Several statistical models for regression of a function $F$ on $\mathbb{R}^d$ without the statistical and computational curse of dimensionality exist, for example by imposing and exploiting geometric assumptions on the distribution of the data (e.g. that its support is low-dimensional), or strong smoothness assumptions on $F$, or a special structure $F$. Among the latter, compositional models assume $F=f\circ g$ with $g$ mapping to $\mathbb{R}^r$ with $r\ll d$, have been studied, and include classical single- and multi-index models and recent works on neural networks. While the case where $g$ is linear is rather well-understood, much less is known when $g$ is nonlinear, and in particular for which $g$'s the curse of dimensionality in estimating $F$, or both $f$ and $g$, may be circumvented. In this paper, we consider a model $F(X):=f(\Pi_\gamma X) $ where $\Pi_\gamma:\mathbb{R}^d\to[0,\rm{len}_\gamma]$ is the closest-point projection onto the parameter of a regular curve $\gamma: [0,\rm{len}_\gamma]\to\mathbb{R}^d$ and $f:[0,\rm{len}_\gamma]\to\mathbb{R}^1$. The input data $X$ is not low-dimensional, far from $\gamma$, conditioned on $\Pi_\gamma(X)$ being well-defined. The distribution of the data, $\gamma$ and $f$ are unknown. This model is a natural nonlinear generalization of the single-index model, which corresponds to $\gamma$ being a line. We propose a nonparametric estimator, based on conditional regression, and show that under suitable assumptions, the strongest of which being that $f$ is coarsely monotone, it can achieve the $one$-$dimensional$ optimal min-max rate for non-parametric regression, up to the level of noise in the observations, and be constructed in time $\mathcal{O}(d^2n\log n)$. All the constants in the learning bounds, in the minimal number of samples required for our bounds to hold, and in the computational complexity are at most low-order polynomials in $d$.

翻译：存在若干用于回归函数$F$在$\mathbb{R}^d$上的统计模型，这些模型避免了统计与计算上的维度灾难，例如通过对数据分布施加并利用几何假设（如其支撑集是低维的），或对$F$施加强光滑性假设，或要求$F$具有特殊结构。在后一类模型中，组合模型假设$F=f\circ g$，其中$g$映射到$\mathbb{R}^r$且$r\ll d$，已被广泛研究，包括经典的单索引与多索引模型以及近期关于神经网络的工作。尽管当$g$为线性时的情况已较为明确，但当$g$为非线性时，尤其是对于哪些$g$可以规避估计$F$或同时估计$f$和$g$时的维度灾难，所知甚少。本文考虑模型$F(X):=f(\Pi_\gamma X)$，其中$\Pi_\gamma:\mathbb{R}^d\to[0,\rm{len}_\gamma]$是到正则曲线$\gamma: [0,\rm{len}_\gamma]\to\mathbb{R}^d$参数上的最近点投影，且$f:[0,\rm{len}_\gamma]\to\mathbb{R}^1$。输入数据$X$并非低维，且远离$\gamma$，条件是其投影$\Pi_\gamma(X)$定义良好。数据分布、$\gamma$和$f$均未知。该模型是单索引模型（对应$\gamma$为直线情形）的自然非线性推广。我们提出一种基于条件回归的非参数估计器，并证明在适当假设下（其中最强假设为$f$是粗单调的），该估计器能够达到非参数回归的一维最优极小极大速率（直至观测噪声水平），且可在$\mathcal{O}(d^2n\log n)$时间内构建。学习界中的所有常数、保证界成立所需的最小样本数以及计算复杂度中的常数至多为$d$的低阶多项式。