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scPRIME.m
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function [X,D,Z,patternZ,stat] = scPRIME(data,params,X0,D0,Z0)
% SCPRIME This function solves the following phase retrieval problem with
% dictionary learing problem:
%
% min 0.5*||Y-abs(A*X*B)||^2_F + 0.5*mu*||X-D*Z||^2_F
% s.t. D contains unit columns
%
% Using BSUM
%
% INPUT:
% data: measurements, measurement parameters, ground-truth
% values
% .A: spatial mixer
% .B: temporal mixer
% .Y: nosiy magnitude-only measurements
% .L: ground-truth sparsity level
% params: algorithm parameters
% .P: no. of columns in dictionary D
% .mu: weight of dictionary learing term
% .lambda: sparsity regularization parameter
% .blockRule: 0: updateInd all blocks X,D,Z in each iteration;
% 1: 1 block not updated in each iteration;
% 2: 2 blocks not updated in each iteration;
% .maxIter: max no. of iterations
% .verb: display algorithm information in each iteration if
% verb = 1
%
% OPTIONAL INPUT:
% X0, D0, Z0 - initialization of variables
%
% OUTPUT:
% X, D, Z - solutions for the variables
% patternZ - support of solution Z
% stat - algorithm statistics
%
% Author: Tianyi Liu
% disp('Algorithm begins...');
%% Initializiation
timer = tic;
format compact;
tol_rational = 1e-9; % tolerance for rational approximation
A = data.A; % spatial mixer
B = data.B; % temporal mixer
Y = data.Y; % noisy magnitude-only measurements
L = data.L; % ground-truth sparsity level
if isfield(params,'verb')
verb = params.verb;
else
verb = 0;
end
if verb
var_str = ['XDZ'];
end
if isfield(params,'trackConvergence')
trackConvergence = params.trackConvergence;
else
trackConvergence = false;
end
% Initializes parameters
N = params.N; % # rows of D
I = params.I; % # columns of Z
P = params.P; % # columns of D
mu = params.mu; % sparse approximation regularization parameter
lambda = max(params.lambda,0); % sparsity regularization parameter
debias = (params.lambda<0); % debiasing mode if params.lambda<0
if mu == 0 % regularization parameter mu must > 0
if verb
disp('Invalid data: mu = 0!');
end
return;
end
if isfield(params,'maxIter')
maxIter = params.maxIter;
else
maxIter = 1000;
end
if isfield(params,'tol')
tol = params.tol;
else
tol = 1e-4;
end
% If A and B are matrices, make function handles
if isnumeric(A)
A = @(XX,forward) op_general_mat(XX,A,forward,1);
end
if isnumeric(B)
B = @(XX,forward) op_general_mat(XX,B,forward,0);
end
% Remove negative measurements if any
Y(Y<0) = 0;
% initialize signals
if( ~exist('X0','var') || isempty(X0) )
X0 = (randn(N,I) + 1i*randn(N,I))/sqrt(2);
end
X = X0;
% initialize dictionary
if( ~exist('D0','var') || isempty(D0) )
% D0 = [eye(s),dict_dct(s1,s)];
D0 = (randn(N,P) + 1i * randn(N,P));
end
D0 = D0./sqrt(sum(abs(D0).^2,1)); % normalize each column
D = D0;
% initialize code
if( ~exist('Z0','var') || isempty(Z0) )
Z0 = D0\X0;
end
Z = Z0;
absZ = abs(Z);
patternZ = (absZ > eps); % indicators of non-zero entries of Z
if lambda > eps
lambdaL1normZ = lambda * sum(absZ(:));
else
lambdaL1normZ = 0;
end
if debias
if sum(patternZ(:)) == 0
if verb
fprintf('Invalid data: debiasing phase with empty support for Z. Aborting.\n');
end
return;
end
Z(~patternZ) = 0;
