Weighted Optimization for CP Tensor Decomposition with Incomplete Data

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We explain how to use the CP Weighted Optimization (CP-WOPT) method implemented in cp_wopt. The method is described in the following article:

Contents

Third-party optimization software

The cp_wopt method uses third-party optimization software to do the optimization. You can use either

The remainder of these instructions assume L-BFGS-B is being used. See here for instructions on using cp_wopt with Poblano.

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');

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.

[M,~,output] = cp_wopt(X, P, R, 'init', M_init);

Running CP-WOPT...
Time for zeroing out masked entries of data tensor is 1.26e-03 seconds.
(If zeroing is done in preprocessing, set 'skip_zeroing' to true.)
Iter    10, f(x) = 3.484657e+01, ||grad||_infty = 1.44e+01
Iter    20, f(x) = 2.686975e+00, ||grad||_infty = 1.68e-01
Iter    30, f(x) = 2.681928e+00, ||grad||_infty = 1.75e-03
Iter    34, f(x) = 2.681928e+00, ||grad||_infty = 2.50e-04

Check the output

It's important to check the output of the optimization method. In particular, it's worthwhile to check the exit message for any problems. The message CONVERGENCE: REL_REDUCTION_OF_F_<=_FACTR*EPSMCH means that it has converged because the function value stopped improving.

exitmsg = output.ExitMsg

exitmsg =
    'CONVERGENCE: REL_REDUCTION_OF_F_<=_FACTR*EPSMCH.'

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

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');

Call the cp_wopt method

[M,~,output] = cp_wopt(X, P, R, 'init', M_init);

Running CP-WOPT...
Time for zeroing out masked entries of data tensor is 4.33e-02 seconds.
(If zeroing is done in preprocessing, set 'skip_zeroing' to true.)
Iter    10, f(x) = 5.028631e+02, ||grad||_infty = 6.58e+00
Iter    20, f(x) = 4.969471e+02, ||grad||_infty = 1.73e+00
Iter    30, f(x) = 4.878141e+02, ||grad||_infty = 1.30e+01
Iter    40, f(x) = 4.563805e+02, ||grad||_infty = 2.24e+01
Iter    50, f(x) = 4.379043e+02, ||grad||_infty = 8.96e+00
Iter    60, f(x) = 4.127368e+02, ||grad||_infty = 2.07e+01
Iter    70, f(x) = 9.426906e+01, ||grad||_infty = 1.29e+01
Iter    80, f(x) = 8.293402e+01, ||grad||_infty = 4.32e-01
Iter    90, f(x) = 8.290516e+01, ||grad||_infty = 7.58e-02
Iter    99, f(x) = 8.290503e+01, ||grad||_infty = 1.25e-02

Check the output

exitmsg = output.ExitMsg

exitmsg =
    'CONVERGENCE: REL_REDUCTION_OF_F_<=_FACTR*EPSMCH.'

Evaluate the output

scr = score(M,M_true)

scr =
    0.9983

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