Weighted optimization for CP tensor decomposition with incomplete data
We explain how to use cp_wopt with the POBLANO toolbox. The method is described in the following article:
- E. Acar, D. M. Dunlavy, T. G. Kolda and M. Mørup, Scalable Tensor Factorizations for Incomplete Data, Chemometrics and Intelligent Laboratory Systems, 106(1):41-56, 2011, http://dx.doi.org/10.1016/j.chemolab.2010.08.004.
Contents
- Important Information
- Create an example problem with missing data.
- Create initial guess using 'nvecs'
- Set up the optimization parameters
- Call the cp_wopt method
- Check the output
- Evaluate the output
- Create a SPARSE example problem with missing data.
- Create initial guess using 'nvecs'
- Set up the optimization parameters
- Call the cp_wopt method
- Check the output
- Evaluate the output
Important Information
It is critical to zero out the values in the missing entries of the data tensor. This can be done by calling cp_wopt(X.*P,P,...). This is a frequent source of errors in using this method.
Create an example problem with missing data.
Here we have 25% missing data and 10% noise.
R = 2; info = create_problem('Size', [15 10 5], 'Num_Factors', R, ... 'M', 0.25, 'Noise', 0.10); X = info.Data; P = info.Pattern; M_true= info.Soln;
Create initial guess using 'nvecs'
M_init = create_guess('Data', X, 'Num_Factors', R, ... 'Factor_Generator', 'nvecs');
Set up the optimization parameters
It's genearlly a good idea to consider the parameters of the optimization method. The default options may be either too stringent or not stringent enough. The most important options to consider are detailed here.
% Get the defaults ncg_opts = ncg('defaults'); % Tighten the stop tolerance (norm of gradient). This is often too large. ncg_opts.StopTol = 1.0e-6; % Tighten relative change in function value tolearnce. This is often too large. ncg_opts.RelFuncTol = 1.0e-20; % Increase the number of iterations. ncg_opts.MaxIters = 10^4; % Only display every 10th iteration ncg_opts.DisplayIters = 10; % Display the final set of options ncg_opts
ncg_opts =
struct with fields:
Display: 'iter'
DisplayIters: 10
LineSearch_ftol: 1.0000e-04
LineSearch_gtol: 0.0100
LineSearch_initialstep: 1
LineSearch_maxfev: 20
LineSearch_method: 'more-thuente'
LineSearch_stpmax: 1.0000e+15
LineSearch_stpmin: 1.0000e-15
LineSearch_xtol: 1.0000e-15
MaxFuncEvals: 10000
MaxIters: 10000
RelFuncTol: 1.0000e-20
RestartIters: 20
RestartNW: 0
RestartNWTol: 0.1000
StopTol: 1.0000e-06
TraceFunc: 0
TraceFuncEvals: 0
TraceGrad: 0
TraceGradNorm: 0
TraceRelFunc: 0
TraceX: 0
Update: 'PR'
Call the cp_wopt method
Here is an example call to the cp_opt method. By default, each iteration prints the least squares fit function value (being minimized) and the norm of the gradient. The meaning of any line search warnings can be checked via doc cvsrch.
[M,~,output] = cp_wopt(X, P, R, 'init', M_init, ... 'opt', 'ncg', 'opt_options', ncg_opts);
Running CP-WOPT...
Time for zeroing out masked entries of data tensor is 4.20e-04 seconds.
(If zeroing is done in preprocessing, set 'skip_zeroing' to true.)
