This work explores the dimension reduction problem for Bayesian nonparametric regression and density estimation. More precisely, we are interested in estimating a functional parameter $f$ over the unit ball in $\mathbb{R}^d$, which depends only on a $d_0$-dimensional subspace of $\mathbb{R}^d$, with $d_0 < d$.It is well-known that rescaled Gaussian process priors over the function space achieve smoothness adaptation and posterior contraction with near minimax-optimal rates. Moreover, hierarchical extensions of this approach, equipped with subspace projection, can also adapt to the intrinsic dimension $d_0$ (\cite{Tokdar2011DimensionAdapt}).When the ambient dimension $d$ does not vary with $n$, the minimax rate remains of the order $n^{-\beta/(2\beta +d_0)}$.%When $d$ does not vary with $n$, the order of the minimax rate remains the same regardless of the ambient dimension $d$. However, this is up to multiplicative constants that can become prohibitively large when $d$ grows. The dependences between the contraction rate and the ambient dimension have not been fully explored yet and this work provides a first insight: we let the dimension $d$ grow with $n$ and, by combining the arguments of \cite{Tokdar2011DimensionAdapt} and \cite{Jiang2021VariableSelection}, we derive a growth rate for $d$ that still leads to posterior consistency with minimax rate.The optimality of this growth rate is then discussed.Additionally, we provide a set of assumptions under which consistent estimation of $f$ leads to a correct estimation of the subspace projection, assuming that $d_0$ is known.

翻译：本文探讨贝叶斯非参数回归与密度估计中的降维问题。具体而言，我们关注在$\mathbb{R}^d$单位球面上估计一个仅依赖于$d_0$维子空间的函数参数$f$（其中$d_0 < d$）。众所周知，函数空间上的重标度高斯过程先验能够实现光滑性适应及后验收缩，并具备近乎极小最大最优速率。此外，该方法结合子空间投影的分层扩展还能自适应内在维度$d_0$（参见文献\cite{Tokdar2011DimensionAdapt}）。当环境维度$d$不随样本量$n$变化时，极小最大速率保持为$n^{-\beta/(2\beta +d_0)}$阶。然而，这一结论仅适用于乘法常数项——当$d$增长时这些常数可能变得异常庞大。目前收缩率与环境维度之间的依赖关系尚未得到充分研究，本文提供初步洞见：我们令维度$d$随$n$增长，通过融合\cite{Tokdar2011DimensionAdapt}与\cite{Jiang2021VariableSelection}的论证方法，推导出$d$的增长率仍能使后验收敛达到极小最大速率。随后讨论了该增长率的最优性。此外，我们在假定$d_0$已知的条件下，提出一组可保证$f$的一致估计能正确估计子空间投影的假设条件。