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 $$ \sum_{i=1}^k w_i \mathcal{N}(\mu_i,\sigma^2), $$ 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.