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REVIEW 4 major objections 4 minor 40 references

Auxiliary Learning and its Statistical Understanding

T0 review · 4 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Auxiliary learning improves primary regression whenever the task coefficient matrix has rank below the number of tasks; the paper derives the optimal weight and proves the feasible estimator attains it.

desk verdict Correct oracle analysis and a plausible Theorem 1, but Corollary 1 does not follow from the stated rates and the real-data analysis leaks test information. read the letter →

arxiv 2501.03463 v1 pith:BS7YUFUV submitted 2025-01-07 math.ST stat.TH

Hanchao Yan , Feifei Wang , Chuanxin Xia , Hansheng Wang This is my paper · ORCID
classification math.STstat.TH MSC 62J0562F12
keywords auxiliarylearningmulti-taskhigh-dimensionallinearregressionlow-rankcoefficientmatrixweightedestimatorordinaryleastsquaresasymptoticefficiencysmartvendingmachines
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 claims that the statistical efficiency of a primary linear regression estimator can be improved by pooling ordinary least squares estimates from K auxiliary tasks that share the same covariates, provided the (p × (K+1)) matrix B of true task coefficients has rank d < K+1. It derives an optimal weight w* that minimizes the noise factor w^T Σ_ε w among all weights w satisfying Bw = β(0), gives its closed form, and constructs a feasible estimator that estimates both the weight and the noise covariance from data. The main theorem shows that the feasible estimator is within O_p(√((K+d)/N) + p/N) of the oracle weighted estimator with probability at least 1 − C2 exp(−K), and the corollary gives its asymptotic normality with variance no larger than the primary-only OLS variance. A sympathetic reader would care because this is a rare case where auxiliary or multi-task learning is shown to have a concrete statistical payoff with a closed-form weight formula, not just a heuristic loss-balancing scheme. Simulations on linear and logistic models and a smart-vending-machine image classification application support the theoretical claims.

What carries the argument

The load-bearing mechanism is the right null space of the true coefficient matrix B. The paper assumes rank(B)=d<K+1, so the eigenvectors of B^T B associated with the K+1−d smallest eigenvalues span a null-space matrix Θ, and every weight that preserves the primary coefficient is w = e1 + Θu. The optimal u then comes from a finite-dimensional quadratic minimization: minimize u ↦ (e1+Θu)^T Σ_ε(e1+Θu), whose solution is the closed-form projection w* = e1 − Θ(Θ^TΣ_εΘ)^{−1}Θ^TΣ_εe1. The feasible estimator replaces Θ by the corresponding eigenvectors of bB^T bB and Σ_ε by the residual covariance matrix; Theorem 1 controls the error from those plug-in estimates using sub-Gaussian concentration, and Corollary 1 transfers the oracle's asymptotic distribution to the feasible version.

What would settle it

Under the conditions of Corollary 1, build Wald intervals for α^Tβ^(0) from the feasible estimator; if their empirical coverage is far below nominal as N grows, the asymptotic normality claim is wrong.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that when B = (β^(0),...,β^(K)) is rank-deficient, the feasible weighted estimator bβ_{bw*} = bB bw* reproduces the oracle estimator's first-order behavior. The oracle weight is w* = e1 − Θ(Θ^T Σ_ε Θ)^{−1} Θ^T Σ_ε e1, where Θ spans the right null space of B; this is the weight in the feasible set W = {e1 + Θu : u ∈ $R^{{K−d+1}}$} that minimizes the scalar noise factor w^TΣ_εw. Because e1 is always feasible, P(w*) ≤ σ_0², and therefore Corollary 1 states that √N α^T(bβ_{bw*} − β(0)) converges to a normal distribution with variance at most σ_0² α^$TΣ_xx^{{−1}}$α. In plain terms, auxiliary tasks never hurt asymptotically, and they help whenever the low-rank structure gives the weight more freedom than the trivial e1 choice.

Load-bearing premise

The method's advantage exists only when the matrix of true regression coefficients across primary and auxiliary tasks has rank strictly below the number of tasks plus one, and the user knows that rank; a full-rank matrix or a wrongly chosen rank leaves the method with no gain or with bias.

