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XRAY Algorithm for Separable Nonnegative Tensor Factorization

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Advances in Computational Intelligence (IWANN 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10306))

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Abstract

Many computational problems in machine learning can be represented by separable matrix factorization models. In a geometric approach, linear separability means that the whole set of data points can be modeled by a convex combination of a few data points, referred to as the extreme rays. The aim of the XRAY algorithm is to find the extreme rays of the conic hull, generated by observed nonnegative vectors. In this paper, we extend the concept of this algorithm to a multi-linear data representation. Instead of searching into a vector space, we attempt to find the equivalent extreme rays in a space of tensors, under the linear separability assumption of subtensors, ordered along the selected mode. The proposed multi-way XRAY algorithm has been applied to Blind Source Separation (BSS) of natural images. The experiments demonstrate that if multi-way observations are at least one-mode linearly separable, the proposed algorithms can estimate the latent factors with high Signal-to-Interference (SIR) performance. The discussed methods may also be useful for analyzing video sequences.

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Notes

  1. 1.

    The inner product \(<{\mathcal {A}}, \; {\mathcal {B}}>\) between the tensors \({\mathcal {A}} = [a_{i_1,\ldots ,i_N}] \in {\mathbb {R}}^{I_1 \times \ldots \times I_N}\) and \({\mathcal {B}} = [b_{i_1,\ldots ,i_N}] \in {\mathbb {R}}^{I_1 \times \ldots \times I_N}\) is defined as follows: \(<{\mathcal {A}}, \; {\mathcal {B}}> = \sum _{i_1 = 1}^{I_1} \cdots \sum _{i_N = 1}^{I_N} a_{i_1,\ldots ,i_N} b_{i_1,\ldots ,i_N}\).

References

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

    Article  Google Scholar 

  2. 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, Hoboken (2009)

    Book  Google Scholar 

  3. Vavasis, S.A.: On the complexity of nonnegative matrix factorization. SIAM J. Optim. 20(3), 1364–1377 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  4. Donoho, D., Stodden, V.: When does non-negative matrix factorization give a correct decomposition into parts? In: Proceedings of NIPS, vol. 16. MIT Press (2004)

    Google Scholar 

  5. 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 

  6. Ding, C., Li, T., Jordan, M.I.: Convex and semi-nonnegative matrix factorizations. IEEE Trans. Pattern Anal. Mach. Intell. 32(1), 45–55 (2010)

    Article  Google Scholar 

  7. Thurau, C., Kersting, K., Bauckhage, C.: Convex non-negative matrix factorization in the wild. In: Proceedings of 9th IEEE International Conference on Data Mining, ICDM 2009, Washington, D.C., USA, pp. 523–532 (2009)

    Google Scholar 

  8. Esser, E., Möller, M., Osher, S., Sapiro, G., Xin, J.: A convex model for nonnegative matrix factorization and dimensionality reduction on physical space. IEEE Trans. Image Process. 21(7), 3239–3252 (2012)

    Article  MathSciNet  Google Scholar 

  9. Arora, S., Ge, R., Kannan, R., Moitra, A.: Computing a nonnegative matrix factorization - provably. SIAM J. Comput. 45(4), 1582–1611 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  10. 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 

  11. 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. Topics Appl. Earth Obs. Remote Sens. 5(2), 354–379 (2012)

    Article  Google Scholar 

  12. 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 

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

    Article  MathSciNet  MATH  Google Scholar 

  14. 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, vol. 28, pp. 280–288 (2013). www.JMLR.org

  15. 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 

  16. 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 

  17. Gillis, N., Luce, R.: Robust near-separable nonnegative matrix factorization using linear optimization. J. Mach. Learn. Res. 15, 1249–1280 (2014)

    MathSciNet  MATH  Google Scholar 

  18. Huang, K., Sidiropoulos, N., Swami, A.: Non-negative matrix factorization revisited: uniqueness and algorithm for symmetric decomposition. IEEE Trans. Sig. Process. 62(1), 211–224 (2014)

