{"id":"a7474c92-53d4-49c2-87cf-271492af000e","arxiv_id":"2507.14825","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper gives single-letter rate-distortion-perception limits for lossy compression with side information under strong realism constraints, including a complete Gaussian solution.","lead":"A theoretical study determines how much a compressed reconstruction can deviate from the original source when it must also match the source distribution exactly, in the presence of extra side information. It shows when side information helps as a source of shared randomness and when it does not, with exact rate formulas for Gaussian sources.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the central characterization is conditional on a clearly stated uniform-integrability assumption that is used precisely where the proofs require it.","rationale":"The reader's weakest-assumption analysis identifies uniform integrability as the key condition, and my independent review agrees that this is the most load-bearing assumption in the paper. However, the paper uses it exactly where needed: Theorem 7's equivalence proof and the distortion-preserving upgrade in Propositions 53/54. The uniform-integrability supremum is applied to a distribution whose marginals are exactly pX, so there is no gap between the assumption and its use. The paper also honestly discloses the assumption's loss of generality and constructs a non-degenerate counterexample, which strengthens rather than weakens confidence. I checked the converse of Theorem 8 for the finite-Z requirements and the achievability for the soft-covering rate conditions; both line up with the single-letter region. The long measure-theoretic proofs in the appendix are intricate, but I found no specific step that would invalidate the central claim. Since the verdict ACCEPT is consistent with the evidence, I recommend no change.","tokens_in":49841,"tokens_out":14577,"duration_ms":164593,"concrete_test":"Re-derive the coupling step in Proposition 53 with full measure-theoretic rigor, checking whether the distribution of (X,Y) on the error event still has both marginals equal to pX. If a pathological coupling biases the Y marginal on the error event, the supremum in Definition 5 would not apply and the distortion bound would need an additional term; if the marginals remain pX, the proof is sound. This check settles the only potential soft spot I found in the achievability machinery.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I examined Theorem 8 and its proof. The most load-bearing assumption is indeed uniform integrability of (d, pX) (Definition 5), which drives both Theorem 7 (near-perfect versus perfect realism) and the distortion bound in achievability (Propositions 53 and 54). This assumption is not a hidden gap: the paper states it explicitly, proves it for finite alphabets and for MSE with a finite fourth moment (Claim 6), and provides a non-vacuous heavy-tailed counterexample (Appendix C-D). I specifically checked the coupling argument in Proposition 53, where the error term E[d(X,Y) 1_{(X,Y) ≠ (X~,Y~)}] is evaluated on the first marginal of the coupling, whose X- and Y-marginals are exactly pX by construction (perfect realism of P'). Thus the supremum in Definition 5 applies directly; there is no mismatch between the marginals in the bound and the marginals allowed by the uniform-integrability condition. The converse and achievability arguments for Theorem 8 are internally consistent: the finite-alphabet restriction on Z is used exactly to make V finite and the entropy terms finite, and the soft-covering lemma conditions match the single-letter rate inequalities via the chain-rule identities (32) and (34). I find no circular step, no hidden regularity failure, and no unsupported leap in the main theorem. The acknowledged limitation—that sources failing uniform integrability are excluded—is a limitation of the theorem's stated scope, not a flaw in the proof of the theorem as stated.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript studies the rate-distortion-perception trade-off for memoryless sources with side information under strong perfect and near-perfect realism constraints, with emphasis on the role of limited common randomness. The main results are: Theorem 8, a single-letter closure characterization for E-D codes under marginal realism when the side-information alphabet is finite; Theorem 14, an inner bound for D-codes under marginal realism; Theorems 15 and 16, optimality results in two special cases for D-codes; Theorem 17, a closure characterization for E-D codes under joint realism; Theorem 18, a closure characterization for D-codes under joint realism; and Propositions 19 and 20, explicit Gaussian solutions. The proofs use soft-covering lemmas with side information, quantized converses, and a uniform-integrability based equivalence between near-perfect and perfect realism (Theorem 7).","tokens_in":50064,"tokens_out":22072,"duration_ms":289119,"significance":"If correct, the results are significant. Theorem 8 gives the first general-alphabet characterization of the three-way trade-off between rate, common-randomness rate, and distortion under strong marginal realism with side information at both terminals, and the Gaussian results provide concrete, falsifiable formulas. The paper is also careful about its hypotheses: the uniform-integrability assumption is stated explicitly, shown to hold for finite alphabets and for MSE with finite fourth moment, and demonstrated to be non-vacuous by the heavy-tailed