Book Review
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3686326
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Book Information:
Authors:
S. Foucart and
H. Rauhut
Title:
A mathematical introduction to compressive sensing
Additional book information:
Applied and Numeric Harmonic Analysis,
Birkh{\"a}user/Springer,
New York,
2013,
xviii+625 pp.,
ISBN 978-0-8176-4948-7
Dennis Amelunxen, Martin Lotz, Michael B. McCoy, and Joel A. Tropp, Living on the edge: phase transitions in convex programs with random data, Inf. Inference 3 (2014), no. 3, 224–294. MR 3311453, DOI 10.1093/imaiai/iau005
M. Salman Asif, Ali Ayremlou, Aswin C. Sankaranarayanan, Ashok Veeraraghavan, and Richard G. Baraniuk, Flatcam: Thin, bare-sensor cameras using coded aperture and computation, 2015. Available at http://arXiv.org/abs/1509.00116.
Afonso S. Bandeira, Matthew Fickus, Dustin G. Mixon, and Joel Moreira, Derandomizing restricted isometries via the Legendre symbol, Constr. Approx. 43 (2016), no. 3, 409–424. MR 3493967, DOI 10.1007/s00365-015-9310-6
R. G. Baraniuk, E. Candès, M. Elad, and Y. Ma, Applications of sparse representation and compressive sensing, Proceedings of the IEEE 98 (2010June), no. 6. Special issue.
Andrew R. Barron, Universal approximation bounds for superpositions of a sigmoidal function, IEEE Trans. Inform. Theory 39 (1993), no. 3, 930–945. MR 1237720, DOI 10.1109/18.256500
Mohsen Bayati, Marc Lelarge, and Andrea Montanari, Universality in polytope phase transitions and message passing algorithms, Ann. Appl. Probab. 25 (2015), no. 2, 753–822. MR 3313755, DOI 10.1214/14-AAP1010
Stephen R. Becker, Emmanuel J. Candès, and Michael C. Grant, Templates for convex cone problems with applications to sparse signal recovery, Math. Program. Comput. 3 (2011), no. 3, 165–218. MR 2833262, DOI 10.1007/s12532-011-0029-5
Jean Bourgain, Stephen Dilworth, Kevin Ford, Sergei Konyagin, and Denka Kutzarova, Explicit constructions of RIP matrices and related problems, Duke Math. J. 159 (2011), no. 1, 145–185. MR 2817651, DOI 10.1215/00127094-1384809
E. J. Candès and M. A. Davenport, How well can we estimate a sparse vector?, Appl. Comput. Harmonic Anal. 34 (2013Mar.), no. 2, 317–323.
Emmanuel J. Candès, Justin Romberg, and Terence Tao, Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information, IEEE Trans. Inform. Theory 52 (2006), no. 2, 489–509. MR 2236170, DOI 10.1109/TIT.2005.862083
Venkat Chandrasekaran, Benjamin Recht, Pablo A. Parrilo, and Alan S. Willsky, The convex geometry of linear inverse problems, Found. Comput. Math. 12 (2012), no. 6, 805–849. MR 2989474, DOI 10.1007/s10208-012-9135-7
Scott Shaobing Chen, David L. Donoho, and Michael A. Saunders, Atomic decomposition by basis pursuit, SIAM J. Sci. Comput. 20 (1998), no. 1, 33–61. MR 1639094, DOI 10.1137/S1064827596304010
Jon F. Claerbout and Francis Muir, Robust modeling with erratic data, Geophysics 38 (1973), no. 5, 826–844, available at http://geophysics.geoscienceworld.org/content/38/5/826.full.pdf.
R. R. Coifman and M. V. Wickerhauser, Entropy-based algorithms for best basis selection, IEEE Transactions on Information Theory 38 (1992March), no. 2, 713–718.
