pith. sign in

arxiv: 2509.13955 · v2 · submitted 2025-09-17 · 💻 cs.IT · eess.SP· math.IT

Asymptotic Analysis of Nonlinear One-Bit Precoding in Massive MIMO Systems via Approximate Message Passing

Zheyu Wu , Junjie Ma , Ya-Feng Liu , Bruno Clerckx This is my paper

Pith reviewed 2026-05-18 16:44 UTC · model grok-4.3

classification 💻 cs.IT eess.SPmath.IT
keywords one-bit precodingmassive MIMOapproximate message passingsymbol error probabilityasymptotic analysisconvex relaxationquantizationregularization
0
0 comments X

The pith

An approximate message passing framework derives a closed-form symbol error probability for convex-relaxation-based one-bit precoding in large massive MIMO systems.

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

The paper analyzes a common approach to one-bit precoding where a convex problem is solved and then quantized to meet the one-bit constraint. It introduces an auxiliary approximate message passing iteration that tracks the effect of the quantization step in the large-system limit. This leads to an exact expression for the symbol error probability at the receiver. The analysis shows how parameters such as the regularization strength influence performance. It also establishes that the squared infinity-norm regularizer is optimal within a mixed class of regularizers.

Core claim

By developing an auxiliary AMP iteration that incorporates the nonlinear quantization function, the authors obtain a closed-form expression for the symbol error probability in the large-system limit for i.i.d. real Gaussian channels. This expression quantifies the impact of model parameters on performance and supports the optimality of the ℓ_∞² regularizer paired with an optimal parameter among mixed ℓ_∞²-ℓ₂² regularizers.

What carries the argument

An auxiliary approximate message passing iteration that folds the nonlinear quantization step into the state evolution equations to track asymptotic behavior.

If this is right

  • The symbol error probability admits a closed-form expression depending on system parameters in the large limit.
  • The ℓ_∞² regularizer with optimal tuning achieves the best SEP within the considered class of convex regularizers.
  • Performance can be predicted without running full simulations for very large antenna arrays.
  • The framework provides a way to optimize regularization parameters analytically.

Where Pith is reading between the lines

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

  • The same auxiliary AMP technique could be adapted to analyze other nonlinear operations in precoding or detection problems.
  • Results for real-valued systems may serve as a stepping stone toward complex-valued MIMO analysis.
  • Finite-system corrections to the asymptotic SEP could be derived as a next step.
  • Hardware designers could use the formula to choose between different regularizers without extensive testing.

Load-bearing premise

The channel entries are independent and identically distributed real Gaussians and the analysis holds in the large-system limit where the number of antennas and users both grow large.

What would settle it

Run Monte Carlo simulations of the precoding scheme with 200 antennas and 50 users using i.i.d. Gaussian channels and compare the observed symbol error rate against the closed-form formula for several regularization parameters.

Figures

Figures reproduced from arXiv: 2509.13955 by Bruno Clerckx, Junjie Ma, Ya-Feng Liu, Zheyu Wu.

Figure 1
Figure 1. Figure 1: An illustration of f(a), where λ = ρ = 0.2. B. Main Results We begin with the following lemmas, which are important for presenting our main results. Lemma 1. Given ρ > 0, δ > 0, and a ≥ 0, define ηa : R × R>0 → R as ηa(x; γ) = P[−a, a]  x γ + 1 . (8) Then, there exists a unique solution (τ 2 a , γa) to the following equations: τ 2 = 1 + 1 δ E [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: An illustration of the values in Eq. (13) as a function [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Theoretical and numerical SEP, where the number of tr [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Theoretical and numerical SEP versus the regulariza [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Contour plot of the theoretical SEP over the [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Contour plot of the SEP over the (ρ, λ) space for regularizers of the form R(x) = ρ r(x) + λkxk 2∞, where δ = 0.5 and SNR= 15 dB [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗
read the original abstract

