Robust Bregman Clustering

Claire Brécheteau (1, 2, 3, 4) , Aurélie Fischer (5) , Clément Levrard (5, 3)

1 ECN - École Centrale de Nantes
2 LMJL - Laboratoire de Mathématiques Jean Leray
3 DATASHAPE - Understanding the Shape of Data
4 SELECT - Model selection in statistical learning
5 LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation

Using a trimming approach, we investigate a k-means type method based on Bregman divergences for clustering data possibly corrupted with clutter noise. The main interest of Bregman divergences is that the standard Lloyd algorithm adapts to these distortion measures, and they are well-suited for clustering data sampled according to mixture models from exponential families. We prove that there exists an optimal codebook, and that an empirically optimal codebook converges a.s. to an optimal codebook in the distortion sense. Moreover, we obtain the sub-Gaussian rate of convergence for k-means 1 √ n under mild tail assumptions. Also, we derive a Lloyd-type algorithm with a trimming parameter that can be selected from data according to some heuristic, and present some experimental results.

Domaines: Statistiques [math.ST]; Théorie [stat.TH]; Machine Learning [stat.ML]

Claire Brécheteau, Aurélie Fischer, Clément Levrard. Robust Bregman Clustering. 2018. ⟨hal-01948051v1⟩