What is VC dimension of a rectangle?

The VC dimension of rectangles is the cardinality of the maximum set of points that can be shattered by a rectangle. The VC dimension of rectangles is 4 because there exists a set of 4 points that can be shattered by a rectangle and any set of 5 points can not be shattered by a rectangle.

What is VC dimension of lines in a plane?

The VC dimension of a finite projective plane is 2. Proof: (a) For each pair of distinct points, there is one line that contains both of them, lines that contain only one of them, and lines that contain none of them, so every set of size 2 is shattered.

What does VC dimension illustrate?

The VC dimension of a classifier is defined by Vapnik and Chervonenkis to be the cardinality (size) of the largest set of points that the classification algorithm can shatter [1].

What is the VC dimension of the set of hypothesis?

The VC dimension of a set of hypotheses H is the size of the largest set C ⊆ X such that C is shattered by H. If H can shatter arbitrarily sized sets, its VC dimension is infinite. We now study the VC dimension of some finite classes, more in particular: classes of boolean functions.

Is VC dimension useful?

VC dimension is useful in formal analysis of learnability, however. This is because VC dimension provides an upper bound on generalization error. So if we have some notion of how many generalization errors are possible, VC dimension gives an indication of how many could be made in any given context.

Can VC dimension of H be 3 True or false?

The VC dimension of H here is 3 even though there may be sets of size 3 that it cannot shatter. under the definition of the VC dimension, in order to prove that VC(H) is at least d, we need to show only that there’s at least one set of size d that H can shatter.

Why is VC dimension useful?

What is the VC dimension of a rotatable rectangle?

Rotatable rectangles are VC dimension 7. Suppose we have a regular heptagon. It’s easy to get any subset of 0, 1, 2, 6, or 7 points. For 3 points, all configurations are symmetric to ABC, ABD, ABE, or ACE. The figure below shows how can we can rotate rectangles to capture these 3 point configurations.

What is the VC dimension of a triangle?

Now, by Lecture 3, the VC dimension of axis-aligned rectangles in dimension 2 is 4 and the VC dimension of triangles (3-gones) is 7. Thus, the VC dimension of the union of these sets is bounded by 4+7+1 = 12. 1 Figure 1: Illustration of (h∆A)∩S = (h∩S)∆(A∩S). 2.

What is the VC dimension of a hyperplane?

  In the plane (hyperplane = line):   VC(Hiperlpanes) is at least 3   VC(Hiperlpanes)< 4 since there is no set of 4 points, which can be shattered by a line. ⇒VC(H)=3. In general, for a k-dimension space VC(H)=k+1   NB: It is useless selecting a set of linealy independent points Upper Bound on Sample Complexity

How to show that the VC dimension is 4?

In order to show that the VC dimension is 4 (in this case), we need to show two things: 1. There exist 4 points that can be shattered. It’s clear that capturing just 1 point and all 4 points are both trivial. The figure below shows how we can capture 2 points and 3 points. So, yes, there exists an arrangement of 4 points that can be shattered. 2.