Matrix Completion.m
From Wikimization
Matlab demonstration of Cai, Candès, & Shen
A Singular Value Thresholding Algorithm for Matrix Completion, 2008
This Matlab code below is working, complete, debugged, and corresponds to the paper cited above. It is capable of solving smaller matrix completion problems quite well, as the demonstration program shows.
Singular Value Thresholding (SVT) code for solving larger completion problems is presently under development. It is written in C and Fortran and can be compiled in Matlab. Read more...
<pre> % Written by: Emmanuel Candes % Email: emmanuel@acm.caltech.edu % Created: October 2008
%% Setup a matrix randn('state',2008); rand('state',2008);
n = 1000; r = 10; M = randn(n,r)*randn(r,n);
df = r*(2*n-r); oversampling = 5; m = 5*df;
Omega = randsample(n^2,m); data = M(Omega);
%% Set parameters and solve
p = m/n^2; delta = 1.2/p; maxiter = 500; tol = 1e-4;
%% Approximate minimum nuclear norm solution by SVT algorithm
tic [U,S,V,numiter] = SVT(n,Omega,data,delta,maxiter,tol); toc
%% Show results
X = U*S*V';
disp(sprintf('The relative error on Omega is: %d ', norm(data-X(Omega))/norm(data))) disp(sprintf('The relative recovery error is: %d ', norm(M-X,'fro')/norm(M,'fro'))) disp(sprintf('The relative recovery in the spectral norm is: %d ', norm(M-X)/norm(M))) </pre>
SVT()
<pre> function [U,Sigma,V,numiter] = SVT(n,Omega,b,delta,maxiter,tol) % % Finds the minimum of tau ||X||_* + .5 || X ||_F^2 % subject to P_Omega(X) == P_Omega(M) % using linear Bregman iterations % % Usage: [U,S,V,numiter] = SVT(n,Omega,b,delta,maxiter,opts) % % Inputs: % n - size of the matrix X assumed n by n % Omega - set of observed entries % b - data vector of the form M(Omega) % delta - step size % maxiter - maximum number of iterations % % Outputs: matrix X stored in SVD format X = U*diag(S)*V' % U - nxr left singular vectors % S - rx1 singular values % V - nxr right singular vectors % numiter - number of iterations to achieve convergence
% Description: % Reference: % % Cai, Candes and Shen % A singular value thresholding algorithm for matrix completion % Submitted for publication, October 2008. % % Written by: Emmanuel Candes % Email: emmanuel@acm.caltech.edu % Created: October 2008
m = length(Omega); [temp,indx] = sort(Omega); tau = 5*n; incre = 5;
[i, j] = ind2sub([n,n], Omega); ProjM = sparse(i,j,b,n,n,m);
normProjM = normest(ProjM); k0 = ceil(tau/(delta*normProjM));
normb = norm(b);
y = k0*delta*b; Y = sparse(i,j,y,n,n,m); r = 0;
fprintf('\nIteration: '); for k = 1:maxiter
fprintf('\b\b\b%3d',k); s = r + 1; OK = 0; while ~OK [U,Sigma,V] = lansvd(Y,s,'L'); OK = Sigma(s,s) <= tau; s = s + incre; end sigma = diag(Sigma); r = sum(sigma > tau); U = U(:,1:r); V = V(:,1:r); sigma = sigma(1:r) - tau; Sigma = diag(sigma); A = U*diag(sigma)*V'; x = A(Omega); if norm(x-b)/normb < tol break end y = y + delta*(b-x); updateSparse(Y,y,indx);
end
fprintf('\n'); numiter = k; </pre>
subroutines
lansvd()
<pre> function [U,S,V,bnd,j] = lansvd(varargin)
%LANSVD Compute a few singular values and singular vectors. % LANSVD computes singular triplets (u,v,sigma) such that % A*u = sigma*v and A'*v = sigma*u. Only a few singular values % and singular vectors are computed using the Lanczos % bidiagonalization algorithm with partial reorthogonalization (BPRO). % % S = LANSVD(A) % S = LANSVD('Afun','Atransfun',M,N) % % The first input argument is either a matrix or a % string containing the name of an M-file which applies a linear % operator to the columns of a given matrix. In the latter case, % the second input must be the name of an M-file which applies the % transpose of the same operator to the columns of a given matrix, % and the third and fourth arguments must be M and N, the dimensions % of the problem. % % [U,S,V] = LANSVD(A,K,'L',...) computes the K largest singular values. % % [U,S,V] = LANSVD(A,K,'S',...) computes the K smallest singular values. % % The full calling sequence is % % [U,S,V] = LANSVD(A,K,SIGMA,OPTIONS) % [U,S,V] = LANSVD('Afun','Atransfun',M,N,K,SIGMA,OPTIONS) % % where K is the number of singular values desired and % SIGMA is 'L' or 'S'. % % The OPTIONS structure specifies certain parameters in the algorithm. % Field name Parameter Default % % OPTIONS.tol Convergence tolerance 16*eps % OPTIONS.lanmax Dimension of the Lanczos basis. % OPTIONS.p0 Starting vector for the Lanczos rand(n,1)-0.5 % iteration. % OPTIONS.delta Level of orthogonality among the sqrt(eps/K) % Lanczos vectors. % OPTIONS.eta Level of orthogonality after 10*eps^(3/4) % reorthogonalization. % OPTIONS.cgs reorthogonalization method used 0 % '0' : iterated modified Gram-Schmidt % '1' : iterated classical Gram-Schmidt % OPTIONS.elr If equal to 1 then extended local 1 % reorthogonalization is enforced. % % See also LANBPRO, SVDS, SVD
% References: % R.M. Larsen, Ph.D. Thesis, Aarhus University, 1998. % % B. N. Parlett, ``The Symmetric Eigenvalue Problem, % Prentice-Hall, Englewood Cliffs, NJ, 1980. % % H. D. Simon, ``The Lanczos algorithm with partial reorthogonalization, % Math. Comp. 42 (1984), no. 165, 115--142.
