MACHINE LEARNING TECHNIQUES - LASA

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72 Figure 4-3: Two-dimensional illustration of a) left: a Gaussian Mixture Model with 3 Gaussians (in violet) superposed to the original set of pairs (x,y) of datapoints (in grey), b) the Gaussian Mixture Regression { } model resulting from the GMM. The dark line is the regression signal, i.e. E p( Y | X ) envelope around it is one standard devitation computed from var pY ( | X ) . { } . The blue Figure 4-4: Gaussian Mixture Model (top row) and Regression (bottom row) with different amounts of Gaussians in the mixture. The center line shows the regression mean, while the two thinner lines display respectively one and two sigmas. The regression function becomes more detailed when the data is modeled by more Gaussians, and represents better the modulation of its variance. [DEMOS\REGRESSION\GMR-THINNING.ML] © A.G.Billard 2004 – Last Update March 2011
73 5 Kernel Methods These lecture notes have sofar covered linear methods for performing a variety of computation, ranging from dimensionality reduction, clustering and classification to regression. The term linear refers to the assumptions that all of these transformations could be expressed as linear transformation of the form y= Ax (where y and xare given and A is the unknown transformation). Kernel methods relax the assumption of a linear transformation, so as to perform non-linear regression, classification, etc. Kernel methods proceed by first projecting the data through a nonlinear transformation into a feature space and then perform the same type of computation (e.g. classification or regression), as in the linear case, in the feature space. This principle is illustrated in Figure 5-1. Figure 5-1: Illustration of the principle of non-linear classification through Support Vector Machine. The data are first projected from input space to a feature space via a non-linear map φ . Linear separation is then performed in the feature space. Figure from "Learning with Kernels" by B. Scholkopf and A. Smola, MIT Press 2002. In these lecture notes, we will cover solely a subset of the existing Kernel methods, focusing on kernel methods that extend the algorithms see previously for dimensionality reduction (PCA, ICA, CCA to kernel PCA, kernel ICA, kernel CCA), for clustering (from K-Means to kernel K-means), for classification (from linear classifier to support vector machine) and for regression (from probabilistic regression to Gaussian Process Regression). 5.1 The kernel trick All kernel methods we will see here are based on the so-called kernel trick. The kernel trick stems from a variety of fundamental mathematical principles which we will skip here, focusing solely on explaining its basic principle. © A.G.Billard 2004 – Last Update March 2011
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SCHOOL OF ENGINEERING MACHINE LEARN
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3 4. 4 Regression Techniques ......
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5 9.2.2 Probability Distributions,
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7 Journals: • Machine Learning
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9 Performance What would be an opti
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11 1.2.3 Key features for a good le
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13 1.3.2 Crossvalidation To ensure
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15 In particular, we will consider
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17 2.1 Principal Component Analysis
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19 ( ) Xʹ′ = W X − µ (2.6) i
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Page 35 and 36: 35 2.3.5 ICA Ambiguities We cannot
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Page 41 and 42: 41 An agglomerative clustering star
Page 43 and 44: 43 3.1.1.1 The CURE Clustering Algo
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Page 51 and 52: 51 k ( σ j ) 2 = k ∑ i α = r k
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Page 59 and 60: 59 C: X → Y ( ) C x K = arg max
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Page 63 and 64: 63 Figure 3-19: Bayes classificatio
Page 65 and 66: 65 ⎛⎛ min ⎜⎜ w ⎝⎝ N i=
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Page 87 and 88: 87 statistical independence. We def
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Page 91 and 92: 91 A simple pattern recognition alg
Page 93 and 94: 93 ( ) ( , ) f x = sign w x + b (5.
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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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