counterexample in Appendix C-D. I found the main proof architecture internally consistent; in particular, the coupling argument in Proposition 54 applies the uniform-integrability supremum to a distribution whose X- and Y-marginals are exactly p_X, so the stress-test concern about a marginal mismatch does not land.","major_comments":[{"comment":"The statement A_D^(m) = S_D^(m) is not supported by the proof. The converse in Section VIII-A2 only shows that for every ε > 0 there exists a distribution satisfying the defining inequalities of S_D^(m) with slack ε, i.e., Eqs. (111)-(114) hold with R+ε, R+Rc+ε, and Δ+ε. This places (R,Rc,Δ) in the closure of S_D^(m), not necessarily in S_D^(m) itself. Achievability is also only available at the level of closures: the proof says 'Achievability follows from Theorem 14', but Theorem 14 states only that the closure of A_D^(m) contains the closure of S_D^(m), not that S_D^(m) ⊆ A_D^(m). Since S_D^(m) is not shown to be closed, the exact equality is not established. The proof supports equality of closures: closure(A_D^(m)) = closure(S_D^(m)). Please either prove that S_D^(m) is closed under the finite-common-component assumptions, or restate Theorem 15 and the surrounding text with closures, as is already done consistently in Theorems 14, 16, 17, and 18.","section":"Section VIII-A, Theorem 15"}],"minor_comments":[{"comment":"The sentence 'By Theorem 14, we have S_D^(m) ⊆ A_D^(m)' is stronger than what Theorem 14 states; Theorem 14 gives only containment of closures. For Theorem 16 the weaker statement is sufficient, but the citation should be corrected to 'closure(S_D^(m)) ⊆ closure(A_D^(m))'.","section":"Section VIII-B1"},{"comment":"The phrase 'characterize the information theoretic limits under various scenarios' overstates the decoder-only marginal-realism case, where Theorem 14 is only an inner bound and conclusive results are obtained only in two special cases. Please qualify the abstract and introduction accordingly.","section":"Abstract and Section III.B"},{"comment":"The notation R≥0 × [H(Z), ∞] × R≥0 in Eq. (7) is nonstandard because ∞ is not a real number; please clarify that this denotes the set of triples with Rc ≥ H(Z), or use a separate phrase for the unconstrained-common-randomness case.","section":"Corollary 9"},{"comment":"There are several typos, including 'receiever' in the introduction and 'infomration' in Appendix C; also, the caption of Fig. 2 writes Rc = ∞ where the text correctly uses Rc → ∞.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The paper is technically strong overall. I found no problem with the main Theorem 8 or with the uniform-integrability framework; the only substantive issue is the exact-versus-closure gap in Theorem 15. If the authors weaken that theorem to an equality of closures, or prove the required closedness, the paper should be acceptable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a solid, serious information-theory paper and the main characterization holds up. The genuinely new results are Theorem 8 (full single-letter region for marginal realism with side info at both terminals), Theorems 17–18 (joint realism, both code configurations), and the first converse for the finite-common-randomness Gaussian rate in Proposition 19, which completes the achievability half from Saldi et al. I found no hidden gap in the load-bearing arguments.\n\nWhat the paper does well: it gives a clean operational distinction between marginal and joint realism, quantifies the dual role of Z as second look and common randomness via the H(Z|Y) term, and extends the general-alphabet machinery under a uniform-integrability assumption that is honestly stated (Definition 5) and shown to be non-vacuous. The proofs are detailed, and the limitations are disclosed: decoder-only marginal realism is only an inner bound in general, with conclusive results in two special cases, and the Gaussian solution in Proposition 20 shows D/E-D equivalence only when common randomness is sufficiently large.\n\nOn the stress-test: the uniform-integrability assumption is used precisely where it is needed, and the coupling in Proposition 53 matches the definition. The counterexample in Appendix C-D shows the assumption is not vacuous. No circularity—the rate regions come from standard auxiliary-variable optimizations and the Gaussian rho is the fixed point of the derived equation, not a fitted constant.\n\nSoft spots: the appendix is dense and long; a reviewer who is not a measure-theoretic specialist will have to take some steps on faith. The decoder-only marginal case remains open in general, but the paper says so and provides partial results. The uniform-integrability restriction excludes some heavy-tailed sources with finite rate-distortion functions; again, this is a stated scope condition, not a flaw. The writing is clear for the technical level, though the paper is long.