Case Study: Sparse Reconstruction Problems from SPARCO Toolbox

TEST PROBLEMS
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Case study background and problem formulations

Instructions for optimization with PSG Run-File, PSG MATLAB Toolbox, PSG MATLAB Subroutines and PSG R.

PROBLEM: L1 Relaxed
Minimize Meanabs_pen (minimizing L1-error of regression)
subject to
Polynom_abs ≤ Const2 (constraint on the sum of absolute values of the components of decision vector)
Box constraints (bounds on variables)
——————————————————————–
Meanabs_pen = Mean Absolute Penalty
Polynom_abs = Polynomial Absolute
Box constraints = constraints on individual decision variables
——————————————————————–

Problem “problem_601_Relaxed”

# of Variables # of Scenarios Objective Value Solving Time, PC 3.14GHz (sec)
Dataset1 4096 3200 6.48E+00 55.33
Environments
Run-File Problem Statement Data Solution
Matlab Toolbox Data
Matlab Subroutines Matlab Code Data
R R Code Data

Download other datasets in Run-File Environment.
Instructions for importing problems from Run-File to PSG MATLAB.

Sources of Data
  • Takhar, D., Laska, J.N., Wakin, M., Duarte, M., Baron, D., Sarvotham, S., Kelly, K.K., Baraniuk, R.G.: A new camera
    architecture based on optical-domain compression. In: Proceedings of the IS&T/SPIE Symposium on Electronic Imaging:
    Computational Imaging, vol. 6065 (2006).
Problem Datasets # of Variables # of Scenarios Objective Value Solving Time, PC 3.14GHz (sec)
Dataset2 ProblemStatement Data Solution 4096 3200 2.78E+00 1965.0
Dataset3 ProblemStatement Data Solution 4096 3200 8.42E-01 2894.0
Dataset4 ProblemStatement Data Solution 4096 3200 9.18E-08 509.2

 
Problem “problem_602_Relaxed”

# of Variables # of Scenarios Objective Value Solving Time, PC 3.14GHz (sec)
Dataset1 4096 3200 6.80E+00 1874.2
Environments
Run-File Problem Statement Data Solution
Matlab Toolbox Data
Matlab Subroutines Matlab Code Data

Download other datasets in Run-File Environment.
Instructions for importing problems from Run-File to PSG MATLAB.

Sources of Data
  • Takhar, D., Laska, J.N., Wakin, M., Duarte, M., Baron, D., Sarvotham, S., Kelly, K.K., Baraniuk, R.G.: A new camera
    architecture based on optical-domain compression. In: Proceedings of the IS&T/SPIE Symposium on Electronic Imaging:
    Computational Imaging, vol. 6065 (2006).
Problem Datasets # of Variables # of Scenarios Objective Value Solving Time, PC 2.66GHz (sec) Dataset2 ProblemStatement Data Solution 4096 3200 2.99E+00 2906.6 Dataset3 ProblemStatement Data Solution 4096 3200 3.60E-01 2982.0 Dataset4 ProblemStatement Data Solution 4096 3200 9.71E-05 272.6

 

PROBLEM: L1 Relaxed D
Minimize Meanabs_pen (minimizing L1-error of regression)
subject to
Linear ≤ Const1 (constraint on sum of components of decision vector)
Box constraints (bounds on variables)
——————————————————————–
Meanabs_pen = Mean Absolute Penalty
Box constraints = constraints on individual decision variables
——————————————————————–Problem “problem_2_Relaxed_Double”

# of Variables # of Scenarios Objective Value Solving Time, PC 3.14GHz (sec)
Dataset1 1024 1024 1.22E+00 0.5
Environments
Run-File Problem Statement Data Solution
Matlab Toolbox Data
Matlab Subroutines Matlab Code Data

Download other datasets in Run-File Environment.
Instructions for importing problems from Run-File to PSG MATLAB.

Sources of Data
Problem Datasets # of Variables # of Scenarios Objective Value Solving Time, PC 2.66GHz (sec)
Dataset2 ProblemStatement Data Solution 1024 1024 6.63E-01 0.6
Dataset3 ProblemStatement Data Solution 1024 1024 4.42E-02 4.5
Dataset4 ProblemStatement Data Solution 1024 1024 1.24E-014 2.8

 

Problem “problem_3_Relaxed_Double”

# of Variables # of Scenarios Objective Value Solving Time, PC 3.14GHz (sec)
Dataset1 2048 1024 6.38E-01 0.5
Environments
Run-File Problem Statement Data Solution
Matlab Toolbox Data
Matlab Subroutines Matlab Code Data

Download other datasets in Run-File Environment.
Instructions for importing problems from Run-File to PSG MATLAB.

