The Fast Cauchy Transform and Faster Robust Linear Regression

We provide fast algorithms for overconstrained $\ell_p$ regression and related problems: for an $n\times d$ input matrix $A$ and vector $b\in\mathbb{R}^n$, in $O(nd\log n)$ time we reduce the problem $\min_{x\in\mathbb{R}^d} \|Ax-b\|_p$ to the same problem with input matrix $\tilde A$ of dimension $s \times d$ and corresponding $\tilde b$ of dimension $s\times 1$. Here, $\tilde A$ and $\tilde b$ 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 $n\gg d$, for all $p\in[1,\infty)$ except $p=2$. 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 $\ell_p$ problems. We also provide an empirical evaluation of implementations of our algorithms for $p=1$, comparing them with related algorithms. Our empirical results 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 $\ell_p$ spaces and, for $p=1$, a fast subspace embedding of independent interest that we call the Fast Cauchy Transform: a distribution over matrices $\Pi:\mathbb{R}^n\mapsto \mathbb{R}^{O(d\log d)}$, found obliviously to $A$, that approximately preserves the $\ell_1$ norms: that is, with large probability, simultaneously for all $x$, $\|Ax\|_1 \approx \|\Pi Ax\|_1$, with distortion $O(d^{2+\eta})$, for an arbitrarily small constant $\eta>0$; and, moreover, $\Pi A$ can be computed in $O(nd\log d)$ time. The techniques underlying our Fast Cauchy Transform include fast Johnson-Lindenstrauss transforms, low-coherence matrices, and rescaling by Cauchy random variables.