Mining labelled tensors by discovering both their common and discriminative subspaces

Conventional non-negative tensor factorization (NTF) methods assume there is only one tensor that needs to be decomposed to low-rank factors. However, in practice data are usually generated from different time periods or by different class labels, which are represented by a sequence of multiple tensors associated with different labels. This raises the problem that when one needs to analyze and compare multiple tensors, existing NTF is unsuitable for discovering all potentially useful patterns: 1) if one factorizes each tensor separately, the common information shared by the tensors is lost in the factors, and 2) if one concatenates these tensors together and forms a larger tensor to factorize, the intrinsic discriminative subspaces that are unique to each tensor are not captured. The cause of such an issue is from the fact that conventional factorization methods handle data observations in an unsupervised way, which only considers features and not labels of the data. To tackle this problem, in this paper we design a novel factorization algorithm called CDNTF (common and discriminative subspace non-negative tensor factorization), which takes both features and class labels into account in the factorization process. CDNTF uses a set of labelled tensors as input and computes both their common and discriminative subspaces simultaneously as output. We design an iterative algorithm that solves the common and discriminative subspace factorization problem with a proof of convergence. Experiment results on solving graph classification problems demonstrate the power and the effectiveness of the subspaces discovered by our method

Tensor Analysis and Fusion of Multimodal Brain Images

Current high-throughput data acquisition technologies probe dynamical systems with different imaging modalities, generating massive data sets at different spatial and temporal resolutions posing challenging problems in multimodal data fusion. A case in point is the attempt to parse out the brain structures and networks that underpin human cognitive processes by analysis of different neuroimaging modalities (functional MRI, EEG, NIRS etc.). We emphasize that the multimodal, multi-scale nature of neuroimaging data is well reflected by a multi-way (tensor) structure where the underlying processes can be summarized by a relatively small number of components or "atoms". We introduce Markov-Penrose diagrams - an integration of Bayesian DAG and tensor network notation in order to analyze these models. These diagrams not only clarify matrix and tensor EEG and fMRI time/frequency analysis and inverse problems, but also help understand multimodal fusion via Multiway Partial Least Squares and Coupled Matrix-Tensor Factorization. We show here, for the first time, that Granger causal analysis of brain networks is a tensor regression problem, thus allowing the atomic decomposition of brain networks. Analysis of EEG and fMRI recordings shows the potential of the methods and suggests their use in other scientific domains.Comment: 23 pages, 15 figures, submitted to Proceedings of the IEE