Sources of Data
  • Berg, E.van den, Friedlander, M.P.: SPARCO: A toolbox for testing sparse reconstruction algorithms (2008).
    URL http://www.cs.ubc.ca/labs/scl/sparco/
  • Berg, E.van den, Friedlander, M.P., Hennenfent, G., Herrmann, F., Saab, R., Yilmaz, O.:
    Sparco: A testing framework for sparse reconstruction.
    Tech. Rep. TR-2007-20, Dept. Computer Science, University of British Columbia, Vancouver (2007)
Problem Datasets # of Variables # of Scenarios Objective Value Solving Time, PC 2.66GHz (sec) Dataset2 ProblemStatement Data Solution 2048 1024 8.03E-02 4.1 Dataset3 ProblemStatement Data Solution 2048 1024 2.27E-03 130.3 Dataset4 ProblemStatement Data Solution 2048 1024 4.58E-14 460.3

Problem “problem_5_Relaxed_Double”

# of Variables # of Scenarios Objective Value Solving Time, PC 3.14GHz (sec)
Dataset1 2048 300 1.00E+00 0.5
Environments
Run-File Problem Statement Data Solution
Matlab Toolbox Data
Matlab Subroutines Matlab Code Data

Download other datasets in Run-File Environment.
Instructions for importing problems from Run-File to PSG MATLAB.

Sources of Data
  • Berg, E.van den, Friedlander, M.P.: SPARCO: A toolbox for testing sparse reconstruction algorithms (2008).
    URL http://www.cs.ubc.ca/labs/scl/sparco/
  • Berg, E.van den, Friedlander, M.P., Hennenfent, G., Herrmann, F., Saab, R., Yilmaz, O.:
    Sparco: A testing framework for sparse reconstruction.
    Tech. Rep. TR-2007-20, Dept. Computer Science, University of British Columbia, Vancouver (2007)
Problem Datasets # of Variables # of Scenarios Objective Value Solving Time, PC 2.66GHz (sec) Dataset2 ProblemStatement Data Solution 2048 300 2.71E-01 8.0 Dataset3 ProblemStatement Data Solution 2048 300 5.79E-02 1.8 Dataset4 ProblemStatement Data Solution 2048 300 2.43E-14 1.8

Problem “problem_6_Relaxed_Double”

# of Variables # of Scenarios Objective Value Solving Time, PC 3.14GHz (sec)
Dataset1 2048 600 1.41E+02 1.4
Environments
Run-File Problem Statement Data Solution
Matlab Toolbox Data
Matlab Subroutines Matlab Code Data

Download other datasets in Run-File Environment.
Instructions for importing problems from Run-File to PSG MATLAB.

Sources of Data
  • Candes, E.J., Romberg, J.: Practical signal recovery from random projections. In: Wavelet Applications
    in Signal and Image Processing XI, Proc. SPIE Conf. 5914. (2004)
Problem Datasets # of Variables # of Scenarios Objective Value Solving Time, PC 2.66GHz (sec) Dataset2 ProblemStatement Data Solution 2048 600 6.62E+01 56.5 Dataset3 ProblemStatement Data Solution 2048 600 1.59E+00 185.9 Dataset4 ProblemStatement Data Solution 2048 600 3.77E-12 30.9

Problem “problem_7_Relaxed_Double”

# of Variables # of Scenarios Objective Value Solving Time, PC 3.14GHz (sec)
Dataset1 2560 600 5.81E-02 2.3
Environments
Run-File Problem Statement Data Solution
Matlab Toolbox Data
Matlab Subroutines Matlab Code Data

Download other datasets in Run-File Environment.
Instructions for importing problems from Run-File to PSG MATLAB.

Sources of Data Problem Datasets # of Variables # of Scenarios Objective Value Solving Time, PC 2.66GHz (sec) Dataset2 ProblemStatement Data Solution 2560 600 3.39E-02 7.9 Dataset3 ProblemStatement Data Solution 2560 600 1.02E-02 12.8 Dataset4 ProblemStatemen Data Solution 2560 600 1.37E-13 1.6

Problem “problem_8_Relaxed_Double”

# of Variables # of Scenarios Objective Value Solving Time, PC 3.14GHz (sec)
Dataset1 2560 600 4.94E-02 6.0
Environments
Run-File Problem Statement Data Solution
Matlab Toolbox Data
Matlab Subroutines Matlab Code Data

Download other datasets in Run-File Environment.
Instructions for importing problems from Run-File to PSG MATLAB.

