[SOLVED] ECE 449 Assignment 6

1. Consider the following training set

         0                     0                     1                     1           

_{   }0_,t1= 0_,_x2= _1_,t2=1_,_x3= _0_,t3=1_,_x4= _1_,t4=1_ x1= 

1. Plot the training samples in the feature space.
2. Apply the perceptron learning rule to the training samples one-at-a-time to obtain weights w₁, w₂, and bias w₀ that separate the training samples. Use w = [w₀, w₁, w₂] = [0, 0, 0] as initial values (consider bias input x₀ = 1, and learning rate = 1). Write the expression for the resulting decision boundary and draw it in the graph. [Hint: You can use Excel / OO Calc to implement the learning rule for perceptron, such as the spreadsheet of InClass_09 posted on eClass].

2. [5 points] Consider the following training set

         0                     0                     1                     1            

x1= 0_,t1= 0_,_x2= _1_,t2=1_,_x3= _0_,t3=1_,_x4= _1_,t4= 0_

                                 

which describes the exclusive OR (XOR) problem.

1. a) Establish mathematical (not graphical) proof that this problem is not linearly separable. [Hint: Start with assumption that these patterns are linearly separable, write down equations/inequalities corresponding to this assumption and examine them for conflict; first such inequality is provided below as an example.]

Suppose that the problem is linearly separable. The decision boundary can be represented as:

∑²₀ 𝑥_𝑖𝑤_𝑖 = 0  or (expanded)  𝑥₀𝑤₀ + 𝑥₁𝑤₁ + 𝑥₂𝑤₂ = 0

This assumption means that either

1. 𝑥0𝑤₀ + 𝑥₁𝑤₁ + 𝑥₂𝑤₂ < 0𝑓𝑜𝑟 (𝑥₁, 𝑥₂) = (0,1) ∧ (𝑥₁, 𝑥₂) = (1,0),

𝑥₀𝑤₀ + 𝑥₁𝑤₁ + 𝑥₂𝑤₂ ≥ 0𝑓𝑜𝑟 (𝑥₁, 𝑥₂) = (0,0) ∧ (𝑥₁, 𝑥₂) = (1,1)

or

1. 𝑥0𝑤₀ + 𝑥₁𝑤₁ + 𝑥₂𝑤₂ > 0𝑓𝑜𝑟 (𝑥₁, 𝑥₂) = (0,1) ∧ (𝑥₁, 𝑥₂) = (1,0).

𝑥₀𝑤₀ + 𝑥₁𝑤₁ + 𝑥₂𝑤₂ ≤ 0𝑓𝑜𝑟 (𝑥₁, 𝑥₂) = (0,0) ∧ (𝑥₁, 𝑥₂) = (1,1) must be satisfied. Following one of the cases and putting the values (𝑥₁, 𝑥₂) under variables, one obtains

(1) 𝑥₀𝑤₀ + 𝑤₂ < 0

(2)

(3)

(4)

…

1. b) Apply the perceptron learning rule following the same procedure as in Problem 1. Describe your observation.