Digital Signal Processing with Kernel Methods

About the Authors xiii Preface xvii Acknowledgements xxi List of Abbreviations xxiii Part I Fundamentals and Basic Elements 1 1 From Signal Processing to Machine Learning 3 1.1 A New Science is Born: Signal Processing 3 1.1 A New Science is Born: Signal Processing 3 1.1.1 Signal Processing Before Being Coined 3 1.1.2 1948: Birth of the Information Age 4 1.1.3 1950s: Audio Engineering Catalyzes Signal Processing 4 1.2 From Analog to Digital Signal Processing 5 1.2.1 1960s: Digital Signal Processing Begins 5 1.2.2 1970s: Digital Signal Processing Becomes Popular 6 1.2.3 1980s: Silicon Meets Digital Signal Processing 6 1.3 Digital Signal Processing Meets Machine Learning 7 1.3.1 1990s: New Application Areas 7 1.3.2 1990s: Neural Networks, Fuzzy Logic, and Genetic Optimization 7 1.4 Recent Machine Learning in Digital Signal Processing 8 1.4.1 Traditional Signal Assumptions Are No Longer Valid 8 1.4.2 Encoding Prior Knowledge 8 1.4.3 Learning and Knowledge from Data 9 1.4.4 From Machine Learning to Digital Signal Processing 9 1.4.5 From Digital Signal Processing to Machine Learning 10 2 Introduction to Digital Signal Processing 13 2.1 Outline of the Signal Processing Field 13 2.1.1 Fundamentals on Signals and Systems 14 2.1.2 Digital Filtering 21 2.1.3 Spectral Analysis 24 2.1.4 Deconvolution 28 2.1.5 Interpolation 30 2.1.6 System Identification 31 2.1.7 Blind Source Separation 36 2.2.3 Sparsity, Compressed Sensing, and Dictionary Learning 44 2.3 Multidimensional Signals and Systems 48 2.3.1 Multidimensional Signals 49 2.3.2 Multidimensional Systems 51 2.4 Spectral Analysis on Manifolds 52 2.4.1 Theoretical Fundamentals 52 2.4.2 Laplacian Matrices 54 2.5 Tutorials and Application Examples 57 2.5.1 Real and Complex Signal Processing and Representations 57 2.5.2 Convolution, Fourier Transform, and Spectrum 63 2.5.3 Continuous-Time Signals and Systems 67 2.5.4 Filtering Cardiac Signals 70 2.5.5 Nonparametric Spectrum Estimation 74 2.5.6 Parametric Spectrum Estimation 77 2.5.7 Source Separation 81 2.5.8 Time-Frequency Representations and Wavelets 84 2.5.9 Examples for Spectral Analysis on Manifolds 87 2.6 Questions and Problems 94 3 Signal Processing Models 97 3.1 Introduction 97 3.2 Vector Spaces, Basis, and Signal Models 98 3.2.1 Basic Operations for Vectors 98 3.2.2 Vector Spaces 100 3.2.3 Hilbert Spaces 101 3.2.4 Signal Models 102 3.2.5 Complex Signal Models 104 3.2.6 Standard Noise Models in Digital Signal Processing 105 3.2.7 The Role of the Cost Function 107 3.2.8 The Role of the Regularizer 109 3.3 Digital Signal Processing Models 111 3.3.1 Sinusoidal Signal Models 112 3.3.2 System Identification Signal Models 113 3.3.3 Sinc Interpolation Models 116 3.3.4 Sparse Deconvolution 120 3.3.5 Array Processing 121 3.4 Tutorials and Application Examples 122 3.4.1 Examples of Noise Models 123 3.4.2 Autoregressive Exogenous System Identification Models 132 3.4.3 Nonlinear System Identification Using Volterra Models 138 3.4.4 Sinusoidal Signal Models 140 3.4.5 Sinc-based Interpolation 144 3.4.6 Sparse Deconvolution 152 3.4.7 Array Processing 157 3.5 Questions and Problems 160 3.A MATLABsimpleInterp Toolbox Structure 161 4 Kernel Functions and Reproducing Kernel Hilbert Spaces 165 4.1 Introduction 165 4.2 Kernel