The Fast Cauchy Transform and Faster Robust Linear Regression

Abstract

@ are a coreset for the problem, consisting of sampled and rescaled rows of A and b; and s is independent of n and polynomial in d. Our results improve on the best previous algorithms when n G d, for all p e [1, ∞) except p = 2; in particular, they improve the O(nd1.376+) running time of Sohler and Woodruff (STOC, 2011) for p = 1, that uses asymptotically fast matrix multiplication, and the O(nd5 log n) time of Dasgupta et al. (SICOMP, 2009) for general p, that uses ellipsoidal rounding. We also provide a suite of improved results for finding well-conditioned bases via ellipsoidal rounding, illustrating tradeoffs between running time and conditioning quality, including a one-pass conditioning algorithm for general ep problems. To complement this theory, we provide a detailed empirical evaluation of implementations of our algorithms for p = 1, comparing them with several related algorithms. Among other things, our empirical results clearly show that, in the asymptotic regime, the theory is a very good guide to the practical performance of these algorithms. Our algorithms use our faster constructions of well-conditioned bases for ep spaces and, for p = 1, a fast subspace embedding of independent interest that we call the Fast Cauchy Transform: a matrix Π: