Spectral clustering: Difference between revisions

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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 <math>k(x_i,x_j) = \phi^T(x_i)\phi(x_j)</math>. The weighted kernel ''k''-means problem further extends this problem by defining a weight <math>w_r</math> for each cluster as the reciprocal of the number of elements in the cluster,

:<math>

\max_{C_i\{C_s\}} \sum_{r=1}^k w_r \sum_{x_i,x_j \in C_r} k(x_i,x_j).

</math>

Suppose <math>F</math> is a matrix of the normalizing coefficients for each point for each cluster <math>F_{ij} = w_r</math> if <math>i,j \in C_r</math> and zero otherwise. Suppose <math>K</math> is the kernel matrix for all points. The weighted kernel ''k''-means problem with n points and k clusters is given as,