On the Optimality of Functional Sliced Inverse Regression

TL;DR

This work establishes the minimax-rate optimality of FSIR-OT for estimating the functional central space $oldsymbol{S}_{Y|oldsymbol{X}}$ in functional SDR. By introducing the weak sliced stable condition (WSSC), deriving a concentration bound for the FSIR estimator of $oldsymbol{ extGamma}_e$, and proving root-$n$ consistency for the inverse-regression subspace, the authors show that, with an optimally chosen truncation level $m ewcommand{ }{n} extasymp n^{1/(oldsymbol{ extalpha}+2oldsymbol{ extbeta})}$, the FSIR-OT estimator attains the rate $O_pig(n^{-rac{2oldsymbol{ extbeta}-1}{oldsymbol{ extalpha}+2oldsymbol{ extbeta}}}ig)$. They further derive a minimax lower bound over a broad distribution class, demonstrating that FSIR-OT is minimax-optimal for central-space estimation under general $Y$, including non-discrete responses. Numerical studies and real-data analysis validate the optimality of the truncation choice and the efficiency of FSIR-OT relative to competing methods. The results bridge FSIR with classical high-dimensional SDR theory and offer new avenues for functional SDR in practice.

Abstract

In this paper, we prove that functional sliced inverse regression (FSIR) achieves the optimal (minimax) rate for estimating the central space in functional sufficient dimension reduction problems. First, we provide a concentration inequality for the FSIR estimator of the covariance of the conditional mean. Based on this inequality, we establish the root-$n$ consistency of the FSIR estimator of the image of covariance of the conditional mean. Second, we apply the most widely used truncated scheme to estimate the inverse of the covariance operator and identify the truncation parameter that ensures that FSIR can achieve the optimal minimax convergence rate for estimating the central space. Finally, we conduct simulations to demonstrate the optimal choice of truncation parameter and the estimation efficiency of FSIR. To the best of our knowledge, this is the first paper to rigorously prove the minimax optimality of FSIR in estimating the central space for multiple-index models and general $Y$ (not necessarily discrete).

Figures (9)

• Figure 1: Experiments for the optimal choice of truncation parameter $m$ with $\varepsilon\sim N(0,2)$ and $H=15$. Left: average subspace estimation error with increasing $m$ for different $n$. Right: linear trend of $\log(m^*)$ against $\log(n)$, with a slope of $0.2$ and $R^2>0.98$.
• Figure 2: Average subspace estimation error of FSIR-OT, RFSIR and FCSE for various models in the case of $\varepsilon\sim N(0,2)$ and $H=15$. The standard errors are all below $0.01$. Left: FSIR-OT with different truncation parameter $m$; Middle: RFSIR with different values of the regularization parameter $\rho$; Right: FCSE with different truncation parameter $m$.
• Figure 3: Bike sharing data
• Figure 4: Experiments for the optimal choice of truncation parameter $m$ with $\varepsilon\sim N(0,1)$ and $H=10$. Left: average subspace estimation error with increasing $m$ for different $n$. Right: linear trend of $\log(m^*)$ against $\log(n)$, with a slope of $0.183$ and $R^2>0.98$.
• Figure 5: Experiments for the optimal choice of truncation parameter $m$ with $\varepsilon\sim N(0,0.25)$ and $H=10$. Left: average subspace estimation error with increasing $m$ for different $n$. Right: linear trend of $\log(m^*)$ against $\log(n)$, with a slope of $0.2$ and $R^2>0.99$.

Theorems & Definitions (42)

• Definition 1: Weak Sliced Stable Condition
• Lemma 1
• Proposition 1
• Lemma 2
• Theorem 1
• Theorem 2
• Theorem 3
• Remark 1
• Lemma 3
• Lemma 4