Massive multiple-input multiple-output (MIMO) systems employing one-bit digital-to-analog converters offer a hardware-efficient solution for wireless communications. However, the one-bit constraint poses significant challenges for precoding design, as it transforms the problem into a discrete and nonconvex optimization task. In this paper, we investigate a widely adopted ``convex-relaxation-then-quantization" approach for nonlinear symbol-level one-bit precoding. Specifically, we first solve a convex relaxation of the discrete minimum mean square error precoding problem, and then quantize the solution to satisfy the one-bit constraint. Focusing on a real-valued system with an independently and identically distributed (i.i.d.) Gaussian channel, we develop a novel analytical framework based on approximate message passing (AMP) to characterize the high-dimensional asymptotic performance of the considered scheme. The key technical ingredient is an auxiliary AMP iteration that dedicatedly incorporates the nonlinear quantization function into the state evolution analysis. With the proposed framework, we derive a closed-form expression for the symbol error probability (SEP) at the receiver side in the large-system limit, which provides a quantitative characterization of how model and system parameters affect the SEP performance. Our empirical results suggest that the $\ell_\infty^2$ regularizer, when paired with an optimally chosen regularization parameter, achieves optimal SEP performance within a broad class of convex regularization functions. As a first step towards a theoretical justification, we prove the optimality of the $\ell_\infty^2$ regularizer within the mixed $\ell_\infty^2$-$\ell_2^2$ regularization functions.

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 / 1 minor

Summary. The manuscript develops an approximate message passing (AMP) framework for asymptotic analysis of a convex-relaxation-then-quantization scheme for nonlinear symbol-level one-bit precoding in massive MIMO systems. Focusing on real-valued i.i.d. Gaussian channels in the large-system limit, it introduces an auxiliary AMP iteration to incorporate the nonlinear quantization into state evolution analysis. This yields a closed-form expression for the symbol error probability (SEP) at the receiver. The paper also reports empirical results suggesting that the ℓ_∞² regularizer achieves optimal SEP performance within a broad class of convex regularizers when paired with an optimally chosen parameter, and provides a proof of optimality within the mixed ℓ_∞²-ℓ₂² regularization functions.

Significance. If the state-evolution justification holds, the closed-form SEP expression supplies a quantitative tool for understanding the impact of model and system parameters on performance in hardware-efficient one-bit massive MIMO, which is a practically relevant setting. The partial theoretical optimality proof for the ℓ_∞² regularizer within the mixed-norm class is a concrete strength that goes beyond pure empirics, and the auxiliary-AMP construction offers a reusable technique for other post-relaxation nonlinearities.

major comments (1)
  1. [auxiliary AMP iteration and state evolution analysis] The central claim is a closed-form SEP obtained by tracking the auxiliary AMP iteration through state evolution after the convex relaxation is solved and then quantized. Standard AMP state evolution applies to Lipschitz denoisers on linear measurements; here the quantization step is a hard nonlinearity applied post-relaxation. The derivation therefore requires that the effective denoiser (relaxed solution followed by sign quantization) still satisfies the conditions for exact state evolution in the large-system limit. No section supplies the requisite fixed-point analysis or Lipschitz-constant bounds for this composite map, leaving open whether the SEP formula holds exactly or only approximately. This is load-bearing for the validity of the closed-form expression (abstract and introduction).
minor comments (1)
  1. The abstract refers to an 'optimally chosen' regularization parameter in the empirical results; a brief clarification on whether this parameter is selected via a data-independent rule or cross-validation on the same realizations used for SEP evaluation would improve reproducibility.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed and constructive comments. We are pleased that the significance of the work is recognized, particularly the auxiliary-AMP technique and the partial optimality proof. Below we respond to the major comment on the state evolution justification.

read point-by-point responses

  1. Referee: [auxiliary AMP iteration and state evolution analysis] The central claim is a closed-form SEP obtained by tracking the auxiliary AMP iteration through state evolution after the convex relaxation is solved and then quantized. Standard AMP state evolution applies to Lipschitz denoisers on linear measurements; here the quantization step is a hard nonlinearity applied post-relaxation. The derivation therefore requires that the effective denoiser (relaxed solution followed by sign quantization) still satisfies the conditions for exact state evolution in the large-system limit. No section supplies the requisite fixed-point analysis or Lipschitz-constant bounds for this composite map, leaving open whether the SEP formula holds exactly or only approximately. This is load-bearing for the validity of the closed-form expression (abstract and introduction).

