Dimensionality Reduction for Tukey Regression
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Proceedings of the 36th International Conference on Machine Learning, PMLR 97:12621271, 2019.
Abstract
We give the first dimensionality reduction methods for the overconstrained Tukey regression problem. The Tukey loss function $\y\_M = \sum_i M(y_i)$ has $M(y_i) \approx y_i^p$ for residual errors $y_i$ smaller than a prescribed threshold $\tau$, but $M(y_i)$ becomes constant for errors $y_i > \tau$. Our results depend on a new structural result, proven constructively, showing that for any $d$dimensional subspace $L \subset \mathbb{R}^n$, there is a fixed boundedsize subset of coordinates containing, for every $y \in L$, all the large coordinates, with respect to the Tukey loss function, of $y$. Our methods reduce a given Tukey regression problem to a smaller weighted version, whose solution is a provably good approximate solution to the original problem. Our reductions are fast, simple and easy to implement, and we give empirical results demonstrating their practicality, using existing heuristic solvers for the small versions. We also give exponentialtime algorithms giving provably good solutions, and hardness results suggesting that a significant speedup in the worst case is unlikely.
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