Non-iterative low-multilinear-rank tensor approximation with application to decomposition in rank-(1,L,L) terms

Computing low-rank approximations is one of the most important and well-studied problems involving tensors. In particular, approximations of low multilinear rank (mrank) have long been investigated by virtue of their usefulness for subspace analysis and dimensionality reduction purposes. The first part of this paper introduces a novel algorithm which computes a low-mrank tensor approximation non-iteratively. This algorithm, called sequential low-rank approximation and projection (SeLRAP), generalizes a recently proposed scheme aimed at the rank-one case, SeROAP. We show that SeLRAP is always at least as accurate as existing alternatives in the rank-(1,L,L) approximation of third-order tensors. By means of computer simulations with random tensors, such a superiority was actually observed for a range of different tensor dimensions and mranks. In the second part, we propose an iterative deflationary approach for computing a decomposition of a tensor in low-mrank blocks, termed DBTD. It first extracts an initial estimate of the blocks by employing SeLRAP, and then iteratively refines them by recomputing low-mrank approximations of each block plus the residue. Our numerical results show that, in the rank-(1,L,L) case, this remarkably simple scheme outperforms existing algorithms if the blocks are not too correlated. In particular, it is much less sensitive to discrepancies among the block's norms.

[1]  J. Cullum,et al.  Lanczos algorithms for large symmetric eigenvalue computations , 1985 .

[2]  Vin de Silva,et al.  Tensor rank and the ill-posedness of the best low-rank approximation problem , 2006, math/0607647.

[3]  Lieven De Lathauwer,et al.  An enhanced line search scheme for complex-valued tensor decompositions. Application in DS-CDMA , 2008, Signal Process..

[4]  Lieven De Lathauwer,et al.  Blind Deconvolution of DS-CDMA Signals by Means of Decomposition in Rank-$(1,L,L)$ Terms , 2008, IEEE Transactions on Signal Processing.

[5]  Pierre Comon,et al.  A novel non-iterative algorithm for low-multilinear-rank tensor approximation , 2017, 2017 25th European Signal Processing Conference (EUSIPCO).

[6]  Gérard Favier,et al.  Low-Rank Tensor Recovery using Sequentially Optimal Modal Projections in Iterative Hard Thresholding (SeMPIHT) , 2017, SIAM J. Sci. Comput..

[7]  L. Lathauwer,et al.  On the best low multilinear rank approximation of higher-order tensors , 2010 .

[8]  W. Marsden I and J , 2012 .

[9]  Lieven De Lathauwer,et al.  Decompositions of a Higher-Order Tensor in Block Terms - Part II: Definitions and Uniqueness , 2008, SIAM J. Matrix Anal. Appl..

[10]  Paul Van Dooren,et al.  Jacobi Algorithm for the Best Low Multilinear Rank Approximation of Symmetric Tensors , 2013, SIAM J. Matrix Anal. Appl..

[11]  Gene H. Golub,et al.  Matrix computations , 1983 .

[12]  Nikos D. Sidiropoulos,et al.  Tensor Algebra and Multidimensional Harmonic Retrieval in Signal Processing for MIMO Radar , 2010, IEEE Transactions on Signal Processing.

[13]  André Lima Férrer de Almeida,et al.  Enhanced block term decomposition for atrial activity extraction in atrial fibrillation ECG , 2016, 2016 IEEE Sensor Array and Multichannel Signal Processing Workshop (SAM).

[14]  Reinhold Schneider,et al.  Low rank tensor recovery via iterative hard thresholding , 2016, ArXiv.

[15]  Pierre Comon,et al.  A Finite Algorithm to Compute Rank-1 Tensor Approximations , 2016, IEEE Signal Processing Letters.

[16]  Gene H. Golub,et al.  Rank-One Approximation to High Order Tensors , 2001, SIAM J. Matrix Anal. Appl..

[17]  Pierre Comon,et al.  Rank-1 Tensor Approximation Methods and Application to Deflation , 2015, ArXiv.

[18]  Joos Vandewalle,et al.  A Multilinear Singular Value Decomposition , 2000, SIAM J. Matrix Anal. Appl..

[19]  Kristiaan Pelckmans,et al.  Tensor decompositions for the analysis of atomic resolution electron energy loss spectra. , 2017, Ultramicroscopy.

[20]  Raf Vandebril,et al.  A New Truncation Strategy for the Higher-Order Singular Value Decomposition , 2012, SIAM J. Sci. Comput..

[21]  Lieven De Lathauwer,et al.  Decompositions of a Higher-Order Tensor in Block Terms - Part III: Alternating Least Squares Algorithms , 2008, SIAM J. Matrix Anal. Appl..

[22]  Florian Roemer,et al.  Higher-Order SVD-Based Subspace Estimation to Improve the Parameter Estimation Accuracy in Multidimensional Harmonic Retrieval Problems , 2008, IEEE Transactions on Signal Processing.

[23]  Rafal Zdunek,et al.  Electromyography and mechanomyography signal recognition: Experimental analysis using multi-way array decomposition methods , 2017 .

[24]  Lieven De Lathauwer,et al.  Optimization-Based Algorithms for Tensor Decompositions: Canonical Polyadic Decomposition, Decomposition in Rank-(Lr, Lr, 1) Terms, and a New Generalization , 2013, SIAM J. Optim..

[25]  Berkant Savas,et al.  A Newton-Grassmann Method for Computing the Best Multilinear Rank-(r1, r2, r3) Approximation of a Tensor , 2009, SIAM J. Matrix Anal. Appl..