On the optimality of sliced inverse regression in high dimensions (Q2656585)

While the main objective of this paper is to understand the fundamental limits of the sparse SIR problem from a decision-theoretic point of view, its main contribution is the determination of the minimax rate for estimating the central space. Such an investigation is not only interesting in its own right, but will also provide insights for other SDR algorithms developed for high-dimensional problems. More precisely, the authors assume an SIR model in the situation when the dimension \(p\) of the data is much larger than the sample size \(n\). After summarizing main results from the literature needed for further developments, they concentrate on estimating the minimax rate when estimating the central space of the model \[ y=f\big(\boldsymbol{\beta}_1^{\prime}\mathbf{x}, \boldsymbol{\beta}_2^{\prime}\mathbf{x},\dots,\boldsymbol{\beta}_d^{\prime}\mathbf{x},\varepsilon). \] To that purpose the authors first specify a quite large class of models, and over this class, they find the rate at which an aggregated estimator based on the SIR procedure converges. Optimality of this rate is discussed for both simple and multiple index model when the central dimension \(d\) is fixed together with the smallest nonzero eigenvalue of \(\mathrm{var}\big(E(\mathbf{x}|y)\big)\). Couple of oversimplified (w.r.t. the complexity of the theory) are presented.