{"id":"https://openalex.org/W4408808398","doi":"https://doi.org/10.1137/24m1655755","title":"High Curvature Means Low Rank: On the Sectional Curvature of Grassmann and Stiefel Manifolds and the Underlying Matrix Trace Inequalities","display_name":"High Curvature Means Low Rank: On the Sectional Curvature of Grassmann and Stiefel Manifolds and the Underlying Matrix Trace Inequalities","publication_year":2025,"publication_date":"2025-03-25","ids":{"openalex":"https://openalex.org/W4408808398","doi":"https://doi.org/10.1137/24m1655755"},"language":"en","primary_location":{"id":"doi:10.1137/24m1655755","is_oa":false,"landing_page_url":"https://doi.org/10.1137/24m1655755","pdf_url":null,"source":{"id":"https://openalex.org/S16958353","display_name":"SIAM Journal on Matrix Analysis and Applications","issn_l":"0895-4798","issn":["0895-4798","1095-7162"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Matrix Analysis and Applications","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://findresearcher.sdu.dk/ws/files/300015952/Binder1.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5101947183","display_name":"Ralf Zimmermann","orcid":"https://orcid.org/0000-0003-1692-3996"},"institutions":[{"id":"https://openalex.org/I177969490","display_name":"University of Southern Denmark","ror":"https://ror.org/03yrrjy16","country_code":"DK","type":"education","lineage":["https://openalex.org/I177969490"]}],"countries":["DK"],"is_corresponding":true,"raw_author_name":"Ralf Zimmermann","raw_affiliation_strings":["Department of Mathematics and Computer Science, University of Southern Denmark, Odense, Denmark"],"raw_orcid":"https://orcid.org/0000-0003-1692-3996","affiliations":[{"raw_affiliation_string":"Department of Mathematics and Computer Science, University of Southern Denmark, Odense, Denmark","institution_ids":["https://openalex.org/I177969490"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5094087265","display_name":"Jakob Stoye","orcid":"https://orcid.org/0009-0003-9119-0013"},"institutions":[{"id":"https://openalex.org/I94509681","display_name":"Technische Universit\u00e4t Braunschweig","ror":"https://ror.org/010nsgg66","country_code":"DE","type":"education","lineage":["https://openalex.org/I94509681"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Jakob Stoye","raw_affiliation_strings":["Institute for Numerical Analysis, TU Braunschweig, Braunschweig, Germany"],"raw_orcid":"https://orcid.org/0009-0003-9119-0013","affiliations":[{"raw_affiliation_string":"Institute for Numerical Analysis, TU Braunschweig, Braunschweig, Germany","institution_ids":["https://openalex.org/I94509681"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5101947183"],"corresponding_institution_ids":["https://openalex.org/I177969490"],"apc_list":null,"apc_paid":null,"fwci":3.9771,"has_fulltext":true,"cited_by_count":3,"citation_normalized_percentile":{"value":0.93218611,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":95,"max":98},"biblio":{"volume":"46","issue":"1","first_page":"748","last_page":"779"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10792","display_name":"Matrix Theory and Algorithms","score":0.9991000294685364,"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"}},"topics":[{"id":"https://openalex.org/T10792","display_name":"Matrix Theory and Algorithms","score":0.9991000294685364,"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"}},{"id":"https://openalex.org/T11205","display_name":"Numerical methods in inverse problems","score":0.9973999857902527,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"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/T10229","display_name":"Geometric Analysis and Curvature Flows","score":0.987500011920929,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8502417802810669},{"id":"https://openalex.org/keywords/trace","display_name":"TRACE (psycholinguistics)","score":0.7291887998580933},{"id":"https://openalex.org/keywords/sectional-curvature","display_name":"Sectional curvature","score":0.6372911930084229},{"id":"https://openalex.org/keywords/curvature","display_name":"Curvature","score":0.6328380703926086},{"id":"https://openalex.org/keywords/rank","display_name":"Rank (graph theory)","score":0.6052349805831909},{"id":"https://openalex.org/keywords/stiefel-manifold","display_name":"Stiefel manifold","score":0.5346810817718506},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.4703787863254547},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.43541571497917175},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.37620455026626587},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.32971203327178955},{"id":"https://openalex.org/keywords/scalar-curvature","display_name":"Scalar