双凸聚类

摘要：

凸聚类由于其吸引人的理论和计算特性而受到越来越多的关注，但其优点在高维数据面前变得有限。在这样的设置中，依赖于$k$ -最近邻的成对亲和度项变得不好指定，并且欧几里得拟合度量提供较弱的区分能力。为了克服这些问题，我们建议修改凸聚类目标，使特征权重与质心联合优化。由此产生的问题成为双凸，因此在统计和算法上仍然表现良好。特别是，我们推导出一个快速算法与封闭形式的更新和收敛保证，并建立有限样本的预测误差的界限。在可解释的正则条件下，误差界分析意味着所提出的估计的一致性。双凸聚类在整个聚类任务中执行特征选择：由于学习的权重改变了有效的特征表示，成对的亲和力可以在迭代中自适应地更新，而不是在可疑的特征空间中预先计算。我们验证了真实和模拟数据的贡献，表明我们的方法有效地解决了维度的挑战，同时减少了对现有方法中典型的精心调整的几何学的依赖。

英文摘要：

Convex clustering has recently garnered increasing interest due to its attractive theoretical and computational properties, but its merits become limited in the face of high-dimensional data. In such settings, pairwise affinity terms that rely on $k$-nearest neighbors become poorly specified and Euclidean measures of fit provide weaker discriminating power. To surmount these issues, we propose to modify the convex clustering objective so that feature weights are optimized jointly with the centroids. The resulting problem becomes biconvex, and as such remains well-behaved statistically and algorithmically. In particular, we derive a fast algorithm with closed form updates and convergence guarantees, and establish finite-sample bounds on its prediction error. Under interpretable regularity conditions, the error bound analysis implies consistency of the proposed estimator. Biconvex clustering performs feature selection throughout the clustering task: as the learned weights change the effective feature representation, pairwise affinities can be updated adaptively across iterations rather than precomputed within a dubious feature space. We validate the contributions on real and simulated data, showing that our method effectively addresses the challenges of dimensionality while reducing dependence on carefully tuned heuristics typical of existing approaches.

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