Robust Bregman clustering

Abstract

Clustering with Bregman divergences encompasses a wide family of clustering procedures that are well suited to mixtures of distributions from exponential families (J. Mach. Learn. Res. 6 (2005) 1705–1749). However, these techniques are highly sensitive to noise. To address the issue of clustering data with possibly adversarial noise, we introduce a robustified version of Bregman clustering based on a trimming approach. We investigate its theoretical properties, showing for instance that our estimator converges at a sub-Gaussian rate $1/\sqrt{\mathit{n}}$, where n denotes the sample size, under mild tail assumptions. We also show that it is robust to a certain amount of noise, stated in terms of breakdown point. We also derive a Lloyd-type algorithm with a trimming parameter, along with a heuristic to select this parameter and the number of clusters from sample. Some numerical experiments assess the performance of our method on simulated and real datasets.