{"id":"88f38ab0-7524-419d-868b-c7ba9a0a49bd","arxiv_id":"2512.14539","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Compression-based denoising achieves an exact asymptotic loss equal to the expected loss of two independent posterior samples, for any stationary source passed through a discrete memoryless channel with the channel-matched distortion.","lead":"Lossy compression tuned to the channel's noise statistics can serve as a general denoiser, and this paper gives an exact formula for the resulting error. It extends earlier additive-noise results to any discrete memoryless channel and replaces a loose upper bound with a precise limit.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 5 assumes an invertible channel matrix, but its proof and Example 3 only require full row rank; as stated, the theorem excludes the erasure channel it showcases.","rationale":"The reader's verdict identifies both the double-sided mixing assumption and the rank/invertibility condition as fragile. I agree that double-sided mixing is a substantive scope restriction, and the paper's 'arguably includes all those of practical interest' is not proved. However, the more concrete and immediately load-bearing issue is the inconsistency in the rank condition. The central theorem is stated with an invertible channel matrix, while the proof uses only full row rank and the erasure-channel example is non-square. This is not a mere typo in one line: it changes the set of channels to which the main theorem applies. The paper's advertised contribution is generalization to arbitrary DMCs, so the theorem statement should match the proof's actual hypothesis. Because the proof does not appear to need square invertibility, the fix is straightforward: restate Theorem 5 with the weaker rank condition and correct Example 3's alphabet. With that correction, the mathematical core of the paper stands. Hence a conditional acceptance, rather than rejection or unchanged acceptance, is the right verdict: the result is sound in its intended form, but the manuscript as written contains a load-bearing statement-level error that must be fixed before the theorem can be used as stated.","tokens_in":17798,"tokens_out":61141,"duration_ms":899892,"concrete_test":"Take the erasure channel of Example 3: X ∈ {0,1}, Z ∈ {0,1,e}, with matrix (rows X, columns Z) [[1-p_e, 0, p_e], [0, 1-p_e, p_e]]. Verify that this matrix has full row rank (rank 2) but is not invertible. Then re-derive the proof of Theorem 5 step-by-step for this channel, checking every place the channel matrix is used. If the only use is through Corollary 2's full-row-rank condition and no step requires squareness, amend Theorem 5's hypothesis to 'full row rank' (or 'left-invertible') and correct the alphabet declaration in Example 3; if a step does require squareness, identify it as the point where the example fails.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Theorem 5 (Eq. 39) is stated for a channel matrix P_{Z|X} that is invertible. Its proof, however, invokes Corollary 2, whose hypothesis is only 'full row rank' (Eq. 35), and Theorem 4 likewise uses full row rank for uniqueness. With the standard orientation of P_{Z|X} as |Z| × |X|, an erasure channel with |X|=2, |Z|=3 has a 2×3 matrix of full column rank (equivalently, full row rank when rows are indexed by X), but it is not square and therefore not invertible. Example 3 explicitly uses this erasure channel and claims Theorem 5's conclusion. Thus the central theorem as written does not cover one of the paper's own demonstrations, and it excludes every DMC with |Z| ≠ |X|. The mathematical proof itself appears to go through with the weaker rank condition, so this is not a fatal flaw; but the statement of the central claim is internally inconsistent with its proof and examples. A reader who applies the theorem literally cannot use it for erasure channels despite the paper's claim of general DMCs.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies compression-based denoising for a stationary ergodic source X^n observed through a known discrete memoryless channel P_{Z|X}. The authors propose to compress Z^n with a lossy code designed for the channel-matched distortion ρ(z,y) = -log p_{Z|X}(z|y) at distortion level H(Z|X). Using a prior result of Weissman–Ordentlich on the empirical distribution of good rate-distortion codes, they show (Theorem 4/Corollary 2) that the empirical (Z^k,Y^k) distribution of a good code asymptotically coincides with (Z^k,X^k), so the reconstruction is a posterior sample. The main contribution (Theorem 5) adds a double-sided mixing condition on (X,Z) and a rank condition on the channel matrix, and proves the stronger statement that, in the joint empirical distribution, X_0 and Y_0 are asymptotically conditionally independent given Z^k, yielding the exact loss lim_n E[Λ^n(X^n,Y^n)] = E_Z E_{U,V∼P_{X_0|Z}^2}[Λ(U,V)]. Examples give a factor-of-two MSE improvement over the earlier bound and Hamming-loss evaluations. A comparison