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enter image description here

Starting with a).

For the first unit: 0 * x1 + 1 * x2 + 1 > 0 (0, because the threshold is 0) which is the same as x2+1 > 0.

For the second unit: x1 * 1 + x2 * 0 + 1 > 0 (0, because the threshold is 0) which is the same as x1 + 1 > 0.

For the third unit: x1 * 1 + x2 * 1 + 1 > -1 (-1, because the threshold is -1. I might have misunderstood this since the assignment states "Circles indicate threshold units with threshold at zero...") which is the same as x1 + x2 > -2.

I then draw those lines. Is it correct so far? I can't however figure out how to solve b) How do I take the bias on the output node into account?

This is the given solution (but I don't understand how they got it):

enter image description here

Thanks!!

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    $\begingroup$ I think you mixed intercept and threshold. The threshold is set to zero (read the text). The intercept is 1 * 0.0 for the first two nodes and 1 * (-1.0) for third one. This leads to different inequalities. $\endgroup$
    – Broele
    Mar 10 at 13:05
  • $\begingroup$ By the way: this looks like you are asking to do your homework. I know such kind of networks mainly from text books and for educational purposes. $\endgroup$
    – Broele
    Mar 10 at 13:07
  • $\begingroup$ Thanks for your reply! I thought I was supposed to use this equation Σwjxj+bias=threshold for the nodes. But I guess that isn't the case since I get the wrong answer if I just plugg in bias = 1 and threshold = 0, and I don't see anything about any 'intercept'. What equation should I use? I feel a bit lost and I'm having a hard time figuring out how it all fits together. @Broele $\endgroup$
    – kim120
    Mar 10 at 13:27
  • $\begingroup$ Bias and intercept mean the same, here $\endgroup$
    – Broele
    Mar 10 at 13:50
  • $\begingroup$ Ok thanks! But what do the numbers next to the filled in circles stand for? I thought those were the bias. @Broele $\endgroup$
    – kim120
    Mar 10 at 14:01

1 Answer 1

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What you assumed were thresholds appear to instead be weights on the edges from the passthrough nodes. At least, that makes the lines consistent with those in the plot.

I gave some hints to the same problem at Draw(by hand) the decision boundary of an neural network ; for the shading, note the output of each hidden neuron in each region and plug those into the output neuron.

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  • $\begingroup$ Thanks for your answer! But if those numbers I assumed were thresholds actually are weights on the edges from the passthrough nodes, then what are the '1.0's closer to the output node? I thought the '1.0's were the weights... @Ben $\endgroup$
    – kim120
    Mar 10 at 13:47
  • $\begingroup$ These are the weights for the output of the hidden layer when uses as input for the output layer. $\endgroup$
    – Broele
    Mar 10 at 13:54
  • $\begingroup$ @kim120 those are the weights for those edges, giving the output as a linear combination of the (post-activation) output from the hidden neurons. $\endgroup$
    – Ben Reiniger ♦
    Mar 10 at 14:17

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