Editorial extensions

If this is right

  • In the rank-deficient regime d<K+1, the feasible estimator is first-order equivalent to the oracle: its asymptotic variance is P(w*) α^TΣ_xx^{−1}α, which is no larger than the primary-only OLS variance σ_0² α^TΣ_xx^{−1}α.
  • The gap between the feasible and oracle estimators vanishes at rate √((K+d)/N)+p/N, so uncertainty about the weight and noise covariance does not change the leading-order distribution.
  • If B has full rank K+1, the feasible set collapses to {e1}, the estimated weight equals e1, and the method reduces to OLS; auxiliary tasks then provide no efficiency gain.
  • With fixed rank d, increasing the number of auxiliary tasks K expands the feasible set and can only lower P(w*), so additional tasks improve the oracle efficiency as long as K/p→0; simulations show this benefit, while adding tasks outside the low-rank span produces a U-shaped MSE pattern.

Reading between the lines

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

  • The same weight formula would apply to any set of asymptotically normal per-task estimators whose joint covariance is estimable, so the argument is not intrinsically tied to OLS; the paper itself only verifies the logistic case empirically, leaving a theory gap.
  • Because the theorem assumes d is known while the real-data analysis selects d by cross-validation, the reported confidence intervals may in practice undercover; a bias correction or a theory for estimated rank would be needed.
  • If the low-rank condition holds only approximately, one can expect the method to remain approximately beneficial when the smallest nonzero singular value of B is not too small; the current proof does not quantify that regime.
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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

4 major / 4 minor

Summary. The paper proposes a weighted OLS estimator for a primary linear regression that borrows strength from auxiliary tasks sharing the same covariates. Under a low-rank assumption on the combined coefficient matrix B, the set of weight vectors that keep the weighted estimator unbiased is characterized as an affine subspace, and the oracle weight minimizing the noise variance is derived in closed form. The authors state a non-asymptotic bound on the distance between the feasible estimator (which plugs in estimated eigenvectors and residual covariances) and the oracle estimator (Theorem 1), and then claim an asymptotic normality result for the feasible estimator with the same variance as the oracle (Corollary 1). Simulations under linear and logistic models, plus a deep-learning real-data example on smart vending machines, are presented to support the method.

Significance. If the theoretical claims were rigorously established, the paper would offer a simple, interpretable way to exploit auxiliary tasks: the oracle weight (2.3) is elegantly derived, the low-rank assumption is a natural structural condition, and numerical experiments suggest practical gains. The paper also makes a useful connection between auxiliary learning and classical least-squares theory. However, the central asymptotic equivalence between the feasible and oracle estimators is not proven by the arguments in the text, and the real-data implementation is not precisely linked to the theoretical model. These gaps currently prevent the paper from fully delivering on its main claims.

