模糊熵三重k均值
Fuzzy entropic triple k-means
- Sapienza University（罗马第一大学）
机构由 AI 辅助整理，请以论文原文为准。
中文总结 AI 辅助
提出模糊熵三重k均值（FE3KM），通过熵正则化线性嵌入隶属度，实现三维数据的同时聚类，并支持偏差分解与解释，优于传统模糊聚类。
中文摘要 AI 辅助
本文提出模糊熵三重k均值（FE3KM），一种用于三维数据数组的熵正则化模糊划分方法，可同时聚类对象、变量和场合。与通过非线性指数m控制模糊度的标准模糊聚类不同，FE3KM在最小二乘（LS）目标中线性嵌入隶属度，使用行随机隶属度矩阵和熵正则化来表示聚类分配中的不确定性。这种线性结构使得总偏差能够精确分解为簇内和簇间成分，并可进一步归因于每个模式（对象、变量、场合）以及每个模式内的各个簇。这是指数m模糊模型所不具备的解释性属性。我们推导了FE3KM的交替最小二乘算法，包括用于大型数组的加速变体，并正式将该方法与串联聚类程序以及Tucker3/三模式划分模型联系起来。一项模拟研究评估了恢复准确性、对噪声和模糊度错误设定的鲁棒性，以及与串联和三路聚类基线的性能对比。对基准三维电视收视率数据集的应用展示了FE3KM如何恢复可解释的对象-变量-场合结构，并量化每个簇和模式对总体数据变异性的贡献。
英文摘要
This paper proposes fuzzy entropic triple k-means (FE3KM), an entropy-regularized fuzzy partitioning method for three-way data arrays that simultaneously clusters objects, variables, and occasions. Unlike standard fuzzy clustering, which controls fuzziness through a nonlinear exponent m, FE3KM embeds memberships linearly within a Least-Squares (LS) objective, using row-stochastic membership matrices and entropy regularization to represent uncertainty in cluster assignments. This linear structure is what enables an exact decomposition of total deviance into within- and between-cluster components, further attributable to each mode (objects, variables, occasions) and to individual clusters within each mode. This is an interpretive property unavailable to exponent-m fuzzy models. We derive Alternating Least-Squares algorithms for FE3KM, including an accelerated variant for large arrays, and formally relate the method to tandem clustering procedures and to Tucker3/three-mode partitioning models. A simulation study assesses recovery accuracy, robustness to noise and fuzziness mis-specification, and performance against tandem and three-way clustering baselines. An application to a benchmark three-way TV ratings dataset demonstrates how FE3KM recovers interpretable object-variable-occasion structures and quantifies the contribution of each cluster and mode to overall data variability.
