Sketching with Kerdock's crayons: Fast sparsifying transforms for arbitrary linear maps
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Given an arbitrary matrix A∈ℝ^n× n, we consider the fundamental problem of computing Ax for any x∈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 x∈ℝ^n such that Ax is s-sparse, our randomized fast transform uses this representation of A to compute the entrywise ϵ-hard threshold of Ax with high probability in only O(sn + ϵ^-2A_2→∞^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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