Reproducing kernel Hilbert spaces (RKHS)

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Table of contents

1 Introduction 
1.1 Thesis context
1.2 Motivation
1.3 Contributions
1.4 Organization of the thesis
2 Linear and kernel adaptive filtering: a general overview 
2.1 Introduction
2.2 Linear adaptive filtering
2.2.1 Overview of linear adaptive filters
2.2.2 Wiener filter
2.2.3 Least-mean-square algorithm
2.2.4 LMS convergence analysis
2.3 Preliminaries on kernel-based methods
2.3.1 Definition of kernel
2.3.2 Reproducing kernel Hilbert spaces (RKHS)
2.3.3 Examples of kernel
2.3.4 Kernel construction
2.4 The existing kernel adaptive filtering algorithms
2.4.1 Sparsification criteria
2.4.2 Kernel affine projection algorithm
2.4.3 Kernel normalized least-mean-square algorithm
2.4.4 Kernel recursive least-square algorithm
2.5 Conclusion
3 Monokernel LMS algorithm with online dictionary 
3.1 Introduction
3.1.1 Monokernel LMS algorithms
3.2 Mean square error analysis
3.3 Transient behavior analysis
3.3.1 Mean weight behavior
3.3.2 Mean square error behavior
3.4 Steady-state behavior
3.5 Simulation results and discussion
3.5.1 Example 1
3.5.2 Example 2
3.5.3 Discussion
3.6 KLMS algorithm with forward-backward splitting
3.6.1 Forward-backward splitting method in a nutshell
3.6.2 Application to KLMS algorithm
3.6.3 Stability in the mean
3.7 Simulation results of proposed algorithm
3.8 Conclusion
4 Multikernel adaptive filtering algorithm 
4.1 Introduction
4.2 Multikernel LMS algorithms
4.2.1 Single-input multikernel LMS algorithm
4.2.2 Multi-input multikernel LMS algorithm
4.2.3 Optimal solution
4.3 Convergence behavior analysis of MI-MKLMS algorithm
4.3.1 Preliminaries and assumptions
4.3.2 Mean weight error analysis
4.3.3 Mean squared error analysis
4.3.4 Steady-state behavior
4.4 Simulation results and discussion
4.4.1 Example 1
4.4.2 Example 2
4.4.3 Discussion
4.5 Conclusion
5 Complex kernel adaptive filtering algorithm 
5.1 Introduction
5.2 Complex monokernel adaptive filtering algorithms
5.2.1 Complexified kernel LMS algorithm
5.2.2 Pure complex kernel LMS algorithm
5.3 Complex multikernel adaptive filtering
5.3.1 The framework
5.3.2 Augmented complex kernel least-mean-squared algorithm
5.4 Stochastic behavior analysis of ACKLMS algorithm
5.4.1 Mean weight error analysis
5.4.2 Mean-square error analysis
5.4.3 Steady-state behavior
5.5 Simulation results and discussion
5.6 Conclusion
6 Diffusion adaptation over networks with KLMS 
6.1 Introduction
6.2 The kernel least-mean-square algorithm
6.3 Diffusion adaptation with KLMS algorithm
6.3.1 Functional adapt-then-Combine diffusion strategy
6.3.2 Functional Combine-then-adapt diffusion strategy
6.3.3 Implementation
6.3.4 Stability of functional diffusion strategy in the mean
6.4 Simulation results and discussion
6.4.1 Example 1
6.4.2 Example 2
6.5 Conclusion
7 Conclusions and perspectives 
7.1 Thesis summary
7.2 Perspectives