# All-at-once optimization for CP tensor decomposition

We explain how to use `cp_opt` with the POBLANO toolbox.

## Contents

- Poblano Optimization Toolbox
- Create an example problem. Here we have 10% noise.
- Create initial guess using 'nvecs'
- Set up the optimization parameters
- Call the `cp_opt` method
- Check the output
- Evaluate the output
- Overfitting example

## Poblano Optimization Toolbox

Check that you have Poblano 1.1 installed. The output of your 'ver' command should look something like the following.

    ver

    ----------------------------------------------------------------------------------------------------
    MATLAB Version: 8.4.0.150421 (R2014b)
    MATLAB License Number: 192525
    Operating System: Microsoft Windows 7 Enterprise  Version 6.1 (Build 7601: Service Pack 1)
    Java Version: Java 1.7.0_11-b21 with Oracle Corporation Java HotSpot(TM) 64-Bit Server VM mixed mode
    ----------------------------------------------------------------------------------------------------
    MATLAB                                                Version 8.4        (R2014b)
    Optimization Toolbox                                  Version 7.1        (R2014b)
    Poblano Toolbox (Sandia National Labs)                Version 1.1
    Statistics Toolbox                                    Version 9.1        (R2014b)
    Tensor Toolbox (Sandia National Labs)                 Version 2.6

## Create an example problem. Here we have 10% noise.

    R = 5;
    info = create_problem('Size', [50 40 30], 'Num_Factors', R, 'Noise', 0.10);
    X = info.Data;
    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_opt` 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_opt(X, R, 'init', M_init, ...
        'alg', 'ncg', 'alg_options', ncg_opts);

     Iter  FuncEvals       F(X)          ||G(X)||/N
    ------ --------- ---------------- ----------------
         0         1   59446.20284454       0.76018888
        10        80    1154.53391434       1.87155134
        20       131     583.85024670       0.02127047
        30       157     583.79035838       0.00013732
        40       177     583.79035282       0.00000162
        42       181     583.79035282       0.00000071

## 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

The fit is the percentage of the data that is explained by the model. Because we have noise, we do not expect the fit to be perfect.

    fit = output.Fit

    fit =
       99.0183

## 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 =
        0.9984

## Overfitting example

Consider the case where we don't know R in advance. We might guess too high. Here we show a case where we guess R+1 factors rather than R.

    % Generate initial guess of the corret size
    M_plus_init = create_guess('Data', X, 'Num_Factors', R+1, ...
        'Factor_Generator', 'nvecs');

    % Loosen the stop tolerance (norm of gradient).
    ncg_opts.StopTol = 1.0e-2;

    % Run the algorithm
    [M_plus,~,output] = cp_opt(X, R+1, 'init', M_plus_init, ...
        'alg', 'ncg', 'alg_options', ncg_opts);
    exitflag = output.ExitFlag
    fit = output.Fit

     Iter  FuncEvals       F(X)          ||G(X)||/N
    ------ --------- ---------------- ----------------
         0         1   59446.61102123       0.63350080
        10        80    1154.62477976       1.55962386
        20       131     583.89115606       0.01846313
        21       134     583.85651979       0.00932713
    exitflag =
         0
    fit =
       99.0182

    % Check the answer (1 is perfect)
    scr = score(M_plus, M_true)

    scr =
        0.9983