← 研踪 PaperTrailarXiv:2501.10172 · v2
Mean and Variance Estimation Complexity in Arbitrary Distributions via Wasserstein Minimization
首次提交 2025-01-17 21:07
版本更新 2026-11-05 03:30
雷达观测 2026-11-06 11:06
摘要
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 $\boldsymbolμ \in \mathbb{R}^l$ and shrinkage $σ\in \mathbb{R}_{++}$ parameters for a distribution of the form $\frac{1}{σ^l} f_0 \left( \frac{\boldsymbol{x} - \boldsymbolμ}σ \right)$, where $f_0$ is a known density in $\mathbb{R}^l$ 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 $\text{poly} \left( \frac{1}{\varepsilon} \right)$ time using the Wasserstein distance.
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v2 · 2026-11-05 03:30 · 最新观测版本
Mean and Variance Estimation Complexity in Arbitrary Distributions via Wasserstein Minimization
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 $\boldsymbolμ \in \mathbb{R}^l$ and shrinkage $σ\in \mathbb{R}_{++}$ parameters for a distribution of the form $\frac{1}{σ^l} f_0 \left( \frac{\boldsymbol{x} - \boldsymbolμ}σ \right)$, where $f_0$ is a known density in $\mathbb{R}^l$ 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 $\text{poly} \left( \frac{1}{\varepsilon} \right)$ time using the Wasserstein distance.
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