Shattered Set - Example

Example

Suppose we have a set A of four points on the unit circle, and we wish to know if it is shattered by the class C of all discs.

To test this, we attempt to draw a disc around every subset of points in A. First, we draw a disc around the subsets of each isolated point, which turns out to be easy. Next, we try to draw a disc around every subset of point pairs. This turns out to be easy for adjacent points, but impossible for points on opposite sides of the circle. As visualized below:

• Each individual point can be isolated with a disc (showing all four).

• Each subset of adjacent points can be isolated with a disc (showing one of four).

• A subset of points on opposite sides of the unit circle can not be isolated with a disc.

Because there is some subset which can not be isolated by any disc in C, we conclude then that A is not shattered by C. And, with a bit of thought, we can prove that NO set of four points is shattered by this C.

However, if we redefine C to be the class of all elliptical discs, we find that we can still isolate all the subsets from above, as well as the points which were problems. Thus, this specific set of 4 points is shattered by the class of ellipitical discs. Visualized below:

• Opposite points of A are now separable by some ellipse (showing one of two)

• Each subset of three points in A is also separable by some ellipse (showing one of four)

With a bit of thought, we could generalize that any set of finite points on a unit circle could be shattered by the class of all convex sets (visualize connecting the dots).