{"id":"https://openalex.org/W7167828013","doi":"https://doi.org/10.48550/arxiv.2607.07032","title":"Eigenbasis-Independent Learnable Spectral Positional Encodings for Directed Graphs via Hermitian Block Krylov Subspaces","display_name":"Eigenbasis-Independent Learnable Spectral Positional Encodings for Directed Graphs via Hermitian Block Krylov Subspaces","publication_year":2026,"publication_date":"2026-07-08","ids":{"openalex":"https://openalex.org/W7167828013","doi":"https://doi.org/10.48550/arxiv.2607.07032"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2607.07032","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2607.07032","pdf_url":"https://arxiv.org/pdf/2607.07032","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2607.07032","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5010620570","display_name":"Jiaqing Xie","orcid":"https://orcid.org/0000-0001-7634-4457","observed_orcids":["https://orcid.org/0000-0001-7634-4457"]},"institutions":[{"id":"https://openalex.org/I24943067","display_name":"Fudan University","ror":"https://ror.org/013q1eq08","country_code":"CN","type":"education","lineage":["https://openalex.org/I24943067"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Xie, Jiaqing","raw_affiliation_strings":["Fudan University Fudan University"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Fudan University Fudan University","institution_ids":["https://openalex.org/I24943067"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5140310562","display_name":"Yuxin Wang","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I24943067","display_name":"Fudan University","ror":"https://ror.org/013q1eq08","country_code":"CN","type":"education","lineage":["https://openalex.org/I24943067"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Wang, Yuxin","raw_affiliation_strings":["Fudan University Fudan University"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Fudan University Fudan University","institution_ids":["https://openalex.org/I24943067"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I24943067"],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11273","display_name":"Advanced Graph Neural Networks","score":0.9785000085830688,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11273","display_name":"Advanced Graph Neural Networks","score":0.9785000085830688,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11476","display_name":"Graph theory and applications","score":0.006300000008195639,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12292","display_name":"Graph Theory and Algorithms","score":0.0017000000225380063,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/directed-graphs","display_name":"directed graphs","score":0.9929999709129333},{"id":"https://openalex.org/keywords/magnetic-laplacian","display_name":"magnetic Laplacian","score":0.9359999895095825},{"id":"https://openalex.org/keywords/gauge-invariance","display_name":"gauge invariance","score":0.8889999985694885},{"id":"https://openalex.org/keywords/sparse-matrix-vector-product","display_name":"sparse matrix-vector product","score":0.7400000095367432},{"id":"https://openalex.org/keywords/covering-number","display_name":"covering number","score":0.5329999923706055},{"id":"https://openalex.org/keywords/stochastic-block-model","display_name":"stochastic block model","score":0.8029999732971191},{"id":"https://openalex.org/keywords/heterophilous-graphs","display_name":"heterophilous graphs","score":0.8309999704360962},{"id":"https://openalex.org/keywords/graph-neural-networks","display_name":"graph neural networks","score":0.5590000152587891}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7513999938964844},{"id":"https://openalex.org/C94940","wikidata":"https://www.wikidata.org/wiki/Q652941","display_name":"Hermitian matrix","level":2,"score":0.6255999803543091},{"id":"https://openalex.org/C2777210771","wikidata":"https://www.wikidata.org/wiki/Q4927124","display_name":"Block (permutation group theory)","level":2,"score":0.5263000130653381},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.49300000071525574},{"id":"https://openalex.org/C12362212","wikidata":"https://www.wikidata.org/wiki/Q728435","display_name":"Linear subspace","level":2,"score":0.48840001225471497},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.487199991941452},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.40610000491142273},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.400299996137619},{"id":"https://openalex.org/C74193536","wikidata":"https://www.wikidata.org/wiki/Q574844","display_name":"Kernel (algebra)","level":2,"score":0.3756999969482422},{"id":"https://openalex.org/C57691317","wikidata":"https://www.wikidata.org/wiki/Q1289248","display_name":"Scalar (mathematics)","level":2,"score":0.36469998955726624},{"id":"https://openalex.org/C146380142","wikidata":"https://www.wikidata.org/wiki/Q1137726","display_name":"Directed graph","level":2,"score":0.35760000348091125},{"id":"https://openalex.org/C30072841","wikidata":"https://www.wikidata.org/wiki/Q1767080","display_name":"Diagonalizable matrix","level":4,"score":0.3571000099182129},{"id":"https://openalex.org/C165464430","wikidata":"https://www.wikidata.org/wiki/Q1570441","display_name":"Parameterized complexity","level":2,"score":0.31529998779296875},{"id":"https://openalex.org/C169756996","wikidata":"https://www.wikidata.org/wiki/Q194919","display_name":"Eigendecomposition of a matrix","level":3,"score":0.30059999227523804},{"id":"https://openalex.org/C64999384","wikidata":"https://www.wikidata.org/wiki/Q3507967","display_name":"Symmetrization","level":2,"score":0.29809999465942383},{"id":"https://openalex.org/C35958075","wikidata":"https://www.wikidata.org/wiki/Q1425077","display_name":"Spectral