{"id":"8f116af5-2660-4625-b420-db54190a75c7","arxiv_id":"2504.17236","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A single-letter formula characterizes the distortion-rate-perception tradeoff with limited common randomness, with explicit Gaussian evaluations and a universality analysis.","lead":"This paper derives the exact rate-distortion-perception tradeoff for lossy compression when encoder and decoder share a limited random key, using squared error and the squared Wasserstein-2 perception measure. It also gives explicit formulas for Gaussian sources, clarifying when a single code works for all perception levels.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Achievability derandomization (Appendix C, Step 3) claims two codebook realizations can match two expectations; in R² Carathéodory requires up to three points, so the proof of Theorem 1 is incomplete as written.","rationale":"The reader's weakest assumption identifies precisely the same load-bearing concern: the derandomization step in Appendix C, Step 3, attempts to match two expectations with two codebook realizations. This is a concrete mathematical gap because the support lemma / Carathéodory in R² requires up to three points. The achievability proof of Theorem 1 depends on this step to eliminate the extra randomness in the codebook, so as written the proof is incomplete. The concern does not appear fatal: the structure of the problem suggests repairs (three-codebook time-sharing, or a direct probabilistic existence argument), and the Gaussian results in Section IV are consistent with the claimed formula, so conditional acceptance remains the appropriate verdict. My stress-test pass did not uncover a more fundamental flaw in the converse, in Lemma 3, or in the Gaussian derivations.","tokens_in":23305,"tokens_out":12157,"duration_ms":121916,"concrete_test":"","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central theorem rests on the achievability proof in Appendix C. In Step 3, after constructing a random codebook C and showing E_C[f(C)] ≤ Eµ[‖X−X̃‖²] + o(1) and E_C[g(C)] ≤ Eν[‖X−X̃‖²] + o(1), the authors write: “By the support lemma, there exist two realizations c0 and c1 of the codebook C, along with a scalar λ ∈ [0,1], such that E(p,C)[f(C)] = (1−λ)f(c0)+λf(c1) and E(q,C)[g(C)] = (1−λ)g(c0)+λg(c1).” This is a two-dimensional matching problem: the point (E[f], E[g]) lies in the convex hull of the set {(f(c), g(c))}. Carathéodory's theorem guarantees representation with at most d+1 = 3 points, not 2. Two points can only represent points on a line segment, and no argument is given that (f(c), g(c)) is effectively one-dimensional or that the expectations lie on such a segment. Since the final deterministic code is obtained only through this derandomization, the existence of a codebook satisfying both (147) and (148) is not established as written. The gap is localized and likely repairable—for example, by using three codebooks and time-sharing, or by a probabilistic argument showing a single typical codebook works—but the current proof does not supply that argument. This is a genuine proof gap, not merely a stylistic issue, and it directly affects the main coding theorem.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies lossy source coding of a square-integrable source under squared error distortion and squared Wasserstein-2 perception, with limited common randomness at rate C. The main contribution is a claimed single-letter formula D(R,C,P) for the minimal distortion D*(R,C,P), defined as an infimum over two couplings µ and ν sharing the same reconstruction marginal p_{X̃}, with mutual information constraints R and R+C and an MMSE condition on µ (Theorem 1). The authors also provide explicit evaluations for scalar Gaussian (Theorem 2) and vector Gaussian (Theorem 3) sources, and discuss universal representations. The proof of Theorem 1 is based on a new soft-covering lemma for Wasserstein-2 distance and an optimal interpolation decoder.","tokens_in":23595,"tokens_out":12969,"duration_ms":111173,"significance":"If the main theorem is correct, the result is a significant advance: it unifies previously known extreme cases (C=0, C=∞, P=0) and yields closed-form expressions for Gaussian sources, which are likely to be useful in applications. The Wasserstein soft-covering lemma (Lemma 3) is an independently useful technical tool. The discussion of universal