# Mean and Variance Estimation Complexity in Arbitrary Distributions via Wasserstein Minimization

arXiv: https://arxiv.org/abs/2501.10172 · HTML: https://mlsanity.com/paper/2501.10172/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 20 references that are in the map.
- First posted on arXiv 17 Jan 2025.

## Builds on (most cited first, 9 of 9)

- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [Learning with a Wasserstein Loss](https://mlsanity.com/paper/1506.05439/) (2015)
- [Generative Modeling with Optimal Transport Maps](https://mlsanity.com/paper/2110.02999/) (2021)
- [Sliced Wasserstein Distance for Learning Gaussian Mixture Models](https://mlsanity.com/paper/1711.05376/) (2017)
- [Score-based Generative Modeling Secretly Minimizes the Wasserstein Distance](https://mlsanity.com/paper/2212.06359/) (2022)
- [Transformer Fusion with Optimal Transport](https://mlsanity.com/paper/2310.05719/) (2023)
- [Multi-marginal Wasserstein GAN](https://mlsanity.com/paper/1911.00888/) (2019)
- [Deep Optimal Transport: A Practical Algorithm for Photo-realistic Image Restoration](https://mlsanity.com/paper/2306.02342/) (2023)
- [IMRO: a proximal quasi-Newton method for solving l1-regularized least square problem](https://mlsanity.com/paper/1401.4220/) (2014)

## Abstract (arXiv)

Parameter estimation is a fundamental challenge in machine learning, crucial for tasks such as neural network weight fitting and Bayesian inference. This paper focuses on the complexity of estimating translation boldsymbolμ ∈ ℝl and shrinkage σ ∈ ℝ++ parameters for a distribution of the form frac1σl f0 left( fracboldsymbolx - boldsymbolμσ right), where f0 is a known density in ℝl given n samples. We highlight that while the problem is NP-hard for Maximum Likelihood Estimation (MLE), it is possible to obtain varepsilon-approximations for arbitrary varepsilon > 0 within poly left( frac1varepsilon right) time using the Wasserstein distance.