trial_DL

Problem Setup
Build the dictionary
Compute coefficient vectors
Form the output channel
EM BiG AMP
SPAMS
ER-SpUD
K-SVD
Store the options structures in results
Show Results

function results = trial_DL(optIn)

%trial_DL: This function runs several algorithms on a sample
%instance of the dictionary learning problem. The function can be run with no
%arguments. See the nargin==0 block below for the required fields to call
%the function with optional parameters. The function returns a structure of
%algorithm results.
%This code can be used to produce the results for the noise free phase
%plane plots for dictionary learning in the BiG-AMP arXiv submission.
%(This code conducts a single trial at 1 point in the phase plane.)

%Add paths
setup_DL

%Test case
if nargin == 0
    clc

    %Handle random seed
    defaultStream = RandStream.getGlobalStream;
    if 1
        savedState = defaultStream.State;
        save random_state.mat savedState;
    else
        load random_state.mat %#ok<UNRCH>
    end
    defaultStream.State = savedState;

    %Flags to test BiG-AMP variants
    optIn.tryBigampEM = 1;

    %Flags to turn on comparison algorithms
    optIn.tryKsvd = 1;
    optIn.tryErspud = 1;
    optIn.trySpams = 1;


    %Problem dimensions
    optIn.M = 20; %size of signal
    optIn.N = optIn.M; %size of dictionary
    optIn.L = ceil(5*optIn.N*log(optIn.N)); %From Spielman et al.


    %Specify dictionary usage
    optIn.K = 5;

    %Specify maximum allowed trials
    optIn.maxTrials = 1;

    %SNR
    optIn.SNR = inf;

    %Specify coding mode (0 for OMP, 1 for TST)
    %The specified function is used to compute a coding error for the
    %resulting dictionary for each algorithm. TST may be significantly
    %slower.
    optIn.useTST = 0;

    %Precondition option (enable to use a preconditioner for EM-BiG-AMP)
    optIn.precondition = 0;


end

Problem Setup

%Turn algs on/off
tryBigampEM = optIn.tryBigampEM;
tryKsvd = optIn.tryKsvd;
tryErspud = optIn.tryErspud;
trySpams = optIn.trySpams;

%SNR
SNR = optIn.SNR;

%Coding
useTST = optIn.useTST;

%Precondition
precondition = optIn.precondition;

%Max trials
maxTrials = optIn.maxTrials;

%Define problem dimensions
M = optIn.M;
L = optIn.L;
N = optIn.N;
K = optIn.K;

%Set options
opt = BiGAMPOpt; %initialize the options object
opt.nit = 500; %limit iterations

%Set sizes
problem = BiGAMPProblem();
problem.M = M;
problem.N = N;
problem.L = L;

Build the dictionary

%Draw randomly
A = randn(M,N);

%Normalize the columns
A = A*diag(1 ./ sqrt(abs(diag(A'*A))));

%Dictionary error function
dictionary_error_function =...
    @(q) 20*log10(norm(A -...
    q*find_permutation(A,q),'fro')/norm(A,'fro'));

Compute coefficient vectors

%Compute true vectors with exactly K non-zeroes in each
X = randn(N,L);
for ll = 1:L
    yada = randperm(N);
    yada2 = zeros(N,1);
    yada2(yada(1:K)) = 1;
    X(:,ll) = X(:,ll) .* yada2;
end

Form the output channel

%Compute noise free output
Z = A*X;

%Define the error function
error_function = @(qval) 20*log10(norm(qval - Z,'fro') / norm(Z,'fro'));
opt.error_function = error_function;


%Determine nuw
nuw = norm(reshape(Z,[],1))^2/M/L*10^(-SNR/10);

%Noisy output channel
Y = Z + sqrt(nuw)*randn(size(Z));

%Coding error
coding_error_function = @(q) 20*log10(coding_error(Y,q,K,useTST));


%Initialize results as empty
results = [];

EM BiG AMP

if tryBigampEM

    %Silence
    opt.verbose = false;
    disp('Starting EM-BiG-AMP')

    %Coding
    bestError = inf;
    bestSparsity = inf;

    %Compute preconditioner
    if precondition
        Q = chol(inv(Y*Y'));
    else
        Q = 1;
    end

