Skip to main content
SpringerLink
Log in

Structured Sparsity

  • Chapter
  • First Online:
Estimation and Testing Under Sparsity

Part of the book series: Lecture Notes in Mathematics ((LNMECOLE,volume 2159))

Abstract

Oracle results are given for least squares and square-root loss with sparsity inducing norms Ω. A general class of sparsity inducing norms are those generated from cones. Examples are the group sparsity norm, the wedge norm, and the sorted ℓ 1-norm. Bounds for the error in dual norm Ω ∗ are given. De-sparsifying is discussed as well.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 44.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 59.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    For example the least-squares estimator with so-called nuclear norm penalization is formally also a structured sparsity estimator. This will be considered in Sect. 12.5 The topic of this chapter is rather norms which are weakly decomposable as defined in Definition 6.1.

  2. 2.

    If Ω is a weakly decomposable norm on \(\mathbb{R}^{p}\) and J is an allowed set, one may think of choosing Ω −J  = Ω −J. Alternatively, if J is the complement of an allowed set, one might choose Ω −J  = Ω(⋅ | − J).

References

  • F. Bach, Structured sparsity-inducing norms through submodular functions, in Advances in Neural Information Processing Systems (NIPS), vol. 23 (2010)

    Google Scholar 

  • F. Bach, R. Jenatton, J. Mairal, G. Obozinski, Optimization with sparsity-inducing penalties. Found. Trends Mach. Learn. 4, 1–106 (2012)

    Article  MATH  Google Scholar 

  • M. Bogdan, E. van den Berg, W. Su, E. Candes, Statistical estimation and testing via de sorted l1 norm. ArXiv:1310.1969 (2013)

    Google Scholar 

  • F. Bunea, J. Lederer, Y. She, The group square-root lasso: theoretical properties and fast algorithms. IEEE Trans. Inf. Theory 2, 1313–1325 (2014)

    Article  MathSciNet  Google Scholar 

  • R. Jenatton, J.-Y. Audibert, F. Bach, Structured variable selection with sparsity inducing norms. J. Mach. Learn. Res. 12, 2777–2824 (2011)

    MathSciNet  MATH  Google Scholar 

  • K. Lounici, M. Pontil, S. van de Geer, A. Tsybakov, Oracle inequalities and optimal inference under group sparsity. Ann. Stat. 39, 2164–2204 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  • A. Maurer, M. Pontil, Structured sparsity and generalization. J. Mach. Learn. Res. 13, 671–690 (2012)

    MathSciNet  MATH  Google Scholar 

  • C. Micchelli, J. Morales, M. Pontil, A family of penalty functions for structured sparsity, in Advances in Neural Information Processing Systems, NIPS 2010, vol. 23 (2010)

    Google Scholar 

  • B. Stucky, S. van de Geer, Sharp oracle inequalities for square root regularization. ArXiv:1509.04093 (2015)

    Google Scholar 

  • S. van de Geer, Weakly decomposable regularization penalties and structured sparsity. Scand. J. Stat. 41, 72–86 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  • M. Yuan, Y. Lin, Model selection and estimation in regression with grouped variables. J. R. Stat. Soc. Ser. B 68, 49 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  • X. Zeng, A. Mario, The ordered weighted l1 norm: Atomic formulation, dual norm, and projections. ArXiv:1409.4271 (2014)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Seminar für Statistik HGG 24.1, ETH Zentrum, Zürich, Switzerland

    Sara van de Geer

Authors
  1. Sara van de Geer
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this chapter

Cite this chapter

van de Geer, S. (2016). Structured Sparsity. In: Estimation and Testing Under Sparsity. Lecture Notes in Mathematics(), vol 2159. Springer, Cham. https://doi.org/10.1007/978-3-319-32774-7_6

Download citation

Publish with us

Policies and ethics

Search

Navigation