MACHINE LEARNING TECHNIQUES - LASA

More documents

Recommendations

Info

86 1 This implies that det ( A − ) is a constant (and is equal to ± 1 in the particular case of whitenend mixtures). Assuming then that the data is zero-mean and white, that the number of sources equal the dimension of the data, i.e. N , and replacing in (5.20), one can see that maximizing the negentropy is equivalent to minimizing mutual information, as the two quantities differ only by a constant factor, i.e. N I s ,... s cst. J x . ( ) ( ) 1 N = −∑ (5.23) Given that the mutual information of a random vector is nonnegative, and zero if and only if the components of the vector are independent, minimization of mutual information for ICA is interesting in that it has a single global minimum. Unfortunately, the mutual information is difficult to approximate and optimize on the basis of a finite sample of datapoints, and much research on ICA has focused on alternative contrast functions. The earliest approach that optimizes for the negentropy using the fast-ICA algorithm, see Section 2.3.6, relies on an adhoc set of non-linear functions. While not very generic, this approach remains advantageous in that it allows estimating each component separately and in that it does the estimation only on uncorrelated patterns. i= 1 i Non-linear Case We will here assume that the sources and the original datapoints have the same dimension and these are zero-mean and white. In practice this can be achieved by performing first PCA and then reducing to the smallest set of eigenvectors with best representation. Hence, given a set of j j= observation vectors { } 1,... i M X = x , we will look for the sources s 1 ,.... N i= 1,... N s , such that: s and 1 i = Wxi W = A − . As in linear ICA, we do not compute A and then its inverse, but rather estimate directly its inverse by computing the columns of W. While the fast-ICA method in linear ICA relied on chosing one or two predefined non-linear functions for the transformation, Kernel ICA solves ICA by considering not just a single non-linear { 1 ., 2 .,...} function, but an entire function space H:= h ( ) h ( ) of candidate nonlinearities. The hope is that using a function space makes it possible to adapt to a variety of sources and thus would h, h ,... make the algorithms more robust to varying source distributions. Each function 1 2 from goes ° N → ° . We can hence determine a set of N transformations j= M j { } 1... i h : x H, i 1,... N i → = with associated kernels ( ) ( ) K = h ., h . , i= 1,... N. Beware i i i that we are here comparing the projections of the datapoints along the dimensions i= 1,... N of the original dataset, which is thought to be the same dimension as that of the sources. Each matrix K is then M × M . i In Section 5.4, when considering kernel CCA, we saw that one can determine the projections that h, h ,... maximize the correlation across the variables using different sets of projections 1 2 measured by ( K K ) 1 ,.... N . This is ρ . This quantity is comprised between zero and one. While CCA aimed at finding the maximal correlation, kernel ICA will try to find the projections that maximize © A.G.Billard 2004 – Last Update March 2011
87 statistical independence. We define a measure of this independence through λ( K ,.... K ) 1 ρ( K ,.... K ) = − . This quantity is also comprised between 0 and 1 and 1 N 1 N is equal to 1 if the variables X ,..., 1 X are pairwise independent. Optimizing for independence N can be done by maximizing λ ( K K ) function: 1 ,.... N . This is equivalent to minimizing the following objective ( ,.... ) log λ ( ,.... ) J K K K K =− (5.24) 1 N 1 This optimization problem is solved in an iterative method. One starts with an initial guess for W and then iterates by gradient descent on J. Details can be found in Bach & Jordan 2002. Interestingly, an implementation of Kernel-ICA, whose computational complexity is linear in the number of data points, is proposed there. This reduces importantly the computational costs. N © 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
