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Recovery of seismic wavefields by an lq-norm constrained regularization method
Risk estimators for choosing regularization parameters in ill-posed problems - properties and limitations
1. | Centrum Wiskunde & Informatica (CWI), Science Park 123, 1098 XG Amsterdam, The Netherlands |
2. | Department of Computer Science, University College London, WC1E 6BT London, UK |
3. | Universität Göttingen, Institut für Mathematische Stochastik, Goldschmidtstrasse 7, 37077 Göttingen, Germany |
4. | Department of Applied Mathematics, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands |
5. | Fakultät für Mathematik, Ruhr-Universität, Bochum, 44780 Bochum, Germany |
6. | Institut für Numerische und Angewandte Mathematik, Westfälische Wilhelms-Universität (WWU) Münster, Einsteinstr. 62, D 48149 Münster, Germany |
This paper discusses the properties of certain risk estimators that recently regained popularity for choosing regularization parameters in ill-posed problems, in particular for sparsity regularization. They apply Stein's unbiased risk estimator (SURE) to estimate the risk in either the space of the unknown variables or in the data space. We will call the latter PSURE in order to distinguish the two different risk functions. It seems intuitive that SURE is more appropriate for ill-posed problems, since the properties in the data space do not tell much about the quality of the reconstruction. We provide theoretical studies of both approaches for linear Tikhonov regularization in a finite dimensional setting and estimate the quality of the risk estimators, which also leads to asymptotic convergence results as the dimension of the problem tends to infinity. Unlike previous works which studied single realizations of image processing problems with a very low degree of ill-posedness, we are interested in the statistical behaviour of the risk estimators for increasing ill-posedness. Interestingly, our theoretical results indicate that the quality of the SURE risk can deteriorate asymptotically for ill-posed problems, which is confirmed by an extensive numerical study. The latter shows that in many cases the SURE estimator leads to extremely small regularization parameters, which obviously cannot stabilize the reconstruction. Similar but less severe issues with respect to robustness also appear for the PSURE estimator, which in comparison to the rather conservative discrepancy principle leads to the conclusion that regularization parameter choice based on unbiased risk estimation is not a reliable procedure for ill-posed problems. A similar numerical study for sparsity regularization demonstrates that the same issue appears in non-linear variational regularization approaches.
References:
[1] |
R. J. Adler and J. E. Taylor,
Random Fields and Geometry, Springer Monographs in Mathematics, Springer, New York, 2007. |
[2] |
M. S. C. Almeida and M. A. T. Figueiredo,
Parameter estimation for blind and non-blind deblurring using residual whiteness measures, IEEE Transactions on Image Processing, 22 (2013), 2751-2763.
doi: 10.1109/TIP.2013.2257810. |
[3] |
F. Bauer and T. Hohage,
A Lepskij-type stopping rule for regularized Newton methods, Inverse Problems, 21 (2005), 1975-1991.
doi: 10.1088/0266-5611/21/6/011. |
[4] |
G. Blanchard and P. Mathé, Discrepancy principle for statistical inverse problems with application to conjugate gradient iteration Inverse Problems, 28 (2012), 115011, 23pp.
doi: 10.1088/0266-5611/28/11/115011. |
[5] |
P. Blomgren and T. F. Chan,
Modular solvers for image restoration problems using the discrepancy principle, Numerical Linear Algebra with Applications, 9 (2002), 347-358.
doi: 10.1002/nla.278. |
[6] |
S. Boyd, N. Parikh, E. Chu, B. Peleato and J. Eckstein,
Distributed optimization and statistical learning via the alternating direction method of multipliers, Foundations and Trends in Machine Learning, 3 (2011), 1-122.
doi: 10.1561/2200000016. |
[7] |
B. Bringmann, D. Cremers, F. Krahmer and M. Möller, The homotopy method revisited: Computing solution paths of $\ell_1$-regularized problems, Math. Comp., 87 (2018), 2343-2364, arXiv: 1605.00071.
doi: 10.1090/mcom/3287. |
[8] |
M. Burger, A. Sawatzky and G. Steidl, First order algorithms in variational image processing, Splitting Methods in Communication, Imaging, Science, and Engineering, 345-407, Sci. Comput., Springer, Cham, 2016. |
[9] |
E. J. Candes, C. A. Sing-Long and J. D. Trzasko,
Unbiased risk estimates for singular value thresholding and spectral estimators, IEEE Transactions on Signal Processing, 61 (2013), 4643-4657.