Functions and Mappings 169 4.2.1 Measuring Similarity with Kernels 169 4.2.2 Positive-Definite Kernels 169 4.2.3 Reproducing Kernel in Hilbert Space and Reproducing Property 170 4.2.4 Mercer's Theorem 173 4.3 Kernel Properties 174 4.3.1 Tikhonov's Regularization 175 4.3.2 Representer Theorem and Regularization Properties 176 4.3.3 Basic Operations with Kernels 178 4.4 Constructing Kernel Functions 179 4.4.1 Standard Kernels 179 4.4.2 Properties of Kernels 180 4.4.3 Engineering Signal Processing Kernels 181 4.5 Complex Reproducing Kernel in Hilbert Spaces 184 4.6 Support Vector Machine Elements for Regression and Estimation 186 4.6.1 Support Vector Regression Signal Model and Cost Function 186 4.6.2 Minimizing Functional 187 4.7 Tutorials and Application Examples 191 4.7.1 Kernel Calculations and Kernel Matrices 191 4.7.2 Basic Operations with Kernels 194 4.7.3 Constructing Kernels 197 4.7.4 Complex Kernels 199 4.7.5 Application Example for Support Vector Regression Elements 202 4.8 Concluding Remarks 205 4.9 Questions and Problems 205 Part II Function Approximation and Adaptive Filtering 209 5 A Support Vector Machine Signal Estimation Framework 211 5.1 Introduction 211 5.2 A Framework for Support Vector Machine Signal Estimation 213 5.3 Primal Signal Models for Support Vector Machine Signal Processing 216 5.3.1 Nonparametric Spectrum and System Identification 218 5.3.2 Orthogonal Frequency Division Multiplexing Digital Communications 220 5.3.3 Convolutional Signal Models 222 5.3.4 Array Processing 225 5.4 Tutorials and Application Examples 227 5.4.1 Nonparametric Spectral Analysis with Primal Signal Models 227 5.4.2 System Identification with Primal Signal Model gamma-filter 228 5.4.3 Parametric Spectral Density Estimation with Primal Signal Models 230 5.4.4 Temporal Reference Array Processing with Primal Signal Models 231 5.4.5 Sinc Interpolation with Primal Signal Models 233 6 Reproducing Kernel Hilbert Space Models for Signal Processing 241 6.1 Introduction 241 6.2 Reproducing Kernel Hilbert Space Signal Models 242 6.2.1 Kernel Autoregressive Exogenous Identification 244 6.2.2 Kernel Finite Impulse Response and the gamma-Filter 247 6.2.3 Kernel Array Processing with Spatial Reference 248 6.2.4 Kernel Semiparametric Regression 249 6.3 Tutorials and Application Examples 258 6.3.1 Nonlinear System Identification with Support Vector Machine-Autoregressive and Moving Average 258 6.3.2 Nonlinear System Identification with the gamma-filter 260 6.3.3 Electric Network Modeling with Semiparametric Regression 264 6.3.4 Promotional Data 272 6.3.5 Spatial and Temporal Antenna Array Kernel Processing 275 6.4 Questions and Problems 279 7 Dual Signal Models for Signal Processing 281 7.1 Introduction 281 7.2 Dual Signal Model Elements 281 7.3 Dual Signal Model Instantiations 283 7.3.1 Dual Signal Model for Nonuniform Signal Interpolation 283 7.3.2 Dual Signal Model for Sparse Signal Deconvolution 284 7.3.3 Spectrally Adapted Mercer Kernels 285 7.4 Tutorials and Application Examples 289 7.4.1 Nonuniform Interpolation with the Dual Signal Model 290 7.4.2 Sparse Deconvolution with the Dual Signal Model 292 7.4.3 Doppler Ultrasound Processing for Fault Detection 294 7.4.4 Spectrally Adapted Mercer Kernels 296 7.4.5 Interpolation of Heart Rate Variability