[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-127729-en":3,"doc-seo-127729-105":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":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":27,"seo_description":14,"update_tm":28,"read_time":29},127729,962084926284,"Aurora","https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0",8,"Research & Report","Alternating Local Enumeration (TnALE) - Solving Tensor Network Structure Search with Fewer Evaluations","Tensor network (TN) models are powerful in machine learning, yet TN structure search (TN-SS)—choosing an effective TN model topology and ranks—is computationally demanding. The TNLS method offers promising results but still requires too many objective-function evaluations. This work introduces TnALE, which alternately updates structure variables using local enumeration to sharply reduce evaluations. Theoretical analysis proves linear convergence under sufficient neighborhood objective decrease, and comparison shows TNLS typically needs Ω(2^K) evaluations, while TnALE ideally needs O(KR).","Alternating Local Enumeration (TnALE): Solving Tensor Network Structure Search with Fewer Evaluations  \nChao Li 1 Junhua Zeng * 2 1 Chunmei Li * 3 4 Cesar Caiafa 5 1 Qibin Zhao 1  \nAbstract  \nTensor network (TN) is a powerful framework in machine learning, but selecting a good TN model, known as TN structure search (TN-SS), is a challenging and computationally intensive task. The recent approach TNLS (Li et al., 2022) showed promising results for this task. However, its computational efficiency is still unaffordable, requiring too many evaluations of the objective function. We propose TnALE, a surprisingly simple algorithm that updates each structure-related variable alternately by local enumeration, greatly reducing the number of evaluations compared to TNLS. We theoretically investigate the descent steps for TNLS and TnALE, proving that both the algorithms can achieve linear convergence up to a constant if a sufficient reduction of the objective is reached in each neighborhood. We further compare the evaluation efficiency of TNLSand TnALE, revealing that Ω(2K ) evaluations are typically required in TNLS for reaching the objective reduction, while ideally O (KR) evaluations are sufficient in TnALE, where K denotes the dimension of search space and R reflects the “low-rankness” of the neighborhood. Experimental results verify that TnALE can find practically good TN structures with vastly fewer evaluations than the state-of-the-art algorithms. Our code is available at [https://github.com/](https://github.com/)[ ](https://github.com/)ChaoLiAtRIKEN/TnALE.  \n*Equal contribution 1RIKEN-AIP, Tokyo, Japan 2 School of Automation, Guangdong University of Technology, Guangzhou, China 3 College of Information and Communication Engineering, Harbin Engineering University, Harbin, China 4Department of Computer Science and Communications Engineering, WASEDA University, Tokyo, Japan 5Instituto Argentino de Radioastronom´ıa, CONICETCCT La Plata/CIC-PBA/UNLP, V. Elisa, ARGENTINA. Correspondence to: Qibin Zhao \u003C[qibin.zhao@riken.jp](qibin.zhao@riken.jp) >, Chao Li \u003C[chao.li@riken.jp](chao.li@riken.jp) >.  \nProceedings of the 40 th International Conference on Machine Learning, Honolulu, Hawaii, USA. PMLR 202, 2023 . Copyright 2023 by the author(s) .  \n1. Introduction  \nTensor network (TN) has been widely applied to solving high-dimensional problems in both machine learning (Anandkumar et al., 2014 ; Novikov et al., 2015 ; Zheet al., 2015 ; Glasser et al., 2019 ; Kossaifi et al., 2020 ; Miller et al., 2021 ; Richter et al., 2021 ; Malik, 2022) and quantum physics (Or´us, 2019 ; Felser et al., 2021) . The success of AlphaTensor (Fawzi et al., 2022) once again confirmed the usefulness of tensors in various fields. TN practitioners, on the other hand, have to face in practice challenging problems associated with model selection, known as TN structure search (TN-SS), for example: (1) how to determine the TN-ranks?; (2) should we prefer tensor-train (TT, Oseledets 2011), tensor-ring (TR, Zhao et al. 2016) or other TN topology?; (3) how to relate the tensor modes to core tensors of a TN (the permutation problem, Li et al. 2022), and so on. Unfortunately, some of these problems have been proven tobe NP-hard (Hillar & Lim, 2013)1 , and most of them suffer from the “combinatorial explosion”2 so that the brute force search is not a viable option in practice.  \nSeveral works have put effort into different aspects of TNSS (see Section 1.1), but many of the methods are restricted to specific tasks or work poorly in practice, so a general and efficient TN-SS method is needed. Recently, Li et al. (2022) proposed an algorithm dubbed TNLS, which addressed the rank and permutation selection problem for TNs. However, its computational complexity is high since it requires evaluating the objective function on a large number of structure candidates.  \nTo address this issue, we accelerate TNLS by equipping the algorithm with Alternating Local Enumeration—a surprisingly ","cbCaikr7iPe9S7o0","https://ap.wps.com/l/cbCaikr7iPe9S7o0","pdf",2731885,1,28,"English","en",105,"# Introduction\n## Tensor network structure search challenges\n## TNLS and its computational cost\n## TnALE: alternating local enumeration\n## Theoretical convergence and evaluation complexity\n## Experimental validation","[{\"question\":\"What problem does TnALE address in tensor network structure search?\",\"answer\":\"TnALE targets the high computational cost of TN structure search by reducing the number of objective-function evaluations required to select a good tensor network model.\"},{\"question\":\"How does TnALE improve efficiency compared with TNLS?\",\"answer\":\"TnALE alternately updates structure-related variables by enumerating candidates within neighborhoods, avoiding combinatorial explosion and ensuring non-increasing objective values, unlike TNLS which relies on random sampling.\"},{\"question\":\"What do the theoretical results say about convergence and evaluations?\",\"answer\":\"Both TNLS and TnALE achieve linear convergence up to a constant under sufficient objective reduction in each neighborhood; additionally, TNLS evaluation needs typically grow exponentially with search dimension, whereas TnALE grows linearly in the ideal case.\"}]","Alternating Local Enumeration (TnALE) - Solving Tensor Network Structure Search with Fewer Evaluations | PDF",1785941295,71,{"code":4,"msg":31,"data":32},"ok",{"site_id":24,"language":23,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":87,"head_meta":89,"extra_data":91,"updated_unix":28},"alternating-local-enumeration-tnale-solving-tensor-network-structure-search-with-fewer-evaluations","",{"@graph":36,"@context":86},[37,54,69],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":20},"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/alternating-local-enumeration-tnale-solving-tensor-network-structure-search-with-fewer-evaluations/127729/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":23,"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","2026-08-23","2026-08-05",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 problem does TnALE address in tensor network structure search?","Question",{"text":76,"@type":77},"TnALE targets the high computational cost of TN structure search by reducing the number of objective-function evaluations required to select a good tensor network model.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"How does TnALE improve efficiency compared with TNLS?",{"text":81,"@type":77},"TnALE alternately updates structure-related variables by enumerating candidates within neighborhoods, avoiding combinatorial explosion and ensuring non-increasing objective values, unlike TNLS which relies on random sampling.",{"name":83,"@type":74,"acceptedAnswer":84},"What do the theoretical results say about convergence and evaluations?",{"text":85,"@type":77},"Both TNLS and TnALE achieve linear convergence up to a constant under sufficient objective reduction in each neighborhood; additionally, TNLS evaluation needs typically grow exponentially with search dimension, whereas TnALE grows linearly in the ideal case.","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":24},{"code":4,"msg":5,"data":93},[94,98,102,106,111,116,121,124,129,132,136],{"id":20,"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":107,"doc_module":4,"doc_module_name":46,"category_name":108,"show_sort_weight":109,"slug":110},5,"Comic",60,"comic",{"id":112,"doc_module":4,"doc_module_name":46,"category_name":113,"show_sort_weight":114,"slug":115},6,"Technology",50,"technology",{"id":117,"doc_module":4,"doc_module_name":46,"category_name":118,"show_sort_weight":119,"slug":120},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":122,"slug":123},30,"research-report",{"id":125,"doc_module":4,"doc_module_name":46,"category_name":126,"show_sort_weight":127,"slug":128},9,"Religion & Spirituality",20,"religion-spirituality",{"id":127,"doc_module":4,"doc_module_name":46,"category_name":130,"show_sort_weight":127,"slug":131},"World Cup","world-cup",{"id":133,"doc_module":4,"doc_module_name":46,"category_name":134,"show_sort_weight":133,"slug":135},10,"Lifestyle","lifestyle",{"id":137,"doc_module":4,"doc_module_name":46,"category_name":138,"show_sort_weight":107,"slug":139},19,"General","general"]