doi: 10.1109/TSP.2013.2270464. |
[10] |
E. Chernousova and Y. Golubev,
Spectral cut-off regularizations for ill-posed linear models, Math. Methods Statist., 23 (2014), 116-131.
doi: 10.3103/S1066530714020033. |
[11] |
C. Deledalle, S. Vaiter, J. Fadili and G. Peyré,
Stein Unbiased GrAdient estimator of the Risk (SUGAR) for Multiple Parameter Selection, SIAM Journal on Imaging Sciences, 7 (2014), 2448-2487.
doi: 10.1137/140968045. |
[12] |
C. Deledalle, S. Vaiter, G. Peyré, J. Fadili and C. Dossal, Proximal splitting derivatives for risk estimation,
Journal of Physics: Conference Series, 386 (2012), 012003.
doi: 10.1088/1742-6596/386/1/012003. |
[13] |
C. Deledalle, S. Vaiter, G. Peyré, J. Fadili and C. Dossal, Unbiased risk estimation for sparse analysis regularization, in 2012 19th IEEE International Conference on Image Processing, IEEE, 2012, 3053-3056.
doi: 10.1109/ICIP.2012.6467544. |
[14] |
C. Dossal, M. Kachour, J. Fadili, G. Peyré and C. Chesneau,
The degrees of freedom of the lasso for general design matrix, Statistica Sinica, 23 (2013), 809-828.
|
[15] |
Y. C. Eldar,
Generalized SURE for Exponential Families: Applications to Regularization, IEEE Transactions on Signal Processing, 57 (2009), 471-481.
doi: 10.1109/TSP.2008.2008212. |
[16] |
H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, vol. 375, Springer Science & Business Media, 1996. |
[17] |
N. P. Galatsanos and A. K. Katsaggelos,
Methods for choosing the regularization parameter and estimating the noise variance in image restoration and their relation, Trans. Img. Proc., 1 (1992), 322-336.
doi: 10.1109/83.148606. |
[18] |
S. K. Ghoreishi and M. R. Meshkani,
On SURE estimates in hierarchical models assuming heteroscedasticity for both levels of a two-level normal hierarchical model, Journal of Multivariate Analysis, 132 (2014), 129-137.
doi: 10.1016/j.jmva.2014.08.001. |
[19] |
R. Giryes, M. Elad and Y. Eldar,
The projected GSURE for automatic parameter tuning in iterative shrinkage methods, Applied and Computational Harmonic Analysis, 30 (2011), 407-422.
doi: 10.1016/j.acha.2010.11.005. |
[20] |
J. Hadamard,
Lectures on Cauchy's Problem in Linear Partial Differential Equations, New Haven, 1953. |
[21] |
H. Haghshenas Lari and A. Gholami,
Curvelet-TV regularized Bregman iteration for seismic random noise attenuation, Journal of Applied Geophysics, 109 (2014), 233-241.
doi: 10.1016/j.jappgeo.2014.08.005. |
[22] |
P. C. Hansen,
Analysis of discrete ill-posed problems by means of the L-curve, SIAM Review, 34 (1992), 561-580.
doi: 10.1137/1034115. |
[23] |
P. C. Hansen and D. P. OLeary,
The Use of the L-Curve in the Regularization of Discrete Ill-Posed Problems, SIAM Journal on Scientific Computing, 14 (1993), 1487-1503.
doi: 10.1137/0914086. |
[24] |
B. Jin, J. Zou et al., Iterative parameter choice by discrepancy principle, IMA Journal of Numerical Analysis, 32 (2012), 1714-1732.
doi: 10.1093/imanum/drr051. |
[25] |
A. Kneip,
Ordered linear smoothers, Ann. Statist., 22 (1994), 835-866.
doi: 10.1214/aos/1176325498. |
[26] |
O. V. Lepskii,
On a Problem of Adaptive Estimation in Gaussian White Noise, Theory of Probability & Its Applications, 35 (1991), 454-466.
doi: 10.1137/1135065. |
[27] |
H. Li and F. Werner, Empirical risk minimization as parameter choice rule for general linear regularization methods, 2017, arXiv: 1703.07809. |
[28] |
K.-C. Li,
From stein's unbiased risk estimates to the method of generalized cross validation, The Annals of Statistics, 13 (1985), 1352-1377.
