Skip to main content
SpringerLink
Log in

Low-Rank Tensor Recovery

  • Chapter
  • First Online:
Tensor Computation for Data Analysis

  • 2501 Accesses

Abstract

During data acquisition and transmission, some entries of data are missing, which will degrade the performance of subsequent data processing. Missing component analysis, also named matrix completion, can recover the missing data based on the low-rank assumption. However, with the emergence of high-order data, traditional methods directly tackle the high-order data by rearranging it into a matrix, which inevitably lose some structural information. As a generation of matrix completion, tensor completion is proposed to recover the missing entries of high-order data.

In this chapter, we mainly focus on discussing the optimization frameworks of tensor completion and corresponding algorithms. To be specific, we divide the existing tensor completion methods into two groups according to whether the rank is given in advanced. One is tensor factorization-based tensor completion model which needs predefined rank, and the other one is rank minimization-based tensor completion model. For each group, the comparison of different tensor decompositions on tensor completion is considered with respect to the optimization model, computational complexity, and sampling complexity. Finally, we introduce some applications, such as visual data recovery, recommendation system, knowledge graph completion, and traffic flow prediction.

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 99.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 129.99
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

Notes

  1. 1.

    https://zenodo.org/record/1205229#.X5FUXi-1Gu4.

References

  1. Ashraphijuo, M., Wang, X.: Fundamental conditions for low-CP-rank tensor completion. J. Mach. Learn. Res. 18(1), 2116–2145 (2017)

    MathSciNet  MATH  Google Scholar 

  2. Ashraphijuo, M., Wang, X.: Characterization of sampling patterns for low-tt-rank tensor retrieval. Ann. Math. Artif. Intell. 88(8), 859–886 (2020)

    Article  MathSciNet  Google Scholar 

  3. Ashraphijuo, M., Wang, X., Zhang, J.: Low-rank data completion with very low sampling rate using Newton’s method. IEEE Trans. Signal Process. 67(7), 1849–1859 (2019)

    Article  MathSciNet  Google Scholar 

  4. Asif, M.T., Mitrovic, N., Garg, L., Dauwels, J., Jaillet, P.: Low-dimensional models for missing data imputation in road networks. In: 2013 IEEE International Conference on Acoustics, Speech and Signal Processing, pp. 3527–3531. IEEE, New York (2013)

    Google Scholar 

  5. Balazevic, I., Allen, C., Hospedales, T.: TuckER: Tensor factorization for knowledge graph completion. In: Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing (EMNLP-IJCNLP), pp. 5188–5197 (2019)

    Google Scholar 

  6. Balažević, I., Allen, C., Hospedales, T.M.: Hypernetwork knowledge graph embeddings. In: International Conference on Artificial Neural Networks, pp. 553–565. Springer, Berlin (2019)

    Google Scholar 

  7. Bengua, J.A., Phien, H.N., Tuan, H.D., Do, M.N.: Efficient tensor completion for color image and video recovery: low-rank tensor train. IEEE Trans. Image Process. 26(5), 2466–2479 (2017)

    Article  MathSciNet  Google Scholar 

  8. Bollacker, K., Evans, C., Paritosh, P., Sturge, T., Taylor, J.: Freebase: a collaboratively created graph database for structuring human knowledge. In: Proceedings of the 2008 ACM SIGMOD International Conference on Management of Data, pp. 1247–1250 (2008)

    Google Scholar 

  9. Bordes, A., Usunier, N., Garcia-Duran, A., Weston, J., Yakhnenko, O.: Translating embeddings for modeling multi-relational data. In: Neural Information Processing Systems (NIPS), pp. 1–9 (2013)

    Google Scholar 

  10. Candès, E.J., Recht, B.: Exact matrix completion via convex optimization. Found. Comput. Math. 9(6), 717 (2009)

    Article  MathSciNet  Google Scholar 

  11. Candès, E.J., Tao, T.: The power of convex relaxation: near-optimal matrix completion. IEEE Trans. Inf. Theory 56(5), 2053–2080 (2010)

    Article  MathSciNet  Google Scholar 

  12. Conn, A.R., Gould, N.I., Toint, P.L.: Trust Region Methods. SIAM, Philadelphia (2000)

    Book  Google Scholar 

  13. Dettmers, T., Minervini, P., Stenetorp, P., Riedel, S.: Convolutional 2d knowledge graph embeddings. In: Proceedings of the AAAI Conference on Artificial Intelligence, vol. 32 (2018)

    Google Scholar 

  14. Fabian, M., Gjergji, K., Gerhard, W., et al.: Yago: A core of semantic knowledge unifying wordnet and wikipedia. In: 16th International World Wide Web Conference, WWW, pp. 697–706 (2007)

