메뉴 건너뛰기




Volumn 59, Issue 6, 2013, Pages 3396-3433

Accurate prediction of phase transitions in compressed sensing via a connection to minimax denoising

Author keywords

Approximate message passing (AMP); group Lasso; James Stein; joint sparsity; Lasso; minimax risk of firm thresholding; minimax risk of soft thresholding; minimax risk over nearly black objects; minimax shrinkage; monotone regression; nonconvex penalization; state evolution; total variation minimization

Indexed keywords

APPROXIMATE MESSAGE PASSING (AMP); GROUP LASSOS; JAMES-STEIN; JOINT SPARSITY; LASSO; MINIMAX; MINIMAX RISK; MONOTONE REGRESSION; NONCONVEX; SOFT THRESHOLDING; STATE EVOLUTIONS; THRESHOLDING; TOTAL VARIATION MINIMIZATION;

EID: 84877914996     PISSN: 00189448     EISSN: None     Source Type: Journal    
DOI: 10.1109/TIT.2013.2239356     Document Type: Article
Times cited : (153)

References (72)
  • 1
    • 0041067082 scopus 로고
    • Minimizing Fisher information overmixtures of distributions
    • P. Bickel and J. Collins, "Minimizing Fisher information overmixtures of distributions," Sankhya A, vol. 45, pp. 1-19, 1983.
    • (1983) Sankhya A , vol.45 , pp. 1-19
    • Bickel, P.1    Collins, J.2
  • 2
    • 34249957970 scopus 로고
    • Active set algorithms for isotonic regression; A unifying framework
    • M. J. Best and N. Chakravarti, "Active set algorithms for isotonic regression; A unifying framework," Math. Programm., vol. 47, pp. 425-439, 1990.
    • (1990) Math. Programm. , vol.47 , pp. 425-439
    • Best, M.J.1    Chakravarti, N.2
  • 4
    • 79957444557 scopus 로고    scopus 로고
    • Compressed sensing: How sharp is the restricted isometry property?
    • J. D. Blanchard, C. Cartis, and J. Tanner, "Compressed sensing: How sharp is the restricted isometry property?," SIAM Rev., vol. 53, no. 1, pp. 105-125, 2011.
    • (2011) SIAM Rev. , vol.53 , Issue.1 , pp. 105-125
    • Blanchard, J.D.1    Cartis, C.2    Tanner, J.3
  • 5
    • 64549162832 scopus 로고    scopus 로고
    • Combining geometry and combinatorics: A unified approach to sparse signal recovery
    • Monticello, IL, USA Sep.
    • R. Berinde, A. C. Gilbert, P. Indyk, H. Karloff, and M. J. Strauss, "Combining geometry and combinatorics: A unified approach to sparse signal recovery," in 47th Annu. Allerton Conf., Monticello, IL, USA, Sep. 2008, pp. 798-805.
    • (2008) 47th Annu. Allerton Conf. , pp. 798-805
    • Berinde, R.1    Gilbert, A.C.2    Indyk, P.3    Karloff, H.4    Strauss, M.J.5
  • 6
    • 0001073033 scopus 로고
    • Minimax estimation of the mean of a normal distribution when the parameter space is restricted
    • P. Bickel, "Minimax estimation of the mean of a normal distribution when the parameter space is restricted," Ann. Statist., vol. 9, pp. 1301-1309, 1981.
    • (1981) Ann. Statist. , vol.9 , pp. 1301-1309
    • Bickel, P.1
  • 7
    • 0000392073 scopus 로고
    • Variation diminishing transformations: A direct approach to total positivity and its statistical applications
    • L. D. Brown, I. M. Johnstone, and K. B. MacGibbon, "Variation diminishing transformations: A direct approach to total positivity and its statistical applications," J. Amer. Statist. Assoc., vol. 76, no. 376, pp. 824-832, 1981.
    • (1981) J. Amer. Statist. Assoc. , vol.76 , Issue.376 , pp. 824-832
    • Brown, L.D.1    Johnstone, I.M.2    MacGibbon, K.B.3
  • 9
    • 79251496987 scopus 로고    scopus 로고
    • The dynamics of message passing on dense graphs, with applications to compressed sensing
    • Feb.
