# Benefits and Pitfalls of the Exponential Mechanism with Applications to Hilbert Spaces and Functional PCA

arXiv: https://arxiv.org/abs/1901.10864 · HTML: https://mlsanity.com/paper/1901.10864/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 30 Jan 2019.

## Built on by (most cited first, 10 of 10)

- [How to DP-fy ML: A Practical Guide to Machine Learning with Differential Privacy](https://mlsanity.com/paper/2303.00654/) (2023)
- [Private Convex Optimization via Exponential Mechanism](https://mlsanity.com/paper/2203.00263/) (2022)
- [Differential Privacy Over Riemannian Manifolds](https://mlsanity.com/paper/2111.02516/) (2021)
- [Differentially Private Bootstrap: New Privacy Analysis and Inference Strategies](https://mlsanity.com/paper/2210.06140/) (2022)
- [Shape And Structure Preserving Differential Privacy](https://mlsanity.com/paper/2209.12667/) (2022)
- [Federated Learning on Riemannian Manifolds with Differential Privacy](https://mlsanity.com/paper/2404.10029/) (2024)
- [Differentially Private Covariate Balancing Causal Inference](https://mlsanity.com/paper/2410.14789/) (2024)
- [Differentially Private Topological Data Analysis](https://mlsanity.com/paper/2305.03609/) (2023)
- [Performances of Symmetric Loss for Private Data from Exponential Mechanism](https://mlsanity.com/paper/2210.04132/) (2022)
- [Pure Differential Privacy for Functional Summaries with a Laplace-like Process](https://mlsanity.com/paper/2309.00125/) (2023)

## Abstract (arXiv)

The exponential mechanism is a fundamental tool of Differential Privacy (DP) due to its strong privacy guarantees and flexibility. We study its extension to settings with summaries based on infinite dimensional outputs such as with functional data analysis, shape analysis, and nonparametric statistics. We show that one can design the mechanism with respect to a specific base measure over the output space, such as a Guassian process. We provide a positive result that establishes a Central Limit Theorem for the exponential mechanism quite broadly. We also provide an apparent negative result, showing that the magnitude of the noise introduced for privacy is asymptotically non-negligible relative to the statistical estimation error. We develop an ep-DP mechanism for functional principal component analysis, applicable in separable Hilbert spaces. We demonstrate its performance via simulations and applications to two datasets.