# Robust Topological Inference: Distance To a Measure and Kernel Distance

arXiv: https://arxiv.org/abs/1412.7197 · HTML: https://mlsanity.com/paper/1412.7197/

## Key facts

- Cited by 26 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Dec 2014.

## Built on by (most cited first, 26 of 26)

- [On the effectiveness of persistent homology](https://mlsanity.com/paper/2206.10551/) (2022)
- [Learning with Density Matrices and Random Features](https://mlsanity.com/paper/2102.04394/) (2021)
- [Stable Vectorization of Multiparameter Persistent Homology using Signed Barcodes as Measures](https://mlsanity.com/paper/2306.03801/) (2023)
- [TopP&R: Robust Support Estimation Approach for Evaluating Fidelity and Diversity in Generative Models](https://mlsanity.com/paper/2306.08013/) (2023)
- [Uniform Manifold Approximation with Two-phase Optimization](https://mlsanity.com/paper/2205.00420/) (2022)
- [Classes are not Clusters: Improving Label-based Evaluation of Dimensionality Reduction](https://mlsanity.com/paper/2308.00278/) (2023)
- [Cycle Registration in Persistent Homology with Applications in Topological Bootstrap](https://mlsanity.com/paper/2101.00698/) (2021)
- [Topological Deep Learning](https://mlsanity.com/paper/2101.05778/) (2021)
- [A Random Persistence Diagram Generator](https://mlsanity.com/paper/2104.07737/) (2021)
- [Diffeomorphic interpolation for efficient persistence-based topological optimization](https://mlsanity.com/paper/2405.18820/) (2024)
- [Inductive Global and Local Manifold Approximation and Projection](https://mlsanity.com/paper/2406.08097/) (2024)
- [AuToMATo: An Out-Of-The-Box Persistence-Based Clustering Algorithm](https://mlsanity.com/paper/2408.06958/) (2024)
- [Topological Representations of Local Explanations](https://mlsanity.com/paper/2201.02155/) (2022)
- [Learning from Topology: Cosmological Parameter Estimation from the Large-scale Structure](https://mlsanity.com/paper/2308.02636/) (2023)
- [Regularization of Mixture Models for Robust Principal Graph Learning](https://mlsanity.com/paper/2106.09035/) (2021)
- [Differentially Private Topological Data Analysis](https://mlsanity.com/paper/2305.03609/) (2023)
- [Topological data analysis of human vowels: Persistent homologies across representation spaces](https://mlsanity.com/paper/2310.06508/) (2023)
- [Beyond Means: Topological Causal Effects under Persistent-Homology Ignorability](https://mlsanity.com/paper/2603.14169/) (2026)
- [Detection of Small Holes by the Scale-Invariant Robust Density-Aware Distance (RDAD) Filtration](https://mlsanity.com/paper/2204.07821/) (2022)
- [Improved Modeling of Persistence Diagram](https://mlsanity.com/paper/2205.10907/) (2022)
- [Topological Detection of Phenomenological Bifurcations with Unreliable Kernel Densities](https://mlsanity.com/paper/2401.16563/) (2024)
- [Torsion in Persistent Homology and Neural Networks](https://mlsanity.com/paper/2506.03049/) (2025)
- [Unsupervised Learning of Density Estimates with Topological Optimization](https://mlsanity.com/paper/2512.08895/) (2025)
- [Conformal Robust Set Estimation](https://mlsanity.com/paper/2604.18441/) (2026)
- [The Topological Stability Index: A Variance-Based Measure for Persistence Barcodes](https://mlsanity.com/paper/2605.29839/) (2026)
- [The sublevel Flood bifiltration: towards scalable 2-parameter persistent homology](https://mlsanity.com/paper/2610.05441/) (2026)

## Abstract (arXiv)

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.