We have been discussing some simple ideas from statistical learning theory. PR NPTEL course p.1/123
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1 We have been discussing some simple ideas from statistical learning theory. PR NPTEL course p.1/123
2 We have been discussing some simple ideas from statistical learning theory. The risk minimization framework that we discussed gives us a better perspective on understanding the unifying theme in different learning algorithms. PR NPTEL course p.2/123
3 We have been discussing some simple ideas from statistical learning theory. The risk minimization framework that we discussed gives us a better perspective on understanding the unifying theme in different learning algorithms. We will now go back to studying pattern classification algorithms. PR NPTEL course p.3/123
4 We have been discussing some simple ideas from statistical learning theory. The risk minimization framework that we discussed gives us a better perspective on understanding the unifying theme in different learning algorithms. We will now go back to studying pattern classification algorithms. We will first briefly review algorithms for learning linear classifiers and then start looking at methods to learn nonlinear classifiers. PR NPTEL course p.4/123
5 Linear Models In the two class case, the linear classifier is given by h(x) = sign(w T X + w 0 ) PR NPTEL course p.5/123
6 Linear Models In the two class case, the linear classifier is given by h(x) = sign(w T X + w 0 ) We have seen that we can also think of h(x) as h(x) = sign(w T Φ(X) + w 0 ), where Φ(X) = [φ 1 (X),,φ m (X)] T as long as φ i are fixed (possibly non-linear) functions. PR NPTEL course p.6/123
7 We discussed many algorithms for learning W. PR NPTEL course p.7/123
8 We discussed many algorithms for learning W. The Perceptron algorithm is a simple error-correcting method that is guarenteed to find a separating hyperplane if one exists. PR NPTEL course p.8/123
9 We discussed many algorithms for learning W. The Perceptron algorithm is a simple error-correcting method that is guarenteed to find a separating hyperplane if one exists. The perceptron convergence theorem shows that given any training set of linearly separable patterns, the algorithm will find a separating hyperplane. PR NPTEL course p.9/123
10 We discussed many algorithms for learning W. The Perceptron algorithm is a simple error-correcting method that is guarenteed to find a separating hyperplane if one exists. The perceptron convergence theorem shows that given any training set of linearly separable patterns, the algorithm will find a separating hyperplane. Our discussion on statistical learning theory gives us an idea of how many iid examples we should have before we can be confident that the hyperplane that separates the examples will also do well on test data. PR NPTEL course p.10/123
11 We have also seen the least-squares method where we find W to minimize J(W) = 1 n i (W T X i y i ) 2 where, for simplicity of notation, we have assumed augumented feature vectors. PR NPTEL course p.11/123
12 We have also seen the least-squares method where we find W to minimize J(W) = 1 n i (W T X i y i ) 2 where, for simplicity of notation, we have assumed augumented feature vectors. In our risk minimization framework, H is parametrized by W, we take h(x) = W T X and minimize empirical risk under squared-error loss function. PR NPTEL course p.12/123
13 We have seen how to obtain the least-squares solution: W = (A T A) 1 A T Y where rows of matrix A are feature vectors and components of Y are y i. PR NPTEL course p.13/123
14 We have seen how to obtain the least-squares solution: W = (A T A) 1 A T Y where rows of matrix A are feature vectors and components of Y are y i. The least-squares method can also be used to learn linear regression models. PR NPTEL course p.14/123
15 We have seen how to obtain the least-squares solution: W = (A T A) 1 A T Y where rows of matrix A are feature vectors and components of Y are y i. The least-squares method can also be used to learn linear regression models. The only difference is that in a regression model, the y i are real-valued. PR NPTEL course p.15/123
16 We have seen that we can also minimize the empirical risk J(W) using gradient descent. PR NPTEL course p.16/123
17 We have seen that we can also minimize the empirical risk J(W) using gradient descent. We can also run this gradient descent in an incremental fashion by considering one example at a time. PR NPTEL course p.17/123
18 We have seen that we can also minimize the empirical risk J(W) using gradient descent. We can also run this gradient descent in an incremental fashion by considering one example at a time. That gives us another classical algorithm called the LMS algorithm. PR NPTEL course p.18/123
19 We have also seen that we can use the least squares idea to learn a model g(w T X) by redefining J as J(W) = 1 n i (g(w T X i ) y i ) 2 PR NPTEL course p.19/123
