[Machine Learning] from distance to kernel (classification via SVM)

Since for many machine learning techniques, you can use kernel trick. This is important to find a kernel, or explicitly construct a kernel from your data. One possible way to do this is calculate metric distance.
This article only gives some fundamental ideas for this transformation, without any strict provment.

Once you have a matrix of distance to measure the difference or distance between a pair of instances, denote I_i, I_j. You may define a kernel function /rho (I_i, I_j) = f(d(I_i,I_j)), where d(I_i,I_j) is the distance you've obtained. The most common choice is natural exponential.
http://math.stackexchange.com/questions/221704/transforming-a-distance-function-to-a-kernel

Another webpage to list some ways to calculate kernel from original feature space:
http://scikit-learn.org/stable/modules/metrics.html

For the requirement for positive kernel, you may refer to this paper:
http://www.kyb.mpg.de/fileadmin/user_upload/files/publications/attachments/scholkopf00kernel_3781%5b0%5d.pdf