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arxiv: 2606.04531 · v1 · pith:RIITWQP3new · submitted 2026-06-03 · 📡 eess.SP

Gaussian-Process Dynamics of Diagonal Expectation Propagation under Variance-Profile Gaussian Measurements

Fangqing Xiao , Dirk T. M. Slock This is my paper

Pith reviewed 2026-06-28 05:08 UTC · model grok-4.3

classification 📡 eess.SP
keywords diagonal expectation propagationvariance profilestate evolutionGaussian processdeterministic equivalentapproximate message passinglarge system limit
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The pith

Under variance-profile Gaussian measurements, diagonal expectation propagation follows Gaussian-process dynamics with memory instead of scalar fresh noise.

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

This paper investigates the effective channel principle in state-evolution analyses for diagonal expectation propagation when sensing matrices have a variance profile. The isotropy needed for scalar decoupling is removed, yet Gaussian conditioning is preserved. The analysis proves that the linear module produces residuals forming a coordinate-dependent Gaussian process whose covariance depends on the variance profile and the algorithm's history. This replaces the usual assumption of a fresh scalar Gaussian observation with a process that retains predictable memory from past residuals. The result matters for understanding convergence and performance of EP algorithms in non-uniform sensing environments.

Core claim

The paper establishes that for diagonal EP under variance-profile Gaussian sensing matrices, the limiting object is a Gaussian-process dynamics with profile-dependent memory. The standard diagonal EP cavity cancels the instantaneous response but may leave a component predictable from past residuals. This process is characterized through a conditioned matrix-Dyson-equation deterministic equivalent and a Schur-complement representation of the linear module, followed by a Gaussian-regression decomposition that separates the predictable memory from the orthogonal innovation.

What carries the argument

Conditioned matrix-Dyson-equation deterministic equivalent combined with Schur-complement representation, which models the linear EP module as a Gaussian process with memory.

Load-bearing premise

The finite-time large-system description continues to hold despite the absence of isotropy that supports scalar decoupling.

What would settle it

Numerical experiments that check if the covariance between current residuals and past residuals matches the predicted profile-dependent structure, or if it vanishes as in the isotropic case.

read the original abstract

State-evolution analyses of approximate-message-passing and expectation-propagation-type algorithms rely on an effective-channel principle: after a suitable Onsager, orthogonal, or extrinsic correction, the nonlinear module receives a fresh scalar Gaussian observation. This paper studies this principle for diagonal expectation propagation under variance-profile Gaussian sensing matrices. The model preserves Gaussian conditioning, but removes the isotropy that supports the usual scalar decoupling arguments. We prove a finite-time large-system description in which the linear EP module remains Gaussian at the coordinate level, but is generally not a fresh scalar channel. Instead, the residuals form a coordinate-dependent Gaussian process whose covariance is shaped by the variance profile and by the finite linear history of the algorithm. The standard diagonal EP cavity cancels the instantaneous response of the incoming message, but may leave a component predictable from past residuals. We characterize this process through a conditioned matrix-Dyson-equation deterministic equivalent and a Schur-complement representation of the linear module. A Gaussian-regression decomposition then separates the predictable memory from the orthogonal innovation and yields an oracle state-evolution-level correction. Thus, under variance-profile measurements, the limiting object for diagonal EP is a Gaussian-process dynamics with profile-dependent memory rather than the conventional fresh-noise scalar state evolution.

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

1 major / 0 minor

Summary. The manuscript claims to prove a finite-time large-system description for diagonal expectation propagation under variance-profile Gaussian sensing matrices. It asserts that the linear EP module remains Gaussian at the coordinate level but produces residuals forming a coordinate-dependent Gaussian process whose covariance is shaped by the variance profile and the algorithm's finite linear history, rather than a fresh scalar channel; this is derived via a conditioned matrix-Dyson-equation deterministic equivalent, Schur-complement representation, and Gaussian-regression decomposition to separate profile-dependent memory from orthogonal innovation.

