A review of heterogeneous data mining for brain disorder identification <s> Supervised learning <s> 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. <s> BIB001 </s> A review of heterogeneous data mining for brain disorder identification <s> Supervised learning <s> article i nfo Brain connectivity datasets comprise networks of brain regions connected by anatomical tracts or by functional associations. Complex network analysis—a new multidisciplinary approach to the study of complex systems—aims to characterize these brain networks with a small number of neurobiologically meaningful and easily computable measures. In this article, we discuss construction of brain networks from connectivity data and describe the most commonly used network measures of structural and functional connectivity. We describe measures that variously detect functional integration and segregation, quantify centrality of individual brain regions or pathways, characterize patterns of local anatomical circuitry, and test resilience of networks to insult. We discuss the issues surrounding comparison of structural and functional network connectivity, as well as comparison of networks across subjects. Finally, we describe a Matlab toolbox (http://www.brain-connectivity-toolbox.net) accompanying this article and containing a collection of complex network measures and large-scale neuroanatomical connectivity datasets. <s> BIB002 </s> A review of heterogeneous data mining for brain disorder identification <s> Supervised learning <s> 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. <s> BIB003 </s> A review of heterogeneous data mining for brain disorder identification <s> Supervised learning <s> A new generalized multilinear regression model, termed the higher order partial least squares (HOPLS), is introduced with the aim to predict a tensor (multiway array) Y from a tensor X through projecting the data onto the latent space and performing regression on the corresponding latent variables. HOPLS differs substantially from other regression models in that it explains the data by a sum of orthogonal Tucker tensors, while the number of orthogonal loadings serves as a parameter to control model complexity and prevent overfitting. The low-dimensional latent space is optimized sequentially via a deflation operation, yielding the best joint subspace approximation for both X and Y. Instead of decomposing X and Y individually, higher order singular value decomposition on a newly defined generalized cross-covariance tensor is employed to optimize the orthogonal loadings. A systematic comparison on both synthetic data and real-world decoding of 3D movement trajectories from electrocorticogram signals demonstrate the advantages of HOPLS over the existing methods in terms of better predictive ability, suitability to handle small sample sizes, and robustness to noise. <s> BIB004 </s> A review of heterogeneous data mining for brain disorder identification <s> Supervised learning <s> 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. <s> BIB005 </s> A review of heterogeneous data mining for brain disorder identification <s> Supervised learning <s> With the ongoing development of neuroimaging technology, neuroimaging classification has become a popular and challenging topic. The high dimension and small sample size characteristics pose many challenges to neuroimaging classification. The traditional neuroimaging classification solutions are tensor-based models, which may not fully consider the structural information and can’t mine the essential features of the input data. Considering the complicated properties of the neuroimaging data, a deep learning based algorithm—the hierarchical convolutional sparse auto-encoder (HCSAE) considering all dimensional information together is proposed in this paper. The HCSAE treats different convolutional sparse auto-encoder (CSAE) in an unsupervised hierarchical mode, where the CSAE extracts the essential features of the input by the sparse auto-encoder (SAE) and encodes the inputs in a convolutional manner, which helps to extract efficient and robust features and conserve abundant detail information for the neuroimaging classification. The proposed algorithm was verified by three human brain fMRI classification datasets, and showed a great potential compared with the traditional classification algorithms. <s> BIB006 </s> A review of heterogeneous data mining for brain disorder identification <s> Supervised learning <s> The widespread use of multisensor technology and the emergence of big data sets have highlighted the limitations of standard flat-view matrix models and the necessity to move toward more versatile data analysis tools. We show that higher-order tensors (i.e., multiway arrays) enable such a fundamental paradigm shift toward models that are essentially polynomial, the uniqueness of which, unlike the matrix methods, is guaranteed under very mild and natural conditions. Benefiting from the power of multilinear algebra as their mathematical backbone, data analysis techniques using tensor decompositions are shown to have great flexibility in the choice of constraints which match data properties and extract more general latent components in the data than matrix-based methods. <s> BIB007
Suppose we have a set of tensor data D ¼ fðX i ; y i Þg n i¼1 for classification problem, where X i 2 R I 1 ÂÁÁÁÂI m is the Fig. 1 Tensor factorization of a third-order tensor BIB002 A voxel is the smallest three-dimensional point volume referenced in a neuroimaging of the brain. neuroimaging data represented as an mth-order tensor and y i 2 fÀ1; þ1g is the corresponding binary class label of X i . For example, if the i-th subject has Alzheimer's disease, the subject is associated with a positive label, i.e., y i ¼ þ1. Otherwise, if the subject is in the control group, the subject is associated with a negative label, i.e., y i ¼ À1. Supervised tensor learning can be formulated as the optimization problem of support tensor machines (STMs) BIB001 which is a generalization of the standard support vector machines (SVMs) from vector data to tensor data. The objective of such learning algorithms is to learn a hyperplane by which the samples with different labels are divided as wide as possible. However, tensor data may not be linearly separable in the input space. To achieve a better performance on finding the most discriminative biomarkers or identifying infected subjects from the control group, in many neuroimaging applications, nonlinear transformation of the original tensor data should be considered. He et al. study the problem of supervised tensor learning with nonlinear kernels which can preserve the structure of tensor data BIB005 . The proposed kernel is an extension of kernels in the vector space to the tensor space which can take the multidimensional structure complexity into account. However, it cannot automatically consider the abundant and complicated information of the neuroimaging data in an integral manner. Han et al. apply a deep learning-based algorithm, the hierarchical convolutional sparse auto-encoder, to extract efficient and robust features and conserve abundant detail information for the neuroimaging classification BIB006 . Slightly different from classifying disease status (discrete label), another family of problems uses tensor neuroimages to predict cognitive outcome (continuous label). The problems can be formulated in a regression setup by treating clinical outcome as the real label, i.e., y i 2 R, and treating tensor neuroimages as the input. However, most classical regression methods take vectors as input features. Simply reshaping a tensor into a vector is clearly an unsatisfactory solution. Zhou et al. exploit the tensor structure in imaging data and integrate tensor decomposition within a statistical regression paradigm to model multidimensional arrays BIB003 . By imposing a low-rank approximation to the extremely highdimensional complex imaging data, the curse of dimensionality is greatly alleviated, thereby allowing development of a fast estimation algorithm and regularization. Numerical analysis demonstrates its potential applications in identifying ROI in brains that are relevant to a particular clinical response. In scenarios where the objective is to predict a set of dependent variables, Cichocki et al. introduce a generalized multilinear regression model, higher order partial least squares, which projects the electrocorticogram data into a latent space and performs regression on the corresponding latent variables BIB007 BIB004 .