# Neural Networks B, SS06 1

UA Dr. Robert Legenstein, WiMAus Prashant Joshi, M.S.

Institute for Theoretical Computer Science
Technische Universität Graz
A-8010 Graz, Austria
{legi, joshi}@igi.tugraz.at

 NACHNAME Vorname Matrikelnmr Team

# Exercise 1: Pattern generator with perceptrons

NOTE: You can download this exercise in pdf or postscript(ps) format here.

In this exercise you are going to do some pattern generation using a network made of 4 perceptrons which receives output feedback. More precisely, the task is to generate two different periodic sequences as the network output, if the network is started from two different initial conditions.

# Theory

A perceptron is a kind of threshold neuron, which receives inputs and gives the output y. The relation between input and output is given by equation 1:

 (1)

where the neuron is receiving inputs, and the input is connected to this neuron by a synapse of weight , and is the bias term. Intuitively, if the weighted sum of the inputs to a neuron is greater then the bias , then the neuron outputs a value , otherwise it outputs a value .

The hardlims function is defined by equation 2:

 (2)

Suppose you have a network made of 4 perceptrons. Each of these perceptrons is receiving a 4 bit binary input (each bit is either or ). More precisely, the value of input bit at time-step is the output of the neuron at time-step (please see figure 1). Also given are two periodic sequences , and .

Design a network whose perceptron's output is given by equation 3

 (3)

or in matrix notation shown as equation 4:

 (4)

where and .

Find the weight matrix and the bias vector , such that when the network is started at time with input , the network produces the sequence A, and when on the other hand the network is started at time with input , the network produces the sequence B.

The two sequences and desired network behavior are given below:

 x(t) for sequence A x(t) for sequence B 0 -1 -1 -1 +1 -1 +1 +1 -1 1 +1 -1 -1 -1 +1 -1 -1 +1 2 -1 +1 -1 -1 -1 +1 +1 -1 3 -1 -1 +1 -1 +1 -1 -1 +1 4 -1 -1 -1 +1 -1 +1 +1 -1 5 +1 -1 -1 -1 +1 -1 -1 +1 6 -1 +1 -1 -1 -1 +1 +1 -1 7 -1 -1 +1 -1 +1 -1 -1 +1 : : : : : :

• Find the weight matrix and the bias . For this, you can:
1. Use some learning algorithm
2. Calculate the weights and biases needed
3. Intuitively find them (descrive your thoughts precisely)
Describe clearly how you reached the weight and bias settings.
• Write clearly the numerical values of and that you obtained.
• Examine the network behavior when it is initialized at time with a different inital condition.

Neural Networks B, SS06 1

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