Efficiently Computing Sparse Fourier Transforms of $q$-ary Functions
This work addresses a computational bottleneck for analyzing q-ary functions in fields like biology, offering a novel method that is more efficient than existing workarounds.
The paper tackles the problem of computing sparse Fourier transforms for functions defined over q-ary alphabets, which are inefficient with binary encodings, and develops the q-SFT algorithm that provably computes an S-sparse transform with vanishing error in O(Sn) function evaluations and O(S n^2 log q) computations, demonstrating scalability on synthetic and real-world RNA data.
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 $S = q^{nδ}$ for some $δ< 1$. 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.