# Differentially Private Topological Data Analysis

arXiv: https://arxiv.org/abs/2305.03609 · HTML: https://mlsanity.com/paper/2305.03609/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 59 references that are in the map.
- First posted on arXiv 5 May 2023.

## Builds on (most cited first, 18 of 18)

- [Deep Learning with Differential Privacy](https://mlsanity.com/paper/1607.00133/) (2016)
- [Concentrated Differential Privacy: Simplifications, Extensions, and Lower Bounds](https://mlsanity.com/paper/1605.02065/) (2016)
- [Connectivity-Optimized Representation Learning via Persistent Homology](https://mlsanity.com/paper/1906.09003/) (2019)
- [Faster Differentially Private Samplers via R'enyi Divergence Analysis of Discretized Langevin MCMC](https://mlsanity.com/paper/2010.14658/) (2020)
- [Robust Topological Inference: Distance To a Measure and Kernel Distance](https://mlsanity.com/paper/1412.7197/) (2014)
- [Uncovering the Topology of Time-Varying fMRI Data using Cubical Persistence](https://mlsanity.com/paper/2006.07882/) (2020)
- [Privacy-preserving parametric inference: a case for robust statistics](https://mlsanity.com/paper/1911.10167/) (2019)
- [DTM-based Filtrations](https://mlsanity.com/paper/1811.04757/) (2018)
- [Benefits and Pitfalls of the Exponential Mechanism with Applications to Hilbert Spaces and Functional PCA](https://mlsanity.com/paper/1901.10864/) (2019)
- [Topological Data Analysis for Arrhythmia Detection through Modular Neural Networks](https://mlsanity.com/paper/1906.05795/) (2019)
- [KNG: The K-Norm Gradient Mechanism](https://mlsanity.com/paper/1905.09436/) (2019)
- [Exact Privacy Guarantees for Markov Chain Implementations of the Exponential Mechanism with Artificial Atoms](https://mlsanity.com/paper/2204.01132/) (2022)
- [Privacy-Aware Rejection Sampling](https://mlsanity.com/paper/2108.00965/) (2021)
- [Rates of convergence for robust geometric inference](https://mlsanity.com/paper/1505.07602/) (2015)
- [Sharing Social Network Data: Differentially Private Estimation of Exponential-Family Random Graph Models](https://mlsanity.com/paper/1511.02930/) (2015)
- [A Differentially Private Framework for Deep Learning with Convexified Loss Functions](https://mlsanity.com/paper/2204.01049/) (2022)
- [Discovering Distribution Shifts using Latent Space Representations](https://mlsanity.com/paper/2202.02339/) (2022)
- [Differentially Private Kolmogorov-Smirnov-Type Tests](https://mlsanity.com/paper/2208.06236/) (2022)

## Built on by (most cited first, 1 of 1)

- [PTOPOFL: Privacy-Preserving Personalised Federated Learning via Persistent Homology](https://mlsanity.com/paper/2603.04323/) (2026)

## Abstract (arXiv)

This paper is the first to attempt differentially private (DP) topological data analysis (TDA), producing near-optimal private persistence diagrams. We analyze the sensitivity of persistence diagrams in terms of the bottleneck distance, and we show that the commonly used vCech complex has sensitivity that does not decrease as the sample size n increases. This makes it challenging for the persistence diagrams of vCech complexes to be privatized. As an alternative, we show that the persistence diagram obtained by the L1-distance to measure (DTM) has sensitivity O(1/n). Based on the sensitivity analysis, we propose using the exponential mechanism whose utility function is defined in terms of the bottleneck distance of the L1-DTM persistence diagrams. We also derive upper and lower bounds of the accuracy of our privacy mechanism; the obtained bounds indicate that the privacy error of our mechanism is near-optimal. We demonstrate the performance of our privatized persistence diagrams through simulations as well as on a real dataset tracking human movement.