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I've been trying to determine the VC dimension of ellipses which are origin centered and axis aligned. My first approach was to find some equivalence to a threshold classifier function family of the form

$$ H=\big\{\text{ sign}\big(\sum_{x}(Wx)+b\big)\ |\ w \in \mathbb{R}^d, b \in \mathbb{R}\ \big\}$$

For these functions I know the VC dimension is $d+1$, so for axis-aligned origin-centered ellipses I tried

$$ H=\big\{\text{ sign}\big(\sum_{x}(W^2x^2)-R^2\big)\ |\ w \in \mathbb{R}^d, R \in \mathbb{R}\ \big\} $$

I thought from here it would be straightforward but I couldn't figure it out. The limitations on the values that $w,x$ and $b$ (which equals $-R^2$) can get now can affect the dimension, but I'm not sure how.

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  • $\begingroup$ do you only want to know the dimension or also a proof? $\endgroup$
    – oW_ ♦
    Feb 8, 2018 at 0:23
  • $\begingroup$ The focus is on the proof $\endgroup$
    – AlexFink
    Feb 8, 2018 at 7:48

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