• Corpus ID: 7908985

A sparse multidimensional FFT for real positive vectors

@article{Ltourneau2016ASM,
  title={A sparse multidimensional FFT for real positive vectors},
  author={Pierre-David L{\'e}tourneau and Harper Langston and Beno{\^i}t Meister and Richard A. Lethin},
  journal={ArXiv},
  year={2016},
  volume={abs/1604.06682}
}
We present a sparse multidimensional FFT (sMFFT) randomized algorithm for real positive vectors. The algorithm works in any fixed dimension, requires (O(R log(R) log(N)) ) samples and runs in O( R log^2(R) log(N)) complexity (where N is the total size of the vector in d dimensions and R is the number of nonzeros). It is stable to low-level noise and exhibits an exponentially small probability of failure. 

Figures and Tables from this paper

A sparse multi-dimensional Fast Fourier Transform with stability to noise in the context of image processing and change detection

Numerical results show the sparse multidimensional FFT (sMFFT)'s large quantitative and qualitative strengths as compared to ℓ1-minimization for Compressive Sensing as well as advantages in the context of image processing and change detection.

Approximate Inverse Chain Preconditioner: Iteration Count Case Study for Spectral Support Solvers

A new preconditioner for symmetric diagonally dominant systems of linear equations that is both algebraic in nature as well as hierarchically-constrained depending on the condition number of the system to be solved and accelerated by utilizing precomputations to simplify setup and multiplications in the context of an iterative Krylov-subspace solver.

References

SHOWING 1-10 OF 42 REFERENCES

A sparse multi-dimensional Fast Fourier Transform with stability to noise in the context of image processing and change detection

Numerical results show the sparse multidimensional FFT (sMFFT)'s large quantitative and qualitative strengths as compared to ℓ1-minimization for Compressive Sensing as well as advantages in the context of image processing and change detection.

(Nearly) Sample-Optimal Sparse Fourier Transform

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.

Sample-optimal average-case sparse Fourier Transform in two dimensions

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.

A sparse fast Fourier algorithm for real non-negative vectors

Sparse fourier transform in any constant dimension with nearly-optimal sample complexity in sublinear time

This work considers the problem of computing a k-sparse approximation to the Fourier transform of a length N signal and proposes a randomized algorithm for computing such an approximation using Od(klogNloglogN) samples of the signal in time domain and Od( klogd+3 N) runtime.

Adaptive Sub-Linear Time Fourier Algorithms

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 is orders of magnitude faster than competing algorithms.

Computing a k-sparse n-length Discrete Fourier Transform using at most 4k samples and O(k log k) complexity

This work shows how computing the sparse DFT X is equivalent to decoding of these sparse-graph codes and is low in both sample complexity and decoding complexity, and dubs the algorithm the FFAST (Fast Fourier Aliasing-based Sparse Transform) algorithm.

Improved time bounds for near-optimal sparse Fourier representations

A significantly improved algorithm for the problem of finding a Fourier representation R of m terms for a given discrete signal A of length N and a quadratic-in-m algorithm that works for any values of Ni's is given.

Simple and practical algorithm for sparse Fourier transform

This work considers the sparse Fourier transform problem, and proposes a new algorithm, which leverages techniques from digital signal processing, notably Gaussian and Dolph-Chebyshev filters, and is faster than FFT, both in theory and practice.

Deterministic sparse FFT algorithms

The focus is on vectors with small support or sparse vectors for which several new deterministic algorithms are proposed that have a lower complexity than regular FFT algorithms, and sublinear time algorithms for the reconstruction of complex vectors or matrices withSmall support from Fourier data as well as an algorithm for the reconstruct of real nonnegative vectors.