Spectral Clustering - Relationship With k-means

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,

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

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,

\max_{F} \operatorname{ trace } \left(KF\right)

such that,

F = G_{n\times k}G_{n\times k}^T
G^TG = I

such that . In addition, there are identity constrains on given by,

F\cdot \mathbb{I} = \mathbb{I}

where represents a vector of ones.

F^T\mathbb{I} = \mathbb{I}

This problem can be recast as,

\max_G \text{ trace }\left(G^TG\right).

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

Famous quotes containing the words relationship with and/or relationship:

    “Every man is in a state of conflict, owing to his attempt to reconcile himself and his relationship with life to his conception of harmony. This conflict makes his soul a battlefield, where the forces that wish this reconciliation fight those that do not and reject the alternative solutions they offer. Works of art are attempts to fight out this conflict in the imaginative world.”
    —Rebecca West (1892–1983)

    “Friendship is by its very nature freer of deceit than any other relationship we can know because it is the bond least affected by striving for power, physical pleasure, or material profit, most liberated from any oath of duty or of constancy.”
    —Francine Du Plesssix Gray (20th century)