Sources of Data Problem Datasets # of Variables # of Scenarios Objective Value Solving Time, PC 2.66GHz (sec) Dataset2 ProblemStatement Data Solution 2560 600 1.64E-02 11.9 Dataset3 ProblemStatement Data Solution 2560 600 3.29E-03 11.6 Dataset4 ProblemStatement Data Solution 2560 600 1.36E-13 1.2

Problem “problem_9_Relaxed_Double”

# of Variables # of Scenarios Objective Value Solving Time, PC 3.14GHz (sec)
Dataset1 128 128 3.14E-01 0.01
Environments
Run-File Problem Statement Data Solution
Matlab Toolbox Data
Matlab Subroutines Matlab Code Data

Download other datasets in Run-File Environment.
Instructions for importing problems from Run-File to PSG MATLAB.

Sources of Data Problem Datasets # of Variables # of Scenarios Objective Value Solving Time, PC 2.66GHz (sec) Dataset2 ProblemStatement Data Solution 128 128 9.30E-02 0.02 Dataset3 ProblemStatement Data Solution 128 128 7.81E-03 0.02 Dataset4 ProblemStatement Data Solution 128 128 2.92E-13 0.04

Problem “problem_10_Relaxed_Double”

# of Variables # of Scenarios Objective Value Solving Time, PC 3.14GHz (sec)
Dataset1 1024 1024 1.32E-01 0.5
Environments
Run-File Problem Statement Data Solution
Matlab Toolbox Data
Matlab Subroutines Matlab Code Data

Download other datasets in Run-File Environment.
Instructions for importing problems from Run-File to PSG MATLAB.

Sources of Data
Problem Datasets # of Variables # of Scenarios Objective Value Solving Time, PC 2.66GHz (sec) Dataset2 ProblemStatement Data Solution 1024 1024 6.10E-02 0.6 Dataset3 ProblemStatement Data Solution 1024 1024 1.15E-02 0.7 Dataset4 ProblemStatement Data Solution 1024 1024 1.64E-12 143.1

Problem “problem_11_Relaxed_Double”

# of Variables # of Scenarios Objective Value Solving Time, PC 3.14GHz (sec)
Dataset1 1024 256 1.22E+00 3.1
Environments
Run-File Problem Statement Data Solution
Matlab Toolbox Data
Matlab Subroutines Matlab Code Data

Download other datasets in Run-File Environment.
Instructions for importing problems from Run-File to PSG MATLAB.

Sources of Data
  • Berg, E.van den, Friedlander, M.P.: SPARCO: A toolbox for testing sparse reconstruction algorithms (2008).
    URL http://www.cs.ubc.ca/labs/scl/sparco/
  • E.van den, Friedlander, M.P., Hennenfent, G., Herrmann, F., Saab, R., Yilmaz, O.: Sparco: A testing framework for
    sparse reconstruction. Tech. Rep. TR-2007-20, Dept. Computer Science, University of British Columbia, Vancouver (2007)
Problem Datasets # of Variables # of Scenarios Objective Value Solving Time, PC 2.66GHz (sec) Dataset2 ProblemStatement Data Solution 1024 256 5.18E-01 6.5 Dataset3 ProblemStatement Data Solution 1024 256 1.29E-01 35.5 Dataset4 ProblemStatement Data Solution 1024 256 2.53E-14 31.6

Problem “problem_603_Relaxed_Double”

# of Variables # of Scenarios Objective Value Solving Time, PC 3.14GHz (sec)
Dataset1 4096 1024 3.35E-01 2.2
Environments
Run-File Problem Statement Data Solution
Matlab Toolbox Data
Matlab Subroutines Matlab Code Data

Download other datasets in Run-File Environment.
Instructions for importing problems from Run-File to PSG MATLAB.

Sources of Data
  • Figueiredo, M., Nowak, R., Wright, S.: Gradient projection for sparse reconstruction: Application to compressed sensing
    and other inverse problems. Selected Topics in Signal Processing, IEEE Journal of 1(4), 586-597 (2007).
    DOI 10.1109/JSTSP.2007.910281. URL http://www.lx.it.pt/~mtf/GPSR
Problem Datasets # of Variables # of Scenarios Objective Value Solving Time, PC 2.66GHz (sec) Dataset2 ProblemStatement Data Solution 4096 1024 1.75E-01 10.7 Dataset3 ProblemStatement Data Solution 4096 1024 4.16E-02 1150.4 Dataset4 ProblemStatement Data Solution 4096 1024 2.32E-14 391.0

Problem “problem_902_Relaxed_Double”

# of Variables # of Scenarios Objective Value Solving Time, PC 3.14GHz (sec)
Dataset1 1000 200 2.32E-02 0.07
Environments
Run-File Problem Statement Data Solution
Matlab Toolbox Data
Matlab Subroutines Matlab Code Data

Download other datasets in Run-File Environment.
Instructions for importing problems from Run-File to PSG MATLAB.