curvature","score":0.30492162704467773},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.21102136373519897}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8502417802810669},{"id":"https://openalex.org/C75291252","wikidata":"https://www.wikidata.org/wiki/Q1315756","display_name":"TRACE (psycholinguistics)","level":2,"score":0.7291887998580933},{"id":"https://openalex.org/C42448751","wikidata":"https://www.wikidata.org/wiki/Q855622","display_name":"Sectional curvature","level":4,"score":0.6372911930084229},{"id":"https://openalex.org/C195065555","wikidata":"https://www.wikidata.org/wiki/Q214881","display_name":"Curvature","level":2,"score":0.6328380703926086},{"id":"https://openalex.org/C164226766","wikidata":"https://www.wikidata.org/wiki/Q7293202","display_name":"Rank (graph theory)","level":2,"score":0.6052349805831909},{"id":"https://openalex.org/C612670","wikidata":"https://www.wikidata.org/wiki/Q7616373","display_name":"Stiefel manifold","level":2,"score":0.5346810817718506},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.4703787863254547},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.43541571497917175},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.37620455026626587},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.32971203327178955},{"id":"https://openalex.org/C12520029","wikidata":"https://www.wikidata.org/wiki/Q1147161","display_name":"Scalar curvature","level":3,"score":0.30492162704467773},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.21102136373519897},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.0},{"id":"https://openalex.org/C192562407","wikidata":"https://www.wikidata.org/wiki/Q228736","display_name":"Materials science","level":0,"score":0.0},{"id":"https://openalex.org/C41895202","wikidata":"https://www.wikidata.org/wiki/Q8162","display_name":"Linguistics","level":1,"score":0.0},{"id":"https://openalex.org/C159985019","wikidata":"https://www.wikidata.org/wiki/Q181790","display_name":"Composite material","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1137/24m1655755","is_oa":false,"landing_page_url":"https://doi.org/10.1137/24m1655755","pdf_url":null,"source":{"id":"https://openalex.org/S16958353","display_name":"SIAM Journal on Matrix Analysis and Applications","issn_l":"0895-4798","issn":["0895-4798","1095-7162"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Matrix Analysis and Applications","raw_type":"journal-article"},{"id":"pmh:oai:sdu.dk:openaire/e749eef4-b93b-4d2c-8274-847de402751b","is_oa":true,"landing_page_url":"https://portal.findresearcher.sdu.dk/da/publications/e749eef4-b93b-4d2c-8274-847de402751b","pdf_url":"https://findresearcher.sdu.dk/ws/files/300015952/Binder1.pdf","source":{"id":"https://openalex.org/S4306400423","display_name":"University of Southern Denmark Research Portal (University of Southern Denmark)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I177969490","host_organization_name":"University of Southern Denmark","host_organization_lineage":["https://openalex.org/I177969490"],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Zimmermann, R & Stoye, J 2025, 'High Curvature Means Low Rank : On the Sectional Curvature of Grassmann and Stiefel Manifolds and the Underlying Matrix Trace Inequalities', SIAM Journal on Matrix Analysis and Applications, vol. 46, no. 1, pp. 748-779. https://doi.org/10.1137/24m1655755","raw_type":"info:eu-repo/semantics/publishedVersion"},{"id":"pmh:oai:sdu.dk:publications/e749eef4-b93b-4d2c-8274-847de402751b","is_oa":true,"landing_page_url":"https://portal.findresearcher.sdu.dk/files/300015952/Binder1.pdf","pdf_url":null,"source":{"id":"https://openalex.org/S4306400423","display_name":"University of Southern Denmark Research Portal (University of Southern Denmark)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I177969490","host_organization_name":"University of Southern Denmark","host_organization_lineage":["https://openalex.org/I177969490"],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Zimmermann, R & Stoye, J 2025, 'High Curvature Means Low Rank : On the Sectional Curvature of Grassmann and Stiefel Manifolds and the Underlying Matrix Trace Inequalities', SIAM Journal on Matrix Analysis and Applications, vol. 46, no. 1, pp. 748-779. https://doi.org/10.1137/24m1655755","raw_type":"info:eu-repo/semantics/publishedVersion"}],"best_oa_location":{"id":"pmh:oai:sdu.dk:openaire/e749eef4-b93b-4d2c-8274-847de402751b","is_oa":true,"landing_page_url":"https://portal.findresearcher.sdu.dk/da/publications/e749eef4-b93b-4d2c-8274-847de402751b","pdf_url":"https://findresearcher.sdu.dk/ws/files/300015952/Binder1.pdf","source":{"id":"https://openalex.org/S4306400423","display_name":"University of Southern Denmark