Optimal Feedback Schemes for Dirty Paper Channels With State Estimation at the Receiver

Bin Dai; Dengfeng Xia; Fan Cheng; Han Deng; Haonan Zhang; Liuguo Yin

REVIEW 5 major objections 2 minor 14 references

Optimal Feedback Schemes for Dirty Paper Channels With State Estimation at the Receiver

T0 review · 5 major / 2 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read 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…

desk verdict 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. read the letter →

arxiv 2507.00942 v1 pith:X2FUNYFS submitted 2025-07-01 cs.IT math.IT

Dengfeng Xia , Han Deng , Haonan Zhang , Fan Cheng , Bin Dai , Liuguo Yin This is my paper · ORCID

classification cs.ITmath.IT MSC 94A1594A2994A40

keywords dirtypaperchannelstateestimationatthereceiverSchalkwijk-Kailathfeedbackschememultiple-accesswithrate-distortionregionGaussianinterferencejointandcommunicationnoiseless

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

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

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

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.

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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

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.

Referee Report

5 major / 2 minor

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.

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 (5)

[Appendix A, Eq. (A16)] 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 III-C1, Eq. (29)] 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 III-C1, after Eq. (29)] 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 II-B, Eq. (14)] 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 II-B, Eq. (15)] 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.

minor comments (2)

[Section IV] 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 II-B, Eqs. (11)-(14)] The indices in equations (11)-(14) are used inconsistently (t, t-1, and i in µ_i). Standardizing the index notation would help reproducibility.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation is self-contained against external benchmarks and prior theorems.

full rationale

The paper's central claims are not circular in the sense defined by this review pass. The single-user scheme in Section II-B is constructed to achieve the previously known region C^fb_dp given in (4), which is cited from [3]; the scheme does not assume the region, it produces the rate-distortion trade-off from an explicit SK-type recursion, and the distortion calculation (17)-(18) is carried out directly. The DP-MAC achievability in Section III-C1 is admittedly terse: after (29) the paper delegates the SK-type MAC analysis to [10] and [11] (Ozarow and Rosenzweig), and asserts that the region R(rho) for rho < rho* follows by combining random coding with the SK-type scheme. This is a missing-proof or correctness risk, not circularity: the cited works are external, parameter-free theorems for Gaussian MACs with feedback, and the paper's equivalent channel (24) is a stated model transformation rather than a redefinition of the target region. The converse in Section III-C2 and Appendix A is an independent outer bound: it starts from Fano inequalities and differential entropy, introduces the covariance matrix (34) as an assumption on the joint distribution of inputs and state, and derives the rate and distortion bounds (A16)-(A20). No fitted parameter is relabeled as a prediction, no result is defined in terms of the claimed region, and there are no self-citations carrying the argument. The skeptic's objection about the Jensen step in Appendix A is a mathematical correctness concern, not a circularity: even if the step from the per-letter bound to (A16) is invalid, that does not mean the proof assumes its conclusion. The paper also explicitly flags its own limitations, e.g., that the capacity-achieving SK-type scheme for the noisy observation case remains unknown, which is consistent with a non-circular presentation. Accordingly, the circularity score is 0.

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

The paper introduces no fitted constants, no ad hoc entities, and no self-referential definitions. The auxiliary parameters gamma, beta, and rho are union variables that trace the rate-distortion region, not free parameters fitted to data. The main external inputs are the known capacity regions from [3] and [9], and the SK/MAC achievability theorems from [10], [11].

assumptions (4)

standard math Gaussian distributions maximize differential entropy for a given covariance matrix
Used in Appendix A, eqs. (A10)-(A14), to upper-bound the differential entropy terms in the converse proof.
domain assumption Lemma 1 of [9]: any scheme achieving distortion D_n satisfies (1/2)log(Q/D_n) <= (1/n)I(S^n;Y^n)
Invoked at eq. (A4) in the converse proof of Theorem 1; this is an external information-theoretic lemma from the cited prior work.
domain assumption Ozarow's characterization of the two-user Gaussian MAC with feedback and the existence of SK-type schemes achieving it [11], extended to state-dependent MACs by [10]
The achievability of Theorem 1 for the DP-MAC is asserted by reference to [10], [11] around eqs. (27)-(29).
domain assumption The optimal R-D region of the DPC with SE-R and feedback given in [3, Remark 3], eq. (4)
Used as the target region for the SK-type scheme in Section II; not re-derived in this paper.

