Robust Topological Inference: Distance To a Measure and Kernel Distance

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.

References in corpus (6)

• Statistical topological data analysis using persistence landscapes
• Confidence sets for persistence diagrams
• Topological consistency via kernel estimation
• Geometric Inference on Kernel Density Estimates
• The structure and stability of persistence modules
• Rates of convergence for robust geometric inference

Cited by in corpus (22)

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• DTM-based Filtrations
• Uniform Manifold Approximation with Two-phase Optimization
• Noise robustness of persistent homology on greyscale images, across filtrations and signatures
• Cosmology with Persistent Homology: a Fisher Forecast
• Stabilizing the unstable output of persistent homology computations
• Random Simplicial Complexes: Models and Phenomena
• Uniform Convergence Rate of the Kernel Density Estimator Adaptive to Intrinsic Volume Dimension
• Nonparametric Clustering of Functional Data Using Pseudo-Densities
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• A Comprehensive Approach to Mode Clustering
• Statistical Inference with Local Optima
• Finding the Mode of a Kernel Density Estimate
• Statistical Inference using the Morse-Smale Complex
• Orientation-Preserving Vectorized Distance Between Curves