2022/09/24 by L. Betti, Betti, L., I. Chio +29
Mathematics · Physics and Astronomy · #28A75 #62R07 #Advanced Mathematical Theories and Applications #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical Dynamics and Fractals #Statistics Theory (math.ST) #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.2209.12079
openalex publication_date 2022/09/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.