major comments (4)
  1. [Section 2.3, Corollary 1] The derivation of Corollary 1 is not valid as written. Theorem 1 gives ||β̂_bŵ* − β̂_w*||_2 = O_p(√((K+d)/N) + p/N). For a fixed nonzero linear functional α, the √N-scaled difference satisfies √N α'(β̂_bŵ* − β̂_w*) = O_p(√(K+d) + p/√N). Under Condition (C4), K may diverge, so this quantity need not be o_p(1). The text compares the unscaled bound with α'(β̂_w* − β(0)) = O_p(√(p/N)), but a fixed linear combination of the oracle estimator's error is actually O_p(1/√N), not O_p(√(p/N)). Thus the Slutsky-type argument in the paragraph before Corollary 1 does not establish the claimed CLT, and the variance-reduction claim P(w*) ≤ σ₀² is not proved for the feasible estimator. The authors need to supply a direct proof of Corollary 1, or state and prove the stronger rate conditions under which it holds, and reconcile those conditions with their simulation settings.
  2. [Section 4.3 and 4.4] The real-data analysis does not specify how the weighted estimator is computed from the DenseNet softmax outputs. The theory in Section 2 applies to OLS estimators with a common covariate vector X and a residual covariance Σ_ε. For a multi-class softmax model, it is unclear what bB, bΣ_xx, and bΣ_ε represent, and what the 'rank' of the coefficient matrix means for the final fully connected layer. The authors should spell out the exact estimation steps, or state clearly that the deep-learning application is a heuristic extension that is not covered by Theorem 1 and Corollary 1.
  3. [Condition (C3) and Section 4.4] Condition (C3) assumes the rank d of B is known and fixed, but Section 4.4 selects d by cross-validation (d = 51). The auxiliary task set S_opt is also selected by a search procedure. The theoretical results do not cover data-adaptive choices of d or S_opt, so the reported error rates and standard deviations in the real-data experiment are not directly supported by Corollary 1. The paper should acknowledge this gap or provide additional theory for the selection step.
  4. [Theorem 1] The proof of Theorem 1 is relegated to an appendix that is not included in the extracted text provided for review. Since Theorem 1 is the foundation for the subsequent asymptotic claims, the authors must provide the complete proof in the revised version so that the non-asymptotic bound in Eq. (2.4) can be verified.
minor comments (4)
  1. [Section 2.2, Remark 1] The expression for w*_k is written in a somewhat confusing way; it would be clearer to write w*_k = (1/σ_k²) / (∑_{k'=0}^K 1/σ_{k'}^2), which is what the current formula reduces to.
  2. [Section 3.1] The condition defining Φ2 is ambiguous: 'φ(2)_ij = 0 if i ≤ j ≤ i + K − d + 1 and φ(2)_ij = 1 otherwise'. Please clarify the intended band structure, since d < K+1 in general and the range of j depends on i.
  3. [Section 4.4, Figure 9] The left panel of Figure 9 plots AvgErr for d in [0,71] but the text focuses on d ∈ [40,71]; the axis label and the range should be made consistent, and the location of d=51 should be marked.
  4. [Introduction and Condition (C4)] The paper uses the term 'high-dimensional' broadly, but the theory requires p/N → 0, so p is smaller than N in the asymptotic regime. The title and abstract should be qualified to avoid overstating the range of applicability, particularly because the introduction mentions settings with p > N.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the feasible estimator plugs independently estimated covariance and eigenstructure quantities into an analytically derived oracle weight, and no fitted constant or self-citation creates the headline result.

full rationale

The paper's derivation chain is self-contained and does not reduce to its inputs by construction. The oracle weight w* is obtained by minimizing P(w) = w'Σ_εw over the feasible set W = {e1 + Θu}, which is a well-defined optimization problem under the low-rank assumption rank(B) = d; the feasible version replaces Σ_ε and Θ with bΣ_ε and bΘ obtained from the data, and Theorem 1 bounds the feasible-oracle discrepancy using concentration and eigenvector perturbation arguments. The claim that the weighted estimator has variance no larger than OLS follows from e1 ∈ W and w* being the minimizer of P(w); that is the defining property of the optimal weight, not a fitted parameter renamed as a prediction. The self-citations in the paper (Wang 2009, 2012; Wang et al. 2021; Zhu et al. 2022) are contextual literature reviews or standard technical references, and they are not load-bearing for the main theorem. The rank d is assumed known in the theory and is selected by cross-validation in the application, so misspecification of d is a robustness or correctness concern, not circularity. The skeptic's objection that Corollary 1 may not follow from Theorem 1 under Condition (C4) is a logical-validity issue: the unscaled bound in Theorem 1 does not automatically become o_p(1) after √N normalization when K grows, but that is not a circularity pattern. No uniqueness theorem, ansatz, or known result is imported from the authors' prior work and then presented as an independent derivation. Therefore no circular step is present; the appropriate finding is score 0.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The central derivation is self-contained given standard linear-model and low-rank assumptions. No new physical or mathematical entities are introduced. The main data-dependent choices are the rank and task set in the deep-learning application, which are selected using test error and are not covered by the theorem.

free parameters (2)
  • rank d of coefficient matrix B = 51 (selected by cross-validation in the real data); fixed in simulations
    Assumed known in Theorem 1. In Section 4.4, d is chosen over [1,71] by minimizing average test error, so the final efficiency claim does not account for this selection.
  • auxiliary task set Sopt = Sopt = {1,2,3,4}
    Selected by backward elimination using test error. The theory assumes the set of K auxiliary tasks is fixed and given.
assumptions (5)
  • domain assumption Linear models (2.1) and (2.2) hold for the primary and all auxiliary responses with the same covariate vector X_i and independent errors with mean zero.
    Section 2.1. The unbiasedness and covariance calculations for the OLS estimators rest on this model.
  • domain assumption rank(B)=d<K+1 with λ_d(B'B) bounded away from zero and λ_1(B'B) bounded above (Condition C3).
    The feasible set W is non-trivial only under this low-rank condition. If B is full rank, W={e1} and auxiliary tasks provide no benefit.
  • standard math X_i and ε_i are sub-Gaussian with bounded sub-Gaussian norm, and Σ_xx and Σ_ε have eigenvalues bounded away from 0 and infinity (Conditions C1 and C2).
    Used for concentration inequalities and the asymptotic normality result; standard in high-dimensional statistics.
  • domain assumption p/N→0 and K/p→0 as N→∞ (Condition C4).
    Needed for OLS invertibility and for the eigenvector perturbation to be negligible. This excludes the p>N motivating case.
  • ad hoc to paper In the vending-machine example, the DenseNet feature representation and softmax heads fit the same linear low-rank structure with one response column per task.
    Section 4 never derives this equivalence explicitly; the theoretical weighted estimator is not concretely computed from the CNN outputs.

