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

2025-01-21

AI Summary

This work addresses the sparse Fourier transform (SFT) problem for generalized $q$-ary functions defined over heterogeneous modulus spaces $mathbb{Z}_{q_1} imes cdots imes mathbb{Z}_{q_n}$, where moduli $q_i$ are not necessarily equal. Existing SFT algorithms assume uniform moduli, leading to excessive sampling and high computational overhead. We propose GFast—the first efficient SFT algorithm supporting arbitrary heterogeneous modulus structures. GFast integrates divide-and-conquer sampling, multidimensional spectral isolation, and adaptive sparse recovery, leveraging number-theoretic constructions and compressed sensing theory to guarantee near-zero failure probability and robustness to noise. Theoretically grounded and empirically validated, GFast reduces sample complexity by 16× and accelerates computation by 8× over state-of-the-art methods on synthetic benchmarks. On real-world protein fitness modeling, it improves interpretability of neural network interactions, reducing normalized mean square error (NMSE) by over 25%.

Technology Category

• Machine Learning: Other Foundations of Machine Learning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

Abstract

Computing the Fourier transform of a $q$-ary function $f:mathbb{Z}_{q}^n ightarrow mathbb{R}$, 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 many practical settings, the function is defined over a more general space -- the space of generalized $q$-ary sequences $mathbb{Z}_{q_1} imes mathbb{Z}_{q_2} imes cdots imes mathbb{Z}_{q_n}$ -- where each $mathbb{Z}_{q_i}$ corresponds to integers modulo $q_i$. A naive approach involves setting $q=max_i{q_i}$ and treating the function as $q$-ary, which results in heavy computational overheads. Herein, we develop GFast, an 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 $N=prod_{i=1}^n q_i ightarrow infty$ with $S = N^delta$ for some $0 leq delta<1$. In the presence of noise, we further demonstrate that a 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. Using large-scale synthetic experiments, we demonstrate that GFast computes the sparse Fourier transform of generalized $q$-ary functions using $16 imes$ fewer samples and running $8 imes$ faster than existing algorithms. In real-world protein fitness datasets, GFast explains the predictive interactions of a neural network with $>25%$ smaller normalized mean-squared error compared to existing algorithms.

Problem

• Fourier Transform
• Sparse Data
• Efficient Computation

Innovation

• GFast Algorithm
• Sparse Fourier Transform
• Protein Data Analysis