MACHINE LEARNING TECHNIQUES - LASA

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2 1. I. Introduction .............................................................................................................. 6 1.1 What is Machine Learning? - Definitions ...................................................................... 6 1.1.1 ML Resources: ........................................................................................................... 6 1.2 What is Learning? ........................................................................................................... 7 1.2.1 Taxonomy of Learning Algorithms ........................................................................... 10 1.2.2 Other important terms in machine learning .............................................................. 10 1.2.3 Key features for a good learning system ................................................................. 11 1.2.4 Exercise ................................................................................................................... 12 1.3 Best Practices in ML ..................................................................................................... 12 1.3.1 Training, validation and testing sets ........................................................................ 12 1.3.2 Crossvalidation ........................................................................................................ 13 1.3.3 Performance Measures in ML .................................................................................. 13 1.3.4 In Practice ................................................................................................................ 14 1.4 Focus of this course ..................................................................................................... 14 2. 2 Methods for Correlation Analysis PCA, CCA, ICA .............................................. 16 2.1 Principal Component Analysis .................................................................................... 17 2.1.1 Dimensionality Reduction ........................................................................................ 18 2.1.2 Solving PCA as an optimization under constraint problem ...................................... 20 2.1.3 PCA limitations ........................................................................................................ 21 2.1.4 Projection Pursuit ..................................................................................................... 22 2.1.5 Probabilistic PCA ..................................................................................................... 24 2.2 Canonical Correlation Analysis ................................................................................... 26 2.2.1 CCA for more than two variables ............................................................................. 27 2.2.2 Limitations ................................................................................................................ 27 2.3 Independent Component Analysis .............................................................................. 28 2.3.1 Illustration of ICA ..................................................................................................... 28 2.3.2 Why Gaussian variables are forbidden .................................................................... 31 2.3.3 Definition of ICA ....................................................................................................... 31 2.3.4 Whitening ................................................................................................................. 33 2.3.5 ICA Ambiguities ....................................................................................................... 35 2.3.6 ICA by maximizing non-gaussianity ......................................................................... 35 2.4 Further Readings .......................................................................................................... 38 3. 3 Clustering and Classification ............................................................................... 39 3.1 Clustering Techniques ................................................................................................. 39 3.1.1 Hierarchical Clustering ............................................................................................. 40 3.1.2 K-means clustering .................................................................................................. 45 3.1.3 Soft K-means ........................................................................................................... 47 3.1.4 Clustering with Mixtures of Gaussians ..................................................................... 49 3.1.5 Gaussian Mixture Models ........................................................................................ 51 3.2 Linear Classifiers .......................................................................................................... 56 3.2.1 Linear Discriminant Analysis .................................................................................... 56 3.2.2 Fisher Linear Discriminant ....................................................................................... 57 3.2.3 Mixture of linear classifiers (boosting and bagging) ................................................. 58 3.3 Bayes Classifier ............................................................................................................ 61 3.4 Linear classification with Gaussian Mixture Models ................................................. 62 3.5 Further Readings .......................................................................................................... 63 © A.G.Billard 2004 – Last Update March 2011
3 4. 4 Regression Techniques ........................................................................................ 64 4.1 Linear Regression ......................................................................................................... 64 4.2 Partial Least Square Methods ...................................................................................... 64 4.3 Probabilistic Regression .............................................................................................. 66 4.4 Gaussian Mixture Regression ..................................................................................... 69 4.4.1 One Gaussian Case ................................................................................................ 70 4.4.2 Multi-Gaussian Case ............................................................................................... 71 5. 5 Kernel Methods ...................................................................................................... 73 5.1 The kernel trick ............................................................................................................. 73 5.2 Which kernel, when? .................................................................................................... 75 5.3 Kernel PCA .................................................................................................................... 76 5.4 Kernel CCA .................................................................................................................... 81 5.5 Kernel ICA ...................................................................................................................... 84 5.6 Kernel K-Means ............................................................................................................. 88 5.7 Support Vector Machines ............................................................................................. 90 5.7.1 Support Vector Machine for Linearly Separable Datasets ....................................... 92 5.7.2 Support Vector Machine for Non-linearly Separable Datasets ................................ 96 5.7.3 Non-Linear Support Vector Machines ...................................................................... 97 5.7.4 n-SVM ...................................................................................................................... 98 5.8 Support Vector Regression ......................................................................................... 99 5.8.1 n-SVR .................................................................................................................... 106 5.9 Gaussian Process Regression .................................................................................. 109 5.9.1 What is a Gaussian Process .................................................................................. 109 5.9.2 Equivalence of Gaussian Process Regression and Gaussian Mixture Regression 113 5.9.3 Curse of dimensionality, choice of hyperparameters ............................................. 115 5.10 Gaussian Process Classification .............................................................................. 116 6. 6 Artificial Neural Networks ................................................................................... 120 6.1 Applications of ANN ................................................................................................... 120 6.2 Biological motivation .................................................................................................. 120 6.2.1 The Brain as an Information Processing System ................................................... 