On linear separability of data sets in feature space

In this paper we focus our topic on linear separability of two data sets in feature space, including finite and infinite data sets. We first develop a method to construct a mapping that maps original data set into a high dimensional feature space, on which inner product is defined by a dot product kernel. Our method can also be applied to the Gaussian kernels. Via this mapping, structure of features in the feature space is easily observed, and the linear separability of data sets in feature space could be studied. We obtain that any two finite sets of data with empty overlap in original input space will become linearly separable in an infinite dimensional feature space. For two infinite data sets, we present several sufficient and necessary conditions for their linear separability in feature space. We also obtain a meaningful formula to judge linear separability of two infinite data sets in feature space by information in original input space.