While the computational benefits are certainly nice, I think that Mercer's theorem is much deeper than that. It says that any [positive definite](http://en.wiki

While the computational benefits are certainly nice, I think that Mercer's theorem is much deeper than that. It says that any [positive definite](http://en.wikipedia.org/wiki/Positive-definite http://en.wikipedia.org/wiki/Positive-definite) function is an inner product, though the space of that inner product might be infinite. The consequences of this go much further than just speeding up algorithms-- we can take plain old linear methods like SVMs and apply them to any data type on which we can define a kernel. This is huge, because it allows machine learning to break free of the vector world in which the algorithms are classically defined. Want to classify gene sequences? You don't have to hack them into a fixed-width vector, just define a string kernel for genes. Want to classify fragments of lambda calculus? Just define an appropriate tree kernel. It's wonderfully powerful.