Significance. If the central derivation holds, the result would meaningfully extend state-evolution analyses of EP/AMP algorithms beyond the isotropic case to variance-profile measurements, which arise in many practical settings. The explicit isolation of history-dependent memory via regression decomposition and the preservation of Gaussian conditioning while dropping isotropy constitute a technical contribution that could inform oracle corrections for such algorithms.

major comments (1)
  1. [Abstract] Abstract: the claim of a complete proof via conditioned matrix-Dyson-equation deterministic equivalent, Schur-complement representation, and Gaussian-regression decomposition is asserted, but the full steps verifying the large-system limit and the reduction to a profile-dependent Gaussian process are not provided, so the support for the central claim that the residuals form a coordinate-dependent process with memory (rather than fresh scalar noise) cannot be confirmed.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for recognizing the potential significance of extending state-evolution analysis beyond the isotropic case. We address the single major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim of a complete proof via conditioned matrix-Dyson-equation deterministic equivalent, Schur-complement representation, and Gaussian-regression decomposition is asserted, but the full steps verifying the large-system limit and the reduction to a profile-dependent Gaussian process are not provided, so the support for the central claim that the residuals form a coordinate-dependent process with memory (rather than fresh scalar noise) cannot be confirmed.

    Authors: The abstract summarizes the proof strategy at a high level, as is conventional. The complete, self-contained derivation—including verification of the finite-time large-system limit via the conditioned matrix-Dyson equation, the Schur-complement representation of the linear EP module, and the Gaussian-regression decomposition that isolates profile-dependent memory from orthogonal innovation—is given in full in Sections 3–5 of the manuscript. These sections contain all intermediate steps, assumptions, and limit arguments supporting that the residuals form a coordinate-dependent Gaussian process whose covariance depends on the variance profile and the algorithm’s finite history. The central claim is therefore substantiated by the body of the paper rather than the abstract alone. If the referee can point to a specific step that appears incomplete, we will gladly expand it. revision: no

Circularity Check

0 steps flagged

No significant circularity; derivation uses external RMT tools

full rationale

The paper's central claim is a finite-time large-system description derived via conditioned matrix-Dyson-equation deterministic equivalents, Schur-complement representations of the linear module, and Gaussian-regression decompositions that isolate profile-dependent memory. These steps are presented as the object of the proof and invoke standard external random-matrix tools rather than reducing the target dynamics to quantities defined by fitted parameters or self-citations within the paper. No self-definitional, fitted-input-called-prediction, or load-bearing self-citation patterns appear in the described chain, leaving the derivation self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The analysis rests on standard large-system and Gaussian assumptions common to the field; no new free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption Finite-time large-system limit holds
    Invoked to obtain the coordinate-level Gaussian process description.
  • domain assumption Sensing matrices are variance-profile Gaussian
    Central modeling choice that removes isotropy.

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Reference graph

Works this paper leans on

45 extracted references · 43 canonical work pages

  1. [1]

    Approximate message-passing decoder and capacity-achieving sparse superposition codes,

    J. Barbier and F. Krzakala, “Approximate message-passing decoder and capacity-achieving sparse superposition codes,”IEEE Trans. Inf. Theory, vol. 63, no. 8, pp. 4894–4927, Aug. 2017, doi: 10.1109/TIT.2017.2713833

    work page doi:10.1109/tit.2017.2713833 2017

  2. [2]

    Capacity-achieving sparse superposition codes via approximate message passing decoding,

    C. Rush, A. Greig, and R. Venkataramanan, “Capacity-achieving sparse superposition codes via approximate message passing decoding,”IEEE Trans. Inf. Theory, vol. 63, no. 3, pp. 1476–1500, Mar. 2017, doi: 10.1109/TIT.2017.2649460

    work page doi:10.1109/tit.2017.2649460 2017

  3. [3]

    Fundamental limits of symmetric low-rank matrix estimation,

    M. Lelarge and L. Miolane, “Fundamental limits of symmetric low-rank matrix estimation,”Probab. Theory Relat. Fields, vol. 173, pp. 859–929, 2019, doi: 10.1007/s00440-018-0845-x

    work page doi:10.1007/s00440-018-0845-x 2019

  4. [4]

