Proof of Growth Function when Points are not binary

[udaykamath]

Prof Dr Yaser
I understand the step by step process in bounding the growth function B(N,k) in terms of recursion using x1,x2...xN data points using structural arrangements. I however started thinking, in a numerical data where x1,x2...xN is real numbers we cannot use this alpha + 2* Beta trick and use the polynomial bound by combination? What is the bounds when data is not binary ? Is it still Polynomial in N, and how?
Thanks

Uday Kamath
PhD candidate
GMU

[yaser]

Originally Posted by udaykamath

I understand the step by step process in bounding the growth function B(N,k) in terms of recursion using x1,x2...xN data points using structural arrangements. I however started thinking, in a numerical data where x1,x2...xN is real numbers we cannot use this alpha + 2* Beta trick and use the polynomial bound by combination? What is the bounds when data is not binary ? Is it still Polynomial in N, and how?

You are correct. For non-binary functions, the argument is more elaborate. This is addressed in Vapnik's book "Statistical Learning Theory" in much detail, and also appears in other books and papers.