Integral Probability Metrics on submanifolds: interpolation inequalities and optimal inference

We study interpolation inequalities between Hölder Integral Probability Metrics (IPMs) in the case where the measures have densities on closed submanifolds. Precisely, it is shown that if two probability measures $\mu$ and ${\mu }^{\star }$ have $\beta$-smooth densities with respect to the volume measure of some submanifolds $\mathcal{M}$ and ${\mathcal{M}}^{\star }$ respectively, then the Hölder IPMs $d_{{\mathcal{H}}_1^{\gamma }}$ of smoothness $\gamma \ge 1$ and $d_{{\mathcal{H}}_1^{\eta }}$ of smoothness $\eta >\gamma$, satisfy $d_{{\mathcal{H}}_1^{\gamma }}(\mu ,{\mu }^{\star })\lesssim d_{{\mathcal{H}}_1^{\eta }}(\mu ,{\mu }^{\star }{)}^{\frac{\beta +\gamma }{\beta +\eta }}$, up to logarithmic factors. We provide an application of this result to high-dimensional inference. These functional inequalities turn out to be a key tool for density estimation on unknown submanifold. In particular, it allows to build the first estimator attaining optimal rates of estimation for all the distances $d_{{\mathcal{H}}_1^{\gamma }}$, $\gamma \in [1,\mathrm{\infty })$ simultaneously.