Abstract
Although several methods are available to compute the multilinear rank of multi-way arrays, they are not sufficiently robust with respect to the noise. In this paper, we propose a novel Multilinear Tensor Decomposition (MTD) method, namely R-MTD (Robust MTD), which is able to identify the multilinear rank even in the presence of noise. The latter is based on sparsity and group sparsity of the core tensor imposed by means of the \(l_1\) norm and the mixed-norm, respectively. After several iterations of R-MTD, the underlying core tensor is sufficiently well estimated, which allows for an efficient use of the minimum description length approach and an accurate estimation of the multilinear rank. Computer results show the good behavior of R-MTD.
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References
Wax, M., Kailath, T.: Detection of signals by information theoretic criteria. IEEE Trans. Acoust. Speech Signal Process. 33(2), 387–392 (1985)
Schwarz, G.: Estimating the dimension of a model. Ann. Stat. 6(2), 461–464 (1978)
Rissanen, J.: Modeling by shortest data description. Automatica 14(5), 465–471 (1978)
Tucker, L.R.: Implications of factor analysis of three-way matrices for measurement of change. In: Problems in Measuring Change, pp. 122–137 (1963)
De Lathauwer, L., De Moor, B., Vandewalle, J.: A multilinear singular value decomposition. SIAM J. Matrix Anal. Appl. 21(4), 1253–1278 (2000)
De Lathauwer, L., De Moor, B., Vandewalle, J.: On the best rank-1 and rank-(\(\mathit{R}_{1}, \mathit{R}_{2},\ldots, \mathit{R}_{n}\)) approximation of higher-order tensors. SIAM J. Matrix Anal. 21(4), 1324–1342 (2000)
Yokota, T., Lee, N., Cichocki, A.: Robust multilinear tensor rank estimation using higher order singular value decomposition and information criteria. IEEE Trans. Signal Process. 65(5), 1196–1206 (2017)
Xie, Q., Zhao, Q., Meng, D., Xu, Z.: Kronecker-basis-representation based tensor sparsity and its applications to tensor recovery. IEEE Trans. Pattern Anal. Mach. Intell. (2017)
Han, X., Albera, L., Kachenoura, A., Senhadji, L., Shu, H.: Low rank canonical polyadic decomposition of tensors based on group sparsity. In: IEEE 25th European Signal Processing Conference (EUSIPCO), pp. 668–672 (2017)
Han, X., Albera, L., Kachenoura, A., Senhadji, L., Shu, H.: Block term decomposition with rank estimation using group sparsity. In: Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP) (2017)
Shu, X., Porikli, F., Ahuja, N.: Robust orthonormal subspace learning: efficient recovery of corrupted low-rank matrices. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 3874–3881 (2014)
Fazel, M.: Matrix rank minimization with applications. Ph.D. dissertation, Ph.D. thesis, Stanford University (2002)
Donoho, D.L., Johnstone, I.M.: Adapting to unknown smoothness via wavelet shrinkage. J. Am. Stat. Assoc. 90(432), 1200–1224 (1995)
Boyd, S., Parikh, N., Chu, E., Peleato, B., Eckstein, J.: Distributed optimization and statistical learning via the alternating direction method of multipliers. Found. \(Trends^{\textregistered }\) Mach. Learn. 3(1), 1–122 (2011)
Petersen, K.B., Pedersen, M.S., et al.: The Matrix Cookbook. Technical University of Denmark, pp. 7–15 (2008)
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Han, X., Albera, L., Kachenoura, A., Shu, H., Senhadji, L. (2018). Robust Multilinear Decomposition of Low Rank Tensors. 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_1
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