2024/03/06 by Odin, Elie, Bachoc, François, Lagnoux, Agnès
#FOS: Mathematics #Statistics Theory (math.ST)
paper · doi:10.48550/arxiv.2403.03540
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 ℝd, which depends only on a d^*-dimensional subspace of ℝd, 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^* (\citeTokdar2011DimensionAdapt). 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 \citeTokdar2011DimensionAdapt and \citeJiang2021VariableSelection, 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.