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625.740 Homework for Module on Multilayer Neural Networks

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Applied and Computational Mathematics
Data Mining: 625.740
Homework for Module on Multilayer Neural Networks
In problems 1 through 5, the perceptrons and neural networks sought each have three inputs:
x1, x2, x3. Let pk represent the parity function of k inputs:
p2 = x1 XOR x2 and p3 = x1 XOR x2 XOR x3.
1. Show a perceptron that calculates x1x2x3.
2. Show a perceptron that calculates x2x3.
3. Show a neural network that calculates x1x2x3 + x1x2x3.
4. Show a neural network that calculates p1 + x1x2x3.
5. Show a neural network that calculates p2(x1, x2) · p3(x1, x2, x3).
6. Generate two-dimensional samples for each of two Gaussians, p(x|ωi) ∼ N(µi
, Σi) with
µ1 =

1
−1

, µ2 =

0
0

, Σ1 =

0.2 0
0 0.2

= Σ2.
Produce 100 samples from each distribution, but to guarantee linear separability, produce
a replacement for a Class 1 vector whenever x1 − x2 < 1 and produce a replacement for a
Class 2 vector whenever x1 − x2 > 1.
Use these vectors to design a linear classifier using the perceptron algorithm. After convergence, draw the decision boundary.

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