Question

Consider the space of instances X corresponding to all points in the x, y plane. Give the VC dimension of the following hypothesis spaces:

(a) H = the set of all rectangles in the x, y plane. That is, H = (a < x < b) and (c < y < d) .

(b) H = circles in the x, y plane. Points inside the circle are classified as positive examples.

(c) H = triangles in the x, y plane. Points inside the triangle are classified as positive examples.

Answer 1

(a) Rectangles in the x, y plane have a VC dimension of 4. (b) Circles have a VC dimension of 3. (c) Triangles in the x, y plane have a VC dimension of 7.

The VC dimension of a hypothesis space is the maximum number of points that can be shattered by the hypothesis space. Let's analyze each hypothesis space:

(a) For rectangles in the x, y plane, the VC dimension is 4. It can shatter any set of four points but fails for five points arranged in a convex or concave configuration.

(b) For circles in the x, y plane, the VC dimension is 3. It can shatter any set of three points but fails for four points arranged in a convex or concave configuration.

(c) For triangles in the x, y plane, the VC dimension is 7. It can shatter any set of seven points but fails for eight points arranged in certain convex or concave configurations.

In summary:

(a) VC dimension of rectangles in the x, y plane is 4.

(b) VC dimension of circles in the x, y plane is 3.

(c) VC dimension of triangles in the x, y plane is 7.