end
% Allocate memory
subgradZ = zeros(P,I);
% Initial intermediate variables, gradient and objective function
lambda_mu = lambda / mu;
% hessianXpart = sum( abs( A(eye(N),1) ).^2 ).' * sum( abs( B(eye(I),1) ).^2, 2 ).' + mu; % second-order partial derivatives of smooth part of objective w.r.t. each entry of X
hessianXmajor = ( svds(A(eye(N),1),1) * svds(B(eye(I),1),1) )^2 + mu;
Yhat = B( A(X,1), 1 );
signYhat = sign(Yhat);
signYhat(signYhat == 0) = 1;
Yt = Y .* signYhat;
residualData = Yt - Yhat; % residual of data fitting term
residualApprox = X - D*Z; % residual of sparse approximation term
obj = 0.5*norm( residualData ,'fro')^2 + 0.5*mu*norm(residualApprox,'fro')^2 + lambdaL1normZ;
% compute gradients of smooth part of upper bound of objective
gradientX = - B( A(residualData,0), 0 ) + mu * residualApprox; % HessianX*X - A'*Yt - mu*DZ; % gradient of smooth part of objective w.r.t. X
gradientD = - mu * residualApprox * Z'; % mu*(D*Z-X)*Z'; % gradient of smooth part of objective w.r.t. D
gradientZ = - mu * D' * residualApprox; % mu*D'*(D*Z-X); % gradient of smooth part of objective w.r.t. Z
stationaryInd = false(3,1); % indicate the variable blocks (X,D,Z) that have achieved stationarity with the given tolerance
% Initialize struct to record statistics:
recstats = (nargout > 4) || verb;
if( recstats )
stat.runtime = zeros(maxIter+1,1);
stat.objVal = zeros(maxIter+1,1);
% stat.distance = zeros(maxIter,1);
stat.improve = zeros(maxIter,1);
stat.error_gradX = zeros(maxIter,1);
stat.error_subgradD = zeros(maxIter,1);
stat.error_subgradZ = zeros(maxIter,1);
stat.sparsityZ = zeros(maxIter+1,1); % record average sparstiy level of columns of Z
stat.rmse_sparsityZ = zeros(maxIter+1,1);
stat.objVal(1) = obj;
stat.sparsityZ(1) = sum(patternZ(:)) / I;
stat.rmse_sparsityZ(1) = sqrt( mean( abs( sum(patternZ) - L ) .^2 ) );
stat.runtime(1) = toc(timer);
end
if trackConvergence
% Evaluate optimality of current point
% minimum-norm sub-gradient
% error_gradX = max( abs(gradientX(:)) );
% error_gradX = norm(gradientX,'fro') * distanceX / abs(obj);
error_gradX = norm(gradientX,'fro') / sqrt(numel(X)) / numel(Y);
if lambda > eps
subgradZ(:) = 0;
subgradZ(patternZ) = gradientZ(patternZ) + lambda * sign(Z(patternZ));
subgradZ(~patternZ) = max( abs(gradientZ(~patternZ))-lambda, 0); % sign(gradientZ(~nnzInd)) .* max( abs(gradientZ(~nnzInd))-lambda, 0); phase can be dropped since we need only its norm
% error_subgradZ = max( abs(subgradZ(:)) );
% error_subgradZ = norm(subgradZ,'fro') * distanceZ / abs(obj);
error_subgradZ = norm(subgradZ,'fro') / sqrt(numel(Z)) / numel(Y);
else
if debias
% error_subgradZ = max( abs(gradientZ(patternZ)) );
% error_subgradZ = norm(gradientZ(patternZ)) * distanceZ / abs(obj);
error_subgradZ = norm(gradientZ(patternZ)) / sqrt(sum(patternZ,'all')) / numel(Y);
else
% error_subgradZ = max(abs(gradientZ(:)));
% error_subgradZ = norm(gradientZ,'fro') * distanceZ / abs(obj);
error_subgradZ = norm(gradientZ,'fro') / sqrt(numel(Z)) / numel(Y);
end
end
subgradD = gradientD + sqrt( sum( abs(gradientD).^2, 1 ) ) .* D;
% error_subgradD = max( abs(subgradD(:)) );
% error_subgradD = norm(subgradD,'fro') * distanceD / abs(obj);
