  The K-means algorithm is among the most commonly used data clustering
methods. However, the regular K-means can only be applied in the input space
and it is applicable when clusters are linearly separable. The kernel K-means,
which extends K-means into the kernel space, is able to capture nonlinear
structures and identify arbitrarily shaped clusters. However, kernel methods
often operate on the kernel matrix of the data, which scale poorly with the
size of the matrix or suffer from the high clustering cost due to the
repetitive calculations of kernel values. Another issue is that algorithms
access the data only through evaluations of $K(x_i, x_j)$, which limits many
processes that can be done on data through the clustering task. This paper
proposes a method to combine the advantages of the linear and nonlinear
approaches by using driven corresponding approximate finite-dimensional feature
maps based on spectral analysis. Applying approximate finite-dimensional
feature maps were only discussed in the Support Vector Machines (SVM) problems
before. We suggest using this method in kernel K-means era as alleviates
storing huge kernel matrix in memory, further calculating cluster centers more
efficiently and access the data explicitly in feature space. These explicit
feature maps enable us to access the data in the feature space explicitly and
take advantage of K-means extensions in that space. We demonstrate our Explicit
Kernel Minkowski Weighted K-mean (Explicit KMWK-mean) method is able to be more
adopted and find best-fitting values in new space by applying additional
Minkowski exponent and feature weights parameter. Moreover, it can reduce the
impact of concentration on nearest neighbour search by suggesting investigate
among other norms instead of Euclidean norm, includes Minkowski norms and
fractional norms (as an extension of the Minkowski norms with p<1).
