Low rank tensor deconvolution

In this paper, we propose a low-rank tensor deconvolution problem which seeks multiway replicative patterns and corresponding activating tensors of rank-1. An alternating least squares (ALS) algorithm has been derived for the model to sequentially update loading components and the patterns. In addition, together with a good initialisation method using tensor diagonalization, the update rules have been implemented with a low cost using fast inversion of block Toeplitz matrices as well as an efficient update strategy. Experiments show that the proposed model and the algorithm are promising in feature extraction and clustering.

[1]  Pierre Comon,et al.  Tensors : A brief introduction , 2014, IEEE Signal Processing Magazine.

[2]  Alexey Ozerov,et al.  Multichannel Nonnegative Matrix Factorization in Convolutive Mixtures for Audio Source Separation , 2010, IEEE Transactions on Audio, Speech, and Language Processing.

[3]  Pierre Comon,et al.  Blind Multilinear Identification , 2012, IEEE Transactions on Information Theory.

[4]  José M. Badía,et al.  Solving the block–Toeplitz least‐squares problem in parallel , 2005, Concurr. Pract. Exp..

[5]  Andrzej Cichocki,et al.  Low-rank blind nonnegative matrix deconvolution , 2012, 2012 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).

[6]  Lars Kai Hansen,et al.  Shift-invariant multilinear decomposition of neuroimaging data , 2008, NeuroImage.

[7]  Andrzej Cichocki,et al.  Nonnegative Matrix and Tensor Factorization T , 2007 .

[8]  A. Cichocki,et al.  Non-Orthogonal Tensor Diagonalization, a Tool for Block Tensor Decompositions , 2014 .

[9]  Paris Smaragdis,et al.  Convolutive Speech Bases and Their Application to Supervised Speech Separation , 2007, IEEE Transactions on Audio, Speech, and Language Processing.

[10]  Paris Smaragdis,et al.  Non-negative Matrix Factor Deconvolution; Extraction of Multiple Sound Sources from Monophonic Inputs , 2004, ICA.

[11]  H. Akaike.  Block Toeplitz Matrix Inversion , 1973 .

[12]  Ting-Zhu Huang,et al.  The Inverses of Block Toeplitz Matrices , 2013 .

[13]  Andrzej Cichocki,et al.  On Fast algorithms for orthogonal Tucker decomposition , 2014, 2014 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).

[14]  Ali Taylan Cemgil,et al.  Probabilistic latent tensor factorization framework for audio modeling , 2011, 2011 IEEE Workshop on Applications of Signal Processing to Audio and Acoustics (WASPAA).

[15]  Derry Fitzgerald,et al.  Extended Nonnegative Tensor Factorisation Models for Musical Sound Source Separation , 2008, Comput. Intell. Neurosci..

[16]  Rafal Zdunek,et al.  Improved Convolutive and Under-Determined Blind Audio Source Separation with MRF Smoothing , 2012, Cognitive Computation.

[17]  Eric Moulines,et al.  Subspace methods for the blind identification of multichannel FIR filters , 1994, Proceedings of ICASSP '94. IEEE International Conference on Acoustics, Speech and Signal Processing.

[18]  R. Harshman,et al.  Shifted factor analysis—Part I: Models and properties , 2003 .

[19]  Andrzej Cichocki,et al.  Tensor Decompositions for Signal Processing Applications: From two-way to multiway component analysis , 2014, IEEE Signal Processing Magazine.

[20]  Joos Vandewalle,et al.  On the Best Rank-1 and Rank-(R1 , R2, ... , RN) Approximation of Higher-Order Tensors , 2000, SIAM J. Matrix Anal. Appl..

[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]  Andrzej Cichocki,et al.  Fast Alternating LS Algorithms for High Order CANDECOMP/PARAFAC Tensor Factorizations , 2013, IEEE Transactions on Signal Processing.