20 We have also seen that we can use the least squares idea to learn a model g(w T X) by redefining J as J(W) = 1 n i (g(w T X i ) y i ) 2 An important example is the logistic regression where we take g as the sigmoid function. PR NPTEL course p.20/123
21 We have also seen that we can use the least squares idea to learn a model g(w T X) by redefining J as J(W) = 1 n i (g(w T X i ) y i ) 2 An important example is the logistic regression where we take g as the sigmoid function. We minimize J by incremental version of gradient descent. PR NPTEL course p.21/123
22 Another important method for learning linear classifiers is the Fisher Linear Discriminant. PR NPTEL course p.22/123
23 Another important method for learning linear classifiers is the Fisher Linear Discriminant. Here, we look for a direction W such that the patterns of the two classes get well-separated when projected onto this one-dimensional subspace. PR NPTEL course p.23/123
24 Another important method for learning linear classifiers is the Fisher Linear Discriminant. Here, we look for a direction W such that the patterns of the two classes get well-separated when projected onto this one-dimensional subspace. As we mentioned, Fisher Linear Discriminant can be thought of as a special case of least-squares method of learning a linear regression model with special target values. PR NPTEL course p.24/123
25 Beyond linear models Learning linear models (classifiers) is generally efficient. PR NPTEL course p.25/123
26 Beyond linear models Learning linear models (classifiers) is generally efficient. However, linear models are not always sufficient. PR NPTEL course p.26/123
27 Beyond linear models Learning linear models (classifiers) is generally efficient. However, linear models are not always sufficient. Best linear functions may still be a poor fit. PR NPTEL course p.27/123
28 Beyond linear models Learning linear models (classifiers) is generally efficient. However, linear models are not always sufficient. Best linear functions may still be a poor fit. We have looked at three broad approaches to learning nonlinear classifiers. PR NPTEL course p.28/123
29 Beyond linear models Learning linear models (classifiers) is generally efficient. However, linear models are not always sufficient. Best linear functions may still be a poor fit. We have looked at three broad approaches to learning nonlinear classifiers. We now discuss neural network models. PR NPTEL course p.29/123
30 Neural network models We need a good parameterized class of nonlinear functions to learn nonlinear classifiers. PR NPTEL course p.30/123
31 Neural network models We need a good parameterized class of nonlinear functions to learn nonlinear classifiers. Artificial neural networks are one such class PR NPTEL course p.31/123
32 Neural network models We need a good parameterized class of nonlinear functions to learn nonlinear classifiers. Artificial neural networks are one such class Nonlinear functions are built up through composition of summation and sigmoids. PR NPTEL course p.32/123
33 Neural network models We need a good parameterized class of nonlinear functions to learn nonlinear classifiers. Artificial neural networks are one such class Nonlinear functions are built up through composition of summation and sigmoids. Useful for both classification and Regression. PR NPTEL course p.33/123
34 In this course we will study only multilayer feedforward networks. PR NPTEL course p.34/123
35 In this course we will study only multilayer feedforward networks. They are useful because they offer good parameterized class of nonlinear functions and there are some efficient algorithms to learn them. PR NPTEL course p.35/123
36 In this course we will study only multilayer feedforward networks. They are useful because they offer good parameterized class of nonlinear functions and there are some efficient algorithms to learn them. However, historically, development of (artificial) neural network models was motivated by some ideas on the structure of human brain. PR NPTEL course p.36/123
37 In this course we will study only multilayer feedforward networks. They are useful because they offer good parameterized class of nonlinear functions and there are some efficient algorithms to learn them. However, historically, development of (artificial) neural network models was motivated by some ideas on the structure of human brain. We briefly look at this perspective of neural networks as an approach to engineering intelligent systems. PR NPTEL course p.37/123
38 What is an Artificial Neural Network? PR NPTEL course p.38/123
39 What is an Artificial Neural Network? "A parallel distributed information processor made up of simple processing units that has a propensity for acquiring problem solving knowledge through experience" PR NPTEL course p.39/123