Sources of Data Problem Datasets # of Variables # of Scenarios Objective Value Solving Time, PC 2.66GHz (sec) Dataset2 ProblemStatement Data Solution 1000 200 1.86E-02 0.12 Dataset3 ProblemStatement Data Solution 1000 200 3.60E-03 0.12 Dataset4 ProblemStatement Data Solution 1000 200 3.12E-14 26.3

Problem “problem_903_Relaxed_Double”

# of Variables # of Scenarios Objective Value Solving Time, PC 3.14GHz (sec)
Dataset1 1024 1024 6.17E-01 0.6
Environments
Run-File Problem Statement Data Solution
Matlab Toolbox Data
Matlab Subroutines Matlab Code Data

Download other datasets in Run-File Environment.
Instructions for importing problems from Run-File to PSG MATLAB.

Sources of Data
  • Dossal, C., Mallat, S.: Sparse spike deconvolution with minimum scale. In: Proceedings of Signal Processing with
    Adaptive Sparse Structured Representations, pp. 123-126. Rennes, France (2005).
    URL http://spars05.irisa.fr/ACTES/PS2-11.pdf
Problem Datasets # of Variables # of Scenarios Objective Value Solving Time, PC 2.66GHz (sec) Dataset2 ProblemStatement Data Solution 1024 1024 4.06E-01 1.4 Dataset3 ProblemStatement Data Solution 1024 1024 9.92E-02 29.7 Dataset4 ProblemStatement Data Solution 1024 1024 9.18E-05 10.7

 

PROBLEM: L2 D
Minimize Meansquare + Linear (minimizing L2-error of regression)
subject to
Linear ≤ Const3 (constraint on sum of components of decision vector)
Box constraints (bounds on variables)
——————————————————————–
Meansquare = Mean Square Penalty
Meanabs_pen = Mean Absolute Penalty
Box constraints = constraints on individual decision variables
——————————————————————–Problem “problem_2_L2_dbl”

# of Variables # of Scenarios Objective Value Solving Time, PC 3.14GHz (sec)
Dataset1 1024 1024 2.98E+03 0.32
Environments
Run-File Problem Statement Data Solution
Matlab Toolbox Data
Matlab Subroutines Matlab Code Data

Download other datasets in Run-File Environment.
Instructions for importing problems from Run-File to PSG MATLAB.

Sources of Data Problem Datasets # of Variables # of Scenarios Objective Value Solving Time, PC 2.66GHz (sec) Dataset2 ProblemStatement Data Solution 1024 1024 2.31E+03 0.32 Dataset3 ProblemStatement Data Solution 1024 1024 4.15E+02 0.32 Dataset4 ProblemStatement Data Solution 1024 1024 4.47E+01 0.32 Dataset5 ProblemStatement Data Solution 1024 1024 4.50E+00 0.31

Problem “problem_3_L2_dbl”

# of Variables # of Scenarios Objective Value Solving Time, PC 3.14GHz (sec)
Dataset1 2048 1024 2.37E+03 1.4
Environments
Run-File Problem Statement Data Solution
Matlab Toolbox Data
Matlab Subroutines Matlab Code Data

Download other datasets in Run-File Environment.
Instructions for importing problems from Run-File to PSG MATLAB.

Sources of Data
  • Berg, E.van den, Friedlander, M.P.: SPARCO: A toolbox for testing sparse reconstruction algorithms (2008).
    URL http://www.cs.ubc.ca/labs/scl/sparco/
  • Berg, E.van den, Friedlander, M.P., Hennenfent, G., Herrmann, F., Saab, R., Yilmaz, O.:
    Sparco: A testing framework for sparse reconstruction. Tech. Rep. TR-2007-20,
    Dept. Computer Science, University of British Columbia, Vancouver (2007)
Problem Datasets # of Variables # of Scenarios Objective Value Solving Time, PC 2.66GHz (sec) Dataset2 ProblemStatement Data Solution 2048 1024 1.31E+03 1.4 Dataset3 ProblemStatement Data Solution 2048 1024 1.78E+02 2.7 Dataset4 ProblemStatement Data Solution 2048 1024 2.16E+01 10.8 Dataset5 ProblemStatement Data Solution 2048 1024 2.22E+00 102.8

Problem “problem_5_L2_dbl”

# of Variables # of Scenarios Objective Value Solving Time, PC 3.14GHz (sec)
Dataset1 2048 300 2.10E+03 1.6
Environments
Run-File Problem Statement Data Solution
Matlab Toolbox Data
Matlab Subroutines Matlab Code Data

Download other datasets in Run-File Environment.
Instructions for importing problems from Run-File to PSG MATLAB.

