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I have a question about Gaussian kernel. I read the following site.

https://datascience.stackexchange.com/questions/17352/why-do-we-use-a-gaussian-kernel-as-a-similarity-metric

My question is whether we can extend this to other distances or dissimilarity measures (e.g., Bray-Curtis). When we would like to use spectral clustering, we need to use an affinity matrix. Is Gaussian kernel specific to Euclidian distance? It seems that this package supports Bray-Curtis dissimilarity. However, BC dissimilarity is different from distance.

https://www.rdocumentation.org/packages/clusterSim/versions/0.47-1/topics/speccl

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Distance metrics have well defined requirements: identity, symmetry and triangle inequality in addition to the distance being nonnegative.

Unfortunately, similarity measures don't have one commonly accepted list of requirements. Usually, some kind of inverse of distance is used. For instance, the Gaussian kernel is an inverse of Euclidean distance, and so is a simple negative and a reciprocal.

So, answering your question: no; Gaussian kernel is not bound to be used with Euclidean distance. You can plug any distance into a Gaussian kernel.

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  • $\begingroup$ Thank you very much for your clear explanation! >You can plug any distance into Gaussian kernel. So do you mean we can use Gaussian kernel even for Bray-Curtis dissimilarity (distance)? $\endgroup$
    – user224050
    Commented Feb 16, 2021 at 4:06
  • $\begingroup$ @user224050 Bray-Curtis dissimilarity isn't a distance because it doesn't satisfy the triangle inequality. $\endgroup$
    – Sycorax ♦
    Commented Sep 27, 2023 at 13:30

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