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function [c] = mvn_bregmancentroid_kl_left(models)
%MVN_BREGMANCENTROID_KL_LEFT Compute the left sided Kullback-Leibler (KL) centroid given mvn models. (GRADIENT SPACE CENTER OF MASS)
%
% [c] = MVN_BREGMANCENTROID_KL_LEFT(models) computes the left sided KL
% centroid for the given multivariate normals.
% (c) 2010-2011, Dominik Schnitzer, <dominik.schnitzer@ofai.at>
% http://www.ofai.at/~dominik.schnitzer/mvn
%
% This file is part of the MVN Octave/Matlab Toolbox
% MVN is free software: you can redistribute it and/or modify
% it under the terms of the GNU General Public License as published by
% the Free Software Foundation, either version 3 of the License, or
% (at your option) any later version.
%
% MVN 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 General Public License for more details.
%
% You should have received a copy of the GNU General Public License
% along with MVN. If not, see <http://www.gnu.org/licenses/>.
c.m = 0 .* models(1).m;
% new means
len = length(models);
for i=1:len
c.m = c.m + 1/len .* models(i).m;
end
% new covariance
% % original:
% c.cov = c.m*c.m';
% for i=1:len
% c.cov = c.cov + 1/len .* (models(i).cov +...
% models(i).m*models(i).m' - models(i).m*c.m' -...
% c.m*models(i).m');
c.cov = -1 .* c.m*c.m';
for i=1:len
c.cov = c.cov + 1/len.*(models(i).cov + models(i).m*models(i).m');
end
% compute inverse covariance
% speedup for
%
% c.logdet = log(det(c.cov));
% c.icov = inv(c.cov);
%
% using Cholesky:
c_chol = chol(c.cov);
c.logdet = 2*sum(log(diag(c_chol)));
c_ui = c_chol\eye(length(c.m));
c.icov = c_ui*c_ui';
end