\n\nBottom line: for information theorists working on rate-distortion-perception and output-constrained coding, this is a paper they will cite. It deserves a serious referee; I would send it to review.","headline":"First full characterization of the RDP tradeoff with side info under strong realism; the Gaussian converse is the real news, and the proofs hold up on inspection.","tokens_in":50644,"tokens_out":2399,"would_cite":true,"duration_ms":27472,"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":"Perfect-realism compression with side information has a complete rate-randomness-distortion characterization.","keywords":["rate-distortion-perception trade-off","strong realism","perfect realism","common randomness","side information","lossy source coding","soft covering","Gaussian source"],"falsifier":"Construct a source and distortion pair that is not uniformly integrable but has a finite rate-distortion function (the paper supplies such a pair) and check whether near-perfect and perfect realism give the same asymptotic region under marginal realism. If a gap appears, the equivalence theorem, and with it the main characterization, fails outside the uniform-integrability assumption.","tokens_in":49601,"feed_emoji":"🎲","tokens_out":9682,"duration_ms":103694,"temperature":0.7,"pith_summary":"The paper studies lossy compression of a memoryless source when a correlated side-information sequence is available, under a strong realism constraint: the reconstructed sequence must be statistically indistinguishable from the original, either marginally or jointly with the side information. It aims to determine, for each distortion level, the minimum compression rate and the minimum rate of common randomness shared by encoder and decoder. The main result is a single-letter characterization of the achievable (rate, common-randomness, distortion) region when side information is available at both terminals and the realism constraint is on the marginal distribution of the output. A companion result shows that near-perfect realism costs nothing asymptotically: any code whose output is nearly indistinguishable from the source can be upgraded to a code whose output is exactly indistinguishable, under a mild uniform-integrability condition. The paper also solves the quadratic-Gaussian case explicitly, showing when decoder-only side information costs the same as encoder-plus-decoder side information.","feed_headline":"Perfect-realism compression: side info doubles as shared randomness","feed_subtitle":"New theorem fixes the rate-versus-randomness trade-off when reconstructions must match the source distribution.","key_machinery":"The carrying object is the single-letter region $S_{E-D}^{(m)}$: distributions on $(X,Z,V,Y)$ with $(X,Z)$ matching the source, $Y$ matching the source marginal, and the Markov chain $X - (Z,V) - Y$, whose rate inequalities are evaluated in mutual information. The proof machinery has three parts: a soft-covering lemma with side information, a concentration result showing that random codebooks indexed by side-information sequences make the output nearly indistinguishable from the target distribution; an identity rewriting the rate sum as $I_p(Y;V,Z) - H_p(Z)$, which displays the effective codebook rate $R + R_c + H(Z)$; and a coupling argument under uniform integrability that upgrades near-perfect to perfect realism without changing rates or distortion.","core_discovery":"On its own terms, the paper's central claim is Theorem 8: for a Polish source alphabet, a finite side-information alphabet, and a uniformly integrable distortion, the closure of the set of triplets $(R, R_c, \\Delta)$ achievable with perfect or near-perfect marginal realism under encoder–decoder side information equals the closure of the single-letter region $S_{E-D}^{(m)}$, defined by $R \\geq I_p(X;V|Z)$, $R + R_c \\geq I_p(Y;V|Z) - H_p(Z|Y)$, and $\\Delta \\geq E_p[d(X,Y)]$, over distributions satisfying $(X,Z) \\sim p_{X,Z}$, $p_Y = p_X$, and the Markov chain $X - (Z,V) - Y$. The extra entropy term $-H_p(Z|Y)$ is what lets the side information double as a source of common randomness: when $Z$ is independent of $X$, it contributes exactly $H(Z)$ to the common-randomness budget. For joint realism, where $(Y^n,Z^n)$ must match $(X^n,Z^n)$, the same machinery yields a region with no entropy term, meaning the side information no longer acts as common randomness. The paper further shows that in the quadratic-Gaussian case with enough common randomness, decoder-only side information achieves the same rate-distortion trade-off as encoder–decoder side information, and that this equivalence fails when common randomness is absent.","pith_inferences":["If the characterization extends to non-uniformly-integrable pairs, the Gaussian and heavy-tailed examples suggest the entropy term $H(Z|Y)$ may still be the right correction, but the achievable scheme would need a different argument in the tail.","The open gap for decoder-only side information under marginal realism suggests that the true region there depends on how much of $Z$ can be converted into common randomness; a common-information decomposition of $(X,Z)$ is the natural next object to try.","For learned video codecs, the paper's entropy term quantifies a previously implicit cost: using previously-compressed frames as side information may create a hidden common-randomness requirement that a rate-distortion-optimized encoder must budget for.","A testable extension would be to measure, for a fixed finite common-randomness rate, whether the rate-distortion curve of a neural codec with a strong perceptual loss matches the paper's Gaussian formula; the $R_c=0$ case in particular gives a definite quantitative prediction."],"forward_implications":["With encoder–decoder side information and marginal realism, the side information contributes $H(Z|Y)$ bits of common randomness in addition to its rate-reduction role, so common-randomness budgets can be smaller than they would be without side information.","Under joint realism the side information provides no common randomness; the achievable region matches the no-side-information region with every mutual information conditioned on $Z$.","Near-perfect and perfect realism are