A Sparse Johnson-Lindenstrauss Transform Using Fast Hashing

The Sparse Johnson-Lindenstrauss Transform of Kane and Nelson (SODA 2012) provides a linear dimensionality-reducing map A ∈ R^m×^u in ℓ2 that preserves distances up to distortion of 1 + ε with probability 1 − δ, where m = O(ε⁻² log 1/δ) and each column of A has O(εm) non-zero entries. The previous analyses of the Sparse Johnson-Lindenstrauss Transform all assumed access to a Ω(log 1/δ)-wise independent hash function. The main contribution of this paper is a more general analysis of the Sparse Johnson-Lindenstrauss Transform with less assumptions on the hash function. We also show that the Mixed Tabulation hash function of Dahlgaard, Knudsen, Rotenberg, and Thorup (FOCS 2015) satisfies the conditions of our analysis, thus giving us the first analysis of a Sparse Johnson-Lindenstrauss Transform that works with a practical hash function.