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see https://github.com/yorkerlin/approxKLVB for detail information

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function [alpha, sW, L, nlZ, dnlZ] = approxLogKLWithLBFGS(hyper, covfunc, lik, x, y)
 
% Approximation to the posterior Gaussian Process by minimization of the
% KL-divergence. The function takes a specified covariance function (see
% covFunction.m) and likelihood function (see likelihoods.m), and is designed to
% be used with binaryGP.m. See also approximations.m.
%
% Written by Hannes Nickisch, 2007-03-29
% Modified by Wu Lin, 2014
 
n = size(x,1);
K = feval(covfunc{:}, hyper.cov, x); % evaluate the covariance matrix
 
alla_init{1} = [zeros(n,1); ones(n,1)*log(0.5)]; % stack alpha/lambda together
 
alla_init=alla_init([1]);
 
for alla_id = 1:length(alla_init) % iterate over initial conditions
 
alla = alla_init{alla_id};
 
use_pinv=false; check_cond=true;
nlZ_old = Inf; nlZ_new = 1e100; it=0; % make sure the while loop starts
 
[alla nlZ_new] = lbfgs(alla, K, y, lik, hyper); %using L-BFGS to find the opt alla
 
% save results
alla_result{alla_id} = alla;
nlZ_result( alla_id) = nlZ_new;
end
 
alla_id = find(nlZ_result==min(nlZ_result)); alla_id = alla_id(1);
alla = alla_result{alla_id}; % extract best result
 
 
%display the result
nlZ_new = min(nlZ_result)
alla(end/2+1:end,1) = -exp(alla(end/2+1:end,1)); %convert log_neg_lambda to lambda
 
 
alpha = alla(1:end/2,1)
W = -2*alla(end/2+1:end,1)
 
% recalculate L
sW = sqrt(W);
L = chol(eye(n)+sW*sW'.*K) % L'*L=B=eye(n)+sW*K*sW
 
 
% bound on neg log marginal likelihood
nlZ = nlZ_result( alla_id);
 
%estimate the hpyer parameter
% do we want derivatives?
if nargout >=4
 
dnlZ = zeros(size(hyper.cov)); % allocate space for derivatives
% parameters after optimization
alpha = alla( 1:end/2,1);
lambda = alla(end/2+1:end ,1);
A = inv( eye(n)-2*K*diag(lambda) );
 
Sigma = A*K
mu = K*alpha
v=abs(diag(A*K))
[a,dm,dC] = a_related2(K*alpha,v,y,lik,hyper);
 
for j=1:length(hyper.cov)
dK = feval(covfunc{:},hyper.cov,x,j);
% from the paper
% -alpha'*dK*dm +(alpha'*dK*alpha)/2 -diag(A*dK*A')'*dC
% -trace(A'*diag(lambda)*dK) +trace(A*dK*diag(lambda)*A)
% Note that lambda == dC
AdK = A*dK;
dnlZ(j) = -(alpha'*dK*(dm-alpha/2) +sum(A.*AdK,2)'*dC ...
+(diag(AdK)'-sum(A'.*AdK,1))*lambda);
end
dnlZ = hyper.cov; % allocate space for derivatives
 
for j=1:length(hyper.cov) % covariance hypers
dK = feval(covfunc{:},hyper.cov,x,j)
%dK = feval(cov{:},hyp.cov,x,[],j);
AdK = A*dK;
tmp1=sum(A.*AdK,2)
tmp2=sum(A'.*AdK,1)
 
z = diag(AdK) + sum(A.*AdK,2) - sum(A'.*AdK,1)';
%dnlZ(j) = alpha'*dK*(alpha/2-df) - z'*dv;
dnlZ(j) = alpha'*dK*(alpha/2-dm) - z'*dC;
end
 
 
dnlZ_lik=zeros(size(hyper.lik));
for j=1:length(hyper.lik) % likelihood hypers
lp_dhyp = likKL(v,lik,hyper.lik,y,K*alpha,[],[],j);
dnlZ_lik(j) = -sum(lp_dhyp);
end
disp('dnlZ_lik=')
sprintf('%.15f\n',dnlZ_lik)
 
dnlZ_mean=zeros(size(hyper.mean));
for j=1:length(hyp.mean) % mean hypers
dm_t = feval(mean{:}, hyper.mean, x, j);
dnlZ_mean(j) = -alpha'*dm_t;
end
 
end
 
 
%% evaluation of current negative log marginal likelihood depending on the
% parameters alpha (al) and lambda (la)
function [nlZ,dnlZ] = margLik_log(alla,K,y,lik,hpyer)
% extract single parameters
alpha = alla(1:end/2,1);
log_neg_lambda = alla(end/2+1:end,1);
 
lambda = -exp(log_neg_lambda);
% dimensions
n = length(y);
 
