arXiv2025
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 $d^*$-dimensional subspace of $\mathbb{R}^d$, with $d^* < d$. 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 $d^*$ (\cite{Tokdar2011DimensionAdapt}). When the ambient dimension $d$ does not vary with $n$, the minimax rate remains of the order $n^{-β/(2β+d^*)}$, where $β$ denotes the smoothnes 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 \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 $d^*$ is known.