Sketching with Kerdock's Crayons: Fast Sparsifying Transforms for Arbitrary Linear Maps

Given an arbitrary matrix $A\in\mathbb{R}^{m\times n}$, we consider the fundamental problem of computing $Ax$ for any $x\in\mathbb{R}^n$ such that is $s$-sparse. While fast algorithms exist particular choices $A$, as discrete Fourier transform, there are hardly approaches beat standard matrix-vector multiplication realistic dimensions without structural assumptions. In this paper, devise a randomized approach to tackle unstructured case. Our method relies on representation $A$ in terms certain real-valued mutually unbiased bases derived from Kerdock sets. preprocessing phase our algorithm, compute $O(mn^2\log n + n^2 \log^2n)$ operations. Next, given unit vector $s$-sparse, transform uses entrywise $\epsilon$-hard threshold with high probability only $O(sn \epsilon^{-2}\|A\|_{2\to\infty}^2 (m+n)\log m)$ addition performance guarantee, provide numerical results demonstrate plausibility real-world implementation algorithm.