[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81482-en":3,"doc-seo-81482-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},81482,1099513958607,"Jiven","https://ap-avatar.wpscdn.com/avatar/100002390cf8733938c?x-image-process=image/resize,m_fixed,w_180,h_180&k=1778829742770036399",8,"Research & Report","A Fourier Analytic Approach to Gaussian Mixture Learning","Independent identical samples from a uniform mixture of spherical Gaussians in R^d are considered, with known common coordinate variance and well-separated component centers under an l2 distance condition. A randomized algorithm is developed to learn the Gaussian centers within l2 error  k^{-ε0}, with polynomial samples and runtime in the number of components and dimension. For sufficiently large k, it also learns unknown weights within specified accuracy and succeeds with high probability. The work provides new sample and computational complexity guarantees beyond constant dimension.","arXiv :2004 .058 13v 3 [ cs .DS] 10 Jul 2026  \nA Fourier analytic approach to Gaussian mixture learning  \nSomnath Chakraborty (􀀰), Hariharan Narayanan (􀀱)  \nAbstract Suppose that we are given independent, identically distributed random samples 􀁇1 , · · · , 􀁇 􀀽 from a mixture at most 􀀺 many 􀀳-dimensional spherical Gaussian distributions 􀁠 1 , · · · , 􀁠 􀀺0 of identical and known variance 􀁦2 in each coordinate, such that the minimum ℓ2 distance between two distinct centers 􀁈 􀀻 and 􀁈 􀀹 is greater than 2Δ􀁦 min{ √􀀳  , √􀀺  }, where Δ > 􀀘0 , and 􀀘0 is a sufficiently large universal constant. We develop a randomized algorithm that lea˜rns the centers 􀁈 􀀻’s of the Gaussian components to within an ℓ2 distance of 􀀺 −􀀘0 — in presence of arbitrarily large number of components and in arbitrary dimension, when the weights are known to be uniform. Furthermore, if the number of components is 􀀺 = Ω (2􀀳 ), then for arbitrary universal constant 􀀲 > 0, even for˜ unknown weights, the algorithm learns the centers to within an ℓ2 distance of 􀀳 −􀀘0 and the weights up to an accuracy of 􀀲􀁼 􀀼􀀸􀀽, with probability greater than 1 − exp(−􀀺/􀀲), provided that the weights lie in [􀀲/􀀺, 1/􀀲􀀺], and the minimum separation is just 2􀀲 √􀀳 . The number of samples and the computational time is bounded above by poly (􀀺, 􀀳) in either case. Such abound on the sample and computational complexity was previously unknown in the regime of non-constant dimension, and in particular, when 􀀳 is not 􀀤 (1), When 􀀳 = 􀀤 (1), this complexity bound follows from [16], where it has also been shown that the sample complexity of learning a random mixture of Gaussians in a ball of radius 􀀾( √􀀳) in 􀀳 dimensions, when 􀀳 is Θ (log 􀀺), is at least super-polynomial in 􀀺, 􀀳, showing that our result is tight in this case.  \nKey words: mixtures of Gaussians, Lipschitz function, Hausdorff distance, sample complexity bound, approximate log-concave functions, PCA, Fourier transform, Johnson-Lindenstrauss lemma,(de)convolution, Hoeffding’s inequality  \nSchool of Technology and Computer Science, Tata Institute of Fundamental Research, Mumbai 400005, India  \nORCIDs: (􀀰) 0000-0002-9854-0181,(􀀱) 0000-0002-4331-8794  \nEmail: (􀀰) [somnath.chakraborty@alumni.iu.edu](somnath.chakraborty@alumni.iu.edu), (􀀱) [hariharan.narayanan@tifr.res.in](hariharan.narayanan@tifr.res.in)  \n1 Introduction  \nDesigning efficient algorithms that estimate the parameters of an underlying probability distribution is a central theme in statistical learning theory. An important special instance of this learning problem is the case when the underlying distribution is known to be a finite mixture of Gaussian distributions in 􀀳-dimensional Euclidean space. Such mixtures are popular models for high-dimensional data clustering, and learning mixture of Gaussiansin an unsupervised setting has been a topic of intensive research for the last few decades.  \nIn its most general form, the underlying problem is as follows: we have access to random samples drawn independently from some Gaussian mixture 􀁠 := 􀁬 1􀁠 1 + · · · + 􀁬 􀀺0 􀁠 􀀺0 , where (􀁬1 , · · · , 􀁬 􀀺0 ) is a probability vector with strictly positive components, and each 􀁠 􀀻 is a Gaussian density in R􀀳, with mean 􀁈 􀀻 ∈ R􀀳 and covariance Σ 􀀻 . The algorithmic task is to estimate each component of the parameter set {(􀁬1, 􀁈 1 , Σ 1 ) , · · · ,(􀁬 􀀺0 , 􀁈 􀀺0 , Σ 􀀺0 )} of the density function 􀁠, within a pre-specified accuracy 􀁮 > 0, with success probability at least 1−􀁘 for a pre-specified 􀁘 > 0. For the purposes of this paper, we will restrict ourselves to the case where all the Gaussian components are spherical, with identical variance in each coordinates. Recently,[7] and [8] devised polynomial time learning algorithms that work with minimum separation of 􀀺 􀁮 (for any 􀁮 > 0), although, the sucess of these algorithms appear to be guaranteed only in some restricted region in the (􀀳, 􀀺)-space (see table 1.1) . In [16], Regev-Vijayaraghavan considered the question of obtaining a lower bound on the se","cbCaiagugqwiRK6o","https://ap.wps.com/l/cbCaiagugqwiRK6o","pdf",396562,4,1,35,"English","en",105,"# Introduction\n## Problem setup and mixture learning objective\n## Prior work and known separation regimes\n## Contributions and algorithmic approach","[{\"question\":\"What learning task does the paper address for Gaussian mixtures?\",\"answer\":\"It targets parameter estimation for a finite mixture of spherical Gaussians, focusing on recovering the component centers (and, in a stronger regime, the weights) to a prescribed l2 accuracy with high probability.\"},{\"question\":\"What separation condition between Gaussian centers is assumed?\",\"answer\":\"The centers must be separated in minimum l2 distance by at least 2Δ min{√d, √k} (and a corresponding scaling in the logarithmic-dimension regime), where Δ exceeds a universal constant threshold.\"},{\"question\":\"How does the proposed algorithm achieve its guarantees?\",\"answer\":\"It uses randomized learning combined with deconvolution in the Fourier domain, applying a carefully chosen cutoff to avoid instability, and derives polynomial bounds on sample and computation complexity.\"}]",1784173732,88,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"a-fourier-analytic-approach-to-gaussian-mixture-learning","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":20},"https://docshare.wps.com/document/a-fourier-analytic-approach-to-gaussian-mixture-learning/81482/",{"url":52,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-25","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What learning task does the paper address for Gaussian mixtures?","Question",{"text":75,"@type":76},"It targets parameter estimation for a finite mixture of spherical Gaussians, focusing on recovering the component centers (and, in a stronger regime, the weights) to a prescribed l2 accuracy with high probability.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What separation condition between Gaussian centers is assumed?",{"text":80,"@type":76},"The centers must be separated in minimum l2 distance by at least 2Δ min{√d, √k} (and a corresponding scaling in the logarithmic-dimension regime), where Δ exceeds a universal constant threshold.",{"name":82,"@type":73,"acceptedAnswer":83},"How does the proposed algorithm achieve its guarantees?",{"text":84,"@type":76},"It uses randomized learning combined with deconvolution in the Fourier domain, applying a carefully chosen cutoff to avoid instability, and derives polynomial bounds on sample and computation complexity.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & 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