Contents

function [X,out] = lbreg_mtx_bbls(A,AT,b,alpha,opts)
% lbreg_mtx_bbls: linearized Bregman iteration with Barzilai-Borwen (BB) stepsize and nonmonotone line search
%   minimize |X|_* + 1/(2*alpha) |X|_F^2
%   subject to A(X) = b
%
% input:
%       A: linear matrix operator, as a function handle
%       AT: the ajoint operator of A, as a function handle
%       b: constraint vector
%       alpha: smoothing parameter, typical value: 1 to 10 times estimated norm(X), the spectral norm of X
%       opts.
%           stepsize: dual ascent default stepsize, see below
%           tol: stop iteration once norm(Ax-b)<tol*norm(b), default: 1e-4
%           maxit: max number of iterations, default: 3000
%           maxT:  max running time in second, default: 1000
%           X_ref: if provided, norm(X^k - X_ref,'fro') is computed and saved in out.hist_err, default: []
%
% output:
%       X: solution matrix
%       out.
%           iter: number of iterations
%           hist_obj: objective value at each iteration
%           hist_res: |A(X)-b|_2 at each iteration
%           hist_err: if opts.X_ref is provided, contains norm(X^k - X_ref,'fro'); otherwise, will be set to []
%
% Algorithm:
%   Linearized Bregman is the dual gradient ascent iteration.
%   The dual problem is: b'y - alpha/2 |shrink(A'y,1)|^2,  where shrink(z,1) = z - proj_[-1,1](z) = sign(z).*max(abs(z)-1,0)
%   Let y be the dual variable. The iteration is
%     y^{k+1} = y^k + stepsize (b - alpha A shrink(A'y^k,1))
%   Promal variable X is obtained as X^k = alpha shrink(A'y^k,1)
%
% How to set alpha:
%   There exists alpha0 so that any alpha >= alpha0 gives the solution to minimize |X|_* subject to A(X) = b.
%   The alpha depends on the data, but a typical value is 1 to 10 times the estimate of norm(solution_X,2), the spectral norm of solution X
%
% How to set opts.stepsize:
%   Too large will cause divergence; too small will cause slow convergence.
%   A conservative value is 1.99/alpha/norm(A)^2, which guarantees convergence.
%   If norm(A)^2 is expensive to compute, one can compute norm(A*A') or norm(A'*A) (choose the easier one!),
%   or use the method in arXiv:1104.2076.
%
% Stepsize:
%   Since iteration 2, the dual ascent stepsize is determined by Barzilai-Borwein's method and nonmonotone line search
%
% How is the algorithm stopped: see "% stopping" below
%
% More information can be found at
% http://www.caam.rice.edu/~optimization/linearized_bregman

Parameters and defaults

if isfield(opts,'stepsize'),  stepsize0 = opts.stepsize;  else stepsize0 = 2/alpha; end % the default one assumes norm(A)=1
if isfield(opts,'tol'),    tol = opts.tol;     else tol = 1e-4;   end
if isfield(opts,'maxit'),  maxit = opts.maxit; else maxit = 500;  end
if isfield(opts,'maxT'),   maxT = opts.maxT;   else maxT = 1e3;   end
if isfield(opts,'X_ref'),  X_ref = opts.X_ref; else X_ref = []; out.out.hist_err = [];  end

Data preprocessing and initialization

m = length(b);

y = zeros(m,1);
ATy = AT(y); % keeps A'y, which is frequently used in the algorithm
res = b; % residual (b - Ax)
norm_b = norm(b);

% parameters for nonmonotone line search
lsObj = 0;
lsQ = 1;

Main iterations

start_time = tic;

for k = 1:maxit

    % --- Barzilai-Borwen (BB) stepsize ---
    if k > 1;
        y_diff = y - yp;
        stepsize = y_diff.'*y_diff / (y_diff.'*(resp-res));
        if isinf(stepsize); stepsize = stepsize0; end % in case resp-res=0, use the default stepsize
    else
        stepsize = stepsize0 + 1/max(abs(reshape(AT(b),[],1))); % the 2nd part is the stepsize to make X nonzero
    end

    % --- y-update ---
    yp = y; ATyp = ATy; % save prior-update y and A'y for future computation of BB stepize and line search
    y = y + stepsize * res; % res = (b - Ax) = dual objective gradient
    ATy = AT(y);
    ATy_diff = ATy - ATyp; % created to save computation during backtracking

    % --- X-update ---
    X = alpha * singular_value_softthresholding(AT(y),1);
    obj = b.'*y - norm(X,'fro')^2/alpha/2; % dual objective

    % --- nonmonotone line search ---
    % initialization
    lsRho = 1; % discount factor
    lsConst = 1e-3*stepsize*(norm(res)^2); % discounted expected amount of ascent
    % back tracking
    while obj < lsObj + lsRho * lsConst % while (not enough ascent)
        lsRho = lsRho * 0.5; % reduce discount factor
        y = yp + (lsRho*stepsize) * res; % apply reduced stepsize
        % computing updated dual objective
        ATy = ATyp + lsRho * ATy_diff;
        X = alpha * singular_value_softthresholding(ATy,1);
        obj = b.'*y - norm(X,'fro')^2/alpha/2;
    end
    % update averaged objective - lsObj - for next line search
    lsQp = lsQ; lsQ = 0.85*lsQp + 1;
    lsObj = (0.85*lsQp*lsObj + obj)/lsQ;

    % --- update other quantities ---
    AX = A(X);
    resp = res; % save prior-update res for next BB
    res = b - AX; % res will be used in next BB and y-update

    % --- diagnostics, reporting, stopping checks ---
    % reporting
    out.hist_obj(k) = obj; % dual objective
    out.hist_res(k) = norm(res); % residual size |b - Ax|_2
    out.hist_stp(k) = stepsize;
    if ~isempty(X_ref); out.hist_err(k) = norm(X - X_ref,'fro'); end

    % stopping
    if toc(start_time) > maxT; break; end; % running time check
    if k > 1 && norm(res) < tol*norm_b; break; end % relative primal residual check
end

out.iter = k;
end % end of main function

Subfunction for singular value soft-thresholding

function res = singular_value_softthresholding(X, mu)

    [U,S,V] = svd(X,'econ');
    s = max(0,abs(diag(S))-mu);
    res = U*diag(s)*V';

end

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