Learning Mixture Model with Missing Values and its Application to Rankings

We consider the question of learning mixtures of generic sub-gaussian distributions based on observations with missing values. To that end, we utilize a matrix estimation method from literature (soft- or hard- singular value thresholding). Specifically, we stack the observations (with missing values) to form a data matrix and learn a low-rank approximation of it so that the row indices can be correctly clustered to belong to appropriate mixture component using a simple distance-based algorithm. To analyze the performance of this algorithm by quantifying finite sample bound, we extend the result for matrix estimation methods in the literature in two important ways: one, noise across columns is correlated and not independent across all entries of matrix as considered in the literature; two, the performance metric of interest is the maximum l2 row norm error, which is stronger than the traditional mean-squared-error averaged over all entries. Equipped with these advances in the context of matrix estimation, we are able to connect matrix estimation and mixture model learning in the presence of missing data.