Intrinsic Hölder classes of density functions on Riemannian manifolds and lower bounds to convergence rates

Abstract

We consider Hölder classes of density functions on Riemannian manifolds in intrinsic perspectives. We develop a theorem that links such Hölder classes on Riemannian manifolds to those on Euclidean spaces. Using the theorem, we derive lower bounds to $L^p$-convergence rates $(1\le p\le \infty )$ for the estimation of the underlying density on a Riemannian manifold.

Introduction

Recent years have seen many data types that assume values on manifolds. Standard examples include data lying on Euclidean spheres ${\mathcal{S}}^2$, directional data on ${\mathcal{S}}^1$ and square-root compositional data. Symmetric positive-definite matrices (SPD) that arise from computer vision, signal processing, medical imaging and neuroscience, etc., are also an important type of manifold-valued data. Some examples of statistical analysis with manifold-valued data may be found in Yuan et al., 2012, Lin and Yao, 2019 and Jeon et al. (2020) among others. In this paper, we are concerned about the construction of Hölder classes of density functions on oriented Riemannian manifolds. Our definition of Hölder smoothness is based on the parallel transport of covariant derivatives respecting the geometric structure of the underlying manifold, so named as intrinsic Hölder smoothness. It is different from the conventional one based on the Euclidean Hölder smoothness via local chart (Triebel, 1987, Berenfeld and Hoffmann, 2020). We refer to the remark immediately below Theorem 2 for the difference, and also to Berger and Gostiaux (1988), e.g., for the definitions of local chart and various other basic notions of differential geometry used in this paper.

We are motivated to study the subject by the problem of deriving lower bounds to the rates of convergence for the estimation of densities on Riemannian manifolds. In deriving lower bounds for the estimation of an unknown function in a Hölder class, one typically constructs a set of basis functions by relocating and scaling a given function that belongs to the Hölder class. The lower bound to the convergence rate depends on how the scaling affects the Lipschitz constants of the basis functions. To get into some details, let $(\phi ,U)$ be a chart map of a Riemannian manifold $\mathcal{M}$ that brings points in $U\subset \mathcal{M}$ to points in ${\mathbb{R}}^d$. Consider a real-valued smooth function $\varphi$ on ${\mathbb{R}}^d$ and let ${\varrho }_c$, for a scalar $c>0$, be a function on $\mathcal{M}$ defined by ${\varrho }_c\left(x\right)=\varphi \left(c\phi \left(x\right)\right),x\in U$. In the case $\mathcal{M}={\mathbb{R}}^d$ and $\phi$ is the identity map, if the $k$th derivative ${\varphi }^{\left(k\right)}$ of $\varphi$ is Hölder continuous with an exponent $\alpha$ and a Lipschitz constant $L$, then $|{\varrho }_c^{\left(k\right)}\left(x_1\right)-{\varrho }_c^{\left(k\right)}\left(x_2\right)|\le L{c}^{k+\alpha }{\Vert x_1-x_2\Vert }^{\alpha }$. However, in the case of general Riemannian manifolds, it is not known how the scalar $c>0$ affects the Lipschitz constant of ${\varrho }_c$.

In this paper, we make precise the scaling effect and present a useful theorem that links the intrinsic Hölder classes on Riemannian manifolds to those on Euclidean spaces. We utilize the theorem to derive lower bounds to the rates of convergence for the estimation of densities on Riemannian manifolds, in general $L^p$-metric including the case $p=\infty$. The problem of deriving lower bounds in the case of density functions on ${\mathbb{R}}^d$ has a long history. Lower bounds for the distance $d(f,g)=|f\left(x_0\right)-g\left(x_0\right)|$ at a fixed point $x_0\in {\mathbb{R}}^d$ may be found in Ibragimov and Khas’minskii (1981) and Stone (1980). Those for the $L^p$-distance with $1\le p\le \infty$ were obtained by Cencov, 1962, Khas’minskii, 1979, and Bretagnolle and Huber (1979) among others. The lower bound was found to be $n^{-\beta ∕(2\beta +d)}$ when $1\le p<\infty$ and ${(n∕logn)}^{-\beta ∕(2\beta +d)}$ when $p=\infty$, where $n$ is the sample size and $\beta$ is the order of smoothness of functions in the Hölder class. The problem has been less well studied for densities on manifolds. A few examples in the literature include Hendriks, 1990, Pelletier, 2005, Kim and Park, 2013 and Berenfeld and Hoffmann (2020) with the references therein. Among them, Hendriks (1990) and Pelletier (2005) are about upper bounds to the convergence rates, i.e., they investigated the rates of convergence for the density estimators they proposed. Kim and Park (2013) gave some results on a lower bound, but in $L^2$-metric and used the property of the Lipschitz constant that is known only for the Euclidean case. Berenfeld and Hoffmann (2020) considered the case where $\mathcal{M}$ is unknown, but derived a lower bound in terms of a pointwise risk and for the classical Hölder class in a Euclidean ambient space (Triebel, 1987), which is different from our framework.

In Section 2, we present our main theorem on the intrinsic Hölder classes. In Section 3, we make use of the theorem to derive lower bound results in general $L^p$-metric including the case $p=\infty$. We note that the technique used by Kim and Park (2013) is not applicable in our general setting, especially for $p=\infty$. A proof of the main theorem and various notions of differential geometry are contained in the Supplement.