Hello, I have trouble understanding the last inequality of the KNN lecture notes.
I don't see how this inequality holds. For example, for: \( \eta(\mathbf{x}) = 0.5 \) and \(\eta(\mathbf{x ^{\prime}}) = 0.6\),
we'd get 0.5 for the first highlighted equation and 0.4 for the second.
Shouldn't the second highlighted equation be \(\leq 2 \eta(\mathbf{x})(1-\eta(\mathbf{x}))+|\eta(\mathbf{x})-\eta(\mathbf{x}^{\prime})|\)
instead?
Hi,
Indeed you are correct: \( 2 \eta(x) - 1 \in [-1, 1] \), therefore \( (2 \eta(x) - 1) ( \eta(x) - \eta(x') ) \leq \vert \eta(x) - \eta(x') \vert \) as you say.
However this does not change the last inequality.
Best,
Scott
KNN Last inequality
Hello, I have trouble understanding the last inequality of the KNN lecture notes.
I don't see how this inequality holds. For example, for:
\( \eta(\mathbf{x}) = 0.5 \) and \(\eta(\mathbf{x ^{\prime}}) = 0.6\),
we'd get 0.5 for the first highlighted equation and 0.4 for the second.
Shouldn't the second highlighted equation be
\(\leq 2 \eta(\mathbf{x})(1-\eta(\mathbf{x}))+|\eta(\mathbf{x})-\eta(\mathbf{x}^{\prime})|\)
instead?
Thanks!
1
Hi,
Indeed you are correct: \( 2 \eta(x) - 1 \in [-1, 1] \), therefore
\( (2 \eta(x) - 1) ( \eta(x) - \eta(x') ) \leq \vert \eta(x) - \eta(x') \vert \) as you say.
However this does not change the last inequality.
Best,
Scott
1
Thank you very much for noticing this typo. I have corrected the lecture notes.
Best,
Nicolas
1
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