theorem","level":3,"score":0.2971999943256378},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.2930000126361847},{"id":"https://openalex.org/C96442724","wikidata":"https://www.wikidata.org/wiki/Q242188","display_name":"Invertible matrix","level":2,"score":0.2921999990940094},{"id":"https://openalex.org/C184898388","wikidata":"https://www.wikidata.org/wiki/Q1435712","display_name":"Pairwise comparison","level":2,"score":0.29100000858306885},{"id":"https://openalex.org/C96214315","wikidata":"https://www.wikidata.org/wiki/Q727103","display_name":"Unitary matrix","level":3,"score":0.289000004529953},{"id":"https://openalex.org/C147060835","wikidata":"https://www.wikidata.org/wiki/Q1757151","display_name":"Krylov subspace","level":3,"score":0.2842999994754791},{"id":"https://openalex.org/C54848796","wikidata":"https://www.wikidata.org/wiki/Q339011","display_name":"Symmetric matrix","level":3,"score":0.2827000021934509},{"id":"https://openalex.org/C67820243","wikidata":"https://www.wikidata.org/wiki/Q179164","display_name":"Unitary state","level":2,"score":0.27410000562667847},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.2646999955177307},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.26339998841285706},{"id":"https://openalex.org/C2779982251","wikidata":"https://www.wikidata.org/wiki/Q25053762","display_name":"Stochastic block model","level":3,"score":0.2599000036716461},{"id":"https://openalex.org/C101901036","wikidata":"https://www.wikidata.org/wiki/Q2269096","display_name":"Semidefinite programming","level":2,"score":0.25949999690055847},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.25940001010894775},{"id":"https://openalex.org/C32834561","wikidata":"https://www.wikidata.org/wiki/Q660730","display_name":"Subspace topology","level":2,"score":0.258899986743927},{"id":"https://openalex.org/C179799912","wikidata":"https://www.wikidata.org/wiki/Q205084","display_name":"Computational complexity theory","level":2,"score":0.25859999656677246},{"id":"https://openalex.org/C2776359362","wikidata":"https://www.wikidata.org/wiki/Q2145286","display_name":"Representation (politics)","level":3,"score":0.25619998574256897},{"id":"https://openalex.org/C6802819","wikidata":"https://www.wikidata.org/wiki/Q1072174","display_name":"Linear system","level":2,"score":0.25589999556541443},{"id":"https://openalex.org/C85817219","wikidata":"https://www.wikidata.org/wiki/Q884772","display_name":"Block matrix","level":3,"score":0.25380000472068787}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2607.07032","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2607.07032","pdf_url":"https://arxiv.org/pdf/2607.07032","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2607.07032","doi":"https://doi.org/10.48550/arxiv.2607.07032","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2607.07032","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2607.07032","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2607.07032","pdf_url":"https://arxiv.org/pdf/2607.07032","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7167828013.pdf","grobid_xml":"https://content.openalex.org/works/W7167828013.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W7220649313","https://openalex.org/W4387389962","https://openalex.org/W4401203330","https://openalex.org/W4407129395","https://openalex.org/W7108209500","https://openalex.org/W3205705101","https://openalex.org/W7116705171","https://openalex.org/W4391462878","https://openalex.org/W7164502608","https://openalex.org/W7140970794"],"abstract_inverted_index":{"Spectral":[0],"positional":[1],"encodings":[2],"(PEs)":[3],"for":[4,113],"\\emph{directed}":[5],"graphs":[6],"face":[7],"two":[8],"obstacles:":[9],"magnetic":[10,51,140],"Laplacians":[11],"require":[12],"an":[13],"$O(n^3)$":[14],"Hermitian":[15,86],"eigendecomposition":[16],"per":[17],"potential,":[18],"and":[19,59,108,163,173],"their":[20],"complex":[21],"eigenvectors":[22],"are":[23],"defined":[24],"only":[25],"up":[26],"to":[27,144],"unitary":[28],"gauge,":[29],"which":[30],"prior":[31],"work":[32],"handles":[33],"with":[34,159],"basis-invariant":[35],"architectures.":[36],"We":[37,81],"propose":[38],"learnable":[39,55],"spectral":[40,57],"PEs":[41,135,142],"of":[42,63,73],"the":[43,67,74,145,149,164],"form":[44],"$h_\u03b8(A_q)\\,R$,":[45],"where":[46,119],"$A_q$":[47],"is":[48,69,77,130],"a":[49,54,61,70,85,110,125],"normalized":[50],"operator,":[52,75],"$h_\u03b8$":[53],"scalar":[56],"response,":[58],"$R$":[60],"block":[62,87,100],"random":[64],"probes.":[65],"Because":[66],"PE":[68],"\\emph{matrix":[71],"function}":[72],"it":[76,83],"gauge-invariant":[78,156],"by":[79,132],"construction.":[80],"compute":[82],"in":[84],"Krylov":[88,141],"subspace":[89],"from":[90],"sparse":[91],"matrix--vector":[92],"products":[93],"only,":[94],"prove":[95],"that":[96],"$k":[97],"=":[98],"O(\\log(1/\\varepsilon))$":[99],"steps":[101],"suffice":[102],"uniformly":[103],"over":[104,171],"heat--resolvent":[105],"response":[106],"families,":[107],"give":[109],"covering-number":[111],"argument":[112],"why":[114],"low-dimensional":[115],"structured":[116],"families":[117],"generalize":[118],"free":[120],"per-eigenvalue":[121],"weights":[122],"overfit.":[123],"On":[124],"directed":[126],"SBM":[127],"whose":[128],"symmetrization":[129],"uninformative":[131],"construction,":[133],"direction-blind":[134],"stay":[136],"at":[137],"chance":[138],"while":[139],"converge":[143],"exact-eigendecomposition":[146],"oracle":[147],"as":[148],"depth":[150],"grows.":[151],"The":[152],"same":[153],"probes":[154],"yield":[155],"pairwise":[157],"features":[158],"$1/\\sqrt{s}$":[160],"Monte-Carlo":[161],"error,":[162],"undirected":[165],"$q{=}0$":[166],"case":[167],"improves":[168],"heterophilous":[169],"benchmarks":[170],"no-PE":[172],"polynomial":[174],"baselines.":[175]},"counts_by_year":[],"updated_date":"2026-10-28T10:33:24.094632","created_date":"2026-07-10T00:00:00"}