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Cite this review

Pith. "Pith review of Auxiliary Learning and its Statistical Understanding." pith.science (2026). https://pith.science/paper/BS7YUFUV

@misc{pith2026250103463,
  author       = {Pith},
  title        = {Pith review of: Auxiliary Learning and its Statistical Understanding},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BS7YUFUV}},
  note         = {Machine review of arXiv:2501.03463}
}
read the original abstract

Modern statistical analysis often encounters high-dimensional problems but with a limited sample size. It poses great challenges to traditional statistical estimation methods. In this work, we adopt auxiliary learning to solve the estimation problem in high-dimensional settings. We start with the linear regression setup. To improve the statistical efficiency of the parameter estimator for the primary task, we consider several auxiliary tasks, which share the same covariates with the primary task. Then a weighted estimator for the primary task is developed, which is a linear combination of the ordinary least squares estimators of both the primary task and auxiliary tasks. The optimal weight is analytically derived and the statistical properties of the corresponding weighted estimator are studied. We then extend the weighted estimator to generalized linear regression models. Extensive numerical experiments are conducted to verify our theoretical results. Last, a deep learning-related real-data example of smart vending machines is presented for illustration purposes.

Figures

Figures reproduced from arXiv: 2501.03463 by the authors.

Figure 1
Figure 1. Boxplots of log-transformed MSE values of the OLS, ORACLE, and FEASIBLE [PITH_FULL_IMAGE:figures/full_fig_p015_1.png] view at source ↗
Figure 2
Figure 2. Boxplots of MSE values of the OLS, ORACLE, and FEASIBLE estimators with [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗
Figure 3
Figure 3. Boxplots of MSE values of the OLS, ORACLE, and FEASIBLE estimators with [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: Boxplots of MSE values of FEASIBLE estimators using different numbers of [PITH_FULL_IMAGE:figures/full_fig_p019_4.png]
Figure 5
Figure 5. Figure 5: Boxplots of MSE values of the MLE, ORACLE, and FEASIBLE estimators for [PITH_FULL_IMAGE:figures/full_fig_p020_5.png]
Figure 6
Figure 6. Figure 6: The left panel is one smart vending machine and the right panel shows 45 types [PITH_FULL_IMAGE:figures/full_fig_p021_6.png]
Figure 7
Figure 7. Figure 7: The illustration of two specific products. The product on the left panel is with [PITH_FULL_IMAGE:figures/full_fig_p023_7.png]
Figure 8
Figure 8. Figure 8: The detailed classification categories of the five auxiliary tasks. Each item has a [PITH_FULL_IMAGE:figures/full_fig_p024_8.png]
Figure 9
Figure 9. Figure 9: The detailed results of AvgErr and StdErr (in logarithm) under different ranks. [PITH_FULL_IMAGE:figures/full_fig_p027_9.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

40 extracted references · 30 canonical work pages

  1. [1]

    Chen, H., X. Wang, C. Guan, Y. Liu, and W. Zhu (2022). Auxiliary learning with joint task and data scheduling. In International Conference on Machine Learning , pp.\ 3634--3647. PMLR

    work page 2022

  2. [2]

    Chen, H., X. Wang, Y. Liu, Y. Zhou, C. Guan, and W. Zhu (2022). Module-aware optimization for auxiliary learning. In 36th Conference on Neural Information Processing Systems (NeurIPS 2022) , pp.\ 145--152

    work page 2022

  3. [3]

