Fractal dimension, approximation and data sets

TL;DR

This paper investigates fractal properties in large data sets and explores alternative dimension reduction techniques beyond PCA, using geometric measure theory tools to identify intrinsic fractal features.

Contribution

It introduces the use of discrete energy from geometric measure theory to analyze data sets where PCA fails to reveal fractal structures.

Findings

Discrete energy effectively identifies fractal features in data.

Classical PCA is limited in capturing fractal dimensions.

Proposes new methods for dimension reduction in complex data sets.

Abstract

The purpose of this paper is to study the fractal phenomena in large data sets and the associated questions of dimension reduction. We examine situations where the classical Principal Component Analysis is not effective in identifying the salient underlying fractal features of the data set. Instead, we employ the discrete energy, a technique borrowed from geometric measure theory, to limit the number of points of a given data set that lie near a $k$-dimensional hyperplane, or, more generally, near a set of a given upper Minkowski dimension. Concrete motivations stemming from naturally arising data sets are described and future directions outlined.