VC Dimension and Degrees of Freedom

[cabjoe]

Firstly, thank you to Professor Yaser for this wonderful course. I am learning a lot from it. I have a question regarding degrees of freedom and their relation to the VC dimension

In Lecture 7, the professor states that the VC dimension of a hypothesis set is equal to the number of degrees of freedom and shows that this indeed holds for positive rays and positive intervals.

However in the case of a perceptron in R2, he has shown that the VC dimension is d+1, i.e. 3 but I can only see 2 degrees of freedom, the slope and intercept of the line.

What am I missing here?

[yaser]

Originally Posted by cabjoe

I have a question regarding degrees of freedom and their relation to the VC dimension

In Lecture 7, the professor states that the VC dimension of a hypothesis set is equal to the number of degrees of freedom and shows that this indeed holds for positive rays and positive intervals.

However in the case of a perceptron in R2, he has shown that the VC dimension is d+1, i.e. 3 but I can only see 2 degrees of freedom, the slope and intercept of the line.

Degrees of freedom are an abstraction of the effective number of parameters. The effective number is based on how many dichotomies one can get, rather than how many real-valued parameters are used. In the case of 2-dimensional perceptron, one can think of slope and intercept (plus a binary degree of freedom for which region goes to ), or one can think of 3 parameters (though the weights can be simultaneously scaled up or down without affecting the resulting hypothesis). The degrees of freedom, however, are 3 because we have the flexibility to shatter 3 points, not because of one way or another of counting the number of parameters.

[Elroch]

Another interesting "angle" on them is as -dimensional vector subspaces of an -dimensional vector space without considering the plane . This has the same VC-dimension () even if the points can be chosen anywhere rather than on the hyperplane . Since no points on a ray can be separated, this is essentially the same as them acting on, say, a unit sphere, which has dimension . Half of this sphere is also the same as our plane by projection along the rays.