Sparse Recovery via Partial Regularization: Models, Theory, and Algorithms

Published Online:https://doi.org/10.1287/moor.2017.0905

References

  • Ahn M, Pang JS, Xin J (2017) Difference-of-convex learning I: Directional stationarity, optimality, and sparsity. SIAM J. Optim. 27(3):1637–1665.CrossrefGoogle Scholar
  • Andreani R, Haeser G, Schuverdt ML, Silva PJ (2012) A relaxed constant positive linear dependence constraint qualification and applications. Math. Programming 135(1–2):255–273.CrossrefGoogle Scholar
  • Baraniuk R, Steeghs P (2007) Compressive radar imaging. Proc. 2007 IEEE Radar Conf., (IEEE, Piscataway, NJ), 128–133.Google Scholar
  • Barzilai J, Borwein J (1988) Two-point step size gradient methods. IMA J. Numer. Anal. 8(1):141–148.CrossrefGoogle Scholar
  • Birgin EG, Martinez JM, Raydan JA (2000) Nonmonotone spectral projected gradient methods on convex sets. SIAM J. Optim. 10(4):1196–1211.CrossrefGoogle Scholar
  • Blumensath T, Davies ME (2008) Iterative thresholding for sparse approximations. J. Fourier Anal. Appl. 14(5):629–654.CrossrefGoogle Scholar
  • Blumensath T, Davies ME (2009) Iterative hard thresholding for compressed sensing. Appl. Comput. Harmonic Anal. 27(3):265–274.CrossrefGoogle Scholar
  • Cai TT, Zhang A (2013) Sharp RIP bound for sparse signal and low-rank matrix recovery. Appl. Comput. Harmonic Anal. 35(1):74–93.CrossrefGoogle Scholar
  • Cai TT, Zhang A (2014) Sparse representation of a polytope and recovery of sparse signals and low-rank matrices. IEEE Trans. Inform. Theory 60(1):122–132.CrossrefGoogle Scholar
  • Candès EJ, Tao T (2005) Decoding by linear programming. IEEE Trans. Inform. Theory 51(12):4203–4215.CrossrefGoogle Scholar
  • Candès EJ, Wakin MB, Boyd SP (2008) Enhancing sparsity by reweighted ℓ1 minimization. J. Fourier Anal. Appl. 14(5–6):877–905.CrossrefGoogle Scholar
  • Chartrand R (2007) Exact reconstruction of sparse signals via nonconvex minimization. IEEE Signal Processing Lett. 14(10):707–710.CrossrefGoogle Scholar
  • Chartrand R, Staneva V (2008) Restricted isometry properties and nonconvex compressive sensing. Inverse Problems 24(3):1–14.CrossrefGoogle Scholar
  • Chen S, Donoho D, Saunders M (1998) Atomic decomposition by basis pursuit. SIAM J. Sci. Comput. 20(1):33–61.CrossrefGoogle Scholar
  • Chen X, Lu Z, Pong TK (2016) Penalty methods for a class of non-Lipschitz optimization problems. SIAM J. Optim. 26(3):1465–1492.CrossrefGoogle Scholar
  • Chen X, Guo L, Lu Z, Ye J (2017) An augmented Lagrangian method for non-Lipschitz nonconvex programming. SIAM J. Numer. Anal. 55(1):168–193.CrossrefGoogle Scholar
  • Clarke FH (1975) Generalized gradients and applications. Trans. Amer. Math. Soc. 205(April):247–262.CrossrefGoogle Scholar
  • Clarke FH (1983) Optimization and Nonsmooth Analysis (John Wiley & Sons, New York).Google Scholar
  • Davenport MA, Laska JN, Treichler JR, Baraniuk RG (2012) The pros and cons of compressive sensing for wideband wignal acquisition: Noise folding versus dynamic range. IEEE Trans. Signal Processing 60(9):4628–4642.CrossrefGoogle Scholar
  • Donoho DL (2006) Compressed sensing. IEEE Trans. Inform. Theory 52(4):1289–1306.CrossrefGoogle Scholar
  • Donoho DL, Elad M, Temlyakov VN (2006) Stable recovery of sparse overcomplete representations in the presence of noise. IEEE Trans. Inform. Theory 52(1):6–18.CrossrefGoogle Scholar
  • Fan J, Li R (2001) Variable selection via nonconcave penalized likelihood and its oracle properties. J. Amer. Statist. Assoc. 96(456):1348–1360.CrossrefGoogle Scholar
  • Foucart S, Lai M (2009) Sparsest solutions of underdetermined linear systems via ℓq-minimization for 0<q < r> ≤ 1. Appl. Comput. Harmonic Anal. 26(3):395–407.CrossrefGoogle Scholar
  • Frank I, Friedman J (1993) A statistical view of some chemometrics regression tools (with discussion). Technometrics 35(2):109–148.CrossrefGoogle Scholar
  • Fu WJ (1998) Penalized regression: The bridge versus the lasso. J. Comput. Graph. Statist. 7(3):397–416.Google Scholar
