{"id":"https://openalex.org/W3128574906","doi":"https://doi.org/10.1007/s10878-021-00729-x","title":"Relative-perfectness of discrete gradient vector fields and multi-parameter persistent homology","display_name":"Relative-perfectness of discrete gradient vector fields and multi-parameter persistent homology","publication_year":2021,"publication_date":"2021-04-13","ids":{"openalex":"https://openalex.org/W3128574906","doi":"https://doi.org/10.1007/s10878-021-00729-x","mag":"3128574906"},"language":"en","primary_location":{"id":"doi:10.1007/s10878-021-00729-x","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10878-021-00729-x","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10878-021-00729-x.pdf","source":{"id":"https://openalex.org/S107925523","display_name":"Journal of Combinatorial Optimization","issn_l":"1382-6905","issn":["1382-6905","1573-2886"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Combinatorial Optimization","raw_type":"journal-article"},"type":"preprint","indexed_in":["arxiv","crossref","datacite"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://link.springer.com/content/pdf/10.1007/s10878-021-00729-x.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5068190896","display_name":"Claudia Landi","orcid":"https://orcid.org/0000-0001-8725-4844"},"institutions":[{"id":"https://openalex.org/I122346577","display_name":"University of Modena and Reggio Emilia","ror":"https://ror.org/02d4c4y02","country_code":"IT","type":"education","lineage":["https://openalex.org/I122346577"]}],"countries":["IT"],"is_corresponding":true,"raw_author_name":"Claudia Landi","raw_affiliation_strings":["University of Modena and Reggio Emilia, Modena, Italy","Univ. of Modena and Reggio Emilia#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of Modena and Reggio Emilia, Modena, Italy","institution_ids":["https://openalex.org/I122346577"]},{"raw_affiliation_string":"Univ. of Modena and Reggio Emilia#TAB#","institution_ids":["https://openalex.org/I122346577"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5039569727","display_name":"Sara Scaramuccia","orcid":"https://orcid.org/0000-0003-4010-6507"},"institutions":[{"id":"https://openalex.org/I177477856","display_name":"Politecnico di Torino","ror":"https://ror.org/00bgk9508","country_code":"IT","type":"education","lineage":["https://openalex.org/I177477856"]},{"id":"https://openalex.org/I4210142780","display_name":"Turin Polytechnic University","ror":"https://ror.org/042xrxv40","country_code":"UZ","type":"education","lineage":["https://openalex.org/I4210142780"]}],"countries":["IT","UZ"],"is_corresponding":false,"raw_author_name":"Sara Scaramuccia","raw_affiliation_strings":["Politecnico di Torino, Torino, Italy","Polytechnic University of Turin"],"raw_orcid":"https://orcid.org/0000-0003-4010-6507","affiliations":[{"raw_affiliation_string":"Politecnico di Torino, Torino, Italy","institution_ids":["https://openalex.org/I177477856"]},{"raw_affiliation_string":"Polytechnic University of Turin","institution_ids":["https://openalex.org/I4210142780","https://openalex.org/I177477856"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5068190896"],"corresponding_institution_ids":["https://openalex.org/I122346577"],"apc_list":{"value":2390,"currency":"EUR","value_usd":2990},"apc_paid":{"value":2390,"currency":"EUR","value_usd":2990},"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.0468362,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"44","issue":"4","first_page":"2347","last_page":"2374"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12536","display_name":"Topological and Geometric Data Analysis","score":0.9998999834060669,"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/T12536","display_name":"Topological and Geometric Data Analysis","score":0.9998999834060669,"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/T10896","display_name":"Homotopy and Cohomology in Algebraic Topology","score":0.9779999852180481,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/persistent-homology","display_name":"Persistent homology","score":0.8279616832733154},{"id":"https://openalex.org/keywords/discrete-morse-theory","display_name":"Discrete Morse theory","score":0.7666468024253845},{"id":"https://openalex.org/keywords/topological-data-analysis","display_name":"Topological data analysis","score":0.7036920785903931},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6433902978897095},{"id":"https://openalex.org/keywords/morse-theory","display_name":"Morse theory","score":0.6337527632713318},{"id":"https://openalex.org/keywords/computational-topology","display_name":"Computational topology","score":0.6148493885993958},{"id":"https://openalex.org/keywords/homology","display_name":"Homology (biology)","score":0.5317249298095703},{"id":"https://openalex.org/keywords/computability","display_name":"Computability","score":0.5218638181686401},{"id":"https://openalex.org/keywords/bounding-overwatch","display_name":"Bounding