    • M. Bayati and A. Montanari, "The dynamics of message passing on dense graphs, with applications to compressed sensing," IEEE Trans. Inf. Theory, vol. 57, no. 2, pp. 764-785, Feb. 2011.
    • (2011) IEEE Trans. Inf. Theory , vol.57 , Issue.2 , pp. 764-785
    • Bayati, M.1    Montanari, A.2
  • 10
    • 84858956033 scopus 로고    scopus 로고
    • The LASSO risk for Gaussianmatrices
    • Apr.
    • M. Bayati and A.Montanari, "The LASSO risk for Gaussianmatrices, " IEEE Trans. Inf. Theory, vol. 58, no. 4, pp. 1997-2017, Apr. 2012.
    • (2012) IEEE Trans. Inf. Theory , vol.58 , Issue.4 , pp. 1997-2017
    • Bayati, M.1    Montanari, A.2
  • 11
    • 68649086910 scopus 로고    scopus 로고
    • Simultaneous analysis of Lasso and Dantzig selector
    • P. J. Bickel, Y. Ritov, and A. B. Tsybakov, "Simultaneous analysis of Lasso and Dantzig selector," Amer. J. Math., vol. 37, pp. 1705-1732, 2009.
    • (2009) Amer. J. Math. , vol.37 , pp. 1705-1732
    • Bickel, P.J.1    Ritov, Y.2    Tsybakov, A.B.3
  • 12
    • 79954529897 scopus 로고    scopus 로고
    • Large scale iterative hard thresholding on a graphical processing unit
    • J. Blanchard and J. Tanner, "Large scale iterative hard thresholding on a graphical processing unit," in Proc. Int. Conf. Numer. Anal. Appl. Math., 2010, vol. 1281, pp. 1730-1734.
    • (2010) Proc. Int. Conf. Numer. Anal. Appl. Math. , vol.1281 , pp. 1730-1734
    • Blanchard, J.1    Tanner, J.2
  • 13
    • 0033248624 scopus 로고    scopus 로고
    • Adaptive wavelet estimation: A block thresholding and oracle inequality approach
    • T. T. Cai, "Adaptive wavelet estimation: A block thresholding and oracle inequality approach," Ann. Statist., vol. 27, no. 3, pp. 898-924, 1999.
    • (1999) Ann. Statist. , vol.27 , Issue.3 , pp. 898-924
    • Cai, T.T.1
  • 14
    • 84878104490 scopus 로고    scopus 로고
    • Compressive sampling
    • Madrid, Spain
    • E. Candès, "Compressive sampling," presented at the Int. Congr. Math., Madrid, Spain, 2006.
    • (2006) Int. Congr. Math.
    • Candès, E.1
  • 15
  • 16
    • 85032751046 scopus 로고    scopus 로고
    • Sparse signal recovery and acquisition with graphical models
    • Nov.
    • V. Cevher, P. Indyk, L. Carin, and R. G. Baraniuk, "Sparse signal recovery and acquisition with graphical models," IEEE Signal Process. Mag., vol. 27, no. 6, pp. 92-103, Nov. 2010.
    • (2010) IEEE Signal Process. Mag. , vol.27 , Issue.6 , pp. 92-103
    • Cevher, V.1    Indyk, P.2    Carin, L.3    Baraniuk, R.G.4
  • 18
    • 0000936351 scopus 로고
    • Estimating a bounded normal mean
    • G. Casella and W. E. Strawderman, "Estimating a bounded normal mean," Ann. Statist., vol. 9, pp. 870-878, 1981.
    • (1981) Ann. Statist. , vol.9 , pp. 870-878
    • Casella, G.1    Strawderman, W.E.2
  • 19
    • 29144439194 scopus 로고    scopus 로고
    • Decoding by linear programming
    • DOI 10.1109/TIT.2005.858979
    • E. J. Candés and T. Tao, "Decoding by linear programming," IEEE Trans. Inf. Theory, vol. 51, no. 12, pp. 4203-4215, Dec. 2005. (Pubitemid 41800353)
    • (2005) IEEE Transactions on Information Theory , vol.51 , Issue.12 , pp. 4203-4215
    • Candes, E.J.1    Tao, T.2
  • 26
    • 61349102591 scopus 로고    scopus 로고
    • Finite size scaling for the core of large random hypergraphs
    • A. Dembo and A. Montanari, "Finite size scaling for the core of large random hypergraphs," Ann. Appl. Prob., vol. 18, pp. 1993-2040, 2008.
