Skip to main content
SpringerLink
Log in

Multivariable Polynomials for the Construction of Binary Sensing Matrices

  • Conference paper
Book cover Forging Connections between Computational Mathematics and Computational Geometry

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 124))

  • 810 Accesses

Abstract

In compressed sensing, the matrices that satisfy restricted isometry property (RIP) play an important role. But to date, very few results for designing such matrices are available. Of interest in several applications is a matrix whose elements are 0’s and 1’s (in short, 0, 1-matrix), excluding column normalization factors. Recently, DeVore (J Complex 23:918–925, 2007) has constructed deterministic 0, 1-matrices that obey sparse recovery properties such as RIP. The present work extends the ideas embedded in DeVore (J Complex 23:918–925, 2007) and shows that the 0, 1-matrices of different sizes can be constructed using multivariable homogeneous polynomials.

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 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 54.99
Price excludes VAT (USA)
  • Durable hardcover 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. Baraniuk, R., Davenpor, M., De Vore, R., Wakin, M.: A simple proof of the restricted isometry property for random matrices. Constr. Approx. 28(3), 253–263 (2008)

    Article  MathSciNet  Google Scholar 

  2. Bourgain, J., Dilworth, S., Ford, K., Konyagin, S., Kutzarova, D.: Explicit constructions of RIP matrices and related problems. Duke Math. J. 159, 145–185 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. Candes, E.: The restricted isometry property and its implications for compressed sensing. Comptes Rendus Mathematique 346, 589–592 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. Candes, E., Romberg, J., Tao, T.: Stable signal recovery from incomplete and inaccurate measurements. Commun. Pure Appl. Math. 59, 1207–1223 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  5. Candes, E., Tao, T.: Decoding by linear programming. IEEE Trans. Inf. Theory 51, 42–4215 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  6. Candes, E.J., Tao, T.: Near-optimal signal recovery from random projections: universal encoding strategies. IEEE Trans. Inf. Theory 52(2), 489–509 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chen, S.S., Donoho, D.L., Saunders, M.A.: Atomic decomposition by basis pursuit. SIAM Rev. 43, 129–159 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cohen, A., Dahmen, W., DeVore, R.: Compressed sensing and best k-term approximation. J. Am. Math. Soc. 22(1), 211–231 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  9. DeVore, R.A.: Deterministic constructions of compressed sensing matrices. J. Complex. 23, 918–925 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  10. Donoho, D.: Compressed sensing. IEEE Trans. Inf. Theory 52, 1289–1306 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  11. Donoho, D.L., Elad, M., Temlyakov, V.N.: Stable recovery of sparse overcomplete representations in the presence of noise. IEEE Trans. Inf. Theory 52, 6–18 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  12. Elad, M.: Sparse and Redundant Representations; from Theory to Applications in Signal and Image Processing. Springer, Berlin (2010)

    Book  MATH  Google Scholar 

  13. Garnaev, A., Gluskin, E.: The widths of a Euclidean ball. Dokl. Akad. Nauk USSR 277, 1048–1052 (1984); English transl. in Soviet Math. Dokl. 30, 200–204 (1984)

    Google Scholar 

  14. Kashin, B.S.: Widths of certain finite-dimensional sets and classes of smooth functions. Izv. Akad. Nauk SSSR, Ser. Mat. 41, 334–351 (1977); English transl. in Math. USSR IZV. 11, 317–333 (1978)

    Google Scholar 

  15. Lidl, R., Niederreiter H.: Finite Fields, 2nd edn., pp. 275–277. Cambridge University Press, New York (1997)

    Google Scholar 

  16. Nelson, J.L., Temlyakov, V.N.: On the size of incoherent systems. J. Approx. Theory 163(9), 1238–1245 (2011)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The first author gratefully acknowledges the support (Ref No. 20-6/2009(i)EU-IV) that he receives from UGC, Government of India. The last author is thankful to DST (SR/FTP/ETA-054/2009) for the partial support that he received.

Author information

Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Technology Hyderabad, Yeddumailaram, A.P. 502 205, India

    R. Ramu Naidu & Sastry Challa

  2. Department of Chemical Engineering, Indian Institute of Technology Hyderabad, Yeddumailaram, A.P. 502 205, India

    Phanindra Jampana

Authors
  1. R. Ramu Naidu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Phanindra Jampana
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Sastry Challa
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to R. Ramu Naidu .

Editor information

Editors and Affiliations

  1. Department of Mathematical Sciences, The University of Liverpool, Liverpool, UK

    Ke Chen

  2. Global Science and Technology Forum, Singapore, Singapore

    Anton Ravindran (Managing Editor, President) (Managing Editor, President)

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this paper

Cite this paper

Naidu, R.R., Jampana, P., Challa, S. (2016). Multivariable Polynomials for the Construction of Binary Sensing Matrices. In: Chen, K., Ravindran, A. (eds) Forging Connections between Computational Mathematics and Computational Geometry. Springer Proceedings in Mathematics & Statistics, vol 124. Springer, Cham. https://doi.org/10.5176/2251-1911_CMCGS14.36_6

Download citation

Publish with us

Policies and ethics

Search

Navigation