    Article  MathSciNet  Google Scholar 

  19. Ouedraogo, W.S.B., Souloumiac, A., Jaïdane, M., Jutten, C.: Simplicial cone shrinking algorithm for unmixing nonnegative sources. In: ICASSP, pp. 2405–2408. IEEE (2012)

    Google Scholar 

  20. 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). doi:10.1007/978-3-642-29347-4_52

    Chapter  Google Scholar 

  21. Araujo, M.C.U., Saldanha, T.C.B., Galvao, R.K.H., Yoneyama, T., Chame, H.C., Visani, V.: The successive projections algorithm for variable selection in spectroscopic multicomponent. Chemometr. Intell. Lab. Syst. 57(2), 65–73 (2001)

    Article  Google Scholar 

  22. Gillis, N.: Robustness analysis of hottopixx, a linear programming model for factoring nonnegative matrices. SIAM J. Mat. Anal. Appl. 34(3), 1189–1212 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  23. Cichocki, A., Zdunek, R.: NMFLAB for signal and image processing. Technical report, Laboratory for Advanced Brain Signal Processing, BSI, RIKEN, Saitama, Japan (2006)

    Google Scholar 

  24. Fu, X., Ma, W.K., Huang, K., Sidiropoulos, N.D.: Blind separation of quasi-stationary sources: exploiting convex geometry in covariance domain. IEEE Trans. Sig. Process. 63(9), 2306–2320 (2015)

    Article  MathSciNet  Google Scholar 

  25. Fu, X., Sidiropoulos, N.D., Ma, W.K.: Power spectra separation via structured matrix factorization. IEEE Trans. Sig. Process. 64(17), 4592–4605 (2016)

    Article  MathSciNet  Google Scholar 

  26. Fu, X., Huang, K., Yang, B., Ma, W.K., Sidiropoulos, N.D.: Robust volume minimization-based matrix factorization for remote sensing and document clustering. IEEE Trans. Sig. Process. 64(23), 6254–6268 (2016)

    Article  MathSciNet  Google Scholar 

  27. Yang, Z., Xiang, Y., Rong, Y., Xie, K.: A convex geometry-based blind source separation method for separating nonnegative sources. IEEE Trans. Neural Netw. Learn. Syst. 26(8), 1635–1644 (2015)

    Article  MathSciNet  Google Scholar 

  28. Yin, P., Sun, Y., Xin, J.: A geometric blind source separation method based on facet component analysis. Sig. Image Video Process. 10(1), 19–28 (2016)

    Article  Google Scholar 

  29. Zhu, Y., Wang, N., Miller, D.J., Wang, Y.: Convex analysis of mixtures for separating non-negative well-grounded sources. Sci. Rep. 6(38350), 1–13 (2016). doi:10.1038/srep38350

    Google Scholar 

  30. Zhou, G., Cichocki, A.: Canonical polyadic decomposition based on a single mode blind source separation. IEEE Sig. Process. Lett. 19(8), 523–526 (2012)

    Article  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.

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Authors and Affiliations

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

    Rafał Zdunek & Tomasz Sadowski

Authors
  1. Rafał Zdunek
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  2. Tomasz Sadowski
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Correspondence to Rafał Zdunek .

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Editors and Affiliations

  1. Universidad de Granada, Granada, Spain

    Ignacio Rojas

  2. University of Malaga, Malaga, Spain

    Gonzalo Joya

  3. Polytechnic University of Catalonia, Vilanova i la Geltrú, Barcelona, Spain

    Andreu Catala

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Zdunek, R., Sadowski, T. (2017). XRAY Algorithm for Separable Nonnegative Tensor Factorization. In: Rojas, I., Joya, G., Catala, A. (eds) Advances in Computational Intelligence. IWANN 2017. Lecture Notes in Computer Science(), vol 10306. Springer, Cham. https://doi.org/10.1007/978-3-319-59147-6_22

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  • DOI: https://doi.org/10.1007/978-3-319-59147-6_22

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  • Online ISBN: 978-3-319-59147-6

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