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I am doing an exam preparation and came across this question from previous exams:

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I cant wrap my head around how to set up the corresponding equations given the thresholds.. (x1 + x2) + (x1 - x2) + (-x1 + x2) but then we have the thresholds... can someone help my thoughtprocess on how to tackle this?

Note that i dont want the drawing, just an systematic way on how to tackle this so i understand the thoughtprocess.

another similar problem but now with an bias(what i assume the value attached to the "pass through" node is):

enter image description here

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It's really best to do the parts in order. After drawing the separating planes (lines here) for each hidden unit, you'll end up with a finite number of cells. In each cell the output of each hidden unit is constant, and you can determine the y value for each.

The second example isn't much different; the biases given on the passthrough nodes together with thresholding-at-zero works essentially the same way as thresholding at a nonzero number without a bias term.

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  • $\begingroup$ how would one draw the separating lines? state the equations please and can you please elaborate a bit more on the "and you can determine the y value for each" $\endgroup$
    – fearloathing121
    Oct 20, 2022 at 17:20
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    $\begingroup$ How much do you want shown? // For the second unit e.g., you've already identified the linear part as $x_1-x_2$; the threshold means you should consider the inequality $x_1-x_2>-1$. I think I would forget the inequality for a moment and just plot the line $x_1-x_2=-1$. Doing the same for the other two, you have three lines in the plane, dividing it into 6 (could've been a different number) regions. In each of those regions, jot down the value of the three hidden units (+-1) using the inequalities we forgot for a moment, plug those into the final unit, and apply the threshold to find $y$. $\endgroup$
    – Ben Reiniger - on strike ♦
    Oct 20, 2022 at 19:37

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