With the advance of sensor technology, it is becoming more commonplace to collect multi-mode data, i.e., tensors, with high dimensionality. To deal with the large amounts of redundancy in tensorial data, different dimensionality reduction methods such as low-rank tensor decomposition have been developed. While low-rank decompositions capture the global structure, there is a need to leverage the underlying local geometry through manifold learning methods. Manifold learning methods have been widely considered in tensor factorization to incorporate the low-dimensional geometry of the underlying data. However, existing techniques focus on only one mode of the data and exploit correlations among the features to reduce the dimension of the feature vectors. Recently, multiway graph signal processing approaches that exploit the correlations among all modes of a tensor have been proposed to learn low-dimensional representations. Inspired by this idea, in this letter we propose a graph regularized robust tensor-train decomposition method where the graph regularization is applied across each mode of the tensor to incorporate the local geometry. As the resulting optimization problem is computationally prohibitive due to the high dimensionality of the graph regularization terms, an equivalence between mode-$n$canonical unfolding and regular mode-$n$unfolding is derived resulting in a computationally efficient optimization algorithm. The proposed method is evaluated on both synthetic and real tensors for denoising and tensor completion.

Anomaly detection in spatiotemporal data is a challenging problem encountered in a variety of applications including hyperspectral imaging, video surveillance, and urban traffic monitoring. Existing anomaly detection methods are most suited for point anomalies in sequence data and cannot deal with temporal and spatial dependencies that arise in spatiotemporal data. In recent years, tensor-based methods have been proposed for anomaly detection to address this problem. These methods rely on conventional tensor decomposition models, not taking the structure of the anomalies into account, and are supervised or semi-supervised. We introduce an unsupervised tensor-based anomaly detection method that takes the sparse and temporally continuous nature of anomalies into account. In particular, the anomaly detection problem is formulated as a robust low-rank + sparse tensor decomposition with a regularization term that minimizes the temporal variation of the sparse part, so that the extracted anomalies are temporally persistent. We also approximate rank minimization with graph total variation minimization to reduce the complexity of the optimization algorithm. The resulting optimization problem is convex, scalable, and is shown to be robust against missing data and noise. The proposed framework is evaluated on both synthetic and real spatiotemporal urban traffic data and compared with baseline methods.

Higher-order data with high dimensionality arise in a diverse set of application areas such as computer vision, video analytics and medical imaging. Tensors provide a natural tool for representing these types of data. Two major challenges that confound current tensor based supervised learning algorithms are storage complexity and computational efficiency. In this paper, we address these problems by introducing a multi-branch tensor network structure. The multi-branch structure is a general tensor decomposition that includes Tucker and tensor-train (TT) as special cases and takes advantage of the flexibility of the tensor network to provide a better balance between storage and computational complexity. We then introduce a supervised discriminative tensor-train subspace learning approach referred to as tensor-train discriminant analysis (TTDA), and its implementations using the multi-branch tensor network structure. Multi-branch implementations of TTDA are shown to achieve lower storage and computational complexity while providing improved classification performance with respect to both Tucker and TT based supervised learning methods.

With the advances in data acquisition technology, tensor objects are collected in a variety of applications including multimedia, medical and hyperspectral imaging. As the dimensionality of tensor objects is usually very high, dimensionality reduction is an important problem. Most of the current tensor dimensionality reduction methods rely on finding low-rank linear representations using different generative models. However, it is well-known that high-dimensional data often reside in a low-dimensional manifold. Therefore, it is important to find a compact representation, which uncovers the low-dimensional tensor structure while respecting the intrinsic geometry. In this paper, we propose a graph regularized tensor train (GRTT) decomposition that learns a low-rank tensor train model that preserves the local relationships between tensor samples. The proposed method is formulated as a non-convex optimization problem on the Stiefel manifold and an efficient algorithm is proposed to solve it. The proposed method is compared to existing tensor based dimensionality reduction methods as well as tensor manifold embedding methods for unsupervised learning applications.

The rapid development of information technology is making it possible to collect massive amounts of multidimensional, multimodal data with high dimensionality in a diverse set of science and engineering disciplines. Although there has been a lot of recent work in the area of unsupervised tensor learning, extensions to supervised learning, feature extraction and classification are still limited. Moreover, most of the existing supervised tensor learning approaches are based on the Tucker model. However, this model has some limitations for large tensors including high memory and execution time costs. In this paper, we introduce a supervised learning approach for tensor classification based on the tensor-train model. In particular, we introduce two computationally efficient implementations of tensor-train discriminant analysis (TT-DA). The proposed approaches are evaluated on image classification tasks with respect to computation time, storage cost and classification accuracy.