The single-index model is a statistical model for intrinsic regression where responses are assumed to depend on a single yet unknown linear combination of the predictors, allowing to express the regression function as $ \mathbb{E} [ Y | X ] = f ( \langle v , X \rangle ) $ for some unknown \emph{index} vector $v$ and \emph{link} function $f$. Conditional methods provide a simple and effective approach to estimate $v$ by averaging moments of $X$ conditioned on $Y$, but depend on parameters whose optimal choice is unknown and do not provide generalization bounds on $f$. In this paper we propose a new conditional method converging at $\sqrt{n}$ rate under an explicit parameter characterization. Moreover, we prove that polynomial partitioning estimates achieve the $1$-dimensional min-max rate for regression of H\"older functions when combined to any $\sqrt{n}$-convergent index estimator. Overall this yields an estimator for dimension reduction and regression of single-index models that attains statistical optimality in quasilinear time.
翻译:单指数模型是内在回归的统计模型, 假设反应取决于预测器单一但未知的线性组合, 从而可以将回归函数表现为$ $ \ mathbb{E} [ Y ⁇ X] = f (\ langle v, X \ rangle) $ 用于某些未知的\ emph{ index} 矢量 $v$ 和\ emph{link} 函数 f$ 。 有条件方法提供了一种简单而有效的方法,通过平均以美元为条件的平均时间以美元为条件估算美元, 但它取决于那些最优选择未知的参数, 并且不以美元为通用约束 。 在本文件中, 我们提议在明确的参数定性下以$\ sqrt{n} = $ = f美元 表示新的有条件方法, 以$\ sqrng =$ 为条件, 新的条件方法在明确的参数描述下, $\ sqrational- restime aminational deminational deminational asionalizational 和 simplainmentalizations