    Authors: We thank the referee for pointing out this critical aspect of our analysis. The auxiliary AMP iteration is specifically designed to incorporate the quantization nonlinearity by defining an effective denoiser that includes both the solution of the convex relaxation and the subsequent sign quantization. In the large-system limit with i.i.d. Gaussian channels, the state evolution is derived by analyzing the asymptotic behavior of this iteration. However, we acknowledge that the manuscript does not explicitly provide the fixed-point analysis or Lipschitz-constant bounds for the composite denoiser. To address this, in the revised manuscript we will include a new appendix section that rigorously verifies the applicability of state evolution to this setting, including the necessary conditions and bounds, thereby confirming that the closed-form SEP expression holds exactly in the asymptotic regime. revision: yes

Circularity Check

0 steps flagged

No significant circularity in AMP-based SEP derivation

full rationale

The paper derives a closed-form SEP expression in the large-system limit by introducing an auxiliary AMP iteration that embeds the post-relaxation quantization step into state evolution under i.i.d. Gaussian channels. This is an extension of standard AMP techniques rather than a redefinition of the target quantity. The statement that the l_infty^2 regularizer with optimally chosen parameter achieves optimal performance is presented as an empirical observation plus a partial proof for the mixed l_infty^2-l_2^2 subclass; neither reduces the central SEP formula to a fitted input or self-citation by construction. No quoted equation or step equates the derived result to its own inputs.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claims rest on the i.i.d. real Gaussian channel model, the large-system limit, and the validity of the state-evolution analysis for the auxiliary AMP; the regularization parameter is treated as a tunable quantity whose optimal value is selected empirically.

free parameters (1)
  • regularization parameter
    Described as 'optimally chosen' for the ℓ∞² regularizer; its value is selected to achieve best SEP and therefore functions as a fitted hyper-parameter.
axioms (2)
  • domain assumption Channel matrix entries are i.i.d. real Gaussian
    Stated explicitly as the setting for the analysis.
  • domain assumption Large-system limit (number of antennas and users both grow to infinity with fixed ratio)
    Required for the state-evolution analysis to yield closed-form SEP.

pith-pipeline@v0.9.0 · 5830 in / 1578 out tokens · 40679 ms · 2026-05-18T16:44:19.124348+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

42 extracted references · 42 canonical work pages

  1. [1]

    What will 5G be?

    J. G. Andrews, S. Buzzi, W. Choi, S. V . Hanly, A. Lozano, A. C. K. Soong, and J. C. Zhang, “What will 5G be?” IEEE J. Sel. Areas Commun. , vol. 32, no. 6, pp. 1065–1082, Jun. 2014

    work page 2014

  2. [2]

    Razavi, Principles of Data Conversion System Design , 1st ed

    B. Razavi, Principles of Data Conversion System Design , 1st ed. Hoboken, NJ, USA: Wiley, 1994

    work page 1994

  3. [3]

    S. C. Cripps, RF Power Amplifiers for Wireless Communications , 2nd ed. Norwood, MA, USA: Artech House, 2006

    work page 2006

  4. [4]

    Downlink achievable rate analysis in massive MIMO systems with one-bit DACs,

    Y . Li, C. Tao, A. L. Swindlehurst, A. Mezghani, and L. Liu, “Downlink achievable rate analysis in massive MIMO systems with one-bit DACs,” IEEE Commun. Lett. , vol. 21, no. 7, pp. 1669–1672, Jul. 2017

    work page 2017

  5. [5]

    Analy sis of one-bit quantized precoding for the multiuser massiv e MIMO downlink,

    A. K. Saxena, I. Fijalkow, and A. L. Swindlehurst, “Analy sis of one-bit quantized precoding for the multiuser massiv e MIMO downlink,” IEEE Trans. Signal Process. , vol. 65, no. 17, pp. 4624–4634, Sept. 2017

    work page 2017

  6. [6]

    MMSE precoder for massive MIMO using 1-bit quantization,

    O. B. Usman, H. Jedda, A. Mezghani, and J. A. Nossek, “MMSE precoder for massive MIMO using 1-bit quantization,” in Proc. IEEE Int. Conf. Acoust., Speech, Signal Process. (ICASSP) , Mar. 2016, pp. 3381–3385

    work page 2016

  7. [7]