with indirect rate–distortion and rate–distortion–perception shows the scheme is generally not equivalent to indirect rate–distortion but can achieve the perfect-perception curve in the scalar Gaussian case.","tokens_in":18110,"tokens_out":12211,"duration_ms":100109,"significance":"If the main theorem is correct, this is a meaningful advance: it replaces the worst-case-coupling upper bound of [7] with an exact characterization for general DMCs, and the conditional-independence structure is a genuine strengthening. The derivation is transparent and rests on standard rate-distortion theory plus the independent prior result of Weissman–Ordentlich; no ad-hoc parameters are introduced. The erasure-channel example and the Gaussian rate-distortion-perception comparison illustrate the applicability and limitations. The main proof (Theorem 5) is essentially correct under the full-row-rank condition, and the claimed factor-of-two MSE improvement over [7] follows from the independence of two posterior samples.","major_comments":[{"comment":"The statement of Theorem 5 requires the channel matrix P_{Z|X} to be invertible, but the proof invokes Corollary 2 (Eq. (35)), whose hypothesis is only full row rank, and the proof of Theorem 4 also uses full row rank for uniqueness. Example 3 (Binary Symmetric Source with Erasures) uses an erasure channel with |X|=2 and |Z|=3; its 2×3 transition matrix has full row rank but is not square, hence not invertible. The theorem as written therefore does not cover the paper's own showcase, nor any DMC with unequal input/output alphabet sizes. The mathematical argument itself goes through under full row rank (this is exactly the condition that forces P_Y=P_X in the uniqueness step), so the fix is local but necessary: restate Theorem 5 with the full-row-rank condition, define the matrix orientation (rows indexed by inputs or outputs), and correct the alphabet declaration in Example 3.","section":"III-B, Theorem 5 (Eq. 39); proof of Corollary 2 (Eq. 35); Example 3"},{"comment":"The abstract, introduction, and conclusion claim the result for any stationary ergodic source, but Theorem 5 assumes (X,Z) are double-sided mixing, i.e. δ_k(X,Z)→0 in Definition 5. This condition is not implied by stationarity/ergodicity; the manuscript itself notes that processes with nonzero δ_k exist, and Proposition 1 establishes the condition only for Markov sources with strictly positive channel probabilities. Since Lemma 1 (Eq. (38)) and hence the exact limit in Eq. (39) collapse without δ_k→0, the advertised scope is broader than the proven statement. Please either prove the result under plain stationarity/ergodicity or amend the abstract, Section I, and Section VI to state the double-sided-mixing assumption.","section":"Abstract / Section VI vs Definition 5 (Eqs. 36–37) and Proposition 1"}],"minor_comments":[{"comment":"The line 'Let X=Z=Y={0,1}' is inconsistent with the erasure channel; Z must include the erasure symbol ε, and indeed the text later uses Z_t≠ε. Please correct the alphabet declaration.","section":"Section IV, Example 3"},{"comment":"Please specify D∈(0,1/2) for the BSC(D) channel. At D=1/2 the channel matrix is singular and the rank/invertibility condition of Theorem 5 fails.","section":"Section IV, Example 2"},{"comment":"These Gaussian examples use continuous alphabets and lie outside the finite-alphabet hypotheses of Theorems 4–5. State explicitly that they are informal infinite-alphabet analogues, or provide the needed extension.","section":"Section IV-A, Examples 4 and 5"},{"comment":"The error notation o_n(1), o_k(1), and o(1/n) is nonstandard and the order of limits is implicit. Since Corollary 2 is applied for fixed k before taking k→∞, please use explicit error bounds η(n,k) with lim_{k→∞} lim_{n→∞} η(n,k)=0, or otherwise clarify the double-limit argument.","section":"Theorem 5 proof, Eqs. (45)–(46)"},{"comment":"The finite-alphabet necessity direction is described only as 'nearly identical' to Proposition 2 and not written out. Since Proposition 3 is used for the optimality claim in Example 5, please include the proof or explicitly state that only the sufficiency direction is needed for the paper's conclusions.","section":"Section IV-A, Proposition 3"}],"recommendation":"major_revision","confidential_remarks":"I am assigning major_revision rather than minor because the mismatch in Theorem 5's hypothesis is in the central theorem and the paper advertises a broader scope than the assumptions warrant. The fix is local and the mathematical core is sound. A careful consistency pass is needed for the theorem statement, the erasure example's alphabet, and the 'stationary ergodic' language in the abstract and conclusion."