    Google Scholar 

  15. Fan, J., Cheng, J.: Matrix completion by deep matrix factorization. Neural Netw. 98, 34–41 (2018)

    Article  Google Scholar 

  16. Filipović, M., Jukić, A.: Tucker factorization with missing data with application to low-rank tensor completion. Multidim. Syst. Sign. Process. 26(3), 677–692 (2015)

    Article  Google Scholar 

  17. Gandy, S., Recht, B., Yamada, I.: Tensor completion and low-n-rank tensor recovery via convex optimization. Inverse Prob. 27(2), 025010 (2011)

    Article  MathSciNet  Google Scholar 

  18. Hu, Y., Zhang, D., Ye, J., Li, X., He, X.: Fast and accurate matrix completion via truncated nuclear norm regularization. IEEE Trans. Pattern Anal. Mach. Intell. 35(9), 2117–2130 (2012)

    Article  Google Scholar 

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

    Article  Google Scholar 

  20. Huang, H., Liu, Y., Long, Z., Zhu, C.: Robust low-rank tensor ring completion. IEEE Trans. Comput. Imag. 6, 1117–1126 (2020)

    Article  MathSciNet  Google Scholar 

  21. Jain, P., Netrapalli, P., Sanghavi, S.: Low-rank matrix completion using alternating minimization. In: Proceedings of the Forty-fifth Annual ACM Symposium on Theory of Computing, pp. 665–674 (2013)

    Google Scholar 

  22. Jannach, D., Resnick, P., Tuzhilin, A., Zanker, M.: Recommender systems—beyond matrix completion. Commun. ACM 59(11), 94–102 (2016)

    Article  Google Scholar 

  23. Kang, Z., Peng, C., Cheng, Q.: Top-n recommender system via matrix completion. In: Proceedings of the AAAI Conference on Artificial Intelligence, vol. 30 (2016)

    Google Scholar 

  24. Kazemi, S.M., Poole, D.: SimplE embedding for link prediction in knowledge graphs. In: Advances in Neural Information Processing Systems, vol. 31 (2018)

    Google Scholar 

  25. Keshavan, R.H., Montanari, A., Oh, S.: Matrix completion from a few entries. IEEE Trans. Inf. Theory 56(6), 2980–2998 (2010)

    Article  MathSciNet  Google Scholar 

  26. Kiefer, J., Wolfowitz, J., et al.: Stochastic estimation of the maximum of a regression function. Ann. Math. Stat. 23(3), 462–466 (1952)

    Article  MathSciNet  Google Scholar 

  27. Lin, Y., Liu, Z., Sun, M., Liu, Y., Zhu, X.: Learning entity and relation embeddings for knowledge graph completion. In: Proceedings of the AAAI Conference on Artificial Intelligence, vol. 29 (2015)

    Google Scholar 

  28. 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 (2012)

    Article  Google Scholar 

  29. Liu, Y., Long, Z., Zhu, C.: Image completion using low tensor tree rank and total variation minimization. IEEE Trans. Multimedia 21(2), 338–350 (2019)

    Article  Google Scholar 

  30. Long, Z., Liu, Y., Chen, L., Zhu, C.: Low rank tensor completion for multiway visual data. Signal Process. 155, 301–316 (2019)

    Article  Google Scholar 

  31. Long Z., Zhu C., Liu, J., Liu, Y.: Bayesian low rank tensor ring for image recovery. IEEE Trans. Image Process. 30, 3568–3580 (2021)

    Article  MathSciNet  Google Scholar 

  32. Lu, C.: A Library of ADMM for Sparse and Low-rank Optimization. National University of Singapore (2016). https://github.com/canyilu/LibADMM

  33. Lukovnikov, D., Fischer, A., Lehmann, J., Auer, S.: Neural network-based question answering over knowledge graphs on word and character level. In: Proceedings of the 26th International Conference on World Wide Web, pp. 1211–1220 (2017)

    Google Scholar 

  34. Miller, G.A.: WordNet: a lexical database for english. Commun. ACM 38(11), 39–41 (1995)

    Article  Google Scholar 

  35. Moré, J.J.: The Levenberg-Marquardt algorithm: implementation and theory. In: Numerical Analysis, pp. 105–116. Springer, Berlin (1978)

    Google Scholar 

  36. Mu, C., Huang, B., Wright, J., Goldfarb, D.: Square deal: lower bounds and improved relaxations for tensor recovery. In: International Conference on Machine Learning, pp. 73–81 (2014)

    Google Scholar 

  37. Nickel, M., Tresp, V., Kriegel, H.P.: A three-way model for collective learning on multi-relational data. In: Proceedings of the 28th International Conference on International Conference on Machine Learning, pp. 809–816 (2011)