Sources of Data
  • Berg, E.van den, Friedlander, M.P.: SPARCO: A toolbox for testing sparse reconstruction algorithms (2008).
    URL http://www.cs.ubc.ca/labs/scl/sparco/
  • Berg, E.van den, Friedlander, M.P., Hennenfent, G., Herrmann, F., Saab, R., Yilmaz, O.:
    Sparco: A testing framework for sparse reconstruction. Tech. Rep. TR-2007-20,
    Dept. Computer Science, University of British Columbia, Vancouver (2007)
Problem Datasets # of Variables # of Scenarios Objective Value Solving Time, PC 2.66GHz (sec) Dataset2 ProblemStatement Data Solution 2048 300 1.23E+03 1.9 Dataset3 ProblemStatement Data Solution 2048 300 1.58E+02 8.8 Dataset4 ProblemStatement Data Solution 2048 300 1.78E+01 85.7 Dataset5 ProblemStatement Data Solution 2048 300 1.82E+00 837.4

Problem “problem_6_L2_dbl”

# of Variables # of Scenarios Objective Value Solving Time, PC 3.14GHz (sec)
Dataset1 2048 600 1.29E+07 1.8
Environments
Run-File Problem Statement Data Solution
Matlab Toolbox Data
Matlab Subroutines Matlab Code Data

Download other datasets in Run-File Environment.
Instructions for importing problems from Run-File to PSG MATLAB.

Sources of Data
  • Candes, E.J., Romberg, J.: Practical signal recovery from random projections. In: Wavelet Applications in Signal and
    Image Processing XI, Proc. SPIE Conf. 5914. (2004)
Problem Datasets # of Variables # of Scenarios Objective Value Solving Time, PC 2.66GHz (sec) Dataset2 ProblemStatement Data Solution 2048 600 5.29E+06 2.4 Dataset3 ProblemStatement Data Solution 2048 600 1.46E+06 9.5 Dataset4 ProblemStatement Data Solution 2048 600 1.70E+05 87.4 Dataset5 ProblemStatement Data Solution 2048 600 1.76E+04 636.5 Dataset6 ProblemStatement Data Solution 2048 600 4.20E+03 112.4 Dataset7 ProblemStatement Data Solution 2048 600 4.60E+02 6.4

Problem “problem_7_L2_dbl”

# of Variables # of Scenarios Objective Value Solving Time, PC 3.14GHz (sec)
Dataset1 2560 600 2.25E+00 2.03
Environments
Run-File Problem Statement Data Solution
Matlab Toolbox Data
Matlab Subroutines Matlab Code Data

Download other datasets in Run-File Environment.
Instructions for importing problems from Run-File to PSG MATLAB.

Sources of Data Problem Datasets # of Variables # of Scenarios Objective Value Solving Time, PC 2.66GHz (sec) Dataset2 ProblemStatement Data Solution 2560 600 1.56E+00 3.00 Dataset3 ProblemStatement Data Solution 2560 600 8.90E-01 2.52 Dataset4 ProblemStatement Data Solution 2560 600 1.96E-01 2.88

Problem “problem_8_L2_dbl”

# of Variables # of Scenarios Objective Value Solving Time, PC 3.14GHz (sec)
Dataset1 2560 600 2.11E+00 2.76
Environments
Run-File Problem Statement Data Solution
Matlab Toolbox Data
Matlab Subroutines Matlab Code Data

Download other datasets in Run-File Environment.
Instructions for importing problems from Run-File to PSG MATLAB.

Sources of Data Problem Datasets # of Variables # of Scenarios Objective Value Solving Time, PC 2.66GHz (sec) Dataset2 ProblemStatement Data Solution 2560 600 1.52E+00 3.00 Dataset3 ProblemStatement Data Solution 2560 600 8.81E-01 3.24 Dataset4 ProblemStatement Data Solution 2560 600 1.95E-01 5.91

Problem “problem_9_L2_dbl”

# of Variables # of Scenarios Objective Value Solving Time, PC 3.14GHz (sec)
Dataset1 128 128 1.68E+02 0.48
Environments
Run-File Problem Statement Data Solution
Matlab Toolbox Data
Matlab Subroutines Matlab Code Data

Download other datasets in Run-File Environment.
Instructions for importing problems from Run-File to PSG MATLAB.

