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Multi-Environment GLAMP: Approximate Message Passing for Transfer Learning with Applications to Lasso-based Estimators

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Multi-environment GLAMP makes the exact risks of three high-dimensional transfer-learning Lasso estimators computable.

desk verdict The precise risk formulas for the three Lasso transfer estimators are new and mostly convincing, but the application theorems rely on unproved fixed-point existence and regularity conditions that need to be verified or weakened. read the letter →

arxiv 2505.22594 v2 pith:4IBYRFPQ submitted 2025-05-28 math.ST stat.MLstat.TH

Longlin Wang , Yanke Song , Kuanhao Jiang , Pragya Sur This is my paper
classification math.STstat.MLstat.TH MSC 62J0762F1260F05
keywords Multi-EnvironmentGLAMPapproximatemessagepassingstateevolutiontransferlearninghigh-dimensionalregressionLassoriskdistributionshiftSecond-Stepestimator
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper introduces Multi-Environment Generalized Long AMP (GLAMP), an approximate-message-passing iteration that tracks several high-dimensional regression datasets at once when sample sizes and feature counts grow proportionally. The authors prove a state-evolution theorem: at every step, the GLAMP iterates in each environment behave like Gaussian matrices whose covariances are computed recursively, so any pseudo-Lipschitz functional of the iterates concentrates on its expectation. They then show that three Lasso-based transfer-learning estimators—the Stacked Lasso, the Model Averaging estimator, and the Second-Step estimator—are exactly tracked by this iteration. The payoff is precise asymptotic mean-squared-error formulas for these estimators, replacing worst-case minimax bounds with constants a practitioner can evaluate and compare under distribution shift.

What carries the argument

The load-bearing object is the multi-environment GLAMP iteration: alternating updates $V^t_e = \kappa_e^{-1}X_e^\top \Psi^{t-1}_e(R^{t-1}_e,W_e) - \kappa_e^{-1}\eta^{t-1}_e(V^{t-1}_1,\dots,V^{t-1}_E,B)[D^{(R)}]^\top$ and $R^t_e = X_e\eta^t_e(V^t_1,\dots,V^t_E,B) - \Psi^{t-1}_e(R^{t-1}_e,W_e)[D^{(V)}]^\top$, with Onsager terms equal to averaged Jacobians evaluated at Gaussian matrices. State evolution converts this recursion into a covariance recursion for the Gaussian limits. The estimator of interest is encoded by choosing the denoisers; for the Lasso cases, the fixed-design denoiser is the weighted Lasso map applied to Gaussian inputs. Distribution shift enters through environment-specific covariance matrices, noise levels, and environment weights, and the fixed-point systems close the recursion at the estimator of interest.

What would settle it

On a two-environment design-shift model with $\Sigma_2=\mathrm{diag}(\chi,1/\chi)$, numerically solve the fixed-point system (7) across a grid of $\lambda$ and $\chi$ and compare the predicted MSE with simulations at increasing $p$; a parameter point where the system has no solution or the solution branch jumps while simulated risk stays smooth would show Theorem 2 does not hold there. Equivalently, check whether the limit function $H$ in Assumption 4(b) is $C^1$ on the simulated region, since a failure of the derivative convergence already violates the theorem's hypotheses.

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Extended reading notes

Core claim

The paper's central claim is that in the proportional asymptotics limit, the risk of these transfer learners is governed by a fixed-design denoiser evaluated at the fixed point of state-evolution equations. Theorem 1 states that for any pseudo-Lipschitz function, the multi-environment GLAMP iterates converge to independent Gaussian matrices with covariances given by the state-evolution recursion. Theorems 2, 3, and 4 then assert that the Stacked Lasso, the Model Averaging estimator, and the Second-Step estimator converge to their fixed-design counterparts $\eta(\tau_1^*Z_1,\dots,\tau_E^*Z_E)$, $\sum_e \pi_e \eta_e(\tau_{\mathrm{ind},e}^* Z_e)$, and $\xi(\tau_{II}^*, \zeta^*, \eta(\tau^*))$, with parameters determined by the fixed-point systems (7), (15), and (23). Consequently, quantities such as $\mathbb{E}\|\eta(\tau^*) - \beta_1\|^2_{\Sigma}$ become exactly computable, and the simulated MSE curves track these formulas at moderate sample sizes.

Load-bearing premise

The paper's central claim stands on the existence of a solution to each fixed-point system (7), (15), and (23) and on $C^1$ convergence of the denoiser moment functions; if any system has no solution or has multiple solution branches, the asymptotic risk formulas are not justified.

Editorial extensions

If this is right

  • Exact MSE curves for the three estimators become computable expressions, so one can compare early fusion, late fusion, and pretrain-then-refine strategies without Monte Carlo.
  • The same state evolution gives the joint law of the estimators, not just marginal risk, so functionals such as support recovery or prediction error under a shifted test distribution are accessible.
  • Because the analysis handles anisotropic covariances and heterogeneous noise, it covers both covariate shift and model shift within one set of equations.
  • The framework's scope is not limited to the three estimators; changing the denoisers can yield precise risk characterizations for other transfer-learning penalties.
  • Cross-over behavior such as the simulated U-shaped MSE as a function of regularization suggests that adaptive tuning can be guided by the formulas.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial extension: the fixed-design representation suggests the formulas may transfer to non-Gaussian covariate distributions via universality, but that requires a separate proof.
  • Editorial extension: comparing the stacked and second-step estimators implies an optimal mixture of source and target information; one could derive oracle environment weights from the fixed-point equations, a selection rule the paper does not spell out.
  • Editorial extension: iterating the second-step refinement could produce a multi-round transfer algorithm, but whether Assumptions 7 and 8 survive iteration is an open question.
  • Editorial extension: the assumption that noise and design are independent of the fixed signal sequences could be relaxed only if the Gaussian conditioning step is preserved; a small simulation study could test whether these formulas are robust to such relaxation.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper introduces Multi-Environment Generalized Long AMP (GLAMP), an approximate message passing framework for transfer learning with multiple Gaussian design matrices. Theorem 1 establishes a state evolution result for the multi-environment GLAMP iterates. The paper then applies this framework to three Lasso-based transfer learning estimators: the Stacked Lasso, the Model Averaging estimator, and the Second-Step estimator. For each, it derives an asymptotic risk characterization expressed through a fixed-design denoiser evaluated at a fixed point of the state evolution equations (Theorems 2, 3, and 4, respectively). The paper also reports finite-sample simulations that support the theoretical predictions.

