Convergence rate of estimated sufficient predictors in CP-form NTSDR

Determine the precise convergence rate of the estimated sufficient predictors produced by the CP-form nonlinear tensor sufficient dimension reduction (NTSDR) method. Specifically, under Assumptions 2.1 (uniqueness of singular vectors), 3.2 (integrability of features), 3.3 (range condition), and 6.1 (smoothness linking Σ_FF and Σ_FY), and using the ridge-regularized regression operator \(\hat{R}_{FY} = (\hat{\Sigma}_{FF} + \epsilon_n I)^{-1} \hat{\Sigma}_{FY}\), establish whether the predictors—i.e., the empirical rank-one sufficient functions returned by the CP envelope extraction—converge at the rate \(\epsilon_n^{\beta} + \epsilon_n^{-1} n^{-1/2}\) in the \(\mathcal{H}_U \otimes \mathcal{H}_V\) norm, thereby confirming or refuting the conjectured order.

Background

In Section 6, the authors establish operator-level convergence rates for the tensor regression operator $R_{FY}$ estimating the stretched-out NTSDR structure, showing that $\|\hat{R}_{FY} - R_{FY}\|$ converges at order $\epsilon_n^{\beta} + \epsilon_n n^{-1/2}$ under a smoothness assumption linking $\Sigma_{FF}$ and $\Sigma_{FY}$.

They note that the ultimate objects of interest are the sufficient predictors obtained via CP-form envelope extraction (rank-one components in $\mathcal{H}_U \otimes \mathcal{H}_V$), and explicitly conjecture that these predictors achieve a convergence rate of order $\epsilon_n^{\beta} + \epsilon_n^{-1} n^{-1/2}$. The paper proves consistency but leaves the precise rate analysis as future work, making the rate determination an explicit open problem.