Denoising data reduction algorithm for Topological Data Analysis

TL;DR

This paper introduces RCLA, a grid-based algorithm that reduces data size and denoises for persistent homology, with theoretical guarantees and improved performance on large, noisy datasets.

Contribution

The paper presents RCLA, a novel integrated data reduction and denoising algorithm with theoretical stability guarantees and automatic parameter selection for topological data analysis.

Findings

RCLA outperforms existing methods in denoising and data reduction.

Theoretical stability bounds are established under Poisson noise.

Experimental validation on 3D shape classification shows improved results.

Abstract

Persistent homology is a central tool in topological data analysis, but its application to large and noisy datasets is often limited by computational cost and the presence of spurious topological features. Noise not only increases data size but also obscures the underlying structure of the data. In this paper, we propose the Refined Characteristic Lattice Algorithm (RCLA), a grid-based method that integrates data reduction with threshold-based denoising in a single procedure. By incorporating a threshold parameter $k$, RCLA removes noise while preserving the essential structure of the data in a single pass. We further provide a theoretical guarantee by proving a stability theorem under a homogeneous Poisson noise model, which bounds the bottleneck distance between the persistence diagrams of the output and the underlying shape with high probability. In addition, we introduce an automatic parameter selection method based on nearest-neighbor statistics. Experimental results demonstrate that RCLA consistently outperforms existing methods, and its effectiveness is further validated on a 3D shape classification task.