Combinatorial Sublinear-Time Fourier Algorithms

We study the problem of estimating the best k term Fourier representation for a given frequency sparse signal (i.e., vector) A of length N ≫ k . More explicitly, we investigate how to deterministically identify k of the largest magnitude frequencies of , and estimate their coefficients, in polynomia...

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Bibliographic Details
Published in:Foundations of computational mathematics Vol. 10; no. 3; pp. 303 - 338
Main Author: Iwen, M. A.
Format: Journal Article
Language:English
Published: New York Springer-Verlag 01.06.2010
Springer Nature B.V
Subjects:
Applications of Mathematics
Computer Science
Economics
Linear and Multilinear Algebras
Math Applications in Computer Science
Mathematics
Mathematics and Statistics
Matrix Theory
Numerical Analysis
05-04
65T50
42A15
65T40
42A10
42A32
Trigonometric approximation and interpolation
Approximation algorithms
Discrete and fast Fourier transforms
68W25
94A12
ISSN:1615-3375, 1615-3383
Online Access:Get full text
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Summary:We study the problem of estimating the best k term Fourier representation for a given frequency sparse signal (i.e., vector) A of length N ≫ k . More explicitly, we investigate how to deterministically identify k of the largest magnitude frequencies of , and estimate their coefficients, in polynomial( k ,log  N ) time. Randomized sublinear-time algorithms which have a small (controllable) probability of failure for each processed signal exist for solving this problem (Gilbert et al. in ACM STOC, pp. 152–161, 2002 ; Proceedings of SPIE Wavelets XI, 2005 ). In this paper we develop the first known deterministic sublinear-time sparse Fourier Transform algorithm which is guaranteed to produce accurate results. As an added bonus, a simple relaxation of our deterministic Fourier result leads to a new Monte Carlo Fourier algorithm with similar runtime/sampling bounds to the current best randomized Fourier method (Gilbert et al. in Proceedings of SPIE Wavelets XI, 2005 ). Finally, the Fourier algorithm we develop here implies a simpler optimized version of the deterministic compressed sensing method previously developed in (Iwen in Proc. of ACM-SIAM Symposium on Discrete Algorithms (SODA’08), 2008 ).
Bibliography:SourceType-Scholarly Journals-1
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ISSN:1615-3375
1615-3383
DOI:10.1007/s10208-009-9057-1