Tensors are generalizations of vectors and matrices for multi-dimensional data representation.Fueled by novel computing technologies, tensors have expanded to various domains, including statistics, data science, signal processing, and machine learning. Comparing to traditional data representation formats, tensor data representation distinguishes itself with its capability of preserving complex structures and multi-way features for multi-dimensional data. In this dissertation, we explore some tensor-based classification models and their statistical properties. In particular, we propose few novel support tensor machine methods for huge-size tensor and multimodal tensor classification problems, and study their classification consistency properties. These methods are applied to different applications for validation.The first piece of work considers classification problems for gigantic size multi-dimensionaldata. Although current tensor-based classification approaches have demonstrated extraordinary performance in empirical studies, they may face more challenges such as long processing time and insufficient computer memory when dealing with big tensors. In chapter 3, we combine tensor-based random projection and support tensor machine, and propose a Tensor Ensemble Classifier (TEC) for ultra-high dimensional tensors, which aggregates multiple support tensor machines estimated from randomly projected CANDECOMP/PARAFAC (CP) tensors. This method utilizes Gaussian and spares random projections to compress high-dimensional tensor CP factors, and predicts their class labels with support tensor machine classifiers. With the well celebrated Johnson-Lindenstrauss Lemma and ensemble techniques, TEC methods are shown to be statistically consistent while having high computational efficiencies for big tensor data. Simulation studies and real data applications including Alzheimer’s Disease MRI Image classification and Traffic Image classification are provided as empirical evidence to validate the performance of TEC models.The second piece of work considers classification problems for multimodal tensor data, whichare particularly common in neuroscience and brain imaging analysis. Utilizing multimodal data is of great interest for machine learning and statistics research in these domains, since it is believed that integration of features from multiple sources can potentially increase model performance while unveiling the interdependence between heterogeneous data. In chapter 4, we propose a Coupled Support Tensor Machine (C-STM) which adopts Advanced Coupled Matrix Tensor Factorization (ACMTF) and Multiple Kernel Learning (MKL) techniques for coupled matrix tensor data classification. The classification risk of C-STM is shown to be converging to the optimal Bayes risk, making itself a statistically consistent rule. The framework can also be easily extended for multimodal tensors with data modalities greater than two. The C-STM is validated through a simulation study as well as a simultaneous EEG-fMRI trial classification problem. The empirical evidence shows that C-STM can utilize information from multiple sources and provide a better performance comparing to the traditional methods.
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