Contents
function [X,out] = lbreg_mtx_fixedstep(A,AT,b,alpha,opts)
% lbreg_mtx_fixedstep: linearized Bregman iteration with fixed stepsize % 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,2), the spectral norm of X % opts. % stepsize: dual ascent 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. % % 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'), stepsize = opts.stepsize; else stepsize = 1.99/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);
res = b; % residual (b - Ax)
norm_b = norm(b);
Main iterations
start_time = tic; for k = 1:maxit % --- y-update --- y = y + stepsize * res; % for k=1, (stepsize + 1/max(abs(b.'*A))) can be used as stepsize % --- X-update --- X = alpha * singular_value_softthresholding(AT(y),1); AX = A(X); res = b - AX; % res will be used in next y-update % --- diagnostics, reporting, stopping checks --- % reporting out.hist_obj(k) = b.'*y - norm(X,'fro')^2/alpha/2; % dual objective out.hist_res(k) = norm(res); % residual size |b - Ax|_2 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 % dual objective check obj_diff = out.hist_obj(k) - out.hist_obj(k-1); if obj_diff < -1e-10; error('dual objective decreases; opts.stepsize is too large!'); end % primal residual check if norm(res) < tol*norm_b; break; end end 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