currently unsupported. If a correct outer bound cannot be supplied, the optimality claim for the DP-MAC should be withdrawn. I saw no evidence of circularity or citation problems; the single-user construction is plausible but also needs the equation corrections noted above."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper has two parts. The single-user DPC with SE-R and feedback is handled by a neat SK-type scheme that achieves the known region from [3]. That part is solid and useful: it gives a low-complexity capacity-achieving construction where only an existence result existed before. The second part, the DP-MAC with SE-R and feedback, is the claimed new result. The achievability extends the single-user idea in a plausible way, and the stated region in Theorem 1 has the right look (it specializes to the no-feedback region at rho=0). But the converse is not established. Appendix A derives the rate bounds (A16) by averaging per-letter expressions. The step claims that the average of (1-b_i^2)(1-rho_i^2) is bounded by gamma(1-rho^2), where gamma and rho are arithmetic means of their respective sequences. That is simply false. I checked a concrete covariance sequence with half the indices having b_i=0.99, rho_i=1 and the rest zero: the average product is 0.5 while gamma(1-rho^2) is about 0.382. So the Jensen step fails, and the outer bound in (A16) does not follow. Since the achievability proof is also mostly delegated to [10] and [11] after equation (29), the main theorem is not proven in this preprint. There are also smaller issues: equations (14) and (15) have typos, and (29) mixes rho and rho*. These are minor and fixable. The paper is honest about its limitations, and the direction is interesting. The authors should be encouraged to fix the converse. The problem deserves a serious referee, but as it stands the central claim is unreliable. I would not cite the DP-MAC region yet. The single-user scheme is worth a close look, though even there the distortion analysis relies on power arguments that are a bit hand-wavy. Send it to peer review, but be prepared for a major revision. The referee should ask for a corrected converse, either by proving a proper bound on the average product or by redefining the union to use empirical correlations.","headline":"A promising SK-type scheme for dirty paper channels with state estimation, but the main DP-MAC theorem's converse has a real Jensen gap that leaves the new region unproved as written.","tokens_in":13180,"tokens_out":1991,"would_cite":false,"duration_ms":25242,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["94A15","94A29","94A40"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that for a two-user dirty paper multiple-access channel with state estimation at the receiver, noiseless feedback strictly expands the optimal rate-distortion region, and a low-complexity SK-style feedback scheme attains…","keywords":["dirty paper channel","state estimation at the receiver","Schalkwijk-Kailath feedback scheme","multiple-access channel with feedback","rate-distortion region","Gaussian state interference","joint state estimation and communication","noiseless feedback"],"falsifier":"For fixed parameters such as $P_1=P_2=1$, $Q=10$, $\\sigma^2=1$, run the recursive scheme of Section III-C1 over a long block and compare the achieved rate pairs and distortion with the claimed region (21); any operating point with $\\rho^*>0$ falling short of (28) or (30) would settle the achievability claim in the negative.","tokens_in":11988,"feed_emoji":"📡","tokens_out":11653,"duration_ms":122892,"temperature":0.7,"pith_summary":"The paper studies the dirty paper channel, a Gaussian channel whose known interference can be pre-cancelled at the transmitter, in the setting where the receiver must both decode a message and estimate the interfering state. It builds feedback schemes of the SK type (a classical recursive feedback method in which the transmitter sends scaled estimation errors so that effective noise variance shrinks geometrically). For a single transmitter the proposed scheme achieves the known optimal rate-distortion region, and feedback does not enlarge that region. The main result is for two transmitters: Theorem 1 characterizes the full region of achievable rate pairs and state-estimation distortion, and this region is strictly larger than the no-feedback region because feedback introduces a correlation coefficient between the two users' error signals. The paper also extends the scheme to noisy state observation, where it is not optimal and the capacity-achieving SK-type scheme remains unknown.","feed_headline":"Feedback expands the dirty-paper MAC region; SK coding achieves it","feed_subtitle":"For two transmitters, feedback adds a correlation knob that strictly grows