# Sparse Nerves in Practice

## TL;DR

This paper introduces algorithms for efficiently approximating sparse nerves in persistent homology, enabling scalable topological data analysis for large and high-dimensional datasets.

## Contribution

It presents novel algorithms for approximate sparse nerves applicable to Dowker dissimilarities, with flexible interleaving functions, and demonstrates their computational advantages over existing methods.

## Key findings

- Algorithms reduce complexity of persistent homology calculations.
- Benchmarks show smaller simplicial complexes in high dimensions.
- Significant improvements over existing methods like SimBa in large, high-dimensional data.

## Abstract

Topological data analysis combines machine learning with methods from algebraic topology. Persistent homology, a method to characterize topological features occurring in data at multiple scales is of particular interest. A major obstacle to the wide-spread use of persistent homology is its computational complexity. In order to be able to calculate persistent homology of large datasets, a number of approximations can be applied in order to reduce its complexity. We propose algorithms for calculation of approximate sparse nerves for classes of Dowker dissimilarities including all finite Dowker dissimilarities and Dowker dissimilarities whose homology is Cech persistent homology. All other sparsification methods and software packages that we are aware of calculate persistent homology with either an additive or a multiplicative interleaving. In dowker_homology, we allow for any non-decreasing interleaving function $\alpha$. We analyze the computational complexity of the algorithms and present some benchmarks. For Euclidean data in dimensions larger than three, the sizes of simplicial complexes we create are in general smaller than the ones created by SimBa. Especially when calculating persistent homology in higher homology dimensions, the differences can become substantial.