# Recursive Minimum Complex Kernel Risk-Sensitive Loss Algorithm

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Fixed Point Algorithm under Minimizing Complex Kernel Risk-Sensitive Loss

#### 2.1. Complex Kernel Risk-Sensitive Loss

#### 2.2. Recursive Minimum Complex Kernel Risk-Sensitive Loss (RMCKRSL)

#### 2.2.1. Cost Function

#### 2.2.2. Recursive Solution

Algorithm 1: RMCKRSL. |

Input:$\sigma $, $\lambda $, $d\left(k\right)$, $x\left(k\right)$1. Initializations: $\delta =0.0001$, $\text{}{p}_{0}=0$, $w\left(0\right)=0$, ${R}_{0}=\delta I$, ${R}_{0}^{-1}={\delta}^{-1}I$ 2. While $\left\{\begin{array}{cc}x\left(k\right)& d\left(k\right)\end{array}\right\}$ available, do 3. $e\left(k\right)=d\left(k\right)-{w}^{H}\left(k\right)x\left(k\right)$ 4. ${\kappa}_{\sigma}^{c}\left(e\left(k\right)\right)=\mathrm{exp}\left(-{\left|e\left(k\right)\right|}^{2}/2{\sigma}^{2}\right)$ 5. $h\left[e\left(k\right)\right]=\mathrm{exp}\left[\lambda \left(1-{\kappa}_{\sigma}^{c}\left(e\left(k\right)\right)\right)\right]{\kappa}_{\sigma}^{c}\left(e\left(k\right)\right)$ 6. $b\left(k\right)={\widehat{R}}_{k-1}^{-1}x\left(k\right){\left({h}^{-1}\left(e\left(k\right)\right)+{x}^{H}\left(k\right){\widehat{R}}_{k-1}^{-1}x\left(k\right)\right)}^{-1}$ 7. $w\left(k\right)=w\left(k-1\right)+b\left(k\right){e}^{*}\left(k\right)$ 8. ${\widehat{R}}_{k}^{-1}={\widehat{R}}_{k-1}^{-1}-{\widehat{R}}_{k-1}^{-1}x\left(k\right){\left({h}^{-1}\left(e\left(k\right)\right)+{x}^{H}\left(k\right){\widehat{R}}_{k-1}^{-1}x\left(k\right)\right)}^{-1}\times {x}^{H}\left(k\right){\widehat{R}}_{k-1}^{-1}$ 9. End while |

10. ${\widehat{w}}_{0}=w\left(k\right)$ |

Output: Estimated filter weight ${\widehat{w}}_{0}$ |

## 3. Convergence Analysis

#### 3.1. Stability Analysis

**Remark**

**1.**

_{0}/k.