overwatch","score":0.4428894519805908},{"id":"https://openalex.org/keywords/simplicial-complex","display_name":"Simplicial complex","score":0.43467360734939575},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.3297746181488037},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.3036497235298157},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.2548569142818451},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.23422014713287354},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.08987265825271606}],"concepts":[{"id":"https://openalex.org/C2874115","wikidata":"https://www.wikidata.org/wiki/Q17099562","display_name":"Persistent homology","level":2,"score":0.8279616832733154},{"id":"https://openalex.org/C170315498","wikidata":"https://www.wikidata.org/wiki/Q5282039","display_name":"Discrete Morse theory","level":3,"score":0.7666468024253845},{"id":"https://openalex.org/C2776477805","wikidata":"https://www.wikidata.org/wiki/Q4460773","display_name":"Topological data analysis","level":2,"score":0.7036920785903931},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6433902978897095},{"id":"https://openalex.org/C156730297","wikidata":"https://www.wikidata.org/wiki/Q2296951","display_name":"Morse theory","level":2,"score":0.6337527632713318},{"id":"https://openalex.org/C181576044","wikidata":"https://www.wikidata.org/wiki/Q4129926","display_name":"Computational topology","level":3,"score":0.6148493885993958},{"id":"https://openalex.org/C165525559","wikidata":"https://www.wikidata.org/wiki/Q224180","display_name":"Homology (biology)","level":3,"score":0.5317249298095703},{"id":"https://openalex.org/C152062344","wikidata":"https://www.wikidata.org/wiki/Q818888","display_name":"Computability","level":2,"score":0.5218638181686401},{"id":"https://openalex.org/C63584917","wikidata":"https://www.wikidata.org/wiki/Q333286","display_name":"Bounding overwatch","level":2,"score":0.4428894519805908},{"id":"https://openalex.org/C187929450","wikidata":"https://www.wikidata.org/wiki/Q994399","display_name":"Simplicial complex","level":2,"score":0.43467360734939575},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3297746181488037},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3036497235298157},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.2548569142818451},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.23422014713287354},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.08987265825271606},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.0},{"id":"https://openalex.org/C110521144","wikidata":"https://www.wikidata.org/wiki/Q193460","display_name":"Scalar field","level":2,"score":0.0},{"id":"https://openalex.org/C37914503","wikidata":"https://www.wikidata.org/wiki/Q156495","display_name":"Mathematical physics","level":1,"score":0.0},{"id":"https://openalex.org/C104317684","wikidata":"https://www.wikidata.org/wiki/Q7187","display_name":"Gene","level":2,"score":0.0},{"id":"https://openalex.org/C55493867","wikidata":"https://www.wikidata.org/wiki/Q7094","display_name":"Biochemistry","level":1,"score":0.0}],"mesh":[],"locations_count":7,"locations":[{"id":"doi:10.1007/s10878-021-00729-x","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10878-021-00729-x","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10878-021-00729-x.pdf","source":{"id":"https://openalex.org/S107925523","display_name":"Journal of Combinatorial Optimization","issn_l":"1382-6905","issn":["1382-6905","1573-2886"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Combinatorial Optimization","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:1904.05081","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1904.05081","pdf_url":"https://arxiv.org/pdf/1904.05081","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"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":"mag:3128574906","is_oa":true,"landing_page_url":"https://arxiv.org/pdf/1904.05081.pdf","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"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":"arXiv (Cornell University)","raw_type":null},{"id":"pmh:oai:RePEc:spr:jcomop:v:44:y:2022:i:4:d:10.1007_s10878-021-00729-x","is_oa":false,"landing_page_url":"http://link.springer.com/10.1007/s10878-021-00729-x","pdf_url":null,"source":{"id":"https://openalex.org/S4306401271","display_name":"RePEc: Research Papers in Economics","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I77793887","host_organization_name":"Federal Reserve Bank of St. Louis","host_organization_lineage":["https://openalex.org/I77793887"],"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":"article"},{"id":"pmh:oai:art.torvergata.it:2108/303256","is_oa":false,"landing_page_url":"http://hdl.handle.net/2108/303256","pdf_url":null,"source":{"id":"https://openalex.org/S4306400993","display_name":"Cineca