Mark A. Davenport, Jason N. Laska, John R. Treichler, and Richard G. Baraniuk, The pros and cons of compressive sensing for wideband signal acquisition: noise folding versus dynamic range, IEEE Trans. Signal Process. 60 (2012), no. 9, 4628–4642. MR 2960550, DOI 10.1109/TSP.2012.2201149
Geoffrey Davis, Stephane Mallat, and Zhifeng Zhang, Adaptive time-frequency approximations with matching pursuits, Wavelets: theory, algorithms, and applications (Taormina, 1993) Wavelet Anal. Appl., vol. 5, Academic Press, San Diego, CA, 1994, pp. 271–293. MR 1321432, DOI 10.1016/B978-0-08-052084-1.50018-1
R. A. DeVore and V. N. Temlyakov, Some remarks on greedy algorithms, Adv. Comput. Math. 5 (1996), no. 2-3, 173–187. MR 1399379, DOI 10.1007/BF02124742
D. L. Donoho and B. F. Logan, Signal recovery and the large sieve, SIAM J. Appl. Math. 52 (1992), no. 2, 577–591. MR 1154788, DOI 10.1137/0152031
David Donoho and Jared Tanner, Observed universality of phase transitions in high-dimensional geometry, with implications for modern data analysis and signal processing, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 367 (2009), no. 1906, 4273–4293. With electronic supplementary materials available online. MR 2546388, DOI 10.1098/rsta.2009.0152
David L. Donoho, Compressed sensing, IEEE Trans. Inform. Theory 52 (2006), no. 4, 1289–1306. MR 2241189, DOI 10.1109/TIT.2006.871582
David L. Donoho and Michael Elad, Optimally sparse representation in general (nonorthogonal) dictionaries via $l^1$ minimization, Proc. Natl. Acad. Sci. USA 100 (2003), no. 5, 2197–2202. MR 1963681, DOI 10.1073/pnas.0437847100
David L. Donoho, Michael Elad, and Vladimir N. Temlyakov, Stable recovery of sparse overcomplete representations in the presence of noise, IEEE Trans. Inform. Theory 52 (2006), no. 1, 6–18. MR 2237332, DOI 10.1109/TIT.2005.860430
David L. Donoho and Xiaoming Huo, Uncertainty principles and ideal atomic decomposition, IEEE Trans. Inform. Theory 47 (2001), no. 7, 2845–2862. MR 1872845, DOI 10.1109/18.959265
David L. Donoho and Iain M. Johnstone, Minimax risk over $l_p$-balls for $l_q$-error, Probab. Theory Related Fields 99 (1994), no. 2, 277–303. MR 1278886, DOI 10.1007/BF01199026
David L. Donoho and Philip B. Stark, Uncertainty principles and signal recovery, SIAM J. Appl. Math. 49 (1989), no. 3, 906–931. MR 997928, DOI 10.1137/0149053
David L. Donoho and Jared Tanner, Counting faces of randomly projected polytopes when the projection radically lowers dimension, J. Amer. Math. Soc. 22 (2009), no. 1, 1–53. MR 2449053, DOI 10.1090/S0894-0347-08-00600-0
David L. Donoho, Martin Vetterli, R. A. DeVore, and Ingrid Daubechies, Data compression and harmonic analysis, IEEE Trans. Inform. Theory 44 (1998), no. 6, 2435–2476. Information theory: 1948–1998. MR 1658775, DOI 10.1109/18.720544
Yonina C. Eldar and Gitta Kutyniok (eds.), Compressed sensing, Cambridge University Press, Cambridge, 2012. Theory and applications. MR 2961961, DOI 10.1017/CBO9780511794308
Y. Erlich, A. Gilbert, H. Ngo, A. Rudra, N. Thierry-Mieg, M. Wootters, D. Zielinski, and O. Zuk, Biological screens from linear codes: theory and tools, 2015. Unpublished.
M. Fazel, Matrix rank minimization with applications, Ph.D. Thesis, Palo Alto, 2002.
J. R. Fienup, Phase retrieval algorithms: a comparison, Appl. Opt. 21 (1982Aug), no. 15, 2758–2769.
Simon Foucart, Alain Pajor, Holger Rauhut, and Tino Ullrich, The Gelfand widths of $\ell _p$-balls for $0<p\leq 1$, J. Complexity 26 (2010), no. 6, 629–640. MR 2735423, DOI 10.1016/j.jco.2010.04.004
Simon Foucart and Holger Rauhut, A mathematical introduction to compressive sensing, Applied and Numerical Harmonic Analysis, Birkhäuser/Springer, New York, 2013. MR 3100033, DOI 10.1007/978-0-8176-4948-7
Jean Jacques Fuchs, Recovery of exact sparse representations in the presence of bounded noise, IEEE Trans. Inform. Theory 51 (2005), no. 10, 3601–3608. MR 2237526, DOI 10.1109/TIT.2005.855614
A. Yu. Garnaev and E. D. Gluskin, The widths of a Euclidean ball, Dokl. Akad. Nauk SSSR 277 (1984), no. 5, 1048–1052 (Russian). MR 759962
R. W. Gerchberg and W. O. Saxton, A practical algorithm for the determination of the phase from image and diffraction plane pictures, Optik 35 (1972), no. 2, 237–246.
