Kac meets Johnson and Lindenstrauss: a memory-optimal, fast Johnson-Lindenstrauss transform
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Based on the Kac random walk on the orthogonal group, we present a fast Johnson-Lindenstrauss transform: given a set X of n point sets in R^d and an error parameter ϵ, this is a linear transformation Ψ: R^d→R^O(ϵ^-2logn) such that Ψ x_2∈ (1- ϵ, 1+ϵ)·x_2 for all x∈ X, and such that for each x∈ X, Ψ x can be computed in time O(dlogd + min{dlogn + ϵ^-2log^3nlog^3(ϵ^-1logn)}) with only a constant amount of memory overhead. In some parameter regimes, our algorithm is best known, and essentially confirms a conjecture of Ailon and Chazelle.
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