A method of explicit mappings for kernel data analysis and applications

The method of kernel data analysis is now a standard tool in modern data mining. An implicit mapping into a high-dimensional feature space is assumed in this method, in other words, an explicit form of the mapping is unknown but their inner product should be known instead. Contrary to this common assumption, we propose a method of explicit mappings. The reason why we use explicit mappings is as follows. (1) The use of these mappings does not lose any fundamental information in kernel data analysis. (2) We have the same formulas in kernel methods with and without the explicit mappings. (2) Usually the derivation becomes simpler by using these mappings. (3) New applications of the kernel methods become possible using these mappings. Two types of the mappings are proposed, one of which uses Φ(x_k) = e_k (k = 1,...,N) while the second type uses Φ(x_k) = K^1/2e_k, where K = (K(x_i,x_j)) is N x N matrix. As an application we consider L₁ space fuzzy c-means clustering and lower dimensional approximation of kernel K. The effectiveness of the proposed method is shown by numerical examples.