LUM\'AWIG: An Efficient Algorithm for Dimension Zero Bottleneck Distance Computation in Topological Data Analysis

TL;DR

LUM'AWIG is a new, efficient algorithm for computing dimension zero bottleneck distance in topological data analysis, significantly faster and more accurate than previous methods, enabling practical applications in data classification.

Contribution

The paper introduces LUM'AWIG, an innovative algorithm that reduces computational complexity and improves accuracy in calculating dimension zero bottleneck distance.

Findings

LUM'AWIG runs with linear empirical complexity.

It provides sharper approximations than existing algorithms.

The algorithm enables practical use of bottleneck distance in real-world data analysis.

Abstract

Stability of persistence diagrams under slight perturbations is a key characteristic behind the validity and growing popularity of topological data analysis in exploring real-world data. Central to this stability is the use of Bottleneck distance which entails matching points between diagrams. Use of this metric in practical studies has, however, been few and sparingly because of the computational obstruction, especially in dimension zero where the computational cost explodes with the growth of data size. We present LUM\'AWIG, a novel efficient algorithm to compute dimension zero bottleneck distance between two persistent diagrams which runs significantly faster and provides significantly sharper approximates with respect to the output of the original algorithm than any other available algorithm. We bypass the overwhelming matching problem in previous implementations of the bottleneck distance, and prove that the zero dimensional bottleneck distance can be recovered from a very small number of matching cases. We show that LUM\'AWIG generally enjoys linear complexity as shown by empirical tests. We also present an application that leverages dimension zero persistence diagrams and the bottleneck distance to produce features for classification tasks.