Abstract
This communication deals with N-th order tensor decompositions. More precisely, we are interested in the (Canonical) Polyadic Decomposition. In our case, this problem is formulated under a variational approach where the considered criterion to be minimized is composed of several terms: one accounting for the fidelity to data and others that can represent not only regularization (such as sparsity prior) but also hard constraints (such as nonnegativity). The resulting optimization problem is solved by using the Block-Coordinate Variable Metric Forward-Backward (BC-VMFB) algorithm. The robustness and efficiency of the suggested approach is illustrated on realistic synthetic data such as those encountered in the context of environmental data analysis and fluorescence spectroscopy. Our simulations are performed on 4-th order tensors.
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Notes
- 1.
In our case, the easiest way to proceed is to consider that each block matches a loading matrix, but other choices could have been made.
- 2.
In practice, elementwise operations are performed instead making it possible to avoid memory issues.
References
Bro, R.: Parafac: tutorial and applications. Chemom. Intell. Lab. Syst. 38(2), 149–171 (1997)
Carroll, P., Chang, J.J.: Analysis of individual differences in multi-dimensional scaling via n-way generalization of Eckart-Young decomposition. Psychometrika 35(3), 283–319 (1970)
Chaux, C., Combettes, P.L., Pesquet, J.C., Wajs, V.R.: A variational formulation for frame based inverse problems. Inverse Probl. 23(4), 1495–1518 (2007)
Chouzenoux, E., Pesquet, J.C., Repetti, A.: Variable metric forward-backward algorithm for minimizing the sum of a differentiable function and a convex function. J. Optim. Theory Appl. 162(1), 107–132 (2014)
Chouzenoux, E., Pesquet, J.C., Repetti, A.: A block coordinate variable metric forward-backward algorithm. J. Global Optim. 66(3), 457–485 November 2016
Cichocki, A., Zdunek, R., Phan, A.H., Amari, S.I.: Non Negative Matrix and Tensor Factorizations: Application to Exploratory Multi-way Data Analysis and Blind Separation. Wiley, Chichester (2009)
Combettes, P.L., Pesquet, J.C.: Proximal splitting methods in signal processing. In: Bauschke, H.H., Burachik, R., Combettes, P.L., Elser, V., Luke, D.R., Wolkowicz, H. (eds.) Fixed-point Algorithms for Inverse Problems in Science and Engineering, pp. 185–212. Springer, New York (2010)
Comon, P., Jutten, C.: Handbook of Blind Source Separation, Independent Component Analysis and Applications. Academic Press, Oxford (2010). ISBN: 978-0-12-374726-6
Franc, A.: Etude algébrique des multi-tableaux: apport de l’algèbre tensorielle. Ph.D. thesis, University of Montepellier II, Montpellier, France (1992)
Harshman, R.A.: Foundation of the Parafac procedure: models and conditions for an explanatory multimodal factor analysis. UCLA Working Papers in Phonetics, vol. 16, pp. 1–84 (1970)
Hiriart-Urruty, J.B., Lemarechal, C.: Convex Analysis and Minimization Algorithms. Springer, Heidelberg (1993)
Huang, K., Sidiropoulos, N.D., Liavas, A.P.: A flexible and efficient algorithmic framework for constrained matrix and tensor factorization. IEEE Trans. Sig. Process. 64(19), 5052–5065 (2016)
Kolda, T.G., Bader, B.W.: Tensor decompositions and applications. SIAM Rev. 51(3), 455–500 (2009)
Kruskal, J.B.: Rank, decomposition and uniqueness for 3-way and n-way arrays. In: Coppi, R., Bolasco, S. (eds.) Multiway Data Analysis, pp. 7–18. North-Holland Publishing Co., Amsterdam (1989)
Lakowicz, J.R., Szmacinski, H., Nowaczyk, K., Berndt, K.W., Johnson, M.: Fluorescence lifetime imaging. Anal. Biochem. 202(2), 316–330 (1992)
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 (2001). http://papers.nips.cc/paper/1861-algorithms-for-non-negative-matrix-factorization.pdf
Moreau, J.J.: Proximité et dualité dans un espace hilbertien. Bull. Soc. Math. France 93, 273–299 (1965)
Phan, A.H., Tichavskỳ, P., Cichocki, A.: Fast alternating is algorithms for high order CANDECOMP\(/\)PARAFAC tensor factorizations. IEEE Trans. Sig. Proc. 61(19), 4834–4846 (2013)
Repetti, A., Chouzenoux, E., Pesquet, J.C.: A preconditioned Forward-Backward approach with application to large-scale nonconvex spectral unmixing problems. In: 39th IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP 2014), Florence, Italie, May 2014
Sidiropoulos, N., Bro, R.: On the uniqueness of multilinear decomposition of \(N\)-way arrays. J. Chemom. 14(3), 229–239 (2000)
Vervliet, N., Lathauwer, L.D.: A randomized block sampling approach to canonical polyadic decomposition of large-scale tensors. IEEE J. Sel. Top. Sig. Proces. 10(2), 284–295 (2016)
Vu, X.T., Chaux, C., Thirion-Moreau, N., Maire, S.: A new penalized nonnegative third order tensor decomposition using a block coordinate proximal gradient approach: application to 3D fluorescence spectroscopy. J. Chemometr., special issue on penalty methods 3 (2017, to appear). CEM. doi:10.1002/cem.2859
Xu, Y., Yin, W.: A block coordinate descent method for regularized multiconvex optimization with applications to nonnegative tensor factorization and completion. SIAM J. Imaging Sci. 6(3), 1758–1789 (2013)
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Vu, X., Chaux, C., Thirion-Moreau, N., Maire, S. (2017). A Proximal Approach for Nonnegative Tensor Decomposition. In: Tichavský, P., Babaie-Zadeh, M., Michel, O., Thirion-Moreau, N. (eds) Latent Variable Analysis and Signal Separation. LVA/ICA 2017. Lecture Notes in Computer Science(), vol 10169. Springer, Cham. https://doi.org/10.1007/978-3-319-53547-0_20
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