Tensor, a multi-dimensional data structure, has been exploited recently in the machine learning commu-nity. Traditional machine learning approaches are vector-or matrix-based, and cannot handle tensorial data directly. In this paper, we propose a tensor train (TT)-based kernel technique for the first time, and apply it to the conventional support vector machine (SVM) for high-dimensional image classifica-tion with very small number of training samples. Specifically, we propose a kernelized support tensor train machine that accepts tensorial input and preserves the intrinsic kernel property. The main con-tributions are threefold. First, we propose a TT-based feature mapping procedure that maintains the TT structure in the feature space. Second, we demonstrate two ways to construct the TT-based kernel func-tion while considering consistency with the TT inner product and preservation of information. Third, we show that it is possible to apply different kernel functions on different data modes. In principle, our method tensorizes the standard SVM on its input structure and kernel mapping scheme. This reduces the storage and computation complexity of kernel matrix construction from exponential to polynomial. The validity proof and computation complexity of the proposed TT-based kernel functions are provided elabo-rately. Extensive experiments are performed on high-dimensional fMRI and color images datasets, which demonstrates the superiority of the proposed scheme compared with the state-of-the-art techniques. (c) 2021 Elsevier Ltd. All rights reserved.
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