Kernel Methods (e.g. SVM, kPCA) are a class of algorithms for pattern analysis, which find different types of relations on the input data (for example classification).
KM approach the problem by using an implicit mapping of the input data into high dimensional feature space , and searching there for relations.
KM operate in the feature space without ever computing the coordinates of the data in that space, but rather by simply computing the inner products between the images of all pairs of data in the feature space .
A kernel is a function such that for all
Positive definite kernel A kernel is positive definite if The matrix G who's elements are is a positive definite matrix. i.e It is sufficient to show that there exist some mapping to a vector space such that