40 What is an Artificial Neural Network? "A parallel distributed information processor made up of simple processing units that has a propensity for acquiring problem solving knowledge through experience" Large number of inter connected units PR NPTEL course p.40/123
41 What is an Artificial Neural Network? "A parallel distributed information processor made up of simple processing units that has a propensity for acquiring problem solving knowledge through experience" Large number of inter connected units Each unit implements simple function, nonlinear PR NPTEL course p.41/123
42 What is an Artificial Neural Network? "A parallel distributed information processor made up of simple processing units that has a propensity for acquiring problem solving knowledge through experience" Large number of inter connected units Each unit implements simple function, nonlinear The knowledge resides in the interconnection strengths. PR NPTEL course p.42/123
43 What is an Artificial Neural Network? "A parallel distributed information processor made up of simple processing units that has a propensity for acquiring problem solving knowledge through experience" Large number of inter connected units Each unit implements simple function, nonlinear The knowledge resides in the interconnection strengths. Problem solving ability is often through learning PR NPTEL course p.43/123
44 What is an Artificial Neural Network? "A parallel distributed information processor made up of simple processing units that has a propensity for acquiring problem solving knowledge through experience" Large number of inter connected units Each unit implements simple function, nonlinear The knowledge resides in the interconnection strengths. Problem solving ability is often through learning An architecture inspired by the structure of Brain PR NPTEL course p.44/123
45 The Human Brain Neuron - the basic computing unit PR NPTEL course p.45/123
46 The Human Brain Neuron - the basic computing unit Brain is a highly organized structure of networks of interconnected neurons PR NPTEL course p.46/123
47 The Human Brain Neuron - the basic computing unit Brain is a highly organized structure of networks of interconnected neurons In the Brain No of neurons (100 billion) PR NPTEL course p.47/123
48 The Human Brain Neuron - the basic computing unit Brain is a highly organized structure of networks of interconnected neurons In the Brain No of neurons (100 billion) Average synapses per neuron ( ) PR NPTEL course p.48/123
49 The Human Brain Neuron - the basic computing unit Brain is a highly organized structure of networks of interconnected neurons In the Brain No of neurons (100 billion) Average synapses per neuron ( ) Total synapses PR NPTEL course p.49/123
50 The Human Brain Neuron - the basic computing unit Brain is a highly organized structure of networks of interconnected neurons In the Brain No of neurons (100 billion) Average synapses per neuron ( ) Total synapses Neuron time constants Milliseconds PR NPTEL course p.50/123
51 The Human Brain Neuron - the basic computing unit Brain is a highly organized structure of networks of interconnected neurons In the Brain No of neurons (100 billion) Average synapses per neuron ( ) Total synapses Neuron time constants Milliseconds Single neuron can send 100 spikes per second PR NPTEL course p.51/123
52 A rough estimate of processing power: One arithmetic operation per synapse 10 4 operations per neuron per spike 10 6 operations per neuron per sec operations per sec!! (A gigaflop is 10 9 operations, teraflop is operations!) PR NPTEL course p.52/123
53 A rough estimate of processing power: One arithmetic operation per synapse 10 4 operations per neuron per spike 10 6 operations per neuron per sec operations per sec!! (A gigaflop is 10 9 operations, teraflop is operations!) Massive parallelism can deliver massive computing power, PR NPTEL course p.53/123
54 A rough estimate of processing power: One arithmetic operation per synapse 10 4 operations per neuron per spike 10 6 operations per neuron per sec operations per sec!! (A gigaflop is 10 9 operations, teraflop is operations!) Massive parallelism can deliver massive computing power, if we know how to manage it PR NPTEL course p.54/123
55 Digital computers: Precise design, highly constrained, not very adaptive or fault tolerant, Centralized control, deterministic, basic switching times 10 9 sec PR NPTEL course p.55/123
56 Digital computers: Precise design, highly constrained, not very adaptive or fault tolerant, Centralized control, deterministic, basic switching times 10 9 sec Natural neural networks: massively parallel, highly adaptive and fault tolerant, self configuring, self repairing, noisy, stochastic, basic switching time 10 3 sec PR NPTEL course p.56/123
57 Digital computers: Precise design, highly constrained, not very adaptive or fault tolerant, Centralized control, deterministic, basic switching times 10 9 sec Natural neural networks: massively parallel, highly adaptive and fault tolerant, self configuring, self repairing, noisy, stochastic, basic switching time 10 3 sec Most capabilities of Brain are LEARNT. PR NPTEL course p.57/123
