With advances in data collection technologies, tensor data is assuming increasing prominence in many applications and the problem of supervised tensor learning has emerged as a topic of critical significance in the data mining and machine learning community. Conventional methods for supervised tensor learning mainly focus on learning kernels by flattening the tensor into vectors or matrices, however structural information within the tensors will be lost. In this paper, we introduce a new scheme to design structure-preserving kernels for supervised tensor learning. Specifically, we demonstrate how to leverage the naturally available structure within the tensorial representation to encode prior knowledge in the kernel. We proposed a tensor kernel that can preserve tensor structures based upon dual-tensorial mapping. The dual-tensorial mapping function can map each tensor instance in the input space to another tensor in the feature space while preserving the tensorial structure. Theoretically, our approach is an extension of the conventional kernels in the vector space to tensor space. We applied our novel kernel in conjunction with SVM to real-world tensor classification problems including brain fMRI classification for three different diseases (i.e., Alzheimer's disease, ADHD and brain damage by HIV). Extensive empirical studies demonstrate that our proposed approach can effectively boost tensor classification performances, particularly with small sample sizes.
: Tensor factorizations are higher-order extensions of matrix factorization that elicit intrinsic multi-way structures and capture the underlying patterns in tensor data. These techniques have been widely used in diverse disciplines to analyze and process tensor data. A thorough survey of these techniques and applications can be found in @cite_20 . The two most commonly factorizations are CP and Tucker. CP is a special case of Tucker decomposition which forces the core array to a (super)diagonal form. It is thus more condensed than that of Tucker. In the supervised tensor learning setting, CP is more frequently applied to explore tensor data because of its properties of uniqueness and simplicity @cite_4 @cite_7 @cite_19 @cite_14 . However, in these applications, CP factorization is used either for exploratory analysis or to deal with linear tensor-based models. In this study, we employ the CP factorization to foster the use of kernel methods for supervised tensor learning.
@cite_14 @cite_4 @cite_7 @cite_19 @cite_20
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Classical regression methods treat covariates as a vector and estimate a corresponding vector of regression coefficients. Modern applications in medical imaging generate covariates of more complex form such as multidimensional arrays (tensors). Traditional statistical and computational methods are proving insufficient for analysis of these high-throughput data due to their ultrahigh dimensionality as well as complex structure. In this article, we propose a new family of tensor regression models that efficiently exploit the special structure of tensor covariates. Under this framework, ultrahigh dimensionality is reduced to a manageable level, resulting in efficient estimation and prediction. A fast and highly scalable estimation algorithm is proposed for maximum likelihood estimation and its associated asymptotic properties are studied. Effectiveness of the new methods is demonstrated on both synthetic and real MRI imaging data. Supplementary materials for this article are available online. We propose a method for unsupervised linear feature extraction through tensor decomposition. The linear feature extraction can be formulated as a canonical polyadic decomposition (CPD) of a third-order tensor when transformation matrix is constrained to be equal to the Khatri-Rao product of two matrices. Therefore, standard algorithms for computing CPD decomposition can be used for feature extraction. The proposed method is validated on publicly available low-resolutionmass spectra of cancerous and non-cancerous samples. Obtained results indicate that this approach could be of practical importance in analysis of protein expression profiles. This paper aims to take general tensors as inputs for supervised learning. A supervised tensor learning (STL) framework is established for convex optimization based learning techniques such as support vector machines (SVM) and minimax probability machines (MPM). Within the STL framework, many conventional learning machines can be generalized to take n sup th -order tensors as inputs. We also study the applications of tensors to learning machine design and feature extraction by linear discriminant analysis (LDA). Our method for tensor based feature extraction is named the tenor rank-one discriminant analysis (TR1DA). These generalized algorithms have several advantages: 1) reduce the curse of dimension problem in machine learning and data mining; 2) avoid the failure to converge; and 3) achieve better separation between the different categories of samples. As an example, we generalize MPM to its STL version, which is named the tensor MPM (TMPM). TMPM learns a series of tensor projections iteratively. It is then evaluated against the original MPM. Our experiments on a binary classification problem show that TMPM significantly outperforms the original MPM. This survey provides an overview of higher-order tensor decompositions, their applications, and available software. A tensor is a multidimensional or @math -way array. Decompositions of higher-order tensors (i.e., @math -way arrays with @math ) have applications in psycho-metrics, chemometrics, signal processing, numerical linear algebra, computer vision, numerical analysis, data mining, neuroscience, graph analysis, and elsewhere. Two particular tensor decompositions can be considered to be higher-order extensions of the matrix singular value decomposition: CANDECOMP PARAFAC (CP) decomposes a tensor as a sum of rank-one tensors, and the Tucker decomposition is a higher-order form of principal component analysis. There are many other tensor decompositions, including INDSCAL, PARAFAC2, CANDELINC, DEDICOM, and PARATUCK2 as well as nonnegative variants of all of the above. The N-way Toolbox, Tensor Toolbox, and Multilinear Engine are examples of software packages for working with tensors.