[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83076-en":3,"doc-seo-83076-105":30,"detail-sidebar-cat-0-en-105":92},{"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},83076,1374391975076,"Riley","https://ap-avatar.wpscdn.com/avatar/14000253ca4ec9f6853?x-image-process=image/resize,m_fixed,w_180,h_180&k=1783305029341752051",8,"Research & Report","Quantitative Gaussian-Process Limits of Tensor Programs","The document studies the infinite-width Gaussian-process limit of random neural networks using the formalism of tensor programs. It establishes a quantitative convergence theory in Wasserstein distance, producing explicit finite-width error bounds between finite-network executions and their Gaussian-process limits with rate on the order of the inverse square root of layer widths. The approach is architecture-agnostic, covering feed-forward models and weight-sharing schemes for recurrent and transformer-type architectures.","arXiv :2607 .06290v 1 [ cs .LG] 7 Jul 2026  \nQUANTITATIVE GAUSSIAN-PROCESS LIMITS OF TENSOR  \nPROGRAMS  \nANDREA AGAZZI, ELOY MOSIG GARC´IA, AND DARIO TREVISAN  \nAbstract. We study the infinite-width Gaussian-process limit of random neural net  \nworks through the lens of tensor programs, and we provide a quantitative convergence  \ntheory in Wasserstein distance. Our main result gives explicit finite-width error bounds,  \nof order inverse square-root of the widths between finite-network executions and their  \nGaussian-process limits. The framework is architecture-agnostic and covers feed-forward  \nmodels together with weight-sharing schemes relevant for recurrent and transformer-type  \narchitectures.  \n1. Introduction  \nA depth-M Multilayer Perceptron (MLP) with input x ∈ Rd and layer widths (n0 , ... , nM ) is defined as the map h (M) : Rn0 → RnM , where h(0)(x) := x and  \n(1.1) h (ℓ+1)(x) := ϕℓ 􀀐 W (ℓ)h (ℓ)(x)􀀑 , for ℓ = 0 ,..., M − 1 ,  \nwith activation functions ϕ ℓ : R → R acting componentwise on their input, and parameters (or weights) W (ℓ) ∈ Rn ℓ+1×n ℓ for ℓ ∈ {0,   M − 1} . These parameters are typically initialized randomly, drawn independently from a common (layer-dependent) distribution. A standard practice is to choose such distribution as Gaussian centered with O (n1 ) variance, for instance  \n(1.2) Wi(ℓj) ∼iid N(0, n1 ) .  \nUnder this choice of scaling, when the widths n 1 , . . . , nM−1 go to infinity, the pre-activationsg (ℓ) := W (ℓ)h (ℓ) converge to a centered Gaussian process whose kernel is given by an explicit recursion in ϕ and in the input Gram matrix. This is the neural-network Gaussian-process (NNGP) limit, first identified by [Nea96] for M = 2 and later studied for generic integer M ≥ 2 by [Lee+20] . Quantitative versions of these Central Limit Theorem (CLT) results were obtained in [BT22; Fav+25; Tre23] and more recently in [Cel26; GKR26] . This work extends such quantitative convergence results to a significantly richer class of functions called tensor programs defined below.  \n1.1. Tensor programs. As discussed in [Yan21], the above function class, as well as many other neural network architectures, can be interpreted as a specific instance of abstract algorithms, called Netsor programs. These are a subclass of the more general tensor programs [Yan20b; YH20], consisting of a sequence of lines where variables are declared and then combined via elementary operations. In particular, a Netsor program T is defined by specifying its inputs and its operations as follows:  \nInputs. The inputs 1 of a Netsor program are associated to two different variable types:  \n• (Input) H-vars are a collection of fixed real vectors  \nX := {xk}k xk ∈ Rd k  \ngeneralizing the input data on which the network acts. The dimensions of this collection are stored in set nin := {dk}k . These variables belong to the class of  \nDate: July 8, 2026 .  \n1a set of variables that are declared at the beginning of the program, and whose definition does not rely on other variables.  \n2 ANDREA AGAZZI, ELOY MOSIG GARC´IA, AND DARIO TREVISAN  \nH-vars, i.e. , real, vector-valued quantities interpretable as hidden states of the network.  \n• A-vars denote an abstract collection of matrix-valued parameters  \nW := {W(j)}j ,  \nrepresenting the weights (and possibly biases) of the network.  \nConforming to the notation in [Yan21], we recall the type and dimension of a newly declared variable through the : notation2 , so that any A-var W of dimension n × m will be declared as W : A (n, m), while H-vars (and G-vars defined below) of dimension n will  \nbe respectively introduced as h : H (n) and g : G (n) .  \nOperations. The variables defined above can be combined through the following operations  \n• MatMul: The Matrix-vector multiplication operation combines a W : A (n, m) and a h : H (m) . The result of this operation is a new n-dimensional vector-valued H-var behaving as a Gaussian when the dimension being contracted diverges. Denoting the subset of H","cbCainNj7uPVprV0","https://ap.wps.com/l/cbCainNj7uPVprV0","pdf",682104,5,1,37,"English","en",105,"# Abstract\n# Introduction\n## Tensor programs","[{\"question\":\"What limit does the paper analyze for random neural networks?\",\"answer\":\"It analyzes the infinite-width Gaussian-process limit, expressing the behavior of randomly initialized networks through tensor programs and their associated Gaussian-process limits.\"},{\"question\":\"What type of convergence guarantee is provided?\",\"answer\":\"The paper provides quantitative convergence in Wasserstein distance, including explicit finite-width error bounds between finite networks and their Gaussian-process limits.\"},{\"question\":\"Which network architectures does the framework cover?\",\"answer\":\"The framework is architecture-agnostic and covers feed-forward models as well as weight-sharing schemes relevant to recurrent and transformer-type architectures.\"}]",1784185034,93,{"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":87,"head_meta":89,"extra_data":91,"updated_unix":28},"quantitative-gaussian-process-limits-of-tensor-programs","",{"@graph":36,"@context":86},[37,54,69],{"@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":53},"https://docshare.wps.com/document/quantitative-gaussian-process-limits-of-tensor-programs/83076/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":24,"description":14,"dateModified":62,"datePublished":63,"encodingFormat":61,"isAccessibleForFree":64,"interactionStatistic":65},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-07-24","2026-07-16",true,{"@type":66,"interactionType":67,"userInteractionCount":20},"InteractionCounter",{"@type":68},"ViewAction",{"@type":70,"mainEntity":71},"FAQPage",[72,78,82],{"name":73,"@type":74,"acceptedAnswer":75},"What limit does the paper analyze for random neural networks?","Question",{"text":76,"@type":77},"It analyzes the infinite-width Gaussian-process limit, expressing the behavior of randomly initialized networks through tensor programs and their associated Gaussian-process limits.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"What type of convergence guarantee is provided?",{"text":81,"@type":77},"The paper provides quantitative convergence in Wasserstein distance, including explicit finite-width error bounds between finite networks and their Gaussian-process limits.",{"name":83,"@type":74,"acceptedAnswer":84},"Which network architectures does the framework cover?",{"text":85,"@type":77},"The framework is architecture-agnostic and covers feed-forward models as well as weight-sharing schemes relevant to recurrent and transformer-type architectures.","https://schema.org",{"og:url":52,"og:type":88,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":90,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":93},[94,98,102,106,110,115,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":95,"show_sort_weight":96,"slug":97},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":99,"show_sort_weight":100,"slug":101},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":103,"show_sort_weight":104,"slug":105},"Exam",70,"exam",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":20,"slug":138},19,"General","general"]