Log-factor-free adaptive contraction on general Minkowski-dimensional domains

Let $D\ge 1$, let $\mathcal X\subset\mathbb R^D$ be compact, and let $N(\mathcal X,r)$ be the covering number by Euclidean balls of radius $r$. Work under an intrinsic-dimension upper-complexity condition of the form

for some $d\in(0,D]$ and $C<\infty$, together with the regression model

where $P_X$ is supported on $\mathcal X$.

This setup follows Tang et al. (2024).

What the source proves (self-contained, with notation aligned to the paper):

1. Prior class (geometry-agnostic GP): use a centered GP on $\mathcal X$ with squared-exponential kernel

and a hierarchical/empirical-Bayes prior $p_n(t)$ on bandwidth $t$. "Geometry-agnostic" means the prior construction does not take intrinsic dimension as input and does not use manifold charts/atlases or Laplace--Beltrami coordinates; it is built from ambient Euclidean data geometry.

1. General assumptions used for the adaptive-rate theorem: (A1) covering-number complexity: for some $\varrho>0$, $C_{\mathcal X}>0$, $r_0\in(0,1)$,

(A2) RKHS approximation: for some $s>0$ and constants $\epsilon_0,\nu_1,\nu_2>0$, for every $0<\epsilon<\epsilon_0$ there exists $F^\epsilon\in\mathbb H_\epsilon(\mathcal X)$ such that

(A3) prior-on-bandwidth condition: $p_n(t)$ places enough mass near $t\asymp n^{-2/(2s+\varrho)}$ and sufficiently little mass at much smaller scales (formal two-sided exponential-tail inequalities in Assumption (A3) of the paper).

1. Posterior contraction definition used in the source: for a semimetric $d_n$, a sequence $\varepsilon_n$ is a contraction rate if

in probability.

1. Main adaptive-rate conclusion under (A1)--(A3): in fixed design, the paper proves

and in random design (with bounded-signal truncation in the theorem statement) obtains the same polynomial exponent in $L_2(P_X)$ up to logarithmic factors.

1. Meaning of "minimax power" and "adaptation": the polynomial exponent is $s/(2s+\varrho)$ (equivalently $\beta/(2\beta+d)$ when writing $s=\beta$, $\varrho=d$), which is the standard nonparametric minimax exponent for $d$-dimensional smoothness-$\beta$ regression classes; "adaptive" means one prior construction attains this exponent across the paper's class of unknown intrinsic dimensions and smoothness levels, with only logarithmic overhead.