{"id":10091,"date":"2026-01-29T07:02:45","date_gmt":"2026-01-29T07:02:45","guid":{"rendered":"https:\/\/mailitics.com\/index.php\/2026\/01\/29\/2601-20047\/"},"modified":"2026-01-29T07:02:45","modified_gmt":"2026-01-29T07:02:45","slug":"2601-20047","status":"publish","type":"post","link":"https:\/\/mailitics.com\/index.php\/2026\/01\/29\/2601-20047\/","title":{"rendered":"Minimax Rates for Hyperbolic Hierarchical Learning"},"content":{"rendered":"

Minimax Rates for Hyperbolic Hierarchical Learning
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arXiv:2601.20047v1 Announce Type: new
\nAbstract: We prove an exponential separation in sample complexity between Euclidean and hyperbolic representations for learning on hierarchical data under standard Lipschitz regularization. For depth-$R$ hierarchies with branching factor $m$, we first establish a geometric obstruction for Euclidean space: any bounded-radius embedding forces volumetric collapse, mapping exponentially many tree-distant points to nearby locations. This necessitates Lipschitz constants scaling as $exp(Omega(R))$ to realize even simple hierarchical targets, yielding exponential sample complexity under capacity control. We then show this obstruction vanishes in hyperbolic space: constant-distortion hyperbolic embeddings admit $O(1)$-Lipschitz realizability, enabling learning with $n = O(mR log m)$ samples. A matching $Omega(mR log m)$ lower bound via Fano’s inequality establishes that hyperbolic representations achieve the information-theoretic optimum. We also show a geometry-independent bottleneck: any rank-$k$ prediction space captures only $O(k)$ canonical hierarchical contrasts.<\/div>\n

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\n Divit Rawal, Sriram Vishwanath
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Minimax Rates for Hyperbolic Hierarchical Learning arXiv:2601.20047v1 Announce Type: new Abstract: We prove an exponential separation in sample complexity between Euclidean and hyperbolic representations for learning on hierarchical data under standard Lipschitz regularization. For depth-$R$ hierarchies with branching factor $m$, we first establish a geometric obstruction for Euclidean space: any bounded-radius embedding forces volumetric collapse, […]<\/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":[1173,4691,199],"class_list":["post-10091","post","type-post","status-publish","format-standard","hentry","category-aimldsaimlds","category-cs-lg","category-stat-ml","tag-hierarchical","tag-hyperbolic","tag-learning"],"_links":{"self":[{"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/posts\/10091"}],"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=10091"}],"version-history":[{"count":0,"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/posts\/10091\/revisions"}],"wp:attachment":[{"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/media?parent=10091"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/categories?post=10091"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/tags?post=10091"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}