{"id":"a8a64d3e-31f8-47aa-960b-c37d7d85551c","arxiv_id":"2607.25997","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For any distortion budget and allowed statistical distance to a chosen target distribution, the minimum compression rate is the solution of an information-theoretic optimization problem; the paper derives it for Bernoulli and Gaussian sources.","lead":"This paper asks how few bits are needed to compress data so the reconstruction stays close to the original but looks statistically like a different target distribution. It defines the rate-distortion-deception tradeoff and gives minimum-rate results for Bernoulli and Gaussian sources under both constraints.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Gaussian 'complete characterization' depends on an unproved optimality ansatz (§V–VI); if a non-Gaussian coupling achieves lower I(X;X̂) for the same (D,P), the closed forms are only upper bounds.","rationale":"The reader's weakest_assumption identifies the same concern: the Gaussian optimality ansatz in Sections V and VI is asserted without proof. I agree that this is the most load-bearing issue. Theorem 1 itself appears sound; the converse and achievability proofs follow the standard rate-distortion-perception pattern and no gap jumps out. The Bernoulli derivations are plausible and the BA algorithm in §IV is a reasonable numerical method for finite alphabets. The Gaussian sections, however, are central to the paper's claim of evaluating the RDD function for Gaussian sources, and they rest entirely on the statement that the optimal reconstruction is jointly Gaussian. Since the cited work [11] addresses the case where the target equals the source, the extension to a different Gaussian target is nontrivial. If a non-Gaussian conditional distribution achieves a lower mutual information for the same distortion and KL constraint, then equations (38)-(39) and (48)-(52) are not the true R_Y(D,P) but only upper bounds. The dimensional inconsistency in (39) reinforces that the Gaussian algebra was not fully verified. Therefore the verdict should remain conditional: accept the central theorem as plausible, but withhold full acceptance of the Gaussian characterizations until the optimality claim is either proved or tested numerically. I would not move the verdict to reject because the concern is localized to the worked Gaussian examples and does not undermine Theorem 1.","tokens_in":13036,"tokens_out":43968,"duration_ms":401834,"concrete_test":"Quantize X~N(0,1) and Y~N(0,2) to K=128 levels each and run the §IV Blahut-Arimoto algorithm (which does not restrict the reconstruction to be Gaussian) over a grid of (D,P), including D=0.5, P=0.1 and the boundary regions. Compare the converged R_Y(D,P) to the §V closed form from (38)-(39). If the BA value is lower than the closed form by more than the quantization tolerance (e.g., 0.01 nats), the Gaussian ansatz is falsified; if it is not lower, the ansatz survives this test. Refine K and repeat to control discretization error.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central coding theorem (Theorem 1) is a credible extension of the RDP theorem and its proof is standard. The load-bearing weakness is confined to the Gaussian claims. Sections V and VI state 'Similar to [11], we show that the optimal reconstruction \\hat X should be jointly Gaussian with X,' but [11] proves this for RDP where the target equals the source P_X. For an arbitrary Gaussian target P_Y, Gaussian optimality does not follow from that reference, and no proof is supplied. All subsequent closed forms—(34)-(42) for the scalar case, (48)-(52) and Appendix D's log-det program for the vector case—parameterize the joint distribution by its covariance (a,b,σ_Z^2 or Σ_hatX,Σ_XhatX), which presumes the Gaussian ansatz. If the true infimum in (5) is achieved by a non-Gaussian output with the same distortion and KL-to-target budget, those expressions are merely upper bounds, so the paper's claimed 'complete characterization' for Gaussian sources is not established. There is also an apparent dimensional inconsistency in (39), σ_X^2 − 2a + σ_hatX^2 = D, where a is dimensionless, which suggests the derivation was not carefully checked. This does not affect Theorem 1, but it does make the Gaussian section the least secure part of the paper.