Turn a distance measure into a kernel function

Question

I have read here that an easy way to turn a distance function $d$ into a similarity function $s$ is to compute: $s = e^{-\gamma * d}$. I believe that this is also what is done with the RBF kernel. There the distance function is the squared eucledian distance $||x - y||^2$ and the corresponding kernel function is: $e^{-\gamma * ||x - y||^2}$. However I'm unable to find a name for this kind of method and I can't find any scientific papers where this method is introduced or applied. Can you point me to some papers with more information ?

(Notice that the distance function I want to turn into a similarity function (kernel function) is not a distance metric. I'm also aware that the resulting kernel function might not be a valid psd kernel.)

Answer 1

Seminal NIPS paper:

• Schölkopf, B. (2000). The kernel trick for distances. In Neural Information Processing Systems, pages 301-307.

There are many kernel design papers and books:

• Schölkopf, B. and Smola, A. J. (2002). Learning with kernels : support vector machines, regularization, optimization, and beyond. Adaptive computation and machine learning. MIT Press.
• Pekalska, E., Paclik, P., and Duin, R. P. W. (2002). A generalized kernel approach to dissimilarity-based classification. J. Mach. Learn. Res., 2:175-211.
• Pekalska, E. and Duin, R. P. W. (2005). The Dissimilarity Representation for Pattern Recognition. World Scientific Publishing Co., Inc., River Edge, NJ, USA.
• Haasdonk, B. and Bahlmann, C. (2004). Learning with distance substitution kernels. Pattern Recognition, volume 3175 of Lecture Notes in Computer Science, pages 220-227.