Artificial Neural Networks (CS5652): Homework 3

Instructor: J.-S. R. Jang

1. (15%) Use the data set x = [0; 1; 2; 3; 4; 5; 6; 7; 8; 9] and y = [0; 1.5; 3.5; 6.5; 7.5; 10.5; 11.5; 14.5; 15.5; 18.5] to do polynomial fitting, with the orders being 6, 7, 8, and 9. Submit hardcopy of your MATLAB script and the following plot generated by the script: . (Hint: try spring.m.)
2. (15%) The following figure shows an MLP involving a single hidden neuron and jumping connections from the inputs to the output directly. (a) Construct a truth table for all variables x1, x2, x3, and x4. Show that the network solves the XOR problem. (b) Plot the decision boundary of x3 in x1-x2 plane; (c) Plot the decision boundary of x4 in x1-x2 plane and explain how you derive it. (Note that your decision boundary should not be limited to the unit square only.) . Note that the number on top of each neuron is the threshold and it is subtracted from the net input. For instance, the equation for x3 is x3 = step(x1+x2-1.5).
3. (30%) An 2-2-1 MLP has the following configuration: Suppose that the above MLP has the following decision boundary for XOR problem: . Find the missing weights of the MLP.
4. (20%) Finish designing the perceptron with step-function threshold units in the following figure; by determining threshold values (i=1,2,3) and six connection weights between the input layer and the hidden layer that enable your perceptron to recognize correctly whether an arbitrary point (x, y) is in the shaded triangular area T or not. The perceptron is supposed to produce the output o:
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   Figure: (a) A classification problem, and (b) the perceptron designed to solve the problem. You should put your answer into a compact matrix: where w_xi and w_yi is the connection weights from inputs x and y, respectively, to the hidden node i. (Note that there are six sets of solutions. You only need to find one of them. If you find all of them, you get extra credits.)