Tensor Regression with Applications in Neuroimaging Data Analysis

Abstract

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.

KEY WORDS:

• Brain imaging
• Dimension reduction
• Generalized linear model
• Magnetic Resonance Imaging
• Multidimensional array
• Tensor regression

Acknowledgments

The work was partially supported by NIH grant HG006139 (Zhou), NSF grant DMS-1106668 (Li), NSF grant BCS-08-26844 and NIH grants RR025747-01, P01CA142538-01, MH086633, and EB005149-01 (Zhu). The authors thank the editor, the associate editor, and three referees for their insightful and constructive comments.