针对稀疏高维张量数据，提出一种稳定聚类算法，可有效抑制噪声并保持聚类一致性。
Consistent spectral clustering in sparse tensor block models
- 设计稀疏整数张量块模型，结合截断谱聚类降低噪声波动。
- 发现密度阈值确保算法一致，理论分析依赖稀疏随机格拉姆矩阵的新浓度不等式。
- 模型在任意维度聚合下仍保持封闭性，适合生物信息与推荐系统等场景。
高阶聚类旨在对多维数据集中的对象进行分类，这类数据广泛存在于生物信息学、推荐系统和社交网络分析中。此类数据通常稀疏且高维，带来显著的统计与计算挑战。本文提出一种专为稀疏整数型张量数据设计的张量块模型。我们提出一种简单的谱聚类算法，并引入截断步骤以减轻噪声波动，同时识别出保证算法一致性的密度阈值。方法采用子泊松噪声集中框架建模稀疏性，能处理重于次高斯尾部的噪声。令人惊讶的是，这一自然的张量块模型类在任意模式下的聚合操作下保持封闭性。因此，我们建立了一个全面框架，用于评估数据聚合带来的信号损失与噪声降低之间的权衡。理论分析基于稀疏随机格拉姆矩阵的新型浓度不等式。数值实验验证了理论结果。
原文摘要 · Abstract (English)
High-order clustering aims to classify objects in multiway datasets that are prevalent in various fields such as bioinformatics, recommendation systems, and social network analysis. Such data are often sparse and high-dimensional, posing significant statistical and computational challenges. This paper introduces a tensor block model specifically designed for sparse integer-valued data tensors. We propose a simple spectral clustering algorithm augmented with a trimming step to mitigate noise fluctuations, and identify a density threshold that ensures the algorithm's consistency. Our approach models sparsity using a sub-Poisson noise concentration framework, accommodating heavier than sub-Gaussian tails. Remarkably, this natural class of tensor block models is closed under aggregation across arbitrary modes. Consequently, we obtain a comprehensive framework for evaluating the tradeoff between signal loss and noise reduction incurred by aggregating data. The analysis is based on a novel concentration bound for sparse random Gram matrices. The theoretical findings are illustrated through numerical experiments.
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