Parsimonious Tensor Response Regression

ABSTRACT

Aiming at abundant scientific and engineering data with not only high dimensionality but also complex structure, we study the regression problem with a multidimensional array (tensor) response and a vector predictor. Applications include, among others, comparing tensor images across groups after adjusting for additional covariates, which is of central interest in neuroimaging analysis. We propose parsimonious tensor response regression adopting a generalized sparsity principle. It models all voxels of the tensor response jointly, while accounting for the inherent structural information among the voxels. It effectively reduces the number of free parameters, leading to feasible computation and improved interpretation. We achieve model estimation through a nascent technique called the envelope method, which identifies the immaterial information and focuses the estimation based upon the material information in the tensor response. We demonstrate that the resulting estimator is asymptotically efficient, and it enjoys a competitive finite sample performance. We also illustrate the new method on two real neuroimaging studies. Supplementary materials for this article are available online.

KEYWORDS:

• Envelope method
• Multidimensional array
• Multivariate linear regression
• Reduced rank regression
• Sparsity principle
• Tensor regression

Acknowledgment

The authors thank the Editor, the Associate Editor, and two referees for their insightful and constructive comments.

Funding

Li’s research was partially supported in part by NSF grants DMS-1310319 and DMS-1613137. Zhang’s research was supported in part by NSF grants DMS-1613154 and CCF-1617691, and the CRC-FYAP grant from Florida State University.