[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-128860-105":59,"doc-detail-128860-en":130},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":123,"head_meta":125,"extra_data":127,"updated_unix":129},105,"en","spectrally-approximating-large-graphs-with-smaller-graphs","Spectrally approximating large graphs with smaller graphs","","Spectrally approximating large graphs with smaller graphs addresses how graph coarsening changes the spectrum of the graph Laplacian. It derives conditions under which principal eigenvalues and eigenvectors of the coarsened and original Laplacians remain close. The approximation depends on graph-theoretic degree and eigenvalue distributions and on the size ratio between coarsened and original graphs. The results yield guarantees for spectral clustering by justifying the quality of lifted coarse eigenvectors without refinement.",{"@graph":69,"@context":122},[70,84,105],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":40,"@type":76,"position":81},"https://docshare.wps.com/document/research-report/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/spectrally-approximating-large-graphs-with-smaller-graphs/128860/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/spectrally-approximating-large-graphs-with-smaller-graphs/128860.png","ImageObject",300,407,{"name":92,"@type":93},"Noah","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-09-18","2026-08-06",true,{"@type":102,"interactionType":103,"userInteractionCount":29},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What property does the paper introduce to relate the spectra of the original and coarsened graphs?","Question",{"text":112,"@type":113},"It introduces the restricted spectral similarity (RSS) property, which states that the Laplacians behave similarly (up to constants) with respect to an appropriate set of vectors.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"How does the approximation quality depend on graph structure?",{"text":117,"@type":113},"The achieved approximation depends on degree and eigenvalue distributions of the graph, and it is further controlled by the ratio between the coarsened and original graph sizes.",{"name":119,"@type":110,"acceptedAnswer":120},"What does the theory imply for spectral clustering?",{"text":121,"@type":113},"It shows that lifted eigenvectors from the coarsened graph can produce high-quality clustering assignments even without refinement, providing formal justification for an effect previously observed empirically.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},128860,1786003994,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":29,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":139,"language":140,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":67,"update_tm":129,"read_time":144},137451207643,"https://ap-avatar.wpscdn.com/davatar_3d24733baf745e90a7e4bdd5f77d97b2","arXiv : 1802 .075 10v 1 [cs .LG] 2 1 Feb 2018  \nSpectrally approximating large graphs with smaller graphs  \nAndreas Loukas, Pierre Vandergheynst  \n􀀓  \nEcole Polytechnique F􀀓ed􀀓erale Lausanne, Switzerland  \nAbstract  \nHow does coarsening a􀀋ect the spectrum of a general graph? We provide conditions such that the principal eigenvalues and eigenspaces of a coarsened and original graph Laplacian matrices are close. The achieved approximation is shown to depend on standard graph-theoretic properties, such as the degree and eigenvalue distributions, as well as on the ratio between the coarsened and actual graph sizes. Our results carry implications for learning methods that utilize coarsening. For the particular case of spectral clustering, they imply that coarse eigenvectors can be used to derive good quality assignments even without re􀀌nement—this phenomenon was previously observed, but lacked formal justi􀀌cation.  \n1 Introduction  \nOne of the most wide-spread techniques for sketching graph-structured data is coarsening. As with most sketching methods, instead of solving a large graph problem in its native domain, coarsening involves solving an akin problem of reduced size at a lower cost; the solution can then be inexpensively lifted and re􀀌ned in the native domain.  \nThe bene􀀌ts of coarsening are well known both in the algorithmic and machine learning communities. There exists a long list of algorithms that utilize it for partitioning [15, 18, 22, 11, 39] and visualizing [20, 38] large graphs in a computationally e􀀎cient manner. In addition, it has been frequently used to create multi-scale representations of graph-structured data, such as coarse-grained di􀀋usion maps [23], multi-scale wavelets [14] and pyramids [29] .  \nMore recently, coarsening is employed as a component of graph convolutional networks analogous to pooling [5, 10, 4] . Combining the values of adjacent vertices reduces the spatial size of each layer’s output, prevents over􀀌tting, and encourages a hierarchical scaling of representations.  \nYet, much remains to be understood about the properties and limitations of graph coarsening.  \nThe majority of theoretical work has so far focused on constructing fast linear solvers using multigrid techniques. These methods are especially relevant for approximating the solution of di􀀋erential equations on grids and 􀀌nite-element meshes. Multigrids were also adapted to arbitrary graphs by Koutis et al. [21] and later on by Livne and Brandt [25] . Based on an optimized version of the Galerkin coarsening, the authors demonstrate an algebraic multi-level approximation scheme that is shown to solve symmetric diagonally dominant linear systems in almost linear time. Similar techniques have also been applied for approximating the Fiedler vector [36, 13] and solving leastsquares problems of the graph Laplacian [16, 8] .  \nDespite this progress, with the exception of certain interlacing results [7, 6], it is currently an open question how coarsening a􀀋ects the spectrum of a general graph. As a consequence, thereis no rigorous way of determining to what extend one may coarsen a graph without signi􀀌cantlya􀀋ecting the performance of spectral methods for graph partitioning and visualization. Moreover,  \nlacking a fundamental understanding of what and how much information is lost, we have little hope of equipping coarsening-based learning algorithms, such as graph neural networks, with the appropriate mechanism of constructing multi-scale representations.  \nThis paper sheds light into some of these questions. Speci􀀌cally, we consider a one-shot coarsening operation and ask how much it a􀀋ects the eigenvalues and eigenvectors of the graph Laplacian. Key to our argument is the introduced restricted spectral similarity (RSS) property, asserting that the Laplacian of the coarsened and actual graphs behave similarly (up to some constants) with respect to an appropriate set of vectors. The RSS property is shown to hold for coarsenings constructed ","cbCaihWyvW43IKGd","https://ap.wps.com/l/cbCaihWyvW43IKGd","pdf",2932511,22,"English","# Introduction\n## Graph coarsening\n## How to coarsen a graph?","[{\"question\":\"What property does the paper introduce to relate the spectra of the original and coarsened graphs?\",\"answer\":\"It introduces the restricted spectral similarity (RSS) property, which states that the Laplacians behave similarly (up to constants) with respect to an appropriate set of vectors.\"},{\"question\":\"How does the approximation quality depend on graph structure?\",\"answer\":\"The achieved approximation depends on degree and eigenvalue distributions of the graph, and it is further controlled by the ratio between the coarsened and original graph sizes.\"},{\"question\":\"What does the theory imply for spectral clustering?\",\"answer\":\"It shows that lifted eigenvectors from the coarsened graph can produce high-quality clustering assignments even without refinement, providing formal justification for an effect previously observed empirically.\"}]","Spectrally approximating large graphs with smaller graphs | PDF",55]