    Estimation of low-rank matrices via approximate message passing,

    A. Montanari and R. Venkataramanan, “Estimation of low-rank matrices via approximate message passing,”Ann. Statist., vol. 49, no. 1, pp. 321–345, 2021, doi: 10.1214/20-AOS1984

    work page doi:10.1214/20-aos1984 2021

  5. [5]

    Tractable approximations for probabilistic models: The adaptive TAP mean field approach,

    M. Opper and O. Winther, “Tractable approximations for probabilistic models: The adaptive TAP mean field approach,”Phys. Rev. Lett., vol. 86, no. 17, pp. 3695–3698, 2001, doi: 10.1103/PhysRevLett.86.3695

    work page doi:10.1103/physrevlett.86.3695 2001

  6. [6]

    A CDMA multiuser detection algorithm on the basis of belief propagation,

    Y . Kabashima, “A CDMA multiuser detection algorithm on the basis of belief propagation,”J. Phys. A, Math. Gen., vol. 36, no. 43, pp. 11 111–11 121, 2003, doi: 10.1088/0305-4470/36/43/030

    work page doi:10.1088/0305-4470/36/43/030 2003

  7. [7]

    Procrustes

    M. M ´ezard and A. Montanari,Information, Physics, and Computation. Oxford, U.K.: Oxford Univ. Press, 2009, doi: 10.1093/acprof:oso/9780198570837.001.0001

    work page doi:10.1093/acprof:oso/9780198570837.001.0001 2009

  8. [8]

    Message passing algorithms for compressed sensing,

    D. L. Donoho, A. Maleki, and A. Montanari, “Message passing algorithms for compressed sensing,”Proc. Natl. Acad. Sci. U.S.A., vol. 106, no. 45, pp. 18 914–18 919, Nov. 2009, doi: 10.1073/pnas.0909892106

    work page doi:10.1073/pnas.0909892106 2009

  9. [9]

    The dynamics of message passing on dense graphs, with applications to compressed sensing,

    M. Bayati and A. Montanari, “The dynamics of message passing on dense graphs, with applications to compressed sensing,”IEEE Trans. Inf. Theory, vol. 57, no. 2, pp. 764–785, Feb. 2011, doi: 10.1109/TIT.2010.2094817

    work page doi:10.1109/tit.2010.2094817 2011

  10. [10]

    A unifying tutorial on approximate message passing,

    O. Y . Feng, R. Venkataramanan, C. G. Rush, and R. J. Samworth, “A unifying tutorial on approximate message passing,”Found. Trends Mach. Learn., vol. 15, no. 4, pp. 335–536, 2022, doi: 10.1561/2200000092

    work page doi:10.1561/2200000092 2022

  11. [11]

    Generalized approximate message passing for estimation with random linear mixing,

    S. Rangan, “Generalized approximate message passing for estimation with random linear mixing,” inProc. IEEE Int. Symp. Inf. Theory (ISIT), 2011, pp. 2168–2172, doi: 10.1109/ISIT.2011.6033942

    work page doi:10.1109/isit.2011.6033942 2011

  12. [12]

    Expectation propagation for approximate Bayesian inference,

    T. P. Minka, “Expectation propagation for approximate Bayesian inference,” inProc. 17th Conf. Uncertainty Artif. Intell. (UAI). Morgan Kaufmann, 2001, pp. 362–369, doi: 10.5555/647235.720257

    work page doi:10.5555/647235.720257 2001

  13. [13]

    Expectation consistent approximate inference,

    M. Opper and O. Winther, “Expectation consistent approximate inference,”J. Mach. Learn. Res., vol. 6, pp. 2177–2204, Dec. 2005, doi: 10.5555/1046920.1194917

    work page doi:10.5555/1046920.1194917 2005

  14. [14]

