I asked a question about forming a valid distance metric yesterday (Link1) and got some very good answer; however, I have got some more questions about forming a proper distance metric for high dimensional data.

  1. Why is triangle inequality so important to make a valid distance metric? Maybe it is too broad to ask this, but I haven't got a simple example in my mind. Not sure if you people can think a simple scenario to explain this with some context?

  2. As mentioned in my previous post (Link1), I think Cosine similarity is the same thing as dot product. Am I right? If so, dot product is not a valid distance metric because it does not have the triangle inequality property and etc. If we transform the similarity measured by dot product into Angular similarity, will it be a proper distance metric?

  3. Regarding to the Euclidean distance, there is another post (Link2) saying that it is not a good metric in high dimensions. As my data vectors are in high dimensional space, I am wondering if some distance metric suffer from the curse of dimensionality?

  4. Regarding to the point C above, considering the dimensionality, will a fractional distance metric be a better distance metric? (Link3)

Thanks very much! A


1 Answer 1


For high-dimensional data, shared-nearest-neighbor distances have been reported to work in

Houle et al., Can Shared-Neighbor Distances Defeat the Curse of Dimensionality? Scientific and Statistical Database Management. Lecture Notes in Computer Science 6187. p. 482. doi:10.1007/978-3-642-13818-8_34

Fractional distances are known to be not metric. $L_p$ is only a metric for $p\geq 1$, you'll find this restriction in every proof of the metric properties of Minkowski norms.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.