{"id":21316,"date":"2022-11-13T01:21:51","date_gmt":"2022-11-12T19:51:51","guid":{"rendered":"https:\/\/jassweb.com\/solved\/solved-jensen-bregman-logdet-divergence-closed\/"},"modified":"2022-11-13T01:21:51","modified_gmt":"2022-11-12T19:51:51","slug":"solved-jensen-bregman-logdet-divergence-closed","status":"publish","type":"post","link":"https:\/\/jassweb.com\/solved\/solved-jensen-bregman-logdet-divergence-closed\/","title":{"rendered":"[Solved] Jensen-Bregman LogDet divergence [closed]"},"content":{"rendered":"

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As I understand it the function (of two symmetric positive definite matrices) is<\/p>\n

JL(X,Y) = log( det( (X+Y)\/2)) - log( det( X*Y))\/2\n        = log( det( (X+Y)\/2)) - (log(det(X)) + log(det(Y)))\/2\n<\/code><\/pre>\n
I reckon the way to do this is to compute the cholesky factorisations of X, Y and (X+Y)\/2, that is find lower triangular matrices L,M,N so that<\/p>\n
X = L*L'\nY = M*M'\n(X+Y)\/2 = N*N'\n<\/code><\/pre>\n
(I’m sorry to say I don’t know matlab, but there is bound to be a function to compute the cholesky factorisation).<\/p>\n
and then, using the properties of determinants and logs we have<\/p>\n
JL(X,Y) = 2*log( det(N)) - log( det(L)) - log(det(M)))\n<\/code><\/pre>\n
For a lower triangular matrix K we have<\/p>\n
log( det( K)) = log ( Prod( K[i,i], i=1..n))\n              = Sum ( log( K[i,i]), i=1..n))\n<\/code><\/pre>\n
I think it might be better to use the second form, even though it is slower, as there could be a risk of underflow in the product if the matrices are large, and perhaps the best thing to do would be to compute<\/p>\n
JL(X,Y) = Sum ( 2*log( N[i,i]) - log( L[i,i]) - log( M[i,i]), i=1..n)\n        = Sum( log( N[i,i]*N[i,i] \/ (L[i,i]*M[i,i])), i=1..n)\n<\/code><\/pre>\n
Note that all of this assumes that the cholesky factorisations can be computed, i.e. that the matrices X and Y are strictly positive definite, but then the function JL is only defined in that case. <\/p>\n<\/p><\/div>\n
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            <\/span> <\/p><\/div>\n<\/div>\n
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solved Jensen-Bregman LogDet divergence [closed] <\/p>\n","protected":false},"excerpt":{"rendered":"
[ad_1] As I understand it the function (of two symmetric positive definite matrices) is JL(X,Y) = log( det( (X+Y)\/2)) – log( det( X*Y))\/2 = log( det( (X+Y)\/2)) – (log(det(X)) + log(det(Y)))\/2 I reckon the way to do this is to compute the cholesky factorisations of X, Y and (X+Y)\/2, that is find lower triangular matrices … Read more<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[320],"tags":[457,357,516],"class_list":["post-21316","post","type-post","status-publish","format-standard","hentry","category-solved","tag-algorithm","tag-math","tag-matlab"],"yoast_head":"\n[Solved] Jensen-Bregman LogDet divergence [closed] - JassWeb<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/jassweb.com\/solved\/solved-jensen-bregman-logdet-divergence-closed\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"[Solved] Jensen-Bregman LogDet divergence [closed] - JassWeb\" \/>\n<meta property=\"og:description\" content=\"[ad_1] As I understand it the function (of two symmetric positive definite matrices) is JL(X,Y) = log( det( (X+Y)\/2)) - log( det( X*Y))\/2 = log( det( (X+Y)\/2)) - (log(det(X)) + log(det(Y)))\/2 I reckon the way to do this is to compute the cholesky factorisations of X, Y and (X+Y)\/2, that is find lower triangular matrices ... Read more\" \/>\n<meta property=\"og:url\" content=\"https:\/\/jassweb.com\/solved\/solved-jensen-bregman-logdet-divergence-closed\/\" \/>\n<meta property=\"og:site_name\" content=\"JassWeb\" \/>\n<meta property=\"article:published_time\" content=\"2022-11-12T19:51:51+00:00\" \/>\n<meta name=\"author\" content=\"Kirat\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"Kirat\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"1 minute\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/jassweb.com\/solved\/solved-jensen-bregman-logdet-divergence-closed\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/jassweb.com\/solved\/solved-jensen-bregman-logdet-divergence-closed\/\"},\"author\":{\"name\":\"Kirat\",\"@id\":\"https:\/\/jassweb.com\/solved\/#\/schema\/person\/65c9c7b7958150c0dc8371fa35dd7c31\"},\"headline\":\"[Solved] Jensen-Bregman LogDet divergence [closed]\",\"datePublished\":\"2022-11-12T19:51:51+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/jassweb.com\/solved\/solved-jensen-bregman-logdet-divergence-closed\/\"},\"wordCount\":176,\"publisher\":{\"@id\":\"https:\/\/jassweb.com\/solved\/#organization\"},\"keywords\":[\"algorithm\",\"math\",\"matlab\"],\"articleSection\":[\"Solved\"],\"inLanguage\":\"en-US\"},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/jassweb.com\/solved\/solved-jensen-bregman-logdet-divergence-closed\/\",\"url\":\"https:\/\/jassweb.com\/solved\/solved-jensen-bregman-logdet-divergence-closed\/\",\"name\":\"[Solved] Jensen-Bregman LogDet divergence [closed] - 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