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I am trying to make sense of the following:

enter image description here

Taken from the MIT website.

So I understand that we want to maximize the distance between the planes H0 and H1, where H0 is defined as wx+b=0 and H1 is defined as wx+b=1. I am following with the definition that the distance betweena point (x0,y0) and a line Ax+By+c=0 is |Ax0+By0+c|/sqrt(A^2+B^2).

What I don't understand is... What is wx+b? So I understand w is the weight vector, x is an input and b is the offset, so wx+b gives an indication's of x's class as predicted using the model with weight vector w. But, when we are looking at the distances between H0 and H1, which are both lines, what 'line' and what 'point' are we referring to?

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  • $\begingroup$ Advice: when you don't understand something, look for a textbook that has an explanation, not just slides. Material often gets left out of slides, so slides are not the best resource when you know you are confused about something. $\endgroup$ – D.W. Dec 20 '15 at 21:51
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First of all, the formula given for the distance between a point and a line is not a definition, it is a calculation. The distance between a point $p$ and a line $\ell$ is the minimum distance between $p$ and a point on $\ell$.

Similarly, if $\ell,\ell'$ are two lines then the distance between them is the minimum distance between a point on $\ell$ to a point on $\ell'$. This is zero unless the lines are parallel, and if they are, say given by $x \bullet w + b = 0$ and $x \bullet w' + b' = 0$, then the distance is (I believe) $|b/\|w\| - b'/\|w'\||$. Again, this is not a formula but rather a calculation.

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