Support Vector Machines (SVM)

Drag either end of the line. As long as it separates the two classes, two dashed lines appear on either side — the "street," bounded by whichever point on each side sits closest. Push either end too far and the street vanishes: the line is now cutting through a class instead of separating it.

1. Maximize the margin

   A support vector machine picks the one separating line that makes this street as wide as possible.

2. Support vectors touch the edges

   Whichever points end up touching the edges of that street — often just one or two per class — are the support vectors: the only points that determine the line at all.

3. Only support vectors matter

   Move any other point further from the boundary and nothing changes; move a support vector and the whole line shifts.

Now both edges are measured separately. An off-center line has a wide gap on one side and a thin one on the other — the street is only ever as wide as its narrower side. Widening the narrow side (even at the cost of the wide side) is how the street actually grows.

1. Find the closest pair and their distance

   $(3,6)$ from one class and $(5,4)$ from the other:

   $$\sqrt{(5-3)^2+(4-6)^2} = \sqrt{8} = 2\sqrt{2} \approx 2.83$$
2. Find their perpendicular bisector

   1. Slope of the segment between them: $(4-6)/(5-3) = -2/2 = -1$
   2. Perpendicular slope (negative reciprocal): $+1$
   3. Midpoint: $((3+5)/2,\ (6+4)/2) = (4,5)$
   4. Line through $(4,5)$ with slope $1$: $y-5=1(x-4) \implies y=x+1$
3. Confirm it's the widest street

   Every other point sits comfortably further from this line than $(3,6)$ and $(5,4)$ do — so the street's width is set entirely by this one pair, and it equals exactly the distance between them, $2\sqrt{2} \approx 2.83$.