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• W2570048418 abstract "We provide fast algorithms for overconstrained $ell_p$ regression and related problems: for an $ntimes d$ input matrix $A$ and vector $binmathbb{R}^n$, in $O(ndlog n)$ time we reduce the problem $min_{xinmathbb{R}^d} |Ax-b|_p$ to the same problem with input matrix $tilde A$ of dimension $s times d$ and corresponding $tilde b$ of dimension $stimes 1$. 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 $ngg d$ for all $pin [1,infty)$ except $p=2$; in particular, they improve the $O(nd^{1.376+})$ running time of Sohler and Woodruff [Proceedings of the 43rd Annual ACM Symposium on Theory of Computing, 2011, pp. 755--764] for $p=1$, which uses asymptotically fast matrix multiplication, and the $O(nd^5log n)$ time of Dasgupta et al. [SIAM J. Comput., 38 (2009), pp. 2060--2078] for general $p$, which 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 $ell_p$ 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 $ell_p$ spaces and, for $p=1$, a fast subspace embedding of independent interest that we call the Fast Cauchy transform: a distribution over matrices $Pi: mathbb{R}^nmapsto mathbb{R}^{O(dlog d)}$, 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 $O(ndlog d)$ time. The techniques underlying our Fast Cauchy transform include Fast Johnson--Lindenstrauss transforms, low-coherence matrices, and rescaling by Cauchy random variables." @default.
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• W2570048418 date "2016-01-01" @default.
• W2570048418 modified "2023-09-25" @default.
• W2570048418 title "The Fast Cauchy Transform and Faster Robust Linear Regression" @default.
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• W2570048418 doi "https://doi.org/10.1137/140963698" @default.
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