{"id":4572,"date":"2025-06-13T07:02:32","date_gmt":"2025-06-13T07:02:32","guid":{"rendered":"https:\/\/mailitics.com\/index.php\/2025\/06\/13\/2506-10101\/"},"modified":"2025-06-13T07:02:32","modified_gmt":"2025-06-13T07:02:32","slug":"2506-10101","status":"publish","type":"post","link":"https:\/\/mailitics.com\/index.php\/2025\/06\/13\/2506-10101\/","title":{"rendered":"Fundamental Limits of Learning High-dimensional Simplices in Noisy Regimes"},"content":{"rendered":"

Fundamental Limits of Learning High-dimensional Simplices in Noisy Regimes
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arXiv:2506.10101v1 Announce Type: new
\nAbstract: In this paper, we establish sample complexity bounds for learning high-dimensional simplices in $mathbb{R}^K$ from noisy data. Specifically, we consider $n$ i.i.d. samples uniformly drawn from an unknown simplex in $mathbb{R}^K$, each corrupted by additive Gaussian noise of unknown variance. We prove an algorithm exists that, with high probability, outputs a simplex within $ell_2$ or total variation (TV) distance at most $varepsilon$ from the true simplex, provided $n ge (K^2\/varepsilon^2) e^{mathcal{O}(K\/mathrm{SNR}^2)}$, where $mathrm{SNR}$ is the signal-to-noise ratio. Extending our prior work~citep{saberi2023sample}, we derive new information-theoretic lower bounds, showing that simplex estimation within TV distance $varepsilon$ requires at least $n ge Omega(K^3 sigma^2\/varepsilon^2 + K\/varepsilon)$ samples, where $sigma^2$ denotes the noise variance. In the noiseless scenario, our lower bound $n ge Omega(K\/varepsilon)$ matches known upper bounds up to constant factors. We resolve an open question by demonstrating that when $mathrm{SNR} ge Omega(K^{1\/2})$, noisy-case complexity aligns with the noiseless case. Our analysis leverages sample compression techniques (Ashtiani et al., 2018) and introduces a novel Fourier-based method for recovering distributions from noisy observations, potentially applicable beyond simplex learning.<\/div>\n

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\n Seyed Amir Hossein Saberi, Amir Najafi, Abolfazl Motahari, Babak H. khalaj
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Fundamental Limits of Learning High-dimensional Simplices in Noisy Regimes arXiv:2506.10101v1 Announce Type: new Abstract: In this paper, we establish sample complexity bounds for learning high-dimensional simplices in $mathbb{R}^K$ from noisy data. Specifically, we consider $n$ i.i.d. samples uniformly drawn from an unknown simplex in $mathbb{R}^K$, each corrupted by additive Gaussian noise of unknown variance. We […]<\/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":[1197,2954,2953],"class_list":["post-4572","post","type-post","status-publish","format-standard","hentry","category-aimldsaimlds","category-cs-lg","category-stat-ml","tag-noisy","tag-simplex","tag-varepsilon"],"_links":{"self":[{"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/posts\/4572"}],"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=4572"}],"version-history":[{"count":0,"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/posts\/4572\/revisions"}],"wp:attachment":[{"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/media?parent=4572"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/categories?post=4572"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/tags?post=4572"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}