{"id":"7f73668c-e9eb-496c-8e27-f28790cb1aea","arxiv_id":"2606.27769","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"CGMT recovers AMP and GAMP fixed-point equations including Onsager correction when AO and PO share the same primal-dual solution in the proportional regime.","lead":"This paper shows that the Convex Gaussian Min-Max Theorem can directly produce the fixed-point equations of Approximate Message Passing for regularized linear regression when its auxiliary and primary optimizations share the same solution. A smart generalist might read it to see how a static optimization tool can generate the iterative algorithms used in high-dimensional signal recovery.","discovery_kind":"unification","skeptic_critique":{"model":"grok-4.3","headline":"Central claim is conditional on AO/PO sharing the same primal-dual solution, but this is asserted rather than shown to hold for the regression problems considered.","rationale":"The reader's weakest_assumption matches the load-bearing conditional exactly. Because the full text makes the derivation explicitly conditional on that premise and supplies no independent verification that the premise holds, the UNVERDICTED status is unchanged.","tokens_in":1674,"tokens_out":302,"duration_ms":15928,"concrete_test":"For ridge regression (quadratic loss + ℓ2 penalty) in the proportional regime, compute the limiting primal-dual pair for both the PO and the AO; check whether the two pairs are asymptotically identical (within o(1) in probability). If they differ by a non-vanishing amount, the premise fails and the derivation does not recover the AMP equations.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The derivation recovers AMP fixed-point equations (including Onsager term) only under the explicit premise that the CGMT auxiliary optimization and primary optimization share the same primal-dual solution. Standard CGMT results equate the asymptotic values of the two programs but do not guarantee that their optimizers (or the associated dual variables) coincide. The manuscript does not supply a separate argument establishing that this stronger optimizer-level equivalence occurs in the proportional regime for regularized M-estimation; without it the claimed direct recovery of the AMP iteration remains conditional on an unverified premise.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript claims to establish a direct connection between the Convex Gaussian Min-Max Theorem (CGMT) and Approximate Message Passing (AMP) for regularized linear regression in the proportional high-dimensional regime. Conditional on the CGMT auxiliary optimization (AO) and primary optimization (PO) sharing the same primal-dual solution, it derives the AMP fixed-point equations (including the Onsager correction) from the CGMT framework and identifies the AO Gaussian vectors with the perturbations appearing in the primal and residual AMP channels. The same viewpoint is used to recover the fixed point of scalar-variance max-sum Generalized AMP (GAMP) for regularized M-estimation.","tokens_in":1813,"tokens_out":541,"duration_ms":41557,"significance":"If the conditional equivalence of optimizers holds and the derivations are correct, the work supplies a unified static-to-iterative perspective that could facilitate construction of AMP-like algorithms in regimes where CGMT applies but conventional AMP derivations are unavailable. The explicit mapping of Gaussian vectors between the two frameworks is a concrete technical contribution.","major_comments":[{"comment":"Abstract and main derivation: the recovery of the AMP fixed-point equations (including Onsager term) is conditioned on AO and PO sharing the same primal-dual solution. Standard CGMT results equate only the asymptotic optimal values of the two programs, not the optimizers or associated dual variables. The manuscript does not supply an independent argument establishing that this stronger optimizer-level equivalence holds for regularized M-estimation in the proportional regime; without such an argument the claimed direct derivation remains conditional on an unverified premise that is load-bearing for the central claim.","section":"Abstract"},{"comment":"The identification of AO Gaussian vectors with the Gaussian perturbations in the primal and residual AMP channels is presented as following from the shared primal-dual solution. It is unclear whether this identification is derived from first principles under the CGMT assumptions or whether it implicitly relies on the target AMP equations themselves; a self-contained verification that does not presuppose the AMP iteration would strengthen the result.","section":"main derivation of AMP fixed-point equations"}],"minor_comments":[{"comment":"Notation for the primal and dual variables in the AO/PO programs should be introduced with explicit reference to the corresponding AMP state variables to make the mapping transparent.","section":null},{"comment":"The extension to GAMP is stated briefly; a short self-contained derivation paralleling the AMP case