OSNAP：通过稀疏子空间嵌入的更快数值线性代数算法
OSNAP: Faster numerical linear algebra algorithms via sparser subspace embeddings
摘要：
给定某些参数eps，d的“不经意子空间嵌入（OSE）”是R^{m x n}中矩阵B上的分布D，使得对于R^n中具有dim（W）= d的任何线性子空间W，它保持Pr_{B ~ D}（对于W中的所有x|| B x||_2 in（1 +/- eps）||x||_2）> 2/3我们证明了一个OSE存在，m = O（d^2/eps^2），并且其中D的支撑中的每个B每列正好有s=1个非零元素。这改进了先前在[Clarkson-Woodruff，arXiv：1207.6365]中最著名的边界。我们对d的二次依赖对于任何s=1的OSE都是最优的[Nelson-Nguyen，2012]。我们还给出了两个OSE，我们称之为不经意稀疏范数近似投影（OSNAP），它们都允许参数设置m = 0（d/eps^2）和s = polylog（d）/eps，或者对于任何常数gamma>0，m = O（d^{1+gamma}/eps^2）和s=O（1/eps）。这个m几乎是最优的，因为m >= d是简单地需要的，W的非零向量没有落在B的核中。这是第一个m=o（d^2）的结构，也是第一个s=o（d）的结构。事实上，我们的OSNAP只不过是[Kane-Nelson，SODA 2012]的稀疏Johnson-Lindenstrauss矩阵。我们的分析都产生OSE的采样使用O（1）明智或O（log d）明智的独立散列函数，这提供了一些效率优势，比以前的工作旋转栅门流应用程序。我们的主要结果实质上是随机矩阵理论中的一个Bai-Yin型定理，并且可能是独立的兴趣：即我们证明了对于R^{nxd}中具有正交列和随机稀疏B的任何U，BU的所有奇异值以良好的概率位于[1-eps，1+eps]中。将OSNAP插入到用于数值线性代数问题的已知算法中，例如近似最小二乘回归、低秩近似和近似杠杆分数，意味着用于所有这些问题的更快算法。
英文摘要：
An "oblivious subspace embedding (OSE)" given some parameters eps,d is a distribution D over matrices B in R^{m x n} such that for any linear subspace W in R^n with dim(W) = d it holds that Pr_{B ~ D}(forall x in W ||B x||_2 in (1 +/- eps)||x||_2) > 2/3 We show an OSE exists with m = O(d^2/eps^2) and where every B in the support of D has exactly s=1 non-zero entries per column. This improves previously best known bound in [Clarkson-Woodruff, arXiv:1207.6365]. Our quadratic dependence on d is optimal for any OSE with s=1 [Nelson-Nguyen, 2012]. We also give two OSE's, which we call Oblivious Sparse Norm-Approximating Projections (OSNAPs), that both allow the parameter settings m = Õ(d/eps^2) and s = polylog(d)/eps, or m = O(d^{1+gamma}/eps^2) and s=O(1/eps) for any constant gamma>0. This m is nearly optimal since m >= d is required simply to no non-zero vector of W lands in the kernel of B. These are the first constructions with m=o(d^2) to have s=o(d). In fact, our OSNAPs are nothing more than the sparse Johnson-Lindenstrauss matrices of [Kane-Nelson, SODA 2012]. Our analyses all yield OSE's that are sampled using either O(1)-wise or O(log d)-wise independent hash functions, which provides some efficiency advantages over previous work for turnstile streaming applications. Our main result is essentially a Bai-Yin type theorem in random matrix theory and is likely to be of independent interest: i.e. we show that for any U in R^{n x d} with orthonormal columns and random sparse B, all singular values of BU lie in [1-eps, 1+eps] with good probability. Plugging OSNAPs into known algorithms for numerical linear algebra problems such as approximate least squares regression, low rank approximation, and approximating leverage scores implies faster algorithms for all these problems.
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