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

12/17/2020

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by   Craig Gross, et al.

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We consider fast, provably accurate algorithms for approximating functions on the d-dimensional torus, f: 𝕋 ^d →ℂ, that are sparse (or compressible) in the Fourier basis. In particular, suppose that the Fourier coefficients of f, {c_ k (f) }_ k∈ℤ^d, are concentrated in a finite set I ⊂ℤ^d so that min_Ω⊂ I s.t. |Ω| =s f - ∑_ k∈Ω c_ k (f) e^ -2 π i k·∘_2 < ϵf _2 holds for s ≪ |I| and ϵ∈ (0,1). We aim to identify a near-minimizing subset Ω⊂ I and accurately approximate the associated Fourier coefficients { c_ k (f) }_ k∈Ω as rapidly as possible. We present both deterministic as well as randomized algorithms using O(s^2 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 Ω⊂ 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 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.