The Statistical Cost of Finding Smooth Coordinates

Being smooth in the right coordinates is not the same as knowing them. We study which frame differences matter for approximation and how unknown coordinates affect function-estimation risk. Our geometric analysis shows that a positive-dimensional frame space can be covered by a single representation neighborhood at coarse resolution. For a fixed two-block matrix-dilation anisotropic Besov profile with rough dimension $k$ and inverse-smoothness gap $\Delta$, let $m=k(d-k)$. At resolution $r$ and a fixed distortion threshold $c>0$, the representation entropy (log covering number) is zero when $\Delta \log (1/r)\le c/2$ and equals $m\Delta \log (1/r)+O_c(m)$ when $\Delta \log (1/r)\ge c$, with constants independent of $k$ and $d$. We derive this transition from an exact finite-resolution metric formula; rotation transfer and explicit witnesses connect the metric to approximation and separation in function space. In a separate regression model with standard Gaussian design and independent Gaussian noise of fixed variance, consider $f(x)=g(W^{\top }x)$ with unknown $W\in \mathrm{St}(d,k)$ and $g$ in the unit $B_{2,q}^s({\mathbb{R}}^k)$-ball, $s>k/2$, $1\le q\le \infty$. We prove matching local entropy bounds and the minimax rate ${\mathcal{R}}_n^{\star }\asymp n^{-2s/(2s+k)}+\frac{m}{n}\log (1+\frac{n}{m})$ for $d>k$ and $m=o(n)$, with constants uniform in the ambient dimension. The first term is the known-subspace estimation rate; the second is the representation cost, whose logarithm is unavoidable when that term dominates. Unknown coordinates can therefore be the leading contribution to function-estimation risk as $d$ grows.