Recently a deterministic method, frequent directions (FD) is proposed to solve the high dimensional low rank approximation problem. It works well in practice, but experiences high computational cost. In this paper, we establish a fast frequent directions algorithm for the low rank approximation problem, which implants a randomized algorithm, sparse subspace embedding (SpEmb) in FD. This new algorithm makes use of FD's natural block structure and sends more information through SpEmb to each block in FD. We prove that our new algorithm produces a good low rank approximation with a sketch of size linear on the rank approximated. Its effectiveness and efficiency are demonstrated by the experimental results on both synthetic and real world datasets, as well as applications in network analysis.

中文翻译：

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低秩逼近的快速方向算法

近来，一种确定性方法，频繁方向（FD）被提出来解决高维低秩逼近问题。它在实践中效果很好，但是计算成本很高。在本文中，我们针对低秩逼近问题建立了一种快速的频繁方向算法，该算法在FD中植入了一种随机算法，即稀疏子空间嵌入（SpEmb）。这种新算法利用了FD的自然块结构，并通过SpEmb向FD中的每个块发送了更多信息。我们证明了我们的新算法产生了一个良好的低秩近似，并且在近似秩上具有线性大小的草图。在合成和真实数据集上的实验结果以及在网络分析中的应用证明了其有效性和效率。