Problem with simple perceptron implmenetation

[lorddoskias]

So I wanted to do a little bit of experimenting with the perceptron algorithm to understand it better and I came up with a simple match making scenario. Essentially i have a 100x2 training data matrix with the first feature being height between 165 and 185 and the second feature is weight (as in physical weight of a person ) between 60 and 80. My target function is a very simple one :

function [res] = target(X)

m = size(X, 1);
res = zeros(100, 1);

for j = 1:length(X)
 if(X(j,1) > 170 && X(j,2) > 65)
    res(j) = 1;
  else
    res(j) =  -1;
  end
end

end

It returns 1 if a person is taller than 170cm and heavier than 65 kilos and -1 otherwise. I use this function to generate the result of a X matrix which consists of random values withing the aforementioned boundaries.

The next logical step is to learn a hypothesis function g(X) which would act much in the same way as the target function - which I assume is unknown. So here is my implementation of the perceptron:

function [w] = perceptron(X, y)

X = [ones(size(X, 1), 1) X]; % add the bias term
w = zeros(size(X, 2), 1); %init weights to zero
m = length(y);
iterations = 0;
wrong = 0;

while(true)
iterations = iterations + 1;
wrong = 0;
for j = 1:m
   if (sign(X(j, :) * w) != y(j)) 
		w = w + (y(j)*X(j, :))';
		wrong = 1;
		break;
   endif
end %inner for 

if(wrong == 0) 
	break;
	endif

end  %outer loop 


end %function

Unfortunately it doesn't converge? Are my assumptions towards the problem wrong or do I have a fault in my implementation?

[JohnH]

I haven't looked at your implementation of the perceptron; however, I think there will be a problem with convergence regardless of the implementation since the target function is not linearly separable.

[lorddoskias]

I think you are mistaken. It is linearly separable since the 170/65 act as the threshold i.e. that's where the linear separation occurs.

[JohnH]

The function, height > 170 and weight > 65, is not linearly separable because it does not bisect the plane but instead defines an infinite rectangular region on the plane. There is no single line that separates this region from the rest of the plane.

[Ubermensch]

John is right. It can't be fit with a linear line. This is what I get with your data (the points are randomly chosen and sorry for not including the axis, legends and labels)



So it isn't a straight line. So it can't converge with a simple 1D perceptron

[lorddoskias]

Thanks for the answers. So if I have understood you correctly in this case I will need to train 2 perceptrons - one for each feature and then when a new sample is given run each of the perceptrons for height/weight and my final hypothesis would return 1 only if the 2 perceptrons return 1 on each respective feature? Is this correct?

[JohnH]

At this point the course hasn't provided us with sufficient tools to address the problem you pose, but I think you are on the right track for a method of resolving the categories. Two perceptrons, one for height and one for weight, could feed a third perceptron that would perform the logical operation (AND is a linearly separable logical operation). This approach probably requires back-propogation in order to train the network.