doi: 10.1214/aos/1176349742. |
[29] |
K.-C. Li,
Asymptotic optimality for $C_p$, $C_L$, cross-validation and generalized cross-validation: Discrete index set, Ann. Statist., 15 (1987), 958-975.
doi: 10.1214/aos/1176350486. |
[30] |
F. Luisier, T. Blu and M. Unser,
Image denoising in mixed Poisson-Gaussian noise, IEEE Transactions on Image Processing, 20 (2011), 696-708.
doi: 10.1109/TIP.2010.2073477. |
[31] |
J.-C. Pesquet, A. Benazza-Benyahia and C. Chaux,
A SURE Approach for Digital Signal/Image Deconvolution Problems, IEEE Transactions on Signal Processing, 57 (2009), 4616-4632.
doi: 10.1109/TSP.2009.2026077. |
[32] |
P. Qu, C. Wang and G. X. Shen,
Discrepancy-based adaptive regularization for grappa reconstruction, Journal of Magnetic Resonance Imaging, 24 (2006), 248-255.
doi: 10.1002/jmri.20620. |
[33] |
S. Ramani, T. Blu and M. Unser,
Monte-Carlo sure: A black-box optimization of regularization parameters for general denoising algorithms, IEEE Transactions on Image Processing, 17 (2008), 1540-1554.
doi: 10.1109/TIP.2008.2001404. |
[34] |
S. Ramani, Z. Liu, J. Rosen, J.-F. Nielsen and J. A. Fessler,
Regularization parameter selection for nonlinear iterative image restoration and MRI reconstruction using GCV and SURE-based methods, IEEE Transactions on Image Processing, 21 (2012), 3659-3672.
doi: 10.1109/TIP.2012.2195015. |
[35] |
J. A. Rice,
Choice of smoothing parameter in deconvolution problems, Contemporary Mathematics, 59 (1986), 137-151.
doi: 10.1090/conm/059/10. |
[36] |
C. M. Stein,
Estimation of the mean of a multivariate normal distribution, The Annals of Statistics, 9 (1981), 1135-1151.
doi: 10.1214/aos/1176345632. |
[37] |
A. M. Thompson, J. C. Brown, J. W. Kay and D. M. Titterington,
A study of methods of choosing the smoothing parameter in image restoration by regularization, IEEE Trans. Pattern Anal. Mach. Intell., 13 (1991), 326-339.
doi: 10.1109/34.88568. |
[38] |
G. M. Vainikko,
The discrepancy principle for a class of regularization methods, USSR Computational Mathematics and Mathematical Physics, 22 (1982), 1-19.
doi: 10.1016/0041-5553(82)90120-3. |
[39] |
S. Vaiter, C. Deledalle and G. Peyré,
The degrees of freedom of partly smooth regularizers, Annals of the Institute of Statistical Mathematics, 69 (2017), 791-832.
doi: 10.1007/s10463-016-0563-z. |
[40] |
S. Vaiter, C. Deledalle, G. Peyré, C. Dossal and J. Fadili,
Local behavior of sparse analysis regularization: Applications to risk estimation, Applied and Computational Harmonic Analysis, 35 (2013), 433-451.
doi: 10.1016/j.acha.2012.11.006. |
[41] |
S. A. vande Geer,
Applications of Empirical Process Theory, vol. 6 of Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge University Press, Cambridge, 2000. |
[42] |
D. Van De Ville and M. Kocher, SURE-Based Non-Local Means, IEEE Signal Processing Letters, 16 (2009), 973-976, URL http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=5165022. |
[43] |
D. Van DeVille and M. Kocher,
Nonlocal means with dimensionality reduction and SURE-based parameter selection, IEEE Transactions on Image Processing, 20 (2011), 2683-2690.
doi: 10.1109/TIP.2011.2121083. |
[44] |
Y.-Q. Wang and J.-M. Morel,
SURE Guided Gaussian Mixture Image Denoising, SIAM Journal on Imaging Sciences, 6 (2013), 999-1034.
doi: 10.1137/120901131. |
[45] |
D. S. Weller, S. Ramani, J.-F. Nielsen and J. A. Fessler, Monte Carlo SURE-based parameter selection for parallel magnetic resonance imaging reconstruction, Magnetic Resonance in Medicine, 71 (2014), 1760-1770. |
[46] |
X. Xie, S. C. Kou and L. D. Brown,
SURE Estimates for a Heteroscedastic Hierarchical Model, Journal of the American Statistical Association, 107 (2012), 1465-1479.