Signals 304 7.4.6 Denoising in Cardiac Motion-Mode Doppler Ultrasound Images 309?m 7.4.7 Indoor Location from Mobile Devices Measurements 316 7.4.8 Electroanatomical Maps in Cardiac Navigation Systems 322 7.5 Questions and Problems 331 8 Advances in Kernel Regression and Function Approximation 333 8.1 Introduction 333 8.2 Kernel-Based Regression Methods 333 8.2.1 Advances in Support Vector Regression 334 8.2.2 Multi-output Support Vector Regression 338 8.2.3 Kernel Ridge Regression 339 8.2.4 Kernel Signal-To-Noise Regression 341 8.2.5 Semisupervised Support Vector Regression 343 8.2.6 Model Selection in Kernel Regression Methods 345 8.4.1 Comparing Support Vector Regression, Relevance Vector Machines, and Gaussian Process Regression 360 8.4.2 Profile-Dependent Support Vector Regression 362 8.4.3 Multi-output Support Vector Regression 364 8.4.4 Kernel Signal-to-Noise Ratio Regression 366 8.4.5 Semisupervised Support Vector Regression 368 8.4.6 Bayesian Nonparametric Model 369 8.4.7 Gaussian Process Regression 370 8.4.8 Relevance Vector Machines 379 8.5 Concluding Remarks 382 8.6 Questions and Problems 383 9 Adaptive Kernel Learning for Signal Processing 387 9.1 Introduction 387 9.2 Linear Adaptive Filtering 387 9.2.1 Least Mean Squares Algorithm 388 9.2.2 Recursive Least-Squares Algorithm 389 9.3 Kernel Adaptive Filtering 392 9.4 Kernel Least Mean Squares 392 9.4.1 Derivation of Kernel Least Mean Squares 393 9.4.2 Implementation Challenges and Dual Formulation 394 9.5.3 Prediction of the Mackey-Glass Time Series with Kernel Recursive Least Squares 401 9.5.4 Beyond the Stationary Model 402 9.5.5 Example on Nonlinear Channel Identification and Reconvergence 405 9.6 Explicit Recursivity for Adaptive Kernel Models 406 9.6.1 Recursivity in Hilbert Spaces 406 9.6.2 Recursive Filters in Reproducing Kernel Hilbert Spaces 408 9.7 Online Sparsification with Kernels 411 9.7.1 Sparsity by Construction 411 9.7.2 Sparsity by Pruning 413 9.8 Probabilistic Approaches to Kernel Adaptive Filtering 414 9.8.1 Gaussian Processes and Kernel Ridge Regression 415 9.8.2 Online Recursive Solution for Gaussian Processes Regression 416 9.8.3 Kernel Recursive Least Squares Tracker 417 9.8.4 Probabilistic Kernel Least Mean Squares 418 9.9 Further Reading 418 9.9.1 Selection of Kernel Parameters 418 9.9.2 Multi-Kernel Adaptive Filtering 419 9.9.3 Recursive Filtering in Kernel Hilbert Spaces 419 9.10 Tutorials and Application Examples 419 9.10.1 Kernel Adaptive Filtering Toolbox 420 9.10.2 Prediction of a Respiratory Motion Time Series 421 9.10.3 Online Regression on the KIN?h?eK Dataset 423 9.10.4 The Mackey-Glass Time Series 425 9.10.5 Explicit Recursivity on Reproducing Kernel in Hilbert Space and Electroencephalogram Prediction 427 9.10.6 Adaptive Antenna Array Processing 428 9.11 Questions and Problems 430 Part III Classification, Detection, and Feature Extraction 433 10 Support Vector Machine and Kernel Classification Algorithms 435 10.1 Introduction 435 10.2 Support Vector Machine and Kernel Classifiers 435 10.2.1 Support Vector Machines 435 10.2.2 Multiclass and Multilabel Support Vector Machines 441 10.2.3 Least-Squares Support Vector Machine 447 10.2.4 Kernel Fisher's Discriminant Analysis 448 10.3 Advances in Kernel-Based Classification 452 10.3.1 Large Margin