A Sparse Johnson--lindenstrauss Transform
Dasgupta Anirban, Kumar Ravi, Sarlós Tamás. Arxiv 2010
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 {\em 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 \(\tilde{O}(\frac{1}{\epsilon})\) 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 \(Ω(\frac{1}{\epsilon^2})\) 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 \(\tilde{O}(\frac{1}{\epsilon})\) update time per non-zero element for a \((1\pm\epsilon)\)-approximate projection, thereby substantially outperforming the \(\tilde{O}(\frac{1}{\epsilon^2})\) update time required by prior approaches. A variant of our method offers the same guarantees for sparse vectors, yet its \(\tilde{O}(d)\) worst case running time matches the best approach of Ailon and Liberty.
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