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We prove the Johnson-Lindenstrauss property for matrices ΦD ξ where Φ has the restricted isometry property and D ξ is a diagonal matrix containing the entries of a Kronecker product d) of d independent Rademacher vectors. Such embeddings have been proposed in recent works for a number of applications concerning compression of tensor structured data, including the oblivious sketching procedure by Ahle et al. for approximate tensor computations. For preserving the norms of p points simultaneously, our result requires Φ to have the restricted isometry property for sparsity C(d)(log p) d . In the case of subsampled Hadamard matrices, this can improve the dependence of the embedding dimension on p to (log p) d while the best previously known result required (log p) d+1 . That is, for the case of d = 2 at the core of the oblivious sketching procedure by Ahle et al., the scaling improves from cubic to quadratic. We provide a counterexample to prove that the scaling established in our result is optimal under mild assumptions.

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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.