Relax and Merge: A Simple Yet Effective Framework for Solving Fair $k$-Means and $k$-sparse Wasserstein Barycenter Problems

• URL: http://arxiv.org/abs/2411.01115v1
• Date: Sat, 02 Nov 2024 02:50:12 GMT
• Title: Relax and Merge: A Simple Yet Effective Framework for Solving Fair $k$-Means and $k$-sparse Wasserstein Barycenter Problems
• Authors: Shihong Song, Guanlin Mo, Qingyuan Yang, Hu Ding,
• Abstract summary: Given a dataset comprising several groups, the fairness constraint requires that each cluster should contain a proportion of points from each group. We propose a novel Relax and Merge'' framework, where $rho$ is the approximate ratio of an off-the-shelf vanilla $k$-means algorithm. If equipped with a PTAS of $k$-means, our solution can achieve an approximation ratio of $(5+O(epsilon))$ with only a slight violation of the fairness constraints.
• Score: 8.74967598360817
• License:
• Abstract: The fairness of clustering algorithms has gained widespread attention across various areas, including machine learning, In this paper, we study fair $k$-means clustering in Euclidean space. Given a dataset comprising several groups, the fairness constraint requires that each cluster should contain a proportion of points from each group within specified lower and upper bounds. Due to these fairness constraints, determining the optimal locations of $k$ centers is a quite challenging task. We propose a novel ``Relax and Merge'' framework that returns a $(1+4\rho + O(\epsilon))$-approximate solution, where $\rho$ is the approximate ratio of an off-the-shelf vanilla $k$-means algorithm and $O(\epsilon)$ can be an arbitrarily small positive number. If equipped with a PTAS of $k$-means, our solution can achieve an approximation ratio of $(5+O(\epsilon))$ with only a slight violation of the fairness constraints, which improves the current state-of-the-art approximation guarantee. Furthermore, using our framework, we can also obtain a $(1+4\rho +O(\epsilon))$-approximate solution for the $k$-sparse Wasserstein Barycenter problem, which is a fundamental optimization problem in the field of optimal transport, and a $(2+6\rho)$-approximate solution for the strictly fair $k$-means clustering with no violation, both of which are better than the current state-of-the-art methods. In addition, the empirical results demonstrate that our proposed algorithm can significantly outperform baseline approaches in terms of clustering cost.

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