representations, especially the negative result for vector Gaussian sources, is conceptually interesting. However, the proof of Theorem 1 contains a gap in the derandomization step, so the stated results are not fully established in the current manuscript.","major_comments":[{"comment":"The derandomization step invokes the support lemma to claim that two codebook realizations c0 and c1 and a scalar λ ∈ [0,1] can match both E_{p,C}[f(C)] and E_{q,C}[g(C)] simultaneously. Since the pair (E[f], E[g]) is a point in the convex hull of {(f(c), g(c))} ⊂ R², Carathéodory's theorem guarantees a representation by at most three points, not two. No argument is provided that the set of attainable pairs lies on a line segment or that two points suffice. Consequently, the existence of a deterministic codebook satisfying both (147) and (148) is not established. This is a load-bearing gap in the achievability proof of Theorem 1. A repair appears feasible—for example, by using three codebooks with time-sharing or by a probabilistic argument showing that a single typical codebook satisfies both constraints—but the current proof is incomplete.","section":"Appendix C, Step 3 (Derandomization)"}],"minor_comments":[{"comment":"In equation (141), the second expectation in the distortion expression should be E_ν[·], not E_µ[·], and the term [(·)⁺] is missing the square; the correct form follows the definition in (30).","section":"Appendix C, Eq. (141)"},{"comment":"The phrase 'a scalar a number λ' contains a typo; it should read 'a scalar λ'.","section":"Appendix C, Step 3"},{"comment":"The proof of Theorem 3 is quite terse; expanding the derivation of (201)–(204), including the entropy lower bounds and the verification of the sum constraints, would improve readability and verifiability.","section":"Appendix D"},{"comment":"There are minor typos, e.g., 'probablility' and 'distriution'; a careful proofread is recommended.","section":"Appendix A"}],"recommendation":"major_revision","confidential_remarks":"The derandomization gap in Appendix C is the main obstacle to acceptance. If the authors can supply a correct derandomization argument (e.g., via three codebooks or a concentration-based selection), the paper would be a strong contribution. The rest of the proof structure appears sound, and the Gaussian evaluations are self-contained and plausible."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe short version: this paper gives a single-letter characterization of the rate-distortion-perception tradeoff with finite common randomness under squared error and squared Wasserstein-2 perception, plus explicit Gaussian formulas. That is a real advance, but the main achievability proof has a localized derandomization gap that needs repair. I would send it to review, not reject it.\n\nWhat's new and good: the quantity D(R,C,P) with two couplings μ and ν, one controlling distortion and one controlling perception, is a sensible interpolation between the known C=0, C=∞, and P=0 extremes. The paper proves the optimization is attained (Prop. 1) and convex/continuous (Prop. 2). The Gaussian evaluations are the cleanest part: Theorem 2 gives a closed-form scalar formula, and Theorem 3 extends to vector Gaussians with a reverse-waterfilling style characterization. The observation that universal representations exist for scalar Gaussians but generally fail for vector sources when C>0 is sharp and new. The special-case consistency checks all line up, and the Gaussian formulas are not circular.\n\nThe soft spot: Appendix C, Step 3. After constructing a random codebook, the authors claim that by the support lemma there exist two realizations c0, c1 and a weight λ matching both expected distortion and expected Wasserstein cost. That is a two-dimensional convex hull representation; Carathéodory gives three points, not two, unless the pair (f(c),g(c)) is effectively one-dimensional. No such argument is given. So the existence of a single deterministic codebook satisfying (147) and (148) is not justified as written. I checked the surrounding steps; nothing before that makes the pair lie on a line. The fix is straightforward—use three codebooks and time-share, or give a concentration argument that a single typical codebook works—but the paper doesn't supply it. This is a genuine gap in the proof of the main theorem.