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

TL;DR

This paper introduces an efficient algorithm for approximate Fourier transforms on subsets of data, achieving $O(k ext{log} n)$ time complexity, suitable for large-scale applications in signal processing and machine learning.

Contribution

The paper presents a novel $O(k ext{log} n)$ time algorithm for approximate Fourier set queries, improving efficiency for large datasets.

Findings

Uses $O(rac{1}{ ext{epsilon}} k ext{log}(rac{n}{ ext{delta}}))$ measurements

Runs in $O(rac{1}{ ext{epsilon}} k ext{log}(rac{n}{ ext{delta}}))$ time

Guarantees approximation with high probability (≥ 90%)

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 $9/10$.