    Weighted sum rate maximization for multi-user MISO systems with low resolution digital to analog converters,

    A. Kakkavas, J. Munir, A. Mezghani, H. Brunner, and J. A. N ossek, “Weighted sum rate maximization for multi-user MISO systems with low resolution digital to analog converters,” in Proc. 20th Int. ITG W orkshop Smart Antennas (WSA) , Mar. 2016

    work page 2016

  8. [8]

    Multi-user do wnlink beamforming using uplink downlink duality with 1-bi t converters for flat fading channels,

    K. U. Mazher, A. Mezghani, and R. W. Heath, “Multi-user do wnlink beamforming using uplink downlink duality with 1-bi t converters for flat fading channels,” IEEE Trans. V eh. Technol., vol. 71, no. 12, pp. 12 885–12 900, Dec. 2022

    work page 2022

  9. [9]

    Asymptot ic SEP analysis and optimization of linear-quantized preco ding in massive MIMO systems,

    Z. Wu, J. Ma, Y .-F. Liu, and A. L. Swindlehurst, “Asymptot ic SEP analysis and optimization of linear-quantized preco ding in massive MIMO systems,” IEEE Trans. Inf. Theory , vol. 70, no. 4, pp. 2566–2589, Apr. 2024

    work page 2024

  10. [10]

    Quantized precoding for massive MU-MIMO,

    S. Jacobsson, G. Durisi, M. Coldrey, T. Goldstein, and C . Studer, “Quantized precoding for massive MU-MIMO,” IEEE Trans. Commun. , vol. 65, no. 11, pp. 4670–4684, Nov. 2017

    work page 2017

  11. [11]

    Massive MIMO downlink 1-bit precoding with linear prog ramming for PSK signaling,

    H. Jedda, A. Mezghani, J. A. Nossek, and A. L. Swindlehur st, “Massive MIMO downlink 1-bit precoding with linear prog ramming for PSK signaling,” in Proc. IEEE W orkshop Signal Process. Adv. Wireless Commun. , Jul. 2017, pp. 1–5

    work page 2017

  12. [12]

    One-bit precoding and constellation range design for massive MIMO with QAM signal ing,

    F. Sohrabi, Y .-F. Liu, and W. Y u, “One-bit precoding and constellation range design for massive MIMO with QAM signal ing,” IEEE J. Sel. Topics Signal Process., vol. 12, no. 3, pp. 557–570, Jun. 2018

    work page 2018

  13. [13]

    Mass ive MIMO 1-bit DAC transmission: A low-complexity symbol sc aling approach,

    A. Li, C. Masouros, F. Liu, and A. L. Swindlehurst, “Mass ive MIMO 1-bit DAC transmission: A low-complexity symbol sc aling approach,” IEEE Trans. Wireless Commun., vol. 17, no. 11, pp. 7559–7575, Nov. 2018

    work page 2018

  14. [14]

    A framework for one-bit and constant-envelope precoding over multiuser ma ssive MISO channels,

    M. Shao, Q. Li, W.-K. Ma, and A. M.-C. So, “A framework for one-bit and constant-envelope precoding over multiuser ma ssive MISO channels,” IEEE Trans. Signal Process. , vol. 67, no. 20, pp. 5309–5324, Oct. 2019

    work page 2019

  15. [15]

    Interf erence exploitation 1-bit massive MIMO precoding: A partia l branch-and-bound solution with near-optimal performance,

    A. Li, F. Liu, C. Masouros, Y . Li, and B. Vucetic, “Interf erence exploitation 1-bit massive MIMO precoding: A partia l branch-and-bound solution with near-optimal performance,” IEEE Trans. Wireless Commun. , vol. 19, no. 5, pp. 3474–3489, May 2020

    work page 2020

  16. [16]

    Effici ent CI-based one-bit precoding for multiuser downlink mass ive MIMO systems with PSK modulation,

    Z. Wu, B. Jiang, Y .-F. Liu, M. Shao, and Y .-H. Dai, “Effici ent CI-based one-bit precoding for multiuser downlink mass ive MIMO systems with PSK modulation,” IEEE Trans. Wireless Commun. , vol. 23, no. 5, pp. 4861–4875, May 2024

    work page 2024

  17. [17]