doi: 10.1080/01621459.2012.728154. |
show all references
References:
[1] |
R. J. Adler and J. E. Taylor,
Random Fields and Geometry, Springer Monographs in Mathematics, Springer, New York, 2007. |
[2] |
M. S. C. Almeida and M. A. T. Figueiredo,
Parameter estimation for blind and non-blind deblurring using residual whiteness measures, IEEE Transactions on Image Processing, 22 (2013), 2751-2763.
doi: 10.1109/TIP.2013.2257810. |
[3] |
F. Bauer and T. Hohage,
A Lepskij-type stopping rule for regularized Newton methods, Inverse Problems, 21 (2005), 1975-1991.
doi: 10.1088/0266-5611/21/6/011. |
[4] |
G. Blanchard and P. Mathé, Discrepancy principle for statistical inverse problems with application to conjugate gradient iteration Inverse Problems, 28 (2012), 115011, 23pp.
doi: 10.1088/0266-5611/28/11/115011. |
[5] |
P. Blomgren and T. F. Chan,
Modular solvers for image restoration problems using the discrepancy principle, Numerical Linear Algebra with Applications, 9 (2002), 347-358.
doi: 10.1002/nla.278. |
[6] |
S. Boyd, N. Parikh, E. Chu, B. Peleato and J. Eckstein,
Distributed optimization and statistical learning via the alternating direction method of multipliers, Foundations and Trends in Machine Learning, 3 (2011), 1-122.
doi: 10.1561/2200000016. |
[7] |
B. Bringmann, D. Cremers, F. Krahmer and M. Möller, The homotopy method revisited: Computing solution paths of $\ell_1$-regularized problems, Math. Comp., 87 (2018), 2343-2364, arXiv: 1605.00071.
doi: 10.1090/mcom/3287. |
[8] |
M. Burger, A. Sawatzky and G. Steidl, First order algorithms in variational image processing, Splitting Methods in Communication, Imaging, Science, and Engineering, 345-407, Sci. Comput., Springer, Cham, 2016. |
[9] |
E. J. Candes, C. A. Sing-Long and J. D. Trzasko,
Unbiased risk estimates for singular value thresholding and spectral estimators, IEEE Transactions on Signal Processing, 61 (2013), 4643-4657.
doi: 10.1109/TSP.2013.2270464. |
[10] |
E. Chernousova and Y. Golubev,
Spectral cut-off regularizations for ill-posed linear models, Math. Methods Statist., 23 (2014), 116-131.
doi: 10.3103/S1066530714020033. |
[11] |
C. Deledalle, S. Vaiter, J. Fadili and G. Peyré,
Stein Unbiased GrAdient estimator of the Risk (SUGAR) for Multiple Parameter Selection, SIAM Journal on Imaging Sciences, 7 (2014), 2448-2487.
doi: 10.1137/140968045. |
[12] |
C. Deledalle, S. Vaiter, G. Peyré, J. Fadili and C. Dossal, Proximal splitting derivatives for risk estimation,
Journal of Physics: Conference Series, 386 (2012), 012003.
doi: 10.1088/1742-6596/386/1/012003. |
[13] |
C. Deledalle, S. Vaiter, G. Peyré, J. Fadili and C. Dossal, Unbiased risk estimation for sparse analysis regularization, in 2012 19th IEEE International Conference on Image Processing, IEEE, 2012, 3053-3056.
doi: 10.1109/ICIP.2012.6467544. |
[14] |
C. Dossal, M. Kachour, J. Fadili, G. Peyré and C. Chesneau,
The degrees of freedom of the lasso for general design matrix, Statistica Sinica, 23 (2013), 809-828.
|
[15] |
Y. C. Eldar,
Generalized SURE for Exponential Families: Applications to Regularization, IEEE Transactions on Signal Processing, 57 (2009), 471-481.
doi: 10.1109/TSP.2008.2008212. |
[16] |
H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, vol. 375, Springer Science & Business Media, 1996. |
[17] |
N. P. Galatsanos and A. K. Katsaggelos,
Methods for choosing the regularization parameter and estimating the noise variance in image restoration and their relation, Trans. Img. Proc., 1 (1992), 322-336.
doi: 10.1109/83.148606. |
[18] |
S. K. Ghoreishi and M. R. Meshkani,
On SURE estimates in hierarchical models assuming heteroscedasticity for both levels of a two-level normal hierarchical model, Journal of Multivariate Analysis, 132 (2014), 129-137.
