MACHINE LEARNING TECHNIQUES - LASA

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4 6.6.1 Weights bounds ..................................................................................................... 135 6.6.2 Weight decay ......................................................................................................... 136 6.6.3 Principal Components ............................................................................................ 137 6.6.4 Oja’s one Neuron Model ........................................................................................ 139 6.6.5 Convergence of the Weights Decay rule ............................................................... 139 6.7 Anti-Hebbian learning ................................................................................................. 140 6.7.1 Foldiak’s models .................................................................................................... 141 6.7.2 CCA Revisited ....................................................................................................... 143 6.7.3 ICA Revisited ......................................................................................................... 145 6.8 The Self-Organizing Map (SOM) ................................................................................ 147 6.8.1 Kohonen Network .................................................................................................. 147 6.8.2 Bayesian self-organizing map ................................................................................ 150 6.9 Static Hopfield Network .............................................................................................. 151 6.9.1 Hopfield Network Structure .................................................................................... 152 6.9.2 Learning Phase ...................................................................................................... 152 6.9.3 Retrieval Phase ..................................................................................................... 153 6.9.4 Capacity of the static Hopfield Network ................................................................. 154 6.9.5 Convergence of the Static Hopfield Network ......................................................... 155 6.10 Continuous Time Hopfield Network and Leaky-Integrator Neurons ...................... 155 7. 7 Markov-Based Models ......................................................................................... 160 7.1 Markov Process .......................................................................................................... 160 7.2 Hidden Markov Models ............................................................................................... 161 7.2.1 Formalism .............................................................................................................. 161 7.2.2 Estimating a HMM ................................................................................................. 163 7.2.3 Determining the number of states .......................................................................... 166 7.2.4 Decoding an HMM ................................................................................................. 167 7.2.5 Further Readings ................................................................................................... 169 7.3 Reinforcement Learning ............................................................................................. 170 7.3.1 Principle ................................................................................................................. 171 7.3.2 Defining the Reward .............................................................................................. 172 7.3.3 Markov World ......................................................................................................... 173 7.3.4 Estimating the policy .............................................................................................. 174 7.3.5 Summary ............................................................................................................... 175 8. 8 Genetic Algorithms .............................................................................................. 177 8.1 Principle ....................................................................................................................... 178 8.2 Encoding ...................................................................................................................... 178 8.3 Breeding and Selection .............................................................................................. 179 8.4 The Algorithm .............................................................................................................. 181 8.5 Convergence ............................................................................................................... 181 9. 9 Annexes ................................................................................................................ 182 9.1 Brief recall of basic transformations from linear algebra ....................................... 182 9.1.1 Eigenvalue Decomposition .................................................................................... 182 9.1.2 Singular Value Decomposition (SVD) .................................................................... 184 9.1.3 Frobenius Norm ..................................................................................................... 184 9.2 Recall of basic notions of statistics and probabilities ............................................ 184 9.2.1 Probabilities ........................................................................................................... 184 © A.G.Billard 2004 – Last Update March 2011
5 9.2.2 Probability Distributions, Probability Density Function ........................................... 186 9.2.3 Distribution Function or Cumulative Distribution Function ..................................... 186 9.2.4 Marginal Probability Distribution or Marginal Density ............................................ 187 9.2.5 Likelihood ............................................................................................................... 188 9.2.6 Probabilistic Independence .................................................................................... 188 9.2.7 Statistical Independence ........................................................................................ 189 9.2.8 Uncorrelatedness ................................................................................................... 189 9.3 Information Theory ..................................................................................................... 189 9.3.1 Entropy .................................................................................................................. 190 9.3.2 Joint and conditional entropy ................................................................................. 191 9.3.3 Mutual Information ................................................................................................. 191 9.3.4 Relation to mutual information ............................................................................... 192 9.3.5 Kullback-Leibler Distance ...................................................................................... 192 9.4 Estimators ................................................................................................................... 193 9.4.1 Gradient descent ................................................................................................... 193 9.4.2 Conjugate Gradient descent .................................................................................. 194 10. 10 References ........................................................................................................... 197 10.1 Other Recommended Textbooks ............................................................................... 197 10.2 Other Recommended Readings ................................................................................ 198 © A.G.Billard 2004 – Last Update March 2011
Page 1 and 2: SCHOOL OF ENGINEERING MACHINE LEARN
Page 3: 3 4. 4 Regression Techniques ......
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
Page 53 and 54: 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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