    A survey of recent advances in optimiza tion methods for wireless communications,

    Y .-F. Liu, T.-H. Chang, M. Hong, Z. Wu, A. M.-C. So, E. A. J orswieck, and W. Y u, “A survey of recent advances in optimiza tion methods for wireless communications,” IEEE J. Sel. Areas Commun. , vol. 42, no. 11, pp. 2992–3031, Nov. 2024

    work page 2024

  18. [18]

    MMSE analysis of certain large isometric random precoded sy stems,

    M. Debbah, W. Hachem, P . Loubaton, and M. de Courville, “ MMSE analysis of certain large isometric random precoded sy stems,” IEEE Trans. Inf. Theory, vol. 49, no. 5, pp. 1293–1311, May 2003

    work page 2003

  19. [19]

    A vector-pe rturbation technique for near-capacity multiantenna mult iuser communication-Part I: channel inversion and regularization,

    C. Peel, B. Hochwald, and A. Swindlehurst, “A vector-pe rturbation technique for near-capacity multiantenna mult iuser communication-Part I: channel inversion and regularization,” IEEE Trans. Commun. , vol. 53, no. 1, pp. 195–202, Jan. 2005

    work page 2005

  20. [20]

    La rge system analysis of linear precoding in correlated MISO b roadcast channels under limited feedback,

    S. Wagner, R. Couillet, M. Debbah, and D. T. M. Slock, “La rge system analysis of linear precoding in correlated MISO b roadcast channels under limited feedback,” IEEE Trans. Inf. Theory , vol. 58, no. 7, pp. 4509–4537, Jul. 2012

    work page 2012

  21. [21]

    Couillet and M

    R. Couillet and M. Debbah, Random Matrix Methods for Wireless Communications . New Y ork, NY , USA: Cambridge University Press, 2011

    work page 2011

  22. [22]

    Crosscorrelation functions of amplitud e-distorted Gaussian signals,

    J. Bussgang, “Crosscorrelation functions of amplitud e-distorted Gaussian signals,” RLE Technical Reports , vol. 216, 1952

    work page 1952

  23. [23]

    Asymptotic analysis of RLS-base d digital precoder with limited PAPR in massive MIMO,

    X. Ma, A. Kammoun, A. M. Alrashdi, T. Ballal, T. Y . Al-Naf fouri, and M.-S. Alouini, “Asymptotic analysis of RLS-base d digital precoder with limited PAPR in massive MIMO,” IEEE Trans. Signal Process. , vol. 70, pp. 5488–5503, Oct. 2022

    work page 2022

  24. [24]

    Precise e rror analysis of regularized M -estimators in high dimensions,

    C. Thrampoulidis, E. Abbasi, and B. Hassibi, “Precise e rror analysis of regularized M -estimators in high dimensions,” IEEE Trans. Inf. Theory , vol. 64, no. 8, pp. 5592–5628, Aug. 2018

    work page 2018

  25. [25]

    Message-pas sing algorithms for compressed sensing,

    D. L. Donoho, A. Maleki, and A. Montanari, “Message-pas sing algorithms for compressed sensing,” Proc. Natl. Acad. Sci. , vol. 106, no. 45, pp. 18 914–18 919, Nov. 2009

    work page 2009

  26. [26]

    The dynamics of message pas sing on dense graphs, with applications to compressed sensi ng,

    M. Bayati and A. Montanari, “The dynamics of message pas sing on dense graphs, with applications to compressed sensi ng,” IEEE Trans. Inf. Theory , vol. 57, no. 2, pp. 764–785, Feb. 2011

    work page 2011

  27. [27]

    A unifying tutorial on approximate message passing,

    O. Y . Feng, R. V enkataramanan, C. Rush, and R. J. Samwort h, “A unifying tutorial on approximate message passing,” F ound. Trends Mach. Learn. , vol. 15, no. 4, pp. 335–536, 2022

    work page 2022

  28. [28]

