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Efficient Algorithm for Sparse Fourier Transform of Generalized qqq-ary Functions

21 January 2025
Darin Tsui
Kunal Talreja
Amirali Aghazadeh
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Abstract

Computing the Fourier transform of a qqq-ary function f:Zqn→Rf:\mathbb{Z}_{q}^n\rightarrow \mathbb{R}f:Zqn​→R, which maps qqq-ary sequences to real numbers, is an important problem in mathematics with wide-ranging applications in biology, signal processing, and machine learning. Previous studies have shown that, under the sparsity assumption, the Fourier transform can be computed efficiently using fast and sample-efficient algorithms. However, in most practical settings, the function is defined over a more general space -- the space of generalized qqq-ary sequences Zq1×Zq2×⋯×Zqn\mathbb{Z}_{q_1} \times \mathbb{Z}_{q_2} \times \cdots \times \mathbb{Z}_{q_n}Zq1​​×Zq2​​×⋯×Zqn​​ -- where each Zqi\mathbb{Z}_{q_i}Zqi​​ corresponds to integers modulo qiq_iqi​. Herein, we develop GFast, a coding theoretic algorithm that computes the SSS-sparse Fourier transform of fff with a sample complexity of O(Sn)O(Sn)O(Sn), computational complexity of O(Snlog⁡N)O(Sn \log N)O(SnlogN), and a failure probability that approaches zero as N=∏i=1nqi→∞N=\prod_{i=1}^n q_i \rightarrow \inftyN=∏i=1n​qi​→∞ with S=NδS = N^\deltaS=Nδ for some 0≤δ<10 \leq \delta < 10≤δ<1. We show that a noise-robust version of GFast computes the transform with a sample complexity of O(Sn2)O(Sn^2)O(Sn2) and computational complexity of O(Sn2log⁡N)O(Sn^2 \log N)O(Sn2logN) under the same high probability guarantees. Additionally, we demonstrate that GFast computes the sparse Fourier transform of generalized qqq-ary functions 8×8\times8× faster using 16×16\times16× fewer samples on synthetic experiments, and enables explaining real-world heart disease diagnosis and protein fitness models using up to 13×13\times13× fewer samples compared to existing Fourier algorithms applied to the most efficient parameterization of the models as qqq-ary functions.

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@article{tsui2025_2501.12365,
  title={ Efficient Algorithm for Sparse Fourier Transform of Generalized $q$-ary Functions },
  author={ Darin Tsui and Kunal Talreja and Amirali Aghazadeh },
  journal={arXiv preprint arXiv:2501.12365},
  year={ 2025 }
}
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