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Operator Approximation for Processing of Large Random Data Sets

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Book cover Concrete Operators, Spectral Theory, Operators in Harmonic Analysis and Approximation

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 236))

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Abstract

Suppose K y and K x are large sets of observed and reference signals, respectively, each containing N signals. Is it possible to construct a filter F : K y → K xthat requires a priori information only on few signals, p << N, from K x but performs better than the known filters based on a priori information on every reference signal from K x ? It is shown that the positive answer is achievable under quite unrestrictive assumptions. The device behind the proposed method is based on a special extension of the piecewise linear interpolation technique to the case of random signal sets. The proposed technique provides a single filter to process any signal from the arbitrarily large signal set. The filter is determined in terms of pseudo-inverse matrices so that it always exists.

Mathematics Subject Classification (2010). Primary 94A12; Secondary 65D05.

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References

  1. J. Chen, J. Benesty, Y. Huang, and S. Doclo, New Insights Into the Noise Reduction Wiener Filter, IEEE Trans. on Audio, Speech, and Language Processing, 14, No. 4, pp. 1218–1234, 2006.

    Google Scholar 

  2. M. Spurbeck and P. Schreier, Causal Wiener filter banks for periodically correlated time series, Signal Processing, 87, 6, pp. 1179–1187, 2007.

    Article  MATH  Google Scholar 

  3. J.S. Goldstein, I. Reed, and L.L. Scharf, “A Multistage Representation of the Wiener Filter Based on Orthogonal Projections,” IEEE Trans. on Information Theory, vol. 44, pp. 2943–2959, 1998.

    Article  MathSciNet  MATH  Google Scholar 

  4. Y. Hua, M. Nikpour, and P. Stoica, “Optimal Reduced-Rank estimation and filtering,” IEEE Trans. on Signal Processing, vol. 49, pp. 457–469, 2001.

    Article  Google Scholar 

  5. A. Torokhti and P. Howlett, Computational Methods for Modelling of Nonlinear Systems, Elsevier, 2007.

    Google Scholar 

  6. E.D. Sontag, Polynomial Response Maps, Lecture Notes in Control and Information Sciences, 13, 1979.

    Google Scholar 

  7. S. Chen and S.A. Billings, Representation of non-linear systems: NARMAX model, Int. J. Control, vol. 49, no. 3, pp. 1013–1032, 1989.

    Article  MathSciNet  MATH  Google Scholar 

  8. V.J. Mathews and G.L. Sicuranza, Polynomial Signal Processing, J. Wiley & Sons, 2001.

    Google Scholar 

  9. A. Torokhti and P. Howlett, Optimal Transform Formed by a Combination of Nonlinear Operators: The Case of Data Dimensionality Reduction, IEEE Trans. on Signal Processing, 54, No. 4, pp. 1431–1444, 2006.

    Article  Google Scholar 

  10. A. Torokhti and P. Howlett, Filtering and Compression for Infinite Sets of Stochastic Signals, Signal Processing, 89, pp. 291–304, 2009.

    Article  MATH  Google Scholar 

  11. J. Vesma and T. Saramaki, Polynomial-Based Interpolation Filters – Part I: Filter Synthesis, Circuits, Systems, and Signal Processing, Volume 26, Number 2, Pages 115–146, 2007.

    Article  MathSciNet  MATH  Google Scholar 

  12. A. Torokhti and J. Manton, Generic Weighted Filtering of Stochastic Signals, IEEE Trans. on Signal Processing, 57, Issue 12, pp. 4675–4685, 2009.

    Article  MathSciNet  Google Scholar 

  13. A. Torokhti and S. Miklavcic, Data Compression under Constraints of Causality and Variable Finite Memory, Signal Processing, 90 , Issue 10, pp. 2822–2834, 2010.

    Google Scholar 

  14. I. Babuska, U. Banerjee, J.E. Osborn, Generalized finite element methods: main ideas, results, and perspective, International Journal of Computational Methods, 1 (1), pp. 67–103, 2004.

    Article  MATH  Google Scholar 

  15. S. Kang and L. Chua, A global representation of multidimensional piecewise-linear functions with linear partitions, IEEE Trans. on Circuits and Systems, 25 Issue:11, pp. 938–940, 1978.

    Google Scholar 

  16. L.O. Chua and A.-C. Deng, Canonical piecewise-linear representation, IEEE Trans. on Circuits and Systems, 35 Issue:1, pp. 101–111, 1988.

