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

Abstract

We consider nonparametric regression with random design and non-Gaussian noise, where the regression function has the multi-index form $F(x)=f(Θx)$ with an unknown link $f:\RR^m\to\RR$ in an anisotropic Hölder class and an unknown matrix $Θ$ 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 $L^{1/(2γ+1)}(n^{-1}\ln n)^{γ/(2γ+1)}$, where $γ$ is the harmonic mean of the smoothness indices of the link divided by $m$. For $m<d$ this extends a Gaussian white noise result to regression. The main new case is $m=d$, where no dimension reduction takes place: the logarithmic factor still appears as soon as the link is anisotropic and the orientation of its axes is unknown. In contrast, when the link is isotropic with smoothness index at most one, the class is rotation invariant up to a constant and the classical rate without logarithm is attained. The logarithmic loss is therefore caused by the unknown orientation of the directions in which the function is least smooth