A. Gilbert and P. Indyk, Sparse recovery using sparse matrices, Proceedings of the IEEE 98 (2010June), no. 6, 937–947.
A. C. Gilbert, S. Guha, P. Indyk, S. Muthukrishnan, and M. Strauss, Near-optimal sparse Fourier representations via sampling, Proceedings of the Thirty-Fourth Annual ACM Symposium on Theory of Computing, ACM, New York, 2002, pp. 152–161. MR 2121138, DOI 10.1145/509907.509933
Anna C. Gilbert, Sudipto Guha, Piotr Indyk, Yannis Kotidis, S. Muthukrishnan, and Martin J. Strauss, Fast, small-space algorithms for approximate histogram maintenance, Proceedings of the Thirty-Fourth Annual ACM Symposium on Theory of Computing, ACM, New York, 2002, pp. 389–398. MR 2121164, DOI 10.1145/509907.509966
Anna C. Gilbert, S. Muthukrishnan, and Martin J. Strauss, Approximation of functions over redundant dictionaries using coherence, Proceedings of the fourteenth annual acm-siam symposium on discrete algorithms, 2003, pp. 243–252.
M. Harwit and N. J. A. Sloane, Hadamard transform optics, Academic Press, 1979.
B. S. Kašin, The widths of certain finite-dimensional sets and classes of smooth functions, Izv. Akad. Nauk SSSR Ser. Mat. 41 (1977), no. 2, 334–351, 478 (Russian). MR 0481792
Felix Krahmer, Shahar Mendelson, and Holger Rauhut, Suprema of chaos processes and the restricted isometry property, Comm. Pure Appl. Math. 67 (2014), no. 11, 1877–1904. MR 3263672, DOI 10.1002/cpa.21504
Eyal Kushilevitz and Yishay Mansour, Learning decision trees using the Fourier spectrum, SIAM J. Comput. 22 (1993), no. 6, 1331–1348. MR 1247193, DOI 10.1137/0222080
Shlomo Levy and Peter K. Fullagar, Reconstruction of a sparse spike train from a portion of its spectrum and application to high-resolution deconvolution, Geophysics 46 (1981), no. 9, 1235–1243, available at http://geophysics.geoscienceworld.org/content/46/9/1235.full.pdf.
B. F. Logan, Properties of high-pass signals, Ph.D. Thesis, New York, 1965.
Michael Lustig, David Donoho, and John M. Pauly, Sparse mri: The application of compressed sensing for rapid mr imaging, Magnetic Resonance in Medicine 58 (2007), no. 6, 1182–1195.
Stéphane Mallat, A wavelet tour of signal processing, 3rd ed., Elsevier/Academic Press, Amsterdam, 2009. The sparse way; With contributions from Gabriel Peyré. MR 2479996
Yishay Mansour, Randomized interpolation and approximation of sparse polynomials, SIAM J. Comput. 24 (1995), no. 2, 357–368. MR 1320215, DOI 10.1137/S0097539792239291
M. McCoy and J. A. Tropp, The achievable performance of convex demixing, 2013. Available at http://arXiv.org/abs/1309.7478.
Michael B. McCoy and Joel A. Tropp, From Steiner formulas for cones to concentration of intrinsic volumes, Discrete Comput. Geom. 51 (2014), no. 4, 926–963. MR 3216671, DOI 10.1007/s00454-014-9595-4
Alan Miller, Subset selection in regression, 2nd ed., Monographs on Statistics and Applied Probability, vol. 95, Chapman & Hall/CRC, Boca Raton, FL, 2002. MR 2001193, DOI 10.1201/9781420035933
B. K. Natarajan, Sparse approximate solutions to linear systems, SIAM J. Comput. 24 (1995), no. 2, 227–234. MR 1320206, DOI 10.1137/S0097539792240406
D. W. Oldenburg, T. Scheuer, and S. Levy, Recovery of the acoustic impedance from reflection seismograms, GEOPHYSICS 48 (1983), no. 10, 1318–1337, available at http://dx.doi.org/10.1190/1.1441413.
S. Oymak and J. A. Tropp, Universality laws for randomized dimension reduction, with applications, 2015. Available at http://arXiv.org/abs/1511.09433.