Research Portal (University of Southern Denmark)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I177969490","host_organization_name":"University of Southern Denmark","host_organization_lineage":["https://openalex.org/I177969490"],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Zimmermann, R & Stoye, J 2025, 'High Curvature Means Low Rank : On the Sectional Curvature of Grassmann and Stiefel Manifolds and the Underlying Matrix Trace Inequalities', SIAM Journal on Matrix Analysis and Applications, vol. 46, no. 1, pp. 748-779. https://doi.org/10.1137/24m1655755","raw_type":"info:eu-repo/semantics/publishedVersion"},"sustainable_development_goals":[{"score":0.5199999809265137,"id":"https://metadata.un.org/sdg/10","display_name":"Reduced inequalities"}],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4408808398.pdf"},"referenced_works_count":24,"referenced_works":["https://openalex.org/W1490644506","https://openalex.org/W1982258673","https://openalex.org/W2045512849","https://openalex.org/W2058590426","https://openalex.org/W2061356302","https://openalex.org/W2088449330","https://openalex.org/W2096610373","https://openalex.org/W2097521761","https://openalex.org/W2147631863","https://openalex.org/W2152896489","https://openalex.org/W2174206810","https://openalex.org/W2597507805","https://openalex.org/W2762042638","https://openalex.org/W2896405912","https://openalex.org/W2964268565","https://openalex.org/W2977511996","https://openalex.org/W3106048390","https://openalex.org/W3137801329","https://openalex.org/W3210764284","https://openalex.org/W4205293427","https://openalex.org/W4206033904","https://openalex.org/W4243055393","https://openalex.org/W4253652099","https://openalex.org/W4407108970"],"related_works":["https://openalex.org/W2005999811","https://openalex.org/W2020635512","https://openalex.org/W4307474388","https://openalex.org/W4298133298","https://openalex.org/W146471306","https://openalex.org/W4381586607","https://openalex.org/W2383363824","https://openalex.org/W2385863434","https://openalex.org/W2950554095","https://openalex.org/W4361215702"],"abstract_inverted_index":{"Methods":[0],"and":[1,60,134,162,196],"algorithms":[2],"that":[3,87,154,186,213],"work":[4],"with":[5],"data":[6],"on":[7],"nonlinear":[8],"manifolds":[9,198],"are":[10,214,235],"collectively":[11],"summarized":[12],"under":[13,80,199],"the":[14,28,43,54,58,64,73,77,81,88,96,99,110,116,139,142,151,155,175,194,200,203,220,225],"term":[15],"``Riemannian":[16],"computing.\"\"":[17],"In":[18,49,145],"practice,":[19],"curvature":[20,35,56,67,90,112,143,176,188,205,222],"can":[21,36],"be":[22,38,104],"a":[23,39,171],"key":[24],"limiting":[25],"factor":[26],"for":[27,131,150,193,230,237],"performance":[29],"of":[30,46,57,119,124,141,158,174,224],"Riemannian":[31,47],"computing":[32],"methods.":[33],"Yet":[34],"also":[37],"powerful":[40],"tool":[41],"in":[42,178],"theoretical":[44],"analysis":[45],"algorithms.":[48],"this":[50,146,169],"work,":[51],"we":[52,148],"investigate":[53],"sectional":[55,89,111],"Stiefel":[59,78,152,195,227],"Grassmann":[61,197],"manifold.":[62],"On":[63,76],"Grassmannian,":[65],"tight":[66],"bounds":[68,157,177],"have":[69],"been":[70],"known":[71],"since":[72],"late":[74],"1960s.":[75],"manifold":[79,153,228],"canonical":[82,201],"metric,":[83,98,202],"it":[84],"was":[85,101],"believed":[86],"does":[91],"not":[92],"exceed":[93],"5/4.":[94],"Under":[95],"Euclidean":[97,226],"maximum":[100,206],"conjectured":[102],"to":[103,138],"at":[105,209],"1.":[106],"For":[107],"both":[108],"manifolds,":[109],"is":[113,182,207],"given":[114],"by":[115,216],"Frobenius":[117],"norm":[118],"certain":[120],"structured":[121],"commutator":[122],"brackets":[123],"skew-symmetric":[125],"matrices.":[126,232],"We":[127,184],"provide":[128],"refined":[129],"inequalities":[130],"such":[132],"terms":[133],"pay":[135],"special":[136],"attention":[137],"maximizers":[140],"bounds.":[144],"way,":[147],"prove":[149],"global":[156,204],"5/4":[159],"(canonical":[160],"metric)":[161,165],"1":[163],"(Euclidean":[164],"hold":[166],"indeed.":[167],"With":[168],"addition,":[170],"complete":[172],"account":[173],"all":[179],"admissible":[180],"dimensions":[181],"obtained.":[183],"observe":[185],"``high":[187],"means":[189],"low-rank\"\";":[190],"more":[191],"precisely,":[192],"attained":[208],"tangent":[210],"plane":[211],"sections":[212],"spanned":[215],"rank-two":[217],"matrices,":[218],"while":[219],"extreme":[221],"cases":[223],"occur":[229],"rank-one":[231],"Numerical":[233],"examples":[234],"included":[236],"illustration":[238],"purposes.":[239]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":2}],"updated_date":"2025-12-28T23:10:05.387466","created_date":"2025-10-10T00:00:00"}