    %Define the error function
    QZ = Q*Y; %Using (possibly) noisy data for error function
    error_function2 = @(qval) 20*log10(norm(qval - QZ,'fro') / norm(QZ,'fro'));
    opt.error_function = error_function2;

    tstart = tic;
    for trial = 1:maxTrials

        %Run EM-BiGAMP
        [estFinTemp,~,~,estHistEMtemp] = ...
            EMBiGAMP_DL(Q*Y,problem,opt);

        %Correct the dictionary
        estFinTemp.Ahat = Q \ estFinTemp.Ahat;

        %Reset error function
        opt.error_function = error_function;

        %If the sparisty is better and the error is very small or better
        if (sum(sum(estHistEMtemp.p1)) < bestSparsity) && ...
                ( (estHistEMtemp.errZ(end) < bestError)  ||...
                (estHistEMtemp.errZ(end) < -100)  )

            %Update
            bestSparsity = sum(sum(estHistEMtemp.p1));
            bestError = estHistEMtemp.errZ(end);
            AhatOptEM = estFinTemp.Ahat;
            estHistEM = estHistEMtemp;
            p1EM = estHistEMtemp.p1;

            %Notify user
            disp(['Accepting new result. Error: ' num2str(bestError)...
                '  Average sparsity: ' num2str(bestSparsity/L)...
                '  Max Sparsity: ' num2str(max(sum(p1EM)))])
        end
        %keyboard
    end
    tEMGAMP = toc(tstart);

    %Save results
    loc = length(results) + 1;
    results{loc}.name = 'EM-BiG-AMP'; %#ok<*AGROW>
    results{loc}.err = estHistEM.errZ(end);
    results{loc}.time = tEMGAMP;
    results{loc}.errHist = estHistEM.errZ;
    results{loc}.timeHist = estHistEM.timing;
    results{loc}.dict = AhatOptEM;
    results{loc}.dictError = dictionary_error_function(results{loc}.dict);
    results{loc}.codingError =...
        coding_error_function(results{loc}.dict);
end

Starting EM-BiG-AMP
Dictionary draw finished after attempts: 1
It 0001 nuX = 1.208e-01 Lam = 0.10 tol = 1.000e-04 SNR = 20.00 Z_e = -29.6376 nuw = 2.417e-03 Avg Spar = 4.5 numIt = 0500
It 0002 nuX = 4.894e-02 Lam = 0.23 tol = 1.000e-04 SNR = 29.61 Z_e = -45.2911 nuw = 2.653e-04 Avg Spar = 4.8 numIt = 0075
It 0003 nuX = 3.324e-02 Lam = 0.24 tol = 2.959e-05 SNR = 45.29 Z_e = -66.1533 nuw = 7.218e-06 Avg Spar = 5.0 numIt = 0064
It 0004 nuX = 2.698e-02 Lam = 0.25 tol = 2.425e-07 SNR = 66.15 Z_e = -98.2798 nuw = 5.918e-08 Avg Spar = 5.0 numIt = 0079
It 0005 nuX = 2.424e-02 Lam = 0.25 tol = 1.000e-08 SNR = 98.28 Z_e = -148.7084 nuw = 3.627e-11 Avg Spar = 5.0 numIt = 0091
It 0006 nuX = 2.282e-02 Lam = 0.25 tol = 1.000e-08 SNR = 148.71 Z_e = -149.6378 nuw = 1.458e-11 Avg Spar = 5.0 numIt = 0091
Accepting new result. Error: -149.6378  Average sparsity: 5  Max Sparsity: 5

SPAMS

if trySpams

    %Build params
    spams_param = [];
    spams_param.K = N;
    spams_param.mode = 2;
    spams_param.lambda = 0.1/sqrt(N);
    spams_param.iter = 1000;


    %Trials
    bestSPAMSerror = inf;

    tstart = tic;
    for trial = 1:maxTrials

        %Do it
        A_spamsTemp = mexTrainDL(Y,spams_param);


        SPAMSerrorTemp = dictionary_error_function(A_spamsTemp);

        %Update the estimate if this result was superior
        if SPAMSerrorTemp < bestSPAMSerror
            SPAMSerror = SPAMSerrorTemp;
            bestSPAMSerror = SPAMSerror;
            A_spams = A_spamsTemp;
            disp(['Updating Solution. Error was: '...
                num2str(SPAMSerror) ' dB'])
        end

    end
    tspams = toc(tstart);


