Johnson–Lindenstrauss Embeddings with Kronecker Structure

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