On consistent estimation of dimension values

TL;DR

This paper investigates the statistical consistency of estimators for the dimension of compact sets in Euclidean space, focusing on Minkowski, correlation, and pointwise dimensions, with theoretical proofs and empirical validation.

Contribution

It provides the first rigorous proof of consistency for estimators of multiple dimension notions and explores the polynomial volume function case, supporting the manifold hypothesis.

Findings

Proved consistency of estimators for Minkowski, correlation, and pointwise dimensions.

Developed an instrumental estimator based on empirical volume functions.

Empirical results support theoretical findings and practical applicability.

Abstract

The problem of estimating, from a random sample of points, the dimension of a compact subset $S$ of the Euclidean space is considered. The emphasis is put on consistency results in the statistical sense. That is, statements of convergence to the true dimension value when the sample size grows to infinity. Among the many available definitions of dimension, we have focused (on the grounds of its statistical tractability) on three notions: the Minkowski dimension, the correlation dimension and the, perhaps less popular, concept of pointwise dimension. We prove the statistical consistency of some natural estimators of these quantities. Our proofs partially rely on the use of an instrumental estimator formulated in terms of the empirical volume function $V_n(r)$, defined as the Lebesgue measure of the set of points whose distance to the sample is at most $r$. In particular, we explore the case in which the true volume function $V(r)$ of the target set $S$ is a polynomial on some interval starting at zero. An empirical study is also included. Our study aims to provide some theoretical support, and some practical insights, for the problem of deciding whether or not the set $S$ has a dimension smaller than that of the ambient space. This is a major statistical motivation of the dimension studies, in connection with the so-called ``Manifold Hypothesis''.