Performance estimation for tensor CP decomposition with structured factors

The Canonical Polyadic tensor decomposition (CPD), also known as Candecomp/Parafac, is very useful in numerous scientific disciplines. Structured CPDs, i.e. with Toeplitz, circulant, or Hankel factor matrices, are often encountered in signal processing applications. As subsequently pointed out, specialized algorithms were recently proposed for estimating the deterministic parameters of structured CP decompositions. A closed-form expression of the Cramér-Rao bound (CRB) is derived, related to the problem of estimating CPD parameters, when the observed tensor is corrupted with an additive circular i.i.d. Gaussian noise. This CRB is provided for arbitrary tensor rank and sizes. Finally, the proposed CRB expression is used to asses the statistical efficiency of the existing algorithms by means of simulation results in the cases of third-order tensors having three circulant factors on one hand, and an Hankel factor on the other hand.

[1]  Rasmus Bro,et al.  Multiway analysis of epilepsy tensors , 2007, ISMB/ECCB.

[2]  Tamara G. Kolda,et al.  Tensor Decompositions and Applications , 2009, SIAM Rev..

[3]  Yonina C. Eldar,et al.  On the Constrained CramÉr–Rao Bound With a Singular Fisher Information Matrix , 2009, IEEE Signal Processing Letters.

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

[5]  Gérard Favier,et al.  An Algebraic Solution for the Candecomp/PARAFAC Decomposition with Circulant Factors , 2014, SIAM J. Matrix Anal. Appl..

[6]  Petre Stoica,et al.  Spectral Analysis of Signals , 2009 .

[7]  A. Kibangou,et al.  Nonlinear system modeling and identification using Volterra‐PARAFAC models , 2012 .

[8]  Philippe Forster,et al.  A new tool for multidimensional low-rank STAP filter: Cross HOSVDs , 2012, 2012 Proceedings of the 20th European Signal Processing Conference (EUSIPCO).

[9]  Nikos D. Sidiropoulos,et al.  Cramer-Rao lower bounds for low-rank decomposition of multidimensional arrays , 2001, IEEE Trans. Signal Process..

[10]  João Marcos Travassos Romano,et al.  Tensor space-time (TST) coding for MIMO wireless communication systems , 2012, Signal Process..

[11]  S. Kay.  Fundamentals of statistical signal processing: estimation theory , 1993 .

[12]  Souleymen Sahnoun,et al.  Tensor polyadic decomposition for antenna array processing , 2014 .

[13]  Nikos D. Sidiropoulos,et al.  Parallel factor analysis in sensor array processing , 2000, IEEE Trans. Signal Process..

[14]  Nikos D. Sidiropoulos,et al.  Blind PARAFAC receivers for DS-CDMA systems , 2000, IEEE Trans. Signal Process..

[15]  J. Landsberg.  Tensors: Geometry and Applications , 2011 .

[16]  André Lima Férrer de Almeida,et al.  PARAFAC-based unified tensor modeling for wireless communication systems with application to blind multiuser equalization , 2007, Signal Process..

[17]  Michael W. Berry,et al.  Discussion Tracking in Enron Email using PARAFAC. , 2008 .

[18]  Louis L. Scharf,et al.  Geometry of the Cramer-Rao bound , 1993, Signal Process..

[19]  Pierre Comon,et al.  Decomposing tensors with structured matrix factors reduces to rank-1 approximations , 2010, 2010 IEEE International Conference on Acoustics, Speech and Signal Processing.

[20]  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.

[21]  Lieven De Lathauwer,et al.  A Block Component Model-Based Blind DS-CDMA Receiver , 2008, IEEE Transactions on Signal Processing.

[22]  Rémy Boyer.  Deterministic asymptotic Cramér-Rao bound for the multidimensional harmonic model , 2008, Signal Process..

[23]  Remy Boyer.  Multidimensional Harmonic Retrieval: Exact, Asymptotic, and Modified Cramér-Rao Bounds , 2011 .

[24]  João Cesar M. Mota,et al.  Blind channel identification algorithms based on the Parafac decomposition of cumulant tensors: The single and multiuser cases , 2008, Signal Process..