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
Autor*in: |
Gross, Craig [verfasserIn] |
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E-Artikel |
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Englisch |
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2021 |
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Anmerkung: |
© The Author(s), under exclusive licence to Springer Nature Switzerland AG 2021 |
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Übergeordnetes Werk: |
Enthalten in: Sampling theory, signal processing, and data analysis - [Cham] : Birkhäuser, 2021, 20(2021), 1 vom: 13. Dez. |
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Übergeordnetes Werk: |
volume:20 ; year:2021 ; number:1 ; day:13 ; month:12 |
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DOI / URN: |
10.1007/s43670-021-00018-y |
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Katalog-ID: |
SPR045794855 |
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520 | |a 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. | ||
650 | 4 | |a Multivariate Fourier approximation |7 (dpeaa)DE-He213 | |
650 | 4 | |a Approximation algorithms |7 (dpeaa)DE-He213 | |
650 | 4 | |a Fast Fourier transforms |7 (dpeaa)DE-He213 | |
650 | 4 | |a Sparse Fourier transforms |7 (dpeaa)DE-He213 | |
650 | 4 | |a Rank-1 lattices |7 (dpeaa)DE-He213 | |
650 | 4 | |a Fast algorithms |7 (dpeaa)DE-He213 | |
700 | 1 | |a Iwen, Mark |4 aut | |
700 | 1 | |a Kämmerer, Lutz |4 aut | |
700 | 1 | |a Volkmer, Toni |4 aut | |
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10.1007/s43670-021-00018-y doi (DE-627)SPR045794855 (SPR)s43670-021-00018-y-e DE-627 ger DE-627 rakwb eng Gross, Craig verfasserin (orcid)0000-0002-1696-6964 aut Sparse Fourier transforms on rank-1 lattices for the rapid and low-memory approximation of functions of many variables 2021 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2021 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. Multivariate Fourier approximation (dpeaa)DE-He213 Approximation algorithms (dpeaa)DE-He213 Fast Fourier transforms (dpeaa)DE-He213 Sparse Fourier transforms (dpeaa)DE-He213 Rank-1 lattices (dpeaa)DE-He213 Fast algorithms (dpeaa)DE-He213 Iwen, Mark aut Kämmerer, Lutz aut Volkmer, Toni aut Enthalten in Sampling theory, signal processing, and data analysis [Cham] : Birkhäuser, 2021 20(2021), 1 vom: 13. Dez. (DE-627)1735681601 (DE-600)3041928-1 2730-5724 nnns volume:20 year:2021 number:1 day:13 month:12 https://dx.doi.org/10.1007/s43670-021-00018-y lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_101 GBV_ILN_105 GBV_ILN_110 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 20 2021 1 13 12 |
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10.1007/s43670-021-00018-y doi (DE-627)SPR045794855 (SPR)s43670-021-00018-y-e DE-627 ger DE-627 rakwb eng Gross, Craig verfasserin (orcid)0000-0002-1696-6964 aut Sparse Fourier transforms on rank-1 lattices for the rapid and low-memory approximation of functions of many variables 2021 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2021 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. Multivariate Fourier approximation (dpeaa)DE-He213 Approximation algorithms (dpeaa)DE-He213 Fast Fourier transforms (dpeaa)DE-He213 Sparse Fourier transforms (dpeaa)DE-He213 Rank-1 lattices (dpeaa)DE-He213 Fast algorithms (dpeaa)DE-He213 Iwen, Mark aut Kämmerer, Lutz aut Volkmer, Toni aut Enthalten in Sampling theory, signal processing, and data analysis [Cham] : Birkhäuser, 2021 20(2021), 1 vom: 13. Dez. (DE-627)1735681601 (DE-600)3041928-1 2730-5724 nnns volume:20 year:2021 number:1 day:13 month:12 https://dx.doi.org/10.1007/s43670-021-00018-y lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_101 GBV_ILN_105 GBV_ILN_110 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 20 2021 1 13 12 |
allfields_unstemmed |
10.1007/s43670-021-00018-y doi (DE-627)SPR045794855 (SPR)s43670-021-00018-y-e DE-627 ger DE-627 rakwb eng Gross, Craig verfasserin (orcid)0000-0002-1696-6964 aut Sparse Fourier transforms on rank-1 lattices for the rapid and low-memory approximation of functions of many variables 2021 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2021 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. Multivariate Fourier approximation (dpeaa)DE-He213 Approximation algorithms (dpeaa)DE-He213 Fast Fourier transforms (dpeaa)DE-He213 Sparse Fourier transforms (dpeaa)DE-He213 Rank-1 lattices (dpeaa)DE-He213 Fast algorithms (dpeaa)DE-He213 Iwen, Mark aut Kämmerer, Lutz aut Volkmer, Toni aut Enthalten in Sampling theory, signal processing, and data analysis [Cham] : Birkhäuser, 2021 20(2021), 1 vom: 13. Dez. (DE-627)1735681601 (DE-600)3041928-1 2730-5724 nnns volume:20 year:2021 number:1 day:13 month:12 https://dx.doi.org/10.1007/s43670-021-00018-y lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_101 GBV_ILN_105 GBV_ILN_110 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 20 2021 1 13 12 |
