NNLS version 0 -- a MATLAB software for nuclear norm regularized linear least squares problems based on an accelerated proximal gradient method

Kim-Chuan Toh and Sangwoon Yun

The software was first released on 14 Oct 2009. It was last updated in 10 Nov 2009 with some minor bugs corrected. The software is designed to solve nuclear norm regularized linear least squares problems of the form:

where $\mu>0$ is a regularization parameter, and $f(X)=\frac{1}{2}\|A(X)-b\|^2$.

Important note: this is a research software. It is not intended nor designed to be a general purpose software at the moment. The solver is expected to work well only for favorable problems such as nuclear norm regularized random matrix completion problems. Being a gradient method, it is quite sensitive to the various parameters used in the algorithm. The selection of the parameters typically depend on the class of problems being solved.

Citation:

Kim-Chuan Toh and Sangwoon Yun, An accelerated proximal gradient algorithm for nuclear norm regularized least squares problems, Pacific J. Optimization, 6 (2010), pp. 615--640.

• Copyright: This version of NNLS is distributed under the GNU General Public License 2.0. For commercial applications that may be incompatible with this license, please contact the authors to discuss alternatives.

Download:

User guide: in preparation

You can download the package here: NNLS-0.zip

Please read. Welcome to NNLS-0! The software requires a few Mex files for execution. You can generate (only need to be done once) these Mex files as follows:

• Firstly, unpack the software:
  

   
  xxxxxxxxxx
  >> unzip NNLS-0.zip
  

• Run Matlab in the directory NNLS-0

• In the Matlab command window, type:

   
  xxxxxxxxxx
  >> Installmex
  

• After that, to see whether you have installed NNLS-0 correctly, type:
  

   
  >> startup
  >> runRandomMatComp
  

• By now, NNLS is ready for you to use.

Some segmentation errors have been reported when running NNLS-0 on Windows 64-bit PCs. We have not observed such errors when running on MATLAB version 7.13.0.564 (R2011b). For example, when running the m-file runRandomMatComp, we get the following successful run:

 
>> runRandomMatComp
​
Create matrix  1 with rank = 10,
 Problem: n = 5000, p = 600614, r = 10, p/dr = 6.01e+000, p/n^2 = 2.40e+000%
 mu = 1.37e-002, noise ratio, sigma = 0.00e+000, 0.00e+000
​
  nr = 5000, nc = 5000, m = 600614
  Lipschitz const = 1.00e+000
  mu_target = 1.37e-002
  matrix storage format = factors
------------------------------------------------------------------------------------------
 it     mu       tau      svp    relDist   relRes  relObjdiff   obj       time
------------------------------------------------------------------------------------------+
  1  1.37e+002 1.00e+000 |      2| 0.00e+000 1.00e+000 1.00e+000| 3.0200e+006| 3.28e-001|+ [0.0e+000]
  2  9.62e+001 1.00e+000 |  1   1| 9.86e-001 9.99e-001 5.56e-004| 3.0183e+006| 5.31e-001|+ [0.0e+000]
  3  6.74e+001 1.00e+000 |  2   2| 6.83e-001 9.96e-001 3.71e-003| 3.0072e+006| 6.55e-001|+ [0.0e+000]
  4  4.72e+001 1.00e+000 |  3   3| 4.70e-001 9.89e-001 1.07e-002| 2.9755e+006| 7.49e-001|+ [0.0e+000]
​
 45  1.37e-002 4.40e-002 | 10  11| 2.61e-004 1.54e-004 1.97e-005| 6.8908e+002| 6.8911e+002| 1.56e+001| [1.1e+004]
 46  1.37e-002 4.40e-002 | 10  11| 2.05e-004 1.48e-004 2.89e-005| 6.8906e+002| 6.8908e+002| 1.58e+001| [1.0e+004]
 47  1.37e-002 4.40e-002 | 10  11| 1.45e-004 1.37e-004 4.40e-005| 6.8903e+002| 6.8904e+002| 1.60e+001| [8.4e+003]
 48  1.37e-002 4.40e-002 | 10  11| 9.63e-005 1.35e-004 3.17e-005| 6.8901e+002|
 [a] relDist < 1.00e-004
 Finished the main algorithm!
 Objective function        = 6.89011e+002
 Number of iterations      = 48
 Number of singular values = 10
 max singular value        = 5.23e+003
 min singular value        = 4.79e+003
 CPU time                  = 1.61e+001
 norm(X-Xold,'fro')/norm(X,'fro') = 9.63e-005
​
 Problem: nr = 5000, nc = 5000, p = 600614, r = 10, p/dr = 6.01e+000, p/n^2 = 2.40e+000%
 mu = 1.37e-002, noise ratio, sigma = 0.00e+000, 0.00e+000
-----------------------------------------------------------------------------
 problem type       :  NNLS
 randstate          :       1
 iterations         :      48
 # singular         :      10
 obj  value         :  6.89011e+002
 cpu   time         :  1.62e+001
 mean square error  :  1.85e-004
------------------------------------------------------------------------------
​