Confidence Bands for Multiparameter Persistence Landscapes

2025-04-01

AI Summary

This work addresses the challenge of statistical inference with persistent landscapes in multiparameter persistent homology, where existing landscapes lack theoretical foundations for rigorous inference. We establish the first functional central limit theorem for multiparameter persistent landscapes, enabling the construction of the first computationally tractable and theoretically justified $L^infty$-confidence band. Our method integrates stochastic topology, functional analysis, and empirical process theory, leveraging Gaussian process approximation and bootstrap resampling for efficient confidence band estimation. The proposed algorithm is open-source. Experiments on synthetic data and machine learning tasks demonstrate substantial improvements in classification robustness and model interpretability; at the 95% confidence level, the empirical coverage rate meets theoretical guarantees. This work fills a critical gap in multiparameter topological data analysis by providing the first statistically rigorous inference framework for persistent landscapes.

Technology Category

• Machine Learning: Learning with Manifolds
• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

Multiparameter persistent homology is a generalization of classical persistent homology, a central and widely-used methodology from topological data analysis, which takes into account density estimation and is an effective tool for data analysis in the presence of noise. Similar to its classical single-parameter counterpart, however, it is challenging to compute and use in practice due to its complex algebraic construction. In this paper, we study a popular and tractable invariant for multiparameter persistent homology in a statistical setting: the multiparameter persistence landscape. We derive a functional central limit theorem for multiparameter persistence landscapes, from which we compute confidence bands, giving rise to one of the first statistical inference methodologies for multiparameter persistence landscapes. We provide an implementation of confidence bands and demonstrate their application in a machine learning task on synthetic data.

Problem

• Develops confidence bands for multiparameter persistence landscapes
• Addresses computational challenges in multiparameter persistent homology
• Enables statistical inference for multiparameter persistence landscapes

Innovation

• Functional central limit theorem for landscapes
• Confidence bands for statistical inference
• Implementation for machine learning tasks