IDEAS home Printed from https://ideas.repec.org/a/wly/jnljam/v2014y2014i1n946241.html

Weaker Regularity Conditions and Sparse Recovery in High‐Dimensional Regression

Author

Listed:
  • Shiqing Wang
  • Yan Shi
  • Limin Su

Abstract

Regularity conditions play a pivotal role for sparse recovery in high‐dimensional regression. In this paper, we present a weaker regularity condition and further discuss the relationships with other regularity conditions, such as restricted eigenvalue condition. We study the behavior of our new condition for design matrices with independent random columns uniformly drawn on the unit sphere. Moreover, the present paper shows that, under a sparsity scenario, the Lasso estimator and Dantzig selector exhibit similar behavior. Based on both methods, we derive, in parallel, more precise bounds for the estimation loss and the prediction risk in the linear regression model when the number of variables can be much larger than the sample size.

Suggested Citation

  • Shiqing Wang & Yan Shi & Limin Su, 2014. "Weaker Regularity Conditions and Sparse Recovery in High‐Dimensional Regression," Journal of Applied Mathematics, John Wiley & Sons, vol. 2014(1).
    • Handle: RePEc:wly:jnljam:v:2014:y:2014:i:1:n:946241
    DOI: 10.1155/2014/946241
    as

    Download full text from publisher

    File URL: https://doi.org/10.1155/2014/946241
    Download Restriction: no
    File URL: https://libkey.io/10.1155/2014/946241?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    References listed on IDEAS

