MACHINE LEARNING TECHNIQUES - LASA

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158 Figure 6-17: To determine the convergence of a leaky-integrator neuron with a self-connection, one must find numerically the value m* for which the derivative of the membrane potential m is zero. Here we have a * single stable point m = 10 for a steady input S=30 with a negative self-connection w 11 =− 20 . The zeros of the derivative correspond to equilibrium points. These may however be stable or unstable. A stable point is such that if it slightly pushed away from the stable point, it will eventually come back to its equilibrium. In contrast, an unstable point is such that a small perturbation in the input may send the system away from its equilibrium. In Leak-Integrator Neurons, this is easily achieved and depends on the value of the different parameters, but especially that of the self-connection. Figure 6-17 and Figure 6-18 illustrate these two cases. Figure 6-18: Example of a leaky-integrator neuron for a steady input S=30 with a positive selfconnection w 11 = 20. The system has three equilibrium points at m=0, m=-10 and m=10. m=0 is an unstable point, whereas m=-10 and m=10 are two stable points. This can be seen by observing that the slope of m(s) around m=0 is positive, whereas it is negative for m=-10 and m=10 (see left figure). Stability of the equilibrium points can be determined by looking at the direction of the slope of the derivative of the membrane potential around the equilibrium point, i.e.: dm 1 f( m) = ( m S w11x) dt = τ − + + (6.80) © A.G.Billard 2004 – Last Update March 2011
159 ∂f If the slope is negative, i.e. ( m) < 0 (note that here we are exploring the function in input space ∂m and not in time), then the equilibrium point is stable. Intuitively, this is easy to see. The slope gives you the direction in which you are pulled. With a negative slope you are pulled back to where you were. With a positive slope you are pulled away from where you were, usually in the direction of another equilibrium point. Given that the dynamics of a single neuron can already become very complex, the dynamics of groups of such neurons becomes rapidly intractable. In a seminal paper, Randall Beer (Beer Adaptive Behavior, Vol 3 No 4, 1995) showed how the dynamics of a group of only two interconnected leaky-integrator neurons could lead to four complex dynamics. Figure 6-19 shows the possible trajectories of values taken by the outputs of the two neurons. Such a plot is called a phase plot. Figure 6-19: Possible dynamics that can be generated by a network composed of two leaky-integrator neurons (R. Beer, Adaptive Behavior, 1995). © 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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MACHINE LEARNING TECHNIQUES - LASA

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