Iter FuncEvals F(X) ||G(X)||/N
------ --------- ---------------- ----------------
0 1 25.10574260 0.17813973
10 35 0.66553174 0.02681209
20 68 0.28614990 0.00668033
30 91 0.27446548 0.00071779
40 111 0.27426663 0.00011737
50 131 0.27425951 0.00003687
60 151 0.27425900 0.00000478
70 171 0.27425899 0.00000078
Check the output
It's important to check the output of the optimization method. In particular, it's worthwhile to check the exit flag. A zero (0) indicates successful termination with the gradient smaller than the specified StopTol, and a three (3) indicates a successful termination where the change in function value is less than RelFuncTol. The meaning of any other flags can be checked via doc poblano_params.
exitflag = output.ExitFlag
exitflag =
0
Evaluate the output
We can "score" the similarity of the model computed by CP and compare that with the truth. The score function on ktensor's gives a score in [0,1] with 1 indicating a perfect match. Because we have noise, we do not expect the fit to be perfect. See doc score for more details.
scr = score(M,M_true)
scr =
0.9427
Create a SPARSE example problem with missing data.
Here we have 95% missing data and 10% noise.
R = 2; info = create_problem('Size', [150 100 50], 'Num_Factors', R, ... 'M', 0.95, 'Sparse_M', true, 'Noise', 0.10); X = info.Data; P = info.Pattern; M_true= info.Soln;
Create initial guess using 'nvecs'
M_init = create_guess('Data', X, 'Num_Factors', R, ... 'Factor_Generator', 'nvecs');
Set up the optimization parameters
It's genearlly a good idea to consider the parameters of the optimization method. The default options may be either too stringent or not stringent enough. The most important options to consider are detailed here.
% Get the defaults ncg_opts = ncg('defaults'); % Tighten the stop tolerance (norm of gradient). This is often too large. ncg_opts.StopTol = 1.0e-6; % Tighten relative change in function value tolearnce. This is often too large. ncg_opts.RelFuncTol = 1.0e-20; % Increase the number of iterations. ncg_opts.MaxIters = 10^4; % Only display every 10th iteration ncg_opts.DisplayIters = 10; % Display the final set of options ncg_opts
ncg_opts =
struct with fields:
Display: 'iter'
DisplayIters: 10
LineSearch_ftol: 1.0000e-04
LineSearch_gtol: 0.0100
LineSearch_initialstep: 1
LineSearch_maxfev: 20
LineSearch_method: 'more-thuente'
LineSearch_stpmax: 1.0000e+15
LineSearch_stpmin: 1.0000e-15
LineSearch_xtol: 1.0000e-15
MaxFuncEvals: 10000
MaxIters: 10000
RelFuncTol: 1.0000e-20
RestartIters: 20
RestartNW: 0
RestartNWTol: 0.1000
StopTol: 1.0000e-06
TraceFunc: 0
TraceFuncEvals: 0
TraceGrad: 0
TraceGradNorm: 0
TraceRelFunc: 0
TraceX: 0
Update: 'PR'
Call the cp_wopt method
Here is an example call to the cp_opt method. By default, each iteration prints the least squares fit function value (being minimized) and the norm of the gradient. The meaning of any line search warnings can be checked via doc cvsrch.
[M,~,output] = cp_wopt(X, P, R, 'init', M_init, ... 'opt', 'ncg', 'opt_options', ncg_opts);
Running CP-WOPT...
Time for zeroing out masked entries of data tensor is 6.03e-02 seconds.
(If zeroing is done in preprocessing, set 'skip_zeroing' to true.)
Iter FuncEvals F(X) ||G(X)||/N
------ --------- ---------------- ----------------
0 1 6245.25896387 0.04522498
10 60 1783.65135517 0.03570645
20 120 622.38293705 0.61597552
30 163 61.17553615 0.00263828
40 185 61.16563723 0.00000672
44 193 61.16563716 0.00000080
Check the output
It's important to check the output of the optimization method. In particular, it's worthwhile to check the exit flag. A zero (0) indicates successful termination with the gradient smaller than the specified StopTol, and a three (3) indicates a successful termination where the change in function value is less than RelFuncTol. The meaning of any other flags can be checked via doc poblano_params.
exitflag = output.ExitFlag
exitflag =
0
Evaluate the output
We can "score" the similarity of the model computed by CP and compare that with the truth. The score function on ktensor's gives a score in [0,1] with 1 indicating a perfect match. Because we have noise, we do not expect the fit to be perfect. See doc score for more details.
scr = score(M,M_true)
scr =
0.9990