    • (2008) Ann. Appl. Prob. , vol.18 , pp. 1993-2040
    • Dembo, A.1    Montanari, A.2
  • 27
    • 73149095169 scopus 로고    scopus 로고
    • Message passing algorithms for compressed sensing
    • D. L. Donoho, A. Maleki, and A. Montanari, "Message passing algorithms for compressed sensing," in Proc. Nat. Acad. Sci., 2009, vol. 106, pp. 18914-18919.
    • (2009) Proc. Nat. Acad. Sci. , vol.106 , pp. 18914-18919
    • Donoho, D.L.1    Maleki, A.2    Montanari, A.3
  • 28
    • 77954825041 scopus 로고    scopus 로고
    • Message passing algorithms for compressed sensing: I. Motivation and construction
    • Cairo, Egypt
    • D. L. Donoho, A. Maleki, and A. Montanari, "Message passing algorithms for compressed sensing: I. Motivation and construction," presented at the IEEE Inf. Theory Workshop, Cairo, Egypt, 2010.
    • (2010) IEEE Inf. Theory Workshop
    • Donoho, D.L.1    Maleki, A.2    Montanari, A.3
  • 29
    • 80053933128 scopus 로고    scopus 로고
    • The noise sensitivity phase transition in compressed sensing
    • Oct.
    • D. L. Donoho, A. Maleki, and A. Montanari, "The noise sensitivity phase transition in compressed sensing," IEEE Trans. Inf. Theory, vol. 57, no. 10, pp. 6920-6941, Oct. 2011.
    • (2011) IEEE Trans. Inf. Theory , vol.57 , Issue.10 , pp. 6920-6941
    • Donoho, D.L.1    Maleki, A.2    Montanari, A.3
  • 30
    • 33646479497 scopus 로고    scopus 로고
    • High-dimensional centrally symmetric polytopes with neighborliness proportional to dimension
    • DOI 10.1007/s00454-005-1220-0
    • D. L. Donoho, "High-dimensional centrally symmetric polytopes with neighborliness proportional to dimension," Discrete Comput. Geom., vol. 35, pp. 617-652, 2006. (Pubitemid 43701356)
    • (2006) Discrete and Computational Geometry , vol.35 , Issue.4 , pp. 617-652
    • Donoho, D.L.1
  • 32
    • 57349138185 scopus 로고    scopus 로고
    • Counting faces of randomly projected polytopes when the projection radically lowers dimension
    • D. L. Donoho and J. Tanner, "Counting faces of randomly projected polytopes when the projection radically lowers dimension," J. Amer. Math. Soc., vol. 22, pp. 1-53, 2009.
    • (2009) J. Amer. Math. Soc. , vol.22 , pp. 1-53
    • Donoho, D.L.1    Tanner, J.2
  • 33
    • 73149095252 scopus 로고    scopus 로고
    • Observed universality of phase transitions in high-dimensional geometry, with implications for modern data analysis and signal processing
    • D. L. Donoho and J. Tanner, "Observed universality of phase transitions in high-dimensional geometry, with implications for modern data analysis and signal processing," Phil. Trans. R. Soc. A, vol. 367, pp. 4273-4293, 2009.
    • (2009) Phil. Trans. R. Soc. A , vol.367 , pp. 4273-4293
    • Donoho, D.L.1    Tanner, J.2
  • 34
    • 77950517814 scopus 로고    scopus 로고
    • Counting the faces of randomly-projected hypercubes and orthants, with applications
    • D. L. Donoho and J. Tanner, "Counting the faces of randomly-projected hypercubes and orthants, with applications," Discrete Comput. Geome., vol. 43, no. 3, pp. 522-541, 2010.
    • (2010) Discrete Comput. Geome. , vol.43 , Issue.3 , pp. 522-541
    • Donoho, D.L.1    Tanner, J.2
  • 35
    • 77953126052 scopus 로고    scopus 로고
    • Precise undersampling theorems
    • Jun.
