Persistent homology for low-complexity models

TL;DR

This paper demonstrates how randomized dimension reduction techniques, based on Gaussian width and doubling dimension, can efficiently simplify persistent homology computations by leveraging the data's intrinsic structure.

Contribution

It introduces a method to apply dimension reduction in persistent homology using Gaussian width, connecting it to the doubling dimension for efficient data analysis.

Findings

Dimension reduction based on Gaussian width is efficient and structure-aware.

Intrinsic data dimensions can replace ambient dimension in persistent homology.

The approach relates to compressed sensing and improves computational complexity.

Abstract

We show that recent results on randomized dimension reduction schemes that exploit structural properties of data can be applied in the context of persistent homology. In the spirit of compressed sensing, the dimension reduction is determined by the Gaussian width of a structure associated to the data set, rather than its size, and such a reduction can be computed efficiently. We further relate the Gaussian width to the doubling dimension of a finite metric space, which appears in the study of the complexity of other methods for approximating persistent homology. We can therefore literally replace the ambient dimension by an intrinsic notion of dimension related to the structure of the data.