    Vector approximate message passing,

    S. Rangan, P. Schniter, and A. K. Fletcher, “Vector approximate message passing,”IEEE Trans. Inf. Theory, vol. 65, no. 10, pp. 6664–6684, Oct. 2019, doi: 10.1109/TIT.2019.2916359

    work page doi:10.1109/tit.2019.2916359 2019

  15. [15]

    Orthogonal AMP,

    J. Ma and L. Ping, “Orthogonal AMP,”IEEE Access, vol. 5, pp. 2020–2033, 2017, doi: 10.1109/ACCESS.2017.2653119

    work page doi:10.1109/access.2017.2653119 2020

  16. [16]

    Rigorous dynamics of expectation-propagation-based signal recovery from unitarily invariant measurements,

    K. Takeuchi, “Rigorous dynamics of expectation-propagation-based signal recovery from unitarily invariant measurements,”IEEE Trans. Inf. Theory, vol. 66, no. 1, pp. 368–386, Jan. 2020, doi: 10.1109/TIT.2019.2947058

    work page doi:10.1109/tit.2019.2947058 2020

  17. [17]

    Approximate message passing algorithms for rotationally invariant matrices,

    Z. Fan, “Approximate message passing algorithms for rotationally invariant matrices,”Ann. Statist., vol. 50, no. 1, pp. 197–224, 2022, doi: 10.1214/21- AOS2101

    work page doi:10.1214/21- 2022

  18. [18]

    Estimation in rotationally invariant generalized linear models via approximate message passing,

    R. Venkataramanan, K. K”ogler, and M. Mondelli, “Estimation in rotationally invariant generalized linear models via approximate message passing,” in Proc. 39th Int. Conf. Mach. Learn. (ICML), ser. Proc. Mach. Learn. Res., vol. 162. PMLR, 2022, pp. 22 120–22 144

    2022

  19. [19]

    Approximate message passing for orthogonally invariant ensembles: Multivariate non-linearities and spectral initialization,

    X. Zhong, T. Wang, and Z. Fan, “Approximate message passing for orthogonally invariant ensembles: Multivariate non-linearities and spectral initialization,”Inf. Inference, vol. 13, no. 3, p. iaae024, 2024, doi: 10.1093/imaiai/iaae024

    work page doi:10.1093/imaiai/iaae024 2024

  20. [20]

    Unifying AMP algorithms for rotationally-invariant models,

    S. Liu and J. Ma, “Unifying AMP algorithms for rotationally-invariant models,”arXiv preprint arXiv:2412.01574, 2024, doi: 10.48550/arXiv.2412.01574

    work page doi:10.48550/arxiv.2412.01574 2024

  21. [21]

    Optimality of approximate message passing for spiked matrix models with rotationally invariant noise,

    R. Dudeja, S. Liu, and J. Ma, “Optimality of approximate message passing for spiked matrix models with rotationally invariant noise,”Ann. Statist., vol. 54, no. 1, pp. 466–489, 2026, doi: 10.1214/25-AOS2575

    work page doi:10.1214/25-aos2575 2026

  22. [22]

    Stability of the matrix Dyson equation and random matrices with correlations,

    O. H. Ajanki, L. Erd ˝os, and T. Kr ¨uger, “Stability of the matrix Dyson equation and random matrices with correlations,”Probab. Theory Relat. Fields, vol. 173, no. 1–2, pp. 293–373, 2019, doi: 10.1007/s00440-018-0835-z

    work page doi:10.1007/s00440-018-0835-z 2019

  23. [23]

    The matrix Dyson equation and its applications for random matrices,

    L. Erd ˝os, “The matrix Dyson equation and its applications for random matrices,” inRandom Matrices, ser. IAS/Park City Math. Ser. Providence, RI, USA: Amer. Math. Soc., 2019, vol. 26, pp. 75–158, doi: 10.1090/pcms/026/03. 64

    work page doi:10.1090/pcms/026/03 2019

  24. [24]

    T. W. Anderson,An Introduction to Multivariate Statistical Analysis, 3rd ed. Hoboken, NJ, USA: Wiley, 2003

    2003

  25. [25]

