MACHINE LEARNING TECHNIQUES - LASA

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76 5.3 Kernel PCA We start our review of kernel methods with kernel PCA, a non-linear extension to Principal Component Analysis (PCA). To recall, PCA is a powerful technique for extracting structure from high-dimensional data. In Section 2.1 and Section 6.6.3, of these lecture notes, we review two ways of computing PCA. We can either do it through an analytical decomposition into eigenvalues and eigenvectors of the data space, or through an iterative decomposition using Hebbian Learning in multiple input-output networks. A key assumption of PCA was that the transformation applied on the data was linear. Kernel PCA is a generalization of the standard PCA, in that it considers any linear or non-linear transformation of the data. We first start by reintroducing the PCA notation and show how we can go from linear PCA to non-linear PCA. Linear PCA: Assume that the dataset is composed of a set of vectors x i ∈ ° N , i = 1... M . Assume further that the M dataset is zero mean, i.e. ∑ x i = 0. i = 1 PCA finds an orthonormal basis such that the projections along each axis maximize the variance of the data. To do so, PCA proceeds by diagonalizing the covariance matrix of the dataset C=XX T . C is positive definite and can thus be diagonalized with non-negative eigenvalues. The principal components are then all the vectors v i , i= 1..., N, solution of: Cv i = λ i iv (5.4) where λ is a scalar and corresponds to the eigenvalue associated to the eigenvector v i i . Non-Linear Case: Observe first that one can rewrite the eigenvalue decomposition problem of linear PCA in terms i of dot product across pairs of the training datapoints. Indeed, each principal component v can be M 1 j j T expressed as a linear combination of the datapoints. Using C= ∑ x ( x ) and replacing in M j = 1 (5.4), we obtain: M 1 T Cv = x ( x ) v = λiv M i j j i i j= 1 which allows us to express each eigenvector as follows: ∑ (5.5) © A.G.Billard 2004 – Last Update March 2011
77 M 1 T v = ∑ x ( x ) v M λ i j j i 1 = M λ i i j= 1 M T ∑( ( ) ) j= 1 x v x j i j (5.6) T ( ( ) ) i j i j Introducing a set of scalars α = x v for each data point x (these correspond to the value of the j j i projection of x onto the eigenvector vector v . i i Then each eigenvector v with non-zero eigenvalue λ lies in the space spanned 1 M by the input vectors x ,...., x : 1 v = ∑ . (5.7) M i i j α i j x Mλ j = 1 Assuming now that the data are projected into a feature space through a non-linear mapφ and that these are centered in feature space, i.e. M i= 1 i ( x ) the Correlation matrix in the feature space is then given by: ∑ φ = 0, (5.8) where each i 1,..., C 1 T M FF φ = (5.9) = M columns of F are composed of the projections ( x i ) φ . © 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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Page 51 and 52: 51 k ( σ j ) 2 = k ∑ i α = r k
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Page 65 and 66: 65 ⎛⎛ min ⎜⎜ w ⎝⎝ N i=
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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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