On Low-Rank Hankel Matrix Denoising

The low-complexity assumption in linear systems can often be expressed as rank deficiency in data matrices with generalized Hankel structure. This makes it possible to denoise the data by estimating the underlying structured low-rank matrix. However, standard low-rank approximation approaches are not guaranteed to perform well in estimating the noise-free matrix. In this paper, recent results in matrix denoising by singular value shrinkage are reviewed. A novel approach is proposed to solve the low-rank Hankel matrix denoising problem by using an iterative algorithm in structured low-rank approximation modified with data-driven singular value shrinkage. It is shown numerically in both the input-output trajectory denoising and the impulse response denoising problems, that the proposed method performs the best in terms of estimating the noise-free matrix among existing algorithms of low-rank matrix approximation and denoising.

[1]  S. Van Huffel,et al.  Exact and Approximate Modeling of Linear Systems: A Behavioral Approach , 2006 .

[2]  D. Donoho,et al.  The Optimal Hard Threshold for Singular Values is 4 / √ 3 , 2013 .

[3]  H. Akaike.  A new look at the statistical model identification , 1974 .

[4]  Geoffrey Ye Li,et al.  A parameter estimation scheme for damped sinusoidal signals based on low-rank Hankel approximation , 1997, IEEE Trans. Signal Process..

[5]  Louis L. Scharf,et al.  The SVD and reduced rank signal processing , 1991, Signal Process..

[6]  Jong Chul Ye,et al.  Annihilating Filter-Based Low-Rank Hankel Matrix Approach for Image Inpainting , 2015, IEEE Transactions on Image Processing.

[7]  Shuaiqi Liu,et al.  Hankel Low-Rank Approximation for Seismic Noise Attenuation , 2019, IEEE Transactions on Geoscience and Remote Sensing.

[8]  S. Chatterjee,et al.  Matrix estimation by Universal Singular Value Thresholding , 2012, 1212.1247.

[9]  Lennart Ljung,et al.  Subspace-based identification of infinite-dimensional multivariable systems from frequency-response data , 1996, Autom..

[10]  C. Eckart,et al.  The approximation of one matrix by another of lower rank , 1936 .

[11]  Bart De Moor,et al.  Algorithms for deterministic balanced subspace identification , 2005, Autom..

[12]  Torsten Söderström,et al.  Model-structure selection by cross-validation , 1986 .

[13]  Stephen P. Boyd,et al.  A rank minimization heuristic with application to minimum order system approximation , 2001, Proceedings of the 2001 American Control Conference. (Cat. No.01CH37148).

[14]  Ivan Markovsky,et al.  Software for weighted structured low-rank approximation , 2014, J. Comput. Appl. Math..

[15]  David L. Donoho,et al.  The Optimal Hard Threshold for Singular Values is 4/sqrt(3) , 2013, 1305.5870.

[16]  Stephen P. Boyd,et al.  Log-det heuristic for matrix rank minimization with applications to Hankel and Euclidean distance matrices , 2003, Proceedings of the 2003 American Control Conference, 2003..

[17]  M. Moonen,et al.  On- and off-line identification of linear state-space models , 1989 .

[18]  Julie Josse,et al.  Adaptive shrinkage of singular values , 2013, Statistics and Computing.

[19]  P. Holmes,et al.  The Proper Orthogonal Decomposition in the Analysis of Turbulent Flows , 1993 .

[20]  V. Marčenko,et al.  DISTRIBUTION OF EIGENVALUES FOR SOME SETS OF RANDOM MATRICES , 1967 .

[21]  Hossein Hassani,et al.  The Empirical Distribution of the Singular Values of a Random Hankel Matrix , 2015 .

[22]  David L. Donoho,et al.  Optimal Shrinkage of Singular Values , 2014, IEEE Transactions on Information Theory.

[23]  Raj Rao Nadakuditi,et al.  OptShrink: An Algorithm for Improved Low-Rank Signal Matrix Denoising by Optimal, Data-Driven Singular Value Shrinkage , 2013, IEEE Transactions on Information Theory.

[24]  H. Abdi,et al.  Principal component analysis , 2010 .

[25]  Paul Tseng,et al.  Hankel Matrix Rank Minimization with Applications to System Identification and Realization , 2013, SIAM J. Matrix Anal. Appl..

[26]  Dietmar Bauer,et al.  Order estimation for subspace methods , 2001, Autom..

[27]  Roy S. Smith,et al.  Frequency Domain Subspace Identification Using Nuclear Norm Minimization and Hankel Matrix Realizations , 2014, IEEE Transactions on Automatic Control.

[28]  Raj Rao Nadakuditi,et al.  The singular values and vectors of low rank perturbations of large rectangular random matrices , 2011, J. Multivar. Anal..

[29]  Ivan Markovsky,et al.  Structured low-rank approximation and its applications , 2008, Autom..

[30]  Ivan Markovsky,et al.  Structured Low-Rank Approximation with Missing Data , 2013, SIAM J. Matrix Anal. Appl..

[31]  J. B. Rosen,et al.  Low Rank Approximation of a Hankel Matrix by Structured Total Least Norm , 1999 .