Skip to main content

Tensor Computation

  • Chapter
  • First Online:
Tensor Computation for Data Analysis

Abstract

Tensor is a natural representation for multiway data. As a generalized form of matrix, tensor has its generalized multilinear operators, which enable tensor computation. In this chapter, some notations, tensor unfoldings, tensor products, and some other related basic operators are illustrated in detail. It can serve as a foundation for the subsequent tensor decompositions and data processing applications.

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

Access this chapter

eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 16.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 129.99
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. Abdelfattah, A., Baboulin, M., Dobrev, V., Dongarra, J., Earl, C., Falcou, J., Haidar, A., Karlin, I., Kolev, T., Masliah, I., et al.: High-performance tensor contractions for GPUs. Procedia Comput. Sci. 80, 108–118 (2016)

    Article  Google Scholar 

  2. Bengua, J.A., Tuan, H.D., Phien, H.N., Do, M.N.: Concatenated image completion via tensor augmentation and completion. In: 2016 10th International Conference on Signal Processing and Communication Systems (ICSPCS), pp. 1–7. IEEE, Piscataway (2016)

    Google Scholar 

  3. Cheng, D., Qi, H., Xue, A.: A survey on semi-tensor product of matrices. J. Syst. Sci. Complex. 20(2), 304–322 (2007)

    Article  MathSciNet  Google Scholar 

  4. Cichocki, A., Mandic, D., De Lathauwer, L., Zhou, G., Zhao, Q., Caiafa, C., Phan, H.A.: Tensor decompositions for signal processing applications: from two-way to multiway component analysis. IEEE Signal Process. Mag. 32(2), 145–163 (2015)

    Article  Google Scholar 

  5. Cichocki, A., Lee, N., Oseledets, I., Phan, A.H., Zhao, Q., Mandic, D.P., et al.: Tensor networks for dimensionality reduction and large-scale optimization: Part 1 low-rank tensor decompositions. Found. Trends Mach. Learn. 9(4–5), 249–429 (2016)

    Article  Google Scholar 

  6. Huang, H., Liu, Y., Liu, J., Zhu, C.: Provable tensor ring completion. Signal Process. 171, 107486 (2020)

    Article  Google Scholar 

  7. Kilmer, M.E., Braman, K., Hao, N., Hoover, R.C.: Third-order tensors as operators on matrices: a theoretical and computational framework with applications in imaging. SIAM J. Matrix Anal. Appl. 34(1), 148–172 (2013)

    Article  MathSciNet  Google Scholar 

  8. Kolda, T.G., Bader, B.W.: Tensor decompositions and applications. SIAM Rev. 51(3), 455–500 (2009)

    Article  MathSciNet  Google Scholar 

  9. Sidiropoulos, n.d., De Lathauwer, L., Fu, X., Huang, K., Papalexakis, E.E., Faloutsos, C.: Tensor decomposition for signal processing and machine learning. IEEE Trans. Signal Process. 65(13), 3551–3582 (2017)

    Google Scholar 

  10. Solomonik, E., Demmel, J.: Fast bilinear algorithms for symmetric tensor contractions. Comput. Methods Appl. Math. 21(1), 211–231 (2020)

    Article  MathSciNet  Google Scholar 

  11. Tucker, L.R.: Implications of factor analysis of three-way matrices for measurement of change. Probl. Meas. Change 15, 122–137 (1963)

    Google Scholar 

  12. Yu, J., Zhou, G., Zhao, Q., Xie, K.: An effective tensor completion method based on multi-linear tensor ring decomposition. In: 2018 Asia-Pacific Signal and Information Processing Association Annual Summit and Conference (APSIPA ASC), pp. 1344–1349. IEEE, Piscataway (2018)

    Google Scholar 

  13. Zhang, Z., Aeron, S.: Exact tensor completion using t-SVD. IEEE Trans. Signal Process. 65(6), 1511–1526 (2016)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

Copyright information

© 2022 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Liu, Y., Liu, J., Long, Z., Zhu, C. (2022). Tensor Computation. In: Tensor Computation for Data Analysis. Springer, Cham. https://doi.org/10.1007/978-3-030-74386-4_1

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-74386-4_1

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-74385-7

  • Online ISBN: 978-3-030-74386-4

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics