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We show that a certain class of distributions, Oblivious Sparse Norm-Approximating Projections (OSNAPs), provides OSE's with m = O(d ^1+γ /ε ² ), and where every matrix Π in the support of the OSE has only s = O _γ (1/ε) non-zero entries per column, for γ > 0 any desired constant. Plugging OSNAPs into known algorithms for approximate least squares regression, ℓ _p regression, low rank approximation, and approximating leverage scores implies faster algorithms for all these problems. Our main result is essentially a Bai-Yin type theorem in random matrix theory and is likely to be of independent interest: we show that for any fixed U ∈ R ^n×d with orthonormal columns and random sparse Π, all singular values of ΠU lie in [1 - ε, 1 + ε] with good probability. This can be seen as a generalization of the sparse Johnson-Lindenstrauss lemma, which was concerned with d = 1. Our methods also recover a slightly sharper version of a main result of [Clarkson-Woodruff, STOC 2013], with a much simpler proof. 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OSNAP: Faster Numerical Linear Algebra Algorithms via Sparser Subspace Embeddings 论文

OSNAP: Faster Numerical Linear Algebra Algorithms via Sparser Subspace Embeddings · 作者