Learning Mixtures of Gaussians with Censored Data

Wai Ming Tai, Bryon Aragam
Proceedings of the 40th International Conference on Machine Learning, PMLR 202:33396-33415, 2023.

Abstract

We study the problem of learning mixtures of Gaussians with censored data. Statistical learning with censored data is a classical problem, with numerous practical applications, however, finite-sample guarantees for even simple latent variable models such as Gaussian mixtures are missing. Formally, we are given censored data from a mixture of univariate Gaussians $ \sum_{i=1}^k w_i \mathcal{N}(\mu_i,\sigma^2), $ i.e. the sample is observed only if it lies inside a set $S$. The goal is to learn the weights $w_i$ and the means $\mu_i$. We propose an algorithm that takes only $\frac{1}{\varepsilon^{O(k)}}$ samples to estimate the weights $w_i$ and the means $\mu_i$ within $\varepsilon$ error.

Cite this Paper

BibTeX
@InProceedings{pmlr-v202-tai23a, title = {Learning Mixtures of {G}aussians with Censored Data}, author = {Tai, Wai Ming and Aragam, Bryon}, booktitle = {Proceedings of the 40th International Conference on Machine Learning}, pages = {33396--33415}, year = {2023}, editor = {Krause, Andreas and Brunskill, Emma and Cho, Kyunghyun and Engelhardt, Barbara and Sabato, Sivan and Scarlett, Jonathan}, volume = {202}, series = {Proceedings of Machine Learning Research}, month = {23--29 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v202/tai23a/tai23a.pdf}, url = {https://proceedings.mlr.press/v202/tai23a.html}, abstract = {We study the problem of learning mixtures of Gaussians with censored data. Statistical learning with censored data is a classical problem, with numerous practical applications, however, finite-sample guarantees for even simple latent variable models such as Gaussian mixtures are missing. Formally, we are given censored data from a mixture of univariate Gaussians $ \sum_{i=1}^k w_i \mathcal{N}(\mu_i,\sigma^2), $ i.e. the sample is observed only if it lies inside a set $S$. The goal is to learn the weights $w_i$ and the means $\mu_i$. We propose an algorithm that takes only $\frac{1}{\varepsilon^{O(k)}}$ samples to estimate the weights $w_i$ and the means $\mu_i$ within $\varepsilon$ error.} }
Endnote
%0 Conference Paper %T Learning Mixtures of Gaussians with Censored Data %A Wai Ming Tai %A Bryon Aragam %B Proceedings of the 40th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2023 %E Andreas Krause %E Emma Brunskill %E Kyunghyun Cho %E Barbara Engelhardt %E Sivan Sabato %E Jonathan Scarlett %F pmlr-v202-tai23a %I PMLR %P 33396--33415 %U https://proceedings.mlr.press/v202/tai23a.html %V 202 %X We study the problem of learning mixtures of Gaussians with censored data. Statistical learning with censored data is a classical problem, with numerous practical applications, however, finite-sample guarantees for even simple latent variable models such as Gaussian mixtures are missing. Formally, we are given censored data from a mixture of univariate Gaussians $ \sum_{i=1}^k w_i \mathcal{N}(\mu_i,\sigma^2), $ i.e. the sample is observed only if it lies inside a set $S$. The goal is to learn the weights $w_i$ and the means $\mu_i$. We propose an algorithm that takes only $\frac{1}{\varepsilon^{O(k)}}$ samples to estimate the weights $w_i$ and the means $\mu_i$ within $\varepsilon$ error.
APA
Tai, W.M. & Aragam, B.. (2023). Learning Mixtures of Gaussians with Censored Data. Proceedings of the 40th International Conference on Machine Learning, in Proceedings of Machine Learning Research 202:33396-33415 Available from https://proceedings.mlr.press/v202/tai23a.html.

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