% Rasmus Munk Larsen, DAIMI, 1998
%%%%%%%%%%%%%%%%%%%%% Parse and check input arguments. %%%%%%%%%%%%%%%%%%%%%%
if nargin<1 | length(varargin)<1
error('Not enough input arguments.');
end
A = varargin{1}; if ~isstr(A)
if ~isreal(A) error('A must be real') end [m n] = size(A); if length(varargin) < 2, k=min(min(m,n),6); else k=varargin{2}; end if length(varargin) < 3, sigma = 'L'; else sigma=varargin{3}; end if length(varargin) < 4, options = []; else options=varargin{4}; end
else
if length(varargin)<4 error('Not enough input arguments.'); end Atrans = varargin{2}; if ~isstr(Atrans) error('Atransfunc must be the name of a function') end m = varargin{3}; n = varargin{4}; if length(varargin) < 5, k=min(min(m,n),6); else k=varargin{5}; end if length(varargin) < 6, sigma = 'L'; else sigma=varargin{6}; end if length(varargin) < 7, options = []; else options=varargin{7}; end
end
if ~isnumeric(n) | real(abs(fix(n))) ~= n | ~isnumeric(m) | ...
real(abs(fix(m))) ~= m | ~isnumeric(k) | real(abs(fix(k))) ~= k error('M, N and K must be positive integers.')
end
% Quick return for min(m,n) equal to 0 or 1 or for zero A.
if min(n,m) < 1 | k<1
if nargout<3 U = zeros(k,1); else U = eye(m,k); S = zeros(k,k); V = eye(n,k); bnd = zeros(k,1); end return
elseif min(n,m) == 1 & k>0
if isstr(A) % Extract the single column or row of A if n==1 A = feval(A,1); else A = feval(Atrans,1)'; end end if nargout==1 U = norm(A); else [U,S,V] = svd(full(A)); bnd = 0; end return
end
% A is the matrix of all zeros (not detectable if A is defined by an m-file) if isnumeric(A)
if nnz(A)==0 if nargout<3 U = zeros(k,1); else U = eye(m,k); S = zeros(k,k); V = eye(n,k); bnd = zeros(k,1); end return end
end
lanmax = min(m,n); tol = 16*eps; p = rand(m,1)-0.5; % Parse options struct if isstruct(options)
c = fieldnames(options); for i=1:length(c) if any(strcmp(c(i),'p0')), p = getfield(options,'p0'); p=p(:); end if any(strcmp(c(i),'tol')), tol = getfield(options,'tol'); end if any(strcmp(c(i),'lanmax')), lanmax = getfield(options,'lanmax'); end end
end
% Protect against absurd options. tol = max(tol,eps); lanmax = min(lanmax,min(m,n)); if size(p,1)~=m
error('p0 must be a vector of length m')
end
lanmax = min(lanmax,min(m,n)); if k>lanmax
error('K must satisfy K <= LANMAX <= MIN(M,N).');
end
%%%%%%%%%%%%%%%%%%%%% Here begins the computation %%%%%%%%%%%%%%%%%%%%%%
if strcmp(sigma,'S')
if isstr(A) error('Shift-and-invert works only when the matrix A is given explicitly.'); else % Prepare for shift-and-invert Lanczos. if issparse(A) pmmd = colmmd(A); A.A = A(:,pmmd); else A.A = A; end if m>=n if issparse(A.A)
A.R = qr(A.A,0); A.Rt = A.R'; p = A.Rt\(A.A'*p); % project starting vector on span(Q1)
else
[A.Q,A.R] = qr(A.A,0); A.Rt = A.R'; p = A.Q'*p; % project starting vector on span(Q1)
end else error('Sorry, shift-and-invert for m<n not implemented yet!') A.R = qr(A.A',0); A.Rt = A.R'; end condR = condest(A.R); if condR > 1/eps error(['A is rank deficient or too ill-conditioned to do shift-and-' ...
' invert.'])
end end
end
ksave = k; neig = 0; nrestart=-1; j = min(k+max(8,k)+1,lanmax); U = []; V = []; B = []; anorm = []; work = zeros(2,2);
while neig < k
%%%%%%%%%%%%%%%%%%%%% Compute Lanczos bidiagonalization %%%%%%%%%%%%%%%%% if ~isstr(A) [U,B,V,p,ierr,w] = lanbpro(A,j,p,options,U,B,V,anorm); else [U,B,V,p,ierr,w] = lanbpro(A,Atrans,m,n,j,p,options,U,B,V,anorm); end work= work + w; if ierr<0 % Invariant subspace of dimension -ierr found. j = -ierr; end
%%%%%%%%%%%%%%%%%% Compute singular values and error bounds %%%%%%%%%%%%%%%% % Analyze B resnrm = norm(p); % We might as well use the extra info. in p. % S = svd(full([B;[zeros(1,j-1),resnrm]]),0); % [P,S,Q] = svd(full([B;[zeros(1,j-1),resnrm]]),0); % S = diag(S); % bot = min(abs([P(end,1:j);Q(end,1:j)]))';
[S,bot] = bdsqr(diag(B),[diag(B,-1); resnrm]); % Use Largest Ritz value to estimate ||A||_2. This might save some % reorth. in case of restart. anorm=S(1); % Set simple error bounds bnd = resnrm*abs(bot); % Examine gap structure and refine error bounds bnd = refinebounds(S.^2,bnd,n*eps*anorm);
%%%%%%%%%%%%%%%%%%% Check convergence criterion %%%%%%%%%%%%%%%%%%%% i=1; neig = 0; while i<=min(j,k) if (bnd(i) <= tol*abs(S(i))) neig = neig + 1; i = i+1; else i = min(j,k)+1; end end
%%%%%%%%%% Check whether to stop or to extend the Krylov basis? %%%%%%%%%% if ierr<0 % Invariant subspace found if j<k warning(['Invariant subspace of dimension ',num2str(j-1),' found.']) end j = j-1; break; end if j>=lanmax % Maximal dimension of Krylov subspace reached. Bail out if j>=min(m,n) neig = ksave; break; end if neig<ksave warning(['Maximum dimension of Krylov subspace exceeded prior',...
' to convergence.']);
end break; end % Increase dimension of Krylov subspace if neig>0 % increase j by approx. half the average number of steps pr. converged % singular value (j/neig) times the number of remaining ones (k-neig). j = j + min(100,max(2,0.5*(k-neig)*j/(neig+1))); else % As long a very few singular values have converged, increase j rapidly. % j = j + ceil(min(100,max(8,2^nrestart*k))); j = max(1.5*j,j+10); end j = ceil(min(j+1,lanmax)); nrestart = nrestart + 1;
end
%%%%%%%%%%%%%%%% Lanczos converged (or failed). Prepare output %%%%%%%%%%%%%%%
k = min(ksave,j);
if nargout>2
j = size(B,2); % Compute singular vectors [P,S,Q] = svd(full([B;[zeros(1,j-1),resnrm]]),0); S = diag(S); if size(Q,2)~=k Q = Q(:,1:k); P = P(:,1:k); end % Compute and normalize Ritz vectors (overwrites U and V to save memory). if resnrm~=0 U = U*P(1:j,:) + (p/resnrm)*P(j+1,:); else U = U*P(1:j,:); end V = V*Q; for i=1:k nq = norm(V(:,i)); if isfinite(nq) & nq~=0 & nq~=1 V(:,i) = V(:,i)/nq; end nq = norm(U(:,i)); if isfinite(nq) & nq~=0 & nq~=1 U(:,i) = U(:,i)/nq; end end
end
% Pick out desired part the spectrum S = S(1:k); bnd = bnd(1:k);
if strcmp(sigma,'S')
[S,p] = sort(-1./S); S = -S; bnd = bnd(p); if nargout>2 if issparse(A.A) U = A.A*(A.R\U(:,p)); V(pmmd,:) = V(:,p); else U = A.Q(:,1:min(m,n))*U(:,p); V = V(:,p); end end
end
if nargout<3
U = S; S = B; % Undocumented feature - for checking B.
else
S = diag(S);
end </pre>
bdsqr()
<pre> function [sigma,bnd] = bdsqr(alpha,beta)
% BDSQR: Compute the singular values and bottom element of % the left singular vectors of a (k+1) x k lower bidiagonal % matrix with diagonal alpha(1:k) and lower bidiagonal beta(1:k), % where length(alpha) = length(beta) = k. % % [sigma,bnd] = bdsqr(alpha,beta) % % Input parameters: % alpha(1:k) : Diagonal elements. % beta(1:k) : Sub-diagonal elements. % Output parameters: % sigma(1:k) : Computed eigenvalues. % bnd(1:k) : Bottom elements in left singular vectors.
% Below is a very slow replacement for the BDSQR MEX-file.
%warning('PROPACK:NotUsingMex','Using slow matlab code for bdsqr.') k = length(alpha); if min(size(alpha)') ~= 1 | min(size(beta)') ~= 1
error('alpha and beta must be vectors')
elseif length(beta) ~= k
error('alpha and beta must have the same lenght')
end B = spdiags([alpha(:),beta(:)],[0,-1],k+1,k); [U,S,V] = svd(full(B),0); sigma = diag(S); bnd = U(end,1:k)'; </pre>
compute_int()
<pre> function int = compute_int(mu,j,delta,eta,LL,strategy,extra) %COMPUTE_INT: Determine which Lanczos vectors to reorthogonalize against. % % int = compute_int(mu,eta,LL,strategy,extra)) % % Strategy 0: Orthogonalize vectors v_{i-r-extra},...,v_{i},...v_{i+s+extra} % with nu>eta, where v_{i} are the vectors with mu>delta. % Strategy 1: Orthogonalize all vectors v_{r-extra},...,v_{s+extra} where % v_{r} is the first and v_{s} the last Lanczos vector with % mu > eta. % Strategy 2: Orthogonalize all vectors with mu > eta. % % Notice: The first LL vectors are excluded since the new Lanczos % vector is already orthogonalized against them in the main iteration.
% Rasmus Munk Larsen, DAIMI, 1998.
if (delta<eta)
error('DELTA should satisfy DELTA >= ETA.')
end switch strategy
case 0 I0 = find(abs(mu(1:j))>=delta); if length(I0)==0 [mm,I0] = max(abs(mu(1:j))); end int = zeros(j,1); for i = 1:length(I0) for r=I0(i):-1:1
if abs(mu(r))<eta | int(r)==1 break; else int(r) = 1; end
end int(max(1,r-extra+1):r) = 1; for s=I0(i)+1:j
if abs(mu(s))<eta | int(s)==1 break; else int(s) = 1; end
end int(s:min(j,s+extra-1)) = 1; end if LL>0 int(1:LL) = 0; end int = find(int); case 1 int=find(abs(mu(1:j))>eta); int = max(LL+1,min(int)-extra):min(max(int)+extra,j); case 2 int=find(abs(mu(1:j))>=eta);
end int = int(:); </pre>
lanbpro()
<pre> function [U,B_k,V,p,ierr,work] = lanbpro(varargin)
%LANBPRO Lanczos bidiagonalization with partial reorthogonalization. % LANBPRO computes the Lanczos bidiagonalization of a real % matrix using the with partial reorthogonalization. % % [U_k,B_k,V_k,R,ierr,work] = LANBPRO(A,K,R0,OPTIONS,U_old,B_old,V_old) % [U_k,B_k,V_k,R,ierr,work] = LANBPRO('Afun','Atransfun',M,N,K,R0, ... % OPTIONS,U_old,B_old,V_old) % % Computes K steps of the Lanczos bidiagonalization algorithm with partial % reorthogonalization (BPRO) with M-by-1 starting vector R0, producing a % lower bidiagonal K-by-K matrix B_k, an N-by-K matrix V_k, an M-by-K % matrix U_k and an M-by-1 vector R such that % A*V_k = U_k*B_k + R % Partial reorthogonalization is used to keep the columns of V_K and U_k % semiorthogonal: % MAX(DIAG((EYE(K) - V_K'*V_K))) <= OPTIONS.delta % and % MAX(DIAG((EYE(K) - U_K'*U_K))) <= OPTIONS.delta. % % B_k = LANBPRO(...) returns the bidiagonal matrix only. % % The first input argument is either a real matrix, or a string % containing the name of an M-file which applies a linear operator % to the columns of a given matrix. In the latter case, the second % input must be the name of an M-file which applies the transpose of % the same linear operator to the columns of a given matrix, % and the third and fourth arguments must be M and N, the dimensions % of then problem. % % The OPTIONS structure is used to control the reorthogonalization: % OPTIONS.delta: Desired level of orthogonality % (default = sqrt(eps/K)). % OPTIONS.eta : Level of orthogonality after reorthogonalization % (default = eps^(3/4)/sqrt(K)). % OPTIONS.cgs : Flag for switching between different reorthogonalization % algorithms: % 0 = iterated modified Gram-Schmidt (default) % 1 = iterated classical Gram-Schmidt % OPTIONS.elr : If OPTIONS.elr = 1 (default) then extended local % reorthogonalization is enforced. % OPTIONS.onesided % : If OPTIONS.onesided = 0 (default) then both the left % (U) and right (V) Lanczos vectors are kept % semiorthogonal. % OPTIONS.onesided = 1 then only the columns of U are % are reorthogonalized. % OPTIONS.onesided = -1 then only the columns of V are % are reorthogonalized. % OPTIONS.waitbar % : The progress of the algorithm is display graphically. % % If both R0, U_old, B_old, and V_old are provided, they must % contain a partial Lanczos bidiagonalization of A on the form % % A V_old = U_old B_old + R0 . % % In this case the factorization is extended to dimension K x K by % continuing the Lanczos bidiagonalization algorithm with R0 as a % starting vector. % % The output array work contains information about the work used in % reorthogonalizing the u- and v-vectors. % work = [ RU PU ] % [ RV PV ] % where % RU = Number of reorthogonalizations of U. % PU = Number of inner products used in reorthogonalizing U. % RV = Number of reorthogonalizations of V. % PV = Number of inner products used in reorthogonalizing V.
% References: % R.M. Larsen, Ph.D. Thesis, Aarhus University, 1998. % % G. H. Golub & C. F. Van Loan, "Matrix Computations", % 3. Ed., Johns Hopkins, 1996. Section 9.3.4. % % B. N. Parlett, ``The Symmetric Eigenvalue Problem, % Prentice-Hall, Englewood Cliffs, NJ, 1980. % % H. D. Simon, ``The Lanczos algorithm with partial reorthogonalization, % Math. Comp. 42 (1984), no. 165, 115--142. %
% Rasmus Munk Larsen, DAIMI, 1998.
% Check input arguments.
global LANBPRO_TRUTH LANBPRO_TRUTH=0;
if LANBPRO_TRUTH==1
global MU NU MUTRUE NUTRUE global MU_AFTER NU_AFTER MUTRUE_AFTER NUTRUE_AFTER
end
if nargin<1 | length(varargin)<2
error('Not enough input arguments.');
end narg=length(varargin);
A = varargin{1}; if isnumeric(A) | isstruct(A)
if isnumeric(A) if ~isreal(A) error('A must be real') end [m n] = size(A); elseif isstruct(A) [m n] = size(A.R); end k=varargin{2}; if narg >= 3 & ~isempty(varargin{3}); p = varargin{3}; else p = rand(m,1)-0.5; end if narg < 4, options = []; else options=varargin{4}; end if narg > 4 if narg<7 error('All or none of U_old, B_old and V_old must be provided.') else U = varargin{5}; B_k = varargin{6}; V = varargin{7}; end else U = []; B_k = []; V = []; end if narg > 7, anorm=varargin{8}; else anorm = []; end
else
if narg<5 error('Not enough input arguments.'); end Atrans = varargin{2}; if ~isstr(Atrans) error('Afunc and Atransfunc must be names of m-files') end m = varargin{3}; n = varargin{4}; if ~isreal(n) | abs(fix(n)) ~= n | ~isreal(m) | abs(fix(m)) ~= m error('M and N must be positive integers.') end k=varargin{5}; if narg < 6, p = rand(m,1)-0.5; else p=varargin{6}; end if narg < 7, options = []; else options=varargin{7}; end if narg > 7 if narg < 10 error('All or none of U_old, B_old and V_old must be provided.') else U = varargin{8}; B_k = varargin{9}; V = varargin{10}; end else U = []; B_k = []; V=[]; end if narg > 10, anorm=varargin{11}; else anorm = []; end
end
% Quick return for min(m,n) equal to 0 or 1. if min(m,n) == 0
U = []; B_k = []; V = []; p = []; ierr = 0; work = zeros(2,2); return
elseif min(m,n) == 1
if isnumeric(A) U = 1; B_k = A; V = 1; p = 0; ierr = 0; work = zeros(2,2); else U = 1; B_k = feval(A,1); V = 1; p = 0; ierr = 0; work = zeros(2,2); end if nargout<3 U = B_k; end return
end
% Set options. %m2 = 3/2*(sqrt(m)+1); %n2 = 3/2*(sqrt(n)+1); m2 = 3/2; n2 = 3/2; delta = sqrt(eps/k); % Desired level of orthogonality. eta = eps^(3/4)/sqrt(k); % Level of orth. after reorthogonalization. cgs = 0; % Flag for switching between iterated MGS and CGS. elr = 2; % Flag for switching extended local
% reorthogonalization on and off.
gamma = 1/sqrt(2); % Tolerance for iterated Gram-Schmidt. onesided = 0; t = 0; waitb = 0;
% Parse options struct if ~isempty(options) & isstruct(options)
c = fieldnames(options); for i=1:length(c) if strmatch(c(i),'delta'), delta = getfield(options,'delta'); end if strmatch(c(i),'eta'), eta = getfield(options,'eta'); end if strmatch(c(i),'cgs'), cgs = getfield(options,'cgs'); end if strmatch(c(i),'elr'), elr = getfield(options,'elr'); end if strmatch(c(i),'gamma'), gamma = getfield(options,'gamma'); end if strmatch(c(i),'onesided'), onesided = getfield(options,'onesided'); end if strmatch(c(i),'waitbar'), waitb=1; end end
end
if waitb
waitbarh = waitbar(0,'Lanczos bidiagonalization in progress...');
end
if isempty(anorm)
anorm = []; est_anorm=1;
else
est_anorm=0;
end
% Conservative statistical estimate on the size of round-off terms. % Notice that {\bf u} == eps/2. FUDGE = 1.01; % Fudge factor for ||A||_2 estimate.
npu = 0; npv = 0; ierr = 0; p = p(:); % Prepare for Lanczos iteration. if isempty(U)
V = zeros(n,k); U = zeros(m,k); beta = zeros(k+1,1); alpha = zeros(k,1); beta(1) = norm(p); % Initialize MU/NU-recurrences for monitoring loss of orthogonality. nu = zeros(k,1); mu = zeros(k+1,1); mu(1)=1; nu(1)=1; numax = zeros(k,1); mumax = zeros(k,1); force_reorth = 0; nreorthu = 0; nreorthv = 0; j0 = 1;
else
j = size(U,2); % Size of existing factorization % Allocate space for Lanczos vectors U = [U, zeros(m,k-j)]; V = [V, zeros(n,k-j)]; alpha = zeros(k+1,1); beta = zeros(k+1,1); alpha(1:j) = diag(B_k); if j>1 beta(2:j) = diag(B_k,-1); end beta(j+1) = norm(p); % Reorthogonalize p. if j<k & beta(j+1)*delta < anorm*eps, fro = 1; ierr = j; end int = [1:j]'; [p,beta(j+1),rr] = reorth(U,p,beta(j+1),int,gamma,cgs); npu = rr*j; nreorthu = 1; force_reorth= 1;
% Compute Gerscgorin bound on ||B_k||_2 if est_anorm anorm = FUDGE*sqrt(norm(B_k'*B_k,1)); end mu = m2*eps*ones(k+1,1); nu = zeros(k,1); numax = zeros(k,1); mumax = zeros(k,1); force_reorth = 1; nreorthu = 0; nreorthv = 0; j0 = j+1;
end
if isnumeric(A)
At = A';
end
if delta==0
fro = 1; % The user has requested full reorthogonalization.
else
fro = 0;
end
if LANBPRO_TRUTH==1
MUTRUE = zeros(k,k); NUTRUE = zeros(k-1,k-1); MU = zeros(k,k); NU = zeros(k-1,k-1); MUTRUE_AFTER = zeros(k,k); NUTRUE_AFTER = zeros(k-1,k-1); MU_AFTER = zeros(k,k); NU_AFTER = zeros(k-1,k-1);
end
% Perform Lanczos bidiagonalization with partial reorthogonalization. for j=j0:k
if waitb waitbar(j/k,waitbarh) end
if beta(j) ~= 0 U(:,j) = p/beta(j); else U(:,j) = p; end
% Replace norm estimate with largest Ritz value. if j==6 B = [[diag(alpha(1:j-1))+diag(beta(2:j-1),-1)]; ... [zeros(1,j-2),beta(j)]]; anorm = FUDGE*norm(B); est_anorm = 0; end %%%%%%%%%% Lanczos step to generate v_j. %%%%%%%%%%%%% if j==1 if isnumeric(A) r = At*U(:,1); elseif isstruct(A) r = A.R\U(:,1); else r = feval(Atrans,U(:,1)); end alpha(1) = norm(r); if est_anorm anorm = FUDGE*alpha(1); end else if isnumeric(A) r = At*U(:,j) - beta(j)*V(:,j-1); elseif isstruct(A) r = A.R\U(:,j) - beta(j)*V(:,j-1); else r = feval(Atrans,U(:,j)) - beta(j)*V(:,j-1); end alpha(j) = norm(r);
% Extended local reorthogonalization if alpha(j)<gamma*beta(j) & elr & ~fro normold = alpha(j); stop = 0; while ~stop
t = V(:,j-1)'*r; r = r - V(:,j-1)*t; alpha(j) = norm(r); if beta(j) ~= 0 beta(j) = beta(j) + t; end if alpha(j)>=gamma*normold stop = 1; else normold = alpha(j); end
end end
if est_anorm if j==2
anorm = max(anorm,FUDGE*sqrt(alpha(1)^2+beta(2)^2+alpha(2)*beta(2)));
else
anorm = max(anorm,FUDGE*sqrt(alpha(j-1)^2+beta(j)^2+alpha(j-1)* ... beta(j-1) + alpha(j)*beta(j)));
end end if ~fro & alpha(j) ~= 0 % Update estimates of the level of orthogonality for the % columns 1 through j-1 in V. nu = update_nu(nu,mu,j,alpha,beta,anorm); numax(j) = max(abs(nu(1:j-1))); end
if j>1 & LANBPRO_TRUTH NU(1:j-1,j-1) = nu(1:j-1); NUTRUE(1:j-1,j-1) = V(:,1:j-1)'*r/alpha(j); end if elr>0 nu(j-1) = n2*eps; end % IF level of orthogonality is worse than delta THEN % Reorthogonalize v_j against some previous v_i's, 0<=i<j. if onesided~=-1 & ( fro | numax(j) > delta | force_reorth ) & alpha(j)~=0 % Decide which vectors to orthogonalize against: if fro | eta==0
int = [1:j-1]';
elseif force_reorth==0
int = compute_int(nu,j-1,delta,eta,0,0,0);
end % Else use int from last reorth. to avoid spillover from mu_{j-1} % to nu_j. % Reorthogonalize v_j [r,alpha(j),rr] = reorth(V,r,alpha(j),int,gamma,cgs); npv = npv + rr*length(int); % number of inner products. nu(int) = n2*eps; % Reset nu for orthogonalized vectors.
