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One sketch for all: fast algorithms for compressed sensing

Published: 11 June 2007 Publication History

Abstract

Compressed Sensing is a new paradigm for acquiring the compressible signals that arise in many applications. These signals can be approximated using an amount of information much smaller than the nominal dimension of the signal. Traditional approaches acquire the entire signal and process it to extract the information. The new approach acquires a small number of nonadaptive linear measurements of the signal and uses sophisticated algorithms to determine its information content. Emerging technologies can compute these general linear measurements of a signal at unit cost per measurement.
This paper exhibits a randomized measurement ensemble and a signal reconstruction algorithm that satisfy four requirements: 1. The measurement ensemble succeeds for all signals, with high probability over the random choices in its construction. 2. The number of measurements of the signal is optimal, except for a factor polylogarithmic in the signal length. 3. The running time of the algorithm is polynomial in the amount of information in the signal and polylogarithmic in the signal length. 4. The recovery algorithm offers the strongest possible type of error guarantee. Moreover, it is a fully polynomial approximation scheme with respect to this type of error bound.
Emerging applications demand this level of performance. Yet no otheralgorithm in the literature simultaneously achieves all four of these desiderata.

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    cover image ACM Conferences
    STOC '07: Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
    June 2007
    734 pages
    ISBN:9781595936318
    DOI:10.1145/1250790
    Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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    Published: 11 June 2007

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    Author Tags

    1. approximation
    2. embedding
    3. group testing
    4. sketching
    5. sparse approximation
    6. sublinear algorithms

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    June 11 - 13, 2007
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    • (2024)Convergence of projected subgradient method with sparse or low-rank constraintsAdvances in Computational Mathematics10.1007/s10444-024-10163-250:4Online publication date: 2-Jul-2024
    • (2023)Comparison of Common Algorithms for Single-Pixel Imaging via Compressed SensingSensors10.3390/s2310467823:10(4678)Online publication date: 11-May-2023
    • (2023)An Efficient Compressive Sensing Event-Detection Scheme for Internet of Things System Based on Sparse-Graph CodesSensors10.3390/s2310462023:10(4620)Online publication date: 10-May-2023
    • (2023)Robust multitask compressive sampling via deep generative models for crack detection in structural health monitoringStructural Health Monitoring10.1177/1475921723118366323:3(1383-1402)Online publication date: 17-Jul-2023
    • (2022)Coded Compressed Sensing With List Recoverable Codes for the Unsourced Random AccessIEEE Transactions on Communications10.1109/TCOMM.2022.321690170:12(7886-7898)Online publication date: Dec-2022
    • (2022)Communication-Efficient Distributed SGD With Compressed SensingIEEE Control Systems Letters10.1109/LCSYS.2021.31378596(2054-2059)Online publication date: 2022
    • (2022)Compressive Spectrum Sensing for 5G Cognitive Radio Networks – LASSO approachHeliyon10.1016/j.heliyon.2022.e09621(e09621)Online publication date: Jun-2022
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    • (2021)On Compressed Sensing of Binary Signals for the Unsourced Random Access ChannelEntropy10.3390/e2305060523:5(605)Online publication date: 14-May-2021
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