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10.1007/s43670-021-00018-y doi (DE-627)SPR045794855 (SPR)s43670-021-00018-y-e DE-627 ger DE-627 rakwb eng Gross, Craig verfasserin (orcid)0000-0002-1696-6964 aut Sparse Fourier transforms on rank-1 lattices for the rapid and low-memory approximation of functions of many variables 2021 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2021 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. Multivariate Fourier approximation (dpeaa)DE-He213 Approximation algorithms (dpeaa)DE-He213 Fast Fourier transforms (dpeaa)DE-He213 Sparse Fourier transforms (dpeaa)DE-He213 Rank-1 lattices (dpeaa)DE-He213 Fast algorithms (dpeaa)DE-He213 Iwen, Mark aut Kämmerer, Lutz aut Volkmer, Toni aut Enthalten in Sampling theory, signal processing, and data analysis [Cham] : Birkhäuser, 2021 20(2021), 1 vom: 13. Dez. (DE-627)1735681601 (DE-600)3041928-1 2730-5724 nnns volume:20 year:2021 number:1 day:13 month:12 https://dx.doi.org/10.1007/s43670-021-00018-y lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_101 GBV_ILN_105 GBV_ILN_110 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 20 2021 1 13 12 |
allfieldsSound |
10.1007/s43670-021-00018-y doi (DE-627)SPR045794855 (SPR)s43670-021-00018-y-e DE-627 ger DE-627 rakwb eng Gross, Craig verfasserin (orcid)0000-0002-1696-6964 aut Sparse Fourier transforms on rank-1 lattices for the rapid and low-memory approximation of functions of many variables 2021 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2021 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. Multivariate Fourier approximation (dpeaa)DE-He213 Approximation algorithms (dpeaa)DE-He213 Fast Fourier transforms (dpeaa)DE-He213 Sparse Fourier transforms (dpeaa)DE-He213 Rank-1 lattices (dpeaa)DE-He213 Fast algorithms (dpeaa)DE-He213 Iwen, Mark aut Kämmerer, Lutz aut Volkmer, Toni aut Enthalten in Sampling theory, signal processing, and data analysis [Cham] : Birkhäuser, 2021 20(2021), 1 vom: 13. Dez. (DE-627)1735681601 (DE-600)3041928-1 2730-5724 nnns volume:20 year:2021 number:1 day:13 month:12 https://dx.doi.org/10.1007/s43670-021-00018-y lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_101 GBV_ILN_105 GBV_ILN_110 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 20 2021 1 13 12 |
language |
English |
source |
Enthalten in Sampling theory, signal processing, and data analysis 20(2021), 1 vom: 13. Dez. volume:20 year:2021 number:1 day:13 month:12 |
sourceStr |
Enthalten in Sampling theory, signal processing, and data analysis 20(2021), 1 vom: 13. Dez. volume:20 year:2021 number:1 day:13 month:12 |
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topic_facet |
Multivariate Fourier approximation Approximation algorithms Fast Fourier transforms Sparse Fourier transforms Rank-1 lattices Fast algorithms |
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container_title |
Sampling theory, signal processing, and data analysis |
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Gross, Craig @@aut@@ Iwen, Mark @@aut@@ Kämmerer, Lutz @@aut@@ Volkmer, Toni @@aut@@ |
publishDateDaySort_date |
2021-12-13T00:00:00Z |
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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. 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Gross, Craig |
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Gross, Craig misc Multivariate Fourier approximation misc Approximation algorithms misc Fast Fourier transforms misc Sparse Fourier transforms misc Rank-1 lattices misc Fast algorithms Sparse Fourier transforms on rank-1 lattices for the rapid and low-memory approximation of functions of many variables |
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Sparse Fourier transforms on rank-1 lattices for the rapid and low-memory approximation of functions of many variables Multivariate Fourier approximation (dpeaa)DE-He213 Approximation algorithms (dpeaa)DE-He213 Fast Fourier transforms (dpeaa)DE-He213 Sparse Fourier transforms (dpeaa)DE-He213 Rank-1 lattices (dpeaa)DE-He213 Fast algorithms (dpeaa)DE-He213 |
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misc Multivariate Fourier approximation misc Approximation algorithms misc Fast Fourier transforms misc Sparse Fourier transforms misc Rank-1 lattices misc Fast algorithms |
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Sparse Fourier transforms on rank-1 lattices for the rapid and low-memory approximation of functions of many variables |
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Sparse Fourier transforms on rank-1 lattices for the rapid and low-memory approximation of functions of many variables |
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sparse fourier transforms on rank-1 lattices for the rapid and low-memory approximation of functions of many variables |
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Sparse Fourier transforms on rank-1 lattices for the rapid and low-memory approximation of functions of many variables |
abstract |
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. © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2021 |
abstractGer |
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. © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2021 |
abstract_unstemmed |
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. © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2021 |
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1 |
title_short |
Sparse Fourier transforms on rank-1 lattices for the rapid and low-memory approximation of functions of many variables |
url |
https://dx.doi.org/10.1007/s43670-021-00018-y |
remote_bool |
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author2 |
Iwen, Mark Kämmerer, Lutz Volkmer, Toni |
author2Str |
Iwen, Mark Kämmerer, Lutz Volkmer, Toni |
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doi_str |
10.1007/s43670-021-00018-y |
up_date |
2024-07-03T18:20:27.875Z |
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|
score |
7.400161 |