error_subgradD = norm(subgradD,'fro') / sqrt(numel(D)) / numel(Y);
error_subgrad = sqrt( (error_subgradD^2*numel(D) + error_subgradZ^2*numel(Z) + error_gradX*numel(X)) / (numel(D) + numel(Z) + numel(X)) );
% evaluate and record estimation quality every 10 iterations
clear sol;
sol.D = D;
sol.Z = Z;
sol.X = X;
quality = eval_quality(sol,data);
stat.quality.iterArray = [0;0];
stat.quality.timeArray = [0; stat.runtime(1)];
stat.quality.FmeasureZ = [quality.FmeasureZ; quality.FmeasureZ];
stat.quality.errorD = [quality.errorD; quality.errorD];
stat.quality.errorZ = [quality.errorZ; quality.errorZ];
stat.quality.errorDZ = [quality.errorDZ; quality.errorDZ];
stat.quality.errorX = [quality.errorX; quality.errorX];
stat.quality.error_subgrad = [error_subgrad; error_subgrad];
stat.quality.objVal = [obj; obj];
end
% distanceX = 1;
% distanceD = 1;
% distanceZ = 1;
% Main algorithm
for t = 1: maxIter
timer = tic;
if verb
fprintf('iter %d:\n',t);
end
%% Update X
updateX = - gradientX ./ hessianXmajor;
X = X + updateX;
% distanceX = norm(updateX,'fro');
%% Update D
Dold = D;
XZh = X*Z';
ZZh = Z*Z';
for p = 1:P
if ZZh(p,p) > 0
newdp = (XZh(:,p)-D*ZZh(:,p))/ZZh(p,p) + D(:,p);
else
newdp = randn(N,1);
end
normNewdp = norm(newdp);
D(:,p) = newdp / max(1,normNewdp);
end
% distanceD = norm(D-Dold,'fro');
%% Update Z
Zold = Z;
if debias % debiasing mode
Z = Z - D'*(D*Z-X) ./ P;
Z(~patternZ) = 0;
else
Z = soft_threshold( lambda_mu, P .* Z - D'*(D*Z-X) ) ./ P;
end
% distanceZ = norm(Z-Zold,'fro');
%% update intermediate variables and objective function value
Yhat = B( A(X,1), 1 );
signYhat = sign(Yhat);
signYhat(signYhat == 0) = 1;
Yt = Y .* signYhat; % Y.*exp(1i*angle(ADZ));
residualData = Yt - Yhat; % residual of data fitting term
residualApprox = X - D*Z; % residual of sparse approximation term
absZ = abs(Z);
if lambda > eps
lambdaL1normZ = lambda * sum(absZ(:));
else
lambdaL1normZ = 0;
end
oldobj = obj;
obj = 0.5*norm( residualData ,'fro')^2 + 0.5*mu*norm(residualApprox,'fro')^2 + lambdaL1normZ;
improve = (oldobj - obj)/oldobj;
% compute gradients of smooth part of upper bound of objective
gradientX = - B( A(residualData,0), 0 ) + mu * residualApprox; % HessianX*X - A'*Yt - mu*DZ; % gradient of smooth part of objective w.r.t. X
runtime = toc(timer);
gradientD = - mu * residualApprox * Z'; % mu*(D*Z-X)*Z'; % gradient of smooth part of objective w.r.t. D
gradientZ = - mu * D' * residualApprox; % mu*D'*(D*Z-X); % gradient of smooth part of objective w.r.t. Z
%% Evaluate optimality of current point
% minimum-norm sub-gradient
% error_gradX = max( abs(gradientX(:)) );
% error_gradX = norm(gradientX,'fro') * distanceX / abs(obj);
error_gradX = norm(gradientX,'fro') / sqrt(numel(X)) / numel(Y);
if lambda > eps
subgradZ(:) = 0;
subgradZ(patternZ) = gradientZ(patternZ) + lambda * sign(Z(patternZ));
subgradZ(~patternZ) = max( abs(gradientZ(~patternZ))-lambda, 0); % sign(gradientZ(~nnzInd)) .* max( abs(gradientZ(~nnzInd))-lambda, 0); phase can be dropped since we need only its norm
% error_subgradZ = max( abs(subgradZ(:)) );
% error_subgradZ = norm(subgradZ,'fro') * distanceZ / abs(obj);
error_subgradZ = norm(subgradZ,'fro') / sqrt(numel(Z)) / numel(Y);
else
if debias
% error_subgradZ = max( abs(gradientZ(patternZ)) );
