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An $O(k\log n)$ Time Fourier Set Query Algorithm
- Publication Year :
- 2022
-
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 $9/10$.
- Subjects :
- Computer Science - Data Structures and Algorithms
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.2208.09634
- Document Type :
- Working Paper