    • D. L. Donoho and J. Tanner, "Precise undersampling theorems," Proc. IEEE, vol. 98, no. 6, pp. 913-924, Jun. 2010.
    • (2010) Proc. IEEE , vol.98 , Issue.6 , pp. 913-924
    • Donoho, D.L.1    Tanner, J.2
  • 36
    • 48349106240 scopus 로고    scopus 로고
    • Iterative thresholding algorithms
    • M. Fornasier and H. Rauhut, "Iterative thresholding algorithms," Appl. Comp. Harmon. Anal., vol. 25, pp. 187-208, 2008.
    • (2008) Appl. Comp. Harmon. Anal. , vol.25 , pp. 187-208
    • Fornasier, M.1    Rauhut, H.2
  • 37
    • 0031312521 scopus 로고    scopus 로고
    • Waveshrink with firm shrinkage
    • H.-Y. Gao and A. G. Bruce, "Waveshrink with firm shrinkage," Statist. Sinica, vol. 7, pp. 855-874, 1997.
    • (1997) Statist. Sinica , vol.7 , pp. 855-874
    • Gao, H.-Y.1    Bruce, A.G.2
  • 38
    • 0032360314 scopus 로고    scopus 로고
    • Block threshold rules for curve estimation using kernel and wavelet methods
    • P. Hall, G. Kerkyacharian, and D. Picard, "Block threshold rules for curve estimation using kernel and wavelet methods," Ann. Statist., vol. 26, pp. 922-942, 1998.
    • (1998) Ann. Statist. , vol.26 , pp. 922-942
    • Hall, P.1    Kerkyacharian, G.2    Picard, D.3
  • 40
    • 0003157339 scopus 로고
    • Robust estimation of a location parameter
    • P. J. Huber, "Robust estimation of a location parameter," Ann. Math. Statist., vol. 35, no. 1, pp. 73-101, 1964.
    • (1964) Ann. Math. Statist. , vol.35 , Issue.1 , pp. 73-101
    • Huber, P.J.1
  • 42
    • 0347311799 scopus 로고
    • Minimax Bayes, asymptotic minimax and sparse wavelet priors
    • S.S. Gupta and J. O. Berger, Eds. New York, NY, USA: Springer-Verlag
    • I. M. Johnstone, "Minimax Bayes, asymptotic minimax and sparse wavelet priors," in Statistical Decision Theory and Related Topics, S.S. Gupta and J. O. Berger, Eds. New York, NY, USA: Springer-Verlag, 1994, vol. V., pp. 303-326.
    • (1994) Statistical Decision Theory and Related Topics , vol.5 , pp. 303-326
    • Johnstone, I.M.1
  • 43
    • 42149112994 scopus 로고
    • On minimax estimation of a sparse normal mean vector
    • I. M. Johnstone, "On minimax estimation of a sparse normal mean vector," Ann. Statist., vol. 22, pp. 271-289, 1994.
    • (1994) Ann. Statist. , vol.22 , pp. 271-289
    • Johnstone, I.M.1
  • 47
    • 84862656358 scopus 로고    scopus 로고
    • Statistical physics-based reconstruction in compressed sensing
    • F. Krzakala,M.Mézard, F. Sausset, Y. Sun, and L. Zdeborova, "Statistical physics-based reconstruction in compressed sensing," Phys. Rev. X, vol. 2, pp. 021005-1-021005-18, 2012.
    • (2012) Phys. Rev. X , vol.2 , pp. 0210051-02100518
    • Krzakala, F.1    Mézard, M.2    Sausset, F.3    Sun, Y.4    Zdeborova, L.5
  • 48
    • 72449137292 scopus 로고    scopus 로고
    • A typical reconstruction limit for compressed sensing based on lp-norm minimization
    • Y. Kabashima, T.Wadayama, and T. Tanaka, "A typical reconstruction limit for compressed sensing based on lp-norm minimization," J. Stat. Mech., vol. 2009, p. L09003, 2009.