Institutional Research Information System (Tor Vergata University)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I116067653","host_organization_name":"University of Rome Tor Vergata","host_organization_lineage":["https://openalex.org/I116067653"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"info:eu-repo/semantics/article"},{"id":"pmh:oai:iris.unimore.it:11380/1241494","is_oa":true,"landing_page_url":"http://hdl.handle.net/11380/1241494","pdf_url":null,"source":{"id":"https://openalex.org/S4306400718","display_name":"IRIS UNIMORE (University of Modena and Reggio Emilia)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I122346577","host_organization_name":"University of Modena and Reggio Emilia","host_organization_lineage":["https://openalex.org/I122346577"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"info:eu-repo/semantics/article"},{"id":"doi:10.48550/arxiv.1904.05081","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.1904.05081","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"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":null,"raw_source_name":null,"raw_type":"article"}],"best_oa_location":{"id":"doi:10.1007/s10878-021-00729-x","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10878-021-00729-x","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10878-021-00729-x.pdf","source":{"id":"https://openalex.org/S107925523","display_name":"Journal of Combinatorial Optimization","issn_l":"1382-6905","issn":["1382-6905","1573-2886"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Combinatorial Optimization","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G199520308","display_name":null,"funder_award_id":"E11G18000350001","funder_id":"https://openalex.org/F4320321873","funder_display_name":"Ministero dell\u2019Istruzione, dell\u2019Universit\u00e0 e della Ricerca"},{"id":"https://openalex.org/G2019402066","display_name":null,"funder_award_id":"E11G18000350001","funder_id":"https://openalex.org/F4320311030","funder_display_name":"Istituto Nazionale di Alta Matematica \"Francesco Severi\""},{"id":"https://openalex.org/G2312790173","display_name":null,"funder_award_id":"E11G18000350001","funder_id":"https://openalex.org/F4320313985","funder_display_name":"Politecnico di Torino"},{"id":"https://openalex.org/G3178914890","display_name":null,"funder_award_id":"2018-2022","funder_id":"https://openalex.org/F4320311030","funder_display_name":"Istituto Nazionale di Alta Matematica \"Francesco Severi\""},{"id":"https://openalex.org/G3866078129","display_name":null,"funder_award_id":"2018-2022","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G4345464418","display_name":null,"funder_award_id":"CUP: E11G18000350001","funder_id":"https://openalex.org/F4320321873","funder_display_name":"Ministero dell\u2019Istruzione, dell\u2019Universit\u00e0 e della Ricerca"},{"id":"https://openalex.org/G4956079712","display_name":null,"funder_award_id":"2018-2022","funder_id":"https://openalex.org/F4320321873","funder_display_name":"Ministero dell\u2019Istruzione, dell\u2019Universit\u00e0 e della Ricerca"},{"id":"https://openalex.org/G7464491239","display_name":null,"funder_award_id":"IIS-1910766","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G7611412312","display_name":null,"funder_award_id":"2018-2022","funder_id":"https://openalex.org/F4320323553","funder_display_name":"Universit\u00e0 di Bologna"},{"id":"https://openalex.org/G8357203474","display_name":null,"funder_award_id":"CUP: E11G18000350001","funder_id":"https://openalex.org/F4320313985","funder_display_name":"Politecnico di Torino"},{"id":"https://openalex.org/G8421614505","display_name":null,"funder_award_id":"Dipartimento di Eccellenza 2018-2022","funder_id":"https://openalex.org/F4320321873","funder_display_name":"Ministero dell\u2019Istruzione, dell\u2019Universit\u00e0 e della Ricerca"},{"id":"https://openalex.org/G904476071","display_name":null,"funder_award_id":"1910766","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320311030","display_name":"Istituto Nazionale di Alta Matematica \"Francesco Severi\"","ror":"https://ror.org/01vx64p53"},{"id":"https://openalex.org/F4320313985","display_name":"Politecnico di Torino","ror":"https://ror.org/00bgk9508"},{"id":"https://openalex.org/F4320321873","display_name":"Ministero dell\u2019Istruzione, dell\u2019Universit\u00e0 e della Ricerca","ror":"https://ror.org/0166hxq48"},{"id":"https://openalex.org/F4320323553","display_name":"Universit\u00e0 di Bologna","ror":"https://ror.org/01111rn36"},{"id":"https://openalex.org/F4320334045","display_name":"Gruppo Nazionale per le Strutture Algebriche, Geometriche e le loro