通过联合探索不同阶次的样本亲和度的均匀张量聚类
Uniform tensor clustering by jointly exploring sample affinities of various orders
摘要：
传统的基于两两相似度的聚类方法在处理高维特征和低样本数据时，往往会受到集中效应的影响，导致样本邻近度编码不准确，聚类性能不理想。为了解决这个问题，我们提出了一个统一的张量聚类方法（UTC），使用多个样本的亲和力来表征样本的接近度，从而补充丰富的空间样本分布，以提高聚类。具体地说，我们发现，三元张量亲和度可以通过两个亲和矩阵的Khari-Rao乘积来构造。此外，我们的早期工作表明，四阶张量亲和度是由克罗内克积定义的。因此，我们利用算术产品，Khatri-Rao和Kronecker产品，在数学上将不同的亲和度整合到一个统一的张量聚类框架中。因此，UTC联合学习联合低维嵌入以组合各种订单。最后，设计了一个数值格式来求解该问题。在人工数据集和真实数据集上的实验表明：1）高阶张量亲和度的使用可以为常用的亲和度矩阵提供样本接近度的补充表征; 2）所提出的UTC方法在处理高维数据时可以利用不同的阶亲和度来增强聚类。
英文摘要：
Conventional clustering methods based on pairwise affinity usually suffer from the concentration effect while processing huge dimensional features yet low sample sizes data, resulting in inaccuracy to encode the sample proximity and suboptimal performance in clustering. To address this issue, we propose a unified tensor clustering method (UTC) that characterizes sample proximity using multiple samples' affinity, thereby supplementing rich spatial sample distributions to boost clustering. Specifically, we find that the triadic tensor affinity can be constructed via the Khari-Rao product of two affinity matrices. Furthermore, our early work shows that the fourth-order tensor affinity is defined by the Kronecker product. Therefore, we utilize arithmetical products, Khatri-Rao and Kronecker products, to mathematically integrate different orders of affinity into a unified tensor clustering framework. Thus, the UTC jointly learns a joint low-dimensional embedding to combine various orders. Finally, a numerical scheme is designed to solve the problem. Experiments on synthetic datasets and real-world datasets demonstrate that 1) the usage of high-order tensor affinity could provide a supplementary characterization of sample proximity to the popular affinity matrix; 2) the proposed method of UTC is affirmed to enhance clustering by exploiting different order affinities when processing high-dimensional data.
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