    Google Scholar 

  38. Robbins, H., Monro, S.: A stochastic approximation method. Ann. Math. Stat. 22(3), 400–407 (1951)

    Article  MathSciNet  Google Scholar 

  39. Singhal, A.: Introducing the knowledge graph: things, not strings. Official Google Blog 5 (2012)

    Google Scholar 

  40. Sun, Z., Deng, Z.H., Nie, J.Y., Tang, J.: RotatE: knowledge graph embedding by relational rotation in complex space. In: International Conference on Learning Representations (2018)

    Google Scholar 

  41. Trouillon, T., Dance, C.R., Gaussier, É., Welbl, J., Riedel, S., Bouchard, G.: Knowledge graph completion via complex tensor factorization. J. Mach. Learn. Res. 18(1), 4735–4772 (2017)

    MathSciNet  MATH  Google Scholar 

  42. Vrandečić, D., Krötzsch, M.: Wikidata: a free collaborative knowledgebase. Commun. ACM 57(10), 78–85 (2014)

    Article  Google Scholar 

  43. Vu, T., Nguyen, T.D., Nguyen, D.Q., Phung, D., et al.: A capsule network-based embedding model for knowledge graph completion and search personalization. In: Proceedings of the 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 1 (Long and Short Papers), pp. 2180–2189 (2019)

    Google Scholar 

  44. Wang, Z., Zhang, J., Feng, J., Chen, Z.: Knowledge graph embedding by translating on hyperplanes. In: Proceedings of the AAAI Conference on Artificial Intelligence, vol. 28 (2014)

    Google Scholar 

  45. Wang, W., Aggarwal, V., Aeron, S.: Tensor completion by alternating minimization under the tensor train (TT) model (2016). Preprint, arXiv:1609.05587

    Google Scholar 

  46. Wang, W., Aggarwal, V., Aeron, S.: Efficient low rank tensor ring completion. In: Proceedings of the IEEE International Conference on Computer Vision, pp. 5697–5705 (2017)

    Google Scholar 

  47. Xu, Y., Yin, W.: A block coordinate descent method for regularized multiconvex optimization with applications to nonnegative tensor factorization and completion. SIAM J. Imag. Sci. 6(3), 1758–1789 (2013)

    Article  MathSciNet  Google Scholar 

  48. Yang, B., Yih, W.T., He, X., Gao, J., Deng, L.: Embedding entities and relations for learning and inference in knowledge bases. In: 3rd International Conference on Learning Representations, ICLR 2015, San Diego, CA, USA, 7–9 May 2015, Conference Track Proceedings (2015)

    Google Scholar 

  49. Yang, Y., Feng, Y., Suykens, J.A.: A rank-one tensor updating algorithm for tensor completion. IEEE Signal Process Lett. 22(10), 1633–1637 (2015)

    Article  Google Scholar 

  50. Yu, J., Li, C., Zhao, Q., Zhao, G.: Tensor-ring nuclear norm minimization and application for visual: Data completion. In: ICASSP 2019-2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 3142–3146. IEEE, New York (2019)

    Google Scholar 

  51. Zeb, A., Haq, A.U., Zhang, D., Chen, J., Gong, Z.: KGEL: a novel end-to-end embedding learning framework for knowledge graph completion. Expert Syst. Appl. 167, 114164 (2020)

    Article  Google Scholar 

  52. Zhang, Z., Aeron, S.: Exact tensor completion using t-SVD. IEEE Trans. Signal Process. 65(6), 1511–1526 (2017)

    Article  MathSciNet  Google Scholar 

  53. Zhang, F., Yuan, N.J., Lian, D., Xie, X., Ma, W.Y.: Collaborative knowledge base embedding for recommender systems. In: Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 353–362 (2016)

    Google Scholar 

  54. Zhao, Q., Zhang, L., Cichocki, A.: Bayesian CP factorization of incomplete tensors with automatic rank determination. IEEE Trans. Pattern Anal. Mach. Intell. 37(9), 1751–1763 (2015)

    Article  Google Scholar 

  55. Zheng, V., Cao, B., Zheng, Y., Xie, X., Yang, Q.: Collaborative filtering meets mobile recommendation: a user-centered approach. In: Proceedings of the AAAI Conference on Artificial Intelligence, vol. 24 (2010)

    Google Scholar 

  56. Zniyed, Y., Boyer, R., de Almeida, A.L., Favier, G.: High-order tensor estimation via trains of coupled third-order CP and Tucker decompositions. Linear Algebra Appl. 588, 304–337 (2020)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Electronic Science and Technology of China (UESTC), Chengdu, China

    Yipeng Liu, Jiani Liu, Zhen Long & Ce Zhu

Authors
  1. Yipeng Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jiani Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zhen Long
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Ce Zhu
    View author publications

    You can also search for this author in PubMed Google Scholar

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). Low-Rank Tensor Recovery. In: Tensor Computation for Data Analysis. Springer, Cham. https://doi.org/10.1007/978-3-030-74386-4_4

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-74386-4_4

  • 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

Search

Navigation