### 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we give a quadratic-in-m algorithm that works for any values of N _{i}'s. This article replaces several earlier, unpublished drafts.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) |
---|---|

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 - Dec 1 2005 |

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

### Fingerprint

### 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

### Cite this

*Proceedings of SPIE - The International Society for Optical Engineering*,

*5914*, 1-15. [59141A]. https://doi.org/10.1117/12.615931

**Improved time bounds for near-optimal sparse Fourier representations.** / Gilbert, A. C.; Muthukrishnan, Shanmugavelayutham; Strauss, M.

Research output: Contribution to journal › Conference article

*Proceedings of SPIE - The International Society for Optical Engineering*, vol. 5914, 59141A, pp. 1-15. https://doi.org/10.1117/12.615931

}

TY - JOUR

T1 - Improved time bounds for near-optimal sparse Fourier representations

AU - Gilbert, A. C.

AU - Muthukrishnan, Shanmugavelayutham

AU - Strauss, M.

PY - 2005/12/1

Y1 - 2005/12/1

N2 - 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.

AB - 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.

KW - Fourier analysis

KW - Randomized approximation algorithms

KW - Sampling

KW - Sparse analysis

UR - http://www.scopus.com/inward/record.url?scp=30844452004&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=30844452004&partnerID=8YFLogxK

U2 - 10.1117/12.615931

DO - 10.1117/12.615931

M3 - Conference article

AN - SCOPUS:30844452004

VL - 5914

SP - 1

EP - 15

JO - Proceedings of SPIE - The International Society for Optical Engineering

JF - Proceedings of SPIE - The International Society for Optical Engineering

SN - 0277-786X

M1 - 59141A

ER -