#### 3.2. Excess Mean Square Error

**Remark**

**2.**

## 4. Simulation

#### 4.1. Example 1

#### 4.2. Example 2

**x**(k) = [s(k − 1) s(k − 2) ⋯ s(k − 6)] and the performance is measured by the mean square error (MSE) with $\mathrm{MSE}\left(k\right)=\frac{1}{N-6}{\displaystyle \sum _{l=7}^{N}\left({\left|s\left(l\right)-{w}^{H}\left(k\right)x\left(l\right)\right|}^{2}\right)}$. The convergence curves of different algorithms on the basis of MSE are compared in Figure 7. One may observe that the RMCKRSL has a faster convergence rate and better filter accuracy than other algorithms. In addition, the RLS behaves the worst since the minimum square error criterion is not robust to the impulse noise.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Principe, J.C. Information Theoretic Learning: Renyi’s Entropy and Kernel Perspectives; Springer: New York, NY, USA, 2010. [Google Scholar]
- Chen, B.; Zhu, Y.; Hu, J.; Principe, J.C. System Parameter Identification: Information Criteria and Algorithms; Newnes: Oxford, UK, 2013. [Google Scholar]
- Liu, W.; Pokharel, P.P.; Pr´ıncipe, J. Correntropy: A localized similarity measure. In Proceedings of the 2006 IEEE International Joint Conference on Neural Network (IJCNN), Vancouver, BC, Canada, 16–21 July 2006; pp. 4919–4924. [Google Scholar]
- Liu, W.; Pokharel, P.P.; Principe, J.C. Correntropy: Properties and applications in non-Gaussian signal processing. IEEE Trans. Signal Process.
**2007**, 55, 5286–5298. [Google Scholar] [CrossRef] - Singh, A.; Principe, J.C. Using correntropy as a cost function in linear adaptive filters. In Proceedings of the 2009 International Joint Conference on Neural Networks (IJCNN), Atlanta, GA, USA, 14–19 June 2009; pp. 2950–2955. [Google Scholar]
- Singh, A.; Principe, J.C. A loss function for classification based on a robust similarity metric. In Proceedings of the 2010 International Joint Conference on Neural Networks (IJCNN), Barcelona, Spain, 18–23 July 2010; pp. 1–6. [Google Scholar]
- Zhao, S.; Chen, B.; Principe, J.C. Kernel adaptive filtering with maximum correntropy criterion. In Proceedings of the 2011 International Joint Conference on Neural Networks (IJCNN), San Jose, CA, USA, 31 July–5 August 2011; pp. 2012–2017. [Google Scholar]
- Chen, B.; Xing, L.; Liang, J.; Zheng, N.; Principe, J.C. Steady-state mean-square error analysis for adaptive filtering under the maximum correntropy criterion. IEEE Signal Process. Lett.
**2014**, 21, 880–884. [Google Scholar] - Wu, Z.; Peng, S.; Chen, B.; Zhao, H. Robust Hammerstein adaptive filtering under maximum correntropy criterion. Entropy
**2015**, 17, 7149–7166. [Google Scholar] [CrossRef] - Chen, B.; Wang, J.; Zhao, H.; Zheng, N.; Principe, J.C. Convergence of a Fixed-Point Algorithm under Maximum Correntropy Criterion. IEEE Signal Process. Lett.
**2015**, 22, 1723–1727. [Google Scholar] [CrossRef] - Wang, W.; Zhao, J.; Qu, H.; Chen, B.; Principe, J.C. Convergence performance analysis of an adaptive kernel width MCC algorithm. AEU-Int. J. Electron. Commun.
**2017**, 76, 71–76. [Google Scholar] [CrossRef] - Liu, X.; Chen, B.; Zhao, H.; Qin, J.; Cao, J. Maximum Correntropy Kalman Filter with State Constraints. IEEE Access
**2017**, 5, 25846–25853. [Google Scholar] [CrossRef] - Wang, F.; He, Y.; Wang, S.; Chen, B. Maximum total correntropy adaptive filtering against heavy-tailed noises. Signal Process.
**2017**, 141, 84–95. [Google Scholar] [CrossRef] - Chen, B.; Liu, X.; Zhao, H.; Principe, J.C. Maximum correntropy Kalman filter. Automatica
**2017**, 76, 70–77. [Google Scholar] [CrossRef] [Green Version] - Wang, S.; Dang, L.; Wang, W.; Qian, G.; Tse, C.K. Kernel Adaptive Filters with Feedback Based on Maximum Correntropy. IEEE Access
**2018**, 6, 10540–10552. [Google Scholar] [CrossRef] - He, Y.; Wang, F.; Yang, J.; Rong, H.; Chen, B. Kernel adaptive filtering under generalized Maximum Correntropy Criterion. In Proceedings of the 2016 International Joint Conference on Neural Networks (IJCNN), Vancouver, BC, Canada, 24–29 July 2016; pp. 1738–1745. [Google Scholar]
- Chen, B.; Xing, L.; Zhao, H.; Zheng, N.; Príncipe, J.C. Generalized correntropy for robust adaptive filtering. IEEE Trans. Signal Process.
