Papers › Low Rank Approximation and Regression in Input Sparsity Time
Low Rank Approximation and Regression in Input Sparsity Time
Kenneth L. Clarkson, David P. Woodruff
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We design a new distribution over (r ⁻¹) ×n matrices S so that for any fixed n ×d matrix A of rank r, with probability at least 9/10, SAx₂ = (1 ±)Ax₂ simultaneously for all x ∈ℝᵈ. Such a matrix S is called a \emph{subspace embedding}. Furthermore, SA can be computed in (A) + (d ⁻¹) time, where (A) is the number of non-zero entries of A. This improves over all previous subspace embeddings, which required at least Ω(nd logd) time to achieve this property. We call our matrices S \emph{sparse embedding matrices}. Using our sparse embedding matrices, we obtain the fastest known algorithms for (1+)-approximation for overconstrained least-squares regression, low-rank approximation, approximating all leverage scores, and ℓₚ-regression. The leading order term in the time complexity of our algorithms is O((A)) or O((A)logn). We optimize the low-order (d/) terms in our running times (or for rank-k approximation, the n*(k/eps) term), and show various tradeoffs. For instance, we also use our methods to design new preconditioners that improve the dependence on $\eps$ in least squares regression to log1/. Finally, we provide preliminary experimental results which suggest that our algorithms are competitive in practice.
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