Optimal Fast Johnson-Lindenstrauss Embeddings for Large Data Sets

We introduce a new fast construction of a Johnson-Lindenstrauss matrix based on the composition of the following two embeddings: A fast construction by the second author joint with Ward [1] maps points into a space of lower, but not optimal dimension. Then a subsequent transformation by a dense matrix with independent entries reaches an optimal embedding dimension. As we show in this note, the computational cost of applying this transform simultaneously to all points in a large data set comes close to the complexity of just reading the data under only very mild restrictions on the size of the data set. Along the way, our construction also yields the least restricted Johnson-Lindenstrauss Transform of order optimal embedding dimension known to date that allows for a fast query step, that is, a fast application to an arbitrary point that is not part of the given data set. [email protected], Department of Mathematics, Technische Universität München [email protected], Department of Mathematics, Technische Universität München