{"id":6801,"date":"2025-09-12T07:03:31","date_gmt":"2025-09-12T07:03:31","guid":{"rendered":"https:\/\/mailitics.com\/index.php\/2025\/09\/12\/2509-09353\/"},"modified":"2025-09-12T07:03:31","modified_gmt":"2025-09-12T07:03:31","slug":"2509-09353","status":"publish","type":"post","link":"https:\/\/mailitics.com\/index.php\/2025\/09\/12\/2509-09353\/","title":{"rendered":"Low-degree lower bounds via almost orthonormal bases"},"content":{"rendered":"

Low-degree lower bounds via almost orthonormal bases
\n \t

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

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

arXiv:2509.09353v1 Announce Type: new
\nAbstract: Low-degree polynomials have emerged as a powerful paradigm for providing evidence of statistical-computational gaps across a variety of high-dimensional statistical models [Wein25]. For detection problems — where the goal is to test a planted distribution $mathbb{P}’$ against a null distribution $mathbb{P}$ with independent components — the standard approach is to bound the advantage using an $mathbb{L}^2(mathbb{P})$-orthonormal family of polynomials. However, this method breaks down for estimation tasks or more complex testing problems where $mathbb{P}$ has some planted structures, so that no simple $mathbb{L}^2(mathbb{P})$-orthogonal polynomial family is available. To address this challenge, several technical workarounds have been proposed [SW22,SW25], though their implementation can be delicate. In this work, we propose a more direct proof strategy. Focusing on random graph models, we construct a basis of polynomials that is almost orthonormal under $mathbb{P}$, in precisely those regimes where statistical-computational gaps arise. This almost orthonormal basis not only yields a direct route to establishing low-degree lower bounds, but also allows us to explicitly identify the polynomials that optimize the low-degree criterion. This, in turn, provides insights into the design of optimal polynomial-time algorithms. We illustrate the effectiveness of our approach by recovering known low-degree lower bounds, and establishing new ones for problems such as hidden subcliques, stochastic block models, and seriation models.<\/div>\n

\t

\n
<\/BR>
\n Alexandra Carpentier (LMO, CELESTE), Simone Maria Giancola (LMO, CELESTE), Christophe Giraud (LMO, CELESTE), Nicolas Verzelen (MISTEA)
\n \t

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

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

Low-degree lower bounds via almost orthonormal bases arXiv:2509.09353v1 Announce Type: new Abstract: Low-degree polynomials have emerged as a powerful paradigm for providing evidence of statistical-computational gaps across a variety of high-dimensional statistical models [Wein25]. For detection problems — where the goal is to test a planted distribution $mathbb{P}’$ against a null distribution $mathbb{P}$ with independent […]<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[62,113,112],"tags":[1764,588,3770],"class_list":["post-6801","post","type-post","status-publish","format-standard","hentry","category-aimldsaimlds","category-cs-lg","category-stat-ml","tag-degree","tag-low","tag-mathbb"],"_links":{"self":[{"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/posts\/6801"}],"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=6801"}],"version-history":[{"count":0,"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/posts\/6801\/revisions"}],"wp:attachment":[{"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/media?parent=6801"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/categories?post=6801"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/tags?post=6801"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}