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Tuesday, February 19, 2013

The Fast Cauchy Transform and Faster Robust Linear Regression

Here is a Randomized Numerical Linear Algebra (RandNLA) paper of note. For some context, the Fast Cauchy Transform (FCT) is an l_1-based analog of the fast Johnson-Lindenstrauss transform (FJLT), in particular the paper addressed the generalization of the Multiple l_1 Regression.

We provide fast algorithms for overconstrained $\ell_p$ regression and related problems: for an $n\times d$ input matrix $A$ and vector $b\in\R^n$, in $O(nd\log n)$ time we reduce the problem $\min_{x\in\R^d} \norm{Ax-b}_p$ 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 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 $\ell_p$ spaces and, for $p=1$, a fast subspace embedding of independent interest that we call the Fast Cauchy Transform: a distribution over matrices $\Pi: \R^n\mapsto \R^{O(d\log d)}$, found obliviously to $A$, that approximately preserves the $\ell_1$ norms: that is, with large probability, simultaneously for all $x$, $\norm{Ax}_1 \approx \norm{\Pi Ax}_1$, 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.

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