would improve readability.","section":null}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the detailed review and valuable feedback on our work connecting CGMT and AMP. We address the major comments below, clarifying the conditional nature of our results and the derivation steps.","responses":[{"response":"We agree with the referee that standard CGMT establishes equivalence of optimal values rather than optimizers, and that our derivation relies on the stronger assumption of shared primal-dual solutions. The manuscript explicitly conditions the result on this assumption (see abstract and Section 3), without claiming to prove it. This conditional connection is still valuable as it shows how AMP equations emerge from the CGMT framework when the assumption holds, potentially guiding derivations in other settings. We will add a discussion in the introduction on scenarios where the optimizer equivalence is known to hold, such as in strictly convex problems or under uniqueness conditions, to better contextualize the premise.","revision_made":"partial","referee_comment":"[Abstract] Abstract and main derivation: the recovery of the AMP fixed-point equations (including Onsager term) is conditioned on AO and PO sharing the same primal-dual solution. Standard CGMT results equate only the asymptotic optimal values of the two programs, not the optimizers or associated dual variables. The manuscript does not supply an independent argument establishing that this stronger optimizer-level equivalence holds for regularized M-estimation in the proportional regime; without such an argument the claimed direct derivation remains conditional on an unverified premise that is load-bearing for the central claim."},{"response":"The derivation begins from the KKT optimality conditions of the AO and PO problems under the shared solution assumption. The AO problem is formulated with explicit Gaussian vectors, and the stationarity conditions directly yield the fixed-point relations, including the Onsager correction term, without referencing the AMP iteration a priori. The Gaussian vectors in the AO are then identified with the effective noise terms in the AMP channels by matching the resulting equations. To address the concern, we will revise the main derivation section to present the steps in a more sequential manner, starting purely from the CGMT AO and deriving the equations before noting the correspondence to AMP.","revision_made":"yes","referee_comment":"[main derivation of AMP fixed-point equations] The identification of AO Gaussian vectors with the Gaussian perturbations in the primal and residual AMP channels is presented as following from the shared primal-dual solution. It is unclear whether this identification is derived from first principles under the CGMT assumptions or whether it implicitly relies on the target AMP equations themselves; a self-contained verification that does not presuppose the AMP iteration would strengthen the result."}],"tokens_in":1425,"tokens_out":549,"duration_ms":52884,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main takeaway is that the authors recover the AMP fixed-point equations, including the Onsager term, directly from the CGMT by conditioning on the auxiliary optimization and primary optimization having identical primal-dual solutions. They also map the AO Gaussian vectors to the perturbations appearing in the primal and residual AMP channels, and they carry the same logic over to scalar-variance max-sum GAMP for regularized M-estimation.\n\nWhat works is the explicit identification of those Gaussian vectors and the clean statement that the CGMT viewpoint can suggest AMP-like iterations in regimes where the usual state-evolution derivations are unavailable. That framing is straightforward and could be practically useful for people who already work with CGMT characterizations.\n\nThe soft spot is exactly the one the stress-test flags. Standard CGMT results equate the asymptotic values of the two programs but do not automatically give matching optimizers or dual variables. The manuscript treats the stronger optimizer-level equivalence as given for proportional high-dimensional regularized regression without a separate argument showing why it holds. Because the recovery of the AMP equations rests on that premise, the claimed direct derivation stays conditional.