Since the time of Gauss, it has been generally accepted that
$\ell_2$-methods of combining observations by minimizing sums of squared errors
have significant computational advantages over earlier $\ell_1$-methods based
on minimization of absolute errors advocated by Boscovich, Laplace and others.
However, $\ell_1$-methods are known to have significant robustness advantages
over $\ell_2$-methods in many applications, and related quantile regression
methods provide a useful, complementary approach to classical least-squares
estimation of statistical models. Combining recent advances in interior point
methods for solving linear programs with a new statistical preprocessing
approach for $\ell_1$-type problems, we obtain a 10- to 100-fold improvement in
computational speeds over current (simplex-based) $\ell_1$-algorithms in large
problems, demonstrating that $\ell_1$-methods can be made competitive with
$\ell_2$-methods in terms of computational speed throughout the entire range of
problem sizes. Formal complexity results suggest that $\ell_1$-regression can
be made faster than least-squares regression for n sufficiently large
and p modest.
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GAUSSIAN HARE, LAPLACIAN TORTOISE 297
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