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I'm studying the knn classification algorithm. Why can the euclidean distance be considered a nice measure of affinity between examples ?
In one dimension (1 attribute) this seems correct, but if I add dimensions, can the euclidean distance still be considerd a good measure of affinity? Why?
You can think of examples as vectors in $\mathbb{R}^p$, where $p$ is the number of features. Two examples will be very similar if the distance between them is close to $0$ (in the extreme case, if two examples are equal their euclidean distance is $0$). One way to measure the distance is using euclidean distance, but other distances can be used, as cosine distance or $L^p$ metrics. In fact, if $p$ is very high, then Euclidean distance is not a good measure, as it tends to make the distances too uniform (see this paper).
Edit: When $p$ is very high:
See this magnificient answer to the issues that very high $p$ may have.
