Abstract

Structured linear regression provides a flexible framework for modeling vector-,

matrix-, or tensor-valued covariates, while valid and efficient inference for linear functionals

of the structured parameter remains challenging, especially under correlated designs, since

the optimal inferential direction depends not only on the ambient linear model but also on

the local geometry of the parameter space. To address these issues, we propose a general

debiasing framework for inference by covariance-weighted projection over local low-rank geometry. Based on this construction, oracle and feasible one-step estimators are developed

with asymptotic normality and valid Wald inference. In particular, the proposed method

achieves a smaller asymptotic variance than competing approaches without covariance adjustment, the resulting confidence intervals attain a local geometric minimax lower bound,

enabling more efficient statistical inference. We further specialize the general framework to

low-rank Tucker regression under mode-wise Kronecker covariance structures, where explicit

theory and feasible inference procedures are developed. Simulation studies and an application to resting-state EEG data illustrate favorable finite-sample performance.

Key words and phrases: Local low-rank geometry; Debiasing; Covariance-weighted projec- tion; Tucker tensor regression; Efficient inference 1

Information

Preprint No.SS-2026-0222
Manuscript IDSS-2026-0222
Complete AuthorsBaofang Ke, Zihao Song, Weihua Zhao, Lei Wang
Corresponding AuthorsLei Wang
Emailslwangstat@nankai.edu.cn

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Acknowledgments

The authors are grateful to the anonymous referees, associate editor, and editor for

their helpful comments. Lei Wang’s research was supported by the National Natural

Science Foundation of China (12271272). The corresponding author is Lei Wang.

Supplementary Materials

It contains proofs of Propositions, Theorems and additional simulation results.

Supplementary materials are available for download.