Significance. If the stated results hold, this is the first exact risk characterization for these transfer learning estimators under proportional asymptotics, going beyond worst-case minimax bounds and complementing ridge-based random matrix analyses. The core theoretical contribution—a multi-environment GLAMP state evolution theorem—is a natural and potentially useful extension of the GLAMP framework, and the paper provides detailed appendices reducing the application theorems to the GLAMP convergence machinery. The paper also ships explicit, machine-checkable formulas for the asymptotic MSE of the three estimators, and the simulations suggest that the finite-sample predictions are accurate. The main weakness is that the application theorems are conditional on fixed-point existence and smooth-limit assumptions that are not verified for the Gaussian Lasso examples, so the strength of the final claims depends on resolving that gap.

major comments (3)
  1. [§4.1–§4.3, Assumptions 4, 6, 8, Eqs. (7), (15), (23)] Theorems 2–4 are stated as unconditional asymptotic characterizations, but they rest on Assumptions 4(a), 6(a), and 8(a), which merely assume existence of solutions to the fixed-point systems (7), (15), and (23), and on Assumptions 3(b,c), 4(b), 5(b,c), 6(b), 7, and 8(b), which assume C^1 convergence of the moment functions. The paper never verifies these conditions for the Gaussian Lasso transfer-learning model of Assumption 2, nor for the specific simulation settings of Figures 1–3. In particular, the proof of the Cauchy property (Lemma C.3) uses the contraction condition κ_e δ_e < 1, which is only guaranteed by the existence part of Assumption 4(a). If (7) has no solution, the limiting denoiser η(τ*,·) in Theorem 2 is not defined; if it has multiple branches, the GLAMP iterates could converge to a different branch and the risk formula would fail. The paper should either prove these fixed-point conditions for the Gaussian Lasso examples (e.g., by reducing to known scalar Lasso state-evolution results and a contraction argument for the multi-environment system), or explicitly present Theorems 2–4 as conditional and verify the assumptions numerically in the simulations, including reporting how (7), (15), and (23) are solved and checking κ_e δ_e < 1 and the derivative bounds.
  2. [Assumption 8(a) and Theorem 4] Assumption 8(a) is self-referential: it requires that the parameter γ_I in Equation (20) be chosen so that γ_I = γ_I, where γ_I is the limit of the Onsager coefficient γ_I^(p) from Proposition 3. The fixed-point system (23) does not include an equation determining γ_I; it treats γ_I as a free parameter and solves for τ_II*, ζ*, θ_II*. The condition γ_I = γ_I is therefore an extra fixed-point equation whose solvability is assumed, not established. As written, Theorem 4 characterizes only a second-step estimator whose penalty is tuned to an unknown asymptotic quantity, and the theorem gives no guidance for constructing such an estimator in finite samples. Please clarify the logical status of this condition and either prove existence of the self-consistent solution or reformulate Theorem 4 as conditional on it.
  3. [Appendix A.2, proof of Theorem 1] The sub-sequence argument in the proof of Theorem 1 asserts that after extracting a sub-sequence on which the cross-environment limits exist, the limiting covariance matrices used to characterize Z_{V,e}^t are 'fully determined by Assumption 1, independently of the choice of sub-sequences.' This is not immediate from the text, because the off-diagonal blocks of the full covariance matrix in the asymmetric GLAMP (Corollary A.1) can a priori depend on the extracted sub-sequence. The claim is, however, correct if one observes that the state evolution matrices Σ_{V}^t are block diagonal for t ≥ 1, since Ψ^t is block diagonal in Definition 3 and Definition 4; this directly yields the mutual independence of the Gaussian limits Z_{V,e}^t. Please add this argument explicitly to justify the independence claim and the sub-sequence independence.
minor comments (5)
  1. [§4.2, Theorem 3] In the first display of Theorem 3, the remainder is written as '+ o(1)', but the convergence is stochastic; elsewhere the paper uses '+ o_P(1)'. Please correct for consistency.
  2. [§4.1, simulation description] In the model-shift simulation, the text states that β̃_j iid ~ N(0, σ̃²π/2) satisfies ∥β̃∥_1/p → σ̃². The correct limit is σ̃, since E|β̃_j| = σ̃. Please fix this formula and the corresponding caption text.
  3. [Throughout] There are several typographical and notational slips, including 'Multi-enrivonment' in Section 1, 'satiesfy' in Assumption 4(b), 'diagnoal' in Appendices C and D, and the overloaded use of η in Equation (4) where the same symbol denotes both the argmin function and its value. These should be cleaned up.
  4. [§4.1, Eq. (2)] The notation β1,j = β2,j i.i.d.∼ N(0, σ_β²) in the simulation setup is ambiguous; presumably the intended meaning is β_e,j i.i.d. across both e and j, not that the two coordinates are identical. Please rephrase.
  5. [Assumption 8(b)] The definition of the family N(ε) and the condition 'for all i ≠ 2, with 1 ≤ i ≤ E+1, Cov(g_i, g_{i+E+1}) ≥ 1−ε' is quite involved and the role of ε_H is not discussed. A few sentences explaining the purpose of this assumption and how it is used would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the state evolution equations and fixed-point systems self-consistently determine the asymptotic risk, and no fitted parameter is renamed as a prediction.

full rationale

The derivation chain is self-contained rather than circular. Theorem 1 is obtained by specializing the symmetric GLAMP state evolution of Gerbelot and Berthier [42], which is an external, independent result. Theorems 2, 3, and 4 then track the GLAMP iterates to the fixed-design denoisers η and ξ via Lemmas C.2, C.6, D.1, and E.7, with the fixed-point systems (7), (15), and (23) used only to determine the constants τ*, ϖ*, θ*, and related parameters. These systems are self-consistency conditions derived from the state evolution, not fits to simulated data: the paper repeatedly states that the theoretical MSE values come from the formulae (e.g., Figure 1 caption: 'The dotted line indicates the theoretical MSE value obtained from our formulae'), and the figures compare, rather than calibrate, those formulae to empirical values. The regularity and fixed-point existence assumptions (Assumptions 3(b),(c), 4(a),(b), 6(a),(b), 8(a),(b)) are conditions under which the theorems hold; they are not the conclusions renamed as assumptions. Any failure to verify these conditions for the simulated Gaussian examples is a technical gap or correctness risk, not circularity. The paper's self-citations (e.g., [25], [29], [30], [31], [61], [64], [71], [73]) appear in background and future-work contexts and are not load-bearing for the main convergence proofs, which rely on external AMP theory, Stein's lemma, and standard random matrix tools. No quoted reduction shows a fitted parameter being called a prediction, and no load-bearing argument reduces to a self-citation. Hence the circularity score is 0.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The central derivation does not fit constants to simulated data. The risk formulas are evaluated at fixed points of state evolution equations derived from the model. The main uncharged inputs are the existing GLAMP theorem, the Gaussian-design domain assumption, and the assumed existence and regularity of fixed points. No new physical or probabilistic entities are postulated.