\n\nThat said, the gap is not fatal to the paper's likely correctness. The converse looks solid; the Gaussian lower bounds are proven directly; and the achievability is plausible via known soft-covering plus MMSE interpolation techniques. I'd bet the theorem is true. But as it stands, the achievability claim is incomplete.\n\nWho benefits: researchers working on rate-distortion-perception theory, common randomness, and Gaussian lossy compression. It deserves a serious referee. My recommendation: have the authors fix the derandomization step with three codebooks or a proper probabilistic argument, then publish. Also ask them to state explicitly why two points suffice if they believe it does.","headline":"A likely correct and valuable generalization of RDP theory to finite common randomness; the achievability proof has a repairable derandomization gap that should be fixed before acceptance.","tokens_in":24150,"tokens_out":2896,"would_cite":true,"duration_ms":27564,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["94A29","94A34"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves an exact single-letter formula for the distortion-rate-perception tradeoff in lossy source coding with limited common randomness, under squared error and squared Wasserstein-2 perception.","keywords":["rate-distortion-perception tradeoff","common randomness","Wasserstein-2 distance","lossy source coding","MMSE estimate","Gaussian source","universal representations","soft-covering lemma"],"falsifier":"Take a concrete source and finite blocklength, and compute the joint law of the two random costs $(f(C),g(C))$ that Appendix C averages over the random codebook; if the mean of that pair is not a convex combination of two codebook realizations, the two-codebook derandomization is impossible. Alternatively, for a non-Gaussian square-integrable source, evaluate $D(R,C,P)$ by numerical optimization and compare it with large-blocklength simulations of the proposed coding scheme; a systematic gap would refute Theorem 1.","tokens_in":23074,"feed_emoji":"📉","tokens_out":13494,"duration_ms":113873,"temperature":0.7,"pith_summary":"The paper targets the fundamental tradeoff in lossy compression when the reconstruction must match the source distribution at the whole-sequence level, not just symbol by symbol. In that setting the amount of common randomness shared by encoder and decoder changes what is achievable, and before this paper only extreme cases were understood: no common randomness, unlimited common randomness, or perfect perception. The paper proposes a single-letter optimization over one representation distribution and two couplings, and proves that it equals the true optimum for every square-integrable source under squared-error distortion and squared Wasserstein-2 perception. It then evaluates the formula explicitly for Gaussian sources, yielding closed-form distortion curves and thresholds that show when extra common randomness stops helping. A sympathetic reader would care because the result turns an infinite-blocklength coding problem into a finite-dimensional optimization and clarifies when one encoder can serve every perception target.","feed_headline":"One formula ties rate, distortion, perception, and shared randomness","feed_subtitle":"The theorem yields explicit Gaussian curves and closed-form thresholds.","key_machinery":"The load-bearing mechanism is the quadratic structure that makes both the distortion and the perception costs expectations of the same squared Euclidean distance. Given any representation $W$, the optimal reconstruction from $W$ under a perception budget $P$ is not a deterministic function of $W$; it is a convex combination of the MMSE estimate $\\tilde S=\\mathbb{E}[S\\mid W]$ and a fresh source-distributed sample $S'$ produced from $\\tilde S$ by optimal transport, with coefficient $1-\\sqrt{P}/W_2(p_S,p_{\\tilde S})$ on the fresh sample and $\\sqrt{P}/W_2(p_S,p_{\\tilde S})$ on the estimate when the budget binds. This identity splits total distortion into the MMSE error plus a perception shortfall term $[(\\sqrt{\\mathbb{E}[\\|S-\\tilde S\\|^2]}-\\sqrt{P})_+]^2$. In the coding theorem, the two couplings $\\mu$ and $\\nu$ do different work: $\\mu$ measures the squared-error cost of the representation, while $\\nu$ measures how close the representation's distribution is to the source in Wasserstein distance. The achievability proof couples both costs through a soft-covering lemma adapted to the Wasserstein-2 metric, which guarantees that a random