A review of heterogeneous data mining for brain disorder identification <s> Definition 1 (Tensor product) <s> Effective diagnosis of Alzheimer's disease (AD) is of primary importance in biomedical research. Recent studies have demonstrated that neuroimaging parameters are sensitive and consistent measures of AD. In addition, genetic and demographic information have also been successfully used for detecting the onset and progression of AD. The research so far has mainly focused on studying one type of data source only. It is expected that the integration of heterogeneous data (neuroimages, demographic, and genetic measures) will improve the prediction accuracy and enhance knowledge discovery from the data, such as the detection of biomarkers. In this paper, we propose to integrate heterogeneous data for AD prediction based on a kernel method. We further extend the kernel framework for selecting features (biomarkers) from heterogeneous data sources. The proposed method is applied to a collection of MRI data from 59 normal healthy controls and 59 AD patients. The MRI data are pre-processed using tensor factorization. In this study, we treat the complementary voxel-based data and region of interest (ROI) data from MRI as two data sources, and attempt to integrate the complementary information by the proposed method. Experimental results show that the integration of multiple data sources leads to a considerable improvement in the prediction accuracy. Results also show that the proposed algorithm identifies biomarkers that play more significant roles than others in AD diagnosis. <s> BIB001 </s> A review of heterogeneous data mining for brain disorder identification <s> Definition 1 (Tensor product) <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> BIB002 </s> A review of heterogeneous data mining for brain disorder identification <s> Definition 1 (Tensor product) <s> We pose the problem of network discovery which involves simplifying spatio-temporal data into cohesive regions (nodes) and relationships between those regions (edges). Such problems naturally exist in fMRI scans of human subjects. These scans consist of activations of thousands of voxels over time with the aim to simplify them into the underlying cognitive network being used. We propose supervised and semi-supervised variations of this problem and postulate a constrained tensor decomposition formulation and a corresponding alternating least squares solver that is easy to implement. We show this formulation works well in controlled experiments where supervision is incomplete, superfluous and noisy and is able to recover the underlying ground truth network. We then show that for real fMRI data our approach can reproduce well known results in neurology regarding the default mode network in resting-state healthy and Alzheimer affected individuals. Finally, we show that the reconstruction error of the decomposition provides a useful measure of the network strength and is useful at predicting key cognitive scores both by itself and with clinical information. <s> BIB003 </s> A review of heterogeneous data mining for brain disorder identification <s> Definition 1 (Tensor product) <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> BIB004 </s> A review of heterogeneous data mining for brain disorder identification <s> Definition 1 (Tensor product) <s> Following the recent advances in neuroimaging technology, the research on brain network analysis becomes an emerging area in data mining community. Brain network data pose many unique challenges for data mining research. For example, in brain networks, the nodes (i.e., the brain regions) and edges (i.e., relationships between brain regions) are usually not given, but should be derived from the neuroimaging data. The network structure can be very noisy and uncertain. Therefore, innovative methods are required for brain network analysis. Many research efforts have been devoted to this area. They have achieved great success in various applications, such as brain network extraction, graph mining, neuroimaging data analysis. In this paper, we review some recent data mining methods which are used in the literature for mining brain network data. <s> BIB005
The tensor product of three vectors a 2 R I 1 , b 2 R I 2 ; and c 2 R I 3 , denoted by a b c, represents a third-order tensor with the elements a b c ð Tensor product is also referred to as outer product in some literature BIB001 BIB003 . An mth-order tensor is a rank-one tensor if it can be defined as the tensor product of m vectors. Definition 2 Given a third-order tensor X 2 R I 1 ÂI 2 ÂI 3 and an integer R, as illustrated in Fig. 1 , a tensor factorization of X can be expressed as One of the major difficulties brought by the tensor data is the curse of dimensionality. The total number of voxels contained in a multi-mode tensor, say, X ¼ ðx i 1 ;...;i m Þ 2 R I 1 ÂÁÁÁÂI m is I 1 Â Á Á Á Â I m which is exponential to the number of modes. If we unfold the tensor into a vector, the number of features will be extremely high BIB002 . This makes traditional data mining methods prone to overfitting, especially with a small sample size. Both computational scalability and theoretical guarantee of the traditional models are compromised by such high dimensionality BIB004 . On the other hand, complex structural information is embedded in the tensor data. For example, in the neuroimaging data, values of adjacent voxels are usually correlated with each other BIB005 . Such spatial relationships among different voxels in a tensor image can be very important in neuroimaging applications. Conventional tensor-based approaches focus on reshaping the tensor data into matrices/ vectors, and thus, the original spatial relationships are lost. The integration of structural information is expected to improve the accuracy and interpretability of tensor models.