free parameters (1)
  • Fixed-point parameters in state evolution = Solutions of systems (7), (15), and (23); no closed form
    The risk formulas in Theorems 2 through 4 are evaluated at fixed points whose existence is assumed in Assumptions 4(a), 6(a), and 8(a). No primitive existence or uniqueness conditions are proven.
assumptions (4)
  • standard math The GLAMP convergence theorem of Gerbelot and Berthier [42] is used as a black box.
    Theorem 1 is proved in Appendix A by embedding the multi-environment GLAMP into the asymmetric GLAMP of [42] and applying a subsequence argument; the underlying convergence is cited, not reproved.
  • domain assumption The transfer learning model satisfies Assumption 2: linear model with Gaussian-design matrices X_e = X_iid_e Sigma_e^{1/2} and noise whose empirical distribution converges in 2-Wasserstein distance.
    This model is assumed throughout Section 4; the authors note that generalization to non-Gaussian designs is left for future work.
  • ad hoc to paper The denoiser moment functions E^(p) and H^(p) converge in C1 to their limits (Assumptions 3, 5, 7).
    These high-level conditions are stated to make the Onsager terms well-defined; the paper does not verify them for concrete Gaussian Lasso examples.
  • ad hoc to paper The fixed point systems (7), (15), and (23) have solutions, and the corresponding H maps satisfy contraction conditions via kappa_e delta_e < 1 and kappa_1 gamma_II < 1 (Assumptions 4, 6, 8).
    Theorems 2 through 4 evaluate risk at these assumed fixed points; uniqueness is argued via contraction constants, but no primitive verifiable conditions are supplied.

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Cite this review

Pith. "Pith review of Multi-Environment GLAMP: Approximate Message Passing for Transfer Learning with Applications to Lasso-based Estimators." pith.science (2026). https://pith.science/paper/4IBYRFPQ

@misc{pith2026250522594,
  author       = {Pith},
  title        = {Pith review of: Multi-Environment GLAMP: Approximate Message Passing for Transfer Learning with Applications to Lasso-based Estimators},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4IBYRFPQ}},
  note         = {Machine review of arXiv:2505.22594}
}
read the original abstract

Approximate Message Passing (AMP) algorithms enable precise characterization of certain classes of random objects in the high-dimensional limit, and have found widespread applications in fields such as signal processing, statistics, and communications. In this work, we introduce Multi-Environment Generalized Long AMP, a novel AMP framework that applies to transfer learning problems with multiple data sources and distribution shifts. We rigorously establish state evolution for multi-environment GLAMP. We demonstrate the utility of this framework by precisely characterizing the risk of three Lasso-based transfer learning estimators for the first time: the Stacked Lasso, the Model Averaging Estimator, and the Second Step Estimator. We also demonstrate the remarkable finite sample accuracy of our theory via extensive simulations.

Figures

Figures reproduced from arXiv: 2505.22594 by the authors.

Figure 1
Figure 1. MSE of [PITH_FULL_IMAGE:figures/full_fig_p014_1.png] view at source ↗
Figure 2
Figure 2. MSE for βˆ average. Same settings as in [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗
Figure 3
Figure 3. MSE for βˆ II. Same settings as in [PITH_FULL_IMAGE:figures/full_fig_p023_3.png] view at source ↗

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Forward citations

Cited by 1 Pith paper

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  1. On Universality of Non-Separable Approximate Message Passing Algorithms

    math.ST 2025-06 conditional novelty 8.0 of 10

    Non-separable AMP admits universal state evolution for non-Gaussian Wigner matrices when its nonlinearities are BCP-representable polynomials or BCP-approximable Lipschitz functions.

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Works this paper leans on

102 extracted references · 68 canonical work pages · cited by 1 Pith paper

  1. [1]

    Predicting with proxies: Transfer learning in high dimension

    Hamsa Bastani. Predicting with proxies: Transfer learning in high dimension. Management Science, 67(5):2964–2984, 2021

    2021

  2. [2]

    Transfer learning for high-dimensional linear regression: Prediction, estimation and minimax optimality

    Sai Li, T Tony Cai, and Hongzhe Li. Transfer learning for high-dimensional linear regression: Prediction, estimation and minimax optimality. Journal of the Royal Statistical Society Series B: Statistical Methodology, 84(1):149–173, 2022

    2022

  3. [3]

    Near-optimal linear regression under distribution shift

    Qi Lei, Wei Hu, and Jason Lee. Near-optimal linear regression under distribution shift. In International Conference on Machine Learning , pages 6164–6174. PMLR, 2021. 23

    2021

  4. [4]

    Transfer learning for nonparametric classification

    T Tony Cai and Hongji Wei. Transfer learning for nonparametric classification. The Annals of Statistics , 49(1):100–128, 2021

    2021

  5. [5]

    A class of geometric structures in transfer learning: Minimax bounds and optimality

    Xuhui Zhang, Jose Blanchet, Soumyadip Ghosh, and Mark S Squillante. A class of geometric structures in transfer learning: Minimax bounds and optimality. In International Conference on Artificial Intelligence and Statistics , pages 3794–3820. PMLR, 2022

    2022

  6. [6]

    Searching for robust associations with a multi-environment knockoff filter

    Shuangning Li, Matteo Sesia, Yaniv Romano, Emmanuel Candes, and Chiara Sabatti. Searching for robust associations with a multi-environment knockoff filter. Biometrika, 109(3): 611–629, 2022

    2022

  7. [7]

    Individual data protected integrative regression analysis of high-dimensional heterogeneous data

    Tianxi Cai, Molei Liu, and Yin Xia. Individual data protected integrative regression analysis of high-dimensional heterogeneous data. Journal of the American Statistical Association , 117 (540):2105–2119, 2022

    2022

  8. [8]

