Sketching with Kerdock's crayons: Fast sparsifying transforms for arbitrary linear maps

Abstract

Given an arbitrary matrix $A\in {\mathbb{R}}^{n\times n}$, we consider the fundamental problem of computing $Ax$ for any $x\in {\mathbb{R}}^n$ such that $Ax$ is $s$-sparse. While fast algorithms exist for particular choices of $A$, such as the discrete Fourier transform, there is currently no $o(n^2)$ algorithm that treats the unstructured case. In this paper, we devise a randomized approach to tackle the unstructured case. Our method relies on a representation of $A$ in terms of certain real-valued mutually unbiased bases derived from Kerdock sets. In the preprocessing phase of our algorithm, we compute this representation of $A$ in $O(n^3\log n)$ operations. Next, given any unit vector $x\in {\mathbb{R}}^n$ such that $Ax$ is $s$-sparse, our randomized fast transform uses this representation of $A$ to compute the entrywise $\epsilon$-hard threshold of $Ax$ with high probability in only $O(sn+{\epsilon }^{-2}\Vert A{\Vert }_{2\to \infty }^{2}n\log n)$ operations. In addition to a performance guarantee, we provide numerical results that demonstrate the plausibility of real-world implementation of our algorithm.