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Structured low-rank approximation for nonlinear matrices

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Abstract

Structured low-rank approximation problems are very popular in the scientific community since they are involved in several applications in different scientific fields. The search for new algorithms which improve the performance of the existing ones is still an active research topic. However, almost all the existing results focus on the class of linearly structured matrices. Motivated by two applications in the field of computer algebra and system identification, respectively, we propose an algorithm for structured low-rank approximation of nonlinearly structured matrices. The idea comes from an extension of the same method studied and tested by the author in the linear case.

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Notes

  1. The algorithm is not expected to find a common factor if two roots of the same polynomial are very close.

References

  1. Eckart, C., Young, G.: The approximation of one matrix by another one of lower rank. Psychometrika 1, 211–218 (1936). https://doi.org/10.1007/BF02288367

    Article  MATH  Google Scholar 

  2. Markovsky, I.: Low-rank Approximation: Algorithms, Implementation, Applications, 2nd edn. Springer, Berlin (2019)

    Book  MATH  Google Scholar 

  3. Chu, M.T., Funderlic, R.E., Plemmons, L.J.: Structured low rank approximation. Linear Algebra Its Appl. 366, 157–172 (2003). https://doi.org/10.1016/S0024-3795(02)00505-0

    Article  MathSciNet  MATH  Google Scholar 

  4. De Moor, B.: Total least squares for affinely structured matrices and the noisy realization problem. IEEE Trans. Signal Process. 42, 3104–3113 (1994). https://doi.org/10.1109/78.330370

    Article  Google Scholar 

  5. Usevich, K., Markovsky, I.: Variable projection for affinely structured low-rank approximation in weighted 2-norms. J. Comput. Appl. Math. 272, 430–448 (2014). https://doi.org/10.1016/j.cam.2013.04.034

    Article  MathSciNet  MATH  Google Scholar 

  6. Shklyar, S., Kukush, A., Markovsky, I., Van Huffel, S.: On the conic section fitting problem. J. Multivar. Anal. 98, 588–624 (2007). https://doi.org/10.1016/j.jmva.2005.12.003

    Article  MathSciNet  MATH  Google Scholar 

  7. Markovsky, I., Kukush, A., Van Huffel, S.: Consistent least squares fitting of ellipsoids. Numer. Math. 98, 177–194 (2004). https://doi.org/10.1007/s00211-004-0526-9

    Article  MathSciNet  MATH  Google Scholar 

  8. Markovsky, I., Usevich, K.: Nonlinearly structured low-rank approximation. In: Fu, Y.R. (ed.) Low-Rank and Sparse Modeling for Visual Analysis, pp 1–22. Springer (2014). https://doi.org/10.1007/978-3-319-12000-3_1

  9. Usevich, K., Markovsky, I.: Adjusted least squares fitting of algebraic hypersurfaces. Linear Algebra Appl. 502, 243–274 (2014). https://doi.org/10.1016/j.laa.2015.07.023

    Article  MathSciNet  MATH  Google Scholar 

  10. Rosen, J.B., Park, H., Glick, J.: Structured total least norm for nonlinear problems. SIAM J. Matrix Anal. Appl. 20(1), 14–30 (1998). https://doi.org/10.1137/S0895479896301662

    Article  MathSciNet  MATH  Google Scholar 

  11. Lemmerling, P., Van Huffel, S., De Moor, B.: The structured total least-squares approach for non-linearly structured matrices. Numer. Linear Algebra Appl. 9, 321–332 (2002). https://doi.org/10.1002/nla.276

    Article  MathSciNet  MATH  Google Scholar 

  12. Fazzi, A., Kukush, A., Markovsky, I.: Bias correction for Vandermonde low-rank approximation. Econ. Stat. In press. https://doi.org/10.1016/j.ecosta.2021.09.001 (2021)

  13. Fazzi, A., Guglielmi, N., Markovsky, I.: An ODE based method for computing the approximate greatest common divisor of polynomials. Numer. Algorithms 81, 719–740 (2019). https://doi.org/10.1007/s11075-018-0569-0

    Article  MathSciNet  MATH  Google Scholar 

  14. Guglielmi, N., Markovsky, I.: An ODE based method for computing the distance of coprime polynomials to common divisibility. SIAM J. Numer. Anal. 55, 1456–1482 (2017). https://doi.org/10.1137/15M1018265

    Article  MathSciNet  MATH  Google Scholar 

  15. Fazzi, A., Guglielmi, N., Markovsky, I.: Generalized algorithms for the approximate matrix polynomial GCD of reducing data uncertainties with application to MIMO system and control. J. Comput. Appl. Math. 393. https://doi.org/10.1016/j.cam.2021.113499 (2021)

  16. Fazzi, A., Guglielmi, N., Markovsky, I.: A gradient system approach for Hankel structured low-rank approximation. Linear Algebra its appl. 623, 236–257 (2021). https://doi.org/10.1016/j.laa.2020.11.016

    Article  MathSciNet  MATH  Google Scholar 

  17. Barkhuijsen, H., De Beer, R., Van Ormondt, D.: Improved algorithm for noniterative time-domain model fitting to exponentially damped magnetic resonance signals. J. Magentic Resonance 73, 553–557 (1987). https://doi.org/10.1016/0022-2364(87)90023-0

