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 $d_0$ ([Tok11]). When the ambient dimension $d$ does not vary with $n$, the minimax rate remains of the order $n^{-\beta/(2\beta +d_0)}$, where $\beta$ denotes the smoothness of $f$. 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 [Tok11] and [CR24], 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：本文探讨了贝叶斯非参数回归与密度估计中的降维问题。具体而言，我们关注在单位球 \(\mathbb{R}^d\) 上估计一个函数参数 \(f\)，该参数仅依赖于 \(\mathbb{R}^d\) 的一个 \(d_0\) 维子空间，其中 \(d_0 < d\)。众所周知，函数空间上经过缩放的高斯过程先验能够实现平滑性自适应，并以近乎极小极大最优的速率达到后验收缩。此外，配备子空间投影的该方法层次化扩展也能自适应于本征维度 \(d_0\)（[Tok11]）。当环境维度 \(d\) 不随 \(n\) 变化时，极小化极大速率仍保持在 \(n^{-\beta/(2\beta +d_0)}\) 的量级，其中 \(\beta\) 表示 \(f\) 的平滑度。然而，这忽略了可能随 \(d\) 增长而变得过大的乘法常数。收缩率与环境维度之间的依赖关系尚未得到充分探索，本文提供了初步见解：我们令维度 \(d\) 随 \(n\) 增长，并通过结合 [Tok11] 与 [CR24] 的论证，推导出仍能导致后验以极小化极大速率相合时 \(d\) 的增长率。随后讨论了该增长率的优性。此外，我们提出了一组假设条件，在已知 \(d_0\) 的前提下，\(f\) 的相合估计能够正确估计子空间投影。