\n\nThis is for readers already comfortable with both CGMT and AMP who want to explore whether the optimization framework can generate new iterative algorithms. A serious referee should see it; the core idea is worth checking even if the current write-up needs work on the equivalence step. I would send it to review rather than desk-reject.","headline":"The paper recovers AMP fixed points from CGMT only under an asserted but unshown condition that AO and PO share the same primal-dual solution.","tokens_in":2243,"tokens_out":363,"would_cite":false,"duration_ms":26603,"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":"The Convex Gaussian Min-Max Theorem recovers the fixed-point equations of Approximate Message Passing when its auxiliary and primary optimizations share the same primal-dual solution.","keywords":["approximate message passing","convex gaussian min-max theorem","high-dimensional regression","onsager correction","generalized approximate message passing","regularized linear regression","asymptotic analysis","state evolution"],"falsifier":"A concrete numerical check for a fixed regularized linear regression instance with Gaussian design matrix showing that the scalar state-evolution equations obtained from CGMT under the matching condition differ from the standard AMP fixed-point equations.","tokens_in":2587,"feed_emoji":"","tokens_out":699,"duration_ms":28159,"temperature":0.7,"pith_summary":"The paper establishes a direct link between the Convex Gaussian Min-Max Theorem and Approximate Message Passing algorithms for high-dimensional signal recovery. It shows that under the condition where the auxiliary and primary optimizations yield the same primal-dual solution, the CGMT framework produces the AMP fixed-point equations, including the Onsager correction term. This connection also identifies the Gaussian vectors in the auxiliary optimization with the perturbations in the AMP channels. For regularized M-estimation, it recovers the fixed point of the generalized AMP. The result suggests that AMP iterations can be derived from the CGMT framework in settings where standard derivations may not apply.","feed_headline":"CGMT recovers AMP fixed-point equations from matching optimizations","feed_subtitle":"When auxiliary and primary optimizations agree on the primal-dual solution, the static CGMT framework directly produces the iterative AMP up","key_machinery":"The Convex Gaussian Min-Max Theorem (CGMT) applied to regularized linear regression, where matching primal-dual solutions between the auxiliary optimization and primary optimization directly produce the AMP iterations and Onsager correction.","core_discovery":"When the CGMT Auxiliary Optimization (AO) and Primary Optimization (PO) give the same primal-dual solution, the CGMT framework recovers the AMP fixed-point equations, including the Onsager correction. The AO Gaussian vectors are identified with the Gaussian perturbations in the primal and residual AMP channels. For regularized M-estimation, the same viewpoint recovers the fixed point of scalar-variance max-sum Generalized AMP (GAMP).","pith_inferences":["The explicit identification of AO Gaussian vectors with AMP channel perturbations may allow a probabilistic view of how static CGMT analysis generates dynamic iterations.","This direct derivation route could extend to derive similar message-passing schemes for problems where CGMT applies but classical AMP analysis does not.","Varying the CGMT formulation might suggest new iterative algorithms whose fixed points match known asymptotic characterizations."],"forward_implications":["AMP fixed-point equations can be obtained directly from CGMT without relying on indirect connections.","The Onsager correction term arises naturally from the structure of the CGMT auxiliary optimization.","The fixed point of scalar-variance max-sum GAMP for regularized M-estimation follows from the same CGMT matching condition.","AMP-like algorithms may be derivable from CGMT in other settings where standard AMP derivations are unavailable."],"fun_headline_variants":["CGMT derives AMP equations from AO PO solution match","AMP fixed points including Onsager from CGMT framework","CGMT framework recovers AMP and GAMP fixed points","AO Gaussians match AMP channel perturbations","CGMT connects static optimization to iterative AMP"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The assumption that the CGMT auxiliary optimization and primary optimization produce the same primal-dual solution for the regularized linear regression problem in the proportional high-dimensional regime.","fun_headline_variants_meta":{"raw":{"variants":["CGMT derives AMP equations from AO PO solution match","AMP fixed points including Onsager from CGMT framework","CGMT framework recovers AMP and GAMP fixed points","AO Gaussians match AMP channel perturbations","CGMT connects static optimization to iterative AMP"]},"model":"grok-4.3","cost_usd":0.005866,"raw_usage":{"total_tokens":2797,"prompt_tokens":686,"num_sources_used":0,"completion_tokens":69,"cost_in_usd_ticks":58662000,"prompt_tokens_details":{"text_tokens":686,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2042,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":686,"tokens_out":69,"duration_ms":51305,"temperature":1.0,"reasoning_tokens":2042,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-29T03:12:42.691006+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A concrete numerical check for a fixed regularized linear regression instance with Gaussian design matrix showing that the scalar state-evolution equations obtained from CGMT under the matching condition differ from the standard AMP fixed-point equations.","supporting_citations":[],"review_version":1}