Integral Probability Metrics on submanifolds: interpolation inequalities and optimal inference

TL;DR

This paper establishes interpolation inequalities between H"older IPMs on submanifolds with densities, providing a new tool for high-dimensional density estimation and inference with optimal convergence rates.

Contribution

It introduces novel interpolation inequalities for H"older IPMs on submanifolds, enabling optimal density estimation across multiple metrics simultaneously.

Findings

Derived inequalities relate different H"older IPMs with smoothness parameters.

Applied inequalities to construct density estimators with optimal rates.

Enabled simultaneous optimal inference for a range of metrics.

Abstract

We study interpolation inequalities between H\"older 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 H\"older IPMs $d_{\mathcal{H}^\gamma_1}$ of smoothness $\gamma\geq 1$ and $d_{\mathcal{H}^\eta_1}$ 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 $d_{\mathcal{H}_1^\gamma}$, $\gamma \in [1,\infty)$ simultaneously.