# Geometry-Adaptive Mechanisms for Private Synthetic Data

arXiv: https://arxiv.org/abs/2609.33363 · HTML: https://mlsanity.com/paper/2609.33363/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 16 of its 39 references that are in the map.
- First posted on arXiv 27 Sep 2026.

## Builds on (most cited first, 16 of 16)

- [Deep Learning with Differential Privacy](https://mlsanity.com/paper/1607.00133/) (2016)
- [A Universal Law of Robustness via Isoperimetry](https://mlsanity.com/paper/2105.12806/) (2021)
- [Minimax Optimal Procedures for Locally Private Estimation](https://mlsanity.com/paper/1604.02390/) (2016)
- [Winning the NIST Contest: A scalable and general approach to differentially private synthetic data](https://mlsanity.com/paper/2108.04978/) (2021)
- [A Dynamical System Perspective for Lipschitz Neural Networks](https://mlsanity.com/paper/2110.12690/) (2021)
- [Private Synthetic Data for Multitask Learning and Marginal Queries](https://mlsanity.com/paper/2209.07400/) (2022)
- [Differentially Private k-Means Clustering](https://mlsanity.com/paper/1504.05998/) (2015)
- [Efficient Algorithms for Privately Releasing Marginals via Convex Relaxations](https://mlsanity.com/paper/1308.1385/) (2013)
- [Algorithmically Effective Differentially Private Synthetic Data](https://mlsanity.com/paper/2302.05552/) (2023)
- [Privacy of synthetic data: a statistical framework](https://mlsanity.com/paper/2109.01748/) (2021)
- [Private sampling: a noiseless approach for generating differentially private synthetic data](https://mlsanity.com/paper/2109.14839/) (2021)
- [Covariance's Loss is Privacy's Gain: Computationally Efficient, Private and Accurate Synthetic Data](https://mlsanity.com/paper/2107.05824/) (2021)
- [Certified private data release for sparse Lipschitz functions](https://mlsanity.com/paper/2302.09680/) (2023)
- [Differentially Private Low-dimensional Synthetic Data from High-dimensional Datasets](https://mlsanity.com/paper/2305.17148/) (2023)
- [Metric geometry of the privacy-utility tradeoff](https://mlsanity.com/paper/2405.00329/) (2024)
- [Differentially Private Synthetic Data with Private Density Estimation](https://mlsanity.com/paper/2405.04554/) (2024)

## Abstract (arXiv)

Generating differentially private synthetic data with meaningful Wasserstein utility guarantees is challenging in high dimensions. For datasets of size (n) on [0,1]d with d≥2, existing pure (varepsilon)-differentially private mechanisms achieve expected 1-Wasserstein error of order (varepsilon n)-1/d, reflecting the curse of dimensionality. While this rate is optimal in the worst case, it can be overly pessimistic when the data are supported on a lower-dimensional set. We formalize this through a multiscale packing-growth dimension k, which captures the geometric complexity of the support via the growth of packing numbers across scales. We propose Adaptive Pruned-PMM, a pure varepsilon-differentially private mechanism that combines private depth selection with our pruned variant of the Private Measure Mechanism (PMM) of He et al. (2023). The mechanism supports deeper, geometry-adapted hierarchies with expected running time O!left(d(n+d)log(varepsilon n)right), which is near-linear in n for fixed dimension and privacy budget. Under an external multiscale packing-growth condition with dimension k, we show that, for fixed positive privacy budgets and fixed geometry, the expected 1-Wasserstein error is of order (varepsilon n)-1/k for k>1 as n grows. We also prove a lower bound under a corresponding internal packing-growth condition, showing that the exponent 1/k is sharp within this framework.