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.
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. Tensor-based techniques for learning allow one to exploit the structure of carefully chosen representations of data. This is a desirable feature in particular when the number of training patterns is small which is often the case in areas such as biosignal processing and chemometrics. However, the class of tensor-based models is somewhat restricted and might suffer from limited discriminative power. On a different track, kernel methods lead to flexible nonlinear models that have been proven successful in many different contexts. Nonetheless, a naive application of kernel methods does not exploit structural properties possessed by the given tensorial representations. The goal of this work is to go beyond this limitation by introducing non-parametric tensor-based models. The proposed framework aims at improving the discriminative power of supervised tensor-based models while still exploiting the structural information embodied in the data. We begin by introducing a feature space formed by multilinear functionals. The latter can be considered as the infinite dimensional analogue of tensors. Successively we show how to implicitly map input patterns in such a feature space by means of kernels that exploit the algebraic structure of data tensors. The proposed tensorial kernel links to the MLSVD and features an interesting invariance property; the approach leads to convex optimization and fits into the same primal-dual framework underlying SVM-like algorithms. We present a new supervised tensor regression method based on multi-way array decompositions and kernel machines. The main issue in the development of a kernel-based framework for tensorial data is that the kernel functions have to be defined on tensor-valued input, which here is defined based on multi-mode product kernels and probabilistic generative models. This strategy enables taking into account the underlying multilinear structure during the learning process. Based on the defined kernels for tensorial data, we develop a kernel-based tensor partial least squares approach for regression. The effectiveness of our method is demonstrated by a real-world application, i.e., the reconstruction of 3D movement trajectories from electrocorticography signals recorded from a monkey brain.  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. In this paper we address the two-class classification problem within the tensor-based framework, by formulating the Support Tucker Machines (STuMs). More precisely, in the proposed STuMs the weights parameters are regarded to be a tensor, calculated according to the Tucker tensor decomposition as the multiplication of a core tensor with a set of matrices, one along each mode. We further extend the proposed STuMs to the Σ Σ w STuMs, in order to fully exploit the information offered by the total or the within-class covariance matrix and whiten the data, thus providing in-variance to affine transformations in the feature space. We formulate the two above mentioned problems in such a way that they can be solved in an iterative manner, where at each iteration the parameters corresponding to the projections along a single tensor mode are estimated by solving a typical Support Vector Machine-type problem. The superiority of the proposed methods in terms of classification accuracy is illustrated on the problems of gait and action recognition.   In this paper, we exploit the advantages of tensorial representations and propose several tensor learning models for regression. The model is based on the canonical parallel-factor decomposition of tensors of multiple modes and allows the simultaneous projections of an input tensor to more than one direction along each mode. Two empirical risk functions are studied, namely, the square loss and e-insensitive loss functions. The former leads to higher rank tensor ridge regression (TRR), and the latter leads to higher rank support tensor regression (STR), both formulated using the Frobenius norm for regularization. We also use the group-sparsity norm for regularization, favoring in that way the low rank decomposition of the tensorial weight. In that way, we achieve the automatic selection of the rank during the learning process and obtain the optimal-rank TRR and STR. Experiments conducted for the problems of head-pose, human-age, and 3-D body-pose estimations using real data from publicly available databases, verified not only the superiority of tensors over their vector counterparts but also the efficiency of the proposed algorithms.