asymptotically equivalent for uniformly integrable distortion–source pairs, so a critic cannot exploit the difference at large blocklengths.","For a standard Normal source with MSE distortion, the minimal rate for a target distortion and common-randomness rate is explicit, and with unbounded common randomness it converges to the known distribution-preserving rate $\\frac{1}{2}\\log\\frac{1}{\\Delta(1-\\Delta/4)}$.","In the jointly Gaussian case with sufficient common randomness, decoder-only side information achieves the same rate-distortion trade-off as encoder–decoder side information, matching classical side-information coding; without common randomness this equivalence fails."],"supporting_citations":[{"why":"Defines the classical rate-distortion problem with side information at the decoder that this paper re-examines under realism constraints.","marker":"[1]"},{"why":"Provides the distribution-preserving quantization construction and the infinite-common-randomness Gaussian rate the paper recovers as a limit.","marker":"[21]"},{"why":"Establishes the achievability half for the quadratic-Gaussian case with limited common randomness, for which this paper proves the matching converse.","marker":"[31]"},{"why":"Characterizes the perfect-realism rate-distortion function with unconstrained common randomness, the baseline that the paper extends to side information.","marker":"[38]"},{"why":"Preliminary version of the no-side-information characterization and the explicit Gaussian rate formula that the paper generalizes.","marker":"[40]"},{"why":"Supplies the soft-covering lemma with side information used in the achievability proofs for both encoder-decoder and decoder-only settings.","marker":"[45]"},{"why":"Provides the Markov-chain converse technique that the paper adapts to general alphabets.","marker":"[50]"},{"why":"Supplies the virtual-message and likelihood-decoder construction used in the decoder-only side-information achievability proofs.","marker":"[51]"}],"fun_headline_variants":["Side info doubles as common randomness in perfect-realism compression","Perfect realism: side info can aid both compression and randomness","Rate-distortion-perception trade-off fixed with strong realism","When realism is perfect, side info becomes shared randomness","Common randomness shortage breaks perfect-realism source coding"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is uniform integrability of the distortion–source pair $(d, p_X)$: roughly, events causing very large distortion must have vanishing total probability. The paper proves this fails for some heavy-tailed sources that still have finite rate-distortion functions, and if it fails the upgrade from near-perfect to perfect realism and the achievability proofs no longer go through.","fun_headline_variants_meta":{"raw":{"variants":["Side info doubles as common randomness in perfect-realism compression","Perfect realism: side info can aid both compression and randomness","Rate-distortion-perception trade-off fixed with strong realism","When realism is perfect, side info becomes shared randomness","Common randomness shortage breaks perfect-realism source coding"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000223,"raw_usage":{"total_tokens":1552,"prompt_tokens":1136,"completion_tokens":416,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":752,"completion_tokens_details":{"reasoning_tokens":338}},"tokens_in":752,"tokens_out":416,"duration_ms":5183,"temperature":1.0,"reasoning_tokens":338,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T15:46:37.991271+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Construct a source and distortion pair that is not uniformly integrable but has a finite rate-distortion function (the paper supplies such a pair) and check whether near-perfect and perfect realism give the same asymptotic region under marginal realism. If a gap appears, the equivalence theorem, and with it the main characterization, fails outside the uniform-integrability assumption.","supporting_citations":[{"cited_title":"The Rate-Distortion Function for Source Coding with Side Information at the Decoder,","cited_arxiv_id":null,"evidence_quote":"Defines the classical rate-distortion problem with side information at the decoder that this paper re-examines under realism constraints."},{"cited_title":"Output Constrained Lossy Source Coding With Limited Common Randomness,","cited_arxiv_id":null,"evidence_quote":"Establishes the achievability half for the quadratic-Gaussian case with limited common randomness, for which this paper proves the matching converse."},{"cited_title":"Randomized Quantization and Source Coding With Constrained Output Distribution,","cited_arxiv_id":null,"evidence_quote":"Characterizes the perfect-realism rate-distortion function with unconstrained common randomness, the baseline that the paper extends to side information."},{"cited_title":"Distributed Channel Synthesis,","cited_arxiv_id":null,"evidence_quote":"Supplies the soft-covering lemma with side information used in the achievability proofs for both encoder-decoder and decoder-only settings."},{"cited_title":"Channel Simulation via Interactive Communications,","cited_arxiv_id":null,"evidence_quote":"Provides the Markov-chain converse technique that the paper adapts to general alphabets."},{"cited_title":"The Likelihood Encoder for Lossy Compression,","cited_arxiv_id":null,"evidence_quote":"Supplies the virtual-message and likelihood-decoder construction used in the decoder-only side-information achievability proofs."}],"review_version":1}