Efficient Inference for Low-rank Regression via Covariance-Weighted Projection

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.