Efficient Algorithm for Sparse Fourier Transform of Generalized $q$-ary Functions

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