[Paper Review] Robust Topological Inference: Distance To a Measure and Kernel Distance

Frédéric Chazal, Brittany Terese Fasy+4 more|arXiv (Cornell University)|Dec 22, 2014

Topological and Geometric Data AnalysisComputer Science41 references105 citations

TL;DR

This paper proposes robust topological inference using the distance-to-a-measure (DTM) and kernel distance (KD) to overcome the sensitivity of standard persistent homology to noise and outliers. It establishes the asymptotic normality of the DTM squared and develops bootstrap-based confidence bands, enabling statistical inference on topological features with rigorous error control.

ABSTRACT

Let P be a distribution with support S. The salient features of S can be quantified with persistent homology, which summarizes topological features of the sublevel sets of the distance function (the distance of any point x to S). Given a sample from P we can infer the persistent homology using an empirical version of the distance function. However, the empirical distance function is highly non-robust to noise and outliers. Even one outlier is deadly. The distance-to-a-measure (DTM), introduced by Chazal et al. (2011), and the kernel distance, introduced by Phillips et al. (2014), are smooth functions that provide useful topological information but are robust to noise and outliers. Chazal et al. (2014) derived concentration bounds for DTM. Building on these results, we derive limiting distributions and confidence sets, and we propose a method for choosing tuning parameters.

Motivation & Objective

• To address the lack of statistical robustness in standard persistent homology, which breaks down under noise and outliers.
• To develop a statistically valid framework for topological inference using the distance-to-a-measure (DTM) and kernel distance (KD) as robust alternatives to the empirical distance function.
• To derive limiting distributions and confidence sets for the DTM to enable statistical inference on topological features.
• To propose a data-driven method for tuning parameter selection in DTM and KD, improving practical applicability.

Proposed method

• Uses the distance-to-a-measure (DTM) as a robust alternative to the empirical distance function, defined as the L2 distance to a probability measure with a mass constraint.
• Establishes the asymptotic normality of √n(δ̂²(x) − δ²(x)) for the DTM, where δ is the true DTM and δ̂ is its empirical estimator.
• Applies the bootstrap to construct asymptotically valid confidence bands for the DTM, enabling identification of topological features that are statistically significant over noise.
• Compares DTM with kernel density estimation (KDE) in persistent homology, showing DTM's superior robustness to high-density structures and outliers.
• Uses boundary correction and data sharpening techniques to improve performance in finite-sample settings.
• Proposes a method for tuning parameter selection based on the bootstrap and stability of topological features.

Experimental results

Research questions

• RQ1 Can the DTM provide a statistically valid, robust alternative to the empirical distance function for persistent homology under contamination and noise?
• RQ2 What is the limiting distribution of the DTM estimator, and can it be used to construct confidence sets for topological features?
• RQ3 How can the bootstrap be used to generate asymptotically valid confidence bands for the DTM to distinguish topological signal from noise?
• RQ4 How does the DTM compare to kernel density estimation in persistent homology, especially in high-density or outlier-prone settings?
• RQ5 What is the impact of tuning parameters on the robustness and accuracy of topological inference using DTM and KD?

Key findings

• The quantity √n(δ̂²(x) − δ²(x)) converges in distribution to a Gaussian process, establishing the asymptotic normality of the DTM estimator.
• The bootstrap provides asymptotically valid confidence bands for the DTM, enabling statistical inference on the significance of topological features.
• The DTM outperforms kernel density estimation in persistent homology, particularly in scenarios with high-density structures and outliers, as demonstrated in the Voronoi foam model experiments.
• In the three-Voronoi model comparison, the DTM persistence diagram correctly identified one connected component and eight voids as significant, while KDE features were obscured by noise.
• The proposed bootstrap-based method successfully separates topological signal from noise, as confirmed by confidence bands in the persistence diagrams.
• The paper provides a principled framework for tuning parameter selection in DTM and KD, enhancing reproducibility and robustness in topological data analysis.