# Optimal Rates for Nonparametric Density Estimation under Communication Constraints

arXiv: https://arxiv.org/abs/2107.10078 · HTML: https://mlsanity.com/paper/2107.10078/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 Jul 2021.

## Built on by (most cited first, 9 of 9)

- [Optimal Federated Learning for Nonparametric Regression with Heterogeneous Distributed Differential Privacy Constraints](https://mlsanity.com/paper/2406.06755/) (2024)
- [On the Statistical Complexity of Estimation and Testing under Privacy Constraints](https://mlsanity.com/paper/2210.02215/) (2022)
- [Order-Optimal Sequential 1-Bit Mean Estimation in General Tail Regimes](https://mlsanity.com/paper/2604.07796/) (2026)
- [Continual Mean Estimation Under User-Level Privacy](https://mlsanity.com/paper/2212.09980/) (2022)
- [Sequential 1-bit Mean Estimation with Near-Optimal Sample Complexity](https://mlsanity.com/paper/2509.21940/) (2025)
- [Private Statistical Estimation of Many Quantiles](https://mlsanity.com/paper/2302.06943/) (2023)
- [Breaking the curse of dimensionality in structured density estimation](https://mlsanity.com/paper/2410.07685/) (2024)
- [About the Cost of Central Privacy in Density Estimation](https://mlsanity.com/paper/2306.14535/) (2023)
- [Optimal Federated Learning for Functional Mean Estimation under Heterogeneous Privacy Constraints](https://mlsanity.com/paper/2412.18992/) (2024)

## Abstract (arXiv)

We consider density estimation for Besov spaces when each sample is quantized to only a limited number of bits. We provide a noninteractive adaptive estimator that exploits the sparsity of wavelet bases, along with a simulate-and-infer technique from parametric estimation under communication constraints. We show that our estimator is nearly rate-optimal by deriving minimax lower bounds that hold even when interactive protocols are allowed. Interestingly, while our wavelet-based estimator is almost rate-optimal for Sobolev spaces as well, it is unclear whether the standard Fourier basis, which arise naturally for those spaces, can be used to achieve the same performance.