Sources of Data Problem Datasets # of Variables # of Scenarios Objective Value Solving Time, PC 2.66GHz (sec) Dataset2 ProblemStatement Data Solution 128 128 1.16E+02 0.98 Dataset3 ProblemStatement Data Solution 128 128 3.65E+01 1.47 Dataset4 ProblemStatement Data Solution 128 128 5.54E+00 2.41 Dataset5 ProblemStatement Data Solution 128 128 3.98E+00 2.40

Problem “problem_10_L2_dbl”

# of Variables # of Scenarios Objective Value Solving Time, PC 3.14GHz (sec)
Dataset1 1024 1024 2.04E+03 150.9
Environments
Run-File Problem Statement Data Solution
Matlab Toolbox Data
Matlab Subroutines Matlab Code Data

Download other datasets in Run-File Environment.
Instructions for importing problems from Run-File to PSG MATLAB.

Sources of Data Problem Datasets # of Variables # of Scenarios Objective Value Solving Time, PC 2.66GHz (sec) Dataset2 ProblemStatement Data Solution 1024 1024 6.66E+02 281.3 Dataset3 ProblemStatement Data Solution 1024 1024 4.15E+02 441.9 Dataset4 ProblemStatement Data Solution 1024 1024 1.01E+02 651.7 Dataset5 ProblemStatement Data Solution 1024 1024 2.07E+01 1848.6

Problem “problem_11_L2_dbl”

# of Variables # of Scenarios Objective Value Solving Time, PC 3.14GHz (sec)
Dataset1 1024 256 1.80E+03 0.5
Environments
Run-File Problem Statement Data Solution
Matlab Toolbox Data
Matlab Subroutines Matlab Code Data

Download other datasets in Run-File Environment.
Instructions for importing problems from Run-File to PSG MATLAB.

Sources of Data
  • Berg,E.van den, Friedlander, M.P.: SPARCO: A toolbox for testing sparse reconstruction algorithms (2008).
    URL http://www.cs.ubc.ca/labs/scl/sparco/
  • Berg, E.van den, Friedlander, M.P., Hennenfent, G., Herrmann, F., Saab, R., Yilmaz, O.:
    Sparco: A testing framework for sparse reconstruction. Tech. Rep. TR-2007-20,
    Dept. Computer Science, University of British Columbia, Vancouver (2007)
Problem Datasets # of Variables # of Scenarios Objective Value Solving Time, PC 2.66GHz (sec) Dataset2 ProblemStatement Data Solution 1024 256 2.33E+02 1.9 Dataset3 ProblemStatement Data Solution 1024 256 2.40E+01 20.0 Dataset4 ProblemStatement Data Solution 1024 256 2.40E+00 179.4 Dataset5 ProblemStatement Data Solution 1024 256 7.19E-01 1.1

Problem “problem_601_L2_dbl”

# of Variables # of Scenarios Objective Value Solving Time, PC 3.14GHz (sec)
Dataset1 4096 3200 8.78E+05 204.8
Environments
Run-File Problem Statement Data Solution
Matlab Toolbox Data
Matlab Subroutines Matlab Code Data

Download other datasets in Run-File Environment.
Instructions for importing problems from Run-File to PSG MATLAB.

Sources of Data
  • Takhar, D., Laska, J.N., Wakin, M., Duarte, M., Baron, D., Sarvotham, S., Kelly, K.K., Baraniuk, R.G.: A new camera
    architecture based on optical-domain compression. In: Proceedings of the IS&T/SPIE Symposium on Electronic Imaging:
    Computational Imaging, vol. 6065 (2006).
Problem Datasets # of Variables # of Scenarios Objective Value Solving Time, PC 2.66GHz (sec) Dataset2 ProblemStatement Data Solution 4096 3200 2.46E+05 800.8 Dataset3 ProblemStatement Data Solution 4096 3200 1.91E+05 1215.5 Dataset4 ProblemStatement Data Solution 4096 3200 5.61E+05 2179.5 Dataset5 ProblemStatement Data Solution 4096 3200 1.05E+06 473.6

Problem “problem_602_L2_dbl”

# of Variables # of Scenarios Objective Value Solving Time, PC 3.14GHz (sec)
Dataset1 4096 3200 3.62E+05 687.5
Environments
Run-File Problem Statement Data Solution
Matlab Toolbox Data
Matlab Subroutines Matlab Code Data

Download other datasets in Run-File Environment.
Instructions for importing problems from Run-File to PSG MATLAB.

