Johnson-Lindenstrauss Embeddings with Kronecker Structure

We prove the Johnson-Lindenstrauss property for matrices Phi D_xi where Phi has the restricted isometry property and D_xi is a diagonal matrix containing the entries of a Kronecker product xi = xi(1) otimes dots otimes xi(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 Phi 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.