    • (2009) J. Stat. Mech. , vol.2009
    • Kabashima, Y.1    Wadayama, T.2    Tanaka, T.3
  • 49
    • 77955703074 scopus 로고    scopus 로고
    • Improved sparse recovery thresholds with two-step reweighted minimization
    • Austin, TX, USA, Jun.
    • M. A. Khajehnejad, W. Xu, A. S. Avestimehr, and B. Hassibi, "Improved sparse recovery thresholds with two-step reweighted minimization," presented at the IEEE Int. Symp. Inf. Theory, Austin, TX, USA, Jun. 2010.
    • (2010) IEEE Int. Symp. Inf. Theory
    • Khajehnejad, M.A.1    Xu, W.2    Avestimehr, A.S.3    Hassibi, B.4
  • 50
    • 84955819845 scopus 로고
    • Minimizing an integral
    • C. L. Mallows, "Minimizing an integral," SIAM Rev., vol. 20, p. 183, 1978.
    • (1978) SIAM Rev. , vol.20 , pp. 183
    • Mallows, C.L.1
  • 52
    • 77949685085 scopus 로고    scopus 로고
    • Optimally tuned iterative reconstruction algorithms for compressed sensing
    • Apr.
    • A. Maleki and D. L. Donoho, "Optimally tuned iterative reconstruction algorithms for compressed sensing," IEEE J. Sel. Topics Signal Process., vol. 4, no. 2, pp. 330-341, Apr. 2010.
    • (2010) IEEE J. Sel. Topics Signal Process. , vol.4 , Issue.2 , pp. 330-341
    • Maleki, A.1    Donoho, D.L.2
  • 54
    • 80054086193 scopus 로고    scopus 로고
    • Graphical models concepts in compressed sensing
    • Y. C. Eldar and G. Kutyniok, Eds. Cambridge, U.K.: Cambridge Univ. Press
    • A. Montanari, "Graphical models concepts in compressed sensing," in Compressed Sensing, Y. C. Eldar and G. Kutyniok, Eds. Cambridge, U.K.: Cambridge Univ. Press, 2012.
    • (2012) Compressed Sensing
    • Montanari, A.1
  • 55
    • 80054799706 scopus 로고    scopus 로고
    • Generalized approximate message passing for estimation with random linear mixing
    • Russia, Aug.
    • S. Rangan, "Generalized approximate message passing for estimation with random linear mixing," presented at the IEEE Int. Symp. Inf. Theory, St. Petersburg, Russia, Aug. 2011.
    • (2011) IEEE Int. Symp. Inf. Theory, St. Petersburg
    • Rangan, S.1
  • 56
    • 84858722178 scopus 로고    scopus 로고
    • Asymptotic analysis of map estimation via the replica method and applications to compressed sensing
    • Vancouver, BC, Canada
    • S. Rangan, A. K. Fletcher, and V. K. Goyal, "Asymptotic analysis of map estimation via the replica method and applications to compressed sensing," presented at the Neural Inf. Process. Syst., Vancouver, BC, Canada, 2009.
    • (2009) Neural Inf. Process. Syst.
    • Rangan, S.1    Fletcher, A.K.2    Goyal, V.K.3
  • 57
    • 44049111982 scopus 로고
    • Nonlinear total variation based noise removal algorithms
    • L. I. Rudin, S. Osher, and E. Fatemi, "Nonlinear total variation based noise removal algorithms," Phys. D: Nonlinear Phenom., vol. 60, pp. 259-268, 1992.
    • (1992) Phys. D: Nonlinear Phenom. , vol.60 , pp. 259-268
    • Rudin, L.I.1    Osher, S.2    Fatemi, E.3
  • 60
    • 77953689056 scopus 로고    scopus 로고
    • Turbo reconstruction of structured sparse signals
    • Princeton, NJ, USA
    • P. Schniter, "Turbo reconstruction of structured sparse signals," presented at the Proc. Conf. Inf. Sci. Syst., Princeton, NJ, USA, 2010.
    • (2010) Proc. Conf. Inf. Sci. Syst.
    • Schniter, P.1
  • 62
    • 79957972819 scopus 로고    scopus 로고
    • Compressive imaging using approximate message passing and a Markov-tree prior
    • Nov.