Benjamin Recht, Maryam Fazel, and Pablo A. Parrilo, Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization, SIAM Rev. 52 (2010), no. 3, 471–501. MR 2680543, DOI 10.1137/070697835
Mark Rudelson and Roman Vershynin, The Littlewood-Offord problem and invertibility of random matrices, Adv. Math. 218 (2008), no. 2, 600–633. MR 2407948, DOI 10.1016/j.aim.2008.01.010
Leonid I. Rudin, Stanley Osher, and Emad Fatemi, Nonlinear total variation based noise removal algorithms, Phys. D 60 (1992), no. 1-4, 259–268. Experimental mathematics: computational issues in nonlinear science (Los Alamos, NM, 1991). MR 3363401, DOI 10.1016/0167-2789(92)90242-F
Fadil Santosa and William W. Symes, Linear inversion of band-limited reflection seismograms, SIAM J. Sci. Statist. Comput. 7 (1986), no. 4, 1307–1330. MR 857796, DOI 10.1137/0907087
M. Stojnic, Regularly random duality, 2013. Available at http://arXiv.org/abs/1303.7295.
Terence Tao, An uncertainty principle for cyclic groups of prime order, Math. Res. Lett. 12 (2005), no. 1, 121–127. MR 2122735, DOI 10.4310/MRL.2005.v12.n1.a11
H. L. Taylor, S. C. Banks, and J. F. McCoy, Deconvolution with the l 1 norm, Geophysics 44 (1979), no. 1, 39–52, available at http://geophysics.geoscienceworld.org/content/44/1/39.full.pdf.
C. Thrampoulidis, S. Oymak, and B. Hassibi, The Gaussian min-max theorem in the presence of convexity, 2014. Available at http://arXiv.org/abs/1408.4837.
Christos Thrampoulidis, Recovering structured signals in high dimensions via non-smooth convex optimization: Precise performance analysis., Ph.D. Thesis, Pasadena, 2016.
Christos Thrampoulidis, Samet Oymak, and Babak Hassibi, Recovering structured signals in noise: least-squares meets compressed sensing, Compressed sensing and its applications, Appl. Numer. Harmon. Anal., Birkhäuser/Springer, Cham, 2015, pp. 97–141. MR 3382104
Robert Tibshirani, Regression shrinkage and selection via the lasso, J. Roy. Statist. Soc. Ser. B 58 (1996), no. 1, 267–288. MR 1379242
J. A. Tropp and S. J. Wright, Computational methods for sparse solution of linear inverse problems, Proceedings of the IEEE 98 (2010June), no. 6, 948–958.
Joel A. Tropp, Greed is good: algorithmic results for sparse approximation, IEEE Trans. Inform. Theory 50 (2004), no. 10, 2231–2242. MR 2097044, DOI 10.1109/TIT.2004.834793
Joel A. Tropp, Just relax: convex programming methods for identifying sparse signals in noise, IEEE Trans. Inform. Theory 52 (2006), no. 3, 1030–1051. MR 2238069, DOI 10.1109/TIT.2005.864420
Joel A. Tropp, User-friendly tail bounds for sums of random matrices, Found. Comput. Math. 12 (2012), no. 4, 389–434. MR 2946459, DOI 10.1007/s10208-011-9099-z
Joel A. Tropp, Jason N. Laska, Marco F. Duarte, Justin K. Romberg, and Richard G. Baraniuk, Beyond Nyquist: efficient sampling of sparse bandlimited signals, IEEE Trans. Inform. Theory 56 (2010), no. 1, 520–544. MR 2589462, DOI 10.1109/TIT.2009.2034811
S. Vasanawala, M. Murphy, M. Alley, P. Lai, K. Keutzer, J. Pauly, and M. Lustig, Practical parallel imaging compressed sensing MRI: Summary of two years of experience in accelerating body MRI of pediatric patients, 2011 ieee international symposium on biomedical imaging: From nano to macro, 2011March, pp. 1039–1043.
References
- Dennis Amelunxen, Martin Lotz, Michael B. McCoy, and Joel A. Tropp, Living on the edge: phase transitions in convex programs with random data, Inf. Inference 3 (2014), no. 3, 224–294. MR 3311453
- M. Salman Asif, Ali Ayremlou, Aswin C. Sankaranarayanan, Ashok Veeraraghavan, and Richard G. Baraniuk, Flatcam: Thin, bare-sensor cameras using coded aperture and computation, 2015. Available at http://arXiv.org/abs/1509.00116.
