Fast approximate L∞ minimization: Speeding up robust regression

Minimization of the L∞ norm, which can be viewed as approximately solving the non-convex least median estimation problem, is a powerful method for outlier removal and hence robust regression. However, current techniques for solving the problem at the heart of L∞ norm minimization are slow, and therefore cannot be scaled to large problems. A new method for the minimization of the L∞ norm is presented here, which provides a speedup of multiple orders of magnitude for data with high dimension. This method, termed Fast   L∞Minimization, allows robust regression to be applied to a class of problems which was previously inaccessible. It is shown how the L∞ norm minimization problem can be broken up into smaller sub-problems, which can then be solved extremely efficiently. Experimental results demonstrate the radical reduction in computation time, along with robustness against large numbers of outliers in a few model-fitting problems.