{"id":"7c318bfa-a924-43bf-a720-21b90a142cc0","arxiv_id":"2606.09698","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"For symmetric discrete channels with feedback, posterior matching is sufficient and essentially necessary to achieve both maximal mutual information and minimal per-step MMSE distortion.","lead":"The paper examines sending a real-valued source over channels with feedback to maximize mutual information to the outputs while also minimizing per-step MMSE estimation error. This dual goal yields a regularization approach that solves the joint optimization for symmetric discrete channels using the posterior matching scheme.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.3","headline":"Necessity claim for posterior matching in joint objective rests on channel symmetries whose role in the MMSE step is not independently verified","rationale":"The reader's weakest_assumption correctly flags the symmetry and input-identifiability requirements. Because the necessity direction for the combined objective is the least secured part of the argument and the abstract provides no further detail on how the MMSE condition propagates into necessity, the UNVERDICTED status is appropriate; the proposed check directly tests whether that necessity step survives isolation of the distortion criterion.","tokens_in":1689,"tokens_out":338,"duration_ms":13694,"concrete_test":"Extract the necessity argument (likely in the section deriving the joint solution) and re-derive the necessity step while holding only the per-step MMSE condition fixed and dropping input-identifiability; check whether posterior matching is still forced or whether counter-example encoders appear for the k-ary symmetric channel.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim requires showing that any scheme achieving both max mutual information and per-step MMSE minimization must coincide with posterior matching (up to equivalence) on k-ary symmetric/erasure channels. The abstract states sufficiency is shown and necessity holds under input-identifiability plus symmetries, yet the joint objective couples the two criteria sequentially. Without an explicit separation of the MMSE-minimization condition from the information-maximization condition in the necessity direction, it remains possible that other encoders satisfy the combined requirement on these channels. The input-identifiable assumption is invoked only for the information-max part; its interaction with the distortion term is not isolated.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper states sufficient conditions for an encoder to maximize mutual information between a real-valued source and channel outputs with feedback; these conditions are also necessary for input-identifiable channels. It then derives the solution to the joint problem of maximizing mutual information while minimizing per-step MMSE distortion for k-ary symmetric and k-ary erasure channels, showing that the posterior matching scheme is sufficient and essentially necessary for achieving both objectives simultaneously on these channels. The work frames information maximization as a regularizer that renders the joint distortion-minimization problem tractable.","tokens_in":1851,"tokens_out":433,"duration_ms":15160,"significance":"If the derivations hold, the result strengthens the understanding of posterior matching by showing its role extends from information maximization alone to the joint objective on symmetric channels, and supplies a concrete regularization technique for otherwise intractable feedback communication problems. The explicit separation of sufficiency from necessity under input-identifiability is a clear strength.","major_comments":[{"comment":"Abstract (necessity claim for joint objective): the statement that posterior matching is 'essentially necessary' for both information maximization and per-step MMSE minimization on k-ary symmetric/erasure channels invokes input-identifiability only for the information-maximization component. The manuscript does not isolate how this assumption interacts with the sequential MMSE-minimization requirement in the necessity direction, leaving open whether other encoders could satisfy the combined criteria; this is load-bearing for the central 'essentially necessary' claim.","section":"Abstract"}],"minor_comments":[{"comment":"The abstract refers to 'derivations exist' for the sufficient conditions and the symmetric-channel case; explicit pointers to the sections containing the full proofs (or statements of the lemmas used) would improve readability.","section":null},{"comment":"Notation for the per-step MMSE estimator and the input-identifiability condition should be introduced with a short definition or reference to prior literature on first use.","section":null}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the careful reading and for identifying a point that merits clarification in the abstract. We address the concern below and will make a targeted revision to improve precision without altering the technical claims.","responses":[{"response":"The referee correctly notes that input-identifiability is invoked for necessity of information maximization. For the k-ary symmetric and erasure channels under consideration, this property holds. The necessity argument for the joint objective proceeds in two steps that are already present in the manuscript: (i) any encoder achieving maximal mutual information must satisfy the posterior-matching conditions (by the necessity result under input-identifiability), and (ii) among all such encoders, only the posterior-matching scheme additionally satisfies the per-step MMSE-minimization requirement at every time. Consequently, the combined criteria are satisfied if and only if the encoder is the posterior-matching scheme. While this logical structure is used in the proofs, the abstract does not explicitly separate the two steps. We will therefore revise the abstract (and the corresponding paragraph in Section IV) to state that input-identifiability is used to characterize the information-maximizing encoders and that the distortion-minimization condition then selects posterior matching from within that class. This revision clarifies the interaction without changing any technical result.","revision_made":"partial","referee_comment":"[Abstract] Abstract (necessity claim for joint objective): the statement that posterior matching is 'essentially necessary' for both information maximization and per-step MMSE minimization on k-ary symmetric/erasure channels invokes input-identifiability only for the information-maximization component. The manuscript does not isolate how this assumption interacts with the sequential MMSE-minimization requirement in the necessity direction, leaving open whether other encoders could satisfy the combined criteria; this is load-bearing for the central 'essentially necessary' claim."