MACHINE LEARNING TECHNIQUES - LASA

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22 It is clear that PCA can be used as the basis of a clustering method, by grouping the points according to how they clump along each projection. Each cluster can then be treated separately. Figure 2-1 illustrates a situation in which PCA is useful in projecting the dataset onto a new frame of reference, from which the cluster of datapoints can be easily inferred. Figure 2-4 on the other hand shows an example in which PCA would fail to segment two clusters. By projecting the data along the principal direction, i.e. the vertical axis, PCA would merge the two clusters. ICA is one alternative method for separating such clusters. Figure 2-4: A classical illustration of problems occurring with variance-based methods such as PCA. 2.1.4 Projection Pursuit Projection–Pursuit (PP) methods aim at finding structures in multivariate data by projecting them on a lower-dimensional subspace. Since there are infinitely many projections from a higher dimension to a lower dimension, PP aims at finding a technique to pursue a finite sequence of projections that can reveal the most “interesting” structures of the data. The interesting projections are often those that depart from that of normal/Gaussian distributions. For instance, these structures can take the form of trends, clusters, hypersurfaces, or anomalies. Figure 2-4 illustrates the ``interestingness'' of non-gaussian projections. The data in this figure is clearly divided into two clusters. However, the principal component, i.e. the direction of maximum variance, would be vertical, providing no separation between the clusters. In contrast, the strongly non-gaussian projection pursuit direction would be horizontal, providing optimal separation of the clusters. Independent Component Analysis (see Section 2.3) is a method for discovering a linear decomposition into directions that maximize non-Gaussianity. In this respect, it provides a first step for PP decomposition. Typically, PP uses a projection index, a functional computed on a projected density (or data set), to measure the “interestingness” of the current projection and then uses a numerical optimizer to move the projection direction to a more interesting position. Interesting projections usually refer to non-linear structures in the data, i.e. structure than cannot be extracted by linear projections such as PCA and ICA. PP can be formalized as follows: Given a dataset P P P P X ∈° and a unit vector a ∈ ° one defines an index Ia : ° → ° that measures the interest of the associated projection P ( ) maximize I a . a X . PP finds the parameters a that © A.G.Billard 2004 – Last Update March 2011
23 PCA is an example of PP approach that takes the variance as the projection index. Further, if we T consider a linear projection of each datapoint x through a of the form z = a xand take the negative Shannon entropy as a measure for the index: =−∫ (2.10) ( ) log ( ) I f z f z dz a z z Where f z is the probability density function of the projected data z, then PP is equivalent to Independent Component Analysis. © 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: 21 2.1.2.2 Reconstruction error min
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
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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=
Page 67 and 68: 67 T ( yi − xi w) 2 M ⎛⎛ ⎞
Page 69 and 70: 69 Figure 4-2: Illustration of the
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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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