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| Main Authors: | , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2501.10172 |
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| _version_ | 1866916569518440448 |
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| author | Iverson, Valentio Vavasis, Stephen |
| author_facet | Iverson, Valentio Vavasis, Stephen |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_10172 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Mean and Variance Estimation Complexity in Arbitrary Distributions via Wasserstein Minimization Iverson, Valentio Vavasis, Stephen Machine Learning 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. |
| title | Mean and Variance Estimation Complexity in Arbitrary Distributions via Wasserstein Minimization |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2501.10172 |