 
% original variables instead of alpha and la
VinvK = inv(eye(n)-2*K*diag(lambda)); % A:=V*inv(K)
V = VinvK*K; V=(V+V')/2; % enforce symmetry
v = abs(diag(V)); % abs prevents numerically negative values
m = K*alpha;
 
% calculate alpha related terms we need
if nargout==1
[a] = a_related2(m,v,y,lik,hpyer);
else
%done
[a,dm,dV] = a_related2(m,v,y,lik,hpyer);
end
 
%res1=trace(VinvK)
%W = abs(-2*lambda);
%sW = sqrt(W); L = chol(eye(n)+sW*sW'.*K);
%L_inv=L\eye(n);
%res2=trace(L_inv'*L_inv)
%Note res1==res2
 
%negative Likelihood
nlZ = -a -logdet(VinvK)/2 -n/2 +(alpha'*K*alpha)/2 +trace(VinvK)/2;
 
if nargout>1 % gradient of Likelihood
dlZ_alpha = K*(dm-alpha);
dlZ_lambda = 2*(V.*V)*dV +v -sum(V.*VinvK,2); % => fast diag(V*VinvK')
dlZ_log_neg_lambda = dlZ_lambda .* lambda;
 
% stack things together
dnlZ = -[dlZ_alpha; dlZ_log_neg_lambda];
end
 
 
 
function [alla2 nlZ] = lbfgs(alla, K, y, lik, hyper)
optMinFunc = struct('Display', 'FULL',...
'Method', 'lbfgs',...
'DerivativeCheck', 'off',...
'LS_type', 1,...
'MaxIter', 1000,...
'LS_interp', 1,...
'MaxFunEvals', 1000000,...
'Corr' , 100,...
'optTol', 1e-15,...
'progTol', 1e-15);
[alla2, nlZ] = minFunc(@margLik_log, alla, optMinFunc, K, y, lik,hyper);
 
 
%% log(det(A)) for det(A)>0
function y = logdet(A)
% 1) y=det(A); if det(A)<=0, error('det(A)<=0'), end, y=log(y);
% => naive implementation, not numerically stable
% 2) U=chol(A); y=2*sum(log(diag(U)));
% => fast, but works for symmetric p.d. matrices only
% 3) det(A)=det(L)*det(U)=det(L)*prod(diag(U))
% => logdet(A)=log(sum(log(diag(U)))) if det(A)>0
[L,U]=lu(A);
u=diag(U);
if prod(sign(u))~=det(L)
error('det(A)<=0')
end
y=sum(log(abs(u))); % slower, but no symmetry needed
% 4) d=eig(A); if prod(sign(d))<1, error('det(A)<=0'), end
% y=sum(log(d)); y=real(y);
% => slowest
 
%% compute all terms related to a
% derivatives w.r.t diag(V) and m, 2nd derivatives w.r.t diag(V) and m
function [a,dm,dV,d2m,d2V,dmdV]=a_related2(m,v,y,lik,hyper)
if nargout<4
[a,dm,d2m,dV] = likKL(v, lik,hyper.lik,y,m);
a = sum(a);
else
[a,dm,d2m,dV,d2V,dmdV] = likKL(v, lik,hyper.lik,y,m)
a = sum(a);
end
%using likelihood function in GPML 3.4
function [ll,df,d2f,dv,d2v,dfdv] = likKL(v, lik, varargin)
N = 20; % number of quadrature points
[t,w] = gauher(N); % location and weights for Gaussian-Hermite quadrature
f = varargin{3}; % obtain location of evaluation
sv = sqrt(v); % smoothing width
ll = 0; df = 0; d2f = 0; dv = 0; d2v = 0; dfdv = 0; % init return arguments
for i=1:N % use Gaussian quadrature
varargin{3} = f + sv*t(i); % coordinate transform of the quadrature points
[lp,dlp,d2lp] = feval(lik{:},varargin{1:3},[],'infLaplace',varargin{6:end});
if nargout>0, ll = ll + w(i)*lp; % value of the integral
if nargout>1, df = df + w(i)*dlp; % derivative w.r.t. mean
if nargout>2, d2f = d2f + w(i)*d2lp; % 2nd derivative w.r.t. mean
if nargout>3 % derivative w.r.t. variance
ai = t(i)./(2*sv+eps); dvi = dlp.*ai; dv = dv + w(i)*dvi; % no 0 div
if nargout>4 % 2nd derivative w.r.t. variance
d2v = d2v + w(i)*(d2lp.*(t(i)^2/2)-dvi)./(v+eps)/2; % no 0 div
if nargout>5 % mixed second derivatives
dfdv = dfdv + w(i)*(ai.*d2lp);
end
end
end
end
end
end
end
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