An axiomatic approach to intrinsic dimension of a dataset

TL;DR

This paper refines an axiomatic framework for understanding dataset intrinsic dimension, linking it to the curse of dimensionality, manifold structure, and noise sensitivity, while discussing computational challenges and potential solutions.

Contribution

It advances the axiomatic approach to intrinsic dimension, clarifies its relation to data properties, and addresses computational and noise-related issues.

Findings

High intrinsic dimension indicates curse of dimensionality.

Sample dimension approximates manifold dimension with high probability.

Computing intrinsic dimension is NP-complete and sensitive to noise.

Abstract

We perform a deeper analysis of an axiomatic approach to the concept of intrinsic dimension of a dataset proposed by us in the IJCNN'07 paper (arXiv:cs/0703125). The main features of our approach are that a high intrinsic dimension of a dataset reflects the presence of the curse of dimensionality (in a certain mathematically precise sense), and that dimension of a discrete i.i.d. sample of a low-dimensional manifold is, with high probability, close to that of the manifold. At the same time, the intrinsic dimension of a sample is easily corrupted by moderate high-dimensional noise (of the same amplitude as the size of the manifold) and suffers from prohibitevely high computational complexity (computing it is an $NP$-complete problem). We outline a possible way to overcome these difficulties.