% error_subgradZ = norm(gradientZ(patternZ)) * distanceZ / abs(obj);
error_subgradZ = norm(gradientZ(patternZ)) / sqrt(sum(patternZ,'all')) / numel(Y);
else
% error_subgradZ = max(abs(gradientZ(:)));
% error_subgradZ = norm(gradientZ,'fro') * distanceZ / abs(obj);
error_subgradZ = norm(gradientZ,'fro') / sqrt(numel(Z)) / numel(Y);
end
end
subgradD = gradientD + sqrt( sum( abs(gradientD).^2, 1 ) ) .* D;
% error_subgradD = max( abs(subgradD(:)) );
% error_subgradD = norm(subgradD,'fro') * distanceD / abs(obj);
error_subgradD = norm(subgradD,'fro') / sqrt(numel(D)) / numel(Y);
% record statistics
if recstats
stat.runtime(t+1) = stat.runtime(t) + runtime;
stat.objVal(t+1) = obj;
% stat.distance(t) = distance;
stat.improve(t) = (stat.objVal(t)-stat.objVal(t+1)) / stat.objVal(t);
if debias
stat.sparsityZ(t+1) = stat.sparsityZ(t);
stat.rmse_sparsityZ(t+1) = stat.rmse_sparsityZ(t);
else
stat.sparsityZ(t+1) = sum(patternZ(:)) / I;
stat.rmse_sparsityZ(t+1) = sqrt( mean( abs( sum(absZ > eps) - L ) .^2 ) );
end
stat.error_gradX(t) = error_gradX;
stat.error_subgradZ(t) = error_subgradZ;
stat.error_subgradD(t) = error_subgradD;
end
if trackConvergence
% evaluate and record estimation quality every 10 iterations
if mod(t,10) == 0
clear sol;
sol.D = D;
sol.Z = Z;
sol.X = X;
quality = eval_quality(sol,data);
stat.quality.iterArray = [stat.quality.iterArray; t];
stat.quality.timeArray = [stat.quality.timeArray; stat.runtime(t+1)];
stat.quality.FmeasureZ = [stat.quality.FmeasureZ; quality.FmeasureZ];
stat.quality.errorD = [stat.quality.errorD; quality.errorD];
stat.quality.errorZ = [stat.quality.errorZ; quality.errorZ];
stat.quality.errorDZ = [stat.quality.errorDZ; quality.errorDZ];
stat.quality.errorX = [stat.quality.errorX; quality.errorX];
error_subgrad = sqrt( (error_subgradD^2*numel(D) + error_subgradZ^2*numel(Z) + error_gradX*numel(X)) / (numel(D) + numel(Z) + numel(X)) );
stat.quality.error_subgrad = [stat.quality.error_subgrad; error_subgrad];
stat.quality.objVal = [stat.quality.objVal; obj];
end
end
if verb
fprintf('\tvalue = %f, improve = %e,\n\tgradientX = %e, subgradD = %e, subgradZ = %e,\n\tavg. code sparsity = %f, time = %f s.\n',...
obj,stat.improve(t),error_gradX, error_subgradD, error_subgradZ, stat.sparsityZ(t+1), stat.runtime(t));
% fprintf('\tdistanceX = %f, distanceD = %f, distanceZ = %f\n',distanceX,distanceD,distanceZ);
end
%% Stopping criterion
stationaryInd(1) = ( error_gradX <= tol );
stationaryInd(2) = ( error_subgradD <= tol );
stationaryInd(3) = ( error_subgradZ <= tol );
if all(stationaryInd)
if verb
fprintf('Tolerance achieved. Terminating.\n');
end
break;
end
if improve < 0
if verb
fprintf('Not descent update. Aborting.');
end
t = t-1;
break;
end
end
if recstats
stat.Niter = t; % # iterations
stat.optVal = stat.objVal(t+1); % optimal value
if t < maxIter
if t > 0
stat.runtime(t+2:end) = stat.runtime(t+1);
end
stat.objVal(t+2:end) = stat.optVal;
stat.error_gradX(t+1:end) = stat.error_gradX(t);
stat.error_subgradD(t+1:end) = stat.error_subgradD(t);
stat.error_subgradZ(t+1:end) = stat.error_subgradZ(t);
stat.sparsityZ(t+2:end) = stat.sparsityZ(t+1);
stat.rmse_sparsityZ(t+2:end) = stat.rmse_sparsityZ(t+1);
end
end
end