An $O(k\log n)$ Time Fourier Set Query Algorithm
Keywords: Fourier transform, set query
TL;DR: We present a new algorithm for the Fourier set query problem that runs in $O(\epsilon^{-1} k \log(n/\delta))$ time and uses $O(\epsilon^{-1} k \log(n/\delta))$ Fourier measurements.
Abstract: 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 $x \in \mathbb{C}^{n}$, approximation parameters $\epsilon, \delta \in (0, 0.1)$, and a query set $S \subset [n]$ of size $k$, we propose an algorithm to compute an approximate Fourier transform result $x'$ which uses $O(\epsilon^{-1} k \log(n/\delta))$ Fourier measurements and runs in $O(\epsilon^{-1} k \log(n/\delta))$ time. For $\hat{x}$ being the Fourier transformation result, our algorithm can output a vector $x'$ such that $\\| ( x' - \widehat{x} )\_S \\|\_2^2 \leq \epsilon \\| \widehat{x}\_{\overline{S}} \\|\_2^2 + \delta \\| \widehat{x} \\|\_1^2 $ holds with probability of at least $9/10$, where $\overline{S}$ denotes the complement of the set $S$, i.e., $\overline{S} := [n] \setminus S$.
Primary Area: learning theory
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Submission Number: 5537
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