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Mat. Zametki, 2007, Volume 82, Issue 6, Pages 829–837 (Mi mz4183)  

This article is cited in 39 scientific papers (total in 39 papers)

A Remark on Compressed Sensing

B. S. Kashina, V. N. Temlyakovb

a Steklov Mathematical Institute, Russian Academy of Sciences
b University of South Carolina

Abstract: Recently, a new direction in signal processing – “Compressed Sensing” is being actively developed. A number of authors have pointed out a connection between the Compressed Sensing problem and the problem of estimating the Kolmogorov widths, studied in the seventies and eighties of the last century. In this paper we make the above mentioned connection more precise.

Keywords: compressed sensing, signal processing, Kolmogorov width, Gelfand width, sparsity, restricted isometry property, combinatorial optimization problem

DOI: https://doi.org/10.4213/mzm4183

Full text: PDF file (501 kB)
References: PDF file   HTML file

English version:
Mathematical Notes, 2007, 82:6, 748–755

Bibliographic databases:

UDC: 517.5
Received: 15.08.2007

Citation: B. S. Kashin, V. N. Temlyakov, “A Remark on Compressed Sensing”, Mat. Zametki, 82:6 (2007), 829–837; Math. Notes, 82:6 (2007), 748–755

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    Citing articles on Google Scholar: Russian citations, English citations
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    1. Xu W., Hassibi B., “Compressed sensing over the Grassmann manifold: A unified analytical framework”, 2008 46th Annual Allerton Conference on Communication, Control, and Computing, v. 1–3, IEEE, 2008, 562–567  crossref  isi  scopus
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    6. Guruswami V., Lee J.R., Razborov A., “Almost Euclidean subspaces of $\ell^N_1$ via expander codes”, Combinatorica, 30:1 (2010), 47–68  crossref  mathscinet  isi  elib  scopus
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    8. Barg A.M., Mazumdar A., “Small ensembles of sampling matrices constructed from coding theory”, 2010 IEEE International Symposium on Information Theory Proceedings (ISIT), IEEE, 2010, 1963–1967  crossref  isi  scopus
    9. Adamczak R., Litvak A.E., Pajor A., Tomczak-Jaegermann N., “Restricted isometry property of matrices with independent columns and neighborly polytopes by random sampling”, Constr. Approx., 34:1 (2011), 61–88  crossref  mathscinet  zmath  isi  elib  scopus
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    16. Minakov E.I., Seregin P.S., “Sravnitelnyi analiz metodov parallelnoi rekonstruktsii izobrazhenii magnitno-rezonansnoi tomografii”, Tsifrovaya obrabotka signalov, 2012, no. 3, 23–28  elib
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    18. S. V. Konyagin, Yu. V. Malykhin, C. S. Rjutin, “On Exact Recovery of Sparse Vectors from Linear Measurements”, Math. Notes, 94:1 (2013), 107–114  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    19. Donoho D.L., Javanmard A., Montanari A., “Information-Theoretically Optimal Compressed Sensing via Spatial Coupling and Approximate Message Passing”, IEEE Trans. Inf. Theory, 59:11 (2013), 7434–7464  crossref  mathscinet  zmath  isi  scopus
    20. Kueng R., Gross D., “Ripless Compressed Sensing From Anisotropic Measurements”, Linear Alg. Appl., 441:SI (2014), 110–123  crossref  mathscinet  zmath  isi  scopus
    21. S. G. Vlăduţ, G. A. Kabatiansky, V. V. Lomakov, “On error correction with errors in both the channel and syndrome”, Problems Inform. Transmission, 51:2 (2015), 132–138  mathnet  crossref  isi  elib
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    23. Chavez-Dominguez J.A., Kutzarova D., “Stability of Low-Rank Matrix Recovery and Its Connections To Banach Space Geometry”, J. Math. Anal. Appl., 427:1 (2015), 320–335  crossref  mathscinet  zmath  isi  scopus
    24. Rostami M., Michailovich O.V., Wang Zh., “Surface Reconstruction in Gradient-Field Domain Using Compressed Sensing”, IEEE Trans. Image Process., 24:5 (2015), 1628–1638  crossref  mathscinet  isi  elib  scopus
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    32. Kaporin I.E., “Deterministic bounds for restricted isometry in compressed sensing matrices”, Dokl. Math., 93:3 (2016), 273–275  crossref  mathscinet  zmath  isi  elib  scopus
    33. I. E. Kaporin, “Bounding the restricted isometry constants for a tight frame”, Sb. Math., 208:11 (2017), 1646–1660  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
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