% * This code was used in the following articles: % * [1] Learning 3-D Scene Structure from a Single Still Image, % * Ashutosh Saxena, Min Sun, Andrew Y. Ng, % * In ICCV workshop on 3D Representation for Recognition (3dRR-07), 2007. % * (best paper) % * [2] 3-D Reconstruction from Sparse Views using Monocular Vision, % * Ashutosh Saxena, Min Sun, Andrew Y. Ng, % * In ICCV workshop on Virtual Representations and Modeling % * of Large-scale environments (VRML), 2007. % * [3] 3-D Depth Reconstruction from a Single Still Image, % * Ashutosh Saxena, Sung H. Chung, Andrew Y. Ng. % * International Journal of Computer Vision (IJCV), Aug 2007. % * [6] Learning Depth from Single Monocular Images, % * Ashutosh Saxena, Sung H. Chung, Andrew Y. Ng. % * In Neural Information Processing Systems (NIPS) 18, 2005. % * % * These articles are available at: % * http://make3d.stanford.edu/publications % * % * We request that you cite the papers [1], [3] and [6] in any of % * your reports that uses this code. % * Further, if you use the code in image3dstiching/ (multiple image version), % * then please cite [2]. % * % * If you use the code in third_party/, then PLEASE CITE and follow the % * LICENSE OF THE CORRESPONDING THIRD PARTY CODE. % * % * Finally, this code is for non-commercial use only. For further % * information and to obtain a copy of the license, see % * % * http://make3d.stanford.edu/publications/code % * % * Also, the software distributed under the License is distributed on an % * "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either % * express or implied. See the License for the specific language governing % * permissions and limitations under the License. % * % */ function [x,status,history] = L1Barrier_wo_Constrain(Para,method,ptol,pmaxi, VERBOSE) %function [x,status,history] = L1Barrier_wo_Constrain(A,b,t_0,method,ptol,pmaxi) % % Fast L1 - norm Solver % % L1 - norm Solver Solves problems of the following form: % % minimize | A*x - b|L1 % % where variable is x and problem data are A and b. % % INPUT % % A : mxn matrix; input data. each column corresponds to each feature % b : m vector; class label % % method : string; search direction method type % 'cg' : conjugate gradients method, 'pcg' % 'pcg' : preconditioned conjugate gradients method % 'exact': exact method (default value) % ptol : scalar; pcg relative tolerance. if empty, use adaptive rule. % pmaxi : scalar: pcg maximum iteration. if empty, use default value (500). % % OUTPUT % % x : n vector; % status : scalar; +1: success, -1: maxiter exceeded % history : % row 1) phi % row 2) norm(gradient of phi) % row 3) cumulative cg iterations % % USAGE EXAMPLE % % [x,status] = l2_logreg(A,b,lambda,'pcg'); % % Written by Kwangmoo Koh % adopted by Min Sun %------------------------------------------------------------ % INITIALIZE %------------------------------------------------------------ global A D p b; % LOG BARRIER METHOD MAX_LOGB_ITER = 100; EPSILON_GAP = 2e-4; MU_t = 100; % for t -- log barrier. Changed ASH %if(isempty(t_0)) t_0 = 1; end t_0 = 1; t = t_0; % NEWTON PARAMETERS MAX_TNT_ITER = 100; % maximum (truncated) Newton iteration ABSTOL = 1e-8; % terminates when the norm of gradient < ABSTOL EPSILON = 1e-7; % terminate when lambdasqr_by_2 < EPSILON StopNorm = 0; % set to 0 using newton decrement % LINE SEARCH PARAMETERS ALPHA = 0.01; % minimum fraction of decrease in norm(gradient) BETA = 0.5; % stepsize decrease factor MAX_LS_ITER = 50; % maximum backtracking line search iteration Eps = -1e-50; % gap for inequality normg_Flag = 1; % if evaluate function set normg_Flag = 1 [m,n] = size(A); % problem size: m examples, n features A2 = A.^2; %if(isempty(pmaxi)) pcgmaxi = 500; else pcgmaxi = pmaxi; end %if(isempty(ptol )) pcgtol = 1e-4; else pcgtol = ptol; end pcgmaxi = 500; pcgtol = 1e-4; % INITIALIZE pobj = Inf; s = inf; pitr = 0 ; pflg = 0 ; prelres = 0; pcgiter = 0; history = []; status = -1; % feasible starting point x = zeros(n,1); dx = zeros(n,1); y = max( abs( A*x-b))-Eps- (-1e-2); dy = zeros(m,1); % check is x y feasible start % if max(A*(x) - (y)-b)< 0 && max(-A*(x) - (y)+b) < 0 % disp('Feasible start'); % end %------------------------------------------------------------ % LOG BARRIER OUTER LOOP %------------------------------------------------------------ %for lbiter = 1:MAX_LOGB_ITER %while true % return; %end for LogBIter = 1:MAX_LOGB_ITER if VERBOSE disp(sprintf('%s %15s %10s %10s %10s %s %s %6s %10s %6s',... 