Sources of Data
  • Takhar, D., Laska, J.N., Wakin, M., Duarte, M., Baron, D., Sarvotham, S., Kelly, K.K., Baraniuk, R.G.: A new camera
    architecture based on optical-domain compression. In: Proceedings of the IS&T/SPIE Symposium on Electronic Imaging:
    Computational Imaging, vol. 6065 (2006).
Problem Datasets # of Variables # of Scenarios Objective Value Solving Time, PC 2.66GHz (sec) Dataset2 ProblemStatement Data Solution 4096 3200 2.66E+05 679.8 Dataset3 ProblemStatement Data Solution 4096 3200 2.94E+05 255.5

Problem “problem_603_L2_dbl”

# of Variables # of Scenarios Objective Value Solving Time, PC 3.14GHz (sec)
Dataset1 4096 1024 1.21E+02 6.6
Environments
Run-File Problem Statement Data Solution
Matlab Toolbox Data
Matlab Subroutines Matlab Code Data

Download other datasets in Run-File Environment.
Instructions for importing problems from Run-File to PSG MATLAB.

Sources of Data
  • Figueiredo, M., Nowak, R., Wright, S.: Gradient projection for sparse reconstruction: Application to compressed sensing
    and other inverse problems. Selected Topics in Signal Processing, IEEE Journal of 1(4), 586-597 (2007).
    DOI 10.1109/JSTSP.2007.910281. URL http://www.lx.it.pt/~mtf/GPSR
Problem Datasets # of Variables # of Scenarios Objective Value Solving Time, PC 2.66GHz (sec) Dataset2 ProblemStatement Data Solution 4096 1024 2.14E+01 18.6 Dataset3 ProblemStatement Data Solution 4096 1024 2.54E+00 168.3

Problem “problem_902_L2_dbl”

# of Variables # of Scenarios Objective Value Solving Time, PC 3.14GHz (sec)
Dataset1 1000 200 1.01E-01 0.3
Environments
Run-File Problem Statement Data Solution
Matlab Toolbox Data
Matlab Subroutines Matlab Code Data

Download other datasets in Run-File Environment.
Instructions for importing problems from Run-File to PSG MATLAB.

Sources of Data Problem Datasets # of Variables # of Scenarios Objective Value Solving Time, PC 2.66GHz (sec) Dataset2 ProblemStatement Data Solution 1000 200 1.67E-02 0.5 Dataset3 ProblemStatement Data Solution 1000 200 1.73E-03 2.0

Problem “problem_903_L2_dbl”

# of Variables # of Scenarios Objective Value Solving Time, PC 3.14GHz (sec)
Dataset1 1024 1024 8.70E+02 68.1
Environments
Run-File Problem Statement Data Solution
Matlab Toolbox Data
Matlab Subroutines Matlab Code Data

Download other datasets in Run-File Environment.
Instructions for importing problems from Run-File to PSG MATLAB.

Sources of Data
  • Dossal, C., Mallat, S.: Sparse spike deconvolution with minimum scale. In: Proceedings of Signal Processing with
    Adaptive Sparse Structured Representations, pp. 123-126. Rennes, France (2005).
    URL http://spars05.irisa.fr/ACTES/PS2-11.pdf
Problem Datasets # of Variables # of Scenarios Objective Value Solving Time, PC 2.66GHz (sec) Dataset2 ProblemStatement Data Solution 1024 1024 1.18E+02 167.2 Dataset3 ProblemStatement Data Solution 1024 1024 1.29E+01 170.9 Dataset4 ProblemStatement Data Solution 1024 1024 1.41E+00 263.0 Dataset5 ProblemStatement Data Solution 1024 1024 4.58E-01 530.4

 

