# Robust Bregman Clustering

## TL;DR

This paper introduces a robust Bregman divergence-based clustering method with trimming, providing theoretical convergence guarantees and a practical algorithm, suitable for noisy data from exponential family distributions.

## Contribution

It develops a new robust clustering approach using Bregman divergences, proves convergence properties, and proposes a data-driven Lloyd-type algorithm with experimental validation.

## Key findings

- Existence of an optimal codebook.
- Almost sure convergence of empirical to optimal codebook.
- Sub-Gaussian convergence rate of √n under mild conditions.

## Abstract

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 $\sqrt$ 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.