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

Abstract

Computing the Fourier transform of a qq-ary function f:ZqnRf:\mathbb{Z}_{q}^n\rightarrow \mathbb{R}, which maps qq-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 qq-ary sequences Zq1×Zq2××Zqn\mathbb{Z}_{q_1} \times \mathbb{Z}_{q_2} \times \cdots \times \mathbb{Z}_{q_n} -- where each Zqi\mathbb{Z}_{q_i} corresponds to integers modulo qiq_i. Herein, we develop GFast, a coding theoretic algorithm that computes the SS-sparse Fourier transform of ff with a sample complexity of O(Sn)O(Sn), computational complexity of O(SnlogN)O(Sn \log N), and a failure probability that approaches zero as N=i=1nqiN=\prod_{i=1}^n q_i \rightarrow \infty with S=NδS = N^\delta for some 0δ<10 \leq \delta < 1. We show that a noise-robust version of GFast computes the transform with a sample complexity of O(Sn2)O(Sn^2) and computational complexity of O(Sn2logN)O(Sn^2 \log N) under the same high probability guarantees. Additionally, we demonstrate that GFast computes the sparse Fourier transform of generalized qq-ary functions 8×8\times faster using 16×16\times fewer samples on synthetic experiments, and enables explaining real-world heart disease diagnosis and protein fitness models using up to 13×13\times fewer samples compared to existing Fourier algorithms applied to the most efficient parameterization of the models as qq-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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