GAUSSMIXK approximate Kullback-Leibler divergence between two GMMs + derivatives Inputs: with kf & kg mixtures, p data dimensions mf(kf,p) mixture means for GMM f vf(kf,p) or vf(p,p,kf) variances (diagonal or full) for GMM f wf(kf,1) weights for GMM f - must sum to 1 mg(kg,p) mixture means for GMM g [g=f if mg,vg,wg omitted] vg(kg,p) or vg(p,p,kg) variances (diagonal or full) for GMM g wg(kg,1) weights for GMM g - must sum to 1 Outputs: d the approximate KL divergence D(f||g) klfg(kf,kg) the exact KL divergence between the components of f and g The Kullback-Leibler (KL) divergence, D(f||g), between two distributions, f(x) and g(x) is also known as the "relative entropy" of f relative to g. It is defined as <log(f(x)) - log(g(x))> where <y(x)> denotes the expectation with respect to f(x), i.e. <y(x)> = Integral(f(x) y(x) dx). The KL divergence is always >=0 and equals 0 if and only if f(x)=g(x) almost everywhere. % It is not a distance because it is not symmetric between f and g and also does not satisfy the triangle inequality. It is normally appropriate for f(x) to be the "true" distribution and g(x) to be an approximation to it. See [1]. This routine calculates the "variational approximation" to the KL divergence from [2] that is exact when f and g are single component gaussians and that is zero if f=g. However, the approximation may be negative if f and g are different. Refs: [1] S. Kullback and R. Leibler. On information and sufficiency. Annals of Mathematical Statistics, 22 (1): 79–86, 1951. doi: 10.1214/aoms/1177729694. [2] J. R. Hershey and P. A. Olsen. Approximating the kullback leibler divergence between gaussian mixture models. In Proc. IEEE Intl Conf. Acoustics, Speech and Signal Processing, volume 4, pages IV–317–IV–320, Apr. 2007. doi: 10.1109/ICASSP.2007.366913.

- logsum LOGSUM logsum(x,d,k)=log(sum(k.*exp(x),d))

