Abstract
The applications of Non-Negative Tensor Factorization (NNTF) is an important tool for brain wave (EEG) analysis. For it to work efficiently, it is essential for NNTF to have a unique solution. In this paper we give a sufficient condition for NNTF to have a unique global optimal solution. For a third-order tensor T we define a matrix by some rearrangement of T and it is shown that the rank of the matrix is less than or equal to the rank of T. It is also shown that if both ranks are equal to r, the decomposition into a sum of r tensors of rank 1 is unique under some assumption.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Cao, B., Shen, D., Sun, J.-T., Wang, X., Yang, Q., Chen, Z.: Detect and Track Latent Factors with Online Nonnegative Matrix Factorization. In: IJCAI 2007, pp. 2689–2694 (2007)
Chichoki, A., Zdunek, R., Amari, S.-i.: Non-Negative Tensor Factorization Using Csiszar’s Divergence. In: Rosca, J., Erdogmus, D., PrÃncipe, J.C., Haykin, S. (eds.) ICA 2006. LNCS, vol. 3889, pp. 32–39. Springer, Heidelberg (2006)
Lee, D.D., Seung, H.S.: Algorithms for Non-negative Matrix Factorization. In: Leen, T.K., Dietterich, T.G., Tresp, V. (eds.) Advances in Neural Information Processing Systems, vol. 13, pp. 556–562. MIT Press, Cambridge (2001)
Miwakeichi, F., MartÃnez-Montes, E., Valdés-Sosa, P.A., Nishiyama, N., Mizuhara, H., Yamaguchi, Y.: Decomposing EEG Data into Space–Time–Frequency Components Using Parallel Factor Analysis. NeuroImage 22, 1035–1045 (2004)
Morup, M., Hansen, L.K., Herman, C.S., Parnas, J., Arnfred, S.M.: Parallel Factor Analysis as an Exploratory Tool for Wavelet Transformed Event-related EEG. NeuroImage 29, 938–947 (2006)
Wang, J., Zhong, W., Zhang, J.: NNMF-Based Factorization techniques for High-Accuracy Privacy Protection on Non-negative-valued Datasets. In: Perner, P. (ed.) ICDM 2006. LNCS (LNAI), vol. 4065, pp. 513–517. Springer, Heidelberg (2006)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Sumi, T., Sakata, T. (2007). A Sufficient Condition for the Unique Solution of Non-Negative Tensor Factorization. In: Davies, M.E., James, C.J., Abdallah, S.A., Plumbley, M.D. (eds) Independent Component Analysis and Signal Separation. ICA 2007. Lecture Notes in Computer Science, vol 4666. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74494-8_15
Download citation
DOI: https://doi.org/10.1007/978-3-540-74494-8_15
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-74493-1
Online ISBN: 978-3-540-74494-8
eBook Packages: Computer ScienceComputer Science (R0)