Mathematical formulation and application of kernel tensor decomposition based unsupervised feature extraction

Abstract

In this work, we extended the recently developed tensor decomposition (TD) based unsupervised feature extraction (FE) to a kernel-based method through a mathematical formulation. Subsequently, the kernel TD (KTD) based unsupervised FE was applied to two synthetic examples as well as real data sets, and the findings were compared with those obtained previously using the TD-based unsupervised FE approaches. The KTD-based unsupervised FE outperformed or performed comparably with the TD-based unsupervised FE in large p small n situations, which are situations involving a limited number of samples with many variables (observations). Nevertheless, the KTD-based unsupervised FE outperformed the TD-based unsupervised FE in non large p small n situations. In general, although the use of the kernel trick can help the TD-based unsupervised FE gain more variations, a wider range of problems may also be encountered. Considering the outperformance or comparable performance of the KTD-based unsupervised FE compared to the TD-based unsupervised FE when applied to large p small n problems, it is expected that the KTD-based unsupervised FE can be applied in the genomic science domain, which involves many large p small n problems, and, in which, the TD-based unsupervised FE approach has been effectively applied.