- Afonso S. Bandeira, Matthew Fickus, Dustin G. Mixon, and Joel Moreira, Derandomizing Restricted Isometries via the Legendre Symbol, Constr. Approx. 43 (2016), no. 3, 409–424. MR 3493967
- R. G. Baraniuk, E. Candès, M. Elad, and Y. Ma, Applications of sparse representation and compressive sensing, Proceedings of the IEEE 98 (2010June), no. 6. Special issue.
- Andrew R. Barron, Universal approximation bounds for superpositions of a sigmoidal function, IEEE Trans. Inform. Theory 39 (1993), no. 3, 930–945. MR 1237720
- Mohsen Bayati, Marc Lelarge, and Andrea Montanari, Universality in polytope phase transitions and message passing algorithms, Ann. Appl. Probab. 25 (2015), no. 2, 753–822. MR 3313755
- Stephen R. Becker, Emmanuel J. Candès, and Michael C. Grant, Templates for convex cone problems with applications to sparse signal recovery, Math. Program. Comput. 3 (2011), no. 3, 165–218. MR 2833262
- Jean Bourgain, Stephen Dilworth, Kevin Ford, Sergei Konyagin, and Denka Kutzarova, Explicit constructions of RIP matrices and related problems, Duke Math. J. 159 (2011), no. 1, 145–185. MR 2817651
- E. J. Candès and M. A. Davenport, How well can we estimate a sparse vector?, Appl. Comput. Harmonic Anal. 34 (2013Mar.), no. 2, 317–323.
- Emmanuel J. Candès, Justin Romberg, and Terence Tao, Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information, IEEE Trans. Inform. Theory 52 (2006), no. 2, 489–509. MR 2236170
- Venkat Chandrasekaran, Benjamin Recht, Pablo A. Parrilo, and Alan S. Willsky, The convex geometry of linear inverse problems, Found. Comput. Math. 12 (2012), no. 6, 805–849. MR 2989474
- Scott Shaobing Chen, David L. Donoho, and Michael A. Saunders, Atomic decomposition by basis pursuit, SIAM J. Sci. Comput. 20 (1998), no. 1, 33–61. MR 1639094
- Jon F. Claerbout and Francis Muir, Robust modeling with erratic data, Geophysics 38 (1973), no. 5, 826–844, available at http://geophysics.geoscienceworld.org/content/38/5/826.full.pdf.
- R. R. Coifman and M. V. Wickerhauser, Entropy-based algorithms for best basis selection, IEEE Transactions on Information Theory 38 (1992March), no. 2, 713–718.
- Mark A. Davenport, Jason N. Laska, John R. Treichler, and Richard G. Baraniuk, The pros and cons of compressive sensing for wideband signal acquisition: noise folding versus dynamic range, IEEE Trans. Signal Process. 60 (2012), no. 9, 4628–4642. MR 2960550
- Geoffrey Davis, Stephane Mallat, and Zhifeng Zhang, Adaptive time-frequency approximations with matching pursuits, Wavelets: theory, algorithms, and applications (Taormina, 1993), 1994, pp. 271–293. MR 1321432
- R. A. DeVore and V. N. Temlyakov, Some remarks on greedy algorithms, Adv. Comput. Math. 5 (1996), no. 2-3, 173–187. MR 1399379
- D. L. Donoho and B. F. Logan, Signal recovery and the large sieve, SIAM J. Appl. Math. 52 (1992), no. 2, 577–591. MR 1154788
- David Donoho and Jared Tanner, Observed universality of phase transitions in high-dimensional geometry, with implications for modern data analysis and signal processing, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 367 (2009), no. 1906, 4273–4293. With electronic supplementary materials available online. MR 2546388
- David L. Donoho, Compressed sensing, IEEE Trans. Inform. Theory 52 (2006), no. 4, 1289–1306. MR 2241189, DOI 10.1109/TIT.2006.871582
- David L. Donoho and Michael Elad, Optimally sparse representation in general (nonorthogonal) dictionaries via $l^ 1$ minimization, Proc. Natl. Acad. Sci. USA 100 (2003), no. 5, 2197–2202 (electronic). MR 1963681