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the paper deserves a serious referee. It extends the Weissman–Ordentlich compression-based denoiser from additive noise to arbitrary DMCs and, more importantly, upgrades their worst-case coupling bound to an exact asymptotic loss formula. The key move is choosing the distortion ρ(z,y)=−log p_{Z|X}(z|y) and operating at level H(Z|X), so a good lossy code for Z behaves as a posterior sampler. Theorem 5's characterization—the limiting loss equals the expected loss of two independent posterior samples—is genuinely new, and the conditional-independence lemma (Lemma 1) is the right technical tool. The derivations are clean, the MSE and Hamming examples show a concrete factor-of-two improvement over [7], and there are no fitted parameters or circular dependencies: [7] is prior published work used as a building block, not as evidence.\n\nSoft spots, in proportion. First, the rank condition is internally inconsistent. Theorem 5 states the channel matrix P_{Z|X} must be invertible, but the proof only uses the full-row-rank condition from Theorem 4 and Corollary 2. The erasure channel in Example 3 is 2×3 and not square, so it doesn't satisfy the stated hypothesis. The math goes through with the weaker rank condition, so this is a fixable statement error, not a fatal flaw—but it has to be addressed. Second, the double-sided mixing condition (Definition 5) is load-bearing. Lemma 1 and Theorem 5 collapse without δ_k→0, and the paper only establishes it for Markov sources with strictly positive channel probabilities (Proposition 1). The claim that this class is 'arguably all of practical interest' is optimistic; it deserves more discussion. Third, minor: Proposition 3's necessity argument is omitted ('nearly identical'), and Example 3 says Z={0,1} but erasure requires a third output symbol. These are small.\n\nBottom line: the core result is solid and the paper is worth engaging with. I'd send it to peer review; with the rank-condition fix, it would be a strong contribution to the source coding literature.","headline":"Genuine advance: exact loss characterization for compression-based denoising over general DMCs, but the main theorem's rank hypothesis is mis-stated and needs a correction.","tokens_in":18519,"tokens_out":3253,"would_cite":true,"duration_ms":28277,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["94A34","94A15"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that a good lossy compressor, when its distortion measure is matched to the observation channel and set to the conditional-entropy level, asymptotically outputs independent posterior samples, making its denoising loss exac","keywords":["compression-based denoising","rate-distortion theory","posterior sampling","conditional independence","discrete memoryless channels","stationary ergodic sources","double-sided mixing","empirical distributions"],"falsifier":"Take a stationary but non-mixing source, e.g., the deterministic alternating sequence X_i=(i mod 2), observed through a binary symmetric channel. Because the posterior P(X0|Z^k) does not stabilize with k, δ_k does not vanish. If one constructs a good code for ρ(z,y)=-log p_{Z|X}(z|y) at D=H(Z|X), the empirical conditional distribution Q^{(n)}_{X0|Z^k,Y^k} should not converge to P_{X0|Z^k}, and the limit in Theorem 5 should differ from E_Z[E_{U,V∼P_{X0|Z}}^2 Λ]. Computing that limit would locate the precise boundary of the theorem's validity.","tokens_in":17706,"feed_emoji":"⚙️","tokens_out":6194,"duration_ms":50217,"temperature":0.7,"pith_summary":"This paper shows that lossy compression of a noisy observation is itself a denoising operation, for any stationary ergodic source passed through any discrete memoryless channel. The key is to choose the compressor's distortion measure as the negative log-likelihood of the channel, ρ(z,y)=-log p_{Z|X}(z|y), and to operate at the distortion level H(Z|X), the conditional entropy of the observation given the source. Under a double-sided mixing condition on the source–observation pair, the reconstruction produced by a good compressor behaves asymptotically like an independent draw from the posterior of the source given the observation. The denoising loss for any loss function therefore converges to the expected loss of two independent posterior samples, an exact expression that improves on earlier upper bounds for additive-noise channels.","feed_headline":"Lossy compression samples the posterior and denoises exactly","feed_subtitle":"A channel-matched distortion measure turns any good compressor into a denoiser; the loss equals two independent posterior draws.","key_machinery":"The engine is the pair (ρ, D) with ρ(z,y)=-log p_{Z|X}(z|y) and D=H(Z|X). For this choice, the rate-distortion function takes the closed form R(Z^k,H(Z|X))=(1/k) I(Z^k;X^k), achieved uniquely by the law (Z^k,Y^k) =_d (Z^k,X^k) when the channel matrix has full row rank. This yields the marginal result that the empirical (Z^k,Y^k) distribution converges to the posterior. The second ingredient is the double-sided mixing coefficient δ_k, which measures how far the finite-window posterior is from the infinite-window posterior; Lemma 1 bounds the total-variation discrepancy of the empirical conditional of X0 given (Z^k,Y^k) from the true