58 Artificial Intelligence (AI) Understanding intelligence in computational terms. PR NPTEL course p.58/123
59 Artificial Intelligence (AI) Understanding intelligence in computational terms. Developing machines that are intelligent. PR NPTEL course p.59/123
60 Artificial Intelligence (AI) Understanding intelligence in computational terms. Developing machines that are intelligent. At least two distinct approaches PR NPTEL course p.60/123
61 Artificial Intelligence (AI) Understanding intelligence in computational terms. Developing machines that are intelligent. At least two distinct approaches Try to model intelligent behavior in terms of processing structured symbols. (Resulting methods, algorithms etc may not resemble brain at the implementation level) PR NPTEL course p.61/123
62 Artificial Intelligence (AI) Understanding intelligence in computational terms. Developing machines that are intelligent. At least two distinct approaches Try to model intelligent behavior in terms of processing structured symbols. (Resulting methods, algorithms etc may not resemble brain at the implementation level) A second approach is based on mimicking human brain at architectural/implementation level PR NPTEL course p.62/123
63 The symbolic AI approach Brain is to be understood in computational terms only PR NPTEL course p.63/123
64 The symbolic AI approach Brain is to be understood in computational terms only Physical symbol system hypothesis PR NPTEL course p.64/123
65 The symbolic AI approach Brain is to be understood in computational terms only Physical symbol system hypothesis A digital computer is a universal symbol manipulator and can be programmed to be intelligent PR NPTEL course p.65/123
66 The symbolic AI approach Brain is to be understood in computational terms only Physical symbol system hypothesis A digital computer is a universal symbol manipulator and can be programmed to be intelligent Many useful engineering applications e.g. Expert systems PR NPTEL course p.66/123
67 The symbolic AI approach Brain is to be understood in computational terms only Physical symbol system hypothesis A digital computer is a universal symbol manipulator and can be programmed to be intelligent Many useful engineering applications e.g. Expert systems An implicit faith: The architecture of Brain per se is irrelevant for engineering intelligent artifacts PR NPTEL course p.67/123
68 Artificial Neural Networks Can be viewed as one approach towards understanding brain/building intelligent machines PR NPTEL course p.68/123
69 Artificial Neural Networks Can be viewed as one approach towards understanding brain/building intelligent machines Computational architectures inspired by brain PR NPTEL course p.69/123
70 Artificial Neural Networks Can be viewed as one approach towards understanding brain/building intelligent machines Computational architectures inspired by brain Computational methods for learning dependencies in data stream PR NPTEL course p.70/123
71 Artificial Neural Networks Can be viewed as one approach towards understanding brain/building intelligent machines Computational architectures inspired by brain Computational methods for learning dependencies in data stream e.g. Pattern Recognition, System identification PR NPTEL course p.71/123
72 Artificial Neural Networks Can be viewed as one approach towards understanding brain/building intelligent machines Computational architectures inspired by brain Computational methods for learning dependencies in data stream e.g. Pattern Recognition, System identification Characteristics: Emergent properties, learning, self adaptation PR NPTEL course p.72/123
73 Artificial Neural Networks Can be viewed as one approach towards understanding brain/building intelligent machines Computational architectures inspired by brain Computational methods for learning dependencies in data stream e.g. Pattern Recognition, System identification Characteristics: Emergent properties, learning, self adaptation Modeling Biology? Mathematically purified neurons!! PR NPTEL course p.73/123
74 Artificial Neural Networks Computing machines that try to mimic brain architecture. PR NPTEL course p.74/123
75 Artificial Neural Networks Computing machines that try to mimic brain architecture. A large network of interconnected units PR NPTEL course p.75/123
76 Artificial Neural Networks Computing machines that try to mimic brain architecture. A large network of interconnected units Each unit has simple input-output mapping PR NPTEL course p.76/123
77 Artificial Neural Networks Computing machines that try to mimic brain architecture. A large network of interconnected units Each unit has simple input-output mapping Each interconnection has numerical weight attached to it PR NPTEL course p.77/123