% If necessary force reorthogonalization of u_{j+1} % to avoid spillover if force_reorth==0
force_reorth = 1;
else
force_reorth = 0;
end nreorthv = nreorthv + 1; end end
% Check for convergence or failure to maintain semiorthogonality if alpha(j) < max(n,m)*anorm*eps & j<k, % If alpha is "small" we deflate by setting it % to 0 and attempt to restart with a basis for a new % invariant subspace by replacing r with a random starting vector: %j %disp('restarting, alpha = 0') alpha(j) = 0; bailout = 1; for attempt=1:3 r = rand(m,1)-0.5; if isnumeric(A)
r = At*r;
elseif isstruct(A)
r = A.R\r;
else
r = feval(Atrans,r);
end nrm=sqrt(r'*r); % not necessary to compute the norm accurately here. int = [1:j-1]'; [r,nrmnew,rr] = reorth(V,r,nrm,int,gamma,cgs); npv = npv + rr*length(int(:)); nreorthv = nreorthv + 1; nu(int) = n2*eps; if nrmnew > 0
% A vector numerically orthogonal to span(Q_k(:,1:j)) was found. % Continue iteration. bailout=0; break;
end end if bailout j = j-1; ierr = -j; break; else r=r/nrmnew; % Continue with new normalized r as starting vector. force_reorth = 1; if delta>0
fro = 0; % Turn off full reorthogonalization.
end end elseif j<k & ~fro & anorm*eps > delta*alpha(j)
% fro = 1;
ierr = j; end
if j>1 & LANBPRO_TRUTH NU_AFTER(1:j-1,j-1) = nu(1:j-1); NUTRUE_AFTER(1:j-1,j-1) = V(:,1:j-1)'*r/alpha(j); end
if alpha(j) ~= 0 V(:,j) = r/alpha(j); else V(:,j) = r; end
%%%%%%%%%% Lanczos step to generate u_{j+1}. %%%%%%%%%%%%% if waitb waitbar((2*j+1)/(2*k),waitbarh) end if isnumeric(A) p = A*V(:,j) - alpha(j)*U(:,j); elseif isstruct(A) p = A.Rt\V(:,j) - alpha(j)*U(:,j); else p = feval(A,V(:,j)) - alpha(j)*U(:,j); end beta(j+1) = norm(p); % Extended local reorthogonalization if beta(j+1)<gamma*alpha(j) & elr & ~fro normold = beta(j+1); stop = 0; while ~stop t = U(:,j)'*p; p = p - U(:,j)*t; beta(j+1) = norm(p); if alpha(j) ~= 0
alpha(j) = alpha(j) + t;
end if beta(j+1) >= gamma*normold
stop = 1;
else
normold = beta(j+1);
end end end
if est_anorm % We should update estimate of ||A|| before updating mu - especially % important in the first step for problems with large norm since alpha(1) % may be a severe underestimate! if j==1 anorm = max(anorm,FUDGE*pythag(alpha(1),beta(2))); else anorm = max(anorm,FUDGE*sqrt(alpha(j)^2+beta(j+1)^2 + alpha(j)*beta(j))); end end if ~fro & beta(j+1) ~= 0 % Update estimates of the level of orthogonality for the columns of V. mu = update_mu(mu,nu,j,alpha,beta,anorm); mumax(j) = max(abs(mu(1:j))); end
if LANBPRO_TRUTH==1 MU(1:j,j) = mu(1:j); MUTRUE(1:j,j) = U(:,1:j)'*p/beta(j+1); end if elr>0 mu(j) = m2*eps; end % IF level of orthogonality is worse than delta THEN % Reorthogonalize u_{j+1} against some previous u_i's, 0<=i<=j. if onesided~=1 & (fro | mumax(j) > delta | force_reorth) & beta(j+1)~=0 % Decide which vectors to orthogonalize against. if fro | eta==0 int = [1:j]'; elseif force_reorth==0 int = compute_int(mu,j,delta,eta,0,0,0); else int = [int; max(int)+1]; end % Else use int from last reorth. to avoid spillover from nu to mu.
% if onesided~=0 % fprintf('i = %i, nr = %i, fro = %i\n',j,size(int(:),1),fro) % end
% Reorthogonalize u_{j+1} [p,beta(j+1),rr] = reorth(U,p,beta(j+1),int,gamma,cgs); npu = npu + rr*length(int); nreorthu = nreorthu + 1;
% Reset mu to epsilon. mu(int) = m2*eps; if force_reorth==0 force_reorth = 1; % Force reorthogonalization of v_{j+1}. else force_reorth = 0; end end % Check for convergence or failure to maintain semiorthogonality if beta(j+1) < max(m,n)*anorm*eps & j<k, % If beta is "small" we deflate by setting it % to 0 and attempt to restart with a basis for a new % invariant subspace by replacing p with a random starting vector: %j %disp('restarting, beta = 0') beta(j+1) = 0; bailout = 1; for attempt=1:3 p = rand(n,1)-0.5; if isnumeric(A)
p = A*p;
elseif isstruct(A)
p = A.Rt\p;
else
p = feval(A,p);
end nrm=sqrt(p'*p); % not necessary to compute the norm accurately here. int = [1:j]'; [p,nrmnew,rr] = reorth(U,p,nrm,int,gamma,cgs); npu = npu + rr*length(int(:)); nreorthu = nreorthu + 1; mu(int) = m2*eps; if nrmnew > 0
% A vector numerically orthogonal to span(Q_k(:,1:j)) was found. % Continue iteration. bailout=0; break;
end end if bailout ierr = -j; break; else p=p/nrmnew; % Continue with new normalized p as starting vector. force_reorth = 1; if delta>0
fro = 0; % Turn off full reorthogonalization.
end end elseif j<k & ~fro & anorm*eps > delta*beta(j+1)
% fro = 1;
ierr = j; end if LANBPRO_TRUTH==1 MU_AFTER(1:j,j) = mu(1:j); MUTRUE_AFTER(1:j,j) = U(:,1:j)'*p/beta(j+1); end
end if waitb
close(waitbarh)
end
if j<k
k = j;
end
B_k = spdiags([alpha(1:k) [beta(2:k);0]],[0 -1],k,k); if nargout==1
U = B_k;
elseif k~=size(U,2) | k~=size(V,2)
U = U(:,1:k); V = V(:,1:k);
end if nargout>5
work = [[nreorthu,npu];[nreorthv,npv]];
end
function mu = update_mu(muold,nu,j,alpha,beta,anorm)
% UPDATE_MU: Update the mu-recurrence for the u-vectors. % % mu_new = update_mu(mu,nu,j,alpha,beta,anorm)
% Rasmus Munk Larsen, DAIMI, 1998.
binv = 1/beta(j+1); mu = muold; eps1 = 100*eps/2; if j==1
T = eps1*(pythag(alpha(1),beta(2)) + pythag(alpha(1),beta(1))); T = T + eps1*anorm; mu(1) = T / beta(2);
else
mu(1) = alpha(1)*nu(1) - alpha(j)*mu(1);
% T = eps1*(pythag(alpha(j),beta(j+1)) + pythag(alpha(1),beta(1)));
T = eps1*(sqrt(alpha(j).^2+beta(j+1).^2) + sqrt(alpha(1).^2+beta(1).^2)); T = T + eps1*anorm; mu(1) = (mu(1) + sign(mu(1))*T) / beta(j+1); % Vectorized version of loop: if j>2 k=2:j-1; mu(k) = alpha(k).*nu(k) + beta(k).*nu(k-1) - alpha(j)*mu(k); %T = eps1*(pythag(alpha(j),beta(j+1)) + pythag(alpha(k),beta(k))); T = eps1*(sqrt(alpha(j).^2+beta(j+1).^2) + sqrt(alpha(k).^2+beta(k).^2)); T = T + eps1*anorm; mu(k) = binv*(mu(k) + sign(mu(k)).*T); end
% T = eps1*(pythag(alpha(j),beta(j+1)) + pythag(alpha(j),beta(j)));
T = eps1*(sqrt(alpha(j).^2+beta(j+1).^2) + sqrt(alpha(j).^2+beta(j).^2)); T = T + eps1*anorm; mu(j) = beta(j)*nu(j-1); mu(j) = (mu(j) + sign(mu(j))*T) / beta(j+1);
end mu(j+1) = 1;
function nu = update_nu(nuold,mu,j,alpha,beta,anorm)
% UPDATE_MU: Update the nu-recurrence for the v-vectors. % % nu_new = update_nu(nu,mu,j,alpha,beta,anorm)
% Rasmus Munk Larsen, DAIMI, 1998.
nu = nuold; ainv = 1/alpha(j); eps1 = 100*eps/2; if j>1
k = 1:(j-1);
% T = eps1*(pythag(alpha(k),beta(k+1)) + pythag(alpha(j),beta(j)));
T = eps1*(sqrt(alpha(k).^2+beta(k+1).^2) + sqrt(alpha(j).^2+beta(j).^2)); T = T + eps1*anorm; nu(k) = beta(k+1).*mu(k+1) + alpha(k).*mu(k) - beta(j)*nu(k); nu(k) = ainv*(nu(k) + sign(nu(k)).*T);
end nu(j) = 1;
function x = pythag(y,z)
%PYTHAG Computes sqrt( y^2 + z^2 ).
%
% x = pythag(y,z)
%
% Returns sqrt(y^2 + z^2) but is careful to scale to avoid overflow.
% Christian H. Bischof, Argonne National Laboratory, 03/31/89.
[m n] = size(y); if m>1 | n>1
y = y(:); z=z(:); rmax = max(abs([y z]'))'; id=find(rmax==0); if length(id)>0 rmax(id) = 1; x = rmax.*sqrt((y./rmax).^2 + (z./rmax).^2); x(id)=0; else x = rmax.*sqrt((y./rmax).^2 + (z./rmax).^2); end x = reshape(x,m,n);
else
rmax = max(abs([y;z])); if (rmax==0) x = 0; else x = rmax*sqrt((y/rmax)^2 + (z/rmax)^2); end
end </pre>
refinebounds()
<pre> function [bnd,gap] = refinebounds(D,bnd,tol1) %REFINEBONDS Refines error bounds for Ritz values based on gap-structure % % bnd = refinebounds(lambda,bnd,tol1) % % Treat eigenvalues closer than tol1 as a cluster.
% Rasmus Munk Larsen, DAIMI, 1998
j = length(D);
if j<=1
return
end % Sort eigenvalues to use interlacing theorem correctly [D,PERM] = sort(D); bnd = bnd(PERM);
% Massage error bounds for very close Ritz values
eps34 = sqrt(eps*sqrt(eps));
[y,mid] = max(bnd);
for l=[-1,1]
for i=((j+1)-l*(j-1))/2:l:mid-l if abs(D(i+l)-D(i)) < eps34*abs(D(i)) if bnd(i)>tol1 & bnd(i+l)>tol1
bnd(i+l) = pythag(bnd(i),bnd(i+l)); bnd(i) = 0;
end end end
end % Refine error bounds gap = inf*ones(1,j); gap(1:j-1) = min([gap(1:j-1);[D(2:j)-bnd(2:j)-D(1:j-1)]']); gap(2:j) = min([gap(2:j);[D(2:j)-D(1:j-1)-bnd(1:j-1)]']); gap = gap(:); I = find(gap>bnd); bnd(I) = bnd(I).*(bnd(I)./gap(I));
bnd(PERM) = bnd; </pre>
reorth()
<pre> function [r,normr,nre,s] = reorth(Q,r,normr,index,alpha,method)
%REORTH Reorthogonalize a vector using iterated Gram-Schmidt % % [R_NEW,NORMR_NEW,NRE] = reorth(Q,R,NORMR,INDEX,ALPHA,METHOD) % reorthogonalizes R against the subset of columns of Q given by INDEX. % If INDEX==[] then R is reorthogonalized all columns of Q. % If the result R_NEW has a small norm, i.e. if norm(R_NEW) < ALPHA*NORMR, % then a second reorthogonalization is performed. If the norm of R_NEW % is once more decreased by more than a factor of ALPHA then R is % numerically in span(Q(:,INDEX)) and a zero-vector is returned for R_NEW. % % If method==0 then iterated modified Gram-Schmidt is used. % If method==1 then iterated classical Gram-Schmidt is used. % % The default value for ALPHA is 0.5. % NRE is the number of reorthogonalizations performed (1 or 2).
