An artificial neural network is a circuit composed of interconnected simple circuit elements called neurons. Each...

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An artificial neural network is a circuit composed of interconnected simple circuit elements called neurons. Each neuron represents a map, typically with multiple inputs and a single output. Specifically, the output of the neuron is a function of the sum of the inputs. The function at the output of the neuron is called the activation function. The following diagram is a model of single neuron. For this particular neuron, the activation function is simply the identity. X1 X2 Xn W1 W2 Wn training points: Xa M = Input vector: x = [x₁, X₂, , Xn]¹ € Rª Output: y E R We wish to find the value of the weights w such that the neuron approximates the map between inputs to output as closely as possible (training). Suppose that we are given X₁ E Rn×p and y₁ € Rº representing p training pairs (p>n). The objective 2 function to be minimized in the training of the neuron is f (w) = ½ || ya – Xaw||² a) Find the gradient of the objective function. b) Write the conjugate gradient algorithm to train the neuron (i.e. to find the weights) c) Suppose that we wish to approximate the function F: R² → R given by F(x) = (sinx₁)(cos x₂) Use the conjugate gradient algorithm from part b to train the linear neuron, using the following -0.5 -0.5 -0.5 0 0 0 0.5 0.5 0.51 -0.5 0 0.5 -0.5 0 0.5 -0.5 0 0.5 d) Plot the level sets of the objective function for the problem in part c, at levels 0.01, 0.1, 0.2, and 0.4. Check if the solution in part-c agrees with the level sets. Weight vector: w = [W₁, W₂, ..., wn]¹ € R¹ An artificial neural network is a circuit composed of interconnected simple circuit elements called neurons. Each neuron represents a map, typically with multiple inputs and a single output. Specifically, the output of the neuron is a function of the sum of the inputs. The function at the output of the neuron is called the activation function. The following diagram is a model of single neuron. For this particular neuron, the activation function is simply the identity. X1 X2 Xn W1 W2 Wn training points: Xa M = Input vector: x = [x₁, X₂, , Xn]¹ € Rª Output: y E R We wish to find the value of the weights w such that the neuron approximates the map between inputs to output as closely as possible (training). Suppose that we are given X₁ E Rn×p and y₁ € Rº representing p training pairs (p>n). The objective 2 function to be minimized in the training of the neuron is f (w) = ½ || ya – Xaw||² a) Find the gradient of the objective function. b) Write the conjugate gradient algorithm to train the neuron (i.e. to find the weights) c) Suppose that we wish to approximate the function F: R² → R given by F(x) = (sinx₁)(cos x₂) Use the conjugate gradient algorithm from part b to train the linear neuron, using the following -0.5 -0.5 -0.5 0 0 0 0.5 0.5 0.51 -0.5 0 0.5 -0.5 0 0.5 -0.5 0 0.5 d) Plot the level sets of the objective function for the problem in part c, at levels 0.01, 0.1, 0.2, and 0.4. Check if the solution in part-c agrees with the level sets. Weight vector: w = [W₁, W₂, ..., wn]¹ € R¹

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c To approximate the function Fx sinx1cosx2 using the linear neuron we will use the conjugate gradient algorithm with the provided training points X d ... View the full answer