CASE STUDY SUMMARY
SPARCO is a suite of problems for testing and benchmarking algorithms for sparse signal reconstruction, Berg et al. (2007, 2008). It is also an environment for creating new test problems. Also a suite of standard linear operators is provided from which new problems can be assembled. SPARCO is implemented entirely in MATLABand is self contained.
This case study presents problem formulations and its solutions for a set of sparse reconstruction problems taken from SPARCO toolbox.
Problems included in the SPARCO toolbox were initially considered by different authors in different application areas: imaging, compressed sensing, geophysics, information compressing, etc. Relevant references can be found in the SPARCO toolbox.
The objective of Sparse Reconstruction is to find a decision vector which has a small number of non-zero components and satisfies exactly or almost exactly a system of linear equations. There are many variants of optimization formulations of such problems.
This case study is described in paper Boyko et al. (2011).
We solved many problems included in SPARCO toolbox problems in so called “L1Relaxed D” formulation. “L1Relaxed D” minimizes L1-error of regression with one linear inequality on the sum of decision vector components; the decision vector components are nonnegative (number of decision variables is doubled to achieve non-negativity). The non-negativity of variables is quite important because an optimal vector contains many zero variables. To investigate property of solution we solved various problems with different values of upper bound in the linear inequality and calculated cardinality and max functions in optimal points.
Some problems were solved in so called “L1Relaxed” formulation with original set of variables (without doubling the number of variables to achieve non-negativity). Variables are bounded by box constraints in this formulation. For these problems “L1Relaxed” formulation is more effective compared to “L1Relaxed D” formulation.
Additionally many problems were solved in so called “L2 D” or LASSO formulation which also has double set of variables but does not have constraints.
Sum of decision variables multiplied by some coefficient is used as regularization term in the objective function.
This problem can be easy solved by methods for unconstrained optimization.
We used SPARCO toolbox software to extract data for the considered problems. SPARCO toolbox provides a set of operators to deal with data.
We converted the problems data to PSG format and solved them in PSG Run-File envoronment.
In the problem formulations we have included several “dummy” functions multiplied by 0 (zero). Such “dummy” functions do not not impact solution process, but values of these functions are printed in the final solution file.

 

For instance, you can view many “dummy” functions in the following problem formulation:
problem: problem_602_Relaxed_700, type = minimize
objective: objective_new, linearize = 1
meanabs_pen_obj(matrix_ab602)
constraint: constraint_card, upper_bound = 700, linearize = 1
polynom_abs_S(matrix_card4096)
0 * cardn_1(1.,matrix_card4096)
0 * cardn_2(0.1,matrix_card4096)
0 * cardn_3(0.01,matrix_card4096)
0 * cardn_4(0.001,matrix_card4096)
0 * cardn_5(0.0001,matrix_card4096)
0 * cardn_6(0.00001,matrix_card4096)
0 * max_comp_pos_7(matrix_card4096)
0 * max_comp_neg_8(matrix_card4096)
box_of_variables: lowerbounds = -40, upperbounds = +40
Solver: van, precision = 4, stages = 6, timelimit = 3600
The corresponding solution file includes values of “dummy” functions on the final optimal point:
Problem: solution_status = optimal
Timing: Data_loading_time = 11.97, Preprocessing_time = 0.69, Solving_time = 272.61
Variables: optimal_point = point_problem_602_Relaxed_700
Objective: objective_new = 9.71029929057e-005
Constraint: constraint_card = 6.968284830013e+002 [-3.171516998665e+000]
Function: meanabs_pen_obj(matrix_ab602) = 9.710299290575e-005
Function: polynom_abs_s(matrix_card4096) = 6.968284830013e+002
Function: cardn_1(0.100000E+01,matrix_card4096) = 1.420000000000e+002
Function: cardn_2(0.100000E+00,matrix_card4096) = 7.890000000000e+002
Function: cardn_3(0.100000E-01,matrix_card4096) = 3.043000000000e+003
Function: cardn_4(0.100000E-02,matrix_card4096) = 3.964000000000e+003
Function: cardn_5(0.100000E-03,matrix_card4096) = 4.079000000000e+003
Function: cardn_6(0.100000E-04,matrix_card4096) = 4.095000000000e+003
Function: max_comp_pos_7(matrix_card4096) = 1.616083034336e+001
Function: max_comp_neg_8(matrix_card4096) = 6.562008879305e+000
References
• Berg, E.V., Friedlander, M.P., Hennenfent, G., Herrmann, F., Saab, R., and O., Yilmaz (2007): SPARCO: A testing framework for sparse reconstruction. Tech. Rep. TR-2007-20, Dept. Computer Science, University of British Columbia, Vancouver.
• Berg, E.V., and M.P., Friedlander (2008): SPARCO: A toolbox for testing sparse reconstruction algorithms. URL http://www.cs.ubc.ca/labs/scl/sparco/
• Boyko, N., Karamemis, G., Kuzmenko, V. and S. Uryasev (2011): Sparse Signal Reconstruction: a Cardinality Approach. Submitted for publication (download http://www.ise.ufl.edu/uryasev/Sparse_Signal_Reconstruction_Cardinality_Approach.pdf).