Clustering algorithms remain valuable tools for grouping and summarizing the most important aspects of data. Example areas where this is the case include image segmentation, dimension reduction, signals analysis, model order reduction, numerical analysis, and others. As a consequence, many clustering approaches have been developed to satisfy the unique needs of each particular field. In this article, we present a family of data-adaptive partitioning algorithms that unifies several well-known methods (e.g., k-means and k-subspaces). Indexed by a single parameter and employing a common minimization strategy, the algorithms are easy to use and interpret, and scale well to large, high-dimensional problems. In addition, we develop an adaptive mechanism that (a) exhibits skill at automatically uncovering data structures and problem parameters without any expert knowledge and, (b) can be used to augment other existing methods. By demonstrating the performance of our methods on examples from disparate fields including subspace clustering, model order reduction, and matrix approximation, we hope to highlight their versatility and potential for extending the boundaries of existing scientific domains. We believe our family's parametrized structure represents a synergism of algorithms that will foster new developments and directions, not least within the data science community.

翻译：聚类算法作为对数据进行分组和总结其最重要方面的工具，仍然具有重要价值。其应用领域包括图像分割、降维、信号分析、模型降阶、数值分析等。因此，为满足各特定领域的独特需求，已发展出多种聚类方法。本文提出一种数据自适应划分算法族，该族统一了多种经典方法（如k-means与k-subspaces）。这些算法通过单一参数索引并采用通用的最小化策略，具有易用、易解释的特点，并能良好地扩展至大规模高维问题。此外，我们开发了一种自适应机制，该机制（a）能够在无需专家知识的情况下自动揭示数据结构和问题参数，且（b）可用于增强其他现有方法。通过在子空间聚类、模型降阶和矩阵逼近等不同领域的示例中展示本方法的性能，我们期望彰显其多功能性以及在拓展现有科学领域边界方面的潜力。我们相信，本算法族的参数化结构代表了一种算法协同，将促进新的发展与研究方向，尤其在数据科学领域内。