    Google Scholar 

  17. J.-N. Lin and R. Unbehauen, Adaptive nonlinear digital filter with canonical piecewise-linear structure, IEEE Trans. on Circuits and Systems, 37 Issue:3, pp. 347–353, 1990.

    Google Scholar 

  18. J.-N. Lin and R. Unbehauen, Canonical piecewise-linear approximations, IEEE Trans. on Circuits and Systems I: Fundamental Theory and Applications, 39 Issue:8, pp. 697–699, 1992.

    Google Scholar 

  19. S.B. Gelfand and C.S. Ravishankar, A tree-structured piecewise linear adaptive filter, IEEE Trans. on Inf. Theory, 39, Issue 6, pp. 1907–1922, 1993.

    Article  MATH  Google Scholar 

  20. E.A. Heredia and G.R. Arce, Piecewise linear system modeling based on a continuous threshold decomposition, IEEE Trans. on Signal Processing, 44 Issue:6, pp. 1440– 1453, 1996.

    Google Scholar 

  21. G. Feng, Robust filtering design of piecewise discrete time linear systems, IEEE Trans. on Signal Processing, 53 Issue:2, pp. 599 - 605, 2005.

    Google Scholar 

  22. F. Russo, Technique for image denoising based on adaptive piecewise linear filters and automatic parameter tuning, IEEE Trans. on Instrumentation and Measurement, 55, Issue:4, pp. 1362–1367, 2006.

    Google Scholar 

  23. J.E. Cousseau, J.L. Figueroa, S. Werner, T.I. Laakso, Efficient Nonlinear Wiener Model Identification Using a Complex-Valued Simplicial Canonical Piecewise Linear Filter, IEEE Trans. on Signal Processing, 55 Issue:5, pp. 1780–1792, 2007.

    Google Scholar 

  24. P. Julian, A. Desages, B. D’Amico, Orthonormal high-level canonical PWL functions with applications to model reduction, IEEE Trans. on Circuits and Systems I: Fundamental Theory and Applications, 47 Issue:5, pp. 702–712, 2000.

    Google Scholar 

  25. T. Wigren, Recursive Prediction Error Identification Using the Nonlinear Wiener Model, Automatica, 29, 4, pp. 1011–1025, 1993.

    Article  MathSciNet  MATH  Google Scholar 

  26. G.H. Golub and C.F. van Loan, Matrix Computations, Johns Hopkins University Press, Baltimore, 1996.

    Google Scholar 

  27. T. Anderson, An Introduction to Multivariate Statistical Analysis, New York, Wiley, 1984.

    Google Scholar 

  28. L.I. Perlovsky and T.L. Marzetta, Estimating a Covariance Matrix from Incomplete Realizations of a Random Vector, IEEE Trans. on Signal Processing, 40, pp. 2097– 2100, 1992.

    Article  Google Scholar 

  29. O. Ledoit and M. Wolf, A well-conditioned estimator for large-dimensional covariance matrices, J. Multivariate Analysis 88, pp. 365–411, 2004.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. School of Mathematics and Statistics, University of South Australia, 1 Mawson Lakes Blv., SA, 5095, Adelaide, Australia

    Anatoli Torokhti

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  1. Anatoli Torokhti
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Correspondence to Anatoli Torokhti .

Editor information

Editors and Affiliations

  1. Departamento de Análisis Matemático, Universidad de Sevilla, Sevilla, Spain

    Manuel Cepedello Boiso

  2. Department of Mathematics, The Royal Institute of Technology, Stockholm, Sweden

    Håkan Hedenmalm

  3. Department of Mathematics & Computer Sci, Free University Amsterdam, Amsterdam, The Netherlands

    Marinus A. Kaashoek

  4. Departamento de Análisis Matemático, Universidad de Sevilla, Sevilla, Spain

    Alfonso Montes Rodríguez

  5. Department of Mathematics, Brown University, Providence, Rhode Island, USA

    Sergei Treil

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Torokhti, A. (2014). Operator Approximation for Processing of Large Random Data Sets. In: Cepedello Boiso, M., Hedenmalm, H., Kaashoek, M., Montes Rodríguez, A., Treil, S. (eds) Concrete Operators, Spectral Theory, Operators in Harmonic Analysis and Approximation. Operator Theory: Advances and Applications, vol 236. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0648-0_31

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