{"id":2802,"date":"2025-04-02T07:03:14","date_gmt":"2025-04-02T07:03:14","guid":{"rendered":"https:\/\/mailitics.com\/index.php\/2025\/04\/02\/2503-22786\/"},"modified":"2025-04-02T07:03:14","modified_gmt":"2025-04-02T07:03:14","slug":"2503-22786","status":"publish","type":"post","link":"https:\/\/mailitics.com\/index.php\/2025\/04\/02\/2503-22786\/","title":{"rendered":"A formula for the area of a triangle: Useless, but explicitly in Deep Sets form"},"content":{"rendered":"<p>    A formula for the area of a triangle: Useless, but explicitly in Deep Sets form<br \/>\n \t<BR><br \/>\n<BR><\/BR><br \/>\n    <!-- no image --><br \/>\n \t<BR><br \/>\n<BR><\/BR><\/p>\n<div>arXiv:2503.22786v1 Announce Type: cross<br \/>\nAbstract: Any permutation-invariant function of data points $vec{r}_i$ can be written in the form $rho(sum_iphi(vec{r}_i))$ for suitable functions $rho$ and $phi$. This form &#8211; known in the machine-learning literature as Deep Sets &#8211; also generates a map-reduce algorithm. The area of a triangle is a permutation-invariant function of the locations $vec{r}_i$ of the three corners $1leq ileq 3$. We find the polynomial formula for the area of a triangle that is explicitly in Deep Sets form. This project was motivated by questions about the fundamental computational complexity of $n$-point statistics in cosmology; that said, no insights of any kind were gained from these results.<\/div>\n<p> \t<BR><br \/>\n <BR><\/BR><br \/>\n    Connor Hainje, David W. Hogg<br \/>\n \t<BR><br \/>\n<BR><\/BR><br \/>\n<a href=\"https:\/\/arxiv.org\/abs\/2503.22786\">Go to original source<\/a><br \/>\n \t<BR><br \/>\n <BR><\/BR><\/p>\n","protected":false},"excerpt":{"rendered":"<p>A formula for the area of a triangle: Useless, but explicitly in Deep Sets form arXiv:2503.22786v1 Announce Type: cross Abstract: Any permutation-invariant function of data points $vec{r}_i$ can be written in the form $rho(sum_iphi(vec{r}_i))$ for suitable functions $rho$ and $phi$. This form &#8211; known in the machine-learning literature as Deep Sets &#8211; also generates a [&hellip;]<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[62,2246,112],"tags":[2247,1216,2248],"class_list":["post-2802","post","type-post","status-publish","format-standard","hentry","category-aimldsaimlds","category-astro-ph-co","category-stat-ml","tag-area","tag-form","tag-triangle"],"_links":{"self":[{"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/posts\/2802"}],"collection":[{"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/comments?post=2802"}],"version-history":[{"count":0,"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/posts\/2802\/revisions"}],"wp:attachment":[{"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/media?parent=2802"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/categories?post=2802"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/tags?post=2802"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}