Skip to main content
SpringerLink
Log in

A Note on the Guarantees of Total Variation Minimization

  • Conference paper
  • First Online:
Intelligent Computing Theories and Application (ICIC 2016)

Part of the book series: Lecture Notes in Computer Science ((LNISA,volume 9772))

Included in the following conference series:

  • 1835 Accesses

Abstract

In this paper, we provide a simplified understanding of the guarantees of 1-dimension total variation minimization. We consider a slightly modified total variation minimization rather than the original one. The modified model can be transformed into an \( \ell_{1} \) minimization problem by several provable mathematical tools. With the techniques developed in random sampling theory, some estimates relative to Gaussian mean width are provided for both Gaussian and sub-Gaussian sampling. We also present a sufficient condition for the exact recovery under Gaussian sampling.

This research is partly supported by the National High Technology Research and Development Program of China (No. 2012AA01A301), National Natural Science Foundation of China (No. 61402495, No. 61170046, No.11401580, No. 61402496, No. 61303189, No.61170049), Science Project of National University of Defense Technology (JC120201) and National Natural Science Foundation of Hunan Province in China (13JJ2001).

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 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
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

References

  1. Tropp, J.A.: Convex recovery of a structured signal from independent random linear measurements (2014). arXiv:1405.1102

  2. Amelunxen, D., Lotz, M., McCoy, M.B., et al.: Living on the edge: phase transitions in convex programs with random data. Inform. Inf. 3(3), 224–294 (2014). iau005

    Article  MathSciNet  MATH  Google Scholar 

  3. McCoy, M.B., Tropp, J.A.: Sharp recovery bounds for convex demixing, with applications. Found. Comput. Math. 14(3), 503–567 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  4. Lerman, G., McCoy, M.B., Tropp, J.A., et al.: Robust computation of linear models by convex relaxation. Found. Comput. Math. 15(2), 363–410 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  5. Foygel, R., Mackey, L.: Corrupted sensing: novel guarantees for separating structured signals. IEEE Trans. Inform. Theory 60(2), 1223–1247 (2014)

    Article  MathSciNet  Google Scholar 

  6. Vershynin, R.: Estimation in high dimensions: a geometric perspective (2014). arXiv:1405.5103

  7. Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60(1), 259–268 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cai, J.F., Dong, B., Osher, S., et al.: Image restoration: total variation, wavelet frames, and beyond. J. Am. Math. Soc. 25, 1033–1089 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  9. Wang, Y., Yang, J., Yin, W., et al.: A new alternating minimization algorithm for total variation image reconstruction. SIAM J. Imag. Sci. 1(3), 248–272 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  10. Cai, J.F., Xu, W.: Guarantees of total variation minimization for signal recovery. Inform. Inf. 4(4), 328–353 (2015)

    MathSciNet  Google Scholar 

  11. Plan, Y., Vershynin, R.: Robust 1-bit compressed sensing and sparse logistic regression: a convex programming approach. IEEE Trans. Inform. Theory 59(1), 482–494 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  12. Rudelson, M., Vershynin, R.: On sparse reconstruction from Fourier and Gaussian measurements. Commun. Pure Appl. Math. 61(8), 1025–1045 (2008). iav009

    Article  MathSciNet  MATH  Google Scholar 

  13. Milman, V.: Surprising geometric phenomena in high-dimensional convexity theory. In: European Congress of Mathematics, pp. 73–91. Birkhauser Basel (1998)

    Google Scholar 

  14. Gordon, Y.: On Milman’s inequality and random subspaces which escape through a mesh in R n. In: Lindenstrauss, J., Milman, V.D. (eds.) Geometric Aspects of Functional Analys, vol. 1317, pp. 84–106. Springer, Berlin (1988)

    Chapter  Google Scholar 

  15. Donoho, D., Tanner, J.: Counting faces of randomly projected polytopes when the projection radically lowers dimension. J. Am. Math. Soc. 22(1), 1–53 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  16. Mendelson, S., Pajor, A., Tomczak-Jaegermann, N.: Reconstruction and subgaussian operators in asymptotic geometric analysis. Geometric Func. Anal. 17(4), 1248–1282 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  17. Vershynin, R.: Introduction to the non-asymptotic analysis of random matrices (2010). arXiv:1011.3027

  18. Oymak, S., Thrampoulidis, C., Hassibi, B.: Simple bounds for noisy linear inverse problems with exact side information (2013). arXiv:1312.0641

  19. Oymak, S.: Convex Relaxation for Low-Dimensional Representation: Phase Transitions and Limitations. California Institute of Technology, Pasadena (2015)

    Google Scholar 

  20. Condat, L.: A direct algorithm for 1D total variation denoising. IEEE Signal Proc. Lett. 20(11), 1054–1057 (2013)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. College of Computer, National University of Defense Technology, Changsha, 410072, China

    Hao Jiang, Sheng-Guo Li & Chun-Jiang Li

  2. College of Science, National University of Defense Technology, Changsha, 410072, China

    Tao Sun, Pei-Bing Du & Li-Zhi Cheng

  3. The State Key Laboratory of High Performance Computation, National University of Defense Technology, Changsha, 410072, China

    Li-Zhi Cheng

Authors
  1. Hao Jiang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tao Sun
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Pei-Bing Du
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Sheng-Guo Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Chun-Jiang Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Li-Zhi Cheng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hao Jiang .

Editor information

Editors and Affiliations

  1. Tongji University , Shanghai, China

    De-Shuang Huang

  2. University of Ulsan , Ulsan, Korea (Republic of)

    Kang-Hyun Jo

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this paper

Cite this paper

Jiang, H., Sun, T., Du, PB., Li, SG., Li, CJ., Cheng, LZ. (2016). A Note on the Guarantees of Total Variation Minimization. In: Huang, DS., Jo, KH. (eds) Intelligent Computing Theories and Application. ICIC 2016. Lecture Notes in Computer Science(), vol 9772. Springer, Cham. https://doi.org/10.1007/978-3-319-42294-7_19

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-42294-7_19

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-42293-0

  • Online ISBN: 978-3-319-42294-7

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation