1维：logistic混沌.rar_chaos_matlab_压缩感知 混沌_混沌_混沌压缩感知

function [x,r,g,info] = spgl1( A, b, tau, sigma, x, options ) %SPGL1 Solve basis pursuit, basis pursuit denoise, and LASSO % % [x, r, g, info] = spgl1(A, b, tau, sigma, x0, options) % % --------------------------------------------------------------------- % Solve the basis pursuit denoise (BPDN) problem % % (BPDN) minimize ||x||_1 subj to ||Ax-b||_2 <= sigma, % % or the l1-regularized least-squares problem % % (LASSO) minimize ||Ax-b||_2 subj to ||x||_1 <= tau. % --------------------------------------------------------------------- % % INPUTS % ====== % A is an m-by-n matrix, explicit or an operator. % If A is a function, then it must have the signature % % y = A(x,mode) if mode == 1 then y = A x (y is m-by-1); % if mode == 2 then y = A'x (y is n-by-1). % % b is an m-vector. % tau is a nonnegative scalar; see (LASSO). % sigma if sigma != inf or != [], then spgl1 will launch into a % root-finding mode to find the tau above that solves (BPDN). % In this case, it's STRONGLY recommended that tau = 0. % x0 is an n-vector estimate of the solution (possibly all % zeros). If x0 = [], then SPGL1 determines the length n via % n = length( A'b ) and sets x0 = zeros(n,1). % options is a structure of options from spgSetParms. Any unset options % are set to their default value; set options=[] to use all % default values. % % OUTPUTS % ======= % x is a solution of the problem % r is the residual, r = b - Ax % g is the gradient, g = -A'r % info is a structure with the following information: % .tau final value of tau (see sigma above) % .rNorm two-norm of the optimal residual % .rGap relative duality gap (an optimality measure) % .gNorm Lagrange multiplier of (LASSO) % .stat = 1 found a BPDN solution % = 2 found a BP sol'n; exit based on small gradient % = 3 found a BP sol'n; exit based on small residual % = 4 found a LASSO solution % = 5 error: too many iterations % = 6 error: linesearch failed % = 7 error: found suboptimal BP solution % = 8 error: too many matrix-vector products % .time total solution time (seconds) % .nProdA number of multiplications with A % .nProdAt number of multiplications with A' % % OPTIONS % ======= % Use the options structure to control various aspects of the algorithm: % % options.fid File ID to direct log output % .verbosity 0=quiet, 1=some output, 2=more output. % .iterations Max. number of iterations (default if 10*m). % .bpTol Tolerance for identifying a basis pursuit solution. % .optTol Optimality tolerance (default is 1e-4). % .decTol Larger decTol means more frequent Newton updates. % .subspaceMin 0=no subspace minimization, 1=subspace minimization. % % EXAMPLE % ======= % m = 120; n = 512; k = 20; % m rows, n cols, k nonzeros. % p = randperm(n); x0 = zeros(n,1); x0(p(1:k)) = sign(randn(k,1)); % A = randn(m,n); [Q,R] = qr(A',0); A = Q'; % b = A*x0 + 0.005 * randn(m,1); % opts = spgSetParms('optTol',1e-4); % [x,r,g,info] = spgl1(A, b, 0, 1e-3, [], opts); % Find BP sol'n. % % AUTHORS % ======= % Ewout van den Berg (ewout78@cs.ubc.ca) % Michael P. Friedlander (mpf@cs.ubc.ca) % Scientific Computing Laboratory (SCL) % University of British Columbia, Canada. % % BUGS % ==== % Please send bug reports or comments to % Michael P. Friedlander (mpf@cs.ubc.ca) % Ewout van den Berg (ewout78@cs.ubc.ca) % 15 Apr 07: First version derived from spg.m. % Michael P. Friedlander (mpf@cs.ubc.ca). % Ewout van den Berg (ewout78@cs.ubc.ca). % 17 Apr 07: Added root-finding code. % 18 Apr 07: sigma was being compared to 1/2 r'r, rather than % norm(r), as advertised. Now immediately change sigma to % (1/2)sigma^2, and changed log output accordingly. % 24 Apr 07: Added quadratic root-finding code as an option. % 24 Apr 07: Exit conditions need to guard against small ||r|| % (ie, a BP solution). Added test1,test2,test3 below. % 15 May 07: Trigger to update tau is now based on relative difference % in objective between consecutive iterations. % 15 Jul 07: Added code to allow a limited number of line-search % errors. % 23 Feb 08: Fixed bug in one-norm projection using weights. Thanks % to Xiangrui Meng for reporting this bug. % 26 May 08: The simple call spgl1(A,b) now solves (BPDN) with sigma=0. % spgl1.m % $Id: spgl1.m 1225 2009-01-30 20:36:31Z ewout78 $ % % ---------------------------------------------------------------------- % This file is part of SPGL1 (Spectral Projected-Gradient for L1). % % Copyright (C) 2007 Ewout van den Berg and Michael P. Friedlander, % Department of Computer Science, University of British Columbia, Canada. % All rights reserved. E-mail: <{ewout78,mpf}@cs.ubc.ca>. % % SPGL1 is free software; you can redistribute it and/or modify it % under the terms of the GNU Lesser General Public License as % published by the Free Software Foundation; either version 2.1 of the % License, or (at your option) any later version. % % SPGL1 is distributed in the hope that it will be useful, but WITHOUT % ANY WARRANTY; without even the implied warranty of MERCHANTABILITY % or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General % Public License for more details. % % You should have received a copy of the GNU Lesser General Public % License along with SPGL1; if not, write to the Free Software % Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 % USA % ---------------------------------------------------------------------- REVISION = '$Revision: 1017 $'; DATE = '$Date: 2008-06-16 22:43:07 -0700 (Mon, 16 Jun 2008) $'; REVISION = REVISION(11:end-1); DATE = DATE(35:50); tic; % Start your watches! m = length(b); %---------------------------------------------------------------------- % Check arguments. %---------------------------------------------------------------------- if ~exist('options','var'), options = []; end if ~exist('x','var'), x = []; end if ~exist('sigma','var'), sigma = []; end if ~exist('tau','var'), tau = []; end if nargin < 2 || isempty(b) || isempty(A) error('At least two arguments are required'); elseif isempty(tau) && isempty(sigma) tau = 0; sigma = 0; singleTau = false; elseif isempty(sigma) % && ~isempty(tau) <-- implied singleTau = true; else if isempty(tau) tau = 0; end singleTau = false; end %---------------------------------------------------------------------- % Grab input options and set defaults where needed. %---------------------------------------------------------------------- defaultopts = spgSetParms(... 'fid' , 1 , ... % File ID for output 'verbosity' , 2 , ... % Verbosity level 'iterations' , 10*m , ... % Max number of iterations 'nPrevVals' , 3 , ... % Number previous func values for linesearch 'bpTol' , 1e-06 , ... % Tolerance for basis pursuit solution 'optTol' , 1e-04 , ... % Optimality tolerance 'decTol' , 1e-04 , ... % Req'd rel. change in primal obj. for Newton 'stepMin' , 1e-16 , ... % Minimum spectral step 'stepMax' , 1e+05 , ... % Maximum spectral step 'rootMethod' , 2 , ... % Root finding method: 2=quad,1=linear (not used). 'activeSetIt', Inf , ... % Exit with EXIT_ACTIVE_SET if nnz same for # its. 'subspaceMin', 0 , ... % Use subspace minimization 'iscomplex' , NaN , ... % Flag set to indicate complex problem 'maxMatvec' , Inf , ... % Maximum matrix-vector multiplies allowed 'weights' , 1 , ... % Weights W in ||Wx||_1 'project'