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International Conference on Latent Variable Analysis and Signal Separation

LVA/ICA 2018: Latent Variable Analysis and Signal Separation pp 116–126Cite as

Image Completion with Nonnegative Matrix Factorization Under Separability Assumption

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10891))

Abstract

Nonnegative matrix factorization is a well-known unsupervised learning method for part-based feature extraction and dimensionality reduction of a nonnegative matrix with a variety of applications. One of them is a matrix completion problem in which missing entries in an observed matrix is recovered on the basis of partially known entries. In this study, we present a geometric approach to the low-rank image completion problem with separable nonnegative matrix factorization of an incomplete data. The proposed method recursively selects extreme rays of a simplicial cone spanned by an observed image and updates the latent factors with the hierarchical alternating least-squares algorithm. The numerical experiments performed on several images with missing entries demonstrate that the proposed method outperforms other algorithms in terms of computational time and accuracy.

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Notes

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    https://www.wcss.pl/en/

References

  1. Cichocki, A., Zdunek, R., Phan, A.H., Amari, S.I.: Nonnegative Matrix and Tensor Factorizations: Applications to Exploratory Multi-way Data Analysis and Blind Source Separation. Wiley, Chichester (2009)

    Book  Google Scholar 

  2. Wang, Y.X., Zhang, Y.J.: Nonnegative matrix factorization: a comprehensive review. IEEE Trans. Knowl. Data Eng. 25(6), 1336–1353 (2013)

    Article  Google Scholar 

  3. Lee, D.D., Seung, H.S.: Learning the parts of objects by non-negative matrix factorization. Nature 401, 788–791 (1999)

    Article  Google Scholar 

  4. Lee, D.D., Seung, H.S.: Algorithms for nonnegative matrix factorization. In: Advances in Neural Information Processing, NIPS, vol. 13, pp. 556–562. MIT Press (2001)

    Google Scholar 

  5. Kumar, A., Sindhwani, V., Kambadur, P.: Fast conical hull algorithms for near-separable non-negative matrix factorization. In: Proceedings of the 30th International Conference on Machine Learning (ICML), Atlanta, Georgia, USA, vol. 28, pp. 231–239 (2013)

    Google Scholar 

  6. Bittorf, V., Recht, B., Re, C., Tropp, J.: Factoring nonnegative matrices with linear programs. In: Pereira, F., Burges, C.J.C., Bottou, L., Weinberger, K.Q. (eds.) Advances in Neural Information Processing Systems, vol. 25, pp. 1214–1222 (2012)

    Google Scholar 

  7. Gillis, N.: Successive nonnegative projection algorithm for robust nonnegative blind source separation. SIAM J. Imaging Sci. 7(2), 1420–1450 (2014)

    Article  MathSciNet  Google Scholar 

  8. Kersting, K., Wahabzada, M., Thurau, C., Bauckhage, C.: Hierarchical convex NMF for clustering massive data. In: Sugiyama, M., Yang, Q. (eds.) Proceedings of 2nd Asian Conference on Machine Learning Research, PMLR, Tokyo, Japan, vol. 13, pp. 253–268, 08–10 November 2010

    Google Scholar 

  9. Zdunek, R.: Initialization of nonnegative matrix factorization with vertices of convex polytope. In: Rutkowski, L., Korytkowski, M., Scherer, R., Tadeusiewicz, R., Zadeh, L.A., Zurada, J.M. (eds.) ICAISC 2012. LNCS (LNAI), vol. 7267, pp. 448–455. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-29347-4_52

    Chapter  Google Scholar 

  10. Bioucas-Dias, J.M., Plaza, A., Dobigeon, N., Parente, M., Du, Q., Gader, P., Chanussot, J.: Hyperspectral unmixing overview: geometrical, statistical, and sparse regression-based approaches. IEEE J. Sel. Top. Appl. Earth Obs. Remote Sens. 5(2), 354–379 (2012)

    Article  Google Scholar 

  11. Chan, T.H., Ma, W.K., Ambikapathi, A.M., Chi, C.Y.: A simplex volume maximization framework for hyperspectral endmember extraction. IEEE Trans. Geosci. Remote Sens. 49(11), 4177–4193 (2011)

