Pointwise minimax lower bounds under a multi-index constraint: the price of an unknown orientation

arXiv:2610.19208v1 Announce Type: new Abstract: We consider nonparametric regression with random design and non-Gaussian noise, where the regression function has the multi-index form $F(x)=f(\Theta x)$ with an unknown link $f:\RR^m\to\RR$ in an anisotropic H\"older class and an unknown matrix $\Theta$ with $m\le d$ orthonormal rows. For estimation of $F$ at a fixed point we prove that the minimax risk is bounded from below by a multiple of $L^{1/(2\gamma+1)}(n^{-1}\ln n)^{\gamma/(2\gamma+1)}$, where $\gamma$ is the harmonic mean of the smoothness indices of the link divided by $m$. For $m