MACHINE LEARNING TECHNIQUES - LASA

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38 In certain applications, however, it may be desired to use a symmetric decorrelation, in which no vectors are ``privileged'' over others. This can be accomplished, e.g., by the classical method involving matrix square roots, Let − T ( ) 1/2 W WW W = (2.33) WW − is where W is the matrix ( ,..., T w1 wq ) of the vectors, and the inverse square root ( ) 1 T 2 obtained from the eigenvalue decomposition of simpler alternative is the following iterative algorithm: WW T 1 − T T = FΛ F as 2 ( ) 1 − 2 T = Λ . A WW F F 1. Let W = W / WW Repeat 2. until convergence 3 3 T 2. Let W = W − WW W 2 2 T (2.34) The norm in step 1 can be almost any ordinary matrix norm, e.g., the 2-norm or the largest absolute row (or column) sum (but not the Frobenius norm). Figure 2-12: (Left) Original distribution; (right) decorrelated distribution after projection through ICA projection. The original axes and the ICA projected axes are the horizontal and vertical axes respectively. 2.4 Further Readings In this chapter, we have focused only on the linear version of ICA, and, on one method for solving ICA, namely fast ICA. Note that there exist also methods for non-linear ICA and for timedependent ICA. The reader can refer to [Hyvarien et al, 2003] for further readings on ICA and its applications. © A.G.Billard 2004 – Last Update March 2011
39 3 Clustering and Classification Classification consists of categorizing data according to some criteria. This can be used as a means to reduce the dimensionality of the dataset, prior to performing other form of data processing on the constructed categories. Classification is a key feature of most learning methods. In this chapter, we will first look at one way to perform classification through data clustering. Then, we will look at a number of linear classification methods that exploit eigenvalue decomposition such as those seen for PCA and CCA in the previous chapter. All of these methods are in effect supervised as they rely on an apriori estimate of the number of classes or clusters. Note that we will see other methods that can be used to perform classification in other chapters of these lecture notes. These are for instance feed-forward neural networks, or Hidden Markov Models. As it is often the case, machine learning methods may be used in various ways and there is thus not a single way to classify these! 3.1 Clustering Techniques Clustering is a type of multivariate statistical analysis also known as cluster analysis, unsupervised classification analysis, or numerical taxonomy. Looking at Figure 3-1, one can easily see that the points group naturally into 3 separate clusters, albeit of different shape and size. While it is obvious to see this when looking at the graph, it would become less so obvious if you were to look at the coordinates of these points. Figure 3-1: Example of a dataset that form 3 clusters There are many ways to cluster data depending on the criteria required for the task. One can, for instance, group the data or segment the data into clusters. Clusters can be either globular or convex, such that any line you can draw between two cluster members stays inside the boundaries of the cluster, or non-globular or concave, i.e. taking any shape. Clusters can contain an equal number of data, or be of equal size (distance between the data). Central to all of the goals of cluster analysis is the notion of degree of similarity (or dissimilarity) between the individual objects being clustered. This notion determines the type of clusters that will be formed. In this chapter, we will see different methods of clustering that belong to two major classes of clustering algorithms, namely hierarchical clustering and partitioning algorithms. We will discuss the limitation of such methods. In particular, we will look at a generalization of K-means clustering, namely soft k-means and mixture of Gaussians that enable to overcome some but no all limitations of simple K-means. © 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 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
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Page 35 and 36: 35 2.3.5 ICA Ambiguities We cannot
Page 37: 37 Denote by g the derivative of th
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 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
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Page 73 and 74: 73 5 Kernel Methods These lecture n
Page 75 and 76: 75 The kernel k provides a metric o
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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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