# Consistent Estimation of Numerical Distributions under Local Differential Privacy by Wavelet Expansion

arXiv: https://arxiv.org/abs/2509.19661 · HTML: https://mlsanity.com/paper/2509.19661/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 59 references that are in the map.
- First posted on arXiv 24 Sep 2025.

## Builds on (most cited first, 25 of 25)

- [RAPPOR: Randomized Aggregatable Privacy-Preserving Ordinal Response](https://mlsanity.com/paper/1407.6981/) (2014)
- [Collecting Telemetry Data Privately](https://mlsanity.com/paper/1712.01524/) (2017)
- [Minimax Optimal Procedures for Locally Private Estimation](https://mlsanity.com/paper/1604.02390/) (2016)
- [Graphical-model based estimation and inference for differential privacy](https://mlsanity.com/paper/1901.09136/) (2019)
- [Discrete Distribution Estimation under Local Privacy](https://mlsanity.com/paper/1602.07387/) (2016)
- [Collecting and Analyzing Multidimensional Data with Local Differential Privacy](https://mlsanity.com/paper/1907.00782/) (2019)
- [Local, Private, Efficient Protocols for Succinct Histograms](https://mlsanity.com/paper/1504.04686/) (2015)
- [Optimal Schemes for Discrete Distribution Estimation under Locally Differential Privacy](https://mlsanity.com/paper/1702.00610/) (2017)
- [Hadamard Response: Estimating Distributions Privately, Efficiently, and with Little Communication](https://mlsanity.com/paper/1802.04705/) (2018)
- [Estimating Numerical Distributions under Local Differential Privacy](https://mlsanity.com/paper/1912.01051/) (2019)
- [Discrete Distribution Estimation under User-level Local Differential Privacy](https://mlsanity.com/paper/2211.03757/) (2022)
- [Answering Range Queries Under Local Differential Privacy](https://mlsanity.com/paper/1812.10942/) (2018)
- [Locally Differentially Private Frequency Estimation with Consistency](https://mlsanity.com/paper/1905.08320/) (2019)
- [A Huber Loss Minimization Approach to Mean Estimation under User-level Differential Privacy](https://mlsanity.com/paper/2405.13453/) (2024)
- [Private Frequency Estimation via Projective Geometry](https://mlsanity.com/paper/2203.00194/) (2022)
- [Instance-Optimal Private Density Estimation in the Wasserstein Distance](https://mlsanity.com/paper/2406.19566/) (2024)
- [Linear Queries Estimation with Local Differential Privacy](https://mlsanity.com/paper/1810.02810/) (2018)
- [Calibrate: Frequency Estimation and Heavy Hitter Identification with Local Differential Privacy via Incorporating Prior Knowledge](https://mlsanity.com/paper/1812.02055/) (2018)
- [Frequency Estimation under Local Differential Privacy [Experiments, Analysis and Benchmarks]](https://mlsanity.com/paper/2103.16640/) (2021)
- [Private Wasserstein Distance](https://mlsanity.com/paper/2404.06787/) (2024)
- [L-SRR: Local Differential Privacy for Location-Based Services with Staircase Randomized Response](https://mlsanity.com/paper/2209.15091/) (2022)
- [Wasserstein Differential Privacy](https://mlsanity.com/paper/2401.12436/) (2024)
- [Learning with User-Level Local Differential Privacy](https://mlsanity.com/paper/2405.17079/) (2024)
- [Enhancing Learning with Label Differential Privacy by Vector Approximation](https://mlsanity.com/paper/2405.15150/) (2024)
- [Numerical Estimation of Spatial Distributions under Differential Privacy](https://mlsanity.com/paper/2412.06541/) (2024)

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

- [Robust Estimation of Sparse Numerical Vectors under Local Differential Privacy](https://mlsanity.com/paper/2607.27815/) (2026)

## Abstract (arXiv)

Distribution estimation under local differential privacy (LDP) is a fundamental and challenging task. Significant progresses have been made on categorical data. However, due to different evaluation metrics, these methods do not work well when transferred to numerical data. In particular, we need to prevent the probability mass from being misplaced far away. In this paper, we propose a new approach that express the sample distribution using wavelet expansions. The coefficients of wavelet series are estimated under LDP. Our method prioritizes the estimation of low-order coefficients, in order to ensure accurate estimation at macroscopic level. Therefore, the probability mass is prevented from being misplaced too far away from its ground truth. We establish theoretical guarantees for our methods. Experiments show that our wavelet expansion method significantly outperforms existing solutions under Wasserstein and KS distances.