# Laplace Transform Interpretation of Differential Privacy

arXiv: https://arxiv.org/abs/2411.09142 · HTML: https://mlsanity.com/paper/2411.09142/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 22 references that are in the map.
- First posted on arXiv 14 Nov 2024.

## Builds on (most cited first, 12 of 12)

- [Renyi Differential Privacy](https://mlsanity.com/paper/1702.07476/) (2017)
- [Concentrated Differential Privacy: Simplifications, Extensions, and Lower Bounds](https://mlsanity.com/paper/1605.02065/) (2016)
- [The Composition Theorem for Differential Privacy](https://mlsanity.com/paper/1311.0776/) (2013)
- [Privacy Amplification by Subsampling: Tight Analyses via Couplings and Divergences](https://mlsanity.com/paper/1807.01647/) (2018)
- [Improving the Gaussian Mechanism for Differential Privacy: Analytical Calibration and Optimal Denoising](https://mlsanity.com/paper/1805.06530/) (2018)
- [Numerical Composition of Differential Privacy](https://mlsanity.com/paper/2106.02848/) (2021)
- [Concentrated Differential Privacy](https://mlsanity.com/paper/1603.01887/) (2016)
- [The Discrete Gaussian for Differential Privacy](https://mlsanity.com/paper/2004.00010/) (2020)
- [Hypothesis Testing Interpretations and Renyi Differential Privacy](https://mlsanity.com/paper/1905.09982/) (2019)
- [Connect the Dots: Tighter Discrete Approximations of Privacy Loss Distributions](https://mlsanity.com/paper/2207.04380/) (2022)
- [Composition of Differential Privacy & Privacy Amplification by Subsampling](https://mlsanity.com/paper/2210.00597/) (2022)
- [A Better Bound Gives a Hundred Rounds: Enhanced Privacy Guarantees via f-Divergences](https://mlsanity.com/paper/2001.05990/) (2020)

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

- [Efficient privacy loss accounting for subsampling and random allocation](https://mlsanity.com/paper/2602.17284/) (2026)

## Abstract (arXiv)

We introduce a set of useful expressions of Differential Privacy (DP) notions in terms of the Laplace transform of the privacy loss distribution. Its bare form expression appears in several related works on analyzing DP, either as an integral or an expectation. We show that recognizing the expression as a Laplace transform unlocks a new way to reason about DP properties by exploiting the duality between time and frequency domains. Leveraging our interpretation, we connect the (q, ρ(q))-R'enyi DP curve and the (ε, δ(ε))-DP curve as being the Laplace and inverse-Laplace transforms of one another. This connection shows that the R'enyi divergence is well-defined for complex orders q = γ + i ω. Using our Laplace transform-based analysis, we also prove an adaptive composition theorem for (ε, δ)-DP guarantees that is exactly tight (i.e., matches even in constants) for all values of ε. Additionally, we resolve an issue regarding symmetry of f-DP on subsampling that prevented equivalence across all functional DP notions.