Rapidly computing sparse Legendre expansions via sparse Fourier transforms

Hu, Xianfeng; Iwen, Mark A.; Kim, Hye-Jin

doi:10.1007/s11075-016-0184-x

Cited by 10 publications

(17 citation statements)

References 50 publications

(117 reference statements)

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“…The support identification algorithm A is assumed to have O (L A ) runtime complexity in line 9. A conjugate gradient least square solver can approximate line 12 with O s 2 K 2 log 4 |I N,d | runtime complexity per iteration (see, e.g., Chapter 7 of [6], and Section 3 of [23]). Furthermore, a constant number of iterations (e.g.…”

Section: Now Assume That the First Conditionmentioning

confidence: 99%

“…As sublinear-time methods for the one-dimensional Fourier basis started to mature, similar algorithms began to be developed for other one-dimensional bases B as well, including for the cosine, Chebyshev, and Legendre polynomial bases [23,4] (see also [37] for traditional compressive sensing methods which focus on the Legendre polynomial basis). Recently these ideas have been extended yet further to produce sublineartime algorithms with reconstruction guarantees for restricted classes of signals exhibiting approximate sparsity in any given one-dimensional Jacobi polynomial basis [18].…”

Section: Introductionmentioning

confidence: 99%

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Sparse Harmonic Transforms II: Best $s$-Term Approximation Guarantees for Bounded Orthonormal Product Bases in Sublinear-Time

Choi¹,

Iwen²,

Volkmer³

2019

Preprint

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In this paper we develop a sublinear-time compressive sensing algorithm for approximating functions of many variables which are compressible in a given Bounded Orthonormal Product Basis (BOPB). The resulting algorithm is shown to both have an associated best s-term recovery guarantee in the given BOPB, and also to work well numerically for solving sparse approximation problems involving functions contained in the span of fairly general sets of as many as ∼ 10 230 orthonormal basis functions. All code is made publicly available.As part of the proof of the main recovery guarantee new variants of the well known CoSaMP algorithm are proposed which can utilize any sufficiently accurate support identification procedure satisfying a Support Identification Property (SIP) in order to obtain strong sparse approximation guarantees. These new CoSaMP variants are then proven to have both runtime and recovery error behavior which are largely determined by the associated runtime and error behavior of the chosen support identification method. The main theoretical results of the paper are then shown by developing a sublinear-time support identification algorithm for general BOPB sets which is robust to arbitrary additive errors. Using this new support identification method to create a new CoSaMP variant then results in a new robust sublinear-time compressive sensing algorithm for BOPB-compressible functions of many variables.

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Section: Now Assume That the First Conditionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Sparse Harmonic Transforms II: Best $s$-Term Approximation Guarantees for Bounded Orthonormal Product Bases in Sublinear-Time

Choi¹,

Iwen²,

Volkmer³

2019

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…This class of block Fourier sparse functions also appears in related numerical methods for the rapid approximation of functions which exhibit sparsity with respect to other orthonormal basis functions. For example, one can rapidly approximate functions which are a sparse combination of highdegree Legendre polynomials by computing the DFT of samples from a related periodic function which is always guaranteed to be approximately block frequency sparse [22].…”

Section: A General Class Of Functions With Structured Frequency Supportmentioning

confidence: 99%

A deterministic sparse FFT for functions with structured Fourier sparsity

2018

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“…The error bound stated in (13) follows. The runtimes follow by observing that c 2 = O α · log We are now ready to empirically evaluate Algorithm 1 with several different SFT algorithms A used in its line 9.…”

Section: Algorithm 1's Operation Count Is Thenmentioning

confidence: 99%

A New Class of Fully Discrete Sparse Fourier Transforms: Faster Stable Implementations with Guarantees

Merhi

Zhang

Iwen

et al. 2018

J Fourier Anal Appl

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In this paper we consider Sparse Fourier Transform (SFT) algorithms for approximately computing the best s-term approximation of the Discrete Fourier Transform (DFT)f ∈ C N of any given input vector f ∈ C N in just (s log N ) O(1) -time using only a similarly small number of entries of f . In particular, we present a deterministic SFT algorithm which is guaranteed to always recover a near best s-term approximation of the DFT of any given input vector f ∈ C N in O s 2 log 11 2 (N )time. Unlike previous deterministic results of this kind, our deterministic result holds for both arbitrary vectors f ∈ C N and vector lengths N . In addition to these deterministic SFT results, we also develop several new publicly available randomized SFT implementations for approximately computingf from f using the same general techniques. The best of these new implementations is shown to outperform existing discrete sparse Fourier transform methods with respect to both runtime and noise robustness for large vector lengths N . 2 ∩Z, by sampling its associated trigonometric polynomial f (x) = ω∈Bf ω e iωx

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Rapidly computing sparse Legendre expansions via sparse Fourier transforms

Cited by 10 publications

References 50 publications

Sparse Harmonic Transforms II: Best $s$-Term Approximation Guarantees for Bounded Orthonormal Product Bases in Sublinear-Time

Sparse Harmonic Transforms II: Best $s$-Term Approximation Guarantees for Bounded Orthonormal Product Bases in Sublinear-Time

A deterministic sparse FFT for functions with structured Fourier sparsity

A New Class of Fully Discrete Sparse Fourier Transforms: Faster Stable Implementations with Guarantees

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