{"id":"https://openalex.org/W7170809197","doi":"https://doi.org/10.48550/arxiv.2607.20578","title":"Fisher Widths: Local Learning Geometry and Anisotropic Recovery","display_name":"Fisher Widths: Local Learning Geometry and Anisotropic Recovery","publication_year":2026,"publication_date":"2026-07-22","ids":{"openalex":"https://openalex.org/W7170809197","doi":"https://doi.org/10.48550/arxiv.2607.20578"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2607.20578","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2607.20578","pdf_url":"https://arxiv.org/pdf/2607.20578","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/2607.20578","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5016120331","display_name":"Vu Khac Ky","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Ky, Vu Khac","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5016120331"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":1,"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/T10500","display_name":"Sparse and Compressive Sensing Techniques","score":0.7523999810218811,"subfield":{"id":"https://openalex.org/subfields/1711","display_name":"Signal Processing"},"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/T10500","display_name":"Sparse and Compressive Sensing Techniques","score":0.7523999810218811,"subfield":{"id":"https://openalex.org/subfields/1711","display_name":"Signal Processing"},"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/T10136","display_name":"Statistical Methods and Inference","score":0.1251000016927719,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"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/T11612","display_name":"Stochastic Gradient Optimization Techniques","score":0.023099999874830246,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/fisher-information-metric","display_name":"Fisher information metric","score":0.6729999780654907},{"id":"https://openalex.org/keywords/statistical-manifolds","display_name":"statistical manifolds","score":0.9430000185966492},{"id":"https://openalex.org/keywords/sparse-signal-recovery","display_name":"sparse signal recovery","score":0.597000002861023},{"id":"https://openalex.org/keywords/statistical-dimension","display_name":"statistical dimension","score":0.8740000128746033}],"concepts":[{"id":"https://openalex.org/C29406490","wikidata":"https://www.wikidata.org/wiki/Q1420659","display_name":"Fisher information","level":2,"score":0.777999997138977},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7709000110626221},{"id":"https://openalex.org/C198082294","wikidata":"https://www.wikidata.org/wiki/Q3399648","display_name":"Position (finance)","level":2,"score":0.5175999999046326},{"id":"https://openalex.org/C195065555","wikidata":"https://www.wikidata.org/wiki/Q214881","display_name":"Curvature","level":2,"score":0.5091000199317932},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.4982999861240387},{"id":"https://openalex.org/C169391604","wikidata":"https://www.wikidata.org/wiki/Q7604402","display_name":"Statistical manifold","level":5,"score":0.486299991607666},{"id":"https://openalex.org/C178650346","wikidata":"https://www.wikidata.org/wiki/Q201984","display_name":"Covariance","level":2,"score":0.48500001430511475},{"id":"https://openalex.org/C163716315","wikidata":"https://www.wikidata.org/wiki/Q901177","display_name":"Gaussian","level":2,"score":0.4336000084877014},{"id":"https://openalex.org/C114289077","wikidata":"https://www.wikidata.org/wiki/Q3284399","display_name":"Statistical model","level":2,"score":0.41339999437332153},{"id":"https://openalex.org/C207798031","wikidata":"https://www.wikidata.org/wiki/Q8563425","display_name":"Fisher kernel","level":5,"score":0.39399999380111694},{"id":"https://openalex.org/C192939610","wikidata":"https://www.wikidata.org/wiki/Q188444","display_name":"Differential geometry","level":2,"score":0.39070001244544983},{"id":"https://openalex.org/C109546454","wikidata":"https://www.wikidata.org/wiki/Q3798604","display_name":"Information geometry","level":4,"score":0.3799999952316284},{"id":"https://openalex.org/C16140857","wikidata":"https://www.wikidata.org/wiki/Q945953","display_name":"Gaussian curvature","level":3,"score":0.37770000100135803},{"id":"https://openalex.org/C85725439","wikidata":"https://www.wikidata.org/wiki/Q466686","display_name":"Anisotropy","level":2,"score":0.3601999878883362},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.36010000109672546},{"id":"https://openalex.org/C129782007","wikidata":"https://www.wikidata.org/wiki/Q162886","display_name":"Euclidean