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper defines a rate-distortion-deception (RDD) function R_Y(D,P) as the infimum of I(X;\\hat X) subject to E[d(X,\\hat X)] ≤ D and D_KL(P_{\\hat X} || P_Y) ≤ P, where P_Y is a target distribution distinct from the source. Theorem 1 states that (R,D,P) is achievable if and only if the feasible set is nonempty and R ≥ R_Y(D,P), proved via the strong functional representation lemma. The paper then evaluates R_Y for Bernoulli sources (binary and ternary reconstructions) using KKT and a Blahut-Arimoto algorithm, and for scalar and vector Gaussian sources using a jointly-Gaussian ansatz, claiming a complete characterization of the Gaussian cases.","tokens_in":13422,"tokens_out":18989,"duration_ms":154837,"significance":"If Theorem 1 and the Gaussian evaluations are correct, the paper offers a useful generalization of the rate-distortion-perception framework, with explicit formulas for Bernoulli sources and numerical procedures for Gaussian sources. The coding theorem is standard and credible; the Bernoulli KKT solution is explicit, and the BA/log-det algorithms are reproducible. However, the Gaussian 'complete characterization' is not fully established: the optimality of a jointly Gaussian reconstruction is asserted without proof for P_Y ≠ P_X, and there is a dimensional error in the scalar Gaussian derivation. These issues are fixable but must be addressed before the Gaussian claims can be accepted.","major_comments":[{"comment":"The claim that the optimal reconstruction \\hat X is jointly Gaussian with X is asserted without proof for P_Y ≠ P_X. Reference [11] treats the RDP case where the target equals the source, P_Y = P_X; for an arbitrary Gaussian target the argument does not follow from that reference. Since all Gaussian closed forms and the log-det program (84)–(86) parameterize the joint distribution by covariance matrices, this ansatz is load-bearing. Please supply a proof (a maximum-entropy argument should suffice) or explicitly label the Gaussian results as upper bounds on R_Y(D,P).","section":"Sections V–VI"},{"comment":"Equation (39) reads σ_X^2 − 2a + σ_{\\hat X}^2 = D. Here a is a dimensionless regression coefficient in \\hat X = aX + Z + b, so the term 2a has incompatible dimensions with the variances. The correct expression is σ_X^2 − 2a σ_X^2 + σ_{\\hat X}^2 = D, obtained from (31) with μ_X = μ_{\\hat X}. This typo propagates into the Region-3 solution and the rate expression and must be corrected.","section":"Eq. (39)"},{"comment":"The sentence 'When these conditions are satisfied, the optimal reconstruction becomes Σ_{\\hat X} = 0' is incorrect. A degenerate zero-covariance Gaussian has infinite KL divergence to N(0,Σ_Y) for finite P, so it is not feasible. The optimal independent reconstruction is the minimizer of the vector analogue of the scalar D_min(P) problem in (41)–(42), not the zero matrix. Please correct this statement and the accompanying characterization.","section":"Section VI, Region 2"}],"minor_comments":[{"comment":"In the achievability part, the encoding function is denoted f', but Eq. (65) writes H(f(X^n,Z')|Z'). It should be H(f'(X^n,Z')|Z').","section":"Appendix B, Eq. (65)"},{"comment":"The statement 'For β=1, p_{x|hat x} depends strictly on the target distribution' is imprecise: the transition probability in (25) also depends on the distortion term exp(−λ d(x,·)) even when β=1.","section":"Section IV, after Eq. (24)"},{"comment":"I checked the algebra leading from (48)–(50) to (51)–(52); it is consistent. My concern is not the algebra but the Gaussian optimality assumption identified in Major Comment 1.","section":"Section VI, Eqs. (48)–(52)"},{"comment":"The paper assumes that the support of \\hat X is contained in the support of P_Y for the KL divergence to be finite. This assumption is stated in Section II but could be made more prominently in the Theorem-1 statement and in the examples.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The paper fits the scope of cs.IT and the central coding theorem is sound. The main risk is the Gaussian optimality gap, which I believe is true and can be closed by a maximum-entropy argument; the remaining issues are local (Eq. (39) typo, Region-2 statement). The paper may merit acceptance after these corrections and a careful proof of the Gaussian ansatz."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Punchline: the RDD formulation is a natural extension of RDP and the coding theorem is sound; the Gaussian examples are the weak spot and should not be described as a complete characterization in the current form. What’s new: replacing the perception target P_X with an arbitrary P_Y gives a unified formulation that covers deception and privacy-style reconstruction. The named RDD function and the single-letter characterization in Theorem 1 are a clean generalization of existing RDP results, and the proof via the strong functional representation lemma is standard and correct. The Bernoulli case with KKT regions and the Blahut-Arimoto scheme for ternary reconstruction are the most developed parts; those derivations hold together and the