78 Artificial Neural Networks Computing machines that try to mimic brain architecture. A large network of interconnected units Each unit has simple input-output mapping Each interconnection has numerical weight attached to it Output of unit depends on outputs and connection weights of units connected to it PR NPTEL course p.78/123
79 Artificial Neural Networks Computing machines that try to mimic brain architecture. A large network of interconnected units Each unit has simple input-output mapping Each interconnection has numerical weight attached to it Output of unit depends on outputs and connection weights of units connected to it Knowledge resides in the weights PR NPTEL course p.79/123
80 Artificial Neural Networks Computing machines that try to mimic brain architecture. A large network of interconnected units Each unit has simple input-output mapping Each interconnection has numerical weight attached to it Output of unit depends on outputs and connection weights of units connected to it Knowledge resides in the weights Problem solving ability is often through learning PR NPTEL course p.80/123
81 Single neuron model PR NPTEL course p.81/123
82 Single neuron model x i are inputs into the (artificial) neuron and w i are the corresponding weights. y is the output of the neuron PR NPTEL course p.82/123
83 Single neuron model x i are inputs into the (artificial) neuron and w i are the corresponding weights. y is the output of the neuron Net input : η = j w jx j output: y = f(η), where f(.) is called activation function PR NPTEL course p.83/123
84 Single neuron model x i are inputs into the (artificial) neuron and w i are the corresponding weights. y is the output of the neuron Net input : η = j w jx j output: y = f(η), where f(.) is called activation function (Perceptron, AdaLinE are such models). PR NPTEL course p.84/123
85 Networks of neurons We can connect a number of such units or neurons to form a network. Inputs to a neuron can be outputs of other neurons (and/or external inputs). PR NPTEL course p.85/123
86 Networks of neurons We can connect a number of such units or neurons to form a network. Inputs to a neuron can be outputs of other neurons (and/or external inputs). PR NPTEL course p.86/123
87 Networks of neurons We can connect a number of such units or neurons to form a network. Inputs to a neuron can be outputs of other neurons (and/or external inputs). Notation: y j output of j th neuron; w ij weight of connection from neuron i to neuron j. PR NPTEL course p.87/123
88 Each neuron computes weighted sum of inputs and passes it through its activation function, to compute output PR NPTEL course p.88/123
89 Each neuron computes weighted sum of inputs and passes it through its activation function, to compute output For example, output of neuron 5 is y 5 = f 5 (w 35 y 3 + w 45 y 4 ) PR NPTEL course p.89/123
90 Each neuron computes weighted sum of inputs and passes it through its activation function, to compute output For example, output of neuron 5 is y 5 = f 5 (w 35 y 3 + w 45 y 4 ) = f 5 (w 35 f 3 (w 13 y 1 + w 23 y 2 ) + w 45 f 4 (w 14 y 1 + w 24 y 2 )) PR NPTEL course p.90/123
91 Each neuron computes weighted sum of inputs and passes it through its activation function, to compute output For example, output of neuron 5 is y 5 = f 5 (w 35 y 3 + w 45 y 4 ) = f 5 (w 35 f 3 (w 13 y 1 + w 23 y 2 ) + w 45 f 4 (w 14 y 1 + w 24 y 2 )) By convention, we take y 1 = x 1 and y 2 = x 2. PR NPTEL course p.91/123
92 Each neuron computes weighted sum of inputs and passes it through its activation function, to compute output For example, output of neuron 5 is y 5 = f 5 (w 35 y 3 + w 45 y 4 ) = f 5 (w 35 f 3 (w 13 y 1 + w 23 y 2 ) + w 45 f 4 (w 14 y 1 + w 24 y 2 )) By convention, we take y 1 = x 1 and y 2 = x 2. Here, x 1, x 2 are inputs and y 5, y 6 are outputs. PR NPTEL course p.92/123
93 A single neuron represents a class of functions from R m to R. PR NPTEL course p.93/123
94 A single neuron represents a class of functions from R m to R. Specific set of weights realise specific functions. PR NPTEL course p.94/123
95 A single neuron represents a class of functions from R m to R. Specific set of weights realise specific functions. By interconnecting many units/neurons, networks can represent more complicated functions from R m to R m. PR NPTEL course p.95/123
96 A single neuron represents a class of functions from R m to R. Specific set of weights realise specific functions. By interconnecting many units/neurons, networks can represent more complicated functions from R m to R m. The architecture constrains the function class that can be represented. Weights define specific function in the class. PR NPTEL course p.96/123
97 A single neuron represents a class of functions from R m to R. Specific set of weights realise specific functions. By interconnecting many units/neurons, networks can represent more complicated functions from R m to R m. The architecture constrains the function class that can be represented. Weights define specific function in the class. To form meaningful networks, nonlinearity of activation function is important. PR NPTEL course p.97/123