    • S. Som, L. C. Potter, and P. Schniter, "Compressive imaging using approximate message passing and a Markov-tree prior," presented at the Asilomar Conf. Signals, Syst., Comput., Nov. 2010.
    • (2010) Asilomar Conf. Signals, Syst., Comput.
    • Som, S.1    Potter, L.C.2    Schniter, P.3
  • 63
    • 79952635553 scopus 로고    scopus 로고
    • On approximatemessage passing for reconstruction of non-uniformly sparse signals
    • Dayton, OH, USA
    • L. C. Potter, S. Som, and P. Schniter, "On approximatemessage passing for reconstruction of non-uniformly sparse signals," presented at the Nat. Aereosp. Electron. Conf., Dayton, OH, USA, 2010.
    • (2010) Nat. Aereosp. Electron. Conf.
    • Potter, L.C.1    Som, S.2    Schniter, P.3
  • 64
    • 78049377842 scopus 로고    scopus 로고
    • Block-length dependent thresholds for 2/1-optimization in block-sparse compressed sensing
    • M. Stojnic, "Block-length dependent thresholds for 2/1-optimization in block-sparse compressed sensing," in Proc. IEEE Int. Conf. Acoust., Speech, Signal Process., 2010, pp. 3918-3921.
    • (2010) Proc. IEEE Int. Conf. Acoust., Speech, Signal Process. , pp. 3918-3921
    • Stojnic, M.1
  • 66
    • 0001556589 scopus 로고    scopus 로고
    • Iterative methods for total variation denoising
    • C. R. Vogel and M. E. Oman, "Iterative methods for total variation denoising," SIAM J. Sci. Comput., vol. 17, pp. 227-238, 1996. (Pubitemid 126621162)
    • (1996) SIAM Journal of Scientific Computing , vol.17 , Issue.1 , pp. 227-238
    • Vogel, C.R.1    Oman, M.E.2
  • 67
    • 65749083666 scopus 로고    scopus 로고
    • Sharp thresholds for high-dimensional and noisy sparsity recovery using-constrained quadratic programming (Lasso)
    • May
    • M. J. Wainwright, "Sharp thresholds for high-dimensional and noisy sparsity recovery using-constrained quadratic programming (Lasso)," IEEE Trans. Inf. Theory, vol. 55, no. 5, pp. 2183-2202, May 2009.
    • (2009) IEEE Trans. Inf. Theory , vol.55 , Issue.5 , pp. 2183-2202
    • Wainwright, M.J.1
  • 68
    • 77954596738 scopus 로고    scopus 로고
    • Rényi information dimension: Fundamental limits of almost lossless analog compression
    • Aug.
    • Y. Wu and S. Verdú, "Rényi information dimension: Fundamental limits of almost lossless analog compression," IEEE Trans. Inf. Theory, vol. 56, no. 8, pp. 3721-3748, Aug. 2010.
    • (2010) IEEE Trans. Inf. Theory , vol.56 , Issue.8 , pp. 3721-3748
    • Wu, Y.1    Verdú, S.2
  • 71
    • 1142284447 scopus 로고
    • Estimation in continuous exponential families: Bayesian estimation subject to risk restrictions and inadmissibility results
    • S. S. Gupta and J. Berger, Eds., 1 ed. New York: Academic
    • J. Berger, "Estimation in continuous exponential families: Bayesian estimation subject to risk restrictions and inadmissibility results," in Statistical Decision Theory and Related Topics III , S. S. Gupta and J. Berger, Eds., 1 ed. New York: Academic, 1982, p. 109.
    • (1982) Statistical Decision Theory and Related Topics , vol.3 , pp. 109
    • Berger, J.1
  • 72
    • 0039789064 scopus 로고
    • Parametric robustness: Small biases can be worthwhile
    • Sep. published by: Institute of Mathematical Statistics Stable URL:
    • P. J. Bickell, "Parametric robustness: Small biases can be worthwhile," Ann. Statist., vol. 12, no. 3, pp. 864-879, Sep. 1984, published by: Institute of Mathematical Statistics Stable URL: http://www.jstor.org/stable/2240965.
    • (1984) Ann. Statist. , vol.12 , Issue.3 , pp. 864-879
    • Bickell, P.J.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.