Concepts leading to Q6 and Q8
I would appreciate some clarification with the concepts leading up to Q6 and Q8. I will build it up, as I understand it, starting with a perceptron. Please correct me where I have erred...
A perceptron learns to separate data that is separable. The breakpoint is associated with data that is separable and it represents the value at which the algorithm has no way of shattering (separating) the case. I understood the k=3 for a line (1D) and k=4 for a plane (2D).
Positive Rays. (Example 2.2 (p. 43) of the book) With the positive rays there seems to be a similar concept to the 1D perceptron in that the data falls on a line and is separable. To the left of "a" h(x) = 1 and to the right of "a" h(x) = +1. It is clearly separable. The hypotheses are for the location of "a".
In the "Positive Intervals" exercise (p. 44 of book, cont. of example 2.2) is where my confusion starts. I understood (from lecture 5) the N+1 regions and the "(N+1) choose 2 + 1" formula. The "choose 2" part refers to the fact that each end point of the interval lands in a region so you need two regions to define your interval. However, I do not see the points as being separable. If they are all on a line, I can not separate with a perceptron the case shown in the book (blue interval in the middle with red regions on either side). I see it as similar to a k = 3 case for case in one dimension. Not separable.
Q6 extends the concept of the "positive intervals" example and asks about two intervals (instead of just one interval as in the example). If I can not see the case of just one interval as being linearly separable then I can not see the possibility of being linearly separable with two intervals. Q8 asks to consider M intervals which I can not see either. It is the same visualization issue as with two or one intervals.
Q6 and Q8 ask for breakpoints so I guess that my issue is that I am not visualizing properly how a break point can be associated with intervals that lie on a line. I associate this case immediately with a break point of k=3.
I hope that my misconception makes sense to you so that you may assist me.
Thank you.
Juan
