Learning Mixtures of Gaussians with Censored Data

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 $$ \sum{i=1}^k wi \mathcal{N}(\mui,\sigma^2), $$ i.e. the sample is observed only if it lies inside a set $S$. The goal is to learn the weights $wi$ and the means $\mui$. We propose an algorithm that takes only $\frac{1}{\varepsilon^{O(k)}}$ samples to estimate the weights $wi$ and the means $\mu_i$ within $\varepsilon$ error.