A sparse Johnson: Lindenstrauss transform
A sparse Johnson: Lindenstrauss transform
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稀疏约翰逊：Lindenstrauss 变换
DOI：
10.1145/1806689.1806737
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发表时间：
2010
期刊：
影响因子：
--
通讯作者：
Tamás Sarlós
中科院分区：
文献类型：
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作者：
Anirban Dasgupta;Ravi Kumar;Tamás Sarlós
尺寸降低是一种关键算法工具，其中包括最近的应用程序搜索，压缩敏感性和线性代数在这项工作中哈希和局部致密化，我们每列构建了一个稀疏的投影矩阵（1/ε）非零条目。 ，鉴于在任何投影矩阵的行数以及使用自然构建体生成的投影矩阵的稀疏性上，ω（1/ε2）的已知下限都在〜O（1/ε）更新时间上。对于A（1ε） - Approximate投影的每个非零元素，从而超过了先验方法所需的〜O（1/ε2）更新时间。 （d）最坏的情况运行时间与艾隆和自由的最佳方法相匹配。
Dimension reduction is a key algorithmic tool with many applications including nearest-neighbor search, compressed sensing and linear algebra in the streaming model. In this work we obtain a sparse version of the fundamental tool in dimension reduction -- the Johnson-Lindenstrauss transform. Using hashing and local densification, we construct a sparse projection matrix with just ~O(1/ε) non-zero entries per column. We also show a matching lower bound on the sparsity for a large class of projection matrices. Our bounds are somewhat surprising, given the known lower bounds of Ω(1/ε2) both on the number of rows of any projection matrix and on the sparsity of projection matrices generated by natural constructions. Using this, we achieve an ~O(1/ε) update time per non-zero element for a (1 ε)-approximate projection, thereby substantially outperforming the ~O(1/ε2) update time required by prior approaches. A variant of our method offers the same guarantees for sparse vectors, yet its ~O(d) worst case running time matches the best approach of Ailon and Liberty.