Imbalanced Mixed Linear Regression

We consider the problem of mixed linear regression (MLR), where each observed sample belongs to one of K unknown linear models. In practical applications, the mixture of the K models may be imbalanced with a significantly different number of samples from each model. Unfortunately, most MLR methods do not perform well in such settings. Motivated by this practical challenge, in this work we propose Mix-IRLS, a novel, simple and fast algorithm for MLR with excellent performance on both balanced and imbalanced mixtures. In contrast to popular approaches that recover the K models simultaneously, Mix-IRLS does it sequentially using tools from robust regression. Empirically, beyond imbalanced mixtures, Mix-IRLS succeeds in a broad range of additional settings where other methods fail, including small sample sizes, presence of outliers, and an unknown number of models K. Furthermore, Mix-IRLS outperforms competing methods on several real-world datasets, in some cases by a large margin. We complement our empirical results by deriving a recovery guarantee for Mix-IRLS, which highlights its advantage on imbalanced mixtures.