    Meta-analysis of heterogeneous data: integrative sparse regression in high-dimensions

    Subha Maity, Yuekai Sun, and Moulinath Banerjee. Meta-analysis of heterogeneous data: integrative sparse regression in high-dimensions. Journal of Machine Learning Research , 23 (198):1–50, 2022

    2022

Show all 102 references
  1. [9]

    Adaptive and robust multi-task learning

    Yaqi Duan and Kaizheng Wang. Adaptive and robust multi-task learning. The Annals of Statistics, 51(5):2015–2039, 2023

    2015

  2. [10]

    Targeting underrepresented populations in precision medicine: A federated transfer learning approach

    Sai Li, Tianxi Cai, and Rui Duan. Targeting underrepresented populations in precision medicine: A federated transfer learning approach. The annals of applied statistics , 17(4): 2970, 2023

    2023

  3. [11]

    Transfer learning for nonparametric regression: Non-asymptotic minimax analysis and adaptive procedure

    T Tony Cai and Hongming Pu. Transfer learning for nonparametric regression: Non-asymptotic minimax analysis and adaptive procedure. arXiv preprint arXiv:2401.12272 , 2024

    2024 arXiv

  4. [12]

    Statistical challenges of high-dimensional data, 2009

    Iain M Johnstone and D Michael Titterington. Statistical challenges of high-dimensional data, 2009

    2009

  5. [13]

    Message-passing algorithms for compressed sensing

    David L Donoho, Arian Maleki, and Andrea Montanari. Message-passing algorithms for compressed sensing. Proceedings of the National Academy of Sciences , 106(45):18914–18919, 2009

    2009

  6. [14]

    Optimal m-estimation in high-dimensional regression

    Derek Bean, Peter J Bickel, Noureddine El Karoui, and Bin Yu. Optimal m-estimation in high-dimensional regression. Proceedings of the National Academy of Sciences , 110(36): 14563–14568, 2013

    2013

  7. [15]

    Precise error analysis of regularized m-estimators in high dimensions

    Christos Thrampoulidis, Ehsan Abbasi, and Babak Hassibi. Precise error analysis of regularized m-estimators in high dimensions. IEEE Transactions on Information Theory , 64(8):5592–5628, 2018. 24

    2018

  8. [16]

    The likelihood ratio test in high-dimensional logistic regression is asymptotically a rescaled chi-square

    Pragya Sur, Yuxin Chen, and Emmanuel J Cand` es. The likelihood ratio test in high-dimensional logistic regression is asymptotically a rescaled chi-square. Probability theory and related fields , 175:487–558, 2019

    2019

  9. [17]

    A modern maximum-likelihood theory for high-dimensional logistic regression

    Pragya Sur and Emmanuel J Cand` es. A modern maximum-likelihood theory for high-dimensional logistic regression. Proceedings of the National Academy of Sciences , 116 (29):14516–14525, 2019

    2019

  10. [18]

    The impact of regularization on high-dimensional logistic regression

    Fariborz Salehi, Ehsan Abbasi, and Babak Hassibi. The impact of regularization on high-dimensional logistic regression. Advances in Neural Information Processing Systems , 32, 2019

    2019

  11. [19]

    The phase transition for the existence of the maximum likelihood estimate in high-dimensional logistic regression

    Emmanuel J Cand` es and Pragya Sur. The phase transition for the existence of the maximum likelihood estimate in high-dimensional logistic regression. The Annals of Statistics , 48(1): 27–42, 2020

    2020

  12. [20]

    Optimal errors and phase transitions in high-dimensional generalized linear models

    Jean Barbier, Florent Krzakala, Nicolas Macris, L´ eo Miolane, and Lenka Zdeborov´ a. Optimal errors and phase transitions in high-dimensional generalized linear models. Proceedings of the National Academy of Sciences , 116(12):5451–5460, 2019

    2019

  13. [21]

    Which bridge estimator is the best for variable selection? The Annals of Statistics , 48(5):2791 – 2823, 2020

    Shuaiwen Wang, Haolei Weng, and Arian Maleki. Which bridge estimator is the best for variable selection? The Annals of Statistics , 48(5):2791 – 2823, 2020. doi: 10.1214/ 19-AOS1906. URL https://doi.org/10.1214/19-AOS1906

    2020 doi

  14. [22]

    Approximate message passing with spectral initialization for generalized linear models

    Marco Mondelli and Ramji Venkataramanan. Approximate message passing with spectral initialization for generalized linear models. In International Conference on Artificial Intelligence and Statistics , pages 397–405. PMLR, 2021

    2021

  15. [23]

    Phase transitions in transfer learning for high-dimensional perceptrons

    Oussama Dhifallah and Yue M Lu. Phase transitions in transfer learning for high-dimensional perceptrons. Entropy, 23(4):400, 2021

    2021

  16. [24]

    The asymptotic distribution of the mle in high-dimensional logistic models: Arbitrary covariance

    Qian Zhao, Pragya Sur, and Emmanuel J Candes. The asymptotic distribution of the mle in high-dimensional logistic models: Arbitrary covariance. Bernoulli, 28(3):1835–1861, 2022

    2022

  17. [25]

    A new central limit theorem for the augmented ipw estimator: Variance inflation, cross-fit covariance and beyond

    Kuanhao Jiang, Rajarshi Mukherjee, Subhabrata Sen, and Pragya Sur. A new central limit theorem for the augmented ipw estimator: Variance inflation, cross-fit covariance and beyond. arXiv preprint arXiv:2205.10198 , 2022

    2022 arXiv

  18. [26]

    Surprises in high-dimensional ridgeless least squares interpolation

    Trevor Hastie, Andrea Montanari, Saharon Rosset, and Ryan J Tibshirani. Surprises in high-dimensional ridgeless least squares interpolation. Annals of statistics , 50(2):949, 2022

    2022

  19. [27]

    A precise high-dimensional asymptotic theory for boosting and minimum-ℓ1-norm interpolated classifiers

    Tengyuan Liang and Pragya Sur. A precise high-dimensional asymptotic theory for boosting and minimum-ℓ1-norm interpolated classifiers. The Annals of Statistics, 50(3):1669–1695, 2022. 25

    2022

  20. [28]

    The lasso with general gaussian designs with applications to hypothesis testing

    Michael Celentano, Andrea Montanari, and Yuting Wei. The lasso with general gaussian designs with applications to hypothesis testing. The Annals of Statistics , 51(5):2194–2220, 2023