}],"tokens_in":1306,"tokens_out":396,"duration_ms":13514,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main point is that this work combines mutual information maximization with per-step MMSE minimization in a feedback setting and derives an explicit solution for k-ary symmetric and erasure channels, where posterior matching turns out to be sufficient and essentially necessary.\n\nWhat is new is the regularization framing that treats information maximization as a way to make the otherwise intractable distortion problem solvable, along with the necessity claim under the joint objective. The sufficient conditions for pure information maximization are also laid out and shown to be necessary for input-identifiable channels. That organizes two objectives that had been treated separately and supplies a concrete scheme for the symmetric cases.\n\nThe soft spot sits in the necessity argument for the combined criterion. The abstract indicates that necessity holds once symmetries and input-identifiability are assumed, yet it is not clear how the MMSE-minimization condition is isolated from the information-maximization condition when proving that other encoders cannot work. The symmetries are invoked for both parts, but their specific role in the distortion step is not separated out, so it remains possible that the necessity result leans more heavily on the information side. Without the full derivations this cannot be confirmed, and the input-identifiability assumption is stated only for the information-max part.\n\nThe paper is aimed at people working on feedback communication, posterior matching, and joint rate-estimation problems. Readers who care about explicit schemes on symmetric channels or about regularizing distortion objectives will get the most out of it. It is worth sending for peer review because the claims are specific and the channel models are standard; a referee can check the necessity separation directly.","headline":"The paper gives a joint info-max plus per-step MMSE-min solution for symmetric feedback channels and claims posterior matching is essentially necessary there, but the necessity step needs close checking on how the distortion term is handled.","tokens_in":2307,"tokens_out":411,"would_cite":false,"duration_ms":13057,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"For symmetric channels with feedback, posterior matching is sufficient and essentially necessary to maximize mutual information while minimizing MMSE distortion at each step.","keywords":["feedback communication","posterior matching","mutual information maximization","MMSE distortion minimization","symmetric channels","channel with feedback"],"falsifier":"On a k-ary symmetric channel, exhibit an encoding scheme other than posterior matching that achieves strictly higher mutual information or strictly lower MMSE at some time step.","tokens_in":2589,"feed_emoji":"📡","tokens_out":562,"duration_ms":13017,"temperature":0.7,"pith_summary":"The paper examines the task of sending a real-valued source over multiple uses of a channel with feedback. It first gives conditions sufficient for an encoder to achieve maximal mutual information between the source and the sequence of channel outputs, and shows these conditions are also necessary when the channel is input-identifiable. It then considers the joint objective of maximizing that mutual information while simultaneously minimizing the MMSE of estimating the source from the outputs received so far. For discrete channels possessing k-ary symmetric or k-ary erasure symmetry, the work derives that the posterior matching scheme meets both requirements at once.","feed_headline":"Posterior matching meets dual goals on symmetric feedback channels","feed_subtitle":"It maximizes mutual information between source and outputs while keeping MMSE estimation error minimal at every step.","key_machinery":"The posterior matching scheme, which at each step selects the channel input so that its distribution matches the current posterior distribution of the source given all previous outputs.","core_discovery":"We show that for such channels the famous posterior matching scheme, while not necessary for information maximization alone, is sufficient and essentially necessary for achieving both information maximization and distortion minimization.","pith_inferences":["The same symmetry-based argument may extend to approximate symmetries in continuous or non-symmetric channels.","Practical systems could adopt posterior matching as a default when both rate and estimation accuracy matter.","Testing the scheme on channels that violate input-identifiability would clarify the boundary of the necessity claim."],"forward_implications":["Encoders for these symmetric channels can be constructed directly from the posterior matching rule to satisfy the joint objective.","Information maximization serves as a regularizer that renders the otherwise intractable distortion-minimization problem solvable.","The necessity result shows that any encoder achieving the joint optimum must satisfy the posterior-matching input selection rule."],"fun_headline_variants":["Posterior matching necessary for dual goals in symmetric feedback channels","Symmetric channels: posterior matching maximizes info while minimizing distortion","Posterior matching essential to optimize feedback communication with info and distortion","For symmetric feedback: posterior matching achieves info max and distortion min"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The channel must have specific symmetries such as k-ary symmetric or k-ary erasure and must be input-identifiable.","fun_headline_variants_meta":{"raw":{"variants":["Posterior matching necessary for dual goals in symmetric feedback channels","Symmetric channels: posterior matching maximizes info while minimizing distortion","Posterior matching essential to optimize feedback communication with info and distortion","For symmetric feedback: posterior matching achieves info max and distortion min"]},"model":"grok-4.3","cost_usd":0.00586,"raw_usage":{"total_tokens":2753,"prompt_tokens":603,"num_sources_used":0,"completion_tokens":66,"cost_in_usd_ticks":58599500,"prompt_tokens_details":{"text_tokens":603,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2084,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":603,"tokens_out":66,"duration_ms":12530,"temperature":1.0,"reasoning_tokens":2084,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-27T14:38:50.846970+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"On a k-ary symmetric channel, exhibit an encoding scheme other than posterior matching that achieves strictly higher mutual information or strictly lower MMSE at some time step.","supporting_citations":[],"review_version":1}