numerical plots look plausible. The paper is also transparent about limitations of its numerical algorithm, which counts in its favor. Where it gets fragile: Sections V and VI. The claim that the optimal reconstruction is jointly Gaussian with X is imported from [11], but [11] proves that for the RDP setting where the target distribution is the source. For an arbitrary Gaussian target P_Y, that conclusion does not follow; the paper supplies no proof. If a non-Gaussian coupling achieves lower mutual information for the same (D,P), then the closed forms in (34)-(42) and the vector formulas in (48)-(52) are upper bounds, not the RDD function. There is also a dimensional inconsistency in (39): σ_X^2 − 2a + σ_hatX^2 = D mixes a variance with a unitless constant; and the algebraic step from (48)-(49) to (50) looks questionable to me. These errors are localized and do not affect Theorem 1, but they matter because the paper claims a complete characterization for Gaussian sources. Bottom line: readers working on rate-distortion-perception, adversarial compression, or privacy will get real value from the problem formulation and the Bernoulli analysis. Those who cite the Gaussian results should wait until they are repaired. I would send this to peer review—the core idea deserves referee time—but as a major revision, with the Gaussian optimality either proved or explicitly downgraded to a Gaussian-ansatz computation.","headline":"RDD coding theorem holds up; Gaussian sections overclaim and need repair.","tokens_in":13811,"tokens_out":2770,"would_cite":true,"duration_ms":25910,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["94A34","94A15","94A17"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper establishes the rate-distortion-deception function R_Y(D,P) as the exact characterization of achievable rates for lossy compression with both a fidelity (distortion) constraint and a statistical deception constraint toward a targe","keywords":["rate-distortion theory","deception constraint","perception constraint","lossy compression","Kullback-Leibler divergence","Gaussian source","Bernoulli source","Blahut-Arimoto algorithm"],"falsifier":"For a scalar Gaussian source with target mean/variance different from the source's, compute the minimum mutual information over all couplings p_{Xhat|X} using a numerical search that does not impose joint Gaussianity (e.g., a fine quantization of the channel or a neural-estimator based optimization). If the resulting minimum I(X;Xhat) for some (D,P) is lower than the paper's expression, the jointly-Gaussian ansatz is false for that regime. Similarly, for a vector case where Σ_X and Σ_Y do not commute, compare the paper's alternating-minimization result with the convex log-det program's output:","tokens_in":12975,"feed_emoji":"🎭","tokens_out":3652,"duration_ms":30101,"temperature":0.7,"pith_summary":"The paper asks what happens when a lossy compressor is required not only to keep distortion below D against the original source, but also to make the reconstruction statistically resemble a different target distribution P_Y, measured by KL divergence at most P. It introduces the rate-distortion-deception (RDD) function R_Y(D,P) and proves a single-letter characterization: a rate R is achievable exactly when a channel p_{Xhat|X} meeting both constraints exists and R ≥ R_Y(D,P). This generalizes the rate-distortion-perception function, setting P_Y equal to the source distribution. The main applications are deception and privacy: an adversary who wants to be perceived as having a different statistical profile, or a user who wants to disguise their data as coming from a generic distribution. A sympathetic reader would care because this gives a fundamental performance limit for a practically motivated class of compression problems that mixes fidelity with statistical disguise.","feed_headline":"Deception constraint sets exact compression-rate limit","feed_subtitle":"For Bernoulli and Gaussian sources, this paper derives the minimum rate to keep distortion low while faking a target distribution.","key_machinery":"The RDD function R_Y(D,P) = inf I(X;Xhat) subject to expected distortion ≤ D and D_KL(P_Xhat || P_Y) ≤ P — a mutual-information minimization whose feasible set is convex in (D,P) because KL divergence is convex and distortion is linear. For the discrete cases, KKT conditions reduce the problem to fixed-point equations for the reconstruction marginal q, with a Lagrange multiplier β controlling deception. For the Gaussian cases, the jointly-Gaussian ansatz allows the optimization to be expressed in covariance terms, leading to closed-form scalar results and, for vector sources, a coupled matrix system that is solved numerically and cross-validated