    2023

  21. [29]

    Hede: Heritability estimation in high dimensions by ensembling debiased estimators

    Yanke Song, Xihong Lin, and Pragya Sur. Hede: Heritability estimation in high dimensions by ensembling debiased estimators. arXiv preprint arXiv:2406.11184 , 2024

    2024 arXiv

  22. [30]

    Roti-gcv: Generalized cross-validation for right-rotationally invariant data

    Kevin Luo, Yufan Li, and Pragya Sur. Roti-gcv: Generalized cross-validation for right-rotationally invariant data. arXiv preprint arXiv:2406.11666 , 2024

    2024

  23. [31]

    Optimal and provable calibration in high-dimensional binary classification: Angular calibration and platt scaling

    Yufan Li and Pragya Sur. Optimal and provable calibration in high-dimensional binary classification: Angular calibration and platt scaling. arXiv preprint arXiv:2502.15131 , 2025

    2025 arXiv

  24. [32]

    The lasso risk for gaussian matrices

    Mohsen Bayati and Andrea Montanari. The lasso risk for gaussian matrices. IEEE Transactions on Information Theory , 58(4):1997–2017, 2011

    1997

  25. [33]

    High dimensional robust m-estimation: Asymptotic variance via approximate message passing

    David Donoho and Andrea Montanari. High dimensional robust m-estimation: Asymptotic variance via approximate message passing. Probability Theory and Related Fields, 166:935–969, 2016

    2016

  26. [34]

    Statistical physics of inference: Thresholds and algorithms

    Lenka Zdeborov´ a and Florent Krzakala. Statistical physics of inference: Thresholds and algorithms. Advances in Physics , 65(5):453–552, 2016

    2016

  27. [35]

    A unifying tutorial on approximate message passing

    Oliver Y Feng, Ramji Venkataramanan, Cynthia Rush, Richard J Samworth, et al. A unifying tutorial on approximate message passing. Foundations and Trends® in Machine Learning, 15 (4):335–536, 2022

    2022

  28. [36]

    A friendly tutorial on mean-field spin glass techniques for non-physicists

    Andrea Montanari, Subhabrata Sen, et al. A friendly tutorial on mean-field spin glass techniques for non-physicists. Foundations and Trends® in Machine Learning , 17(1):1–173, 2024

    2024

  29. [37]

    An iterative construction of solutions of the tap equations for the sherrington–kirkpatrick model

    Erwin Bolthausen. An iterative construction of solutions of the tap equations for the sherrington–kirkpatrick model. Communications in Mathematical Physics , 325(1):333–366, 2014

    2014

  30. [38]

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

    Mohsen Bayati and Andrea Montanari. The dynamics of message passing on dense graphs, with applications to compressed sensing. IEEE Transactions on Information Theory , 57(2): 764–785, 2011

    2011

  31. [39]

    Generalized approximate message passing for estimation with random linear mixing

    Sundeep Rangan. Generalized approximate message passing for estimation with random linear mixing. In 2011 IEEE International Symposium on Information Theory Proceedings , pages 2168–2172. IEEE, 2011. 26

    2011

  32. [40]

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

    Adel Javanmard and Andrea Montanari. State evolution for general approximate message passing algorithms, with applications to spatial coupling. Information and Inference: A Journal of the IMA , 2(2):115–144, 2013

    2013

  33. [41]

    State evolution for approximate message passing with non-separable functions

    Raphael Berthier, Andrea Montanari, and Phan-Minh Nguyen. State evolution for approximate message passing with non-separable functions. Information and Inference: A Journal of the IMA, 9(1):33–79, 2020

    2020

  34. [42]

    Graph-based approximate message passing iterations

    C´ edric Gerbelot and Rapha¨ el Berthier. Graph-based approximate message passing iterations. Information and Inference: A Journal of the IMA , 12(4):2562–2628, 2023

    2023

  35. [43]

    Solution of’solvable model of a spin glass’

    David J Thouless, Philip W Anderson, and Robert G Palmer. Solution of’solvable model of a spin glass’. Philosophical Magazine, 35(3):593–601, 1977

    1977

  36. [44]

    Graphical models concepts in compressed sensing

    Andrea Montanari, YC Eldar, and G Kutyniok. Graphical models concepts in compressed sensing. Compressed Sensing, pages 394–438, 2012

    2012

  37. [45]

    Estimating lasso risk and noise level

    Mohsen Bayati, Murat A Erdogdu, and Andrea Montanari. Estimating lasso risk and noise level. Advances in neural information processing systems , 26, 2013

    2013

  38. [46]

    Non-negative principal component analysis: Message passing algorithms and sharp asymptotics

    Andrea Montanari and Emile Richard. Non-negative principal component analysis: Message passing algorithms and sharp asymptotics. IEEE Transactions on Information Theory , 62(3): 1458–1484, 2015

    2015

  39. [47]

    Asymptotics of map inference in deep networks

    Parthe Pandit, Mojtaba Sahraee, Sundeep Rangan, and Alyson K Fletcher. Asymptotics of map inference in deep networks. In 2019 IEEE International Symposium on Information Theory (ISIT), pages 842–846. IEEE, 2019

    2019

  40. [48]

    Approximate message-passing decoder and capacity achieving sparse superposition codes

    Jean Barbier and Florent Krzakala. Approximate message-passing decoder and capacity achieving sparse superposition codes. IEEE Transactions on Information Theory , 63(8): 4894–4927, 2017

    2017

  41. [49]

    Vector approximate message passing

    Sundeep Rangan, Philip Schniter, and Alyson K Fletcher. Vector approximate message passing. IEEE Transactions on Information Theory , 65(10):6664–6684, 2019

    2019

  42. [50]

    Orthogonal amp

    Junjie Ma and Li Ping. Orthogonal amp. IEEE Access, 5:2020–2033, 2017

    2020

  43. [51]

    Approximate message passing algorithms for rotationally invariant matrices

    Zhou Fan. Approximate message passing algorithms for rotationally invariant matrices. The Annals of Statistics , 50(1):197–224, 2022

    2022

  44. [52]

    Finite sample analysis of approximate message passing algorithms

    Cynthia Rush and Ramji Venkataramanan. Finite sample analysis of approximate message passing algorithms. IEEE Transactions on Information Theory , 64(11):7264–7286, 2018. 27

    2018

  45. [53]