    %Save results
    loc = length(results) + 1;
    results{loc}.name = 'SPAMS'; %#ok<*AGROW>
    results{loc}.err = dictionary_error_function(A_spams);
    results{loc}.time = tspams;
    results{loc}.errHist = results{loc}.err;
    results{loc}.timeHist = zeros(size(results{loc}.errHist));
    results{loc}.dict = A_spams;
    results{loc}.dictError = dictionary_error_function(results{loc}.dict);
    results{loc}.codingError =...
        coding_error_function(results{loc}.dict);


end

num param iterD: 1
Online Dictionary Learning with no parameter 
mode Alpha 2
Cleaning activated 
batch size: 300
L: 20
lambda: 0.0223607
mode: 2
*****Online Dictionary Learning*****
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Iteration: 999
Time elapsed : 1.41524
Updating Solution. Error was: -30.9983 dB

ER-SpUD

if tryErspud


    %Call it
    tic
    %[Aerspud,Xerspud] = ER_SpUD_SC(Y);
    [Aerspud,Xerspud]=dl_spud(Y);   %#ok<NASGU>
    tErspud = toc;

    %Save results
    loc = length(results) + 1;
    results{loc}.name = 'ER-SpUD (proj)'; %#ok<*AGROW>
    results{loc}.err = dictionary_error_function(Aerspud);
    results{loc}.time = tErspud;
    results{loc}.errHist = results{loc}.err;
    results{loc}.timeHist = zeros(size(results{loc}.errHist));
    results{loc}.dict = Aerspud;
    results{loc}.dictError = dictionary_error_function(results{loc}.dict);
    results{loc}.codingError =...
        coding_error_function(results{loc}.dict);



end

1 row(s) of X recovered, nonzero # in the new row:60
2 row(s) of X recovered, nonzero # in the new row:61
3 row(s) of X recovered, nonzero # in the new row:64
4 row(s) of X recovered, nonzero # in the new row:65
5 row(s) of X recovered, nonzero # in the new row:67
6 row(s) of X recovered, nonzero # in the new row:68
7 row(s) of X recovered, nonzero # in the new row:69
8 row(s) of X recovered, nonzero # in the new row:73
9 row(s) of X recovered, nonzero # in the new row:74
10 row(s) of X recovered, nonzero # in the new row:75
11 row(s) of X recovered, nonzero # in the new row:75
12 row(s) of X recovered, nonzero # in the new row:75
13 row(s) of X recovered, nonzero # in the new row:76
14 row(s) of X recovered, nonzero # in the new row:76
15 row(s) of X recovered, nonzero # in the new row:76
16 row(s) of X recovered, nonzero # in the new row:86
17 row(s) of X recovered, nonzero # in the new row:86
18 row(s) of X recovered, nonzero # in the new row:88
19 row(s) of X recovered, nonzero # in the new row:92
20 row(s) of X recovered, nonzero # in the new row:94

K-SVD

if tryKsvd

    %Build params
    ksvd_params = [];
    ksvd_params.data = Y;
    ksvd_params.Tdata = K;
    ksvd_params.dictsize = N;
    ksvd_params.iternum = 100;
    ksvd_params.exacty = 1;

    %Trials
    bestKSVDerror = inf;
    tstart = tic;
    for trial = 1:maxTrials

        %Do it

        [A_ksvdtemp,~,err_ksvdtemp] = ksvd(ksvd_params);


        %Fix error
        err_ksvdtemp = err_ksvdtemp.^2 * numel(Y);

        %Update the estimate if this result was superior
        if err_ksvdtemp(end) < bestKSVDerror
            bestKSVDerror = err_ksvdtemp(end);
            err_ksvd = err_ksvdtemp;
            A_ksvd = A_ksvdtemp;
            disp(['Updating Solution. Error was: '...
                num2str(10*log10(err_ksvd(end)/norm(Z,'fro')^2))])
        end

    end
    tksvd = toc(tstart);

