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arXiv · 2208.09634

An $O(k\log n)$ Time Fourier Set Query Algorithm

Yeqi Gao·Zhao Song·Baocheng Sun
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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 $x \in \mathbb{C}^{n}$, an approximation parameter $\epsilon$ 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, runs in $O(\epsilon^{-1} k \log(n/\delta))$ time and outputs a vector $x'$ such that $\| ( x' - \widehat{x} )_S \|_2^2 \leq \epsilon \| \widehat{x}_{\bar{S}} \|_2^2 + \delta \| \widehat{x} \|_1^2 $ holds with probability of at least $9/10$.

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Yeqi Gao, Zhao Song, Baocheng Sun. 2022-08-20. An $O(k\log n)$ Time Fourier Set Query Algorithm. https://arxiv.org/abs/2208.09634

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