Kernel clustering: Breiman's bias and solutions

Clustering is widely used in data analysis where kernel methods are particularly popular due to their generality and discriminating power. However, kernel clustering has a practically significant bias to small dense clusters, e.g. empirically observed in (Shi & Malik, TPAMI'00). Its causes have never been analyzed and understood theoretically, even though many attempts were made to improve the results. We provide conditions and formally prove this bias in kernel clustering. Moreover, we show a general class of locally adaptive kernels directly addressing these conditions. Previously, (Breiman, ML'96) proved a bias to histogram mode isolation in discrete Gini criterion for decision tree learning. We found that kernel clustering reduces to a continuous generalization of Gini criterion for a common class of kernels where we prove a bias to density mode isolation and call it Breiman's bias. These theoretical findings suggest that a principal solution for the bias should directly address data density inhomogeneity. In particular, our density law shows how density equalization can be done implicitly using certain locally adaptive geodesic kernels. Interestingly, a popular heuristic kernel in (Zelnik-Manor and Perona, NIPS'04) approximates a special case of our Riemannian kernel framework. Our general ideas are relevant to any algorithms for kernel clustering. We show many synthetic and real data experiments illustrating Breiman's bias and its solution. We anticipate that theoretical understanding of kernel clustering limitations and their principled solutions will be important for a broad spectrum of data analysis applications in diverse disciplines.