posterior by |X|δ_k. Vanishing of δ_k turns the marginal pos","core_discovery":"The paper's central claim is Theorem 5: for finite alphabets, a double-sided mixing pair (X,Z), an invertible channel matrix P_{Z|X}, and a bounded loss Λ, any sequence of good lossy codes for Z under the channel-matched distortion at level H(Z|X) satisfies lim_{n→∞} E[Λ^n(X^n,Y^n(Z^n))] = E_Z[E_{U,V∼(P_{X0|Z})^2} Λ(U,V)]. In words, compression samples from the posterior exactly, and the two random variables — the clean symbol and its reconstruction — are asymptotically conditionally independent given the observation. This both generalizes the framework beyond additive noise and replaces a worst-case-coupling upper bound with an exact formula; for squared error, the formula is 2 Var(X0|Z), a","pith_inferences":["The exact loss formula depends on conditional independence; if double-sided mixing fails, the empirical conditional may not converge to the posterior, so the formula would break down even though the marginal posterior result might survive — a boundary worth probing with non-mixing sources like periodic or deterministic signals.","The factor-of-two gap relative to the Bayes-optimal MSE suggests a natural test: if one runs two independent good codes and averages their outputs, the variance term should halve, potentially approaching the Bayes envelope; the paper leaves this multi-code averaging unexplored.","The rate-distortion-perception optimality shown for Gaussian sources hints that the compression-based denoiser might be optimal in the perfect-perception sense for a broader class of sources; a general proof would require extending Proposition 3 beyond the Gaussian case.","Since the distortion measure is exactly the channel's log-likelihood, one could implement the scheme in practice by using any standard lossy codec on a transformed representation of the observations; experiments on real data, which the paper lists as future work, could verify whether the asymptotic formula estimates finite-block performance."],"forward_implications":["For mean-squared-error loss, the compression-based denoiser achieves exactly 2 Var(X0|Z), a factor-of-two improvement over the prior worst-case bound of 4 Var(X0|Z).","For Hamming loss on a binary source through a BSC, the achieved loss is E_Z[2α(1−α)] with α=P(X0=1|Z), which never exceeds the old bound 2φ(α) and coincides with the Bayes envelope at α=0, 1/2, and 1.","The framework applies to arbitrary discrete memoryless channels, including erasure channels, and to sources with memory such as Markov chains, as long as the channel matrix has full row rank.","Because the distortion measure depends only on the channel, the same compressor design works for any downstream loss function; the result thus supplies a universal recipe for posterior sampling via rate-distortion coding.","In the scalar Gaussian case, the scheme achieves the rate-distortion-perception optimal tradeoff, linking compression-based denoising to perception-constrained coding."],"fun_headline_variants":["Compression-based denoisers now handle any discrete channel","Denoising via compression: exact loss from posterior draws","Lossy codes sample the posterior, so denoising is exact","Any channel, any compressor: denoising loss is exactly two posterior samples"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The load-bearing premise is that the source and observation pair is double-sided mixing, δ_k→0, which is not implied by stationarity or ergodicity; the paper establishes it only for Markov sources with strictly positive channel transition probabilities, and the exact loss limit collapses without it.","fun_headline_variants_meta":{"raw":{"variants":["Compression-based denoisers now handle any discrete channel","Denoising via compression: exact loss from posterior draws","Lossy codes sample the posterior, so denoising is exact","Any channel, any compressor: denoising loss is exactly two posterior samples"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000336,"raw_usage":{"total_tokens":1665,"prompt_tokens":679,"completion_tokens":986,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":423,"completion_tokens_details":{"reasoning_tokens":913}},"tokens_in":423,"tokens_out":986,"duration_ms":8703,"temperature":1.0,"reasoning_tokens":913,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T16:00:53.389856+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a stationary but non-mixing source, e.g., the deterministic alternating sequence X_i=(i mod 2), observed through a binary symmetric channel. Because the posterior P(X0|Z^k) does not stabilize with k, δ_k does not vanish. If one constructs a good code for ρ(z,y)=-log p_{Z|X}(z|y) at D=H(Z|X), the empirical conditional distribution Q^{(n)}_{X0|Z^k,Y^k} should not converge to P_{X0|Z^k}, and the limit in Theorem 5 should differ from E_Z[E_{U,V∼P_{X0|Z}}^2 Λ]. Computing that limit would locate the precise boundary of the theorem's validity.","supporting_citations":[],"review_version":1}