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## A Bayesian and Optimization Perspective

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### Sergios Theodoridis

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1st Edition - March 27, 2015

Author: Sergios Theodoridis

Hardback ISBN:

9 7 8 - 0 - 1 2 - 8 0 1 5 2 2 - 3

eBook ISBN:

9 7 8 - 0 - 1 2 - 8 0 1 7 2 2 - 7

This tutorial text gives a unifying perspective on machine learning by covering both probabilistic and deterministic approaches -which are based on optimization techniques –… Read more

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This tutorial text gives a unifying perspective on machine learning by covering both probabilistic and deterministic approaches -which are based on optimization techniques – together with the Bayesian inference approach, whose essence lies in the use of a hierarchy of probabilistic models.The book presents the major machine learning methods as they have been developed in different disciplines, such as statistics, statistical and adaptive signal processing and computer science. Focusing on the physical reasoning behind the mathematics, all the various methods and techniques are explained in depth, supported by examples and problems, giving an invaluable resource to the student and researcher for understanding and applying machine learning concepts.

The book builds carefully from the basic classical methods to the most recent trends, with chapters written to be as self-contained as possible, making the text suitable for different courses: pattern recognition, statistical/adaptive signal processing, statistical/Bayesian learning, as well as short courses on sparse modeling, deep learning, and probabilistic graphical models.

- All major classical techniques: Mean/Least-Squares regression and filtering, Kalman filtering, stochastic approximation and online learning, Bayesian classification, decision trees, logistic regression and boosting methods.
- The latest trends: Sparsity, convex analysis and optimization, online distributed algorithms, learning in RKH spaces, Bayesian inference, graphical and hidden Markov models, particle filtering, deep learning, dictionary learning and latent variables modeling.
- Case studies - protein folding prediction, optical character recognition, text authorship identification, fMRI data analysis, change point detection, hyperspectral image unmixing, target localization, channel equalization and echo cancellation, show how the theory can be applied.
- MATLAB code for all the main algorithms are available on an accompanying website, enabling the reader to experiment with the code.

University Researchers, R&D engineers, graduate students taking a machine learning course.

- Preface
- Acknowledgments
- Notation
- Dedication
- Chapter 1: Introduction
- Abstract
- 1.1 What Machine Learning is About
- 1.2 Structure and a Road Map of the Book

- Chapter 2: Probability and Stochastic Processes
- Abstract
- 2.1 Introduction
- 2.2 Probability and Random Variables
- 2.3 Examples of Distributions
- 2.4 Stochastic Processes
- 2.5 Information Theory
- 2.6 Stochastic Convergence
- Problems

- Chapter 3: Learning in Parametric Modeling: Basic Concepts and Directions
- Abstract
- 3.1 Introduction
- 3.2 Parameter Estimation: The Deterministic Point of View
- 3.3 Linear Regression
- 3.4 Classification
- 3.5 Biased Versus Unbiased Estimation
- 3.6 The Cramér-Rao Lower Bound
- 3.7 Sufficient Statistic
- 3.8 Regularization
- 3.9 The Bias-Variance Dilemma
- 3.10 Maximum Likelihood Method
- 3.11 Bayesian Inference
- 3.12 Curse of Dimensionality
- 3.13 Validation
- 3.14 Expected and Empirical Loss Functions
- 3.15 Nonparametric Modeling and Estimation
- Problems

- Chapter 4: Mean-Square Error Linear Estimation
- Abstract
- 4.1 Introduction
- 4.2 Mean-Square Error Linear Estimation: The Normal Equations

- Chapter 5: Stochastic Gradient Descent: The LMS Algorithm and its Family
- Abstract
- 5.1 Introduction
- 5.2 The Steepest Descent Method
- 5.3 Application to the Mean-Square Error Cost Function
- 5.4 Stochastic Approximation
- 5.5 The Least-Mean-Squares Adaptive Algorithm
- 5.6 The Affine Projection Algorithm
- 5.7 The Complex-Valued Case
- 5.8 Relatives of the LMS
- 5.9 Simulation Examples
- 5.10 Adaptive Decision Feedback Equalization
- 5.11 The Linearly Constrained LMS
- 5.12 Tracking Performance of the LMS in Nonstationary Environments
- 5.13 Distributed Learning: The Distributed LMS
- 5.14 A Case Study: Target Localization
- 5.15 Some Concluding Remarks: Consensus Matrix
- Problems
- MATLAB Exercises

- Chapter 6: The Least-Squares Family
- Abstract
- 6.1 Introduction
- 6.2 Least-Squares Linear Regression: A Geometric Perspective
- 6.3 Statistical Properties of the LS Estimator
- 6.4 Orthogonalizing the Column Space of
*X*: The SVD Method - 6.5 Ridge Regression
- 6.6 The Recursive Least-Squares Algorithm
- 6.7 Newton’s Iterative Minimization Method
- 6.8 Steady-State Performance of the RLS
- 6.9 Complex-Valued Data: The Widely Linear RLS
- 6.10 Computational Aspects of the LS Solution
- 6.11 The Coordinate and Cyclic Coordinate Descent Methods
- 6.12 Simulation Examples
- 6.13 Total-Least-Squares
- Problems