    Badrinarayanan, C.-Y

    Chen, Z., V. Badrinarayanan, C.-Y. Lee, and A. Rabinovich (2018). Gradnorm: Gradient normalization for adaptive loss balancing in deep multitask networks. In International conference on machine learning , pp.\ 794--803. PMLR

    work page 2018

  4. [4]

    Fan, and J

    Fan, J., Y. Fan, and J. Lv (2008). High dimensional covariance matrix estimation using a factor model. Journal of Econometrics\/ 147\/ (1), 186--197

    work page 2008

  5. [5]

    Fan, J. and R. Li (2001). Variable selection via nonconcave penalized likelihood and its oracle properties. Journal of the American statistical Association\/ 96\/ (456), 1348--1360

    2001

  6. [6]

    Li, C.-H

    Fan, J., R. Li, C.-H. Zhang, and H. Zou (2020). Statistical foundations of data science . CRC press

    work page 2020

  7. [7]

    Lou, and M

    Fan, J., Z. Lou, and M. Yu (2024). Are latent factor regression and sparse regression adequate? Journal of the American Statistical Association\/ 119\/ (546), 1076--1088

    2024

  8. [8]

    Fan, J. and J. Lv (2008). Sure independence screening for ultrahigh dimensional feature space. Journal of the Royal Statistical Society: Series B (Statistical Methodology)\/ 70\/ (5), 849--911

    work page 2008

Show all 40 references
  1. [9]

    Gao, Y., W. Liu, H. Wang, X. Wang, Y. Yan, and R. Zhang (2022). A review of distributed statistical inference. Statistical Theory and Related Fields\/ 6\/ (2), 89--99

    2022

  2. [10]

    Howard, J. and S. Ruder (2018). Universal language model fine-tuning for text classification. arXiv preprint arXiv:1801.06146\/

    2018 arXiv

  3. [11]

    Huang, G., Z. Liu, L. Van Der Maaten, and K. Q. Weinberger (2017). Densely connected convolutional networks. In Proceedings of the IEEE conference on computer vision and pattern recognition , pp.\ 4700--4708

    2017

  4. [12]

    He, and H

    Jiang, Y., Y. He, and H. Zhang (2016). Variable selection with prior information for generalized linear models via the prior lasso method. Journal of the American Statistical Association\/ 111\/ (513), 355--376

    2016

  5. [13]

    Guo, and J

    Jiang, Z., H. Guo, and J. Wang (2023). Feature screening for multiple responses. Journal of Multivariate Analysis\/ 198 , 105223

    2023

  6. [14]

    Matsushita, and Y

    Kanezaki, A., Y. Matsushita, and Y. Nishida (2018). Rotationnet: Joint object categorization and pose estimation using multiviews from unsupervised viewpoints. In Proceedings of the IEEE conference on computer vision and pattern recognition , pp.\ 5010--5019

    2018

  7. [15]

    Gal, and R

    Kendall, A., Y. Gal, and R. Cipolla (2018). Multi-task learning using uncertainty to weigh losses for scene geometry and semantics. In Proceedings of the IEEE conference on computer vision and pattern recognition , pp.\ 7482--7491

    2018

  8. [16]

    Kokkinos, I. (2017). Ubernet: Training a universal convolutional neural network for low-, mid-, and high-level vision using diverse datasets and limited memory. In Proceedings of the IEEE conference on computer vision and pattern recognition , pp.\ 6129--6138

    2017

  9. [17]

    Yin, Y.-C

    Kung, P.-N., S.-S. Yin, Y.-C. Chen, T.-H. Yang, and Y.-N. Chen (2021, November). Efficient multi-task auxiliary learning: Selecting auxiliary data by feature similarity. In Proceedings of the 2021 Conference on Empirical Methods in Natural Language Processing , Online and Punt...

