[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-124810-en":3,"doc-seo-124810-105":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":4,"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},124810,4810365810221,"Aurora","https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d",8,"Research & Report","Total Variation Graph Neural Networks","The document presents a graph neural network approach for vertex clustering that improves on prior unsupervised methods trained with a spectral clustering (SC) relaxation of the minimum cut. SC relaxations are described as loose, producing overly smooth cluster assignments that separate vertices poorly. The proposed model optimizes a tighter relaxation using graph total variation (GTV), producing sharp transitions and balanced partitions. It includes a message-passing layer minimizing L1 feature differences and an unsupervised GTV-based loss, supporting direct clustering and graph pooling for classification tasks.","Total Variation Graph Neural Networks  \nJonas Berg Hansen * 1 Filippo Maria Bianchi * 1 2  \narXiv :2211 .06218v2 [ cs .LG] 27 Apr 2023  \nAbstract  \nRecently proposed Graph Neural Networks (GNNs) for vertex clustering are trained with an unsupervised minimum cut objective, approximated by a Spectral Clustering (SC) relaxation.  \nHowever, the SC relaxation is loose and, while itoffers a closed-form solution, it also yields overly smooth cluster assignments that poorly separate the vertices. In this paper, we propose a GNN model that computes cluster assignments by optimizing a tighter relaxation of the minimum cut based on graph total variation (GTV) . The cluster assignments can be used directly to perform vertex clustering or to implement graph pooling in a graph classification framework. Our model consists of two core components: i) a messagepassing layer that minimizes the l 1 distance in the features of adjacent vertices, which is key to achieving sharp transitions between clusters; ii) an unsupervised loss function that minimizes the GTV of the cluster assignments while ensuring balanced partitions. Experimental results show that our model outperforms other GNNs for vertex clustering and graph classification.  \n1. Introduction  \nTraditional clustering techniques partition samples based on their features or on suitable data representations computed, for example, with deep learning models (Tian et al., 2014; Min et al., 2018; Su et al., 2022) . Spectral clustering (SC) (Von Luxburg, 2007) is a popular technique that first encodes the similarity of the data features into a graph and then creates a partition based on the graph topology. Such a graph is just a convenient representation of the similarity among the samples and has no attributes on its vertices. On the other hand, an attributed graph can represent both  \n*Equal contribution 1Department of Mathematics and Statistics, UiT the Arctic University of Norway 2NORCE, The Norwegian Research Centre AS. Correspondence to: Filippo Maria Bianchi \u003Cfi[lippo.m.bianchi@uit.no](lippo.m.bianchi@uit.no)>.  \nProceedings of the 40th International Conference on Machine Learning, Honolulu, Hawaii, USA. PMLR 202, 2023 . Copyright 2023 by the author(s) .  \nFeatures-only  Topology-only  Features + Topology  \n(a)  \n\n| 0.8 | 0.2 |\n| --- | --- |\n\nSmooth assignment  \nSharp assignment  \n(b)  \nFigure 1: a) Most clustering methods partition the data only based on the features (􀂛) . SC partitions the vertices of the graph based on its topology (􀂛) . GNNs account both for the vertex features and the graph topology(􀂛) . b) Difference between smooth and sharp cluster assignments. Especially at the edge of a cluster, the smooth assignments give large weights to more than one cluster.  \nthe relationships among samples and their features. Graph Neural Networks (GNNs) are deep learning architectures specifically designed to process and make inference on such data (Hamilton, 2020) . Therefore, contrarily to traditional clustering methods, a GNN-based approach for clustering can account for both the features and the relationships among samples to generate partitions (see Fig.1a) .  \nSimilarly to other deep learning architectures for clustering (Shaham et al., 2018; Kampffmeyer et al., 2019), GNNs can be trained end-to-end and return soft cluster assignments as part of the output. Several existing GNN clustering approaches compute cluster assignments from the vertex representations generated by message passing (MP) layers and, then, optimize the assignments with an unsupervised loss inspired by SC (Bianchi et al., 2020; Tsitsulin et al., 2020; Duval & Malliaros, 2022) . The SC objective benefits from the smoothing operations performed by the MP layers, which minimize the local quadratic variation of adjacent  \nvertex features. However, this approach produces smooth cluster assignment vectors that are less informative as they do not separate well the samples (see Fig.1b) . Indeed, the SC objective is known to give a l","cbCaiuITLw3vzb7o","https://ap.wps.com/l/cbCaiuITLw3vzb7o","pdf",5015969,1,24,"English","en",105,"# Abstract\n# Introduction\n## Contributions\n# Background","[{\"question\":\"What limitation of spectral clustering relaxation motivates the proposed method?\",\"answer\":\"The SC relaxation is loose and yields overly smooth cluster assignments that do not separate vertices well.\"},{\"question\":\"How does the model compute cluster assignments?\",\"answer\":\"It optimizes a tighter relaxation of the minimum cut based on graph total variation (GTV), using a message-passing layer and an unsupervised GTV loss with balanced partitions.\"},{\"question\":\"How is the proposed approach used beyond vertex clustering?\",\"answer\":\"Learned partitions can be used for hierarchical graph pooling in graph classification frameworks, combining the proposed unsupervised loss with an additional supervised loss such as cross-entropy.\"}]","Total Variation Graph Neural Networks | PDF",1785894777,60,{"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":86,"head_meta":88,"extra_data":90,"updated_unix":28},"total-variation-graph-neural-networks","",{"@graph":36,"@context":85},[37,54,68],{"@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/total-variation-graph-neural-networks/124810/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":23,"description":14,"dateModified":62,"datePublished":62,"encodingFormat":61,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-08-05",true,{"@type":65,"interactionType":66,"userInteractionCount":4},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What limitation of spectral clustering relaxation motivates the proposed method?","Question",{"text":75,"@type":76},"The SC relaxation is loose and yields overly smooth cluster assignments that do not separate vertices well.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the model compute cluster assignments?",{"text":80,"@type":76},"It optimizes a tighter relaxation of the minimum cut based on graph total variation (GTV), using a message-passing layer and an unsupervised GTV loss with balanced partitions.",{"name":82,"@type":73,"acceptedAnswer":83},"How is the proposed approach used beyond vertex clustering?",{"text":84,"@type":76},"Learned partitions can be used for hierarchical graph pooling in graph classification frameworks, combining the proposed unsupervised loss with an additional supervised loss such as cross-entropy.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":89,"canonical":52},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,109,114,119,122,127,130,134],{"id":20,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":29,"slug":108},5,"Comic","comic",{"id":110,"doc_module":4,"doc_module_name":46,"category_name":111,"show_sort_weight":112,"slug":113},6,"Technology",50,"technology",{"id":115,"doc_module":4,"doc_module_name":46,"category_name":116,"show_sort_weight":117,"slug":118},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":120,"slug":121},30,"research-report",{"id":123,"doc_module":4,"doc_module_name":46,"category_name":124,"show_sort_weight":125,"slug":126},9,"Religion & Spirituality",20,"religion-spirituality",{"id":125,"doc_module":4,"doc_module_name":46,"category_name":128,"show_sort_weight":125,"slug":129},"World Cup","world-cup",{"id":131,"doc_module":4,"doc_module_name":46,"category_name":132,"show_sort_weight":131,"slug":133},10,"Lifestyle","lifestyle",{"id":135,"doc_module":4,"doc_module_name":46,"category_name":136,"show_sort_weight":106,"slug":137},19,"General","general"]