- David L. Donoho, Michael Elad, and Vladimir N. Temlyakov, Stable recovery of sparse overcomplete representations in the presence of noise, IEEE Trans. Inform. Theory 52 (2006), no. 1, 6–18. MR 2237332
- David L. Donoho and Xiaoming Huo, Uncertainty principles and ideal atomic decomposition, IEEE Trans. Inform. Theory 47 (2001), no. 7, 2845–2862. MR 1872845
- David L. Donoho and Iain M. Johnstone, Minimax risk over $l_ p$-balls for $l_ q$-error, Probab. Theory Related Fields 99 (1994), no. 2, 277–303. MR 1278886
- David L. Donoho and Philip B. Stark, Uncertainty principles and signal recovery, SIAM J. Appl. Math. 49 (1989), no. 3, 906–931. MR 997928
- David L. Donoho and Jared Tanner, Counting faces of randomly projected polytopes when the projection radically lowers dimension, J. Amer. Math. Soc. 22 (2009), no. 1, 1–53. MR 2449053, DOI 10.1090/S0894-0347-08-00600-0
- David L. Donoho, Martin Vetterli, R. A. DeVore, and Ingrid Daubechies, Data compression and harmonic analysis, IEEE Trans. Inform. Theory 44 (1998), no. 6, 2435–2476. Information theory: 1948–1998. MR 1658775, DOI 10.1109/18.720544
- Y. C. Eldar and G. Kutyniok (eds.), Compressed sensing. Theory and applications, Cambridge University Press, Cambridge, 2012. MR 2961961, DOI 10.1017/CBO9780511794308
- Y. Erlich, A. Gilbert, H. Ngo, A. Rudra, N. Thierry-Mieg, M. Wootters, D. Zielinski, and O. Zuk, Biological screens from linear codes: theory and tools, 2015. Unpublished.
- M. Fazel, Matrix rank minimization with applications, Ph.D. Thesis, Palo Alto, 2002.
- J. R. Fienup, Phase retrieval algorithms: a comparison, Appl. Opt. 21 (1982Aug), no. 15, 2758–2769.
- Simon Foucart, Alain Pajor, Holger Rauhut, and Tino Ullrich, The Gelfand widths of $\ell _ p$-balls for $0<p\leq 1$, J. Complexity 26 (2010), no. 6, 629–640. MR 2735423
- Simon Foucart and Holger Rauhut, A mathematical introduction to compressive sensing, Applied and Numerical Harmonic Analysis, Birkhäuser/Springer, New York, 2013. MR 3100033, DOI 10.1007/978-0-8176-4948-7
- Jean Jacques Fuchs, Recovery of exact sparse representations in the presence of bounded noise, IEEE Trans. Inform. Theory 51 (2005), no. 10, 3601–3608. MR 2237526
- A. Yu. Garnaev and E. D. Gluskin, The widths of a Euclidean ball, Dokl. Akad. Nauk SSSR 277 (1984), no. 5, 1048–1052. MR 759962
- R. W. Gerchberg and W. O. Saxton, A practical algorithm for the determination of the phase from image and diffraction plane pictures, Optik 35 (1972), no. 2, 237–246.
- A. Gilbert and P. Indyk, Sparse recovery using sparse matrices, Proceedings of the IEEE 98 (2010June), no. 6, 937–947.
- A. C. Gilbert, S. Guha, P. Indyk, S. Muthukrishnan, and M. Strauss, Near-optimal sparse Fourier representations via sampling, Proceedings of the Thirty-Fourth Annual ACM Symposium on Theory of Computing, 2002, pp. 152–161. MR 2121138
- Anna C. Gilbert, Sudipto Guha, Piotr Indyk, Yannis Kotidis, S. Muthukrishnan, and Martin J. Strauss, Fast, small-space algorithms for approximate histogram maintenance, Proceedings of the Thirty-Fourth Annual ACM Symposium on Theory of Computing, 2002, pp. 389–398. MR 2121164
- Anna C. Gilbert, S. Muthukrishnan, and Martin J. Strauss, Approximation of functions over redundant dictionaries using coherence, Proceedings of the fourteenth annual acm-siam symposium on discrete algorithms, 2003, pp. 243–252.
- M. Harwit and N. J. A. Sloane, Hadamard transform optics, Academic Press, 1979.
