Covariate-Adjusted Tensor Classification in High Dimensions

ABSTRACT

In contemporary scientific research, it is often of great interest to predict a categorical response based on a high-dimensional tensor (i.e., multi-dimensional array) and additional covariates. Motivated by applications in science and engineering, we propose a comprehensive and interpretable discriminant analysis model, called the CATCH model (short for covariate-adjusted tensor classification in high-dimensions). The CATCH model efficiently integrates the covariates and the tensor to predict the categorical outcome. It also jointly explains the complicated relationships among the covariates, the tensor predictor, and the categorical response. The tensor structure is used to achieve easy interpretation and accurate prediction. To tackle the new computational and statistical challenges arising from the intimidating tensor dimensions, we propose a penalized approach to select a subset of the tensor predictor entries that affect classification after adjustment for the covariates. An efficient algorithm is developed to take advantage of the tensor structure in the penalized estimation. Theoretical results confirm that the proposed method achieves variable selection and prediction consistency, even when the tensor dimension is much larger than the sample size. The superior performance of our method over existing methods is demonstrated in extensive simulated and real data examples. Supplementary materials for this article are available online.

Keywords:

• Discriminant analysis
• Multicategory classification
• Multi-dimensional array
• Sparsity
• Tensor classification and regression

Supplementary Materials

Detailed proofs, technical details and additional simulation studies are provided in the supplement to this article (PDF file). An R package is available at https://cran.r-project.org/web/packages/catch/index.html.

Acknowledgments

The authors are grateful to the editor, associate editor, and two referees for insightful comments that have led to significant improvements of this article. The authors thank Dr. Lexin Li for sharing the ADHD and ASD datasets, Dr. Qun Li and Dr. Dan Schonfeld for sharing their code for CMDA and DGTDA, and Dr. Elizabeth Slate for helpful discussion.

Additional information

Funding

Research for this article was supported in part by grants CCF-1617691 and DMS-1613154 from the U.S. National Science Foundation.