Sparse Fourier transforms on rank-1 lattices for the rapid and low-memory approximation of functions of many variables

Abstract

This paper considers fast and provably accurate algorithms for approximating smooth functions on the d-dimensional torus, \(f: \mathbb {T}^d \rightarrow \mathbb {C}\), that are sparse (or compressible) in the multidimensional Fourier basis. In particular, suppose that the Fourier series coefficients of f, \(\{c_\mathbf{k} (f) \}_{\mathbf{k} \in \mathbb {Z}^d}\), are concentrated in a given arbitrary finite set \(\mathscr {I} \subset \mathbb {Z}^d\) so that $$\begin{aligned} \min _{\text{\O}mega \subset \mathscr {I} ~s.t.~ \left| \text{\O}mega \right| =s }\left\| f - \sum _{\mathbf{k} \in \text{\O}mega } c_\mathbf{k} (f) \, \mathbb {e}^{ -2 \pi \mathbb {i} {{\mathbf {k}}}\cdot \circ } \right\| _2 < \epsilon \Vert f \Vert _2 \end{aligned}$$holds for \(s \ll \left| \mathscr {I} \right| \) and \(\epsilon \in (0,1)\) small. In such cases we aim to both identify a near-minimizing subset \(\text{\O}mega \subset \mathscr {I}\) and accurately approximate its associated Fourier coefficients \(\{ c_\mathbf{k} (f) \}_{\mathbf{k} \in \text{\O}mega }\) as rapidly as possible. In this paper we present both deterministic and explicit as well as randomized algorithms for solving this problem using \(\mathscr {O}(s^2 d \log ^c (|\mathscr {I}|))\)-time/memory and \(\mathscr {O}(s d \log ^c (|\mathscr {I}|))\)-time/memory, respectively. Most crucially, all of the methods proposed herein achieve these runtimes while simultaneously satisfying theoretical best s-term approximation guarantees which guarantee their numerical accuracy and robustness to noise for general functions. These results are achieved by modifying several different one-dimensional Sparse Fourier Transform (SFT) methods to subsample a function along a reconstructing rank-1 lattice for the given frequency set \(\mathscr {I} \subset \mathbb {Z}^d\) in order to rapidly identify a near-minimizing subset \(\text{\O}mega \subset \mathscr {I}\) as above without having to use anything about the lattice beyond its generating vector. This requires the development of new fast and low-memory frequency identification techniques capable of rapidly recovering vector-valued frequencies in \(\mathbb {Z}^d\) as opposed to recovering simple integer frequencies as required in the univariate setting. Two different multivariate frequency identification strategies are proposed, analyzed, and shown to lead to their own best s-term approximation methods herein, each with different accuracy versus computational speed and memory tradeoffs.