- B. S. Kašin, The widths of certain finite-dimensional sets and classes of smooth functions, Izv. Akad. Nauk SSSR Ser. Mat. 41 (1977), no. 2, 334–351, 478. MR 0481792
- Felix Krahmer, Shahar Mendelson, and Holger Rauhut, Suprema of chaos processes and the restricted isometry property, Comm. Pure Appl. Math. 67 (2014), no. 11, 1877–1904. MR 3263672
- Eyal Kushilevitz and Yishay Mansour, Learning decision trees using the Fourier spectrum, SIAM J. Comput. 22 (1993), no. 6, 1331–1348. MR 1247193
- Shlomo Levy and Peter K. Fullagar, Reconstruction of a sparse spike train from a portion of its spectrum and application to high-resolution deconvolution, Geophysics 46 (1981), no. 9, 1235–1243, available at http://geophysics.geoscienceworld.org/content/46/9/1235.full.pdf.
- B. F. Logan, Properties of high-pass signals, Ph.D. Thesis, New York, 1965.
- Michael Lustig, David Donoho, and John M. Pauly, Sparse mri: The application of compressed sensing for rapid mr imaging, Magnetic Resonance in Medicine 58 (2007), no. 6, 1182–1195.
- Stéphane Mallat, A wavelet tour of signal processing, The sparse way, 3rd ed., Elsevier/Academic Press, Amsterdam, 2009. MR 2479996
- Yishay Mansour, Randomized interpolation and approximation of sparse polynomials, SIAM J. Comput. 24 (1995), no. 2, 357–368. MR 1320215
- M. McCoy and J. A. Tropp, The achievable performance of convex demixing, 2013. Available at http://arXiv.org/abs/1309.7478.
- Michael B. McCoy and Joel A. Tropp, From Steiner formulas for cones to concentration of intrinsic volumes, Discrete Comput. Geom. 51 (2014), no. 4, 926–963. MR 3216671
- Alan Miller, Subset selection in regression, Second, Monographs on Statistics and Applied Probability, vol. 95, Chapman & Hall/CRC, Boca Raton, FL, 2002. MR 2001193
- B. K. Natarajan, Sparse approximate solutions to linear systems, SIAM J. Comput. 24 (1995), no. 2, 227–234. MR 1320206
- D. W. Oldenburg, T. Scheuer, and S. Levy, Recovery of the acoustic impedance from reflection seismograms, GEOPHYSICS 48 (1983), no. 10, 1318–1337, available at http://dx.doi.org/10.1190/1.1441413.
- S. Oymak and J. A. Tropp, Universality laws for randomized dimension reduction, with applications, 2015. Available at http://arXiv.org/abs/1511.09433.
- Benjamin Recht, Maryam Fazel, and Pablo A. Parrilo, Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization, SIAM Rev. 52 (2010), no. 3, 471–501. MR 2680543
- Mark Rudelson and Roman Vershynin, The Littlewood-Offord problem and invertibility of random matrices, Adv. Math. 218 (2008), no. 2, 600–633. MR 2407948
- Leonid I. Rudin, Stanley Osher, and Emad Fatemi, Nonlinear total variation based noise removal algorithms, Phys. D 60 (1992), no. 1-4, 259–268. Experimental mathematics: computational issues in nonlinear science, (Los Alamos, NM, 1991). MR 3363401
- Fadil Santosa and William W. Symes, Linear inversion of band-limited reflection seismograms, SIAM J. Sci. Statist. Comput. 7 (1986), no. 4, 1307–1330. MR 857796
- M. Stojnic, Regularly random duality, 2013. Available at http://arXiv.org/abs/1303.7295.
- Terence Tao, An uncertainty principle for cyclic groups of prime order, Math. Res. Lett. 12 (2005), no. 1, 121–127. MR 2122735
- H. L. Taylor, S. C. Banks, and J. F. McCoy, Deconvolution with the l 1 norm, Geophysics 44 (1979), no. 1, 39–52, available at http://geophysics.geoscienceworld.org/content/44/1/39.full.pdf.
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Review Information:
Reviewer:
Joel A. Tropp
Affiliation:
California Institute of Technology, Pasadena, California 91125-5000
Email:
jtropp@cms.caltech.edu
Journal:
Bull. Amer. Math. Soc.
54 (2017), 151-165
DOI:
https://doi.org/10.1090/bull/1546
Keywords:
Image processing,
information theory,
mathematical programming,
probability in Banach spaces,
random matrix theory,
sampling theory,
signal processing
Published electronically:
August 25, 2016
Review copyright:
© Copyright 2016
American Mathematical Society