the region; a simple recursion reaches it.","key_machinery":"The load-bearing object is the SK recursion with state-offset pre-cancellation: the transmitter maps the message to a point $\\theta\\in[0,1)$ and, on the first channel use, sends a signal proportional to $\\theta$ minus a linear offset computed from all future state samples, so the receiver's first estimate contains the not-yet-sent state terms rather than the message error. At each later use the receiver updates its estimate from the last received symbol, and the transmitter sends the current estimation error scaled up; the error variance then decays geometrically with ratio $\\gamma P/(\\gamma P+\\sigma^2)$. For the multiple-access channel, two such recursions run in parallel, with the sign of the cross-correlation $\\rho_{t-1}$ chosen at each step so that the two error processes maintain a controlled correlation $\\rho$; the cited two-user MAC feedback analysis turns that correlation into the rate bounds (28). The distortion bound (30) comes from MMSE estimation of the state from the same received symbols.","core_discovery":"The central claim, on the paper's own terms, is Theorem 1: for the dirty paper multiple-access channel with state estimation at the receiver and noiseless feedback, the optimal rate-distortion region is exactly $\\bigcup_{0\\le\\rho\\le1} R(\\rho)$, where $R(\\rho)$ is the set in (21). In that region, letting $\\rho=0$ recovers the known no-feedback region, so the statement says feedback enlarges the region only by allowing the two users' SK error processes to be correlated. The achievability scheme splits each user's power into a state-carrying part, $(1-\\gamma)P_1$ and $(1-\\beta)P_2$, and a message-carrying part that runs an SK recursion on the equivalent channel $Y=G_1+G_2+\\lambda S+\\eta$; the state-carrying part is what feeds the receiver's MMSE state estimate. The converse bounds $R_1$ and $R_2$ with a $(1-\\rho^2)$ penalty and the sum rate with the positive correlation term $2\\sqrt{\\gamma P_1\\beta P_2}\\rho$, giving the trade-off in (21). For the single-user model, the same machinery gives a scheme achieving the known region (4), so feedback buys a simple capacity-achieving code rather than a larger region.","pith_inferences":["The same correlation-knob picture likely extends to more than two transmitters: each pairwise correlation among the users' error processes would enter the rate bounds, and the optimal region would be a union over a correlation matrix rather than over a single scalar $\\rho$.","A finite-blocklength analysis of the recursion would be a natural next step: the paper's geometric error-variance decay suggests explicit bounds on error probability and distortion as functions of block length, which are not derived here.","In the noisy observation case, the suboptimality appears to stem from estimating the state from the channel output alone rather than from the auxiliary random variable used by the optimal no-feedback scheme; feeding that auxiliary variable into the MMSE estimator might close the gap while preserving the recursive structure.","Because the offset term is a linear function of future state samples, the same pre-cancellation trick should adapt to state sequences with memory, changing only the MMSE predictors inside the recursion."],"forward_implications":["For the single-user dirty paper channel with state estimation at the receiver, feedback does not enlarge the optimal rate-distortion region, yet the proposed SK-type scheme attains the known region (4) with a simple recursive encoder and decoder.","For the two-user dirty paper multiple-access channel, feedback strictly enlarges the region: the no-feedback region is exactly the $\\rho=0$ slice of Theorem 1, and positive $\\rho$ adds the sum-rate term $2\\sqrt{\\gamma P_1\\beta P_2}\\rho$ while adjusting the distortion trade-off.","The converse holds for all codes, however complex, so the union over $\\rho\\in[0,1]$, power splits $\\gamma$ and $\\beta$, is the definitive rate-distortion characterization for the two-user model with noiseless feedback.","For noisy state observation at the transmitter, the optimal region with feedback equals the no-feedback optimal region, and the paper's extended SK-type scheme is suboptimal; the achievable region of the extended scheme is given by (40).","The capacity-achieving SK-type scheme for the noisy observation case remains an open problem, so the low-complexity benefit of feedback is not yet recovered in that setting."],"supporting_citations":[{"why":"Supplies the known single-user region (4) and the result that feedback does not enlarge it; the single-user scheme is measured against this region.","marker":"[3]"},{"why":"Supplies the original recursive feedback scheme whose error-variance decay is reused