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A Sparse Johnson--Lindenstrauss Transform

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 $\tilde{O}(\frac{1}ε)$ 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 $Ω(\frac{1}{ε^2})$ 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 $\tilde{O}(\frac{1}ε)$ update time per non-zero element for a $(1\pmε)$-approximate projection, thereby substantially outperforming the $\tilde{O}(\frac{1}{ε^2})$ 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.