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Adaptive sub-linear Fourier algorithms
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A new deterministic algorithm for the sparse Fourier transform problem, in which the algorithm seeks to identify k << N significant Fourier coefficients from a signal of bandwidth N, which shows that empirically it is orders of magnitude faster than competing algorithms.Abstract:
We present a new deterministic algorithm for the sparse Fourier transform problem, in which we seek to identify k << N significant Fourier coefficients from a signal of bandwidth N. Previous deterministic algorithms exhibit quadratic runtime scaling, while our algorithm scales linearly with k in the average case. Underlying our algorithm are a few simple observations relating the Fourier coefficients of time-shifted samples to unshifted samples of the input function. This allows us to detect when aliasing between two or more frequencies has occurred, as well as to determine the value of unaliased frequencies. We show that empirically our algorithm is orders of magnitude faster than competing algorithms.read more
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(Nearly) sample-optimal sparse Fourier transform
TL;DR: A randomized algorithm that computes a k-sparse approximation to the discrete Fourier transform of an n-dimensional signal in time O(k log2 n(log log n)O(1)), assuming that the entries of the signal are polynomially bounded.
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Sample-Optimal Fourier Sampling in Any Constant Dimension
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TL;DR: An algorithm for sparse recovery from Fourier measurements using O(k log N) samples, matching the lower bound of Do Ba-Indyk-Price-Woodruff'10 for non-adaptive algorithms up to constant factors for any k ≤ N1-δ.
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Sample-Optimal Average-Case Sparse Fourier Transform in Two Dimensions
TL;DR: The first sample-optimal sublinear time algorithms for the sparse Discrete Fourier Transform over a two-dimensional√n × √n grid are presented and match the lower bounds on sample complexity for their respective signal models.
Proceedings ArticleDOI
Sample-optimal average-case sparse Fourier Transform in two dimensions
TL;DR: In this paper, the authors presented the first sample-optimal sublinear time algorithms for the sparse Discrete Fourier Transform over a two-dimensional √ n × √n grid.
References
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Introduction to Algorithms
TL;DR: This chapter provides an overview of the fundamentals of algorithms and their links to self-organization, exploration, and exploitation.
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An Introduction to the Theory of Numbers
TL;DR: The fifth edition of the introduction to the theory of numbers has been published by as discussed by the authors, and the main changes are in the notes at the end of each chapter, where the author seeks to provide up-to-date references for the reader who wishes to pursue a particular topic further and to present a reasonably accurate account of the present state of knowledge.
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The Design and Implementation of FFTW3
Matteo Frigo,Steven G. Johnson +1 more
TL;DR: It is shown that such an approach can yield an implementation of the discrete Fourier transform that is competitive with hand-optimized libraries, and the software structure that makes the current FFTW3 version flexible and adaptive is described.
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