    Article  Google Scholar 

  18. Markovsky, I., Fazzi, A., Guglielmi, N.: Applications of polynomial common factor computation in signal processing. In: et al, Y.D. (ed.) Latent Variable Analysis and Signal Separation. Lecture Notes in Computer Science, vol. 10891, pp 99–106. Springer (2018. https://doi.org/10.1007/978-3-319-93764-9_10)

  19. Fazzi, A., Guglielmi, N., Markovsky, I.: Computing common factors of matrix polynomials with applications in system and control theory. In: Proc. of the IEEE Conf. on Decision and Control, Nice, France, pp. 7721–7726. https://doi.org/10.1109/CDC40024.2019.9030137 (2019)

  20. De Iuliis, V., Germani, A., Manes, C.: Identification of forward and feedback transfer functions in closed-loop systems with feedback delay. IFAC-PapersOnLine 50(1), 12847–12852 (2017). https://doi.org/10.1016/j.ifacol.2017.08.1935

    Article  Google Scholar 

  21. Markovsky, I., Van Huffel, S.: An algorithm for approximate common divisor computation. In: Proceedings of MTNS’06, Kyoto, Japan, pp. 274–279 (2006)

  22. Pan, V.Y.: Computation of approximate polynomial GCDs and an extension. Inf. Comput. 167, 71–85 (2001). https://doi.org/10.1006/inco.2001.3032

    Article  MathSciNet  MATH  Google Scholar 

  23. Qiu, W., Hua, Y., Abed-Meraim, K.: A subspace method for the computation of the GCD of polynomials. Automatica 33, 741–743 (1997). https://doi.org/10.1016/S0005-1098(96)00244-0

    Article  MathSciNet  MATH  Google Scholar 

  24. Terui, A.: GPGCD: An iterative method for calculating approximate GCD of univariate polynomials. Theoret. Comput. Sci. 479, 127–149 (2013). https://doi.org/10.1007/978-3-642-15274-0_22

    Article  MathSciNet  MATH  Google Scholar 

  25. Usevich, K., Markovsky, I.: Variable projection methods for approximate (greatest) common divisor computations. Theor. Comput. Sci. 681, 176–198 (2017). https://doi.org/10.1016/j.tcs.2017.03.028

    Article  MathSciNet  MATH  Google Scholar 

  26. Zeng, Z., Dayton, B.H.: The approximate GCD of inexact polynomials, Part I: a univariate algorithm. In: Proceedings of the 2004 International Symposium on Symbolic and Algebraic Computation, pp 320–327. ACM (2004), https://doi.org/10.1145/1005285.1005331

  27. Markovsky, I., Willems, J.C., Van Huffel, S., De Moor, B.: Exact and approximate modeling of linear systems: a behavioral approach. SIAM (2006)

  28. Berberich, J., Allgöwer, F.: A trajectory-based framework for data-driven system analysis and control. In: Eur. Control Conf, pp 1365–1370 (2020), https://doi.org/10.23919/ECC51009.2020.9143608

  29. Rueda-Escobedo, J.G., Schiffer, J.: Data-driven internal model control of second-order discrete volterra systems. In: IEEE Conf. Decision Control, pp 4572–4579 (2020), https://doi.org/10.1109/CDC42340.2020.9304122

  30. Mishra, V., Markovsky, I., Fazzi, A., Dreesen, P.: Data-driven simulation for NARX Systems. In: 2021 29Th European Signal Processing Conference (EUSIPCO), pp 1055–1059 (2021), https://doi.org/10.23919/EUSIPCO54536.2021.9616226

  31. Vaidya, P.G., Anand, P.S.S., Nagaraj, N.: A nonlinear generalization of singular value decomposition and its applications to mathematical modeling and chaotic cryptanalysis. Acta Appl. Math. 112, 205–221 (2010). https://doi.org/10.1007/s10440-010-9560-z

    Article  MathSciNet  MATH  Google Scholar 

  32. Golub, G.H., Van Loan, C.F.: Matrix Computations, 4th edn. The Johns Hopkins University Press, Baltimore (2013)

    Book  MATH  Google Scholar 

  33. Andrew, A.L., Chu, K. -W. E., Lancaster, P.: Derivatives of eigenvalues and eigenvectors of matrix functions. SIAM J. Matrix Anal. Appl. 14 (4), 903–926 (1993). https://doi.org/10.1137/0614061

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The author is thankful to an anonymous referee for several constructive comments which improved the quality of the paper.

Funding

The research leading to these results has received funding from the Fond for Scientific Research Vlaanderen (FWO) projects G028015N and G090117N and the FNRS-FWO under Excellence of Science (EOS) Project no 30468160 “Structured low-rank matrix/tensor approximation: numerical optimization-based algorithms and applications.”

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Authors and Affiliations

  1. Department ELEC, Vrije Universiteit Brussel, Pleinlaan 2, Brussels, 1050, Belgium

    Antonio Fazzi

  2. Department of Information Engineering, University of Padova, via Gradenigo 6/b, Padova, 35131, Italy

    Antonio Fazzi

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  1. Antonio Fazzi
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Correspondence to Antonio Fazzi.

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Fazzi, A. Structured low-rank approximation for nonlinear matrices. Numer Algor 93, 1561–1580 (2023). https://doi.org/10.1007/s11075-022-01479-5

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