Filtering 452 10.3.2 Semisupervised Learning 454 10.3.3 Multiple Kernel Learning 460 10.3.4 Structured-Output Learning 462 10.3.5 Active Learning 468 10.4 Large-Scale Support Vector Machines 477 10.4.1 Large-Scale Support Vector Machine Implementations 477 10.4.2 Random Fourier Features 478 10.4.3 Parallel Support Vector Machine 480 10.4.4 Outlook 483 10.5 Tutorials and Application Examples 485 10.5.1 Examples of Support Vector Machine Classification 485 10.5.2 Example of Least-Squares Support Vector Machine 492 10.5.3 Kernel-Filtering Support Vector Machine for Brain-Computer Interface Signal Classification 493 10.5.4 Example of Laplacian Support Vector Machine 494 10.5.5 Example of Graph-Based Label Propagation 498 10.5.6 Examples of Multiple Kernel Learning 498 10.6 Concluding Remarks 501 10.7 Questions and Problems 502 11 Clustering and Anomaly Detection with Kernels 503 11.1 Introduction 503 11.2 Kernel Clustering 506 11.2.1 Kernelization of the Metric 506 11.2.2 Clustering in Feature Spaces 508 11.3 Domain Description Via Support Vectors 514 11.3.1 Support Vector Domain Description 514 11.3.2 One-Class Support Vector Machine 515 11.3.3 Relationship Between Support Vector Domain Description and Density Estimation 516 11.3.4 Semisupervised One-Class Classification 517 11.4 Kernel Matched Subspace Detectors 518 11.4.1 Kernel Orthogonal Subspace Projection 518 11.4.2 Kernel Spectral Angle Mapper 520 11.5 Kernel Anomaly Change Detection 522 11.5.1 Linear Anomaly Change Detection Algorithms 522 11.5.2 Kernel Anomaly Change Detection Algorithms 523 11.6 Hypothesis Testing with Kernels 525 11.6.1 Distribution Embeddings 526 11.6.3 Maximum Mean Discrepancy 527 11.6.3 One-Class Support Measure Machine 528 11.7 Tutorials and Application Examples 529 11.7.1 Example on Kernelization of the Metric 529 11.7.2 Example on Kernel k-Means 530 11.7.3 Domain Description Examples 531 11.7.4 Kernel Spectral Angle Mapper and Kernel Orthogonal Subspace Projection Examples 534 11.7.5 Example of Kernel Anomaly Change Detection Algorithms 536 11.7.6 Example on Distribution Embeddings and Maximum Mean Discrepancy 540 11.8 Concluding Remarks 541 11.9 Questions and Problems 542 12 Kernel Feature Extraction in Signal Processing 543 12.1 Introduction 543 12.2 Multivariate Analysis in Reproducing Kernel Hilbert Spaces 545 12.2.1 Problem Statement and Notation 545 12.2.2 Linear Multivariate Analysis 546 12.2.3 Kernel Multivariate Analysis 549 12.2.4 Multivariate Analysis Experiments 551 12.3 Feature Extraction with Kernel Dependence Estimates 555 12.3.1 Feature Extraction Using Hilbert-Schmidt Independence Criterion 556 12.3.2 Blind Source Separation Using Kernels 563 12.4 Extensions for Large-Scale and Semisupervised Problems 570 12.4.2 Efficiency with the Incomplete Cholesky Decomposition 570 12.4.3 Efficiency with Random Fourier Features 570 12.4.3 Sparse Kernel Feature Extraction 571 12.4.4 Semisupervised Kernel Feature Extraction 573 12.5 Domain Adaptation with Kernels 575 12.5.1 Kernel Mean Matching 578 12.5.2 Transfer Component Analysis 579 12.5.3 Kernel Manifold Alignment 581 12.5.4 Relations between Domain Adaptation Methods 585 12.5.5 Experimental Comparison between Domain Adaptation Methods 12.6 Concluding Remarks 587 12.7 Questions and Problems 588 References 589