Sparse Fourier Transforms on Rank-1 Lattices for the Rapid and Low-Memory Approximation of Functions of Many Variables

We consider fast, provably accurate algorithms for approximating functions on the $d$-dimensional torus, $f: \mathbb{ T }^d \rightarrow \mathbb{C}$, that are sparse (or compressible) in the Fourier basis. In particular, suppose that the Fourier coefficients of $f$, ${c{\bf k} (f) }{{\bf k} \in \mathbb{Z}^d}$, are concentrated in a finite set $I \subset \mathbb{Z}^d$ so that $$\min{\Omega \subset I s.t. |\Omega| =s } \left| f - \sum{{\bf k} \in \Omega} c{\bf k} (f) e^{ -2 \pi i {\bf k} \cdot \circ} \right|2 < \epsilon |f |2$$ holds for $s \ll |I|$ and $\epsilon \in (0,1)$. We aim to identify a near-minimizing subset $\Omega \subset I$ and accurately approximate the associated Fourier coefficients ${ c{\bf k} (f) }_{{\bf k} \in \Omega}$ as rapidly as possible. We present both deterministic as well as randomized algorithms using $O(s² d \log^c (|I|))$-time/memory and $O(s d \log^c (|I|))$-time/memory, respectively. Most crucially, all of the methods proposed herein achieve these runtimes while satisfying theoretical best $s$-term approximation guarantees which guarantee their numerical accuracy and robustness to noise for general functions. These are achieved by modifying several one-dimensional Sparse Fourier Transform (SFT) methods to subsample a function along a reconstructing rank-1 lattice for the given frequency set $I$ to rapidly identify a near-minimizing subset $\Omega \subset I$ without using anything about the lattice beyond its generating vector. This requires new fast and low-memory frequency identification techniques capable of rapidly recovering vector-valued frequencies in $\mathbb{Z}^d$ as opposed to simple integer frequencies in the univariate setting. Two different strategies are proposed and analyzed, each with different accuracy versus computational speed and memory tradeoffs.