Two Algorithms for Compressed Sensing of Sparse Tensors

  • Shmuel FriedlandEmail author
  • Qun Li
  • Dan Schonfeld
  • Edgar A. Bernal
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)


Compressed sensing (CS) exploits the sparsity of a signal in order to integrate acquisition and compression. CS theory enables exact reconstruction of a sparse signal from relatively few linear measurements via a suitable nonlinear minimization process. Conventional CS theory relies on vectorial data representation, which results in good compression ratios at the expense of increased computational complexity. In applications involving color images, video sequences, and multi-sensor networks, the data is intrinsically of high order, and thus more suitably represented in tensorial form. Standard applications of CS to higher-order data typically involve representation of the data as long vectors that are in turn measured using large sampling matrices, thus imposing a huge computational and memory burden. In this chapter, we introduce Generalized Tensor Compressed Sensing (GTCS)—a unified framework for compressed sensing of higher-order tensors which preserves the intrinsic structure of tensorial data with reduced computational complexity at reconstruction. We demonstrate that GTCS offers an efficient means for representation of multidimensional data by providing simultaneous acquisition and compression from all tensor modes. In addition, we propound two reconstruction procedures, a serial method (GTCS-S) and a parallelizable method (GTCS-P), both capable of recovering a tensor based on noiseless or noisy observations. We then compare the performance of the proposed methods with Kronecker compressed sensing (KCS) and multi-way compressed sensing (MWCS). We demonstrate experimentally that GTCS outperforms KCS and MWCS in terms of both reconstruction accuracy (within a range of compression ratios) and processing speed. The major disadvantage of our methods (and of MWCS as well) is that the achieved compression ratios may be worse than those offered by KCS.


Singular Value Decomposition Discrete Cosine Transform Inverse Discrete Cosine Transform Reconstruction Accuracy Tensor Mode 
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  1. 1.
    Candes, E.J., Romberg, J.K.: The l 1 magic toolbox, available online: (2006)
  2. 2.
    Candes, E.J., Romberg, J.K., Tao, T.: Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information. IEEE Trans. Inf. Theory 52(2), 489–509 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Candes, E.J., Romberg, J.K., Tao, T.: Stable signal recovery from incomplete and inaccurate measurements. Commun. Pure Appl. Math. 59, 1207–1223 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Candes, E.J.: The restricted isometry property and its implications for compressed sensing. C. R. Math. 346(9–10), 589–592 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cohen, A., Dahmen, W., Devore, R.: Compressed sensing and best k-term approximation. J. Am. Math. Soc. 22(1), 211–231 (2009)MathSciNetCrossRefGoogle Scholar
  6. 6.
    De Lathauwer, L., De Moor, B., Vandewalle, J.: A multilinear singular value decomposition. SIAM J. Matrix Anal. Appl. 21, 1253–1278 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Donoho, D.L.: Compressed sensing. IEEE Trans. Inf. Theory 52(4), 1289–1306 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Donoho, D.L., Tsaig, Y.: Extensions of compressed sensing. Signal Process. 86(3), 533–548 (2006)CrossRefzbMATHGoogle Scholar
  9. 9.
    Donoho, D.L., Tsaig, Y.: Fast solution of l 1-norm minimization problems when the solution may be sparse. IEEE Trans. Inf. Theory 54(11), 4789–4812 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Duarte, M.F., Baraniuk, R.G.: Kronecker compressive sensing. IEEE Trans. Image Process. 21(2), 494–504 (2012)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Friedland, S., Qun, L., Schonfeld, D.: Compressive sensing of sparse tensors. IEEE Trans. Image Process. 23(10), 4438–4447 (2014)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Golub, G.H., Charles, F.V.L.: Matrix Computations, 4th edn. Johns Hopkins University Press, Baltimore (2013)zbMATHGoogle Scholar
  13. 13.
    Kolda, T.G., Bader, B.W.: Tensor decompositions and applications. SIAM Rev. 51(3), 455–500 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Rauhut, H.: On the impossibility of uniform sparse reconstruction using greedy methods. Sampl. Theory Signal Image Process (2008)Google Scholar
  15. 15.
    Sidiropoulos, N.D., Kyrillidis, A.: Multi-way compressed sensing for sparse low-rank tensors. IEEE Signal Process. Lett. 19(11), 757–760 (2012)CrossRefGoogle Scholar
  16. 16.
    Tucker, L.R.: The extension of factor analysis to three-dimensional matrices. In: Contributions to Mathematical Psychology. Holt, Rinehart and Winston, New York (1964)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Shmuel Friedland
    • 1
    Email author
  • Qun Li
    • 2
  • Dan Schonfeld
    • 3
  • Edgar A. Bernal
    • 2
  1. 1.Department of Mathematics, Statistics and Computer ScienceUniversity of Illinois at ChicagoChicagoUSA
  2. 2.PARC, A Xerox CompanyWebsterUSA
  3. 3.Department of Electrical and Computer EngineeringUniversity of Illinois at ChicagoChicagoUSA

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