Correlation Clustering

Synonyms

Clustering with advice; Clustering with constraints; Clustering with qualitative information; Clustering with side information

Definition

In its rawest form, correlation clustering is graph optimization problem. Consider a clusteringC to be a mapping from the elements to be clustered, V, to the set $\{1,\ldots,\vert V \vert \}$, so that u and v are in the same cluster if and only if C[u] = C[v]. Given a collection of items in which each pair (u, v) has two weights $w_{uv}^{+}$ and $w_{uv}^{-}$, we must find a clustering C that minimizes

$$\displaystyle{ \sum _{C[u]=C[v]}w_{uv}^{-} +\sum _{ C[u]\neq C[v]}w_{uv}^{+}, }$$

(1)

or, equivalently, maximizes

$$\displaystyle{ \sum _{C[u]=C[v]}w_{uv}^{+} +\sum _{ C[u]\neq C[v]}w_{uv}^{-}. }$$

(2)

Note that although $w_{uv}^{+}$ and $w_{uv}^{-}$ may be thought of as positive and negative evidence towards coassociation, the actual weights are nonnegative.

Motivation and Background

The notion of clustering with advice, that is...