- Chapter 7: Classification: A Tour of the Classics
- Abstract
- 7.1 Introduction
- 7.2 Bayesian Classification
- 7.3 Decision (Hyper)Surfaces
- 7.4 The Naive Bayes Classifier
- 7.5 The Nearest Neighbor Rule
- 7.6 Logistic Regression
- 7.7 Fisher’s Linear Discriminant
- 7.8 Classification Trees
- 7.9 Combining Classifiers
- 7.10 The Boosting Approach
- 7.11 Boosting Trees
- 7.12 A Case Study: Protein Folding Prediction
- Problems

- Chapter 8: Parameter Learning: A Convex Analytic Path
- Abstract
- 8.1 Introduction
- 8.2 Convex Sets and Functions
- 8.3 Projections onto Convex Sets
- 8.4 Fundamental Theorem of Projections onto Convex Sets
- 8.5 A Parallel Version of POCS
- 8.6 From Convex Sets to Parameter Estimation and Machine Learning
- 8.7 Infinite Many Closed Convex Sets: The Online Learning Case
- 8.8 Constrained Learning
- 8.9 The Distributed APSM
- 8.10 Optimizing Nonsmooth Convex Cost Functions
- 8.11 Regret Analysis
- 8.12 Online Learning and Big Data Applications: A Discussion
- 8.13 Proximal Operators
- 8.14 Proximal Splitting Methods for Optimization
- Problems
- MATLAB Exercises
- 8.15 Appendix to Chapter 8

- Chapter 9: Sparsity-Aware Learning: Concepts and Theoretical Foundations
- Abstract
- 9.1 Introduction
- 9.2 Searching for a Norm
- 9.3 The Least Absolute Shrinkage and Selection Operator (LASSO)
- 9.4 Sparse Signal Representation
- 9.5 In Search of the Sparsest Solution
- 9.6 Uniqueness of the
*ℓ*_{0}Minimizer - 9.7 Equivalence of
*ℓ*_{0}and*ℓ*_{1}Minimizers: Sufficiency Conditions - 9.8 Robust Sparse Signal Recovery from Noisy Measurements
- 9.9 Compressed Sensing: The Glory of Randomness
- 9.10 A Case Study: Image De-Noising
- Problems

- Chapter 10: Sparsity-Aware Learning: Algorithms and Applications
- Abstract
- 10.1 Introduction
- 10.2 Sparsity-Promoting Algorithms
- 10.3 Variations on the Sparsity-Aware Theme
- 10.4 Online Sparsity-Promoting Algorithms
- 10.5 Learning Sparse Analysis Models
- 10.6 A Case Study: Time-Frequency Analysis
- 10.7 Appendix to Chapter 10: Some Hints from the Theory of Frames
- Problems

- Chapter 11: Learning in Reproducing Kernel Hilbert Spaces
- Abstract
- 11.1 Introduction
- 11.2 Generalized Linear Models
- 11.3 Volterra, Wiener, and Hammerstein Models
- 11.4 Cover’s Theorem: Capacity of a Space in Linear Dichotomies
- 11.5 Reproducing Kernel Hilbert Spaces
- 11.6 Representer Theorem
- 11.7 Kernel Ridge Regression
- 11.8 Support Vector Regression
- 11.9 Kernel Ridge Regression Revisited
- 11.10 Optimal Margin Classification: Support Vector Machines
- 11.11 Computational Considerations
- 11.12 Online Learning in RKHS
- 11.13 Multiple Kernel Learning
- 11.14 Nonparametric Sparsity-Aware Learning: Additive Models
- 11.15 A Case Study: Authorship Identification
- Problems

- Chapter 12: Bayesian Learning: Inference and the EM Algorithm
- Abstract
- 12.1 Introduction
- 12.2 Regression: A Bayesian Perspective
- 12.3 The Evidence Function and Occam’s Razor Rule
- 12.4 Exponential Family of Probability Distributions
- 12.5 Latent Variables and the EM Algorithm
- 12.6 Linear Regression and the EM Algorithm
- 12.7 Gaussian Mixture Models
- 12.8 Combining Learning Models: A Probabilistic Point of View
- Problems
- MATLAB Exercises
- 12.9 Appendix to Chapter 12

- Chapter 13: Bayesian Learning: Approximate Inference and Nonparametric Models
- Abstract
- 13.1 Introduction
- 13.2 Variational Approximation in Bayesian Learning
- 13.3 A Variational Bayesian Approach to Linear Regression
- 13.4 A Variational Bayesian Approach to Gaussian Mixture Modeling
- 13.5 When Bayesian Inference Meets Sparsity
- 13.6 Sparse Bayesian Learning (SBL)
- 13.7 The Relevance Vector Machine Framework
- 13.8 Convex Duality and Variational Bounds
- 13.9 Sparsity-Aware Regression: A Variational Bound Bayesian Path
- 13.10 Sparsity-Aware Learning: Some Concluding Remarks
- 13.11 Expectation Propagation
- 13.12 Nonparametric Bayesian Modeling
- 13.13 Gaussian Processes
- 13.14 A Case Study: Hyperspectral Image Unmixing
- Problems

