A Wiener Process Perspective on Local Intrinsic Dimension Estimation Methods

TL;DR

This paper analyzes recent parametric local intrinsic dimension estimation methods through the lens of Wiener processes, revealing their behavior and errors when assumptions are violated, especially in high-dimensional data.

Contribution

It provides an extended mathematical framework for understanding parametric LID methods using Wiener processes and examines their robustness beyond ideal assumptions.

Findings

Error analysis as a function of data density

Behavior of methods under assumption violations

Mathematical description of LID estimation errors

Abstract

Local intrinsic dimension (LID) estimation methods have received a lot of attention in recent years thanks to the progress in deep neural networks and generative modeling. In opposition to old non-parametric methods, new methods use generative models to approximate diffused dataset density to scale the methods to high-dimensional datasets (e.g. images). In this paper, we investigate the recent state-of-the-art parametric LID estimation methods from the perspective of the Wiener process. We explore how these methods behave when their assumptions are not met. We give an extended mathematical description of those methods and their error as a function of the probability density of the data.