Tensor, a multi-dimensional data structure, has been exploited recently in the machine learning community. 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 classification 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 contributions 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 function 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 elaborately. 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.

For many real-world image classification tasks, collecting high-quality labeled image data is challenging. Therefore, a complicated convolutional neural network might not be able to get well trained and traditional machine learning methods would be a better choice. However, traditional vector-based machine learning algorithms cannot achieve a satisfactory performance when dealing with high-dimensional tensorial data. There are mainly two reasons. First, vectorizing tensor data loses useful structural information in the original data, which might be helpful in the classification task. Second, traditional vector-based methods commonly contain a similar number of model parameters as the data size. In this case, when the data dimension is relatively high and the number of training samples is small, an overfitted model would be derived. To address these issues, researchers extend the vector-based classifiers into their tensorial formats, which accept tensorial data as input directly, and at the same time employ much fewer model parameters. In this chapter, two traditional vector-based machine learning algorithms, namely, support vector machine and logistic regression, are generalized to their tensorial counterparts to facilitate the tensor-based classification tasks.

Sum-product networks (SPNs) constitute an emerging class of neural networks with clear probabilistic semantics and superior inference speed over other graphical models. This brief reveals an important connection between SPNs and tensor trains (TTs), leading to a new canonical form which we call tensor SPNs (tSPNs). Specifically, we demonstrate the intimate relationship between a valid SPN and a TT. For the first time, through mapping an SPN onto a tSPN and employing specially customized optimization techniques, we demonstrate improvements up to a factor of 100 on both model compression and inference speedup for various data sets with negligible loss in accuracy.

There has been growing interest in extending traditional vector-based machine learning techniques to their tensor forms. Support tensor machine (STM) and support Tucker machine (STuM) are two typical tensor generalization of the conventional support vector machine (SVM). However, the expressive power of STM is restrictive due to its rank-one tensor constraint, and STuM is not scalable because of the exponentially sized Tucker core tensor. To overcome these limitations, we introduce a novel and effective support tensor train machine (STTM) by employing a general and scalable tensor train as the parameter model. Experiments validate and confirm the superiority of the STTM over SVM, STM and STuM.

A restricted Boltzmann machine (RBM) learns a probability distribution over its input samples and has numerous uses like dimensionality reduction, classification and generative modeling. Conventional RBMs accept vectorized data that dismiss potentially important structural information in the original tensor (multi-way) input. Matrix-variate and tensor-variate RBMs, named MvRBM and TvRBM, have been proposed but are all restrictive by model construction and have weak model expression power. This work presents the matrix product operator RBM (MPORBM) that utilizes a tensor network generalization of Mv/TvRBM, preserves input formats in both the visible and hidden layers, and results in higher expressive power. A novel training algorithm integrating contrastive divergence and an alternating optimization procedure is also developed. Numerical experiments compare the MPORBM with the traditional RBM and MvRBM for data classification and image completion and denoising tasks. The expressive power of the MPORBM as a function of the MPO-rank is also investigated.