doi: 10.1016/j.jmva.2014.08.001. |
[19] |
R. Giryes, M. Elad and Y. Eldar,
The projected GSURE for automatic parameter tuning in iterative shrinkage methods, Applied and Computational Harmonic Analysis, 30 (2011), 407-422.
doi: 10.1016/j.acha.2010.11.005. |
[20] |
J. Hadamard,
Lectures on Cauchy's Problem in Linear Partial Differential Equations, New Haven, 1953. |
[21] |
H. Haghshenas Lari and A. Gholami,
Curvelet-TV regularized Bregman iteration for seismic random noise attenuation, Journal of Applied Geophysics, 109 (2014), 233-241.
doi: 10.1016/j.jappgeo.2014.08.005. |
[22] |
P. C. Hansen,
Analysis of discrete ill-posed problems by means of the L-curve, SIAM Review, 34 (1992), 561-580.
doi: 10.1137/1034115. |
[23] |
P. C. Hansen and D. P. OLeary,
The Use of the L-Curve in the Regularization of Discrete Ill-Posed Problems, SIAM Journal on Scientific Computing, 14 (1993), 1487-1503.
doi: 10.1137/0914086. |
[24] |
B. Jin, J. Zou et al., Iterative parameter choice by discrepancy principle, IMA Journal of Numerical Analysis, 32 (2012), 1714-1732.
doi: 10.1093/imanum/drr051. |
[25] |
A. Kneip,
Ordered linear smoothers, Ann. Statist., 22 (1994), 835-866.
doi: 10.1214/aos/1176325498. |
[26] |
O. V. Lepskii,
On a Problem of Adaptive Estimation in Gaussian White Noise, Theory of Probability & Its Applications, 35 (1991), 454-466.
doi: 10.1137/1135065. |
[27] |
H. Li and F. Werner, Empirical risk minimization as parameter choice rule for general linear regularization methods, 2017, arXiv: 1703.07809. |
[28] |
K.-C. Li,
From stein's unbiased risk estimates to the method of generalized cross validation, The Annals of Statistics, 13 (1985), 1352-1377.
doi: 10.1214/aos/1176349742. |
[29] |
K.-C. Li,
Asymptotic optimality for $C_p$, $C_L$, cross-validation and generalized cross-validation: Discrete index set, Ann. Statist., 15 (1987), 958-975.
doi: 10.1214/aos/1176350486. |
[30] |
F. Luisier, T. Blu and M. Unser,
Image denoising in mixed Poisson-Gaussian noise, IEEE Transactions on Image Processing, 20 (2011), 696-708.
doi: 10.1109/TIP.2010.2073477. |
[31] |
J.-C. Pesquet, A. Benazza-Benyahia and C. Chaux,
A SURE Approach for Digital Signal/Image Deconvolution Problems, IEEE Transactions on Signal Processing, 57 (2009), 4616-4632.
doi: 10.1109/TSP.2009.2026077. |
[32] |
P. Qu, C. Wang and G. X. Shen,
Discrepancy-based adaptive regularization for grappa reconstruction, Journal of Magnetic Resonance Imaging, 24 (2006), 248-255.
doi: 10.1002/jmri.20620. |
[33] |
S. Ramani, T. Blu and M. Unser,
Monte-Carlo sure: A black-box optimization of regularization parameters for general denoising algorithms, IEEE Transactions on Image Processing, 17 (2008), 1540-1554.
doi: 10.1109/TIP.2008.2001404. |
[34] |
S. Ramani, Z. Liu, J. Rosen, J.-F. Nielsen and J. A. Fessler,
Regularization parameter selection for nonlinear iterative image restoration and MRI reconstruction using GCV and SURE-based methods, IEEE Transactions on Image Processing, 21 (2012), 3659-3672.
doi: 10.1109/TIP.2012.2195015. |
[35] |
J. A. Rice,
Choice of smoothing parameter in deconvolution problems, Contemporary Mathematics, 59 (1986), 137-151.
doi: 10.1090/conm/059/10. |
[36] |
C. M. Stein,
Estimation of the mean of a multivariate normal distribution, The Annals of Statistics, 9 (1981), 1135-1151.
doi: 10.1214/aos/1176345632. |
[37] |
A. M. Thompson, J. C. Brown, J. W. Kay and D. M. Titterington,
A study of methods of choosing the smoothing parameter in image restoration by regularization, IEEE Trans. Pattern Anal. Mach. Intell., 13 (1991), 326-339.