    as
    1. Shiqing Wang & Limin Su, 2013. "The Oracle Inequalities on Simultaneous Lasso and Dantzig Selector in High-Dimensional Nonparametric Regression," Mathematical Problems in Engineering, Hindawi, vol. 2013, pages 1-6, June.
    2. Alquier, Pierre & Hebiri, Mohamed, 2011. "Generalization of ℓ1 constraints for high dimensional regression problems," Statistics & Probability Letters, Elsevier, vol. 81(12), pages 1760-1765.
    3. Jun Wang & Changfeng Ge & Dong Sun Lee & Michael A. Sek & Vanee Chonhenchob, 2013. "Mathematical Problems in Packaging Engineering," Mathematical Problems in Engineering, Hindawi, vol. 2013, pages 1-2, November.
    4. Eric Gautier & Alexandre Tsybakov, 2011. "High-Dimensional Instrumental Variables Regression and Confidence Sets," Working Papers 2011-13, Center for Research in Economics and Statistics.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Victor Chernozhukov & Denis Chetverikov & Kengo Kato, 2012. "Central limit theorems and multiplier bootstrap when p is much larger than n," CeMMAP working papers 45/12, Institute for Fiscal Studies.
    2. Yunfeng Liu & Jidong Suo & Hamid Reza Karimi & Xiaoming Liu, 2014. "A Filtering Algorithm for Maneuvering Target Tracking Based on Smoothing Spline Fitting," Abstract and Applied Analysis, John Wiley & Sons, vol. 2014(1).
    3. Hansen, Christian & Liao, Yuan, 2019. "The Factor-Lasso And K-Step Bootstrap Approach For Inference In High-Dimensional Economic Applications," Econometric Theory, Cambridge University Press, vol. 35(3), pages 465-509, June.
    4. Alexandre Belloni & Victor Chernozhukov & Christian Hansen, 2014. "High-Dimensional Methods and Inference on Structural and Treatment Effects," Journal of Economic Perspectives, American Economic Association, vol. 28(2), pages 29-50, Spring.
    5. Hong Jiang & Jinlong Gai & Shukuan Zhao & Peggy E. Chaudhry & Sohail S. Chaudhry, 2022. "Applications and development of artificial intelligence system from the perspective of system science: A bibliometric review," Systems Research and Behavioral Science, Wiley Blackwell, vol. 39(3), pages 361-378, May.
    6. Eliana Barrenho & Eric Gautier & Marisa Miraldo & Carol Propper & Christiern Rose, 2025. "Innovation Diffusion Among Coworkers: Evidence from Senior Doctors," Management Science, INFORMS, vol. 71(10), pages 8109-8126, October.
    7. Qingliang Fan & Yaqian Wu, 2020. "Endogenous Treatment Effect Estimation with some Invalid and Irrelevant Instruments," Papers 2006.14998, arXiv.org.
    8. Michal Kolesár & Raj Chetty & John Friedman & Edward Glaeser & Guido W. Imbens, 2015. "Identification and Inference With Many Invalid Instruments," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 33(4), pages 474-484, October.
    9. Gai-Ge Wang & Lihong Guo & Amir Hossein Gandomi & Amir Hossein Alavi & Hong Duan, 2013. "Simulated Annealing‐Based Krill Herd Algorithm for Global Optimization," Abstract and Applied Analysis, John Wiley & Sons, vol. 2013(1).
    10. Victor Chernozhukov & Christian Hansen & Martin Spindler, 2015. "Post-Selection and Post-Regularization Inference in Linear Models with Many Controls and Instruments," American Economic Review, American Economic Association, vol. 105(5), pages 486-490, May.
    11. Xu Cheng & Zhipeng Liao, 2012. "Select the Valid and Relevant Moments: A One-Step Procedure for GMM with Many Moments," PIER Working Paper Archive 12-045, Penn Institute for Economic Research, Department of Economics, University of Pennsylvania.
    12. Eric Gautier & Christiern Rose, 2022. "Fast, Robust Inference for Linear Instrumental Variables Models using Self-Normalized Moments," Papers 2211.02249, arXiv.org, revised Nov 2022.
    13. Victor Chernozhukov & Denis Chetverikov & Mert Demirer & Esther Duflo & Christian Hansen & Whitney Newey & James Robins, 2018. "Double/debiased machine learning for treatment and structural parameters," Econometrics Journal, Royal Economic Society, vol. 21(1), pages 1-68, February.
    14. Alexandre Belloni & Mathieu Rosenbaum & Alexandre Tsybakov, 2016. "An {l1, l2, l-infinity} Regularization Approach to High-Dimensional Errors-in-variables Models," Working Papers 2016-12, Center for Research in Economics and Statistics.
    15. Chang, Jinyuan & Chen, Cheng & Qiao, Xinghao & Yao, Qiwei, 2023. "An autocovariance-based learning framework for high-dimensional functional time series," LSE Research Online Documents on Economics 117910, London School of Economics and Political Science, LSE Library.
    16. Qinqin Hu & Lu Lin, 2022. "Feature Screening in High Dimensional Regression with Endogenous Covariates," Computational Economics, Springer;Society for Computational Economics, vol. 60(3), pages 949-969, October.
    17. Emilio Jiménez-Macías & Angel Sánchez-Roca & Hipólito Carvajal-Fals & Julio Blanco-Fernández & Eduardo Martínez-Cámara, 2014. "Wavelets Application in Prediction of Friction Stir Welding Parameters of Alloy Joints from Vibroacoustic ANN‐Based Model," Abstract and Applied Analysis, John Wiley & Sons, vol. 2014(1).
    18. Chatterjee, A. & Gupta, S. & Lahiri, S.N., 2015. "On the residual empirical process based on the ALASSO in high dimensions and its functional oracle property," Journal of Econometrics, Elsevier, vol. 186(2), pages 317-324.
    19. Ouyang, Fu & Yang, Thomas T., 2025. "High dimensional binary choice model with unknown heteroskedasticity or instrumental variables," Journal of Econometrics, Elsevier, vol. 251(C).
    20. Victor Chernozhukov & Denis Chetverikov & Mert Demirer & Esther Duflo & Christian Hansen & Whitney Newey & James Robins, 2016. "Double/Debiased Machine Learning for Treatment and Causal Parameters," Papers 1608.00060, arXiv.org, revised Nov 2024.

    More about this item

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:wly:jnljam:v:2014:y:2014:i:1:n:946241. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Wiley Content Delivery (email available below). General contact details of provider: https://onlinelibrary.wiley.com/journal/4185 .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.