# Near-Optimal Mean Estimation with Unknown, Heteroskedastic Variances

arXiv: https://arxiv.org/abs/2312.02417 · HTML: https://mlsanity.com/paper/2312.02417/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 30 references that are in the map.
- First posted on arXiv 5 Dec 2023.

## Builds on (most cited first, 14 of 14)

- [Robust Estimators in High Dimensions without the Computational Intractability](https://mlsanity.com/paper/1604.06443/) (2016)
- [Agnostic Estimation of Mean and Covariance](https://mlsanity.com/paper/1604.06968/) (2016)
- [Learning from Untrusted Data](https://mlsanity.com/paper/1611.02315/) (2016)
- [Mixture Models, Robustness, and Sum of Squares Proofs](https://mlsanity.com/paper/1711.07454/) (2017)
- [Efficient Algorithms and Lower Bounds for Robust Linear Regression](https://mlsanity.com/paper/1806.00040/) (2018)
- [Meta-learning for mixed linear regression](https://mlsanity.com/paper/2002.08936/) (2020)
- [Maximum Likelihood Estimation for Learning Populations of Parameters](https://mlsanity.com/paper/1902.04553/) (2019)
- [Finite-Sample Maximum Likelihood Estimation of Location](https://mlsanity.com/paper/2206.02348/) (2022)
- [Learning Entangled Single-Sample Distributions via Iterative Trimming](https://mlsanity.com/paper/2004.09563/) (2020)
- [On Mean Estimation for Heteroscedastic Random Variables](https://mlsanity.com/paper/2010.11537/) (2020)
- [High-dimensional Location Estimation via Norm Concentration for Subgamma Vectors](https://mlsanity.com/paper/2302.02497/) (2023)
- [Estimating location parameters in entangled single-sample distributions](https://mlsanity.com/paper/1907.03087/) (2019)
- [Learning Entangled Single-Sample Gaussians in the Subset-of-Signals Model](https://mlsanity.com/paper/2007.05557/) (2020)
- [Near Optimal Heteroscedastic Regression with Symbiotic Learning](https://mlsanity.com/paper/2306.14288/) (2023)

## Built on by (most cited first, 5 of 5)

- [Efficient Multivariate Robust Mean Estimation Under Mean-Shift Contamination](https://mlsanity.com/paper/2502.14772/) (2025)
- [Entangled Mean Estimation in High-Dimensions](https://mlsanity.com/paper/2501.05425/) (2025)
- [Attainability of Two-Point Testing Rates for Finite-Sample Location Estimation](https://mlsanity.com/paper/2502.05730/) (2025)
- [Performance of the empirical median for location estimation in heteroscedastic settings](https://mlsanity.com/paper/2501.16956/) (2025)
- [Learning Mixture Models via Efficient High-dimensional Sparse Fourier Transforms](https://mlsanity.com/paper/2601.05157/) (2026)

## Abstract (arXiv)

Given data drawn from a collection of Gaussian variables with a common mean but different and unknown variances, what is the best algorithm for estimating their common mean? We present an intuitive and efficient algorithm for this task. As different closed-form guarantees can be hard to compare, the Subset-of-Signals model serves as a benchmark for heteroskedastic mean estimation: given n Gaussian variables with an unknown subset of m variables having variance bounded by 1, what is the optimal estimation error as a function of n and m? Our algorithm resolves this open question up to logarithmic factors, improving upon the previous best known estimation error by polynomial factors when m = nc for all 0<c<1. Of particular note, we obtain error o(1) with m = tildeO(n1/4) variance-bounded samples, whereas previous work required m = tildeOmega(n1/2). Finally, we show that in the multi-dimensional setting, even for d=2, our techniques enable rates comparable to knowing the variance of each sample.