doi: 10.1109/34.88568. |
[38] |
G. M. Vainikko,
The discrepancy principle for a class of regularization methods, USSR Computational Mathematics and Mathematical Physics, 22 (1982), 1-19.
doi: 10.1016/0041-5553(82)90120-3. |
[39] |
S. Vaiter, C. Deledalle and G. Peyré,
The degrees of freedom of partly smooth regularizers, Annals of the Institute of Statistical Mathematics, 69 (2017), 791-832.
doi: 10.1007/s10463-016-0563-z. |
[40] |
S. Vaiter, C. Deledalle, G. Peyré, C. Dossal and J. Fadili,
Local behavior of sparse analysis regularization: Applications to risk estimation, Applied and Computational Harmonic Analysis, 35 (2013), 433-451.
doi: 10.1016/j.acha.2012.11.006. |
[41] |
S. A. vande Geer,
Applications of Empirical Process Theory, vol. 6 of Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge University Press, Cambridge, 2000. |
[42] |
D. Van De Ville and M. Kocher, SURE-Based Non-Local Means, IEEE Signal Processing Letters, 16 (2009), 973-976, URL http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=5165022. |
[43] |
D. Van DeVille and M. Kocher,
Nonlocal means with dimensionality reduction and SURE-based parameter selection, IEEE Transactions on Image Processing, 20 (2011), 2683-2690.
doi: 10.1109/TIP.2011.2121083. |
[44] |
Y.-Q. Wang and J.-M. Morel,
SURE Guided Gaussian Mixture Image Denoising, SIAM Journal on Imaging Sciences, 6 (2013), 999-1034.
doi: 10.1137/120901131. |
[45] |
D. S. Weller, S. Ramani, J.-F. Nielsen and J. A. Fessler, Monte Carlo SURE-based parameter selection for parallel magnetic resonance imaging reconstruction, Magnetic Resonance in Medicine, 71 (2014), 1760-1770. |
[46] |
X. Xie, S. C. Kou and L. D. Brown,
SURE Estimates for a Heteroscedastic Hierarchical Model, Journal of the American Statistical Association, 107 (2012), 1465-1479.
doi: 10.1080/01621459.2012.728154. |















1.27e+0 | 1.75e+0 | 2.79e+0 | 6.77e+0 | 2.31e+2 | |
1.75e+0 | 6.77e+0 | 6.94e+1 | 6.88e+2 | 2.30e+2 | |
6.77e+0 | 6.88e+2 | 6.42e+2 | 1.51e+3 | 4.22e+3 | |
6.88e+2 | 1.51e+3 | 1.51e+4 | 4.29e+3 | 4.29e+4 | |
1.70e+3 | 4.70e+4 | 1.87e+6 | 4.07e+6 | 1.79e+6 | |
4.70e+4 | 1.11e+7 | 1.22e+7 | 2.12e+7 | 3.70e+7 |
1.27e+0 | 1.75e+0 | 2.79e+0 | 6.77e+0 | 2.31e+2 | |
1.75e+0 | 6.77e+0 | 6.94e+1 | 6.88e+2 | 2.30e+2 | |
6.77e+0 | 6.88e+2 | 6.42e+2 | 1.51e+3 | 4.22e+3 | |
6.88e+2 | 1.51e+3 | 1.51e+4 | 4.29e+3 | 4.29e+4 | |
1.70e+3 | 4.70e+4 | 1.87e+6 | 4.07e+6 | 1.79e+6 | |
4.70e+4 | 1.11e+7 | 1.22e+7 | 2.12e+7 | 3.70e+7 |
min | max | mean | median | std | |
optimal | 4.78 | 9.63 | 8.04 | 8.05 | 0.43 |
DP | 6.57 | 10.81 | 8.82 | 8.87 | 0.34 |
PSURE | 6.10 | 277.24 | 8.38 | 8.23 | 1.53 |
SURE | 6.08 | 339.80 | 27.71 | 8.95 | 37.26 |
min | max | mean | median | std | |
optimal | 4.78 | 9.63 | 8.04 | 8.05 | 0.43 |
DP | 6.57 | 10.81 | 8.82 | 8.87 | 0.34 |
PSURE | 6.10 | 277.24 | 8.38 | 8.23 | 1.53 |
SURE | 6.08 | 339.80 | 27.71 | 8.95 | 37.26 |
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