    %Save results
    loc = length(results) + 1;
    results{loc}.name = 'K-SVD'; %#ok<*AGROW>
    results{loc}.err = 10*log10(err_ksvd(end)/norm(Z,'fro')^2);
    results{loc}.time = tksvd;
    results{loc}.errHist = 10*log10(err_ksvd/norm(Z,'fro')^2);
    results{loc}.timeHist = zeros(size(err_ksvd));
    results{loc}.dict = A_ksvd;
    results{loc}.dictError = dictionary_error_function(results{loc}.dict);
    results{loc}.codingError =...
        coding_error_function(results{loc}.dict);


end

Iteration 1 / 100 complete, RMSE = 0.1769
Iteration 2 / 100 complete, RMSE = 0.1624
Iteration 3 / 100 complete, RMSE = 0.1527
Iteration 4 / 100 complete, RMSE = 0.1468
Iteration 5 / 100 complete, RMSE = 0.1423
Iteration 6 / 100 complete, RMSE = 0.1403
Iteration 7 / 100 complete, RMSE = 0.1375
Iteration 8 / 100 complete, RMSE = 0.1353
Iteration 9 / 100 complete, RMSE = 0.1325
Iteration 10 / 100 complete, RMSE = 0.1292
Iteration 11 / 100 complete, RMSE = 0.1268
Iteration 12 / 100 complete, RMSE = 0.1241
Iteration 13 / 100 complete, RMSE = 0.1215
Iteration 14 / 100 complete, RMSE = 0.1199
Iteration 15 / 100 complete, RMSE = 0.1196
Iteration 16 / 100 complete, RMSE = 0.1183
Iteration 17 / 100 complete, RMSE = 0.1176
Iteration 18 / 100 complete, RMSE = 0.1164
Iteration 19 / 100 complete, RMSE = 0.1157
Iteration 20 / 100 complete, RMSE = 0.1158
Iteration 21 / 100 complete, RMSE = 0.115
Iteration 22 / 100 complete, RMSE = 0.1139
Iteration 23 / 100 complete, RMSE = 0.1138
Iteration 24 / 100 complete, RMSE = 0.1132
Iteration 25 / 100 complete, RMSE = 0.1127
Iteration 26 / 100 complete, RMSE = 0.1121
Iteration 27 / 100 complete, RMSE = 0.1116
Iteration 28 / 100 complete, RMSE = 0.1107
Iteration 29 / 100 complete, RMSE = 0.1097
Iteration 30 / 100 complete, RMSE = 0.1093
Iteration 31 / 100 complete, RMSE = 0.1082
Iteration 32 / 100 complete, RMSE = 0.1068
Iteration 33 / 100 complete, RMSE = 0.1058
Iteration 34 / 100 complete, RMSE = 0.1058
Iteration 35 / 100 complete, RMSE = 0.1047
Iteration 36 / 100 complete, RMSE = 0.1039
Iteration 37 / 100 complete, RMSE = 0.1047
Iteration 38 / 100 complete, RMSE = 0.1045
Iteration 39 / 100 complete, RMSE = 0.1041
Iteration 40 / 100 complete, RMSE = 0.1038
Iteration 41 / 100 complete, RMSE = 0.1041
Iteration 42 / 100 complete, RMSE = 0.1033
Iteration 43 / 100 complete, RMSE = 0.1024
Iteration 44 / 100 complete, RMSE = 0.1017
Iteration 45 / 100 complete, RMSE = 0.1
Iteration 46 / 100 complete, RMSE = 0.0986
Iteration 47 / 100 complete, RMSE = 0.09788
Iteration 48 / 100 complete, RMSE = 0.09664
Iteration 49 / 100 complete, RMSE = 0.09654
Iteration 50 / 100 complete, RMSE = 0.0943
Iteration 51 / 100 complete, RMSE = 0.09365
Iteration 52 / 100 complete, RMSE = 0.0934
Iteration 53 / 100 complete, RMSE = 0.09396
Iteration 54 / 100 complete, RMSE = 0.09411
Iteration 55 / 100 complete, RMSE = 0.09357
Iteration 56 / 100 complete, RMSE = 0.0935
Iteration 57 / 100 complete, RMSE = 0.09325
Iteration 58 / 100 complete, RMSE = 0.09194
Iteration 59 / 100 complete, RMSE = 0.09192
Iteration 60 / 100 complete, RMSE = 0.09176
Iteration 61 / 100 complete, RMSE = 0.09112
Iteration 62 / 100 complete, RMSE = 0.09049
Iteration 63 / 100 complete, RMSE = 0.08966
Iteration 64 / 100 complete, RMSE = 0.08892
Iteration 65 / 100 complete, RMSE = 0.08815
Iteration 66 / 100 complete, RMSE = 0.08766
Iteration 67 / 100 complete, RMSE = 0.0882
Iteration 68 / 100 complete, RMSE = 0.08798
Iteration 69 / 100 complete, RMSE = 0.08665
Iteration 70 / 100 complete, RMSE = 0.08556
Iteration 71 / 100 complete, RMSE = 0.08462
Iteration 72 / 100 complete, RMSE = 0.08439
Iteration 73 / 100 complete, RMSE = 0.08386
Iteration 74 / 100 complete, RMSE = 0.0826
Iteration 75 / 100 complete, RMSE = 0.08168
Iteration 76 / 100 complete, RMSE = 0.08112
Iteration 77 / 100 complete, RMSE = 0.08164
Iteration 78 / 100 complete, RMSE = 0.08146
Iteration 79 / 100 complete, RMSE = 0.08026
Iteration 80 / 100 complete, RMSE = 0.08035
Iteration 81 / 100 complete, RMSE = 0.07992
Iteration 82 / 100 complete, RMSE = 0.07981
Iteration 83 / 100 complete, RMSE = 0.08039
Iteration 84 / 100 complete, RMSE = 0.08032
Iteration 85 / 100 complete, RMSE = 0.08001
Iteration 86 / 100 complete, RMSE = 0.07926
Iteration 87 / 100 complete, RMSE = 0.07942
Iteration 88 / 100 complete, RMSE = 0.07912
Iteration 89 / 100 complete, RMSE = 0.07971
Iteration 90 / 100 complete, RMSE = 0.07968
Iteration 91 / 100 complete, RMSE = 0.07943
Iteration 92 / 100 complete, RMSE = 0.0794
Iteration 93 / 100 complete, RMSE = 0.0791
Iteration 94 / 100 complete, RMSE = 0.07834
Iteration 95 / 100 complete, RMSE = 0.07821
Iteration 96 / 100 complete, RMSE = 0.07833
Iteration 97 / 100 complete, RMSE = 0.07779
Iteration 98 / 100 complete, RMSE = 0.07752
Iteration 99 / 100 complete, RMSE = 0.0772
Iteration 100 / 100 complete, RMSE = 0.07701
Updating Solution. Error was: -16.1448

