Denoising data reduction algorithm for Topological Data Analysis

2026-03-31

AI Summary

This study addresses the challenges posed by large-scale noisy data in topological data analysis, where high computational complexity and spurious topological features often obscure underlying structures. To overcome these limitations, the authors propose the Refined Cell Lattice Algorithm (RCLA), which uniquely integrates dimensionality reduction, denoising, and data sparsification into a single unified step, effectively filtering noise while preserving essential topological information. The method incorporates an automatic parameter selection mechanism based on local neighbor statistics and provides theoretical stability guarantees under a Poisson noise model. Experimental results demonstrate that RCLA outperforms existing approaches across multiple benchmark datasets and significantly enhances both accuracy and efficiency in extracting topological features for 3D shape classification tasks.

Technology Category

Machine Learning: Learning with ManifoldsComputer Vision: Other Foundations of Computer VisionData Mining & Knowledge Management: Anomaly/Outlier Detection

Application Category

Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphsWeb Mining and Content Analysis: Web data quality in the era of algorithmically-generated contentResponsible Web: Data and user privacy-enhancing technologies for the Web

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.

Problem

• Topological Data Analysis
• Persistent Homology
• Denoising
• Data Reduction
• Noisy Data

Innovation

• Topological Data Analysis
• Persistent Homology
• Denoising
• Data Reduction
• Stability Theorem