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What is the problem with an asymmetric distance measure in k-NN classifier?

I think it will not cause problem, so long as I compute the distance consistently, say always from test_data to labeled_data.

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It has to be symmetric. The reason is because, KNN can be viewed as a non-parametric kernel density estimation problem. In the estimation problem you get a term of the form $K(x-x_{i})$ and $K$ is a Kernel which has to be symmetric (see http://en.wikipedia.org/wiki/Variable_kernel_density_estimation)

Intutively I think this can be explained by noting that two "extremely close" points can get different classifications which is not desirable

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  • $\begingroup$ Ok, but what about the other criteria for distance---triangle inequality? It is much more difficult to prove than symmetry especially my distance function is highly nonlinear $\endgroup$ – Sibbs Gambling Mar 22 '15 at 7:55
  • $\begingroup$ It is a distance metric therefore has to satisfy the triangle inequality.. $\endgroup$ – Sid Mar 22 '15 at 8:04

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