{"id":26,"date":"2018-12-18T20:22:55","date_gmt":"2018-12-18T20:22:55","guid":{"rendered":"https:\/\/math-sites.uncg.edu\/comgrouptom\/?p=26"},"modified":"2025-01-24T19:59:36","modified_gmt":"2025-01-24T19:59:36","slug":"multi-radius-persistent-homology","status":"publish","type":"post","link":"https:\/\/math-sites.uncg.edu\/comgrouptom\/multi-radius-persistent-homology\/","title":{"rendered":"Multi-radius Persistent Homology"},"content":{"rendered":"<p>Associate Professors\u00a0<a href=\"https:\/\/math-sites.uncg.edu\/directory\/bell\/\">Greg Bell<\/a>\u00a0and\u00a0<a href=\"https:\/\/math-sites.uncg.edu\/directory\/smyth\/\">Clifford Smyth<\/a>\u00a0worked with Graduate Students\u00a0<a href=\"https:\/\/math-sites.uncg.edu\/students\/austin-lawson\/\">Austin Lawson<\/a>,\u00a0<a href=\"https:\/\/math-sites.uncg.edu\/students\/joshua-martin\/\">Joshua Martin<\/a>, and\u00a0<a href=\"https:\/\/math-sites.uncg.edu\/directory\/rudzinski\/\">James Rudzinski<\/a>\u00a0to consider a weighted persistence method that seeks to solve the outlier problem and the minimal cost coverage problem using the same mechanism.<\/p>\n<p>The method assigns a scaling to the radii of the balls in the persistence algorithm. For the coverage problem, low density points are assigned large scales to increase radius, while denser points have slower growing radii. For the outlier problem, the situation is dual: we minimize the contribution of sparse points by assigning small scales to their radii.<\/p>\n<p>The details and the algorithm are developed in a forthcoming paper.<\/p>\n<table class=\"tbl-pad\" width=\"100%\">\n<tbody>\n<tr>\n<td width=\"60%\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-27 size-full\" src=\"https:\/\/math-sites.uncg.edu\/comgrouptom\/wp-content\/uploads\/sites\/14\/2018\/12\/1-topology.gif\" alt=\"Multi-radius Persistent Homology\" width=\"436\" height=\"344\" \/><\/td>\n<td>This is the evolution of the usual persistence for the covering problem. It can be seen that the radii of the balls grow at the same rate.<\/td>\n<\/tr>\n<tr>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-28 size-full\" src=\"https:\/\/math-sites.uncg.edu\/comgrouptom\/wp-content\/uploads\/sites\/14\/2018\/12\/2-topology.gif\" alt=\"Multi-radius Persistent Homology\" width=\"436\" height=\"344\" \/><\/td>\n<td>This is the evolution of balls with radii proportional to the codensity. It can be seen that the region is covered more efficiently.<\/td>\n<\/tr>\n<tr>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-29 size-full\" src=\"https:\/\/math-sites.uncg.edu\/comgrouptom\/wp-content\/uploads\/sites\/14\/2018\/12\/3-movie.gif\" alt=\"Multi-radius Persistent Homology\" width=\"436\" height=\"344\" \/><\/td>\n<td>Here the usual persistence is used to recognize points sampled from an annulus. The radii grow at equal rates.<\/td>\n<\/tr>\n<tr>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-30 size-full\" src=\"https:\/\/math-sites.uncg.edu\/comgrouptom\/wp-content\/uploads\/sites\/14\/2018\/12\/4-topology-m.gif\" alt=\"Multi-radius Persistent Homology\" width=\"436\" height=\"344\" \/><\/td>\n<td>Here the radii of the balls grow in proportion to the density so that sparse points are ignored. The annulus is recognizable for a longer time period.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n","protected":false},"excerpt":{"rendered":"<p>Associate Professors\u00a0Greg Bell\u00a0and\u00a0Clifford Smyth\u00a0worked with Graduate Students\u00a0Austin Lawson,\u00a0Joshua Martin, and\u00a0James Rudzinski\u00a0to consider a weighted persistence method that seeks to solve the outlier&#8230;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[8],"tags":[],"class_list":["post-26","post","type-post","status-publish","format-standard","hentry","category-combinatorics"],"acf":[],"_links":{"self":[{"href":"https:\/\/math-sites.uncg.edu\/comgrouptom\/wp-json\/wp\/v2\/posts\/26","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/math-sites.uncg.edu\/comgrouptom\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/math-sites.uncg.edu\/comgrouptom\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/math-sites.uncg.edu\/comgrouptom\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/math-sites.uncg.edu\/comgrouptom\/wp-json\/wp\/v2\/comments?post=26"}],"version-history":[{"count":2,"href":"https:\/\/math-sites.uncg.edu\/comgrouptom\/wp-json\/wp\/v2\/posts\/26\/revisions"}],"predecessor-version":[{"id":79,"href":"https:\/\/math-sites.uncg.edu\/comgrouptom\/wp-json\/wp\/v2\/posts\/26\/revisions\/79"}],"wp:attachment":[{"href":"https:\/\/math-sites.uncg.edu\/comgrouptom\/wp-json\/wp\/v2\/media?parent=26"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/math-sites.uncg.edu\/comgrouptom\/wp-json\/wp\/v2\/categories?post=26"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/math-sites.uncg.edu\/comgrouptom\/wp-json\/wp\/v2\/tags?post=26"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}