    The LASSO risk for Gaussian matrices,

    M. Bayati and A. Montanari, “The LASSO risk for Gaussian matrices,” IEEE Trans. Inf. Theory , vol. 58, no. 4, pp. 1997–2017, Apr. 2012

    work page 1997

  29. [29]

    Universality of the ela stic net error,

    A. Montanari and P .-M. Nguyen, “Universality of the ela stic net error,” in Proc. IEEE Int. Symp. Inf. Theory (ISIT) , Jun. 2017, pp. 2338–2342

    work page 2017

  30. [30]

    Asymptotic analysis of one-bit quantized box-constraine d precoding in large-scale multi-user systems,

    X. Ma, A. Kammoun, M.-S. Alouini, and T. Y . Al-Naffouri, “Asymptotic analysis of one-bit quantized box-constraine d precoding in large-scale multi-user systems,” 2025. [Online]. Available: https:// arxiv.org/abs/2502.02953

    work page arXiv 2025

  31. [31]

    Orthogonal AMP,

    J. Ma and L. Ping, “Orthogonal AMP,” IEEE Access , vol. 5, pp. 2020–2033, 2017

    work page 2020

  32. [32]

    V ector appr oximate message passing,

    S. Rangan, P . Schniter, and A. K. Fletcher, “V ector appr oximate message passing,” IEEE Trans. Inf. Theory , vol. 65, no. 10, pp. 6664–6684, Oct. 2019

    work page 2019

  33. [33]

    Approximate message passing algorithms for ro tationally invariant matrices,

    Z. Fan, “Approximate message passing algorithms for ro tationally invariant matrices,” Ann. Stat. , vol. 50, no. 1, pp. 197–224, Feb. 2022

    work page 2022

  34. [34]

    Unifying AMP algorithms for rotationa lly-invariant models,

    S. Liu and J. Ma, “Unifying AMP algorithms for rotationa lly-invariant models,” arXiv preprint arXiv:2412.01574 , 2024

    work page arXiv 2024

  35. [35]

    The asymptotic distri bution of the MLE in high-dimensional logistic models: Arbi trary covariance,

    Q. Zhao, P . Sur, and E. J. Candes, “The asymptotic distri bution of the MLE in high-dimensional logistic models: Arbi trary covariance,” Bernoulli, vol. 28, no. 3, pp. 1835–1861, 2022

    work page 2022

  36. [36]

    Universality of approxi mate message passing with semirandom matrices,

    R. Dudeja, Y . M. Lu, and S. Sen, “Universality of approxi mate message passing with semirandom matrices,” Ann. Probab., vol. 51, no. 5, pp. 1616–1683, Sept. 2023

    work page 2023

  37. [37]

    Universality of approxim ate message passing algorithms and tensor networks,

    T. Wang, X. Zhong, and Z. Fan, “Universality of approxim ate message passing algorithms and tensor networks,” Ann. Appl. Probab. , vol. 34, no. 4, pp. 3943–3994, Aug. 2024

    work page 2024

  38. [38]

    Positively correlated normal variables ar e associated,

    L. D. Pitt, “Positively correlated normal variables ar e associated,” Ann. Probab., vol. 10, no. 2, pp. 496–499, May 1982

    work page 1982

  39. [39]

    Error bounds, quadrat ic growth, and linear convergence of proximal methods,

    D. Drusvyatskiy and A. S. Lewis, “Error bounds, quadrat ic growth, and linear convergence of proximal methods,” Math. Oper . Res., vol. 43, no. 3, pp. 919–948, Aug. 2018

    work page 2018

  40. [40]

    S. P . Boyd and L. V andenberghe, Convex Optimization. Cambridge Univ. Press, 2004

    work page 2004

  41. [41]

    Large sample estimation an d hypothesis testing,

    W. K. Newey and D. McFadden, “Large sample estimation an d hypothesis testing,” Handb. Econom. , vol. 4, pp. 2111–2245, 1994

    work page 1994

  42. [42]

    Universality laws for high-dimensio nal learning with random features,

    H. Hu and Y . M. Lu, “Universality laws for high-dimensio nal learning with random features,” IEEE Trans. Inf. Theory , vol. 69, no. 3, pp. 1932–1964, Mar. 2023

    work page 1932