120 6.2.2 Neural Networks in the Brain ................................................................................. 121 6.2.3 Neurons and Synapses ......................................................................................... 122 6.2.4 Synaptic Learning .................................................................................................. 122 6.2.5 Summary ............................................................................................................... 123 6.3 Perceptron ................................................................................................................... 124 6.3.1 Learning rule for the Perceptron ............................................................................ 126 6.3.2 Information Theory and the Neuron ....................................................................... 127 6.4 The Backpropagation Learning Rule ........................................................................ 129 6.4.1 The Adaline ............................................................................................................ 130 6.4.2 The Backpropagation Network .............................................................................. 131 6.4.3 The Backpropagation Algorithm ............................................................................ 132 6.5 Willshaw net ................................................................................................................ 133 6.6 Hebbian Learning ........................................................................................................ 134 © A.G.Billard 2004 – Last Update March 2011
Page 1: SCHOOL OF ENGINEERING MACHINE LEARN
Page 5 and 6: 5 9.2.2 Probability Distributions,
Page 7 and 8: 7 Journals: • Machine Learning
Page 9 and 10: 9 Performance What would be an opti
Page 11 and 12: 11 1.2.3 Key features for a good le
Page 13 and 14: 13 1.3.2 Crossvalidation To ensure
Page 15 and 16: 15 In particular, we will consider
Page 17 and 18: 17 2.1 Principal Component Analysis
Page 19 and 20: 19 ( ) Xʹ′ = W X − µ (2.6) i
Page 21 and 22: 21 2.1.2.2 Reconstruction error min
Page 23 and 24: 23 PCA is an example of PP approach
Page 25 and 26: 25 Algorithm: If one further assume
Page 27 and 28: 27 The CCA algorithm consists thus
Page 29 and 30: 29 Figure 2-6: Mixture of variables
Page 31 and 32: 31 2.3.2 Why Gaussian variables are
Page 33 and 34: 33 • In our general definition of
Page 35 and 36: 35 2.3.5 ICA Ambiguities We cannot
Page 37 and 38: 37 Denote by g the derivative of th
Page 39 and 40: 39 3 Clustering and Classification
Page 41 and 42: 41 An agglomerative clustering star
Page 43 and 44: 43 3.1.1.1 The CURE Clustering Algo
Page 45 and 46: 45 Disadvantages of hierarchical cl
Page 47 and 48: 47 Cases where K-means might be vie
Page 49 and 50: 49 3.1.4 Clustering with Mixtures o
Page 51 and 52: 51 k ( σ j ) 2 = k ∑ i α = r k
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53 Theα are the so-called mixing c
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55 Figure 3-16: Clustering with 3 G
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57 When the transformation A is lin
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59 C: X → Y ( ) C x K = arg max
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61 Figure 3-18: Linear combination
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63 Figure 3-19: Bayes classificatio
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65 ⎛⎛ min ⎜⎜ w ⎝⎝ N i=
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67 T ( yi − xi w) 2 M ⎛⎛ ⎞
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69 Figure 4-2: Illustration of the
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71 4.4.2 Multi-Gaussian Case It is
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73 5 Kernel Methods These lecture n
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75 The kernel k provides a metric o
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77 M 1 T v = ∑ x ( x ) v M λ i j
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79 1 M The solutions to the dual ei
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81 5.4 Kernel CCA The linear versio
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83 additional ridge parameter induc
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85 Figure 5-3: TOP: Marginal (left)
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87 statistical independence. We def
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89 J j ( µ 1,...., µ K) = ∑∑
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91 A simple pattern recognition alg
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93 ( ) ( , ) f x = sign w x + b (5.
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95 Figure 5-6: A binary classificat
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97 where N is the number of support
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99 5.8 Support Vector Regression In
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101 The optimization problem given
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103 Note that since we never have t
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105 Figure 5-13: Effect of the kern
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107 To better understand the effect
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109 5.9 Gaussian Process Regression
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111 One can then use the above expr
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113 5.9.2 Equivalence of Gaussian P
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115 5.9.3 Curse of dimensionality,
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117 The weight w determines the slo
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119 Figure 5-21: Example of success
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121 • its performance tends to de
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123 neurons. Furthermore, they lear
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125 The sigmoid f x ( x) ( ) = tanh
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127 6.3.2 Information Theory and th
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129 ( R) ⎛⎛det ⎞⎞ I( x, y)
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131 y= ∑ w x ) of Because of the
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133 6.5 Willshaw net David Willshaw
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135 6.6.1 Weights bounds One of the
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137 Figure 6-11: The weight vector
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139 6.6.4 Oja’s one Neuron Model
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141 If y i and y are highly correla
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143 Foldiak’s second model allows
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145 ∂ ∂ J 1 = fy − 1 2 λ 1yf
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147 6.8 The Self-Organizing Map (SO
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149 6. Decrease the size of the nei
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151 the resulting distribution is a
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153 To simplify the description of
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155 C µν −1 where ( ) is the µ
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157 The continuous time Hopfield ne
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159 ∂f If the slope is negative,
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161 7.2 Hidden Markov Models Hidden
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163 Figure 7-2: Schematic illustrat
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165 these two quantities to compute
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167 7.2.4 Decoding an HMM There are
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169 7.2.5 Further Readings Rabiner,
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171 7.3.1 Principle In reinforcemen
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173 general, acting to maximize imm
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175 of reinforcement learning makes
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177 8 Genetic Algorithms We conclud
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179 However, you must define geneti
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181 Often the crossover operator an
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183 ( A λI) x 0 − = (8.5) where
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185 Joint probability: The joint pr
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187 The two most classical distribu
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189 9.2.7 Statistical Independence
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191 1 1 1 1 h x ∫ a b a b 0 a a
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193 9.4 Estimators 9.4.1 Gradient d
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195 9.4.2.1 Maximum Likelihood Mach
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197 10 References • Machine Learn
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MACHINE LEARNING TECHNIQUES - LASA

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