'iter','primal obj','stepsize','norg(g)','lambdasqr','LSiter', 'LSiter_fea''p_flg','p_res','p_itr')); end %------------------------------------------------------------ % MAIN LOOP %------------------------------------------------------------ status = -1; % initalized to -1; % nt_hist = []; Ax_b = A*x - b; Ax_b_y = Ax_b + y; Neg_Ax_b_y = -Ax_b + y; g_1 = (1./Neg_Ax_b_y - 1./Ax_b_y); g_2 = (t - 1./Ax_b_y - 1./Neg_Ax_b_y); gradphi_x = (g_1'*A)'; for ntiter = 0:MAX_TNT_ITER D_1 = (1./Neg_Ax_b_y).^2; D_2 = (1./Ax_b_y).^2; D = 2./(y.^2 + Ax_b.^2 ); g = g_1 +(D_1-D_2).*(1./(D_1+D_2)).*g_2; gradphi_x_eli = (g'*A)'; %------------------------------------------------------------ % CALCULATE NEWTON STEP %------------------------------------------------------------ % switch lower(method) % case 'pcg' % p = 1./(A2'*D); % if (isempty(ptol)) pcgtol = min(0.1,norm(gradphi_x_eli)); end % [dx, pflg, prelres, pitr, presvec] = ... % pcg(@AXfunc,-gradphi_x_eli,pcgtol,pcgmaxi,@Mfunc,[],[]); % % A,D,[],1./(A2'*D)); % if (pitr == 0) pitr = pcgmaxi; end % % case 'cg' % if (isempty(ptol)) pcgtol = min(0.1,norm(gradphi_x_eli)); end % [dx, pflg, prelres, pitr, presvec] = ... % pcg(@AXfunc,-gradphi_x_eli,pcgtol,pcgmaxi,[],[],[]); % % A,D,[],[]); % if (pitr == 0) pitr = pcgmaxi; end % % otherwise % exact method % hessphi_x = A'*sparse(1:m,1:m,D)*A; hessphi_x = (sparse(1:m,1:m, D)*A)' * A; dx = -hessphi_x\gradphi_x_eli; % end dy = (1./(D_1 + D_2)) .*(-g_2 + (D_1 - D_2).*(A*dx)); %pcgiter = pcgiter+pitr; % function value and normg for back tracking line search or stoping critera normg = norm([gradphi_x; g_2]); phi = sum(y) - sum( log( [Ax_b_y; Neg_Ax_b_y]))/t; lambda_sqr = -gradphi_x'*dx - g_2'*dy; %------------------------------------------------------------ % BACKTRACKING LINE SEARCH %------------------------------------------------------------ s = 1; % if false % debug Delta = A*(dx) - (dy); Delta_Negdy = -A*(dx) - (dy); LSiter = 0; while any( (s*Delta - Neg_Ax_b_y) >= Eps) || any(( s*Delta_Negdy - Ax_b_y) >= Eps) s = BETA*s; LSiter = LSiter + 1; end for lsiter = 1:MAX_LS_ITER new_x = x + s*dx; new_y = y + s*dy; Ax_b = A*new_x - b; Ax_b_y = Ax_b + new_y; Neg_Ax_b_y = -Ax_b + new_y; % if normg_Flag g_1 = (1./Neg_Ax_b_y - 1./Ax_b_y); g_2 = (t - 1./Ax_b_y - 1./Neg_Ax_b_y); gradphi_x = (g_1'*A)'; if (norm([gradphi_x; g_2])<=(1-ALPHA*s)*normg) break; end s = BETA*s; % else % % evaluate function value % new_phi = sum(y) - sum( log( [Anew_x_b_y; Neg_Anew_x_b_y]))/t; % if new_phi <= phi +ALPHA*s*[gradphi_x; g_2]'*[dx; dy] break; end % s = BETA*s; % end end % end x = new_x; % dx; % x'; y = new_y; % if VERBOSE % disp(sprintf('%4d %15.6e %10.2e %10.2e %10.2e %4d %3d %6d %10.2e %6d',... % ntiter,phi,s,normg, lambda_sqr/2, lsiter, LSiter, pflg,prelres,pitr)); % nt_hist = [nt_hist [phi; normg; pcgiter]]; % end %------------------------------------------------------------ % STOPPING CRITERION %------------------------------------------------------------ if (lsiter == MAX_LS_ITER) disp('MaxLSIter'); break; end if StopNorm if (normg < ABSTOL) status = 1; %disp('Absolute normg tolerance reached.'); % disp(sprintf('%d/%d',sum(abs((A2'*h)./(2*lambda))<0.5),n)); break; end else if (lambda_sqr/2 <= EPSILON) status = 1; %disp('Absolute Lambda tolerance reached.'); break; end end end if status == -1 disp('Error status -1'); end %-------------- decreasing the gap ------------- gap = m/t; disp(gap); %history=[history [length(nt_hist); gap]]; if gap< EPSILON_GAP break; end t = MU_t*t; end return; %------------------------------------------------------------ % CALL BACK FUNCTIONS FOR PCG %------------------------------------------------------------ function y = AXfunc(x) global A D; y = A'*(D.*(A*x)); function y = Mfunc(x) global p; y = x.*p;