Multidimensional Harmonic Retrieval via Coupled Canonical Polyadic Decomposition—Part I: Model and Identifiability

Multidimensional Harmonic Retrieval (MHR) is a fundamental problem in signal processing. We make a connection with coupled Canonical Polyadic Decomposition (CPD), which allows us to better exploit the rich MHR structure than existing approaches in the derivation of uniqueness results. We discuss both deterministic and generic conditions. We obtain a deterministic condition that is both necessary and sufficient but which may be difficult to check in practice. We derive mild deterministic relaxations that are easy to verify. We also discuss the variant in which the generators have unit norm. We narrow the transition zone between generic uniqueness and generic non-uniqueness to two values of the number of harmonics. We explain differences with one-dimensional HR.

[1]  T. Kailath,et al.  A subspace rotation approach to signal parameter estimation , 1986, Proceedings of the IEEE.

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

[3]  Eberhard Freitag,et al.  Analytic Functions of Several Complex Variables , 2011 .

[4]  Lieven De Lathauwer,et al.  Multidimensional Harmonic Retrieval via Coupled Canonical Polyadic Decomposition—Part II: Algorithm and Multirate Sampling , 2017, IEEE Transactions on Signal Processing.

[5]  Xiaoli Ma,et al.  Ieee Transactions on Signal Processing Multidimensional Frequency Estimation with Finite Snapshots in the Presence of Identical Frequencies , 2022 .

[6]  Yonina C. Eldar,et al.  An algorithm for exact super-resolution and phase retrieval , 2013, 2014 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).

[7]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[8]  H. Kiers.  Towards a standardized notation and terminology in multiway analysis , 2000 .

[9]  Thomas Kailath,et al.  ESPRIT-estimation of signal parameters via rotational invariance techniques , 1989, IEEE Trans. Acoust. Speech Signal Process..

[10]  Nikos D. Sidiropoulos,et al.  Generalizing Carathéodory's uniqueness of harmonic parameterization to N dimensions , 2001, IEEE Trans. Inf. Theory.

[11]  Jun Liu,et al.  An Eigenvector-Based Approach for Multidimensional Frequency Estimation With Improved Identifiability , 2006, IEEE Transactions on Signal Processing.

[12]  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.

[13]  Thomas Kailath,et al.  On spatial smoothing for direction-of-arrival estimation of coherent signals , 1985, IEEE Trans. Acoust. Speech Signal Process..

[14]  Charles R. Johnson,et al.  Matrix Analysis, 2nd Ed , 2012 .

[15]  Lieven De Lathauwer,et al.  Generic Uniqueness of a Structured Matrix Factorization and Applications in Blind Source Separation , 2016, IEEE Journal of Selected Topics in Signal Processing.

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

[17]  Martin Vetterli,et al.  Exact sampling results for some classes of parametric nonbandlimited 2-D signals , 2004, IEEE Transactions on Signal Processing.

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

[19]  Lieven De Lathauwer,et al.  Blind Signal Separation via Tensor Decomposition With Vandermonde Factor: Canonical Polyadic Decomposition , 2013, IEEE Transactions on Signal Processing.

[20]  Lieven De Lathauwer,et al.  Canonical Polyadic Decomposition of Third-Order Tensors: Reduction to Generalized Eigenvalue Decomposition , 2013, SIAM J. Matrix Anal. Appl..

[21]  Lieven De Lathauwer,et al.  New Uniqueness Conditions for the Canonical Polyadic Decomposition of Third-Order Tensors , 2015, SIAM J. Matrix Anal. Appl..

[22]  Lieven De Lathauwer,et al.  Blind Separation of Exponential Polynomials and the Decomposition of a Tensor in Rank-(Lr, Lr, 1) Terms , 2011, SIAM J. Matrix Anal. Appl..

[23]  Lieven De Lathauwer,et al.  Multidimensional ESPRIT: A coupled canonical polyadic decomposition approach , 2014, 2014 IEEE 8th Sensor Array and Multichannel Signal Processing Workshop (SAM).

[24]  V. Strassen.  Rank and optimal computation of generic tensors , 1983 .

