book

Machine Learning

Name: Machine Learning
Author: Sergios Theodoridis
ISBN: 9780128017227

by Sergios Theodoridis

April 2015

Intermediate to advanced

1062 pages

40h 35m

English

Academic Press

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Cover image
Title page
Table of Contents
Copyright
Preface
Acknowledgments
Notation
Dedication
Chapter 1: Introduction
Abstract1.1 What Machine Learning is About1.2 Structure and a Road Map of the Book
Chapter 2: Probability and Stochastic Processes
Abstract2.1 Introduction2.2 Probability and Random Variables2.3 Examples of Distributions2.4 Stochastic Processes

2.5 Information Theory
2.6 Stochastic ConvergenceProblems
Chapter 3: Learning in Parametric Modeling: Basic Concepts and Directions
Abstract3.1 Introduction3.2 Parameter Estimation: The Deterministic Point of View3.3 Linear Regression3.4 Classification3.5 Biased Versus Unbiased Estimation3.6 The Cramér-Rao Lower Bound3.7 Sufficient Statistic3.8 Regularization3.9 The Bias-Variance Dilemma
3.10 Maximum Likelihood Method
3.11 Bayesian Inference3.12 Curse of Dimensionality3.13 Validation3.14 Expected and Empirical Loss Functions3.15 Nonparametric Modeling and EstimationProblems
Chapter 4: Mean-Square Error Linear Estimation
Abstract4.1 Introduction4.2 Mean-Square Error Linear Estimation: The Normal Equations
Chapter 5: Stochastic Gradient Descent: The LMS Algorithm and its Family
Abstract5.1 Introduction5.2 The Steepest Descent Method5.3 Application to the Mean-Square Error Cost Function5.4 Stochastic Approximation5.5 The Least-Mean-Squares Adaptive Algorithm
5.6 The Affine Projection Algorithm
5.7 The Complex-Valued Case5.8 Relatives of the LMS5.9 Simulation Examples5.10 Adaptive Decision Feedback Equalization5.11 The Linearly Constrained LMS5.12 Tracking Performance of the LMS in Nonstationary Environments5.13 Distributed Learning: The Distributed LMS
5.14 A Case Study: Target Localization
5.15 Some Concluding Remarks: Consensus MatrixProblemsMATLAB Exercises
Chapter 6: The Least-Squares Family
Abstract6.1 Introduction6.2 Least-Squares Linear Regression: A Geometric Perspective6.3 Statistical Properties of the LS Estimator6.4 Orthogonalizing the Column Space of X: The SVD Method6.5 Ridge Regression6.6 The Recursive Least-Squares Algorithm6.7 Newton’s Iterative Minimization Method6.8 Steady-State Performance of the RLS6.9 Complex-Valued Data: The Widely Linear RLS6.10 Computational Aspects of the LS Solution
6.11 The Coordinate and Cyclic Coordinate Descent Methods
6.12 Simulation Examples6.13 Total-Least-SquaresProblems
Chapter 7: Classification: A Tour of the Classics
Abstract7.1 Introduction7.2 Bayesian Classification7.3 Decision (Hyper)Surfaces7.4 The Naive Bayes Classifier7.5 The Nearest Neighbor Rule7.6 Logistic Regression7.7 Fisher’s Linear Discriminant7.8 Classification Trees7.9 Combining Classifiers
7.10 The Boosting Approach
7.11 Boosting Trees7.12 A Case Study: Protein Folding PredictionProblems
Chapter 8: Parameter Learning: A Convex Analytic Path
Abstract8.1 Introduction8.2 Convex Sets and Functions8.3 Projections onto Convex Sets8.4 Fundamental Theorem of Projections onto Convex Sets8.5 A Parallel Version of POCS8.6 From Convex Sets to Parameter Estimation and Machine Learning8.7 Infinite Many Closed Convex Sets: The Online Learning Case8.8 Constrained Learning8.9 The Distributed APSM
8.10 Optimizing Nonsmooth Convex Cost Functions
8.11 Regret Analysis8.12 Online Learning and Big Data Applications: A Discussion
8.13 Proximal Operators
8.14 Proximal Splitting Methods for OptimizationProblemsMATLAB Exercises8.15 Appendix to Chapter 8
Chapter 9: Sparsity-Aware Learning: Concepts and Theoretical Foundations
Abstract9.1 Introduction9.2 Searching for a Norm9.3 The Least Absolute Shrinkage and Selection Operator (LASSO)9.4 Sparse Signal Representation9.5 In Search of the Sparsest Solution9.6 Uniqueness of the ℓ0 Minimizer9.7 Equivalence of ℓ0 and ℓ1 Minimizers: Sufficiency Conditions9.8 Robust Sparse Signal Recovery from Noisy Measurements9.9 Compressed Sensing: The Glory of Randomness9.10 A Case Study: Image De-Noising
Problems
Chapter 10: Sparsity-Aware Learning: Algorithms and Applications
Abstract10.1 Introduction10.2 Sparsity-Promoting Algorithms10.3 Variations on the Sparsity-Aware Theme10.4 Online Sparsity-Promoting Algorithms
10.5 Learning Sparse Analysis Models
10.6 A Case Study: Time-Frequency Analysis10.7 Appendix to Chapter 10: Some Hints from the Theory of FramesProblems
Chapter 11: Learning in Reproducing Kernel Hilbert Spaces
Abstract11.1 Introduction11.2 Generalized Linear Models11.3 Volterra, Wiener, and Hammerstein Models11.4 Cover’s Theorem: Capacity of a Space in Linear Dichotomies11.5 Reproducing Kernel Hilbert Spaces11.6 Representer Theorem11.7 Kernel Ridge Regression11.8 Support Vector Regression
11.9 Kernel Ridge Regression Revisited
11.10 Optimal Margin Classification: Support Vector Machines11.11 Computational Considerations11.12 Online Learning in RKHS
