# Estimating Numerical Distributions under Local Differential Privacy

arXiv: https://arxiv.org/abs/1912.01051 · HTML: https://mlsanity.com/paper/1912.01051/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Dec 2019.

## Built on by (most cited first, 17 of 17)

- [Fine-grained Poisoning Attack to Local Differential Privacy Protocols for Mean and Variance Estimation](https://mlsanity.com/paper/2205.11782/) (2022)
- [Nonparametric extensions of randomized response for private confidence sets](https://mlsanity.com/paper/2202.08728/) (2022)
- [On the Robustness of LDP Protocols for Numerical Attributes under Data Poisoning Attacks](https://mlsanity.com/paper/2403.19510/) (2024)
- [Privacy-Preserving Graph Embedding based on Local Differential Privacy](https://mlsanity.com/paper/2310.11060/) (2023)
- [How Much is Left? LLMs Linearly Encode Their Remaining Output Length](https://mlsanity.com/paper/2607.05316/) (2026)
- [Locally Differentially Private Bayesian Inference](https://mlsanity.com/paper/2110.14426/) (2021)
- [Beyond Statistical Estimation: Differentially Private Individual Computation via Shuffling](https://mlsanity.com/paper/2406.18145/) (2024)
- [Data Poisoning Attacks to Locally Differentially Private Range Query Protocols](https://mlsanity.com/paper/2503.03454/) (2025)
- [Mitigating Data Poisoning Attacks to Local Differential Privacy](https://mlsanity.com/paper/2506.02156/) (2025)
- [Devil's Hand: Data Poisoning Attacks to Locally Private Graph Learning Protocols](https://mlsanity.com/paper/2506.09803/) (2025)
- [Consistent Estimation of Numerical Distributions under Local Differential Privacy by Wavelet Expansion](https://mlsanity.com/paper/2509.19661/) (2025)
- [Communication Cost Reduction for Subgraph Counting under Local Differential Privacy via Hash Functions](https://mlsanity.com/paper/2312.07055/) (2023)
- [A Survey on Differential Privacy for SpatioTemporal Data in Transportation Research](https://mlsanity.com/paper/2407.15868/) (2024)
- [Robust Estimation of Sparse Numerical Vectors under Local Differential Privacy](https://mlsanity.com/paper/2607.27815/) (2026)
- [ABC: Numerical Data Collection under Local Differential Privacy without Prior Knowledge](https://mlsanity.com/paper/2608.05737/) (2026)
- [Auditing Privacy Risks in LLM-Enhanced Graph Neural Networks](https://mlsanity.com/paper/2608.25727/) (2026)
- [Trust-But-Verify: Poisoning-Resilient Locally Private Graph Learning Protocols](https://mlsanity.com/paper/2609.07063/) (2026)

## Abstract (arXiv)

When collecting information, local differential privacy (LDP) relieves the concern of privacy leakage from users' perspective, as user's private information is randomized before sent to the aggregator. We study the problem of recovering the distribution over a numerical domain while satisfying LDP. While one can discretize a numerical domain and then apply the protocols developed for categorical domains, we show that taking advantage of the numerical nature of the domain results in better trade-off of privacy and utility. We introduce a new reporting mechanism, called the square wave SW mechanism, which exploits the numerical nature in reporting. We also develop an Expectation Maximization with Smoothing (EMS) algorithm, which is applied to aggregated histograms from the SW mechanism to estimate the original distributions. Extensive experiments demonstrate that our proposed approach, SW with EMS, consistently outperforms other methods in a variety of utility metrics.