codebook of size $2^{nR}$ placed on the representation distribution approximately realizes $p_{\\tilde X}^n$ after conditioning.","core_discovery":"The central discovery is that the asymptotic problem collapses to a finite-dimensional two-coupling optimization. For a source $X$ with $\\mathbb{E}[\\|X\\|^2]<\\infty$, the paper defines $D(R,C,P)$ as the infimum over a representation variable $\\tilde X$ and two couplings $\\mu,\\nu$ between $p_X$ and $p_{\\tilde X}$ of $$\\mathbb{E}_\\mu[\\|X-\\tilde X\\|^2] + \\bigl[(\\sqrt{\\mathbb{E}_\\nu[\\|X-\\tilde X\\|^2]}-\\sqrt{P})_+\\bigr]^2,$$ subject to the MMSE condition $\\mathbb{E}_\\mu[X\\mid \\tilde X]=\\tilde X$, the rate constraint $I_\\mu(X;\\tilde X)\\le R$, and the combined-rate constraint $I_\\nu(X;\\tilde X)\\le R+C$. Theorem 1 asserts that this quantity equals $D^*(R,C,P)$, the minimum distortion achievable by any length-$n$ system with code rate $R$, common-randomness rate $C$, and sequence-level perception constraint $\\frac1n W_2^2(p_{X^n},p_{\\hat X^n})\\le P$. The proof builds a random codebook with distribution $p_{\\tilde X}^n$, uses $\\mu$ to control the MMSE error and $\\nu$ to control the Wasserstein distance between the source and the representation, and finishes with a linear-interpolation decoder that optimally trades perception against distortion.","pith_inferences":["Editorial extension: if the two-codebook derandomization in Appendix C cannot be repaired by a three-codebook argument, the formula may still be true but the achievability proof as written would need revision; the fix is a proof-technical question that a counterexample to the support step would settle.","The perception-inactive threshold for scalar Gaussians suggests a practical design rule: a system can stop spending common randomness once $C$ reaches the saturation point in (81), and further seed rate is wasted; this rule could be tested by implementing the described codebook scheme on Gaussian-like data.","The vector-source non-universality result points to a qualitative conclusion not stated in the paper: with intermediate common randomness, the encoder must know the perception target, so practical perception-aware codecs with shared randomness should be trained target-dependently rather than as universal representations.","A natural next step, not taken in the paper, is to solve the coupling optimization numerically for small-support non-Gaussian sources and compare the resulting $D(R,C,P)$ with finite-blocklength simulations of the coding scheme; agreement would indicate that the quadratic-Wasserstein structure, rather than Gaussianity, is what drives tractability."],"forward_implications":["The formula recovers all previously known extreme cases: $C=0$ gives $D(R)+[(\\sqrt{D(R)}-\\sqrt{P})_+]^2$, $C=\\infty$ gives the standard distortion-rate-perception function $D(R,P)$, and $P=0$ matches the output-constrained lossy source coding characterization.","For a scalar Gaussian source, $D^*(R,C,P)=\\gamma 2^{-2R} + [(\\sqrt{\\gamma(2-2^{-2R}-2\\psi(R,R+C))}-\\sqrt{P})_+]^2$ with $\\psi(a,b)=\\sqrt{(1-2^{-2a})(1-2^{-2b})}$, and closed-form thresholds on $C$ or $R$ identify when the perception constraint is inactive.","The scalar Gaussian minimizer does not depend on $P$, so one asymptotically optimal coded representation works for every perception level; this extends known universality results beyond the unlimited-common-randomness case.","For vector Gaussian sources, the evaluation reduces to a waterfilling-style rate allocation, and in general the optimal allocation depends on $P$ when $C>0$, so asymptotically universal representations need not exist with limited common randomness.","The infimum in the single-letter formula is attained, and $D(R,C,P)$ is decreasing, convex, and continuous in $(R,C,P)$, so the tradeoff surface is well-behaved for numerical computation."],"supporting_citations":[{"why":"Supplies the distortion-perception decomposition and the interpolation decoder for MMSE estimates used in the achievability scheme.","marker":"[12]"},{"why":"Defines the rate-distortion-perception function and proves its coding theorem, which the paper recovers in the C=∞ limit.","marker":"[4]"},{"why":"Gives the no-common-randomness characterization that Theorem 1 recovers when C=0.","marker":"[14]"},{"why":"Establishes the role of common randomness and the output-constrained formulation behind the P=0 extreme