Benefits and Pitfalls of the Exponential Mechanism with Applications to Hilbert Spaces and Functional PCA
This work addresses privacy challenges in functional data analysis, a domain-specific area, but is incremental as it builds on the existing exponential mechanism framework.
The authors extended the exponential mechanism for differential privacy to infinite-dimensional settings like functional data analysis, establishing a central limit theorem but showing that privacy noise remains asymptotically non-negligible relative to estimation error. They developed and tested an ε-DP mechanism for functional principal component analysis in separable Hilbert spaces, demonstrating its performance through simulations and two real-world datasets.
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.