{"id":6174,"date":"2025-08-19T07:02:37","date_gmt":"2025-08-19T07:02:37","guid":{"rendered":"https:\/\/mailitics.com\/index.php\/2025\/08\/19\/2508-12627\/"},"modified":"2025-08-19T07:02:37","modified_gmt":"2025-08-19T07:02:37","slug":"2508-12627","status":"publish","type":"post","link":"https:\/\/mailitics.com\/index.php\/2025\/08\/19\/2508-12627\/","title":{"rendered":"On computing and the complexity of computing higher-order $U$-statistics, exactly"},"content":{"rendered":"

On computing and the complexity of computing higher-order $U$-statistics, exactly
\n \t

\n
<\/BR>
\n
\n \t

\n
<\/BR><\/p>\n

arXiv:2508.12627v1 Announce Type: new
\nAbstract: Higher-order $U$-statistics abound in fields such as statistics, machine learning, and computer science, but are known to be highly time-consuming to compute in practice. Despite their widespread appearance, a comprehensive study of their computational complexity is surprisingly lacking. This paper aims to fill that gap by presenting several results related to the computational aspect of $U$-statistics. First, we derive a useful decomposition from an $m$-th order $U$-statistic to a linear combination of $V$-statistics with orders not exceeding $m$, which are generally more feasible to compute. Second, we explore the connection between exactly computing $V$-statistics and Einstein summation, a tool often used in computational mathematics, quantum computing, and quantum information sciences for accelerating tensor computations. Third, we provide an optimistic estimate of the time complexity for exactly computing $U$-statistics, based on the treewidth of a particular graph associated with the $U$-statistic kernel. The above ingredients lead to a new, much more runtime-efficient algorithm of exactly computing general higher-order $U$-statistics. We also wrap our new algorithm into an open-source Python package called $texttt{u-stats}$. We demonstrate via three statistical applications that $texttt{u-stats}$ achieves impressive runtime performance compared to existing benchmarks. This paper aspires to achieve two goals: (1) to capture the interest of researchers in both statistics and other related areas further to advance the algorithmic development of $U$-statistics, and (2) to offer the package $texttt{u-stats}$ as a valuable tool for practitioners, making the implementation of methods based on higher-order $U$-statistics a more delightful experience.<\/div>\n

\t

\n
<\/BR>
\n Xingyu Chen, Ruiqi Zhang, Lin Liu
\n \t

\n
<\/BR>
\nGo to original source<\/a>
\n \t

\n
<\/BR><\/p>\n","protected":false},"excerpt":{"rendered":"

On computing and the complexity of computing higher-order $U$-statistics, exactly arXiv:2508.12627v1 Announce Type: new Abstract: Higher-order $U$-statistics abound in fields such as statistics, machine learning, and computer science, but are known to be highly time-consuming to compute in practice. Despite their widespread appearance, a comprehensive study of their computational complexity is surprisingly lacking. This paper […]<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[62,413,450,451,482,183,112],"tags":[97,1424,2530],"class_list":["post-6174","post","type-post","status-publish","format-standard","hentry","category-aimldsaimlds","category-cs-ds","category-cs-na","category-math-na","category-stat-co","category-stat-me","category-stat-ml","tag-computing","tag-order","tag-statistics"],"_links":{"self":[{"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/posts\/6174"}],"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=6174"}],"version-history":[{"count":0,"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/posts\/6174\/revisions"}],"wp:attachment":[{"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/media?parent=6174"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/categories?post=6174"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/tags?post=6174"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}