  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.
