Weighted optimization for CP tensor decomposition with incomplete data
We explain how to use cp_wopt with the POBLANO toolbox.
Contents
- 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
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 = 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, ... 'alg', 'ncg', 'alg_options', ncg_opts);
Iter FuncEvals F(X) ||G(X)||/N ------ --------- ---------------- ---------------- 0 1 185.00205958 0.48027652 10 47 1.68251600 0.03869586 20 74 1.51902470 0.00179193 30 95 1.51868289 0.00009479 40 115 1.51868158 0.00000174 45 125 1.51868158 0.00000050
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 = 0.9849
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 = 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, ... 'alg', 'ncg', 'alg_options', ncg_opts);
Iter FuncEvals F(X) ||G(X)||/N ------ --------- ---------------- ---------------- 0 1 1170.62424530 0.02422437 10 52 190.11900267 0.00893357 20 96 162.40135695 0.10203161 30 137 11.42949623 0.00295760 40 160 11.38681461 0.00002031 48 176 11.38681239 0.00000096
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 = 0.9977