case.","marker":"[8]"},{"why":"Provides the output-constrained lossy source coding theorem with limited common randomness used in the P=0 characterization.","marker":"[16]"},{"why":"Gives Gaussian rate-distortion-perception representations under unlimited common randomness, extended here to finite C.","marker":"[17]"},{"why":"Gives the vector Gaussian rate-distortion-perception tradeoff for C=∞ that Theorem 3 generalizes.","marker":"[18]"},{"why":"Supplies the soft-covering bound used to prove the Wasserstein-2 covering lemma.","marker":"[23]"},{"why":"Provides the distributed channel synthesis/soft-covering result used to control total variation in the covering lemma.","marker":"[24]"}],"fun_headline_variants":["Two-coupling formula settles rate-distortion-perception tradeoff","Wasserstein tradeoff collapses to a two-coupling optimization","Single formula gives Gaussian rate-distortion-perception curves","Explicit Gaussian tradeoff in quadratic Wasserstein rate-distortion"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The achievability proof assumes in Appendix C that the randomness used to generate the codebook can be removed by blending two codebook realizations with one time-sharing weight so that both the expected squared-error cost and the expected Wasserstein cost are matched exactly; matching two expectations simultaneously generally requires three realizations, so this step is not justified as written.","fun_headline_variants_meta":{"raw":{"variants":["Two-coupling formula settles rate-distortion-perception tradeoff","Wasserstein tradeoff collapses to a two-coupling optimization","Single formula gives Gaussian rate-distortion-perception curves","Explicit Gaussian tradeoff in quadratic Wasserstein rate-distortion"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001539,"raw_usage":{"total_tokens":6125,"prompt_tokens":878,"completion_tokens":5247,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":494,"completion_tokens_details":{"reasoning_tokens":5178}},"tokens_in":494,"tokens_out":5247,"duration_ms":30813,"temperature":1.0,"reasoning_tokens":5178,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T10:45:51.429681+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a concrete source and finite blocklength, and compute the joint law of the two random costs $(f(C),g(C))$ that Appendix C averages over the random codebook; if the mean of that pair is not a convex combination of two codebook realizations, the two-codebook derandomization is impossible. Alternatively, for a non-Gaussian square-integrable source, evaluate $D(R,C,P)$ by numerical optimization and compare it with large-blocklength simulations of the proposed coding scheme; a systematic gap would refute Theorem 1.","supporting_citations":[{"cited_title":"A theory of the di stortion-perception tradeoff in Wasserstein space,","cited_arxiv_id":null,"evidence_quote":"Supplies the distortion-perception decomposition and the interpolation decoder for MMSE estimates used in the achievability scheme."},{"cited_title":"On the ra te-distortion-perception function,","cited_arxiv_id":null,"evidence_quote":"Defines the rate-distortion-perception function and proves its coding theorem, which the paper recovers in the C=∞ limit."},{"cited_title":"Optimally controllable perc eptual lossy compression,","cited_arxiv_id":null,"evidence_quote":"Gives the no-common-randomness characterization that Theorem 1 recovers when C=0."},{"cited_title":"Output constraine d lossy source coding with limited common randomness,","cited_arxiv_id":null,"evidence_quote":"Provides the output-constrained lossy source coding theorem with limited common randomness used in the P=0 characterization."},{"cited_title":"Rate-distortion-perception tradeo ff for vector Gaussian sources,","cited_arxiv_id":null,"evidence_quote":"Gives the vector Gaussian rate-distortion-perception tradeoff for C=∞ that Theorem 3 generalizes."},{"cited_title":"General nonasymptotic and asymptotic for mulas in channel resolvability and identification capacity and their application to the wiretap channel,","cited_arxiv_id":null,"evidence_quote":"Supplies the soft-covering bound used to prove the Wasserstein-2 covering lemma."},{"cited_title":"Distributed channel synthesis,","cited_arxiv_id":null,"evidence_quote":"Provides the distributed channel synthesis/soft-covering result used to control total variation in the covering lemma."}],"review_version":1}