{"id":"https://openalex.org/W7169842760","doi":"https://doi.org/10.48550/arxiv.2607.17673","title":"Beyond Objective Expressivity: Geometry Preservation in Multimodal Contrastive Learning","display_name":"Beyond Objective Expressivity: Geometry Preservation in Multimodal Contrastive Learning","publication_year":2026,"publication_date":"2026-07-20","ids":{"openalex":"https://openalex.org/W7169842760","doi":"https://doi.org/10.48550/arxiv.2607.17673"},"language":null,"primary_location":{"id":"doi:10.48550/arxiv.2607.17673","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2607.17673","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.2607.17673","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5121665171","display_name":"Tillmann Rheude","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Rheude, Tillmann","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5141300185","display_name":"Roland Eils","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Eils, Roland","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5141306938","display_name":"Benjamin Wild","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Wild, Benjamin","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"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/T11307","display_name":"Domain Adaptation and Few-Shot Learning","score":0.30329999327659607,"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/T11307","display_name":"Domain Adaptation and Few-Shot Learning","score":0.30329999327659607,"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/T11273","display_name":"Advanced Graph Neural Networks","score":0.17870000004768372,"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/T11448","display_name":"Face recognition and analysis","score":0.13899999856948853,"subfield":{"id":"https://openalex.org/subfields/1707","display_name":"Computer Vision and Pattern Recognition"},"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/jacobian-matrix-and-determinant","display_name":"Jacobian matrix and determinant","score":0.7272999882698059},{"id":"https://openalex.org/keywords/benchmark","display_name":"Benchmark (surveying)","score":0.5443000197410583},{"id":"https://openalex.org/keywords/representation","display_name":"Representation (politics)","score":0.5110999941825867},{"id":"https://openalex.org/keywords/encoder","display_name":"Encoder","score":0.47620001435279846},{"id":"https://openalex.org/keywords/pairwise-comparison","display_name":"Pairwise comparison","score":0.447299987077713},{"id":"https://openalex.org/keywords/feature-learning","display_name":"Feature learning","score":0.4165000021457672},{"id":"https://openalex.org/keywords/residual","display_name":"Residual","score":0.40139999985694885},{"id":"https://openalex.org/keywords/minification","display_name":"Minification","score":0.3978999853134155}],"concepts":[{"id":"https://openalex.org/C200331156","wikidata":"https://www.wikidata.org/wiki/Q506041","display_name":"Jacobian matrix and determinant","level":2,"score":0.7272999882698059},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.5669000148773193},{"id":"https://openalex.org/C185798385","wikidata":"https://www.wikidata.org/wiki/Q1161707","display_name":"Benchmark (surveying)","level":2,"score":0.5443000197410583},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.5418000221252441},{"id":"https://openalex.org/C2776359362","wikidata":"https://www.wikidata.org/wiki/Q2145286","display_name":"Representation (politics)","level":3,"score":0.5110999941825867},{"id":"https://openalex.org/C118505674","wikidata":"https://www.wikidata.org/wiki/Q42586063","display_name":"Encoder","level":2,"score":0.47620001435279846},{"id":"https://openalex.org/C184898388","wikidata":"https://www.wikidata.org/wiki/Q1435712","display_name":"Pairwise comparison","level":2,"score":0.447299987077713},{"id":"https://openalex.org/C59404180","wikidata":"https://www.wikidata.org/wiki/Q17013334","display_name":"Feature learning","level":2,"score":0.4165000021457672},{"id":"https://openalex.org/C155512373","wikidata":"https://www.wikidata.org/wiki/Q287450","display_name":"Residual","level":2,"score":0.40139999985694885},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.4009000062942505},{"id":"https://openalex.org/C147764199","wikidata":"https://www.wikidata.org/wiki/Q6865248","display_name":"Minification","level":2,"score":0.3978999853134155},{"id":"https://openalex.org/C2776135515","wikidata":"https://www.wikidata.org/wiki/Q17143721","display_name":"Regularization (linguistics)","level":2,"score":0.3968999981880188},{"id":"https://openalex.org/C2776502983","wikidata":"https://www.wikidata.org/wiki/Q690182","display_name":"Contrast (vision)","level":2,"score":0.38339999318122864},{"id":"https://openalex.org/C137836250","wikidata":"https://www.wikidata.org/wiki/Q984063","display_name":"Optimization problem","level":2,"score":0.36419999599456787},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.36059999465942383},{"id":"https://openalex.org/C2780586882","wikidata":"https://www.wikidata.org/wiki/Q7520643","display_name":"Simple (philosophy)","level":2,"score":0.32589998841285706},{"id":"https://openalex.org/C31972630","wikidata":"https://www.wikidata.org/wiki/Q844240","display_name":"Computer vision","level":1,"score":0.3046000003814697},{"id":"https://openalex.org/C119857082","wikidata":"https://www.wikidata.org/wiki/Q2539","display_name":"Machine learning","level":1,"score":0.30399999022483826},{"id":"https://openalex.org/C2780695315","wikidata":"https://www.wikidata.org/wiki/Q3799040","display_name":"Unobservable","level":2,"score":0.30219998955726624},{"id":"https://openalex.org/C2779903281","wikidata":"https://www.wikidata.org/wiki/Q6888026","display_name":"Modalities","level":2,"score":0.287200003862381},{"id":"https://openalex.org/C153180895","wikidata":"https://www.wikidata.org/wiki/Q7148389","display_name":"Pattern recognition (psychology)","level":2,"score":0.2863999903202057},{"id":"https://openalex.org/C26517878","wikidata":"https://www.wikidata.org/wiki/Q228039","display_name":"Key (lock)","level":2,"score":0.28060001134872437},{"id":"https://openalex.org/C80444323","wikidata":"https://www.wikidata.org/wiki/Q2878974","display_name":"Theoretical computer science","level":1,"score":0.2782000005245209},{"id":"https://openalex.org/C204321447","wikidata":"https://www.wikidata.org/wiki/Q30642","display_name":"Natural language processing","level":1,"score":0.2736000120639801},{"id":"https://openalex.org/C75553542","wikidata":"https://www.wikidata.org/wiki/Q178161","display_name":"A priori and a posteriori","level":2,"score":0.26350000500679016}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.48550/arxiv.2607.17673","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2607.17673","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.2607.17673","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2607.17673","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":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":[],"abstract_inverted_index":{"Contrastive":[0],"learning":[1,131],"is":[2],"increasingly":[3],"moving":[4],"toward":[5],"settings":[6],"with":[7],"three":[8],"or":[9,49],"more":[10],"modalities":[11],"instead":[12],"of":[13,146],"image-text":[14],"pairs.":[15],"Yet,":[16],"extending":[17],"models":[18],"from":[19],"pairwise":[20],"to":[21,54],"higher-order":[22],"multimodal":[23,61,129],"alignment":[24],"can":[25],"introduce":[26,64],"optimization":[27,144],"and":[28,59,75,83,94,106,143],"representation":[29],"challenges.":[30],"We":[31,63],"identify":[32],"encoder":[33],"Jacobian":[34,56,72,102],"conditioning":[35,70,103],"as":[36],"a":[37,91],"key":[38],"factor":[39],"in":[40,120],"trimodal":[41],"contrastive":[42,112,130],"learning:":[43],"poorly":[44],"conditioned":[45],"encoders":[46,66],"exhibit":[47],"collapsing":[48],"amplified":[50],"singular-value":[51],"spectra,":[52],"leading":[53],"exploding":[55],"condition":[57],"numbers":[58],"degraded":[60],"alignment.":[62],"geometry-preserving":[65],"(GPEs)":[67],"by":[68],"directly":[69],"the":[71,141,147],"through":[73],"regularization":[74],"demonstrating":[76],"that":[77,128],"simple":[78],"modifications":[79],"like":[80],"LeakyReLU":[81],"activations":[82],"residual":[84],"paths":[85],"recover":[86],"these":[87],"geometric":[88,142],"benefits.":[89],"Across":[90],"synthetic":[92],"benchmark":[93],"four":[95],"real-world":[96],"datasets":[97],"including":[98],"missing":[99],"modalities,":[100],"improving":[101],"boosts":[104],"retrieval":[105],"linear":[107,121],"probe":[108],"performance":[109],"across":[110],"multiple":[111],"objectives,":[113],"whereas":[114],"expressive":[115],"objectives":[116],"yield":[117],"little":[118],"benefit":[119],"probes.":[122],"More":[123],"broadly,":[124],"our":[125],"results":[126],"show":[127],"depends":[132],"not":[133],"only":[134],"on":[135,140],"objective":[136],"expressivity,":[137],"but":[138],"also":[139],"properties":[145],"underlying":[148],"encoders.":[149]},"counts_by_year":[],"updated_date":"2026-07-28T07:46:37.118299","created_date":"2026-07-22T00:00:00"}