I see following slides in a video on YouTube:
I ran into a doubt. VC dimension of a line that is parallel to one of axis (X or Y axis) is equal to $2$. can I tell $(-\infty, x]$ is equal to case $H1$ in the mentioned image? why not?
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$\begingroup$ BTW, {x| a<x<b} would normally be called an "open interval", and {x| a<=x<=b} a closed interval. $\endgroup$– chrishmorrisCommented Dec 7, 2020 at 12:11
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$\begingroup$ @chrishmorris Exactly. I need these hint. would you please provide a bit more to understanding as an answer? $\endgroup$– M KCommented Dec 7, 2020 at 12:14
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VC dimension is about the complexity of the class of functions, and intuitively the higher the dimension is, the more complex the class of cuntions is. Here, we can show that in fact "VC dimension of a line that is parallel to one of axis" depends really a lot on the dimension, in dimension $n$, the VC dimension is $n$. Aply this to your problem, H1 in the slides is dimension 1, hence VC dimension 1 whereas when you say "parallel to one of axis (X or Y axis)" I assume you are talking about dimension 2, hence VC dimension 2.
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$\begingroup$ why in slide 12 here cs.cmu.edu/~epxing/Class/10701/slides/lecture16-VC.pdf tell us VC dimension is 3 !! $\endgroup$– M KCommented Dec 7, 2020 at 11:18
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$\begingroup$ because in slide 12, this is not "line parallel to one of axis" we are talking about, this is "any line". Hence, this is more complex than just the lines parallel to on of the axis. $\endgroup$– TMatCommented Dec 7, 2020 at 12:00