Skip to main content
SpringerLink
Log in

Kernel Least Mean Square Based on the Nyström Method

  • Published:
Circuits, Systems, and Signal Processing Aims and scope Submit manuscript

Abstract

The kernel least mean square (KLMS) algorithm is the simplest algorithm in kernel adaptive filters. However, the network growth of KLMS is still an issue for preventing its online applications, especially when the length of training data is large. The Nyström method is an efficient method for curbing the growth of the network size. In this paper, we apply the Nyström method to the KLMS algorithm, generating a novel algorithm named kernel least mean square based on the Nyström method (NysKLMS). In comparison with the KLMS algorithm, the proposed NysKLMS algorithm can reduce the computational complexity, significantly. The NysKLMS algorithm is proved to be convergent in the mean square sense when its step size satisfies some conditions. In addition, the theoretical steady-state excess mean square error of NysKLMS supported by simulations is calculated to evaluate the filtering accuracy. Simulations on system identification and nonlinear channel equalization show that the NysKLMS algorithm can approach the filtering performance of the KLMS algorithm by using much lower computational complexity, and outperform the KLMS with the novelty criterion, the KLMS with the surprise criterion, the quantized KLMS, the fixed-budget QKLMS, and the random Fourier features KLMS.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

References

  1. T.Y. Al-Naffouri, A.H. Sayed, Transient analysis of adaptive filters with error nonlinearities. IEEE Trans. Signal Process. 51(3), 653–663 (2003)

    Article  Google Scholar 

  2. A.Z.U.M. Al-Saggaf, M. Moinuddin, M. Arif, The q-least mean squares algorithm. Signal Process. 111, 50–60 (2015)

    Article  Google Scholar 

  3. N. Aronszajn, Theory of reproducing kernels. Trans. Am. Math. Soc. 68(3), 337–404 (1950)

    Article  MathSciNet  MATH  Google Scholar 

  4. P. Bouboulis, S. Pougkakiotis, S. Theodoridis, Efficient KLMS and KRLS algorithms: a random Fourier feature perspective, in IEEE Statistical Signal Processing Workshop (SSP), Palma de Mallorca, Spain, 26–29 June 2016, pp. 1–5

  5. C. Burges, A tutorial on support vector machines for pattern recognition. Data Min. Knowl. Disc. 2(2), 121–167 (1998)

    Article  Google Scholar 

  6. B. Chen, L. Li, W. Liu, J.C. Príncipe, Nonlinear adaptive filtering in kernel spaces (Springer, Berlin, 2014), pp. 715–734

    Google Scholar 

  7. B. Chen, S. Zhao, P. Zhu, J.C. Príncipe, Mean square convergence analysis for kernel least mean square algorithm. Signal Process. 92(11), 2624–2632 (2012)

    Article  Google Scholar 

  8. B. Chen, S. Zhao, P. Zhu, J.C. Príncipe, Quantized kernel least mean square algorithm. IEEE Trans. Neural Netw. Learn. Syst. 23, 22–32 (2012)

    Article  Google Scholar 

  9. S. Craciun, D. Cheney, K. Gugel, J.C. Sanchez, J.C. Príncipe, Wireless transmission of neural signals using entropy and mutual information compression. IEEE Trans. Neural Syst. Rehabil. Eng. 19(1), 35–44 (2011)

    Article  Google Scholar 

  10. L. Csato, M. Opper, Sparse on-line Gaussian processes. Neural Comput. 14(3), 641–668 (2002)

    Article  MATH  Google Scholar 

  11. P. Drineas, M.W. Mahoney, On the nystrom method for approximating a Gram matrix for improved kernel-based learning. J. Mach. Learn. Res. 6, 2153–2175 (2005)

    MathSciNet  MATH  Google Scholar 

  12. Y. Engel, S. Mannor, R. Meir, The kernel recursive least-squares algorithm. IEEE Trans. Signal Process. 52(8), 2275–2285 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  13. H. Fan, Q. Song, A linear recurrent kernel online learning algorithm with sparse updates. Neural Netw. 50(2), 142–153 (2014)

    Article  MATH  Google Scholar 

  14. Z. Hu, M. Lin, C. Zhang, Dependent online kernel learning with constant number of random fourier features. IEEE Trans. Neural Netw. Learn. Syst. 26(10), 2464–2476 (2015)

    Article  MathSciNet  Google Scholar 

  15. A. Khalili, A. Rastegarnia, M.K. Islam, T.Y. Rezaii, Codebook design for vector quantization based on a kernel fuzzy learning algorithm. Circuits Syst. Signal Process. 30(5), 999–1010 (2011)

    Article  Google Scholar 

  16. A. Khalili, A. Rastegarnia, M.K. Islam, T.Y. Rezaii, Steady-state tracking analysis of adaptive filter with maximum correntropy criterion. Circuits Syst. Signal Process. 36(4), 1725–1734 (2017)

    Article  MATH  Google Scholar 

  17. S. Khan, I. Naseem, R. Togneri, M. Bennamoun, A novel adaptive kernel for the RBF neural networks. Circuits Syst. Signal Process. 36(4), 1639–1653 (2017)

