Reduced-Rank Tensor-on-Tensor Regression and Tensor-Variate Analysis of Variance

Area of Science:

• Statistics
• Machine Learning
• Neuroimaging

Background:

• Multivariate regression with many responses and covariates is challenging.
• Tensor-variate structures in data offer opportunities for advanced modeling.
• Classical methods may not fully exploit these complex data structures.

Purpose of the Study:

• To extend classical multivariate regression to accommodate tensor-variate structures.
• To develop a robust tensor-variate analysis of variance (TANOVA) methodology.
• To enable identification of significant interactions in complex datasets.

Main Methods:

• Imposing low-rank tensor formats on regression coefficients.
• Modeling errors using tensor-variate normal distribution with Kronecker separable covariance.
• Utilizing block-relaxation algorithms for maximum likelihood estimation.
• Deriving computational complexity and asymptotic distributions.

Main Results:

• Developed a novel regression framework for tensor-variate data.
• Formulated and applied tensor-variate analysis of variance (TANOVA).
• Successfully identified significant interactions in functional Magnetic Resonance Imaging (fMRI) data and the Labeled Faces in the Wild dataset.
• Implemented the methodology in an R package named 'totr'.

Conclusions:

• The proposed tensor-variate regression model effectively handles complex, high-dimensional data.
• TANOVA provides a powerful tool for interaction analysis in neuroimaging and computer vision.
• The 'totr' R package facilitates the application of these advanced statistical methods.