Given an arbitrary matrix $Ainmathbb{R}^{ntimes n}$, we consider the fundamental problem of computing $Ax$ for any $xinmathbb{R}^n$ 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 $O(n^3log n)$ operations. Next, given any unit vector $xinmathbb{R}^n$ 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 $O(sn + epsilon^{-2}|A|_{2toinfty}^2nlog n)$ operations. In addition to a performance guarantee, we provide numerical results that demonstrate the plausibility of real-world implementation of our algorithm.
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