Applicazioni","ror":null}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W3128574906.pdf","grobid_xml":"https://content.openalex.org/works/W3128574906.grobid-xml"},"referenced_works_count":32,"referenced_works":["https://openalex.org/W151084626","https://openalex.org/W1491346501","https://openalex.org/W1520338983","https://openalex.org/W1600748172","https://openalex.org/W1991566301","https://openalex.org/W1991874394","https://openalex.org/W1999826929","https://openalex.org/W2002165972","https://openalex.org/W2014209644","https://openalex.org/W2039564144","https://openalex.org/W2064020260","https://openalex.org/W2072701060","https://openalex.org/W2081349787","https://openalex.org/W2110369778","https://openalex.org/W2134678736","https://openalex.org/W2146019744","https://openalex.org/W2160639462","https://openalex.org/W2171287813","https://openalex.org/W2185147444","https://openalex.org/W2475195788","https://openalex.org/W2558363557","https://openalex.org/W2750288705","https://openalex.org/W2900648503","https://openalex.org/W2952523613","https://openalex.org/W2963097828","https://openalex.org/W2963887930","https://openalex.org/W3005844668","https://openalex.org/W3039671159","https://openalex.org/W3140579943","https://openalex.org/W3141871949","https://openalex.org/W4213199372","https://openalex.org/W4251980118"],"related_works":["https://openalex.org/W2138017022","https://openalex.org/W2914300301","https://openalex.org/W2115896258","https://openalex.org/W2148561979","https://openalex.org/W2740764576","https://openalex.org/W2766081397","https://openalex.org/W2020556227","https://openalex.org/W2963734607","https://openalex.org/W2611102007","https://openalex.org/W2948194493","https://openalex.org/W2765925170","https://openalex.org/W636421395","https://openalex.org/W2767077668","https://openalex.org/W3043360678","https://openalex.org/W2973473267","https://openalex.org/W619100130","https://openalex.org/W2995866785","https://openalex.org/W2763954298","https://openalex.org/W3212134443","https://openalex.org/W2576312643"],"abstract_inverted_index":{"Abstract":[0],"The":[1,58],"combination":[2],"of":[3,29,46,61,84,104,123,142,156,171,200,210,216,226],"persistent":[4,134],"homology":[5,182],"and":[6,16,19,26,36,102,119,221],"discrete":[7,48,126],"Morse":[8,146,175],"theory":[9,147],"has":[10],"proven":[11],"very":[12,53],"effective":[13],"in":[14,114,145,203],"visualizing":[15],"analyzing":[17],"big":[18],"heterogeneous":[20],"data.":[21],"Indeed,":[22],"topology":[23],"provides":[24],"computable":[25],"coarse":[27],"summaries":[28],"data":[30,63,85,109],"independently":[31],"from":[32,219],"specific":[33,59],"coordinate":[34],"systems":[35],"does":[37],"so":[38],"robustly":[39],"to":[40,79,86,132,177,189,195,251],"noise.":[41],"Moreover,":[42],"the":[43,70,81,90,96,100,105,153,178,224,233],"geometric":[44],"content":[45],"a":[47,121,139,164],"gradient":[49,127],"vector":[50,128],"field":[51],"is":[52,77,232],"useful":[54],"for":[55,66,73,93,125,148,160,197,244],"visualization":[56],"purposes.":[57],"case":[60,98],"multivariate":[62,97],"still":[64],"demands":[65],"further":[67],"investigations,":[68],"on":[69,239],"one":[71],"hand,":[72,92],"computational":[74],"reasons,":[75,95],"it":[76],"important":[78],"reduce":[80],"necessary":[82],"amount":[83],"be":[87],"processed.":[88],"On":[89],"other":[91],"analysis":[94],"requires":[99],"detection":[101],"interpretation":[103,199],"possible":[106],"interdepedance":[107],"among":[108],"components.":[110],"To":[111],"this":[112,115],"end,":[113],"paper":[116],"we":[117,167],"introduce":[118],"study":[120],"notion":[122],"perfectness":[124,144],"fields":[129],"with":[130,187],"respect":[131,188],"multi-parameter":[133,161],"homology,":[135,149],"called":[136],"relative-perfectness.":[137],"As":[138,163],"natural":[140],"generalization":[141],"usual":[143],"relative-perfectness":[150,172],"entails":[151],"having":[152],"least":[154],"number":[155,225],"critical":[157,201,227],"cells":[158,202],"relevant":[159],"persistence.":[162],"first":[165],"contribution,":[166],"support":[168],"our":[169,206],"definition":[170],"by":[173],"generalizing":[174],"inequalities":[176,212],"filtration":[179],"structure":[180],"where":[181],"groups":[183],"involved":[184],"are":[185],"relative":[186],"subsequent":[190],"sublevel":[191],"sets.":[192],"In":[193],"order":[194],"allow":[196,243],"an":[198],"2-parameter":[204],"persistence,":[205],"second":[207],"contribution":[208],"consists":[209],"two":[211],"bounding":[213],"Betti":[214],"tables":[215],"persistence":[217],"modules":[218],"above":[220],"below,":[222],"via":[223],"cells.":[228],"Our":[229],"last":[230],"result":[231],"proof":[234],"that":[235],"existing":[236],"algorithms":[237],"based":[238],"local":[240],"homotopy":[241],"expansions":[242],"efficient":[245],"computability":[246],"over":[247],"simplicial":[248],"complexes":[249],"up":[250],"dimension":[252],"2.":[253]},"counts_by_year":[],"updated_date":"2026-05-13T08:25:38.343686","created_date":"2025-10-10T00:00:00"}
