Efficiently Computing Sparse Fourier Transforms of $q$-ary Functions

TL;DR

This paper introduces a novel sparse Fourier transform algorithm tailored for $q$-ary functions, enabling efficient analysis of high-dimensional functions over larger alphabets, with proven accuracy and demonstrated scalability.

Contribution

The paper develops the first $q$-ary sparse Fourier transform algorithm, $q$-SFT, with provable efficiency and accuracy, extending sparse Fourier techniques beyond binary functions.

Findings

Algorithm computes $S$-sparse transforms with vanishing error as $q^n$ grows.

Achieves $O(Sn)$ function evaluations and $O(S n^2 ext{log} q)$ computations.

Numerical results show scalability to high-dimensional $q$-ary functions.

Abstract

Fourier transformations of pseudo-Boolean functions are popular tools for analyzing functions of binary sequences. Real-world functions often have structures that manifest in a sparse Fourier transform, and previous works have shown that under the assumption of sparsity the transform can be computed efficiently. But what if we want to compute the Fourier transform of functions defined over a $q$-ary alphabet? These types of functions arise naturally in many areas including biology. A typical workaround is to encode the $q$-ary sequence in binary, however, this approach is computationally inefficient and fundamentally incompatible with the existing sparse Fourier transform techniques. Herein, we develop a sparse Fourier transform algorithm specifically for $q$-ary functions of length $n$ sequences, dubbed $q$-SFT, which provably computes an $S$-sparse transform with vanishing error as $q^n \rightarrow \infty$ in $O(Sn)$ function evaluations and $O(S n^2 \log q)$ computations, where $S = q^{n\delta}$ for some $\delta < 1$. Under certain assumptions, we show that for fixed $q$, a robust version of $q$-SFT has a sample complexity of $O(Sn^2)$ and a computational complexity of $O(Sn^3)$ with the same asymptotic guarantees. We present numerical simulations on synthetic and real-world RNA data, demonstrating the scalability of $q$-SFT to massively high dimensional $q$-ary functions.