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arxiv: 2606.27769 · v1 · pith:BZ6DHD5X · submitted 2026-06-26 · cs.IT · math.IT

Deriving Approximate Message Passing from the Convex Gaussian Min-Max Theorem

Vikrant Malik , Babak Hassibi This is my paper

Reviewed by Pith2026-06-29 03:12 UTCgrok-4.3pith:BZ6DHD5Xopen to challenge →

classification cs.IT math.IT
keywords approximate message passingconvex gaussian min-max theoremhigh-dimensional regressiononsager correctiongeneralized approximate message passingregularized linear regressionasymptotic analysisstate evolution
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The pith

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.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

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.

Core claim

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).

What carries the argument

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.

If this is right

  • 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.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • 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.

Load-bearing premise

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.

What would settle it

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.

read the original abstract

Approximate message passing (AMP) provides fast iterative algorithms for high-dimensional signal recovery with Gaussian design matrices, while the Convex Gaussian Min-max Theorem (CGMT) gives a static optimization framework for obtaining sharp asymptotic characterizations of convex estimators. Although these two frameworks often lead to the same scalar state-evolution equations, their connection is usually indirect. In this paper, we establish a direct connection between the two for regularized linear regression in the proportional high-dimensional regime. When the CGMT Auxiliary Optimization (AO) and Primary Optimization (PO) give the same primal-dual solution, we show that the CGMT framework recovers the AMP fixed-point equations, including the Onsager correction. We further identify the AO Gaussian vectors 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). These results show that the AMP (and GAMP) iterations are suggested, and can be derived, from the CGMT framework, and may further suggest a way to derive AMP-like algorithms in settings where CGMT applies but standard AMP derivations are unavailable.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

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.

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 (2)
  1. [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.
  2. [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.
minor comments (2)
  1. 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.
  2. The extension to GAMP is stated briefly; a short self-contained derivation paralleling the AMP case would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

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.

read point-by-point responses

  1. Referee: [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.

    Authors: 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: partial

  2. Referee: [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.

    Authors: 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: yes

Circularity Check

0 steps flagged

No significant circularity; conditional derivation is self-contained.

full rationale

The paper explicitly states its central result as conditional on the premise that CGMT AO and PO share the same primal-dual solution, then shows that this premise implies recovery of the AMP fixed-point equations (including Onsager term) from the CGMT framework. This is an implication under a stated assumption rather than any reduction of the target equations to the inputs by construction. No self-citations, fitted parameters renamed as predictions, ansatzes smuggled via citation, or uniqueness theorems imported from prior author work appear in the provided text. The derivation chain is therefore self-contained against external benchmarks under the explicit condition.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Review based on abstract only; no explicit free parameters, axioms, or invented entities can be identified.

pith-pipeline@v0.9.1-grok · 5728 in / 1027 out tokens · 49055 ms · 2026-06-29T03:12:42.691006+00:00 · methodology

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