{"id":5791,"date":"2013-02-22T21:27:40","date_gmt":"2013-02-22T20:27:40","guid":{"rendered":"http:\/\/djalil.chafai.net\/blog\/?p=5791"},"modified":"2014-02-19T23:45:21","modified_gmt":"2014-02-19T22:45:21","slug":"mean-of-a-random-variable-on-a-metric-space","status":"publish","type":"post","link":"https:\/\/djalil.chafai.net\/blog\/2013\/02\/22\/mean-of-a-random-variable-on-a-metric-space\/","title":{"rendered":"Mean of a random variable on a metric space"},"content":{"rendered":"<p><a href=\"http:\/\/en.wikipedia.org\/wiki\/Maurice_Ren%C3%A9_Fr%C3%A9chet\"><img loading=\"lazy\" class=\"aligncenter size-medium wp-image-5796\" title=\"Maurice Fr\u00e9chet (1878 - 1973)\" alt=\"\" src=\"\/blog\/wp-content\/uploads\/2013\/02\/Frechet-257x300.jpg\" width=\"257\" height=\"300\" srcset=\"https:\/\/djalil.chafai.net\/blog\/wp-content\/uploads\/2013\/02\/Frechet-257x300.jpg 257w, https:\/\/djalil.chafai.net\/blog\/wp-content\/uploads\/2013\/02\/Frechet.jpg 769w\" sizes=\"(max-width: 257px) 100vw, 257px\" \/><\/a><\/p>\n<p style=\"text-align: justify;\">In this short post, we recall the pleasant notion of <b>Fr\u00e9chet mean<\/b> (or <a href=\"http:\/\/en.wikipedia.org\/wiki\/Frechet_mean\">Karcher mean<\/a>) of a probability measure on a metric space, a concept already considered in an <a href=\"\/blog\/2010\/05\/01\/mean-of-random-variable-on-manifold\">old previous post<\/a>. Let \\( {(E,d)} \\) be a metric space, such as a graph (with vertices and edges) or a Riemannian manifold, equipped with its Borel \\( {\\sigma} \\)-field. Let \\( {\\mu} \\) be a probability measure on \\( {E} \\). How can we define the mean \\( {m_\\mu} \\) and the variance \\( {v_\\mu} \\) of \\( {\\mu} \\)? A very natural idea is to consider the variational definition:<\/p>\n<p style=\"text-align: center;\">\\[ v_\\mu:=\\inf_{x\\in E}\\mathbb{E}(d(x,Y)^2), \\]<\/p>\n<p style=\"text-align: justify;\">where \\( {Y\\sim\\mu} \\). The set \\( {m_\\mu:=\\arg\\inf_{x\\in E}\\mathbb{E}(d(x,Y)^2)} \\) where this infimum is achieved plays the role of a mean (which is not necessarily unique), while the value of the infimum plays the role of the variance. Note that<\/p>\n<p style=\"text-align: justify;\">\\[v_\\mu=\\inf_{x\\in E}W_2(\\delta_x,\\mu)^2\\]<\/p>\n<p style=\"text-align: justify;\">where $W_2$ is the so-called Wasserstein-Fr\u00e9chet-Kantorovich coupling distance<\/p>\n<p style=\"text-align: justify;\">\\[ W_2(\\nu,\\mu)^2=\\inf_{(X,Y),X\\sim\\nu,Y\\sim\\mu}\\mathbb{E}(d(X,Y)^2).\\]<\/p>\n<p style=\"text-align: justify;\">From this observation, the Fr\u00e9chet means of $\\mu$ are the atoms of the closest Dirac masses to $\\mu$ in $W_2$ distance. We may replace the exponent \\( {2} \\) by any real \\( {p\\geq1} \\) to get a more general notion of moments of \\( {\\mu} \\) (this leads by the way to moments problems on metric spaces!). The notion of Fr\u00e9chet mean is so natural that it has been studied by various authors in pure and applied contexts. Here are some recent examples:<\/p>\n<ul>\n<li><a href=\"http:\/\/arxiv.org\/abs\/1204.3183\">Strong and weak laws of large numbers for Fr\u00e9chet sample means in bounded metric spaces<\/a><\/li>\n<li><a href=\"http:\/\/arxiv.org\/abs\/1102.0228\">Limit theorems for empirical Fr\u00e9chet means of independent and non-identically distributed manifold-valued random variables<\/a><\/li>\n<li><a href=\"http:\/\/arxiv.org\/abs\/1211.7046\">Averaging metric phylogenetic trees<\/a><\/li>\n<li><a href=\"http:\/\/dx.doi.org\/10.1214\/EJP.v18-2201\">Central limit theorems for Fr\u00e9chet means in the space of phylogenetic trees<\/a><\/li>\n<li><a href=\"http:\/\/arxiv.org\/abs\/1301.7156\">A stochastic algorithm finding \\( {p} \\)-means on the circle<\/a><\/li>\n<li><a href=\"http:\/\/arxiv.org\/abs\/1207.3232\">Means in complete manifolds: uniqueness and approximation<\/a><\/li>\n<li><a href=\"http:\/\/arxiv.org\/abs\/1212.2562\">Consistent estimation of a population barycenter in the Wasserstein space<\/a><\/li>\n<li><a href=\"http:\/\/arxiv.org\/abs\/1210.0771\">Minimax properties of Fr\u00e9chet means of discretely sampled curves<\/a><\/li>\n<li><a href=\"http:\/\/arxiv.org\/abs\/1111.1855\">Fr\u00e9chet means of curves for signal averaging and application to ECG data analysis<\/a><\/li>\n<li><a href=\"http:\/\/arxiv.org\/abs\/1010.0427\">On the consistency of Fr\u00e9chet means in deformable models for curve and image analysis<\/a><\/li>\n<li><a href=\"http:\/\/arxiv.org\/abs\/1111.5927\">Distribution's template estimate with Wasserstein metrics<\/a><\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>In this short post, we recall the pleasant notion of Fr&eacute;chet mean (or Karcher mean) of a probability measure on a metric space, a concept&#8230;<\/p>\n<div class=\"more-link-wrapper\"><a class=\"more-link\" href=\"https:\/\/djalil.chafai.net\/blog\/2013\/02\/22\/mean-of-a-random-variable-on-a-metric-space\/\">Continue reading<span class=\"screen-reader-text\">Mean of a random variable on a metric space<\/span><\/a><\/div>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":142},"categories":[1],"tags":[],"_links":{"self":[{"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/posts\/5791"}],"collection":[{"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/comments?post=5791"}],"version-history":[{"count":23,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/posts\/5791\/revisions"}],"predecessor-version":[{"id":6860,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/posts\/5791\/revisions\/6860"}],"wp:attachment":[{"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/media?parent=5791"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/categories?post=5791"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/tags?post=5791"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}