**2016**, 64, 3376–3387. [Google Scholar] [CrossRef] - Chen, B.; Wang, R. Risk-sensitive loss in kernel space for robust adaptive filtering. In Proceedings of the 2015 IEEE International Conference on Digital Signal Processing (DSP), Singapore, 21–24 July 2015; pp. 921–925. [Google Scholar]
- Chen, B.; Xing, L.; Xu, B.; Zhao, H.; Zheng, N.; Príncipe, J.C. Kernel Risk-Sensitive Loss: Definition, Properties and Application to Robust Adaptive Filtering. IEEE Trans. Signal Process.
**2017**, 65, 2888–2901. [Google Scholar] [CrossRef] [Green Version] - Guimaraes, J.P.F.; Fontes, A.I.R.; Rego, J.B.A.; Martins, A.M.; Principe, J.C. Complex correntropy: Probabilistic interpretation and application to complex-valued data. IEEE Signal Process. Lett.
**2017**, 24, 42–45. [Google Scholar] [CrossRef] - Guimaraes, J.P.F.; Fontes, A.I.R.; Rego, J.B.A.; Martins, A.M.; Principe, J.C. Complex Correntropy Function: Properties, and application to a channel equalization problem. Expert Syst. Appl.
**2018**, 107, 173–181. [Google Scholar] [CrossRef] - Alliney, S.; Ruzinsky, S.A. An algorithm for the minimization of mixed l1 and l2 norms with application to Bayesian estimation. IEEE Trans. Signal Process.
**1994**, 42, 618–627. [Google Scholar] [CrossRef] - Mandic, D.; Goh, V. Complex Valued Nonlinear Adaptive Filters: Noncircularity, Widely Linear and Neural Models (ser. Adaptive and Cognitive Dynamic Systems: Signal Processing, Learning, Communications and Control); John Wiley & Sons: New York, NY, USA, 2009. [Google Scholar]
- Diniz, P.S.R. Adaptive Filtering: Algorithms and Practical Implementation, 4th ed.; Springer-Verlag: New York, NY, USA, 2013. [Google Scholar]
- Qian, G.; Wang, S.; Wang, L.; Duan, S. Convergence Analysis of a Fixed Point Algorithm under Maximum Complex Correntropy Criterion. IEEE Signal Process. Lett.
**2018**, 24, 1830–1834. [Google Scholar] [CrossRef] - Qian, G.; Wang, S. Generalized Complex Correntropy: Application to Adaptive Filtering of Complex Data. IEEE Access
**2018**, 6, 19113–19120. [Google Scholar] [CrossRef] - Qian, G.; Wang, S. Complex Kernel Risk-Sensitive Loss: Application to Robust Adaptive Filtering in Complex Domain. IEEE Access
**2018**, 6, 2169–3536. [Google Scholar] [CrossRef] - Wirtinger, W. Zur formalen theorie der funktionen von mehr complexen veränderlichen. Math. Ann.
**1927**, 97, 357–375. [Google Scholar] [CrossRef] - Bouboulis, P.; Theodoridis, S. Extension of Wirtinger’s calculus to reproducing Kernel Hilbert spaces and the complex kernel LMS. IEEE Trans. Signal Process.
**2011**, 59, 964–978. [Google Scholar] - Zhang, X. Matrix Analysis and Application, 2nd ed.; Tsinghua University Press: Beijing, China, 2013. [Google Scholar]
- Picinbono, B. On circularity. IEEE Trans. Signal Process.
**1994**, 42, 3473–3482. [Google Scholar] [CrossRef]

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Qian, G.; Luo, D.; Wang, S.
Recursive Minimum Complex Kernel Risk-Sensitive Loss Algorithm. *Entropy* **2018**, *20*, 902.
https://doi.org/10.3390/e20120902

**AMA Style**

Qian G, Luo D, Wang S.
Recursive Minimum Complex Kernel Risk-Sensitive Loss Algorithm. *Entropy*. 2018; 20(12):902.
https://doi.org/10.3390/e20120902

**Chicago/Turabian Style**

Qian, Guobing, Dan Luo, and Shiyuan Wang.
2018. "Recursive Minimum Complex Kernel Risk-Sensitive Loss Algorithm" *Entropy* 20, no. 12: 902.
https://doi.org/10.3390/e20120902