% References: % Aake Bjorck, "Numerical Methods for Least Squares Problems", % SIAM, Philadelphia, 1996, pp. 68-69. % % J.~W. Daniel, W.~B. Gragg, L. Kaufman and G.~W. Stewart, % ``Reorthogonalization and Stable Algorithms Updating the % Gram-Schmidt QR Factorization, Math. Comp., 30 (1976), no. % 136, pp. 772-795. % % B. N. Parlett, ``The Symmetric Eigenvalue Problem, % Prentice-Hall, Englewood Cliffs, NJ, 1980. pp. 105-109
% Rasmus Munk Larsen, DAIMI, 1998.
% Check input arguments. %warning('PROPACK:NotUsingMex','Using slow matlab code for reorth.') if nargin<2
error('Not enough input arguments.')
end [n k1] = size(Q); if nargin<3 | isempty(normr) % normr = norm(r);
normr = sqrt(r'*r);
end if nargin<4 | isempty(index)
k=k1; index = [1:k]'; simple = 1;
else
k = length(index); if k==k1 & index(:)==[1:k]' simple = 1; else simple = 0; end
end if nargin<5 | isempty(alpha)
alpha=0.5; % This choice garanties that % || Q^T*r_new - e_{k+1} ||_2 <= 2*eps*||r_new||_2, % cf. Kahans ``twice is enough statement proved in % Parletts book.
end if nargin<6 | isempty(method)
method = 0;
end if k==0 | n==0
return
end if nargout>3
s = zeros(k,1);
end
normr_old = 0;
nre = 0;
while normr < alpha*normr_old | nre==0
if method==1 if simple t = Q'*r; r = r - Q*t; else t = Q(:,index)'*r; r = r - Q(:,index)*t; end else for i=index, t = Q(:,i)'*r; r = r - Q(:,i)*t; end end if nargout>3 s = s + t; end normr_old = normr;
% normr = norm(r);
normr = sqrt(r'*r); nre = nre + 1; if nre > 4 % r is in span(Q) to full accuracy => accept r = 0 as the new vector. r = zeros(n,1); normr = 0; return end
end </pre>
updateSparse.c
A precompiled Intel Windows-32bit Matlab mex file is here.
Otherwise, compile this C program in Matlab via command <pre>mex updateSparse.c</pre> <pre> /*
* Stephen Becker, 11/10/08 * Updates a sparse vector very quickly * calling format: * updateSparse(Y,b) * which updates the values of Y to be b * * Modified 11/12/08 to allow unsorted omega * (omega is the implicit index: in Matlab, what * we are doing is Y(omega) = b. So, if omega * is unsorted, then b must be re-ordered appropriately * */
- include "mex.h"
- ifndef true
#define true 1
- endif
- ifndef false
#define false 0
- endif
void printUsage() {
mexPrintf("usage:\tupdateSparse(Y,b)\nchanges the sparse Y matrix"); mexPrintf(" to have values b\non its nonzero elements. Be careful:\n\t"); mexPrintf("this assumes b is sorted in the appropriate order!\n"); mexPrintf("If b (i.e. the index omega, where we want to perform Y(omega)=b)\n"); mexPrintf(" is unsorted, then call the command as follows:\n"); mexPrintf("\tupdateSparse(Y,b,omegaIndx)\n"); mexPrintf("where [temp,omegaIndx] = sort(omega)\n");
}
void mexFunction(
int nlhs, mxArray *plhs[], int nrhs, const mxArray *prhs[] )
{
/* Declare variable */ int M, N, i, j, m, n; double *b, *S, *omega; int SORTED = true; /* Check for proper number of input and output arguments */ if ( (nrhs < 2) || (nrhs > 3) ) { printUsage(); mexErrMsgTxt("Needs 2 or 3 input arguments"); } if ( nrhs == 3 ) SORTED = false; if(nlhs > 0){ printUsage(); mexErrMsgTxt("No output arguments!"); } /* Check data type of input argument */ if (!(mxIsSparse(prhs[0])) || !((mxIsDouble(prhs[1]))) ){ printUsage(); mexErrMsgTxt("Input arguments wrong data-type (must be sparse, double)."); }
/* Get the size and pointers to input data */ /* Check second input */ N = mxGetN( prhs[1] ); M = mxGetM( prhs[1] ); if ( (M>1) && (N>1) ) { printUsage(); mexErrMsgTxt("Second argument must be a vector"); } N = (N>M) ? N : M;
/* Check first input */ M = mxGetNzmax( prhs[0] ); if ( M != N ) { printUsage(); mexErrMsgTxt("Length of second argument must match nnz of first argument"); }
/* if 3rd argument provided, check that it agrees with 2nd argument */ if (!SORTED) { m = mxGetM( prhs[2] ); n = mxGetN( prhs[2] ); if ( (m>1) && (n>1) ) { printUsage(); mexErrMsgTxt("Third argument must be a vector"); } n = (n>m) ? n : m; if ( n != N ) { printUsage(); mexErrMsgTxt("Third argument must be same length as second argument"); } omega = mxGetPr( prhs[2] ); }
b = mxGetPr( prhs[1] ); S = mxGetPr( prhs[0] );
if (SORTED) { /* And here's the really fancy part: */ for ( i=0 ; i < N ; i++ ) S[i] = b[i]; } else { for ( i=0 ; i < N ; i++ ) { /* this is a little slow, but I should really check * to make sure the index is not out-of-bounds, otherwise * Matlab could crash */ j = (int)omega[i]-1; /* the -1 because Matlab is 1-based */ if (j >= N){ printUsage(); mexErrMsgTxt("Third argument must have values < length of 2nd argument"); }
/* S[ j ] = b[i]; */ /* this is incorrect */
S[ i ] = b[j]; /* this is the correct form */ } }
} </pre>