  • Gao Y (2010) Structured low rank matrix optimization problems: A penalized approach. Unpublished Ph.D. thesis, Department of Mathematics, National University of Singapore, Singapore.Google Scholar
  • Gao Y, Sun D (2010) A majorized penalty approach for calibrating rank constrained correlation matrix problems. Technical report, Department of Mathematics, National University of Singapore, Singapore.Google Scholar
  • Golub GH, Van Loan CF (1996) Matrix Computations, Studies in Applied Mathematics, 3rd ed., Vol. 13 (John Hopkins University Press, Baltimore).Google Scholar
  • Gotoh J, Takeda A, Tono K (2015) DC formulations and algorithms for sparse optimization problems. Mathematical engineering technical report, Department of Mathematical Informatics, University of Tokyo, Tokyo.Google Scholar
  • Herman M, Strohmer T (2009) High-resolution radar via compressed sensing. IEEE Trans. Signal Processing 57(6):2275–2284.CrossrefGoogle Scholar
  • Hoffman AJ (1952) Nonlinear analysis, differential equations and control. J. Res. Natl. Bureau Standards 49(4):263–265.CrossrefGoogle Scholar
  • Huang XL, Shi L, Yan M (2015) Nonconvex sorted ℓ1 minimization for sparse approximation. J. Oper. Res. Soc. China 3(2):207–229.CrossrefGoogle Scholar
  • Koh K, Kim SJ, Boyd S (2007) An interior-point method for large-scale l1-regularized logistic regression. J. Machine Learn. Res. 8(December):1519–1555.Google Scholar
  • Lichman M (2013) UCI Machine Learning Repository. Accessed May 8, 2015, http://archive.ics.uci.edu/ml.Google Scholar
  • Lu Z (2014) Iterative hard thresholding methods for ℓ0 regularized convex cone programming. Math. Programming 147(1–2):125–154.CrossrefGoogle Scholar
  • Lu Z, Zhang Y (2012) An augmented Lagrangian approach for sparse principal component analysis. Math. Programming 135(1–2):149–193.CrossrefGoogle Scholar
  • Lu Z, Zhang Y (2013) Sparse approximation via penalty decomposition methods. SIAM J. Optim. 23(4):2448–2478.CrossrefGoogle Scholar
  • Lustig M, Donoho DL, Pauly JM (2008) Sparse MRI: The application of compressed sensing for rapid MR imaging. Magnetic Resonance Medicine 58(6):1182–1195.CrossrefGoogle Scholar
  • Lustig M, Donoho DL, Santos JM, Pauly JM (2008) Compressed sensing MRI. IEEE Signal Processing Magazine 25(2):72–82.CrossrefGoogle Scholar
  • Pang JS (2016) Difference-of-convex statistical learning: Directional stationarity, optimality, and sparsity. Plenary talk at the Fifth International Conference on Continuous Optimization, August 11, Tokyo, Japan.Google Scholar
  • Rockafellar RT, Wets RJ-B (1998) Variational Analysis (Springer, Berlin).CrossrefGoogle Scholar
  • Song C, Xia S (2014) Sparse signal recovery by ℓq minimization under restricted isometry property. IEEE Signal Processing Lett. 21(9):1154–1158.CrossrefGoogle Scholar
  • Tibshirani R (1996) Regression shrinkage and selection via the lasso. J. Roy. Statist. Soc. Ser. B 58(1):267–288.CrossrefGoogle Scholar
  • Tropp JA, Laska JN, Duarte MF, Romberg JK, Baraniuk RG (2010) Beyond Nyquist: Efficient sampling of sparse bandlimited signals. IEEE Trans. Inform. Theory 56(1):520–544.CrossrefGoogle Scholar
  • Weston J, Elisseeff A, Scholkopf B, Tipping M (2003) The use of zero-norm with linear models and kernel methods. J. Machine Learn. Res. 3(March):1439–1461.Google Scholar
  • Wright SJ, Nowak RD, Figueiredo M (2009) Sparse reconstruction by separable approximation. IEEE Trans. Signal Processing 57(7):2479–2493.CrossrefGoogle Scholar
  • Zhang CH (2010a) Nearly unbiased variable selection under minimax concave penalty. Ann. Statist. 38(2):894–942.CrossrefGoogle Scholar
  • Zhang T (2010b) Analysis of multi-stage convex relaxation for sparse regularization. J. Machine Learn. Res. 11(March):1081–1107.Google Scholar
INFORMS site uses cookies to store information on your computer. Some are essential to make our site work; Others help us improve the user experience. By using this site, you consent to the placement of these cookies. Please read our Privacy Statement to learn more.