MACHINE LEARNING TECHNIQUES - LASA

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28 2.3 Independent Component Analysis Adapted from Independent Component Analysis, A. Hyvarinen, J. Karhunen and E. Oja, Wiley Inter-Sciences. 2001 Similarly to PCA, ICA is a linear transformation that projects the dataset on a sub-manifold, that best represent the data at hand. While PCA looks for the principal components, ICA looks for the directions along which statistical dependence of the data is minimal. ICA is particularly useful in order to decorrelate and denoise data. ICA is very closely related to the method called blind source separation (BSS) or blind signal separation. The ``source'' is the original signal, i.e. the independent components, (e.g. the speaker in a cocktail party). ``Blind'' means that neither the mixing matrix nor the independent components are known to start with. ICA is one method, perhaps the most widely used, for performing blind source separation. It has been used successfully in a wide range of signal processing applications, e.g. for de-correlating multiple sound sources or extracting invariant features in image processing. 2.3.1 Illustration of ICA , To illustrate the idea of ICA, consider that you have observed four instances of a two-dimensional distribution S of independent components s 1 and s 2 and that the four instances form the four corners of a rectangle, as shown in Figure 2-5: Figure 2-5; Distribution of two independent components Note that the range of values for these data was chosen so as to make the mean zero and the variance equal to one, an important constraint of ICA. Now let us mix these two independent components, using the mixing matrix ⎛⎛x1 ⎞⎞ ⎛⎛s1 ⎞⎞ ⎜⎜ ⎟⎟= A⋅⎜⎜ ⎟⎟ x s ⎝⎝ 2⎠⎠ ⎝⎝ 2⎠⎠ ⎛⎛1 1⎞⎞ A = ⎜⎜ ⎟⎟ ⎝⎝1 1⎠⎠ . This gives us two mixed variables, x 1 and x 2 . © A.G.Billard 2004 – Last Update March 2011
29 Figure 2-6: Mixture of variables Note that the random variables x 1 and x 2 are not independent any more, see Figure 2-6; an easy way to see this is to consider, whether it is possible to predict the value of one of them, say x 2 , from the value of the other. Clearly it is the case, as they line up along a line of slope 1. The problem of estimating the data model of ICA is now to estimate the mixing matrix A using only information contained in the mixtures x 1 and x 2 . In the above example, it is easy to estimate A by simply solving the inverse for each of the four data points, assuming that we have at least four points. The problem is less trivial, once we consider a mixture of two arbitrary continuous distributions. Let us now assume that s 1 and s 2 were generated by the following uniform distribution: ( ) p s i ⎧⎧ 1 ⎪⎪ if si ≤ 3 = ⎨⎨2 3 (2.20) ⎪⎪⎩⎩ 0 otherwise Figure 2-7: joint distributions of S1 and S2 © A.G.Billard 2004 – Last Update March 2011
Page 1 and 2: SCHOOL OF ENGINEERING MACHINE LEARN
Page 3 and 4: 3 4. 4 Regression Techniques ......
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: 27 The CCA algorithm consists thus
Page 31 and 32: 31 2.3.2 Why Gaussian variables are
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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
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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
Page 55 and 56: 55 Figure 3-16: Clustering with 3 G
Page 57 and 58: 57 When the transformation A is lin
Page 59 and 60: 59 C: X → Y ( ) C x K = arg max
Page 61 and 62: 61 Figure 3-18: Linear combination
Page 63 and 64: 63 Figure 3-19: Bayes classificatio
Page 65 and 66: 65 ⎛⎛ min ⎜⎜ w ⎝⎝ N i=
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Page 69 and 70: 69 Figure 4-2: Illustration of the
Page 71 and 72: 71 4.4.2 Multi-Gaussian Case It is
Page 73 and 74: 73 5 Kernel Methods These lecture n
Page 75 and 76: 75 The kernel k provides a metric o
Page 77 and 78: 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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