98 Typical activation functions 1. Hard limiter: f(x) = 1 if x > τ = 0 otherwise PR NPTEL course p.98/123
99 Typical activation functions 1. Hard limiter: f(x) = 1 if x > τ = 0 otherwise We can keep the τ to be zero and add one more input line to the neuron. An example of a single neuron with this activation function is Perceptron. PR NPTEL course p.99/123
100 Activation functions (cont) Sigmoid function: f(x) = a 1 + exp ( bx), a, b > 0 PR NPTEL course p.100/123
101 Activation functions (Contd.) 3. tanh f(x) = atanh(bx), a, b > 0 PR NPTEL course p.101/123
102 Why study such models? A belief that the architecture of Brain is critical to intelligent behavior. PR NPTEL course p.102/123
103 Why study such models? A belief that the architecture of Brain is critical to intelligent behavior. Models can implement highly nonlinear functions. They are adaptive and can be trained. PR NPTEL course p.103/123
104 Why study such models? A belief that the architecture of Brain is critical to intelligent behavior. Models can implement highly nonlinear functions. They are adaptive and can be trained. Useful in many applications PR NPTEL course p.104/123
105 Why study such models? A belief that the architecture of Brain is critical to intelligent behavior. Models can implement highly nonlinear functions. They are adaptive and can be trained. Useful in many applications Time series prediction PR NPTEL course p.105/123
106 Why study such models? A belief that the architecture of Brain is critical to intelligent behavior. Models can implement highly nonlinear functions. They are adaptive and can be trained. Useful in many applications Time series prediction system identification and control PR NPTEL course p.106/123
107 Why study such models? A belief that the architecture of Brain is critical to intelligent behavior. Models can implement highly nonlinear functions. They are adaptive and can be trained. Useful in many applications Time series prediction system identification and control pattern recognition and Regression PR NPTEL course p.107/123
108 Why study such models? A belief that the architecture of Brain is critical to intelligent behavior. Models can implement highly nonlinear functions. They are adaptive and can be trained. Useful in many applications Time series prediction system identification and control pattern recognition and Regression Model can help us understand Brain function Computational neuroscience PR NPTEL course p.108/123
109 Many different models are possible PR NPTEL course p.109/123
110 Many different models are possible Evolution: Discrete time / continuous time synchronous / asynchronous deterministic / stochastic PR NPTEL course p.110/123
111 Many different models are possible Evolution: Discrete time / continuous time synchronous / asynchronous deterministic / stochastic Interconnections: Feedforward / having feedback PR NPTEL course p.111/123
112 Many different models are possible Evolution: Discrete time / continuous time synchronous / asynchronous deterministic / stochastic Interconnections: Feedforward / having feedback States or outputs of units: binary / finitely many / continuous PR NPTEL course p.112/123
113 Recurrent networks The network we saw earlier has no feedback. PR NPTEL course p.113/123
114 Recurrent networks The network we saw earlier has no feedback. Here is an example of a network with feedback PR NPTEL course p.114/123
115 Recurrent networks The network we saw earlier has no feedback. Here is an example of a network with feedback Can model a dynamical system: y(k) = f(y(k 1), x 1 (k), x 2 (k)) PR NPTEL course p.115/123
116 We will consider only feedforward networks which provide a general class of nonlinear functions. PR NPTEL course p.116/123
117 We will consider only feedforward networks which provide a general class of nonlinear functions. These can always be organized as a layered network. PR NPTEL course p.117/123
118 We will consider only feedforward networks which provide a general class of nonlinear functions. These can always be organized as a layered network. This network represents a class of functions from R 2 to R 2. PR NPTEL course p.118/123
119 Each unit can also have a bias input. PR NPTEL course p.119/123
120 Each unit can also have a bias input. This is shown for a single unit below. PR NPTEL course p.120/123
121 Each unit can also have a bias input. This is shown for a single unit below. One can always think of bias as an extra input y = f ( d i=1 w i x i + w 0 ) PR NPTEL course p.121/123
122 Each unit can also have a bias input. This is shown for a single unit below. One can always think of bias as an extra input ( d ) ( d y = f w i x i + w 0 = f i=1 i=0 ) w i x i, x 0 = +1 PR NPTEL course p.122/123
123 Multilayer feedforward networks Here is a general multilayer feedforward network. PR NPTEL course p.123/123
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