11.13 Multiple Kernel Learning
11.14 Nonparametric Sparsity-Aware Learning: Additive Models11.15 A Case Study: Authorship IdentificationProblems
Chapter 12: Bayesian Learning: Inference and the EM Algorithm
Abstract12.1 Introduction12.2 Regression: A Bayesian Perspective12.3 The Evidence Function and Occam’s Razor Rule12.4 Exponential Family of Probability Distributions
12.5 Latent Variables and the EM Algorithm
12.6 Linear Regression and the EM Algorithm12.7 Gaussian Mixture Models12.8 Combining Learning Models: A Probabilistic Point of ViewProblemsMATLAB Exercises
12.9 Appendix to Chapter 12
Chapter 13: Bayesian Learning: Approximate Inference and Nonparametric Models
Abstract13.1 Introduction13.2 Variational Approximation in Bayesian Learning13.3 A Variational Bayesian Approach to Linear Regression13.4 A Variational Bayesian Approach to Gaussian Mixture Modeling13.5 When Bayesian Inference Meets Sparsity
13.6 Sparse Bayesian Learning (SBL)
13.7 The Relevance Vector Machine Framework13.8 Convex Duality and Variational Bounds13.9 Sparsity-Aware Regression: A Variational Bound Bayesian Path13.10 Sparsity-Aware Learning: Some Concluding Remarks13.11 Expectation Propagation
13.12 Nonparametric Bayesian Modeling
13.13 Gaussian Processes13.14 A Case Study: Hyperspectral Image UnmixingProblems
Chapter 14: Monte Carlo Methods
Abstract14.1 Introduction14.2 Monte Carlo Methods: The Main Concept14.3 Random Sampling Based on Function Transformation14.4 Rejection Sampling14.5 Importance Sampling14.6 Monte Carlo Methods and the EM Algorithm14.7 Markov Chain Monte Carlo Methods14.8 The Metropolis Method14.9 Gibbs Sampling14.10 In Search of More Efficient Methods: A Discussion
14.11 A Case Study: Change-Point Detection
Problems
Chapter 15: Probabilistic Graphical Models: Part I
Abstract15.1 Introduction15.2 The Need for Graphical Models15.3 Bayesian Networks and the Markov Condition15.4 Undirected Graphical Models15.5 Factor Graphs15.6 Moralization of Directed Graphs15.7 Exact Inference Methods: Message-Passing Algorithms
Problems
Chapter 16: Probabilistic Graphical Models: Part II
Abstract16.1 Introduction16.2 Triangulated Graphs and Junction Trees16.3 Approximate Inference Methods16.4 Dynamic Graphical Models16.5 Hidden Markov Models
16.6 Beyond HMMs: A Discussion
16.7 Learning Graphical ModelsProblems
Chapter 17: Particle Filtering
Abstract17.1 Introduction17.2 Sequential Importance Sampling17.3 Kalman and Particle Filtering17.4 Particle Filtering
Problems
Chapter 18: Neural Networks and Deep Learning
Abstract18.1 Introduction18.2 The Perceptron18.3 Feed-Forward Multilayer Neural Networks18.4 The Backpropagation Algorithm18.5 Pruning the Network18.6 Universal Approximation Property of Feed-Forward Neural Networks18.7 Neural Networks: A Bayesian Flavor18.8 Learning Deep Networks
18.9 Deep Belief Networks
18.10 Variations on the Deep Learning Theme18.11 Case Study: A Deep Network for Optical Character Recognition18.12 CASE Study: A Deep Autoencoder18.13 Example: Generating Data via a DBNProblemsMATLAB Exercises
Chapter 19: Dimensionality Reduction and Latent Variables Modeling
Abstract19.1 Introduction19.2 Intrinsic Dimensionality19.3 Principle Component Analysis19.4 Canonical Correlation Analysis19.5 Independent Component Analysis19.6 Dictionary Learning: The k-SVD Algorithm
19.7 Nonnegative Matrix Factorization
19.8 Learning Low-Dimensional Models: A Probabilistic Perspective19.9 Nonlinear Dimensionality Reduction19.10 Low-Rank Matrix Factorization: A Sparse Modeling Path19.11 A Case Study: fMRI Data AnalysisProblems
Appendix A: Linear Algebra
A.1 Properties of MatricesA.2 Positive Definite and Symmetric MatricesA.3 Wirtinger Calculus
Appendix B: Probability Theory and Statistics
B.1 Cramér-Rao BoundB.2 Characteristic FunctionsB.3 Moments and CumulantsB.4 Edgeworth Expansion of a pdf
Appendix C: Hints on Constrained Optimization
C.1 Equality ConstraintsC.2 Inequality Constrains
Index

Content preview from Machine Learning

3.10 Maximum Likelihood Method

So far, we have approached the estimation problem as an optimization task around a set of training examples, without paying any attention to the underlying statistics that generates these points. We only used statistics in order to check under which conditions the estimators were efficient. However, the optimization step did not involve any statistical information. For the rest of the chapter, we are going to involve statistics more and more. In this section, the ML method is introduced. It is not an exaggeration to say that ML and LS are two of the major pillars on which parameter estimation is based and new methods are inspired from. The ML method was suggested by Sir Ronald Aylmer Fisher.

Once more, we will ...

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Machine Learning

by Sergios Theodoridis

3.10 Maximum Likelihood Method

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