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Pith. "Pith review of Optimal Feedback Schemes for Dirty Paper Channels With State Estimation at the Receiver." pith.science (2026). https://pith.science/paper/X2FUNYFS

@misc{pith2026250700942,
  author       = {Pith},
  title        = {Pith review of: Optimal Feedback Schemes for Dirty Paper Channels With State Estimation at the Receiver},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/X2FUNYFS}},
  note         = {Machine review of arXiv:2507.00942}
}

read the original abstract

In the literature, it has been shown that feedback does not increase the optimal rate-distortion region of the dirty paper channel with state estimation at the receiver (SE-R). On the other hand, it is well-known that feedback helps to construct low-complexity coding schemes in Gaussian channels, such as the elegant Schalkwijk-Kailath (SK) feedback scheme. This motivates us to explore capacity-achieving SK-type schemes in dirty paper channels with SE-R and feedback. In this paper, we first propose a capacity-achieving feedback scheme for the dirty paper channel with SE-R (DPC-SE-R), which combines the superposition coding and the classical SK-type scheme. Then, we extend this scheme to the dirty paper multiple-access channel with SE-R and feedback, and also show the extended scheme is capacity-achieving. Finally, we discuss how to extend our scheme to a noisy state observation case of the DPC-SE-R. However, the capacity-achieving SK-type scheme for such a case remains unknown.

Figures

Figures reproduced from arXiv: 2507.00942 by the authors.

**Figure 2.** The DP-MAC with SE-R and feedback where Xi,t (i ∈ {1, 2}) is the output signal of Encoder i, ηt ∼ N (0, σ2 ) is an i.i.d. channel noise, and St ∼ N (0, Q) is an i.i.d. state interference. Both transmitters know the state sequence S n = (S1, S2, . . . , Sn) in a non-causal manner. In addition, we assume that ηt, St and the messages W1, W2 are independent of each other. An (n, R1, R2, P1, P2)-code for the model of [P… view at source ↗

**Figure 3.** The comparison of C fb dp, the optimal region of [13], and our SK-type scheme for P = 7.7, Q = 10, σ 2 = 5 and σ 2 z = 1 [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages

[3]

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[6]

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Pith tools

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

[1] [3]

The rate-and-state capacity with feedback,

S. I. Bross and A. Lapidoth, “The rate-and-state capacity with feedback,” IEEE Trans. Inf. Theory , vol. 64, no. 3, pp. 1893-1918, Mar. 2018

work page 1918

[2] [10]

The capacity of Gaussian multi-user channels with state and feedback,

A. Rosenzweig, “The capacity of Gaussian multi-user channels with state and feedback,” IEEE Trans. Inf. Theory , vol. 53, no. 11, pp. 4349-4355, Nov. 2007

work page 2007

[3] [11]

The capacity of the white Gaussian multiple access channel with feedback,

L. Ozarow, “The capacity of the white Gaussian multiple access channel with feedback,” IEEE Trans. Inf. Theory , vol. 30, no. 4, pp. 623-629, Jul. 1984

work page 1984

[4] [9]

Joint state estimation and communication over a state-dependent Gaussian multiple access channel,

V . Ramachandran, S. R. B. Pillai, and V . M. Prabhakaran, “Joint state estimation and communication over a state-dependent Gaussian multiple access channel,” IEEE Trans. Commun. , vol. 67, no. 10, pp. 6743-6752, Oct. 2019

work page 2019

[5] [1]

Writing on dirty paper,

M. H. M. Costa, “Writing on dirty paper,” IEEE Trans. Inf. Theory , vol. 29, no. 3, pp. 439-441, May 1983

work page 1983

[6] [2]

Channel capacity and state estimation for state-dependent Gaussian channels,

A. Sutivong, M. Chiang, T. M. Cover, and Y .-H. Kim, “Channel capacity and state estimation for state-dependent Gaussian channels,” IEEE Trans. Inf. Theory , vol. 51, no. 4, pp. 1486–1495, Apr. 2005

work page 2005

[7] [4]

Source coding with a side information ‘vending machine’,

H. Permuter and T. Weissman, “Source coding with a side information ‘vending machine’,” IEEE Trans. Inf. Theory , vol. 57, no. 7, pp. 4530- 4544, Jul. 2011

work page 2011

[8] [5]

Distributed and cascade lossy source coding with a side information ‘vending machine’,

B. Ahmadi and O. Simeone, “Distributed and cascade lossy source coding with a side information ‘vending machine’,” IEEE Trans. Inf. Theory , vol. 59, no. 10, pp. 6807-6819, Oct. 2013

work page 2013