June 2024 Tensor quantile regression with low-rank tensor train estimation
Zihuan Liu, Cheuk Yin Lee, Heping Zhang
Author Affiliations +
Ann. Appl. Stat. 18(2): 1294-1318 (June 2024). DOI: 10.1214/23-AOAS1835


Neuroimaging studies often involve predicting a scalar outcome from an array of images collectively called tensor. The use of magnetic resonance imaging (MRI) provides a unique opportunity to investigate the structures of the brain. To learn the association between MRI images and human intelligence, we formulate a scalar-on-image quantile regression framework. However, the high dimensionality of the tensor makes estimating the coefficients for all elements computationally challenging. To address this, we propose a low-rank coefficient array estimation algorithm, based on tensor train (TT) decomposition, which we demonstrate can effectively reduce the dimensionality of the coefficient tensor to a feasible level while ensuring adequacy to the data. Our method is more stable and efficient compared to the commonly used canonic polyadic rank approximation-based method. We also propose a generalized lasso penalty on the coefficient tensor to take advantage of the spatial structure of the tensor, further reduce the dimensionality of the coefficient tensor, and improve the interpretability of the model. The consistency and asymptotic normality of the TT estimator are established under some mild conditions on the covariates and random errors in quantile regression models. The rate of convergence is obtained with regularization under the total variation penalty. Extensive numerical studies, including both synthetic and real MRI imaging data, are conducted to examine the empirical performance of the proposed method and its competitors.

Funding Statement

This research is supported in part by U.S. National Institutes of Health (R01HG010171 and R01MH116527) and National Science Foundation (DMS-2112711). Data were provided by the Human Connectome Project (1U54MH091657) funded by the 16 NIH Institutes and Centers that Support the NIH Blueprint for Neuroscience Research, and by the McDonnell Center for Systems Neuroscience at Washington University.


The authors thank the Editor, Professor Yufeng Liu, and two reviewers for many constructive comments and helpful suggestions, which greatly improved the article. We thank the Yale Center for Research Computing for guidance and use of the research computing infrastructure.


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Zihuan Liu. Cheuk Yin Lee. Heping Zhang. "Tensor quantile regression with low-rank tensor train estimation." Ann. Appl. Stat. 18 (2) 1294 - 1318, June 2024. https://doi.org/10.1214/23-AOAS1835


Received: 1 April 2023; Revised: 1 September 2023; Published: June 2024
First available in Project Euclid: 5 April 2024

Digital Object Identifier: 10.1214/23-AOAS1835

Keywords: Conditional quantile , tensor regression , tensor train (TT) decomposition , Total variation

Rights: Copyright © 2024 Institute of Mathematical Statistics


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Vol.18 • No. 2 • June 2024
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