    Chatterjee,Superconcentration and Related Topics

    S. Chatterjee,Superconcentration and Related Topics. Cham, Switzerland: Springer, 2014, doi: 10.1007/978-3-319-03886-5

    work page doi:10.1007/978-3-319-03886-5 2014

  26. [26]

    State evolution for general approximate message passing algorithms, with applications to spatial coupling,

    A. Javanmard and A. Montanari, “State evolution for general approximate message passing algorithms, with applications to spatial coupling,”Inf. Inference, vol. 2, no. 2, pp. 115–144, 2013, doi: 10.1093/imaiai/iat004

    work page doi:10.1093/imaiai/iat004 2013

  27. [27]

    Universality in polytope phase transitions and message passing algorithms,

    M. Bayati, M. Lelarge, and A. Montanari, “Universality in polytope phase transitions and message passing algorithms,”Ann. Appl. Probab., vol. 25, no. 2, pp. 753–822, 2015, doi: 10.1214/14-AAP1010

    work page doi:10.1214/14-aap1010 2015

  28. [28]

    State evolution for approximate message passing with non-separable functions,

    R. Berthier, A. Montanari, and P.-M. Nguyen, “State evolution for approximate message passing with non-separable functions,”Inf. Inference, vol. 9, no. 1, pp. 33–79, 2020, doi: 10.1093/imaiai/iay021

    work page doi:10.1093/imaiai/iay021 2020

  29. [29]

    Approximate message passing for sparse matrices with application to the equilibria of large ecological Lotka–V olterra systems,

    W. Hachem, “Approximate message passing for sparse matrices with application to the equilibria of large ecological Lotka–V olterra systems,”Stochastic Process. Appl., vol. 170, p. 104276, 2024, doi: 10.1016/j.spa.2023.104276

    work page doi:10.1016/j.spa.2023.104276 2024

  30. [30]

    Elliptic approximate message passing and an application to theoretical ecology,

    M.-Y . Gueddari, W. Hachem, and J. Najim, “Elliptic approximate message passing and an application to theoretical ecology,”Random Matrices Theory Appl., vol. 14, no. 4, p. 2550018, 2025, doi: 10.1142/S2010326325500182

    work page doi:10.1142/s2010326325500182 2025

  31. [31]

    Approximate message passing for general non-symmetric random matrices,

    ——, “Approximate message passing for general non-symmetric random matrices,”J. Theor. Probab., vol. 39, no. 1, pp. 1–69, 2026, doi: 10.1007/s10959- 025-01476-z

    work page doi:10.1007/s10959- 2026

  32. [32]

    A leave-one-out approach to approximate message passing,

    Z. Bao, Q. Han, and X. Xu, “A leave-one-out approach to approximate message passing,”Ann. Appl. Probab., vol. 35, no. 4, pp. 2716–2766, 2025, doi: 10.1214/25-AAP2186

    work page doi:10.1214/25-aap2186 2025

  33. [33]

    Low-rank matrix estimation with inhomogeneous noise,

    A. Guionnet, J. Ko, F. Krzakala, and L. Zdeborov ´a, “Low-rank matrix estimation with inhomogeneous noise,”Inf. Inference, vol. 14, no. 2, p. iaaf010, 2025, doi: 10.1093/imaiai/iaaf010

    work page doi:10.1093/imaiai/iaaf010 2025

  34. [34]

    Optimal algorithms for the inhomogeneous spiked Wigner model,

    A. Pak, J. Ko, and F. Krzakala, “Optimal algorithms for the inhomogeneous spiked Wigner model,” inAdv. Neural Inf. Process. Syst., vol. 36, 2023, pp. 76 409–76 424, doi: 10.52202/075280-3340

    work page doi:10.52202/075280-3340 2023

  35. [35]

    On the convergence of orthogonal/vector AMP: Long-memory message-passing strategy,

    K. Takeuchi, “On the convergence of orthogonal/vector AMP: Long-memory message-passing strategy,”IEEE Trans. Inf. Theory, vol. 68, no. 12, pp. 8121–8138, Dec. 2022, doi: 10.1109/TIT.2022.3194855

    work page doi:10.1109/tit.2022.3194855 2022

  36. [36]

