Analysis of high dimensional data in modern applications, such as neuroscience, text mining, spectral analysis or chemometrices naturally requires tensor decomposition methods. The Tucker decompositions allow us to extract hidden factors (component matrices) with a different dimension in each mode and investigate interactions among various modes. The Alternating Least Squares (ALS) algorithms have been confirmed effective and efficient in most of tensor decompositions, especially, Tucker with orthogonality constraints. However, for nonnegative Tucker decomposition (NTD), standard ALS algorithms suffer from unstable convergence properties, demand high computational cost for large scale problems due to matrix inversion and often return suboptimal solutions. Moreover, they are quite sensitive with respect to noise, and can be relatively slow in the special case when the data are nearly collinear. In this paper, we propose a new algorithm for nonnegative Tucker decomposition based on constrained minimization of a set of local cost functions and Hierarchical Alternating Least Squares (HALS). The developed HALS NTD algorithm sequentially updates components, hence avoids matrix inversion, and is suitable for large-scale problems. The proposed algorithm is also regularized with additional constraint terms such as sparseness, orthogonality, smoothness, and especially discriminant constraints for classification problems. Extensive experiments confirm the validity and higher performance of the developed algorithm in comparison with other existing algorithms.

α-divergence based nonnegative tensor factorization (NTF) is applied to blind multi-spectral image (MSI) decomposition. Matrix of spectral profiles and matrix of spatial distributions of the materials resident in the image are identified from the factors in Tucker3 and PARAFAC models. NTF preserves local structure in the MSI that is lost, due to vectorization of the image, with nonnegative matrix factorization (NMF)- or independent component analysis (ICA)-based decompositions. Moreover, NTF based on PARAFAC model is unique up to permutation and scale under mild conditions. To achieve this, NMF- and ICA-based factorizations respectively require enforcement of sparseness (orthogonality) and statistical independence constraints on the spatial distributions of the materials resident in the MSI, and that is not true. We demonstrate 1 efficiency of the NTF-based factorization in relation to NMF- and ICA-based factorizations on blind decomposition of the experimental MSI with the known ground truth. OCIS codes: 100.6890; 100.3190; 150.6910; 100.2960; 170.3880. Blind or unsupervised multi-spectral and hyper-spectral image (HSI) decomposition attracts increased attention due to its capability to discriminate materials resident in the MSI/HSI without

Tensor decompositions for feature extraction and classification of high

Matrix factorization is an important and unifying topic in signal processing and linear algebra, which has found numerous applications in many other areas. This chapter introduces basic linear and multi-linear 1 models for matrix and tensor factorizations and decompositions, and formulates the analysis framework for

In addition to helping better understand how the human brain works, the brain-computer interface neuroscience paradigm allows researchers to develop a new class of bioengineering control devices and robots, offering promise for rehabilitation and other medical applications as well as exploring possibilities for advanced human-computer interfaces. Brain computer interfaces (BCIs) are systems that use electric, magnetic, or hemodynamic brain signals to control external devices such as computers, switches, wheelchairs, or neuroprostheses. While BCI research endeavors to create new communication channels for severely handicapped people using their brain signals, recent efforts also have been focused on developing potential applications in rehabilitation, multimedia communication, virtual reality, and entertainment/relaxation. 1-14 The three major components of BCIs are: 2

Abstract—Tensor representation and tensor decompositions are natural approaches to deal with large amounts of data with multiple aspects and high dimensionality in modern applications, such as environmental analysis, chemometrices, pharmaceutical analysis, spectral analysis, neuroscience. The two most popular decomposition/factorization models for N-th order tensors are the Tucker model and the more restricted PARAFAC model. The Tucker decomposition allows for the extraction of different numbers of factors in each of the modes, and permits interactions within each modality while PARAFAC does not. This advantage, however, is also one of the weakness of this decomposition. The difficult problem is to identify the dominant relationships between components, and to establish unique representation. In this paper, we will introduce a new measure index which is called the Joint Rate (JR) index, in order to evaluate interactions among various components in the general Tucker decomposition. The Hinton diagram is also extended to 3-D visualization. The use of the JR index will be illustrated with the analysis of EEG data for classification and BCI applications.