with a convex log-det program.","core_discovery":"The central claim is Theorem 1: the triplet (R,D,P) is achievable if and only if the feasible set S_{D,P} is nonempty and R ≥ R_Y(D,P), where R_Y(D,P) is the infimum of I(X;Xhat) over all conditional distributions satisfying the distortion and deception constraints. This is a complete characterization, meaning the tradeoff surface is exactly described by the single-letter mutual information optimization. The paper also evaluates the function for Bernoulli sources with Bernoulli and ternary targets, giving closed-form KKT solutions and a Blahut-Arimoto algorithm, and for scalar and vector Gaussian sources, where the optimal reconstruction is assumed jointly Gaussian.","pith_inferences":["The paper's Gaussian analysis restricts to jointly Gaussian coupling; if a non-Gaussian coupling could achieve lower mutual information for the same (D,P), the closed-form RDD expressions would be only upper bounds. The paper does not rule this out for P_Y ≠ P_X.","The fixed-point iteration for non-commuting covariance pairs is empirically stable but not proven to converge globally; the paper itself notes this and offers a convex program instead. A practical implementation would need to rely on the convex solver for reliability.","The deception framework suggests an immediate testable extension to task-oriented compression where the target distribution is derived from the receiver's decision boundary, not a fixed P_Y.","The sharp distortion floor might be exploitable in reverse: a receiver could detect deception by checking whether the observed reconstruction statistics are impossible at the claimed rate."],"forward_implications":["The RDD function provides a fundamental limit for any compression system that must simultaneously meet a fidelity requirement and a statistical camouflage requirement; no scheme can beat the characterized rate.","Unlike rate-distortion-perception, RDD has a sharp distortion floor: below a certain D, no rate, not even infinite, can satisfy the deception constraint; the paper identifies this via the non-emptiness of S_{D,P}.","Because the target distribution can be chosen freely, the same machinery covers privacy-by-disguise scenarios, where the 'target' is a generic or vendor distribution rather than the user's own.","For Gaussian sources, the derived equations give explicit operational meaning to the tradeoff: the reconstruction is a linearly-scaled and noise-added version of the source, with coefficients determined by λ and β.","The convex log-det program in Appendix D provides a polynomial-time way to compute R_Y(D,P) for vector Gaussian sources and a ground truth to verify the faster alternating algorithm."],"fun_headline_variants":["Exact rate-distortion-deception tradeoff derived","Minimum rate to fake data while limiting error","Deception constraint sets precise compression bound","Rate limits when reconstructions must deceive","Full tradeoff surface for rate, distortion, deception"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The closed-form Gaussian RDD expressions rest on the unproved assumption that the optimal reconstruction is jointly Gaussian with the source for a target distribution different from the source; if that assumption fails, those expressions could overestimate the true minimum rate.","fun_headline_variants_meta":{"raw":{"variants":["Exact rate-distortion-deception tradeoff derived","Minimum rate to fake data while limiting error","Deception constraint sets precise compression bound","Rate limits when reconstructions must deceive","Full tradeoff surface for rate, distortion, deception"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000201,"raw_usage":{"total_tokens":1156,"prompt_tokens":626,"completion_tokens":530,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":370,"completion_tokens_details":{"reasoning_tokens":462}},"tokens_in":370,"tokens_out":530,"duration_ms":5239,"temperature":1.0,"reasoning_tokens":462,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T00:55:13.152626+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a scalar Gaussian source with target mean/variance different from the source's, compute the minimum mutual information over all couplings p_{Xhat|X} using a numerical search that does not impose joint Gaussianity (e.g., a fine quantization of the channel or a neural-estimator based optimization). If the resulting minimum I(X;Xhat) for some (D,P) is lower than the paper's expression, the jointly-Gaussian ansatz is false for that regime. Similarly, for a vector case where Σ_X and Σ_Y do not commute, compare the paper's alternating-minimization result with the convex log-det program's output:","supporting_citations":[],"review_version":1}