[25]  P. P. Vaidyanathan,et al.  Nested Arrays in Two Dimensions, Part II: Application in Two Dimensional Array Processing , 2012, IEEE Transactions on Signal Processing.

[26]  A. Stegeman.  On uniqueness conditions for Candecomp/Parafac and Indscal with full column rank in one mode , 2009 .

[27]  Lieven De Lathauwer,et al.  On the Uniqueness of the Canonical Polyadic Decomposition of Third-Order Tensors - Part II: Uniqueness of the Overall Decomposition , 2013, SIAM J. Matrix Anal. Appl..

[28]  Lieven De Lathauwer,et al.  Coupled tensor decompositions for applications in array signal processing , 2013, 2013 5th IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP).

[29]  Lieven De Lathauwer,et al.  Coupled Canonical Polyadic Decompositions and (Coupled) Decompositions in Multilinear Rank- (Lr, n, Lr, n, 1) Terms - Part II: Algorithms , 2015, SIAM J. Matrix Anal. Appl..

[30]  Roland Badeau,et al.  High-resolution spectral analysis of mixtures of complex exponentials modulated by polynomials , 2006, IEEE Transactions on Signal Processing.

[31]  Lieven De Lathauwer,et al.  Structured Data Fusion , 2015, IEEE Journal of Selected Topics in Signal Processing.

[32]  L. Lathauwer,et al.  Sufficient conditions for uniqueness in Candecomp/Parafac and Indscal with random component matrices , 2006, Psychometrika.

[33]  Chandra Sekhar Seelamantula,et al.  Exact Phase Retrieval for a Class of 2-D Parametric Signals , 2015, IEEE Transactions on Signal Processing.

[34]  Giorgio Ottaviani,et al.  On Generic Identifiability of 3-Tensors of Small Rank , 2011, SIAM J. Matrix Anal. Appl..

[35]  Lieven De Lathauwer,et al.  A Link between the Canonical Decomposition in Multilinear Algebra and Simultaneous Matrix Diagonalization , 2006, SIAM J. Matrix Anal. Appl..

[36]  S. Unnikrishna Pillai,et al.  Forward/backward spatial smoothing techniques for coherent signal identification , 1989, IEEE Trans. Acoust. Speech Signal Process..

[37]  Nikos D. Sidiropoulos,et al.  Almost sure identifiability of constant modulus multidimensional harmonic retrieval , 2002, IEEE Trans. Signal Process..

[38]  S. Leurgans,et al.  A Decomposition for Three-Way Arrays , 1993, SIAM J. Matrix Anal. Appl..

[39]  Lieven De Lathauwer,et al.  On the Uniqueness of the Canonical Polyadic Decomposition of Third-Order Tensors - Part I: Basic Results and Uniqueness of One Factor Matrix , 2013, SIAM J. Matrix Anal. Appl..

[40]  Lieven De Lathauwer,et al.  Coupled Canonical Polyadic Decompositions and (Coupled) Decompositions in Multilinear Rank-(Lr, n, Lr, n, 1) Terms - Part I: Uniqueness , 2015, SIAM J. Matrix Anal. Appl..

[41]  Lieven De Lathauwer,et al.  Generic Uniqueness Conditions for the Canonical Polyadic Decomposition and INDSCAL , 2014, SIAM J. Matrix Anal. Appl..

[42]  Nikos D. Sidiropoulos,et al.  Kruskal's permutation lemma and the identification of CANDECOMP/PARAFAC and bilinear models with constant modulus constraints , 2004, IEEE Transactions on Signal Processing.

[43]  Ed F. Deprettere,et al.  Analysis of joint angle-frequency estimation using ESPRIT , 2003, IEEE Trans. Signal Process..

[44]  Nikos D. Sidiropoulos,et al.  Almost sure identifiability of multidimensional harmonic retrieval , 2001, 2001 IEEE International Conference on Acoustics, Speech, and Signal Processing. Proceedings (Cat. No.01CH37221).

[45]  K. J. Liu,et al.  A high-resolution technique for multidimensional NMR spectroscopy , 1998, IEEE Transactions on Biomedical Engineering.