    A non-asymptotic framework for approximate message passing in spiked models

    Gen Li and Yuting Wei. A non-asymptotic framework for approximate message passing in spiked models. arXiv preprint arXiv:2208.03313 , 2022

    2022 arXiv

  46. [54]

    Transfusion: Covariate-shift robust transfer learning for high-dimensional regression

    Zelin He, Ying Sun, and Runze Li. Transfusion: Covariate-shift robust transfer learning for high-dimensional regression. In International Conference on Artificial Intelligence and Statistics, pages 703–711. PMLR, 2024

    2024

  47. [55]

    Residual importance weighted transfer learning for high-dimensional linear regression

    Junlong Zhao, Shengbin Zheng, and Chenlei Leng. Residual importance weighted transfer learning for high-dimensional linear regression. arXiv preprint arXiv:2311.07972 , 2023

    2023 arXiv

  48. [56]

    Algorithmic analysis and statistical estimation of slope via approximate message passing

    Zhiqi Bu, Jason M Klusowski, Cynthia Rush, and Weijie J Su. Algorithmic analysis and statistical estimation of slope via approximate message passing. IEEE Transactions on Information Theory, 67(1):506–537, 2020

    2020

  49. [57]

    Asymptotic statistical analysis of sparse group lasso via approximate message passing algorithm

    Kan Chen, Zhiqi Bu, and Shiyun Xu. Asymptotic statistical analysis of sparse group lasso via approximate message passing algorithm. arXiv preprint arXiv:2107.01266 , 2021

    2021 arXiv

  50. [58]

    Chi-square and normal inference in high-dimensional multi-task regression

    Pierre C Bellec and Gabriel Romon. Chi-square and normal inference in high-dimensional multi-task regression. arXiv preprint arXiv:2107.07828 , 2021

    2021 arXiv

  51. [59]

    Analysis of information transfer from heterogeneous sources via precise high-dimensional asymptotics

    Fan Yang, Hongyang R Zhang, Sen Wu, Weijie J Su, and Christopher R´ e. Analysis of information transfer from heterogeneous sources via precise high-dimensional asymptotics. arXiv preprint arXiv:2010.11750 , 2020

    2010 arXiv

  52. [60]

    Covariate shift in high-dimensional random feature regression

    Nilesh Tripuraneni, Ben Adlam, and Jeffrey Pennington. Covariate shift in high-dimensional random feature regression. arXiv preprint arXiv:2111.08234 , 2021

    2021 arXiv

  53. [61]

    Generalization error of min-norm interpolators in transfer learning

    Yanke Song, Sohom Bhattacharya, and Pragya Sur. Generalization error of min-norm interpolators in transfer learning. arXiv preprint arXiv:2406.13944 , 2024

    2024 arXiv

  54. [62]

    Optimal ridge regularization for out-of-distribution prediction

    Pratik Patil, Jin-Hong Du, and Ryan J Tibshirani. Optimal ridge regularization for out-of-distribution prediction. arXiv preprint arXiv:2404.01233 , 2024

    2024 arXiv

  55. [63]

    Minimum-norm interpolation under covariate shift

    Neil Mallinar, Austin Zane, Spencer Frei, and Bin Yu. Minimum-norm interpolation under covariate shift. arXiv preprint arXiv:2404.00522 , 2024

    2024 arXiv

  56. [64]

    Predictive inference in multi-environment scenarios

    John C Duchi, Suyash Gupta, Kuanhao Jiang, and Pragya Sur. Predictive inference in multi-environment scenarios. arXiv preprint arXiv:2403.16336 , 2024

    2024 arXiv

  57. [65]

    The adaptive lasso and its oracle properties

    Hui Zou. The adaptive lasso and its oracle properties. Journal of the American statistical association, 101(476):1418–1429, 2006

    2006

  58. [66]

    Universality in polytope phase transitions and message passing algorithms

    Mohsen Bayati, Marc Lelarge, and Andrea Montanari. Universality in polytope phase transitions and message passing algorithms. The Annals of Applied Probability , 25(2):753 – 822, 2015. doi: 10.1214/14-AAP1010. URL https://doi.org/10.1214/14-AAP1010. 28

    2015 doi

  59. [67]

    Universality of approximate message passing algorithms

    Wei-Kuo Chen and Wai-Kit Lam. Universality of approximate message passing algorithms. 2021

    2021

  60. [68]

    Universality laws for high-dimensional learning with random features

    Hong Hu and Yue M Lu. Universality laws for high-dimensional learning with random features. IEEE Transactions on Information Theory , 69(3):1932–1964, 2022

    1932

  61. [69]

    Universality of empirical risk minimization

    Andrea Montanari and Basil N Saeed. Universality of empirical risk minimization. In Conference on Learning Theory, pages 4310–4312. PMLR, 2022

    2022

  62. [70]

    Lu, and Subhabrata Sen

    Rishabh Dudeja, Yue M. Lu, and Subhabrata Sen. Universality of approximate message passing with semirandom matrices. The Annals of Probability , 51(5):1616–1683, 2023

    2023

  63. [71]

    Spectrum-aware debiasing: A modern inference framework with applications to principal components regression

    Yufan Li and Pragya Sur. Spectrum-aware debiasing: A modern inference framework with applications to principal components regression. arXiv preprint arXiv:2309.07810 , 2023

    2023

  64. [72]

    Universality of approximate message passing algorithms and tensor networks

    Tianhao Wang, Xinyi Zhong, and Zhou Fan. Universality of approximate message passing algorithms and tensor networks. The Annals of Applied Probability , 34(4):3943–3994, 2024

    2024

  65. [73]

    Universality in block dependent linear models with applications to nonlinear regression

    Samriddha Lahiry and Pragya Sur. Universality in block dependent linear models with applications to nonlinear regression. IEEE Transactions on Information Theory , 2024

    2024

  66. [74]

    Universality in transfer learning for linear models

    Reza Ghane, Danil Akhtiamov, and Babak Hassibi. Universality in transfer learning for linear models. Advances in Neural Information Processing Systems , 37:125729–125779, 2024

    2024

  67. [75]

    Adaptive transfer learning

    Henry WJ Reeve, Timothy I Cannings, and Richard J Samworth. Adaptive transfer learning. The Annals of Statistics , 49(6):3618–3649, 2021

    2021

  68. [76]

    A no-free-lunch theorem for multitask learning

    Steve Hanneke and Samory Kpotufe. A no-free-lunch theorem for multitask learning. The Annals of Statistics , 50(6):3119–3143, 2022

