An O(klog n) Time Fourier Set Query Algorithm
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Fourier transformation is an extensively studied problem in many research fields. It has many applications in machine learning, signal processing, compressed sensing, and so on. In many real-world applications, approximated Fourier transformation is sufficient and we only need to do the Fourier transform on a subset of coordinates. Given a vector x ∈ℂ^n, an approximation parameter ϵ and a query set S ⊂ [n] of size k, we propose an algorithm to compute an approximate Fourier transform result x' which uses O(ϵ^-1 k log(n/δ)) Fourier measurements, runs in O(ϵ^-1 k log(n/δ)) time and outputs a vector x' such that ( x' - x )_S _2^2 ≤ϵx_S̅_2^2 + δx_1^2 holds with probability of at least 9/10.
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