discrep

(PublicToolbox/regutools/discrep.m in BrainStorm 2.0 (Alpha))

Function Synopsis

[x_delta,lambda] = discrep(U,s,V,b,delta,x_0)

Help Text

DISCREP Discrepancy principle criterion for choosing the reg. parameter.

 [x_delta,lambda] = discrep(U,s,V,b,delta,x_0)
 [x_delta,lambda] = discrep(U,sm,X,b,delta,x_0)  ,  sm = [sigma,mu]

 Least squares minimization with a quadratic inequality constraint:
    min || x - x_0 ||       subject to   || A x - b || <= delta
    min || L (x - x_0) ||   subject to   || A x - b || <= delta
 where x_0 is an initial guess of the solution, and delta is a
 positive constant.  Requires either the compact SVD of A saved as
 U, s, and V, or part of the GSVD of (A,L) saved as U, sm, and X.
 The regularization parameter lambda is also returned.

 If delta is a vector, then x_delta is a matrix such that
    x_delta = [ x_delta(1), x_delta(2), ... ] .

 If x_0 is not specified, x_0 = 0 is used.

Cross-Reference Information

newton                             C:\BrainStorm_2001\PublicToolbox\regutools\newton.m

Listing of function C:\BrainStorm_2001\PublicToolbox\regutools\discrep.m

function [x_delta,lambda] = discrep(U,s,V,b,delta,x_0)
%DISCREP Discrepancy principle criterion for choosing the reg. parameter.
%
% [x_delta,lambda] = discrep(U,s,V,b,delta,x_0)
% [x_delta,lambda] = discrep(U,sm,X,b,delta,x_0)  ,  sm = [sigma,mu]
%
% Least squares minimization with a quadratic inequality constraint:
%    min || x - x_0 ||       subject to   || A x - b || <= delta
%    min || L (x - x_0) ||   subject to   || A x - b || <= delta
% where x_0 is an initial guess of the solution, and delta is a
% positive constant.  Requires either the compact SVD of A saved as
% U, s, and V, or part of the GSVD of (A,L) saved as U, sm, and X.
% The regularization parameter lambda is also returned.
%
% If delta is a vector, then x_delta is a matrix such that
%    x_delta = [ x_delta(1), x_delta(2), ... ] .
%
% If x_0 is not specified, x_0 = 0 is used.

% Reference: V. A. Morozov, "Methods for Solving Incorrectly Posed
% Problems", Springer, 1984; Chapter 26.

% Per Christian Hansen, UNI-C, 02/20/92.

% Initialization.
[n,p] = size(V);    [p,ps] = size(s);      ld  = length(delta);
x_k = zeros(n,ld);  lambda = zeros(1,ld);  rho = zeros(p,1);
if (min(delta)<0)
  error('Illegal inequality constraint delta')
end
if (nargin==5), x_0 = zeros(n,1); end
if (ps == 1), omega = V'*x_0; else, omega = V\x_0; end

% Compute residual norms corresponding to TSVD/TGSVD.
beta = U'*b;
nb   = norm(b);
snz  = length(find(s(:,1)>0));
if (ps == 1)
  rho(n) = max(nb^2 - beta'*beta,0);
  for i=n:-1:2
    rho(i-1) = rho(i) + (beta(i) - s(i)*omega(i))^2;
  end
  delta_0 = sqrt(rho(n));
else
  rho(1) = max(nb^2 - beta'*beta,0);
  for i=1:p-1
    rho(i+1) = rho(i) + (beta(i) - s(i,1)*omega(i))^2;
  end
  delta_0 = sqrt(rho(1));
end

% Check input.
if (min(delta) < delta_0)
  error('Irrelevant delta < || (I - U*U'')*b ||')
end

if (ps == 1)
  s2 = s.^2;
  for k=1:ld
    if (nb < delta(k))
      x_delta(:,k) = x_0;
    else
      [dummy,kmin] = min(abs(rho - delta(k)^2));
      lambda_0 = s(kmin);
      lambda(k) = newton(lambda_0,delta(k),s,beta,omega,delta_0);
      e = s./(s2 + lambda(k)^2); f = s.*e;
      x_delta(:,k) = V(:,1:p)*(e.*beta + (1-f).*omega);
    end
  end
else
  omega = V\x_0; omega = omega(1:p); gamma = s(:,1)./s(:,2);
  x_u   = V(:,p+1:n)*beta(p+1:n);
  for k=1:ld
    if (nb < delta(k))
      x_delta(:,k) = x_0;
    else
      [dummy,kmin] = min(abs(rho - delta(k)^2));
      lambda_0 = gamma(kmin);
      lambda(k) = newton(lambda_0,delta(k),s,beta(1:p),omega,delta_0);
      e = gamma./(gamma.^2 + lambda(k)^2); f = gamma.*e;
      x_delta(:,k) = V(:,1:p)*(e.*beta(1:p)./s(:,2) + ...
                               (1-f).*s(:,2).*omega) + x_u;
    end
  end
end