in both proposed schemes.","marker":"[6]"},{"why":"Supplies the offset pre-cancellation idea for SK-type coding on dirty paper channels.","marker":"[7]"},{"why":"Supplies the power analysis and proof pattern for SK-type codes with non-causally known state.","marker":"[8]"},{"why":"Supplies the no-feedback multiple-access region (23) that Theorem 1 contains at $\\rho=0$, and the distortion-to-mutual-information lemma used in the converse.","marker":"[9]"},{"why":"Supplies the multi-user Gaussian feedback construction that the paper transplants to the equivalent channel $Y=G_1+G_2+\\lambda S+\\eta$.","marker":"[10]"},{"why":"Supplies the correlation-coefficient recursion and the $R(\\rho)$ structure for the two-user multiple-access channel with feedback.","marker":"[11]"},{"why":"Supplies the optimal noisy-observation region used to show that the paper's extended scheme is not optimal in that case.","marker":"[13]"}],"fun_headline_variants":["Feedback grows dirty-paper MAC region; SK recursion achieves it","SK-type feedback hits dirty-paper MAC capacity with state estimation","Feedback adds correlation knob to dirty-paper MAC, SK scheme optimal","Dirty-paper MAC: feedback enlarges region, SK scheme reaches cap"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The main result's reachability relies on the assertion, made after equation (29), that the existing two-user Gaussian MAC feedback construction carries over to the equivalent channel $Y=G_1+G_2+\\lambda S+\\eta$ and that random coding fills the region $0\\le\\rho<\\rho^*$; the paper states this rather than proving it.","fun_headline_variants_meta":{"raw":{"variants":["Feedback grows dirty-paper MAC region; SK recursion achieves it","SK-type feedback hits dirty-paper MAC capacity with state estimation","Feedback adds correlation knob to dirty-paper MAC, SK scheme optimal","Dirty-paper MAC: feedback enlarges region, SK scheme reaches cap"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000969,"raw_usage":{"total_tokens":4154,"prompt_tokens":1011,"completion_tokens":3143,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":627,"completion_tokens_details":{"reasoning_tokens":3068}},"tokens_in":627,"tokens_out":3143,"duration_ms":22891,"temperature":1.0,"reasoning_tokens":3068,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T21:02:33.163398+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For fixed parameters such as $P_1=P_2=1$, $Q=10$, $\\sigma^2=1$, run the recursive scheme of Section III-C1 over a long block and compare the achieved rate pairs and distortion with the claimed region (21); any operating point with $\\rho^*>0$ falling short of (28) or (30) would settle the achievability claim in the negative.","supporting_citations":[{"cited_title":"The rate-and-state capacity with feedback,","cited_arxiv_id":null,"evidence_quote":"Supplies the known single-user region (4) and the result that feedback does not enlarge it; the single-user scheme is measured against this region."},{"cited_title":"A coding scheme for additive noise channels with feedback. part I: No bandwidth constraint,","cited_arxiv_id":null,"evidence_quote":"Supplies the original recursive feedback scheme whose error-variance decay is reused in both proposed schemes."},{"cited_title":"Writing on dirty paper with feedback,","cited_arxiv_id":null,"evidence_quote":"Supplies the offset pre-cancellation idea for SK-type coding on dirty paper channels."},{"cited_title":"Coding for the feedback Gel’fand-Pinsker channel and the feedforward Wyner-Ziv source,","cited_arxiv_id":null,"evidence_quote":"Supplies the power analysis and proof pattern for SK-type codes with non-causally known state."},{"cited_title":"Joint state estimation and communication over a state-dependent Gaussian multiple access channel,","cited_arxiv_id":null,"evidence_quote":"Supplies the no-feedback multiple-access region (23) that Theorem 1 contains at $\\rho=0$, and the distortion-to-mutual-information lemma used in the converse."},{"cited_title":"The capacity of Gaussian multi-user channels with state and feedback,","cited_arxiv_id":null,"evidence_quote":"Supplies the multi-user Gaussian feedback construction that the paper transplants to the equivalent channel $Y=G_1+G_2+\\lambda S+\\eta$."},{"cited_title":"The capacity of the white Gaussian multiple access channel with feedback,","cited_arxiv_id":null,"evidence_quote":"Supplies the correlation-coefficient recursion and the $R(\\rho)$ structure for the two-user multiple-access channel with feedback."},{"cited_title":"Capacity-Distortion tradeoff of noisy Gaussian state amplification,","cited_arxiv_id":null,"evidence_quote":"Supplies the optimal noisy-observation region used to show that the paper's extended scheme is not optimal in that case."}],"review_version":1}