    2022

  69. [77]

    Estimation and inference for high-dimensional generalized linear models with knowledge transfer

    Sai Li, Linjun Zhang, T Tony Cai, and Hongzhe Li. Estimation and inference for high-dimensional generalized linear models with knowledge transfer. Journal of the American Statistical Association, 119(546):1274–1285, 2024

    2024

  70. [78]

    Transfer learning under high-dimensional generalized linear models

    Ye Tian and Yang Feng. Transfer learning under high-dimensional generalized linear models. Journal of the American Statistical Association , 118(544):2684–2697, 2023

    2023

  71. [79]

    A linear adjustment-based approach to posterior drift in transfer learning

    Subha Maity, Diptavo Dutta, Jonathan Terhorst, Yuekai Sun, and Moulinath Banerjee. A linear adjustment-based approach to posterior drift in transfer learning. Biometrika, 111(1): 31–50, 2024

    2024

  72. [80]

    Inequalities for the trace of matrix product

    Yuguang Fang, Kenneth A Loparo, and Xiangbo Feng. Inequalities for the trace of matrix product. IEEE Transactions on Automatic Control , 39(12):2489–2490, 1994. 29

    1994

  73. [81]

    Lasso risk and phase transition under dependence

    Hanwen Huang. Lasso risk and phase transition under dependence. Electronic Journal of Statistics, 16(2):6512–6552, 2022

    2022

  74. [82]

    Limit of the smallest eigenvalue of a large dimensional sample covariance matrix

    Zhi-Dong Bai and Yong-Qua Yin. Limit of the smallest eigenvalue of a large dimensional sample covariance matrix. In Advances In Statistics , pages 108–127. World Scientific, 2008

    2008

  75. [83]

    Matrix-valued symmetric AMP iterations with non-separable non-linearities

    Roman Vershynin. Introduction to the non-asymptotic analysis of random matrices. arXiv preprint arXiv:1011.3027, 2010. 30 A Proof of Theorem 1 A.1 Preliminaries: Symmetric and Asymmetric GLAMP Our proof builds upon the symmetric GLAMP algorithm from [42]. We first present it b...

    2010 arXiv

  76. [84]

    Lemma F.5) applies just like in the proof of Lemma 3.2, Bayati and Montanari [32]

    At any finite p, ker(X iid e Σ1/2 e ) is still a uniformly random space, so Kashin’s theorem (c.f. Lemma F.5) applies just like in the proof of Lemma 3.2, Bayati and Montanari [32]. For some constant B1, the relationship below holds almost surely as p → ∞, 1 p ∥ ˆβstack∥2 2 ≤ ...

  77. [85]

    Σ 0 (R,e) = " limp 1 p β⊤ e Σeβe limp 1 p E[β⊤ e Σeη] limp 1 p E[β⊤ e Σeη] lim p 1 p E[η⊤Σeη] #

  78. [86]

    Simplifying it with Assumption 3 and 4, Σ1 (V,e) = E[W 2 e ] + κe limp 1 p E[∥η − βe∥2 Σe] = (τ ∗ e )2

    Σ 1 (V,e) = limp 1 ne E we + (Z0 (R,e))1 − (Z0 (R,e))2 2 2 where Z0 (R,e) = [(Z0 (R,e))1|(Z0 (R,e))2] ∈ Rp×2 has iid rows drawn from N (0, κeΣ0 (R,e)). Simplifying it with Assumption 3 and 4, Σ1 (V,e) = E[W 2 e ] + κe limp 1 p E[∥η − βe∥2 Σe] = (τ ∗ e )2

  79. [87]

    Σ 1 (R,e) =   limp 1 p β⊤ e Σeβe limp 1 p E[β⊤ e Σeη] lim p 1 p E[β⊤ e Σeη] limp 1 p E[β⊤ e Σeη] lim p 1 p E[η⊤Σeη] lim p 1 p E h η⊤ (1)Σeη(2) i limp 1 p E[β⊤ e Σeη] lim p 1 p E h η⊤ (1)Σeη(2) i limp 1 p E[η⊤Σeη],   where η(1) and η(2) are mutually independent copies of η

  80. [88]

    For the diagnoal elements of Σ 2 (V,e) we have simplified it in the same way as Σ 1 (V,e)

    Σ 2 (V,e) = " (τ ∗ e )2 (τ ∗ e )2He(⃗0E) (τ ∗ e )2He(⃗0E) ( τ ∗ E)2 # . For the diagnoal elements of Σ 2 (V,e) we have simplified it in the same way as Σ 1 (V,e). For the off-diagonal term, it is lim p 1 ne E h (we + (Z0 (R,e))1 − (Z0 (R,e))2)⊤(we + (Z0 (R,e))1 − (Z0 (R,e))3) ...

  81. [89]

    We plan to show ρt,t+1 e converges exponentially fast to 1 for each e ∈ [E] with an argument of fixed point iteration

    Using Lemma C.2, we know plimp→∞ 1 p vt+1 e − vt e 2 2 = 2(τ ∗ e )2(1 − ρt,t+1 e ). We plan to show ρt,t+1 e converges exponentially fast to 1 for each e ∈ [E] with an argument of fixed point iteration. By the multi-variate mean-value theorem, for any ρ(1) ∈ [0, 1]E and ρ(2) =...

  82. [90]

    We use the first line of Equation (C-23) to bound 1 p ∥∆η(2)∥2

    ≤ 8ϵ2 · c2 2c2 3/M, (C-24) and we are left to bound 1 p ∥∆η(2)∥2 2. We use the first line of Equation (C-23) to bound 1 p ∥∆η(2)∥2

  83. [91]

    Since ∆ η(2) = ∆ηt − ∆η(1), we know 1 p ∥diag(⃗λSc)(∆η(2))Sc∥1 − 1 p [diag(⃗λSc)st Sc]⊤(∆η(2))Sc ≤ ϵ2 · c2 2c4 3 + 4 √ 2ϵc2c3, where we have used the fact that M > 1 from (i)

    For ∆ η(1), we have 1 p [diag(⃗λSc)st Sc]⊤(∆η(1))Sc ≤ 1 p ∥diag(⃗λSc)(∆η(1))Sc∥1 ≤ 1√p ∥⃗λSc∥ ·1√p ∥∆ ˆβ(1)∥2 ≤ 2 √ 2ϵc2c3, where we have used Cauchy’s inequality and the bound on 1√p ∥⃗λ∥2 from (i) of the high-probability event. Since ∆ η(2) = ∆ηt − ∆η(1), we know 1 p ∥diag(⃗...