- Chapter 14: Monte Carlo Methods
- Abstract
- 14.1 Introduction
- 14.2 Monte Carlo Methods: The Main Concept
- 14.3 Random Sampling Based on Function Transformation
- 14.4 Rejection Sampling
- 14.5 Importance Sampling
- 14.6 Monte Carlo Methods and the EM Algorithm
- 14.7 Markov Chain Monte Carlo Methods
- 14.8 The Metropolis Method
- 14.9 Gibbs Sampling
- 14.10 In Search of More Efficient Methods: A Discussion
- 14.11 A Case Study: Change-Point Detection
- Problems

- Chapter 15: Probabilistic Graphical Models: Part I
- Abstract
- 15.1 Introduction
- 15.2 The Need for Graphical Models
- 15.3 Bayesian Networks and the Markov Condition
- 15.4 Undirected Graphical Models
- 15.5 Factor Graphs
- 15.6 Moralization of Directed Graphs
- 15.7 Exact Inference Methods: Message-Passing Algorithms
- Problems

- Chapter 16: Probabilistic Graphical Models: Part II
- Abstract
- 16.1 Introduction
- 16.2 Triangulated Graphs and Junction Trees
- 16.3 Approximate Inference Methods
- 16.4 Dynamic Graphical Models
- 16.5 Hidden Markov Models
- 16.6 Beyond HMMs: A Discussion
- 16.7 Learning Graphical Models
- Problems

- Chapter 17: Particle Filtering
- Abstract
- 17.1 Introduction
- 17.2 Sequential Importance Sampling
- 17.3 Kalman and Particle Filtering
- 17.4 Particle Filtering
- Problems

- Chapter 18: Neural Networks and Deep Learning
- Abstract
- 18.1 Introduction
- 18.2 The Perceptron
- 18.3 Feed-Forward Multilayer Neural Networks
- 18.4 The Backpropagation Algorithm
- 18.5 Pruning the Network
- 18.6 Universal Approximation Property of Feed-Forward Neural Networks
- 18.7 Neural Networks: A Bayesian Flavor
- 18.8 Learning Deep Networks
- 18.9 Deep Belief Networks
- 18.10 Variations on the Deep Learning Theme
- 18.11 Case Study: A Deep Network for Optical Character Recognition
- 18.12 CASE Study: A Deep Autoencoder
- 18.13 Example: Generating Data via a DBN
- Problems
- MATLAB Exercises

- Chapter 19: Dimensionality Reduction and Latent Variables Modeling
- Abstract
- 19.1 Introduction
- 19.2 Intrinsic Dimensionality
- 19.3 Principle Component Analysis
- 19.4 Canonical Correlation Analysis
- 19.5 Independent Component Analysis
- 19.6 Dictionary Learning: The
*k*-SVD Algorithm - 19.7 Nonnegative Matrix Factorization
- 19.8 Learning Low-Dimensional Models: A Probabilistic Perspective
- 19.9 Nonlinear Dimensionality Reduction
- 19.10 Low-Rank Matrix Factorization: A Sparse Modeling Path
- 19.11 A Case Study: fMRI Data Analysis
- Problems

- Appendix A: Linear Algebra
- A.1 Properties of Matrices
- A.2 Positive Definite and Symmetric Matrices
- A.3 Wirtinger Calculus

- Appendix B: Probability Theory and Statistics
- B.1 Cramér-Rao Bound
- B.2 Characteristic Functions
- B.3 Moments and Cumulants
- B.4 Edgeworth Expansion of a pdf

- Appendix C: Hints on Constrained Optimization
- C.1 Equality Constraints
- C.2 Inequality Constrains

- Index

- No. of pages: 1062
- Language: English
- Published: March 27, 2015
- Imprint: Academic Press
- Hardback ISBN: 9780128015223
- eBook ISBN: 9780128017227

ST

Sergios Theodoridis is professor of machine learning and signal processing with the National and Kapodistrian University of Athens, Athens, Greece and with the Chinese University of Hong Kong, Shenzhen, China.
He has received a number of prestigious awards, including the 2014 IEEE Signal Processing Magazine Best Paper Award, the 2009 IEEE Computational Intelligence Society Transactions on Neural Networks Outstanding Paper Award, the 2017 European Association for Signal Processing
(EURASIP) Athanasios Papoulis Award, the 2014 IEEE Signal Processing Society Education Award, and the 2014 EURASIP Meritorious Service Award. He has served as president of EURASIP and vice president for the IEEE Signal Processing Society and as Editor-in-Chief IEEE Transactions on Signal processing. He is a Fellow of EURASIP and a Life Fellow of IEEE.
He is the coauthor of the best selling book Pattern Recognition, 4th edition, Academic Press, 2009 and of the book Introduction to Pattern Recognition: A MATLAB Approach, Academic Press, 2010.

Affiliations and expertise

professor of machine learning and signal processing with the National and Kapodistrian University of Athens, Athens, Greece and with the Chinese University of Hong Kong, Shenzhen, China.