{"id":3175,"date":"2011-09-29T13:37:39","date_gmt":"2011-09-29T11:37:39","guid":{"rendered":"http:\/\/djalil.chafai.net\/blog\/?p=3175"},"modified":"2011-10-02T01:05:39","modified_gmt":"2011-10-01T23:05:39","slug":"two-open-problems-about-concentation","status":"publish","type":"post","link":"https:\/\/djalil.chafai.net\/blog\/2011\/09\/29\/two-open-problems-about-concentation\/","title":{"rendered":"Two open problems about concentation"},"content":{"rendered":"<p style=\"text-align: justify;\">Talagrand has shown that there exists universal constants \\( {c,C&gt;0} \\) such that for any independent and identically distributed random variables \\( {X_1,\\ldots,X_n} \\) in the unit disk \\( {D} \\) of \\( {\\mathbb{C}} \\), any convex and Lipschitz function \\( {f:D^n\\rightarrow\\mathbb{R}} \\), and any real number \\( {r&gt;0} \\),<\/p>\n<p style=\"text-align: center;\">\\[ \\mathbb{P}(|f(X_1,\\ldots,X_n)-\\mathbb{E}f(X_1,\\ldots,X_n)|\\geq r) \\leq C\\exp(-cr^2). \\]<\/p>\n<p style=\"text-align: justify;\">An accessible proof can be found in <a href= \"http:\/\/www.ams.org\/mathscinet-getitem?mr=1849347\">Ledoux's monograph on the concentration phenomenon<\/a> (chapter 4). This inequality is useful for instance in order to control the distance of a random vector to a sub-vector space of controlled dimension. In many situations, one would like a similar concentration result, beyond these restrictive assumptions. Let us consider for instance a \\( {n\\times n} \\) random matrix \\( {M} \\) with i.i.d. entries (standard Gaussian or symmetric Bernoulli \\( {\\pm 1} \\)). Here are two open questions about concentration of measure for which the Talagrand inequality is not enough as is due to the lack of one of the assumptions:<\/p>\n<ul>\n<li>concentration for the function \\( {\\log|\\det(M)|=\\log\\det\\sqrt{MM^*}=\\sum_{i=1}^n\\log(s_i(M))} \\)<\/li>\n<li>concentration for the least singular value \\( {s_n(M)=\\min_{\\left\\Vert x\\right\\Vert=1}\\left\\Vert Mx\\right\\Vert_2} \\)<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Talagrand has shown that there exists universal constants \\( {c,C&gt;0} \\) such that for any independent and identically distributed random variables \\( {X_1,\\ldots,X_n} \\) in&#8230;<\/p>\n<div class=\"more-link-wrapper\"><a class=\"more-link\" href=\"https:\/\/djalil.chafai.net\/blog\/2011\/09\/29\/two-open-problems-about-concentation\/\">Continue reading<span class=\"screen-reader-text\">Two open problems about concentation<\/span><\/a><\/div>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":124},"categories":[1],"tags":[],"_links":{"self":[{"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/posts\/3175"}],"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=3175"}],"version-history":[{"count":11,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/posts\/3175\/revisions"}],"predecessor-version":[{"id":3190,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/posts\/3175\/revisions\/3190"}],"wp:attachment":[{"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/media?parent=3175"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/categories?post=3175"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/tags?post=3175"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}