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 coe...
Ausführliche Beschreibung

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 minΩ⊂Is.t.Ω=sf-∑k∈Ωck(f)e-2πik·∘2<ϵ‖f‖2%$\begin{aligned} \min _{\Omega \subset \mathscr {I} ~s.t.~ \left| \Omega \right| =s }\left\| f - \sum _{\mathbf{k} \in \Omega } c_\mathbf{k} (f) \, \mathbb {e}^{ -2 \pi \mathbb {i} {{\mathbf {k}}}\cdot \circ } \right\| _2 < \epsilon \Vert f \Vert _2 \end{aligned}%$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 %$\Omega \subset \mathscr {I}%$ and accurately approximate its associated Fourier coefficients %$\{ c_\mathbf{k} (f) \}_{\mathbf{k} \in \Omega }%$ 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 %$\Omega \subset \mathscr {I}%$ 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. Ausführliche Beschreibung

Autor*in:	Gross, Craig [verfasserIn] Iwen, Mark Kämmerer, Lutz Volkmer, Toni

Format:	E-Artikel
Sprache:	Englisch

Erschienen:	2021

Schlagwörter:	Multivariate Fourier approximation Approximation algorithms Fast Fourier transforms Sparse Fourier transforms Rank-1 lattices Fast algorithms

Anmerkung:	© The Author(s), under exclusive licence to Springer Nature Switzerland AG 2021

Übergeordnetes Werk:	Enthalten in: Sampling theory, signal processing, and data analysis - [Cham] : Birkhäuser, 2021, 20(2021), 1 vom: 13. Dez.
Übergeordnetes Werk:	volume:20 ; year:2021 ; number:1 ; day:13 ; month:12

Links:	Volltext

DOI / URN:	10.1007/s43670-021-00018-y

Katalog-ID:	SPR045794855