    Article  Google Scholar 

  12. Arora, S., Ge, R., Halpern, Y., Mimno, D.M., Moitra, A., Sontag, D., Wu, Y., Zhu, M.: A practical algorithm for topic modeling with provable guarantees. In: Proceedings of ICML, JMLR Workshop and Conference Proceedings, JMLR.org, vol. 28, pp. 280–288 (2013)

    Google Scholar 

  13. Ding, W., Rohban, M.H., Ishwar, P., Saligrama, V.: Topic discovery through data dependent and random projections. In: Proceedings of ICML, vol. 28, pp. 471–479 (2013)

    Google Scholar 

  14. Guo, X., Ma, Y.: Generalized tensor total variation minimization for visual data recovery. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), June 2015

    Google Scholar 

  15. Yu, X., Bian, W., Tao, D.: Scalable completion of nonnegative matrix with separable structure. In: Proceedings of the Thirtieth AAAI Conference on Artificial Intelligence, Phoenix, Arizona, USA, 12–17 February 2016, pp. 2279–2285 (2016)

    Google Scholar 

  16. Cichocki, A., Zdunek, R., Amari, S.: Hierarchical ALS algorithms for nonnegative matrix and 3D tensor factorization. In: Davies, M.E., James, C.J., Abdallah, S.A., Plumbley, M.D. (eds.) ICA 2007. LNCS, vol. 4666, pp. 169–176. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-74494-8_22

    Chapter  MATH  Google Scholar 

  17. Wang, F.Y., Chi, C.Y., Chan, T.H., Wang, Y.: Nonnegative least-correlated component analysis for separation of dependent sources by volume maximization. IEEE Trans. Pattern Anal. Mach. Intell. 32(5), 875–888 (2010)

    Article  Google Scholar 

  18. Yokota, T., Zhao, Q., Cichocki, A.: Smooth PARAFAC decomposition for tensor completion. IEEE Trans. Signal Process. 64(20), 5423–5436 (2016)

    Article  MathSciNet  Google Scholar 

  19. Cai, J.F., Candès, E.J., Shen, Z.: A singular value thresholding algorithm for matrix completion. SIAM J. Optim. 20(4), 1956–1982 (2010)

    Article  MathSciNet  Google Scholar 

  20. Wen, Z., Yin, W., Zhang, Y.: Solving a low-rank factorization model for matrix completion by a nonlinear successive over-relaxation algorithm. Math. Program. Comput. 4(4), 333–361 (2012)

    Article  MathSciNet  Google Scholar 

  21. Komodakis, N., Tziritas, G.: Image completion using efficient belief propagation via priority scheduling and dynamic pruning. IEEE Trans. Image Process. 16(11), 2649–2661 (2007)

    Article  MathSciNet  Google Scholar 

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Acknowledgment

This work was partially supported by the grant 2015/17/B/ST6/01865 funded by National Science Center (NCN) in Poland. Calculations were performed at the Wroclaw Centre for Networking and Supercomputing under grant no. 127.

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Authors and Affiliations

  1. Faculty of Electronics, Wroclaw University of Science and Technology, Wybrzeze Wyspianskiego 27, 50-370, Wroclaw, Poland

    Tomasz Sadowski & Rafał Zdunek

Authors
  1. Tomasz Sadowski
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  2. Rafał Zdunek
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Corresponding authors

Correspondence to Tomasz Sadowski or Rafał Zdunek .

Editor information

Editors and Affiliations

  1. Paul Sabatier University, Toulouse, France

    Yannick Deville

  2. Bar-Ilan University, Ramat Gan, Israel

    Sharon Gannot

  3. University of Surrey, Guildford, United Kingdom

    Russell Mason

  4. University of Surrey, Guildford, United Kingdom

    Mark D. Plumbley

  5. University of Surrey, Guildford, United Kingdom

    Dominic Ward

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Sadowski, T., Zdunek, R. (2018). Image Completion with Nonnegative Matrix Factorization Under Separability Assumption. In: Deville, Y., Gannot, S., Mason, R., Plumbley, M., Ward, D. (eds) Latent Variable Analysis and Signal Separation. LVA/ICA 2018. Lecture Notes in Computer Science(), vol 10891. Springer, Cham. https://doi.org/10.1007/978-3-319-93764-9_12

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  • DOI: https://doi.org/10.1007/978-3-319-93764-9_12

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-93763-2

  • Online ISBN: 978-3-319-93764-9

  • eBook Packages: Computer ScienceComputer Science (R0)

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