geometry","level":2,"score":0.3416999876499176},{"id":"https://openalex.org/C61326573","wikidata":"https://www.wikidata.org/wiki/Q1496376","display_name":"Gaussian process","level":3,"score":0.33239999413490295},{"id":"https://openalex.org/C2776799497","wikidata":"https://www.wikidata.org/wiki/Q484298","display_name":"Surface (topology)","level":2,"score":0.32190001010894775},{"id":"https://openalex.org/C87007009","wikidata":"https://www.wikidata.org/wiki/Q210832","display_name":"Statistical hypothesis testing","level":2,"score":0.321399986743927},{"id":"https://openalex.org/C207467116","wikidata":"https://www.wikidata.org/wiki/Q4385666","display_name":"Inverse","level":2,"score":0.3093999922275543},{"id":"https://openalex.org/C135252773","wikidata":"https://www.wikidata.org/wiki/Q1567213","display_name":"Inverse problem","level":2,"score":0.2773999869823456},{"id":"https://openalex.org/C135692309","wikidata":"https://www.wikidata.org/wiki/Q111124","display_name":"Square (algebra)","level":2,"score":0.2632000148296356},{"id":"https://openalex.org/C185142706","wikidata":"https://www.wikidata.org/wiki/Q1134404","display_name":"Covariance matrix","level":2,"score":0.2572000026702881}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2607.20578","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2607.20578","pdf_url":"https://arxiv.org/pdf/2607.20578","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.2607.20578","doi":"https://doi.org/10.48550/arxiv.2607.20578","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2607.20578","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:2607.20578","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2607.20578","pdf_url":"https://arxiv.org/pdf/2607.20578","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/W7170809197.pdf","grobid_xml":"https://content.openalex.org/works/W7170809197.grobid-xml"},"referenced_works_count":1,"referenced_works":["https://openalex.org/W7165110421"],"related_works":["https://openalex.org/W7165110421","https://openalex.org/W6942301466","https://openalex.org/W4295274053","https://openalex.org/W7166357610","https://openalex.org/W4307995878","https://openalex.org/W1035391226","https://openalex.org/W1525725047","https://openalex.org/W2913466177","https://openalex.org/W3049028083","https://openalex.org/W3114928620"],"abstract_inverted_index":{"We":[0,130],"study":[1],"Gaussian-width":[2],"complexity":[3,185],"on":[4,78,115,119],"statistical":[5,42,138],"manifolds":[6],"through":[7],"a":[8,132,146,158],"pair":[9],"of":[10,54,93,122,149],"functionals:":[11],"the":[12,21,25,33,45,48,52,59,63,72,84,87,91,102,109,120,123,127,136,162,190,199],"primal":[13,163],"Fisher":[14,22,35,49,64,81,104,128,181],"width":[15,27,50,89],"$w_G(T)":[16],"=":[17,29],"w(G^{1/2}T)$,":[18],"induced":[19,31,61],"by":[20,32,62,101],"metric,":[23],"and":[24,145,164],"inverse-Fisher":[26,88,165],"$w_{G^{-1}}(T)":[28],"w(G^{-1/2}T)$,":[30],"inverse":[34,103],"metric.":[36],"The":[37],"two":[38],"widths":[39,196],"play":[40],"complementary":[41],"roles.":[43],"On":[44,83,167],"learning":[46],"side,":[47,86],"measures":[51],"size":[53],"local":[55],"parameter":[56],"fluctuations":[57],"in":[58,126],"geometry":[60,111,188],"information.":[65,105],"For":[66,106],"Fisher-regular":[67],"losses,":[68],"we":[69,156],"prove":[70],"that":[71],"scale":[73],"\\(w_G(H_r)/\\sqrt":[74],"n\\)":[75],"is":[76,99],"attained":[77],"sufficiently":[79],"small":[80],"balls.":[82],"recovery":[85,143],"captures":[90],"effect":[92],"anisotropic":[94],"Gaussian":[95],"measurements":[96],"whose":[97],"covariance":[98],"determined":[100],"sparse":[107],"recovery,":[108],"resulting":[110],"depends":[112],"not":[113],"only":[114],"sparsity":[116],"but":[117,192],"also":[118],"position":[121],"active":[124],"coordinates":[125],"spectrum.":[129],"obtain":[131],"two-sided":[133],"estimate":[134],"for":[135],"corresponding":[137],"dimension,":[139],"together":[140],"with":[141,151],"support-sensitive":[142],"estimates":[144],"natural":[147],"ordering":[148],"supports":[150],"different":[152],"curvature":[153],"profiles.":[154],"Finally,":[155],"establish":[157],"sharp":[159],"relation":[160],"between":[161],"widths.":[166],"any":[168],"common":[169],"compact":[170],"coordinate":[171],"set":[172],"$T$,":[173],"they":[174],"satisfy":[175],"\\[":[176],"w_G(T)w_{G^{-1}}(T)\\geq":[177],"w(T)^2.":[178],"\\]":[179],"Thus,":[180],"anisotropy":[182],"may":[183],"transfer":[184],"from":[186],"one":[187],"to":[189,198],"other,":[191],"cannot":[193],"reduce":[194],"both":[195],"relative":[197],"Euclidean":[200],"scale.":[201]},"counts_by_year":[{"year":2026,"cited_by_count":1}],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-07-25T00:00:00"}