Store the options structures in results

results{1}.optIn = optIn;
results{1}.trueDict = A;
results{1}.trueEncoding = X;

Show Results

if nargin == 0
    results{:} %#ok<NOPRT>
    disp('Note that dictError is the normalized error in dB for recovering')
    disp('the dictionary for each algorithm')
end

ans = 

            name: 'EM-BiG-AMP'
             err: -149.6378
            time: 2.8104
         errHist: [900x1 double]
        timeHist: [900x1 double]
            dict: [20x20 double]
       dictError: -146.4313
     codingError: -26.7985
           optIn: [1x1 struct]
        trueDict: [20x20 double]
    trueEncoding: [20x300 double]


ans = 

           name: 'SPAMS'
            err: -30.9983
           time: 1.4600
        errHist: -30.9983
       timeHist: 0
           dict: [20x20 double]
      dictError: -30.9983
    codingError: -27.4858


ans = 

           name: 'ER-SpUD (proj)'
            err: -122.5402
           time: 155.2519
        errHist: -122.5402
       timeHist: 0
           dict: [20x20 double]
      dictError: -122.5402
    codingError: -26.7985


ans = 

           name: 'K-SVD'
            err: -16.1448
           time: 2.9766
        errHist: [1x100 double]
       timeHist: [1x100 double]
           dict: [20x20 double]
      dictError: -6.5899
    codingError: -16.1796

Note that dictError is the normalized error in dB for recovering
the dictionary for each algorithm

ans = 

    [1x1 struct]    [1x1 struct]    [1x1 struct]    [1x1 struct]

Contents

Problem Setup

Build the dictionary

Compute coefficient vectors

Form the output channel

EM BiG AMP

SPAMS

ER-SpUD

K-SVD

Store the options structures in results

Show Results