    Article  Google Scholar 

  18. J. Kivinen, A. Smola, R. Williamson, Online learning with kernels. IEEE Trans. Signal Process. 52(8), 2165–2176 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  19. T. Lehn-Schiøler, A. Hegde, D. Erdogmus, J.C. Príncipe, Vector quantization using information theoretic concepts. Natural Comput. 4(1), 39–51 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  20. S. Li, L. Song, T. Qiu, Steady-state and tracking analysis of fractional lower-order constant modulus algorithm. Circuits Syst. Signal Process. 30(6), 1275–1288 (2011)

    Article  MATH  Google Scholar 

  21. M. Lin, L. Zhang, R. Jin, S. Weng, C. Zhang, Online kernel learning with nearly constant support vectors. Neurocomputing 179, 26–36 (2016)

    Article  Google Scholar 

  22. Y.Y. Linde, A. Buzo, R.M. Gray, An algorithm for vector quantizer design. IEEE Trans. Commun. 28(1), 84–95 (1980)

    Article  Google Scholar 

  23. W. Liu, I. Park, J.C. Príncipe, An information theoretic approach of designing sparse kernel adaptive filters. IEEE Trans. Neural Netw. 20(12), 1950–1961 (2009)

    Article  Google Scholar 

  24. W. Liu, P. Pokharel, J.C. Príncipe, The kernel least-mean-square algorithm. IEEE Trans. Signal Process. 56(2), 543–554 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  25. W. Liu, J.C. Príncipe, Kernel affine projection algorithms. EURASIP J. Adv. Signal Process. 2008(1), 1–13 (2008)

    MATH  Google Scholar 

  26. S. Nan, L. Sun, B. Chen, Z. Lin, K.A. Toh, Density-dependent quantized least squares support vector machine for large data sets. IEEE Trans. Neural Netw. Learn. Syst. 28(1), 94–106 (2015)

    Article  Google Scholar 

  27. J. Platt, A resource-allocating network for function interpolation. Neural Comput. 3(2), 213–225 (1991)

    Article  MathSciNet  Google Scholar 

  28. J.C. Príncipe, W. Liu, S.S. Haykin, Kernel adaptive filtering: a comprehensive introduction (Wiley, New York, 2010)

    Google Scholar 

  29. A. Rahimi, B. Recht, Random features for large scale kernel machines, in Advances in Neural Information Processing Systems, Vancouver, Canada, 3–6 December 2007, pp. 1177–1184

  30. A.H. Sayed, Fundamentals of adaptive filtering (Wiley, New York, 2003)

    Google Scholar 

  31. K. Slavakis, S. Theodoridis, I. Yamada, Online kernel-based classification using adaptive projection algorithms. IEEE Trans. Signal Process. 56(7), 2781–2796 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  32. S. Smale, D. Zhou, Geometry on probability spaces. Constr. Approx. 30(3), 311–323 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  33. S. Wang, Y. Zheng, S. Duan, L. Wang, Simplified quantised kernel least mean square algorithm with fixed budget. Electron Lett. 52, 1453–1455 (2016)

    Article  Google Scholar 

  34. S. Wang, Y. Zheng, S. Duan, L. Wang, H. Tan, Quantized kernel maximum correntropy and its mean square convergence analysis. Digital Signal Process. 63(3), 164–176 (2017)

    Article  Google Scholar 

  35. C. Williams, M. Seeger, Using the nystrom method to speed up kernel machines, in Advances in Neural Information Processing Systems, Vancouver, Canada, 3–8 December 2001, pp. 682–688

  36. T. Yang, Y. Li, M. Mahdavi, R. Jin, Z. Zhou, Nystrom method vs random Fourier features: a theoretical and empirical comparison, in Advances in Neural Information Processing Systems, Lake Tahoe, USA, 3–6 December 2012, pp. 485–493

  37. N.R. Yousef, A.H. Sayed, A unified approach to the steady-state and tracking analyses of adaptive filters. IEEE Trans. Signal Process. 49(2), 314–324 (2001)

    Article  Google Scholar 

  38. S. Zhao, B. Chen, P. Zhu, J.C. Príncipe, Fixed budget quantized kernel least-mean-square algorithm. Signal Process. 93(9), 2759–2770 (2013)

    Article  Google Scholar 

Download references

Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant No. 61671389), China Postdoctoral Science Foundation Funded Project (Grant No. 2017M620779), Chongqing Postdoctoral Science Foundation Special Funded Project (Grant No. Xm2017107), and the Fundamental Research Funds for the Central Universities (Grant No. XDJK2016C096).

Author information

Authors and Affiliations

  1. College of Electronic and Information Engineering, Southwest University, Chongqing, 400715, China

    Shi-Yuan Wang, Wen-Yue Wang & Lu-Juan Dang

  2. Chongqing Key Laboratory of Nonlinear Circuits and Intelligent Information Processing, Chongqing, 400715, China

    Shi-Yuan Wang, Wen-Yue Wang & Lu-Juan Dang

  3. Graduate School at Shenzhen of Tsinghua University, Shenzhen, 518055, China

    Yun-Xiang Jiang

Authors
  1. Shi-Yuan Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Wen-Yue Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Lu-Juan Dang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Yun-Xiang Jiang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shi-Yuan Wang.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, SY., Wang, WY., Dang, LJ. et al. Kernel Least Mean Square Based on the Nyström Method. Circuits Syst Signal Process 38, 3133–3151 (2019). https://doi.org/10.1007/s00034-018-1006-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00034-018-1006-2

Keywords

Search

Navigation