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
Page 49 and 50: 49 3.1.4 Clustering with Mixtures o
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
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
Page 79 and 80: 79 1 M The solutions to the dual ei
Page 81 and 82: 81 5.4 Kernel CCA The linear versio
Page 83 and 84: 83 additional ridge parameter induc
Page 85: 85 Figure 5-3: TOP: Marginal (left)
Page 89 and 90: 89 J j ( µ 1,...., µ K) = ∑∑
Page 91 and 92: 91 A simple pattern recognition alg
Page 93 and 94: 93 ( ) ( , ) f x = sign w x + b (5.
Page 95 and 96: 95 Figure 5-6: A binary classificat
Page 97 and 98: 97 where N is the number of support
Page 99 and 100: 99 5.8 Support Vector Regression In
Page 101 and 102: 101 The optimization problem given
Page 103 and 104: 103 Note that since we never have t
Page 105 and 106: 105 Figure 5-13: Effect of the kern
Page 107 and 108: 107 To better understand the effect
Page 109 and 110: 109 5.9 Gaussian Process Regression
Page 111 and 112: 111 One can then use the above expr
Page 113 and 114: 113 5.9.2 Equivalence of Gaussian P
Page 115 and 116: 115 5.9.3 Curse of dimensionality,
Page 117 and 118: 117 The weight w determines the slo
Page 119 and 120: 119 Figure 5-21: Example of success
Page 121 and 122: 121 • its performance tends to de
Page 123 and 124: 123 neurons. Furthermore, they lear
Page 125 and 126: 125 The sigmoid f x ( x) ( ) = tanh
Page 127 and 128: 127 6.3.2 Information Theory and th
Page 129 and 130: 129 ( R) ⎛⎛det ⎞⎞ I( x, y)
Page 131 and 132: 131 y= ∑ w x ) of Because of the
Page 133 and 134: 133 6.5 Willshaw net David Willshaw
Page 135 and 136: 135 6.6.1 Weights bounds One of the
Page 137 and 138:
137 Figure 6-11: The weight vector
Page 139 and 140:
139 6.6.4 Oja’s one Neuron Model
Page 141 and 142:
141 If y i and y are highly correla
Page 143 and 144:
143 Foldiak’s second model allows
Page 145 and 146:
145 ∂ ∂ J 1 = fy − 1 2 λ 1yf
Page 147 and 148:
147 6.8 The Self-Organizing Map (SO
Page 149 and 150:
149 6. Decrease the size of the nei
Page 151 and 152:
151 the resulting distribution is a
Page 153 and 154:
153 To simplify the description of
Page 155 and 156:
155 C µν −1 where ( ) is the µ
Page 157 and 158:
157 The continuous time Hopfield ne
Page 159 and 160:
159 ∂f If the slope is negative,
Page 161 and 162:
161 7.2 Hidden Markov Models Hidden
Page 163 and 164:
163 Figure 7-2: Schematic illustrat
Page 165 and 166:
165 these two quantities to compute
Page 167 and 168:
167 7.2.4 Decoding an HMM There are
Page 169 and 170:
169 7.2.5 Further Readings Rabiner,
Page 171 and 172:
171 7.3.1 Principle In reinforcemen
Page 173 and 174:
173 general, acting to maximize imm
Page 175 and 176:
175 of reinforcement learning makes
Page 177 and 178:
177 8 Genetic Algorithms We conclud
Page 179 and 180:
179 However, you must define geneti
Page 181 and 182:
181 Often the crossover operator an
Page 183 and 184:
183 ( A λI) x 0 − = (8.5) where
Page 185 and 186:
185 Joint probability: The joint pr
Page 187 and 188:
187 The two most classical distribu
Page 189 and 190:
189 9.2.7 Statistical Independence
Page 191 and 192:
191 1 1 1 1 h x ∫ a b a b 0 a a
Page 193 and 194:
193 9.4 Estimators 9.4.1 Gradient d
Page 195 and 196:
195 9.4.2.1 Maximum Likelihood Mach
Page 197 and 198:
197 10 References • Machine Learn
show all

MACHINE LEARNING TECHNIQUES - LASA

Create successful ePaper yourself

Delete template?

Save as template?