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 β†’β„‚, that are sparse (or compressible) in the Fourier basis. In particular, suppose that the Fourier coefficients of f, {c_ k (f) }_ kβˆˆβ„€^d, are concentrated in a finite set I βŠ‚β„€^d so that min_Ξ©βŠ‚ I s.t. |Ξ©| =s f - βˆ‘_ k∈Ω c_ k (f) e^ -2 Ο€ i k·∘_2 < Ο΅f _2 holds for s β‰ͺ |I| and ϡ∈ (0,1). We aim to identify a near-minimizing subset Ξ©βŠ‚ 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 Ξ©βŠ‚ 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 β„€^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.

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