0001 function [d,klfg]=gaussmixk(mf,vf,wf,mg,vg,wg) 0002 %GAUSSMIXK approximate Kullback-Leibler divergence between two GMMs + derivatives 0003 % 0004 % Inputs: with kf & kg mixtures, p data dimensions 0005 % 0006 % mf(kf,p) mixture means for GMM f 0007 % vf(kf,p) or vf(p,p,kf) variances (diagonal or full) for GMM f 0008 % wf(kf,1) weights for GMM f - must sum to 1 0009 % mg(kg,p) mixture means for GMM g [g=f if mg,vg,wg omitted] 0010 % vg(kg,p) or vg(p,p,kg) variances (diagonal or full) for GMM g 0011 % wg(kg,1) weights for GMM g - must sum to 1 0012 % 0013 % Outputs: 0014 % d the approximate KL divergence D(f||g) 0015 % klfg(kf,kg) the exact KL divergence between the components of f and g 0016 % 0017 % The Kullback-Leibler (KL) divergence, D(f||g), between two distributions, 0018 % f(x) and g(x) is also known as the "relative entropy" of f relative to g. 0019 % It is defined as <log(f(x)) - log(g(x))> where <y(x)> denotes the 0020 % expectation with respect to f(x), i.e. <y(x)> = Integral(f(x) y(x) dx). 0021 % The KL divergence is always >=0 and equals 0 if and only if f(x)=g(x) 0022 % almost everywhere. % It is not a distance because it is not symmetric 0023 % between f and g and also does not satisfy the triangle inequality. 0024 % It is normally appropriate for f(x) to be the "true" distribution and 0025 % g(x) to be an approximation to it. See [1]. 0026 % 0027 % This routine calculates the "variational approximation" to the KL divergence 0028 % from [2] that is exact when f and g are single component gaussians and that is zero 0029 % if f=g. However, the approximation may be negative if f and g are different. 0030 % 0031 % Refs: 0032 % [1] S. Kullback and R. Leibler. On information and sufficiency. 0033 % Annals of Mathematical Statistics, 22 (1): 79–86, 1951. doi: 10.1214/aoms/1177729694. 0034 % [2] J. R. Hershey and P. A. Olsen. 0035 % Approximating the kullback leibler divergence between gaussian mixture models. 0036 % In Proc. IEEE Intl Conf. Acoustics, Speech and Signal Processing, volume 4, 0037 % pages IV–317–IV–320, Apr. 2007. doi: 10.1109/ICASSP.2007.366913. 0038 0039 % Copyright (C) Mike Brookes 2012 0040 % Version: $Id: gaussmixk.m 3231 2013-07-04 16:54:05Z dmb $ 0041 % 0042 % VOICEBOX is a MATLAB toolbox for speech processing. 0043 % Home page: http://www.ee.ic.ac.uk/hp/staff/dmb/voicebox/voicebox.html 0044 % 0045 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 0046 % This program is free software; you can redistribute it and/or modify 0047 % it under the terms of the GNU General Public License as published by 0048 % the Free Software Foundation; either version 2 of the License, or 0049 % (at your option) any later version. 0050 % 0051 % This program is distributed in the hope that it will be useful, 0052 % but WITHOUT ANY WARRANTY; without even the implied warranty of 0053 % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 0054 % GNU General Public License for more details. 0055 % 0056 % You can obtain a copy of the GNU General Public License from 0057 % http://www.gnu.org/copyleft/gpl.html or by writing to 0058 % Free Software Foundation, Inc.,675 Mass Ave, Cambridge, MA 02139, USA. 0059 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 0060 [kf,p]=size(mf); 0061 if isempty(wf) 0062 wf=repmat(1/kf,kf,1); 0063 end 0064 if isempty(vf) 0065 vf=ones(kf,p); 0066 end 0067 fvf=ndims(vf)>2 || size(vf,1)>kf; % full covariance matrix vf is supplied 0068 0069 % First calculate vf covariance matrix determinants and precision matrices 0070 % and then the individual KL divergences between the components of f 0071 0072 klff=zeros(kf,kf); % space for intra-a KL negative divergence 0073 ixdf=1:kf+1:kf*kf; % indexes of diagonal elements in kf*kf matrix 0074 ixdp=(1:p+1:p*p)'; % indexes of diagonal elements in p*p matrix 0075 wkf=ones(kf,1); 0076 if fvf % vf is a full matrix 0077 dvf=zeros(kf,1); % space for log(det(vf)) 0078 for i=1:kf 0079 dvf(i)=log(det(vf(:,:,i))); 0080 end 0081 for j=1:kf % calculate KL divergence between all mixtures and mixture j 0082 pfj=inv(vf(:,:,j)); 0083 mffj=mf-mf(j(wkf),:); 0084 pfjvf=reshape(pfj*reshape(vf,p,p*kf),p^2,kf); % pf(:,:,j)* all the vf matices 0085 klff(:,j)=0.5*(dvf(j)-p-dvf+sum(pfjvf(ixdp,:),1)'+sum((mffj*pfj).*mffj,2)); 0086 end 0087 else % vf is diagonal 0088 dvf=log(prod(vf,2)); 0089 pf=1./vf; 0090 mf2p=mf.^2*pf'; 0091 mf2pd=mf2p(ixdf); % get diagonal elements 0092 klff=0.5*(dvf(:,wkf)'-dvf(:,wkf)+vf*pf'-p+mf2p+mf2pd(wkf,:)-2*mf*(mf.*pf)'); 0093 end 0094 klff(ixdf)=0; % force the diagonal elements to exact zero 0095 if nargin<4 0096 d=0; 0097 klfg=klff; 0098 else 0099 [kg,pg]=size(mg); 0100 if pg~=p 0101 error('GMMs must have the same data dimension (%d versus %d)',p,pg); 0102 end 0103 if nargin<6 || isempty(wg) 0104 wg=repmat(1/kg,kg,1); 0105 end 0106 if nargin<5 || isempty(vg) 0107 vg=ones(kg,p); 0108 end 0109 fvb=ndims(vg)>2 || size(vg,1)>kg; % full covariance matrix vg is supplied 0110 0111 % Calculate vg covariance matrix determinants and precision matrices 0112 % and then the individual inter-component KL divergences between components of f and g 0113 0114 klfg=zeros(kf,kg); % space for inter-a-b KL negative divergence 0115 wkg=ones(kg,1); 0116 if fvb % vg is a full matrix 0117 dvg=zeros(kg,1); % space for log(det(vg)) 0118 pg=zeros(p,p,kg); % space for inv(vg) 0119 for j=1:kg 0120 dvgj=log(det(vg(:,:,j))); 0121 dvg(j)=dvgj; 0122 pgj=inv(vg(:,:,j)); 0123 pg(:,:,j)=pgj; 0124 mfgj=mf-mg(j(wkf),:); 0125 if fvf % vf and vg are both full matrices 0126 pgjvf=reshape(pgj*reshape(vf,p,p*kf),p^2,kf); % pg(:,:,j)* all the vf matices 0127 klfg(:,j)=0.5*(dvgj-p-dvf+sum(pgjvf(ixdp,:),1)'+sum((mfgj*pgj).*mfgj,2)); 0128 else % vf diagonal but vg is full 0129 klfg(:,j)=0.5*(dvgj-p-dvf+vf*pgj(ixdp)+sum((mfgj*pgj).*mfgj,2)); 0130 end 0131 end 0132 else % vg is diagonal 0133 dvg=log(prod(vg,2)); % log(det(vg)) 0134 pg=1./vg; % precision matrix pg = inv(vg) 0135 mg2p=sum(mg.^2.*pg,2)'; 0136 if fvf % vf a full matrix, vg diagonal 0137 vav=reshape(vf,p^2,kf); 0138 klfg=0.5*(dvg(:,wkf)'-dvf(:,wkg)+vav(ixdp,:)'*pg'-p+mf.^2*pg'+mg2p(wkf,:)-2*mf*(mg.*pg)'); 0139 else % vf and vg both diagonal 0140 klfg=0.5*(dvg(:,wkf)'-dvf(:,wkg)+vf*pg'-p+mf.^2*pg'+mg2p(wkf,:)-2*mf*(mg.*pg)'); 0141 end 0142 end 0143 d=wf'*(logsum(-klff,2,wf)-logsum(-klfg,2,wg)); 0144 end 0145 0146

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