Effective data reduction algorithm for topological data analysis

TL;DR

This paper introduces the Characteristic Lattice Algorithm (CLA), a preprocessing method that reduces dataset size while preserving topological features, enabling feasible and faster topological data analysis on large, high-dimensional data.

Contribution

The paper presents CLA, a novel preprocessing algorithm that maintains topological integrity during data reduction, improving the efficiency of TDA for large datasets.

Findings

CLA effectively reduces data size while preserving topology.

The stability theorem guarantees minimal barcode errors.

CLA accelerates TDA computations on high-dimensional data.

Abstract

One of the most interesting tools that have recently entered the data science toolbox is topological data analysis (TDA). With the explosion of available data sizes and dimensions, identifying and extracting the underlying structure of a given dataset is a fundamental challenge in data science, and TDA provides a methodology for analyzing the shape of a dataset using tools and prospects from algebraic topology. However, the computational complexity makes it quickly infeasible to process large datasets, especially those with high dimensions. Here, we introduce a preprocessing strategy called the Characteristic Lattice Algorithm (CLA), which allows users to reduce the size of a given dataset as desired while maintaining geometric and topological features in order to make the computation of TDA feasible or to shorten its computation time. In addition, we derive a stability theorem and an upper bound of the barcode errors for CLA based on the bottleneck distance.