Relationship With k-means
The kernel k-means problem is an extension of the k-means problem where the input data points are mapped non-linearly into a higher-dimensional feature space via a kernel function . The weighted kernel k-means problem further extends this problem by defining a weight for each cluster as the reciprocal of the number of elements in the cluster,
Suppose is a matrix of the normalizing coefficients for each point for each cluster if and zero otherwise. Suppose is the kernel matrix for all points. The weighted kernel k-means problem with n points and k clusters is given as,
such that,
such that . In addition, there are identity constrains on given by,
where represents a vector of ones.
This problem can be recast as,
This problem is equivalent to the spectral clustering problem when the identity constraints on are relaxed. In particular, the weighted kernel k-means problem can be reformulated as a spectral clustering (graph partitioning) problem and vice-versa. The output of the algorithms are eigenvectors which do not satisfy the identity requirements for indicator variables defined by . Hence, post-processing of the eigenvectors is required for the equivalence between the problems. Transforming the spectral clustering problem into a weighted kernel k-means problem greatly reduces the computational burden.
Read more about this topic: Spectral Clustering
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