Advertisement

Yet Another Schatten Norm for Tensor Recovery

  • Chao Li
  • Lili Guo
  • Yu Tao
  • Jinyu Wang
  • Lin Qi
  • Zheng DouEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9949)

Abstract

In this paper, we introduce a new class of Schatten norms for tensor recovery. In the new norm, unfoldings of a tensor along not only every single order but also all combinations of orders are taken into account. Additionally, we prove that the proposed tensor norm has similar properties to matrix Schatten norm, and also provides several propositions which is useful in the recovery problem. Furthermore, for reliable recovery of a tensor with Gaussian measurements, we show the necessary size of measurements using the new norm. Compared to using conventional overlapped Schatten norm, the new norm results in less measurements for reliable recovery with high probability. Finally, experimental results demonstrate the efficiency of the new norm in video in-painting.

Keywords

Tensor completion Video in-painting Tensor norm Low-rank decomposition 

References

  1. 1.
    Amelunxen, D., Lotz, M., McCoy, M.B., Tropp, J.A.: Living on the edge: a geometric theory of phase transitions in convex optimization. arXiv preprint arXiv:1303.6672 (2013)
  2. 2.
    Cai, J.F., Cands, E.J., Shen, Z.: A singular value thresholding algorithm for matrix completion. SIAM J. Optim. 20(4), 1956–1982 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Cichocki, A.: Era of big data processing: a new approach via tensor networks and tensor decompositions. arXiv preprint arXiv:1403.2048 (2014)
  4. 4.
    De Lathauwer, L., De Moor, B., Vandewalle, J.: A multilinear singular value decomposition. SIAM J. Matrix Anal. Appl. 21(4), 1253–1278 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Kolda, T.G., Bader, B.W.: Tensor decompositions and applications. SIAM Rev. 51(3), 455–500 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Liu, J., Musialski, P., Wonka, P., Ye, J.: Tensor completion for estimating missing values in visual data. IEEE Trans. Pattern Anal. Mach. Intell. 35(1), 208–220 (2013)CrossRefGoogle Scholar
  7. 7.
    Mu, C., Huang, B., Wright, J., Goldfarb, D.: Square deal: Lower bounds and improved relaxations for tensor recovery. arXiv preprint arXiv:1307.5870 (2013)
  8. 8.
    Nickel, M.: Tensor factorization for relational learning. Ph.D. thesis (2013)Google Scholar
  9. 9.
    Signoretto, M., De Lathauwer, L., Suykens, J.A.: Nuclear norms for tensors and their use for convex multilinear estimation. Submitted to Linear Algebra and Its Applications, vol. 43 (2010)Google Scholar
  10. 10.
    Signoretto, M., Dinh, Q.T., De Lathauwer, L., Suykens, J.A.: Learning with tensors: a framework based on convex optimization and spectral regularization. Mach. Learn. 94(3), 303–351 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Tomioka, R., Suzuki, T.: Convex tensor decomposition via structured schatten norm regularization. In: Advances in Neural Information Processing Systems, pp. 1331–1339Google Scholar

Copyright information

© Springer International Publishing AG 2016

Authors and Affiliations

  • Chao Li
    • 1
  • Lili Guo
    • 1
  • Yu Tao
    • 1
  • Jinyu Wang
    • 1
  • Lin Qi
    • 1
  • Zheng Dou
    • 1
    Email author
  1. 1.College of Information and Communication EngineeringHarbin Engineering UniversityHarbinChina

Personalised recommendations