The widespread use of multisensor technology and the emergence of big data sets have brought the necessity to develop more versatile tools to represent higher order data with multiple aspects and high dimensionality. Data in the form of multidimensional arrays, also referred to as tensors, arise in a variety of applications including chemometrics, hyperspectral imaging, high-resolution videos, neuroimaging, biometrics, and social network analysis. Early multiway data analysis approaches reformatted such tensor data as large vectors or matrices and then resorted to dimensionality reduction methods developed for classical two-way analysis such as principal component analysis (PCA). However, one cannot discover hidden components within multiway data using conventional PCA. To this end, tensor decomposition methods which are flexible in the choice of the constraints and that extract more general latent components have been proposed. In this paper, we review the major tensor decomposition methods with a focus on problems targeted by classical PCA. In particular, we present tensor methods that aim to solve three important challenges typically addressed by PCA: dimensionality reduction, i.e., low-rank tensor approximation; supervised learning, i.e., learning linear subspaces for feature extraction; and robust low-rank tensor recovery. We also provide experimental results to compare different tensor models for both dimensionality reduction and supervised learning applications.

Advances in information technology are making it possible to collect increasingly massive amounts of multidimensional, multi-modal neuroimaging data such as functional magnetic resonance imaging (fMRI). Current fMRI datasets involve multiple variables including multiple subjects, as well as both temporal and spatial data. These high dimensional datasets pose a challenge to the signal processing community to develop data reduction methods that can exploit their rich structure and extract meaningful summarizations. In this paper, we propose a tensor-based framework for data reduction and low-dimensional structure learning with a particular focus on reducing high dimensional fMRI data sets into physiologically meaningful network components. We develop a multiscale tensor factorization method for higher order data inspired by hybrid linear modeling and subspace clustering techniques. In particular, we develop a multi-scale HoSVD approach where a given tensor is first permuted and then partitioned into several sub-tensors each of which can be represented more efficiently. This multi-scale framework is applied to resting state fMRI data to identify the default mode network from compressed data.

Recent developments In Information technology have enabled us to collect and analyze high dimensional and higher order data such as tensors. High dimensional data usually lies in a lower dimensional subspace and identifying this low-dimensional structure is important in many signal and information processing applications. Traditional subspace estimation approaches have been limited to vector-type data and cannot effectively deal with these high order datasets. Moreover, most of the existing methods are batch algorithms which can't handle streaming data. In this paper, we propose a new tensor subspace tracking approach to identify changes in dynamic networks. The proposed approach recursively estimates low-rank subspace of higher order data and decomposes it into low-rank and sparse components. The proposed approach is evaluated on both simulated and real dynamic networks.