    2021

  10. [18]

    Li, B. and A. Dong (2021). Multi-task learning with attention : Constructing auxiliary tasks for learning to learn. In 2021 IEEE 33rd International Conference on Tools with Artificial Intelligence (ICTAI) , pp.\ 145--152

    2021

  11. [19]

    Zhong, and L

    Li, R., W. Zhong, and L. Zhu (2012). Feature screening via distance correlation learning. Journal of the American Statistical Association\/ 107\/ (499), 1129

    2012

  12. [20]

    Li, X., Y. Gao, H. Chang, D. Huang, Y. Ma, R. Pan, H. Qi, F. Wang, S. Wu, K. Xu, et al. (2024). A selective review on statistical methods for massive data computation: distributed computing, subsampling, and minibatch techniques. Statistical Theory and Related Fields\/ , 1--23

    2024

  13. [21]

    Liebel, L. and M. K \"o rner (2018). Auxiliary tasks in multi-task learning. arXiv preprint arXiv:1805.06334\/

    2018 arXiv

  14. [22]

    Shrivastava, A

    Misra, I., A. Shrivastava, A. Gupta, and M. Hebert (2016). Cross-stitch networks for multi-task learning. In Proceedings of the IEEE conference on computer vision and pattern recognition , pp.\ 3994--4003

    2016

  15. [23]

    Murray, J. D. (2012). Asymptotic analysis , Volume 48. Springer Science & Business Media

    2012

  16. [24]

    Achituve, H

    Navon, A., I. Achituve, H. Maron, G. Chechik, and E. Fetaya (2020). Auxiliary learning by implicit differentiation. arXiv preprint arXiv:2007.02693\/

    2020 arXiv

  17. [25]

    Gross, S

    Paszke, A., S. Gross, S. Chintala, G. Chanan, E. Yang, Z. DeVito, Z. Lin, A. Desmaison, L. Antiga, and A. Lerer (2017). Automatic differentiation in pytorch

    2017

  18. [26]

    Bingel, I

    Ruder, S., J. Bingel, I. Augenstein, and A. S gaard (2019). Latent multi-task architecture learning. In Proceedings of the AAAI Conference on Artificial Intelligence , Volume 33, pp.\ 4822--4829

    2019

  19. [27]

    Tian, Y. and Y. Feng (2023). Transfer learning under high-dimensional generalized linear models. Journal of the American Statistical Association\/ 118\/ (544), 2684--2697

    2023

  20. [28]

    Tibshirani, R. (1996). Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology\/ 58\/ (1), 267--288

    1996

  21. [29]

    Van der Vaart, A. W. (2000). Asymptotic statistics , Volume 3. Cambridge university press

    2000

  22. [30]

    Georgoulis, W

    Vandenhende, S., S. Georgoulis, W. Van Gansbeke, M. Proesmans, D. Dai, and L. Van Gool (2021). Multi-task learning for dense prediction tasks: A survey. IEEE transactions on pattern analysis and machine intelligence\/ 44\/ (7), 3614--3633

    2021

  23. [31]

    Vershynin, R. (2018). High-dimensional probability: An introduction with applications in data science , Volume 47. Cambridge university press

    2018

  24. [32]

    Wainwright, M. J. (2019). High-dimensional statistics: A non-asymptotic viewpoint , Volume 48. Cambridge university press

    2019

  25. [33]

    Liu, and H

    Wang, F., J. Liu, and H. Wang (2021). Sequential text-term selection in vector space models. Journal of Business & Economic Statistics\/ 39\/ (1), 82--97

    2021

  26. [34]

    Wang, H. (2009). Forward regression for ultra-high dimensional variable screening. Journal of the American Statistical Association\/ 104\/ (488), 1512--1524

    2009

  27. [35]

    Wang, H. (2012). Factor profiled sure independence screening. Biometrika\/ 99\/ (1), 15--28

    2012

  28. [36]

    Wang, X., T. Li, J. Chen, S. Zong, and Z. Kai (2022). Design and implementation of intelligent unmanned vending container system. Modern Electronics Technique\/ 6 , 163--168,

    2022

  29. [37]

    Yuan, M. and Y. Lin (2005). Model selection and estimation in regression with grouped variables. Journal of the Royal Statistical Society Series B: Statistical Methodology\/ 68\/ (1), 49--67

    2005

  30. [38]

    Huang, R

    Zhu, X., D. Huang, R. Pan, and H. Wang (2020). Multivariate spatial autoregressive model for large scale social networks. Journal of Econometrics\/ 215\/ (2), 591--606

    2020

  31. [39]

    Zhu, X., R. Pan, S. Wu, and H. Wang (2022). Feature screening for massive data analysis by subsampling. Journal of Business & Economic Statistics\/ 40\/ (4), 1892--1903

    2022

  32. [40]

    Zou, H. (2006). The adaptive lasso and its oracle properties. Journal of the American statistical association\/ 101\/ (476), 1418--1429

    2006

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