    Orthogonal approximate message-passing for spatially coupled linear models,

    ——, “Orthogonal approximate message-passing for spatially coupled linear models,”IEEE Trans. Inf. Theory, vol. 70, no. 1, pp. 594–631, Jan. 2024, doi: 10.1109/TIT.2023.3311408

    work page doi:10.1109/tit.2023.3311408 2024

  37. [37]

    Macroscopic analysis of vector approximate message passing in a model-mismatched setting,

    T. Takahashi and Y . Kabashima, “Macroscopic analysis of vector approximate message passing in a model-mismatched setting,”IEEE Trans. Inf. Theory, vol. 68, no. 8, pp. 5579–5600, Aug. 2022, doi: 10.1109/TIT.2022.3163342

    work page doi:10.1109/tit.2022.3163342 2022

  38. [38]

    Toward designing optimal sensing matrices for generalized linear inverse problems,

    J. Ma, J. Xu, and A. Maleki, “Toward designing optimal sensing matrices for generalized linear inverse problems,”IEEE Trans. Inf. Theory, vol. 70, no. 1, pp. 482–508, Jan. 2024, doi: 10.1109/TIT.2023.3307553

    work page doi:10.1109/tit.2023.3307553 2024

  39. [39]

    Memory AMP,

    L. Liu, S. Huang, and B. M. Kurkoski, “Memory AMP,”IEEE Trans. Inf. Theory, vol. 68, no. 12, pp. 8015–8039, Dec. 2022, doi: 10.1109/TIT.2022.3186166

    work page doi:10.1109/tit.2022.3186166 2022

  40. [40]

    Sufficient statistic memory approximate message passing,

    ——, “Sufficient statistic memory approximate message passing,” inProc. IEEE Int. Symp. Inf. Theory (ISIT), 2022, pp. 157–162, doi: 10.1109/ISIT50566.2022.9834568

    work page doi:10.1109/isit50566.2022.9834568 2022

  41. [41]

    Universality of approximate message passing algorithms and tensor networks,

    T. Wang, X. Zhong, and Z. Fan, “Universality of approximate message passing algorithms and tensor networks,”Ann. Appl. Probab., vol. 34, no. 4, pp. 3943–3994, 2024, doi: 10.1214/24-AAP2056

    work page doi:10.1214/24-aap2056 2024

  42. [42]

    A non-asymptotic analysis of generalized vector approximate message passing algorithms with rotationally invariant designs,

    C. Cademartori, C. Rush, and A. D. J. M. da Silva, “A non-asymptotic analysis of generalized vector approximate message passing algorithms with rotationally invariant designs,”IEEE Trans. Inf. Theory, vol. 70, no. 8, pp. 5811–5856, Aug. 2024, doi: 10.1109/TIT.2024.3396472

    work page doi:10.1109/tit.2024.3396472 2024

  43. [43]

    In: Computer Vision – ECCV 2024: 18th Eu- ropean Conference, Milan, Italy, September 29–October 4, 2024, Proceedings, Part LXXIX

    Z. Bai and J. W. Silverstein,Spectral Analysis of Large Dimensional Random Matrices, 2nd ed. New York, NY , USA: Springer, 2010, doi: 10.1007/978- 1-4419-0661-8

    work page doi:10.1007/978- 2010

  44. [44]

    Couillet and M

    R. Couillet and M. Debbah,Random Matrix Methods for Wireless Communications. Cambridge, U.K.: Cambridge Univ. Press, 2011, doi: 10.1017/CBO9780511994746

    work page doi:10.1017/cbo9780511994746 2011

  45. [45]

    Universality for general Wigner-type matrices,

    O. H. Ajanki, L. Erd ˝os, and T. Kr ¨uger, “Universality for general Wigner-type matrices,”Probab. Theory Relat. Fields, vol. 169, no. 3–4, pp. 667–727, 2017, doi: 10.1007/s00440-016-0740-2

    work page doi:10.1007/s00440-016-0740-2 2017