{"id":"https://openalex.org/W7162408036","doi":"https://doi.org/10.48550/arxiv.2605.24608","title":"Lattice theory and algebraic models for deep convolutional learning based on mathematical morphology","display_name":"Lattice theory and algebraic models for deep convolutional learning based on mathematical morphology","publication_year":2026,"publication_date":"2026-05-23","ids":{"openalex":"https://openalex.org/W7162408036","doi":"https://doi.org/10.48550/arxiv.2605.24608"},"language":null,"primary_location":{"id":"doi:10.48550/arxiv.2605.24608","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2605.24608","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,"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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"},"type":"preprint","indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.48550/arxiv.2605.24608","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5137076860","display_name":"Gustavo","orcid":null},"institutions":[{"id":"https://openalex.org/I4210107427","display_name":"Jesus University","ror":"https://ror.org/01nm0nj94","country_code":"KR","type":"education","lineage":["https://openalex.org/I4210107427"]}],"countries":["KR"],"is_corresponding":false,"raw_author_name":"Gustavo","raw_affiliation_strings":["Jesus"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Jesus","institution_ids":["https://openalex.org/I4210107427"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5137056401","display_name":"Angulo","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Angulo","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":false,"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":0.23999999463558197,"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":0.23999999463558197,"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/T11612","display_name":"Stochastic Gradient Optimization Techniques","score":0.20759999752044678,"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/T11206","display_name":"Model Reduction and Neural Networks","score":0.07240000367164612,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematical-morphology","display_name":"Mathematical morphology","score":0.483599990606308},{"id":"https://openalex.org/keywords/dilation","display_name":"Dilation (metric space)","score":0.42730000615119934},{"id":"https://openalex.org/keywords/pointwise","display_name":"Pointwise","score":0.41670000553131104},{"id":"https://openalex.org/keywords/lattice","display_name":"Lattice (music)","score":0.40790000557899475},{"id":"https://openalex.org/keywords/algebraic-number","display_name":"Algebraic number","score":0.38989999890327454},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.38029998540878296},{"id":"https://openalex.org/keywords/convolution","display_name":"Convolution (computer science)","score":0.3546999990940094},{"id":"https://openalex.org/keywords/idempotence","display_name":"Idempotence","score":0.3402000069618225}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6157000064849854},{"id":"https://openalex.org/C185568154","wikidata":"https://www.wikidata.org/wiki/Q530242","display_name":"Mathematical morphology","level":4,"score":0.483599990606308},{"id":"https://openalex.org/C2780757906","wikidata":"https://www.wikidata.org/wiki/Q5276676","display_name":"Dilation (metric space)","level":2,"score":0.42730000615119934},{"id":"https://openalex.org/C2777984123","wikidata":"https://www.wikidata.org/wiki/Q9248237","display_name":"Pointwise","level":2,"score":0.41670000553131104},{"id":"https://openalex.org/C2781204021","wikidata":"https://www.wikidata.org/wiki/Q6497091","display_name":"Lattice (music)","level":2,"score":0.40790000557899475},{"id":"https://openalex.org/C9376300","wikidata":"https://www.wikidata.org/wiki/Q168817","display_name":"Algebraic number","level":2,"score":0.38989999890327454},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.38029998540878296},{"id":"https://openalex.org/C45347329","wikidata":"https://www.wikidata.org/wiki/Q5166604","display_name":"Convolution (computer science)","level":3,"score":0.3546999990940094},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.35030001401901245},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.3409000039100647},{"id":"https://openalex.org/C172252984","wikidata":"https://www.wikidata.org/wiki/Q368988","display_name":"Idempotence","level":2,"score":0.3402000069618225},{"id":"https://openalex.org/C2780580338","wikidata":"https://www.wikidata.org/wiki/Q577835","display_name":"Adjunction","level":2,"score":0.31119999289512634},{"id":"https://openalex.org/C182419690","wikidata":"https://www.wikidata.org/wiki/Q205464","display_name":"Algebraic structure","level":2,"score":0.3107999861240387},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3093000054359436},{"id":"https://openalex.org/C9973445","wikidata":"https://www.wikidata.org/wiki/Q8349849","display_name":"Congruence