Fast Approximation of Persistence Diagrams with Guarantees

TL;DR

This paper introduces a new hierarchical approximation algorithm for persistence diagrams that guarantees a user-controlled error bound, significantly speeding up computations while maintaining accuracy for scalar field analysis.

Contribution

It presents a novel approximation method with theoretical error guarantees based on hierarchical data representation and local simplifications, improving speed and accuracy over existing approaches.

Findings

Achieves up to 48% faster runtime on large datasets.

Provides a controlled approximation error with a 5% relative Bottleneck distance.

Produces more accurate outputs, 5 times better in L2-Wasserstein distance than naive methods.

Abstract

This paper presents an algorithm for the efficient approximation of the saddle-extremum persistence diagram of a scalar field. Vidal et al. introduced recently a fast algorithm for such an approximation (by interrupting a progressive computation framework). However, no theoretical guarantee was provided regarding its approximation quality. In this work, we revisit the progressive framework of Vidal et al. and we introduce in contrast a novel approximation algorithm, with a user controlled approximation error, specifically, on the Bottleneck distance to the exact solution. Our approach is based on a hierarchical representation of the input data, and relies on local simplifications of the scalar field to accelerate the computation, while maintaining a controlled bound on the output error. The locality of our approach enables further speedups thanks to shared memory parallelism. Experiments conducted on real life datasets show that for a mild error tolerance (5% relative Bottleneck distance), our approach improves runtime performance by 18% on average (and up to 48% on large, noisy datasets) in comparison to standard, exact, publicly available implementations. In addition to the strong guarantees on its approximation error, we show that our algorithm also provides in practice outputs which are on average 5 times more accurate (in terms of the L2-Wasserstein distance) than a naive approximation baseline. We illustrate the utility of our approach for interactive data exploration and we document visualization strategies for conveying the uncertainty related to our approximations.