Tensor-on-tensor regression

We propose a framework for the linear prediction of a multi-way array (i.e., a tensor) from another multi-way array of arbitrary dimension, using the contracted tensor product. This framework generalizes several existing approaches, including methods to predict a scalar outcome from a tensor, a matrix from a matrix, or a tensor from a scalar. We describe an approach that exploits the multiway structure of both the predictors and the outcomes by restricting the coefficients to have reduced CP-rank. We propose a general and efficient algorithm for penalized least-squares estimation, which allows for a ridge (L_2) penalty on the coefficients. The objective is shown to give the mode of a Bayesian posterior, which motivates a Gibbs sampling algorithm for inference. We illustrate the approach with an application to facial image data. An R package is available at https://github.com/lockEF/MultiwayRegression .Comment: 33 pages, 3 figure

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

Damage to Association Fiber Tracts Impairs Recognition of the Facial Expression of Emotion

An array of cortical and subcortical structures have been implicated in the recognition of emotion from facial expressions. It remains unknown how these regions communicate as parts of a system to achieve recognition, but white matter tracts are likely critical to this process. We hypothesized that (1) damage to white matter tracts would be associated with recognition impairment and (2) the degree of disconnection of association fiber tracts [inferior longitudinal fasciculus (ILF) and/or inferior fronto-occipital fasciculus (IFOF)] connecting the visual cortex with emotion-related regions would negatively correlate with recognition performance. One hundred three patients with focal, stable brain lesions mapped onto a reference brain were tested on their recognition of six basic emotional facial expressions. Association fiber tracts from a probabilistic atlas were coregistered to the reference brain. Parameters estimating disconnection were entered in a general linear model to predict emotion recognition impairments, accounting for lesion size and cortical damage. Damage associated with the right IFOF significantly predicted an overall facial emotion recognition impairment and specific impairments for sadness, anger, and fear. One subject had a pure white matter lesion in the location of the right IFOF and ILF. He presented specific, unequivocal emotion recognition impairments. Additional analysis suggested that impairment in fear recognition can result from damage to the IFOF and not the amygdala. Our findings demonstrate the key role of white matter association tracts in the recognition of the facial expression of emotion and identify specific tracts that may be most critical

Nonconvex Nonsmooth Low-Rank Minimization via Iteratively Reweighted Nuclear Norm

The nuclear norm is widely used as a convex surrogate of the rank function in compressive sensing for low rank matrix recovery with its applications in image recovery and signal processing. However, solving the nuclear norm based relaxed convex problem usually leads to a suboptimal solution of the original rank minimization problem. In this paper, we propose to perform a family of nonconvex surrogates of $L_0$-norm on the singular values of a matrix to approximate the rank function. This leads to a nonconvex nonsmooth minimization problem. Then we propose to solve the problem by Iteratively Reweighted Nuclear Norm (IRNN) algorithm. IRNN iteratively solves a Weighted Singular Value Thresholding (WSVT) problem, which has a closed form solution due to the special properties of the nonconvex surrogate functions. We also extend IRNN to solve the nonconvex problem with two or more blocks of variables. In theory, we prove that IRNN decreases the objective function value monotonically, and any limit point is a stationary point. Extensive experiments on both synthesized data and real images demonstrate that IRNN enhances the low-rank matrix recovery compared with state-of-the-art convex algorithms