Computational Intelligence, SS08
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Circles

What is the VC-dimension of circles in the plane $\mathbb{R}^2$? I.e., examples are points in $\mathbb{R}^2, C=\{c_{p,r}:p\in \mathbb{R}^2,r\in \mathbb{R}\}$ and cp,r=1 if x is within distance r of p. Or, in words, a legal target function is specified by a circle, and labels any example positive iff it lies inside that circle.





VC-dim = 3. It is easy to see the VC-dimension is at least 3 since any 3 points that make up a non-degenerate triangle can be shattered.It is a bit trickier to prove that the VC-dimension is less than 4. Given 4 points, the easy case is when one is inside the convex hull of th others. In that case, because circles are convex, it is not possible to label the inside point - and the outside points +. Otherwise, call the points a,b,c,d in clockwise order. the claim is that it is not possible for one circle c1 to achieve labeling +,-,+,- and for some other c2 to achieve labeling -,+,-,+,-. If such c1, c2 existed, then their symmetric difference would consist of 4 disjoint regions, which is impossible for circles.