## Abstract

We study the problem of finding a Fourier representation R of m terms for a given discrete signal A of length N. The Fast Fourier Transform (FFT) can find the optimal N-term representation in time O(N log N), but our goal is to get sublinear time algorithms when m " N. Suppose ∥A∥ _{2} ≤ M ∥A - R _{opt}∥ _{2}, where R _{opt} is the optimal output. The previously best known algorithms output R such that ∥ A - R∥ _{2} ^{2} ≤ (1 + ε)∥ A - R _{opt} ∥ _{2} ^{2} with probability at least 1 - δ in time* polym,(m, log(1/δ), log N, log M, 1/ε). Although this is sublinear in the input size, the dominating expression is the polynomial factor in m which, for published algorithms, is greater than or equal to the bottleneck at m ^{2} that we identify below. Our experience with these algorithms shows that this is serious limitation in theory and in practice. Our algorithm beats this m ^{2} bottleneck. Our main result is a significantly improved algorithm for this problem and the d-dimensional analog. Our algorithm outputs an R with the same approximation guarantees but it runs in time m · poly(log(1/δ), log N, log M, 1/ε). A version of the algorithm holds for all N, though the details differ slightly according to the factorization of N. For the d-dimensional problem of size N _{1} × N _{2} × ⋯× N _{d}, the linear-in-m algorithm extends efficiently to higher dimensions for certain factorizations of the N _{i}'s; we give a quadratic-in-m algorithm that works for any values of N _{i}'s. This article replaces several earlier, unpublished drafts.

Original language | English (US) |
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Article number | 59141A |

Pages (from-to) | 1-15 |

Number of pages | 15 |

Journal | Proceedings of SPIE - The International Society for Optical Engineering |

Volume | 5914 |

DOIs | |

State | Published - 2005 |

Event | Wavelets XI - San Diego, CA, United States Duration: Jul 31 2005 → Aug 3 2005 |

## Keywords

- Fourier analysis
- Randomized approximation algorithms
- Sampling
- Sparse analysis

## ASJC Scopus subject areas

- Electronic, Optical and Magnetic Materials
- Condensed Matter Physics
- Computer Science Applications
- Applied Mathematics
- Electrical and Electronic Engineering