Tucker Tensor Regression and Neuroimaging Analysis

TL;DR

This paper introduces a novel generalized linear tensor regression model using Tucker decomposition, enabling efficient analysis of high-dimensional neuroimaging data while preserving spatial structure.

Contribution

It proposes a new tensor regression framework based on Tucker decomposition that effectively manages ultrahigh dimensional neuroimaging data and demonstrates consistent estimation.

Findings

Model accurately recovers high-rank signals

Provides asymptotic consistency in estimating Tucker structure

Outperforms CP-based tensor regression models

Abstract

Large-scale neuroimaging studies have been collecting brain images of study individuals, which take the form of two-dimensional, three-dimensional, or higher dimensional arrays, also known as tensors. Addressing scientific questions arising from such data demands new regression models that take multidimensional arrays as covariates. Simply turning an image array into a long vector causes extremely high dimensionality that compromises classical regression methods, and, more seriously, destroys the inherent spatial structure of array data that possesses wealth of information. In this article, we propose a family of generalized linear tensor regression models based upon the Tucker decomposition of regression coefficient arrays. Effectively exploiting the low rank structure of tensor covariates brings the ultrahigh dimensionality to a manageable level that leads to efficient estimation. We demonstrate, both numerically that the new model could provide a sound recovery of even high rank signals, and asymptotically that the model is consistently estimating the best Tucker structure approximation to the full array model in the sense of Kullback-Liebler distance. The new model is also compared to a recently proposed tensor regression model that relies upon an alternative CANDECOMP/PARAFAC (CP) decomposition.