Iterative Least Trimmed Squares for Mixed Linear Regression
This addresses robust regression in noisy data settings, offering a simple method with theoretical guarantees, though it appears incremental as it builds on existing least trimmed squares ideas.
The paper tackles the problem of mixed linear regression with corruptions by proposing Iterative Least Trimmed Squares (ILTS), which alternates between selecting low-loss samples and refitting the model, and shows it converges linearly under deterministic conditions and matches or improves sample complexity for isotropic Gaussian features.
Given a linear regression setting, Iterative Least Trimmed Squares (ILTS) involves alternating between (a) selecting the subset of samples with lowest current loss, and (b) re-fitting the linear model only on that subset. Both steps are very fast and simple. In this paper we analyze ILTS in the setting of mixed linear regression with corruptions (MLR-C). We first establish deterministic conditions (on the features etc.) under which the ILTS iterate converges linearly to the closest mixture component. We also provide a global algorithm that uses ILTS as a subroutine, to fully solve mixed linear regressions with corruptions. We then evaluate it for the widely studied setting of isotropic Gaussian features, and establish that we match or better existing results in terms of sample complexity. Finally, we provide an ODE analysis for a gradient-descent variant of ILTS that has optimal time complexity. Our results provide initial theoretical evidence that iteratively fitting to the best subset of samples -- a potentially widely applicable idea -- can provably provide state of the art performance in bad training data settings.