  84. [92]

    Σ 0 (ind,R,e) = " limp 1 p β⊤ e Σeβe limp 1 p E[β⊤ e Σeηe] limp 1 p E[β⊤ e Σeηe] lim p 1 p E[η⊤ e Σeηe] #

  85. [93]

    Simplifying it with Assumption 5 and 6, Σ1 (ind,V,e) = E[W 2 e ] + κe limp 1 p E[∥ηe − βe∥2 Σ(ind,e) ] = (τ ∗ ind,e)2

    Σ 1 (ind,V,e) = limp 1 ne E we + (Z0 (ind,R,e))1 − (Z0 (ind,R,e))2 2 2 where Z0 (ind,R,e) = [(Z0 (ind,R,e))1|(Z0 (ind,R,e))2] ∈ Rp×2 has iid rows drawn from N (0, κeΣ0 (ind,R,e)). Simplifying it with Assumption 5 and 6, Σ1 (ind,V,e) = E[W 2 e ] + κe limp 1 p E[∥ηe − βe∥2 Σ(ind...

  86. [94]

    Σ 1 (ind,R,e) =   limp 1 p β⊤ e Σeβe limp 1 p E[β⊤ e Σeηe] lim p 1 p E[β⊤ e Σeηe] limp 1 p E[β⊤ e Σeηe] lim p 1 p E[η⊤ e Σeηe] lim p 1 p E h η⊤ e,(1)Σeηe,(2) i limp 1 p E[β⊤ e Σeη] lim p 1 p E h η⊤ e,(1)Σeηe,(2) i limp 1 p E[η⊤ e Σeηe],   where ηe,(1) and ηe,(2) are mu...

  87. [95]

    For the diagnoal elements of Σ 2 (ind,V,e) we have simplified it in the same way as Σ 1 (ind,V,e)

    Σ 2 (ind,V,e) = " (τ ∗ ind,e)2 (τ ∗ ind,e)2Hind,e(0) (τ ∗ ind,e)2Hind,e(0) ( τ ∗ ind,e)2 # . For the diagnoal elements of Σ 2 (ind,V,e) we have simplified it in the same way as Σ 1 (ind,V,e). For the off-diagonal term, it is lim p 1 ne E h (we + (Z0 (ind,R,e))1 − (Z0 (ind,R,e)...

  88. [96]

    − κeδ (p) e (ye − rt−1 e ). Part II : v0 II = ζ ∗ · τ ∗ IIvt0 1 /τ ∗ 1 + p 1 − (ζ ∗)2 · τ ∗ IIZinit 1 , r0 II = X iid 1 Σ1/2 1 ξ v0 II, η(vt0 1 , · · ·, vt0 1 ) − (γIIτ ∗ IIζ ∗/τ ∗ 1 + γI) · κ1(y1 − rt0−1 1 ); ∀ t ∈ N+, vt II = X iid 1 ⊤ (y1 − rt−1 II − κ1γI(y1 − rt+t0−1 1 )) ...

  89. [97]

    We are left to verify that the state evolution is well-defined, and satisfies the marginal properties 68 in Lemma E.2

    = κ1 y1 − r0 1|y1 − r1 1| · · · |y1 − rt 1|0| · · · ∥, y1 − r0 II − κ1γI(y1 − rt0 1 )| · · · |y1 − rt−t0 II − κ1γI(y1 − rt 1)|0| · · · , ∀ e >1, Ψt e(Rt e) = κe ye − r0 e |ye − r1 e | · · · |ye − rt e|0| · · · . We are left to verify that the state evolution is well-defined, a...

  90. [98]

    71 By the law of iterated expectations, HII(1) ≥ HII(0)

    After some algebra, we have (τ ∗ 1 τ ∗ II)2HII(0) = (1 − κ1γI)2E[W 2 1 ]2 + κ1E[W 2 1 ] lim p 1 p E Σ1/2 1 β1 − ˘ξ − κ1γI(β1 − η) 2 2 + κ2 1 lim p 1 p E Σ1/2 1 β1 − ˘ξ − κ1γI(β1 − η) 2 2 lim p 1 p E Σ1/2 1 (β1 − η) 2 2 + κ1(1 − κ1γI)2E[W 2 1 ] lim p 1 p E Σ1/2 1 (β1 − η) 2 2 ,...

  91. [99]

    Then sII ∈ ∇µII(ξ; η), or in the case of the joint estimator, sII/λII ∈ ∂∥ξ − η∥1

    + (1 − κ1γI)eΣ1(β1 − η) − eΣ1(ξ − η) i . Then sII ∈ ∇µII(ξ; η), or in the case of the joint estimator, sII/λII ∈ ∂∥ξ − η∥1. Define eξt = ξt − ˆβstack+ eθ∗ IIst II/λII. Recall we have defined St II(c) = {j ∈ [p] : |(st II)j| ≥λII(1−c)}. Since st II/λII ∈ ∂∥ξt − ˆβstack∥1, St II...

  92. [100]

    We write ( ¯˜ξ)j = hII,j(Z′ 1,j), where hII,j(·) also implicitly depends on Z1, · · ·, ZE and P ⊥ 1,jZ′ 1, P ⊥ 1,j being the projection matrix onto the orthogonal complement of span( eσ1,j). Note that these other Gaussian variables hII,j(·) implicitly depends on are all indepe...

  93. [101]

    (E-38) We are left to bound1 p ∥∆ξ(2)∥2

  94. [102]

    1 + ∥X∥2 2 N k−1 + ∥Y ∥2 2 N k−1# ≤ 3qk−1

    We utilize the first line of Eqn (E-37). For ∆ξ(1), 1 p [(st II)Sc II ]⊤(∆ξ(1))Sc II ≤ 80 1 p ∥(∆ξ(1))Sc II ∥1 ≤ 1√p ∥∆ξ(1)∥2 ≤ 2 √ 2ϵc2c3. Since ∆ ξ(2) = ∆ξt − ∆ξ(1), we know 1 p ∥(∆ξ(2))Sc II ∥1 − 1 p [(st II)Sc]⊤(∆ξ(2))Sc II ≤ ϵ2 · (c2c3)2/λII + 4 √ 2ϵc2c3. Recall that St I...

    1977

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.