Tight Bounds for $ell_p$ Oblivious Subspace Embeddings
arXiv: Data Structures and Algorithms(2019)
摘要
An $ell_p$ oblivious subspace embedding is a distribution over $r times n$ matrices $Pi$ such that for any fixed $n times d$ matrix $A$, $Pr_{Pi}[textrm{for all }x, |Ax|_p leq |Pi Ax|_p leq kappa |Ax|_p] geq 9/10,$ where $r$ is the dimension of the embedding, $kappa$ is the distortion of the embedding, and for an $n$-dimensional vector $y$, $|y|_p$ is the $ell_p$-norm. Another important property is the sparsity of $Pi$, that is, the maximum number of non-zero entries per column, as this determines the running time of computing $Pi cdot A$. While for $p = 2$ there are nearly optimal tradeoffs in terms of the dimension, distortion, and sparisty, for the important case of $1 leq p u003c 2$, much less was known. In this paper we obtain nearly optimal tradeoffs for $ell_p$ oblivious subspace embeddings for every $1 leq p u003c 2$. show for every $1 leq p u003c 2$, any oblivious subspace embedding with dimension $r$ has distortion $kappa = Omega left(frac{1}{left(frac{1}{d}right)^{1 / p} cdot log^{2 / p}r + left(frac{r}{n}right)^{1 / p - 1 / 2}}right).$ When $r = mathrm{poly}(d)$ in applications, this gives a $kappa = Omega(d^{1/p}log^{-2/p} d)$ lower bound, and shows the oblivious subspace embedding of Sohler and Woodruff (STOC, 2011) for $p = 1$ and the oblivious subspace embedding of Meng and Mahoney (STOC, 2013) for $1 u003c p u003c 2$ are optimal up to $mathrm{poly}(log(d))$ factors. We also give sparse oblivious subspace embeddings for every $1 leq p u003c 2$ which are optimal in dimension and distortion, up to $mathrm{poly}(log d)$ factors. Oblivious subspace embeddings are crucial for distributed and streaming environments, as well as entrywise $ell_p$ low rank approximation. Our results give improved algorithms for these applications.
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