relation","level":2,"score":0.2928999960422516},{"id":"https://openalex.org/C2779193601","wikidata":"https://www.wikidata.org/wiki/Q20026918","display_name":"Mathematical theory","level":2,"score":0.28700000047683716},{"id":"https://openalex.org/C204707403","wikidata":"https://www.wikidata.org/wiki/Q1152398","display_name":"Canonical form","level":2,"score":0.2806999981403351},{"id":"https://openalex.org/C192939610","wikidata":"https://www.wikidata.org/wiki/Q188444","display_name":"Differential geometry","level":2,"score":0.27810001373291016},{"id":"https://openalex.org/C187834632","wikidata":"https://www.wikidata.org/wiki/Q188804","display_name":"Factorization","level":2,"score":0.2678999900817871},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.26649999618530273},{"id":"https://openalex.org/C102519508","wikidata":"https://www.wikidata.org/wiki/Q6520159","display_name":"Fourier transform","level":2,"score":0.2506999969482422},{"id":"https://openalex.org/C110484373","wikidata":"https://www.wikidata.org/wiki/Q264398","display_name":"Adjacency list","level":2,"score":0.2502000033855438}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.48550/arxiv.2605.24608","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2605.24608","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,"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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"doi:10.48550/arxiv.2605.24608","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2605.24608","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,"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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"},"sustainable_development_goals":[{"display_name":"Sustainable cities and communities","score":0.5780581831932068,"id":"https://metadata.un.org/sdg/11"}],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":[],"abstract_inverted_index":{"We":[0,197],"develop":[1],"a":[2,46,65,80,86,97,117,199],"rigorous":[3],"algebraic":[4,149],"framework":[5,206],"for":[6,35],"deep":[7,48],"convolutional":[8],"architectures,":[9],"CNNs,":[10],"ResNets,":[11],"and":[12,22,83,93,127,176,192,202,212,222],"encoder--decoder":[13],"networks":[14],"such":[15],"as":[16],"UNet,":[17],"grounded":[18],"in":[19,73,88,100,112,153],"lattice":[20,115],"theory":[21,34],"mathematical":[23],"morphology.":[24],"The":[25,50,205],"central":[26],"tool":[27],"is":[28,53,64,70,79,85,96,105,116,163],"the":[29,55,68,74,89,107,113,121,147,160,179,186,193,213,217,224],"Matheron--Maragos--Banon--Barrera":[30],"(MMBB)":[31],"universal":[32],"representation":[33],"translation-invariant":[36],"operators,":[37],"which":[38],"we":[39],"apply":[40],"systematically":[41],"to":[42],"every":[43],"layer":[44,162,167,183,189],"of":[45,110],"standard":[47,56,154],"network.":[49],"principal":[51],"finding":[52,104],"that":[54,106,169],"CNN":[57],"pipeline":[58],"(linear":[59],"convolution~$+$":[60],"ReLU~$+$":[61],"flat":[62],"max-pooling)":[63],"cross-lattice":[66],"operator:":[67],"convolution":[69],"an":[71,137],"erosion":[72,134],"Fourier":[75],"inf-semilattice":[76],"while":[77],"ReLU":[78,111],"lattice-join":[81],"closing":[82],"max-pooling":[84],"dilation":[87],"pointwise":[90,114],"max-plus":[91,181],"lattice,":[92],"their":[94],"composition":[95],"morphological":[98,133,182,195],"opening":[99],"neither.":[101],"A":[102],"second":[103],"upper":[108],"adjoint":[109,219],"global":[118],"(non-local)":[119],"operator,":[120],"identity":[122],"on":[123],"globally":[124],"non-negative":[125],"functions":[126],"$-\\infty$":[128],"otherwise,":[129],"so":[130],"no":[131],"local":[132],"can":[135],"form":[136],"adjunction":[138],"pair":[139],"with":[140,229],"ReLU.":[141],"These":[142],"two":[143],"results":[144],"together":[145],"provide":[146],"precise":[148],"reason":[150],"why":[151],"depth":[152],"CNNs":[155],"introduces":[156],"genuine":[157,171],"representational":[158],"power:":[159],"composed":[161],"not":[164],"idempotent.":[165],"Three":[166],"designs":[168],"are":[170,174],"idempotent":[172],"openings":[173],"identified":[175],"fully":[177],"characterised:":[178],"pure":[180],"(pointwise":[184],"lattice),":[185,191],"spectral":[187],"Wiener":[188],"(Fourier":[190],"self-dual":[194],"layer.":[196],"establish":[198],"complete":[200],"fixed-point":[201],"convergence":[203],"theory.":[204],"also":[207],"unifies":[208],"max-pooling,":[209],"strided":[210],"convolution,":[211],"Laplacian":[214],"pyramid":[215,220],"under":[216],"Goutsias--Heijmans":[218],"theory,":[221],"gives":[223],"Activation--Pooling":[225],"Dilation":[226],"(APD)":[227],"factorisation":[228],"its":[230],"correct":[231],"adjoint.":[232]},"counts_by_year":[],"updated_date":"2026-07-28T07:46:37.118299","created_date":"2026-05-27T00:00:00"}