MACHINE LEARNING TECHNIQUES - LASA

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192 ( ) ( ) ( ) H( x| y) ( ) ( ) ( , ) I( x, y) = H x −H y| x = H y − = H x + H y −H x y (8.34) Mutual information is a natural measure of the dependence between random variables. If the variables are independent, they provide no information on each other. An important property of mutual information is that one can define an invertible linear transformation y= Wx, s.t. I( y) = H( y) −H( x) − log detW 9.3.4 Relation to mutual information Mutual information can be related to the notion of correlation through canonical correlation analysis. Since information is additive for statistically independent variables and since canonical variates are uncorrelated, the mutual information between two independent variate x and y is the sum of mutual information between all the canonical variates xi and y (i.e. the canonical i projections of x and y). For Gaussian variables this means: ⎛⎛ ⎞⎞ 1 ⎜⎜ 1 ⎟⎟ 1 ⎛⎛ 1 ⎞⎞ I( x; y) = log log 2 2 2 ⎜⎜ ( 1 ρ ⎟⎟= ∑ ⎜⎜ ⎟⎟ ⎜⎜∏ − ) 2 i ⎟⎟ i ⎜⎜( 1−ρi ) ⎟⎟ ⎝⎝ ⎝⎝ ⎠⎠ i ⎠⎠ (8.35) 9.3.5 Kullback-Leibler Distance The most common method to measure the difference between two probability distributions p and q is to us the Kullback-Leibler distance (DKL), sometimes known as relative entropy: DKL is always positive ( ) ( ) ( ) ( ) ( ) pi ⎛⎛ pi⎞⎞ D( p|| q) = p( i) log = E log i qi ⎜⎜ qi ⎟⎟ ⎝⎝ ⎠⎠ D p|| q ≥ 0 ∑ (8.36) . It is zero if and only if p=q, i.e. if the variables are statistically independent. Note that in general the relative entropy or DKL is not symmetric under interchange of the distributions p and q: in general D( p|| q) ≠ D( q|| p) . Hence, DKL is not strictly a distance. The measure of relative entropy is very important in pattern recognition and neural networks, as well as in information theory, as will be shown in other parts of this class. © A.G.Billard 2004 – Last Update March 2011
193 9.4 Estimators 9.4.1 Gradient descent (from Mathworld) Gradient descent is an incremental hill-climbing algorithm that approaches a minimum or maximum of a function by taking steps proportional to the gradient at the current point. An algorithm for finding the nearest local minimum of a function, which presupposes that the gradient of the function can be computed. The method of steepest descent, also called the gradient descent method, starts at a point P i + and, as many times as needed, moves from P 1 i to −∇ the local downhill gradient. P + by minimizing along the line extending from ( ) i 1 f P i When applied to a 1-dimensional function f(x), the method takes the form of iterating ( ) x = x − ε f x (8.37) i i−1 i−1 from a starting point 0 x for small ε > 0 above for the functions ( ) 3 2 until a fixed point is reached. The results are illustrated f x = x − 2x + 2 with ε = 0.1and starting point x 0 = 2 and 0.01, respectively. This method has the severe drawback of requiring a great many iterations for functions, which have long, narrow valley structures. In such cases, a conjugate gradient method is preferable. © 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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21 2.1.2.2 Reconstruction error min
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23 PCA is an example of PP approach
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25 Algorithm: If one further assume
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27 The CCA algorithm consists thus
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29 Figure 2-6: Mixture of variables
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31 2.3.2 Why Gaussian variables are
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33 • In our general definition of
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35 2.3.5 ICA Ambiguities We cannot
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37 Denote by g the derivative of th
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39 3 Clustering and Classification
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41 An agglomerative clustering star
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43 3.1.1.1 The CURE Clustering Algo
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45 Disadvantages of hierarchical cl
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47 Cases where K-means might be vie
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49 3.1.4 Clustering with Mixtures o
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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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