论文标题
单索引模型的条件回归
Conditional regression for single-index models
论文作者
论文摘要
单个索引模型是用于内在回归的统计模型，其中假定响应取决于预测变量的单个但未知的线性组合，从而使回归函数表示为$ \ Mathbb {e} [y | x] = f（\ langle v，x \ rangle）$对于某些未知\ emph {index} vector $ v $和\ emph {link}函数$ f $。条件方法通过平均$ x $以$ y $为条件的$ x $提供了一种简单有效的方法来估算$ v $，但取决于其最佳选择未知的参数，并且不提供$ f $的概括。在本文中，我们提出了一种在明确的参数表征下以$ \ sqrt {n} $速率收敛的新条件方法。此外，我们证明，当组合到任何$ \ sqrt {n} $ - 融合索引估算器时，多项式分区估计值达到了Hölder函数回归的$ 1 $二维最小值率。总体而言，这产生了一个缩小尺寸的估计器，并在准线性时间内达到统计最佳性的单个单个模型的回归。
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 Hölder 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.
