{"id":"https://openalex.org/W7164919563","doi":"https://doi.org/10.48550/arxiv.2606.15669","title":"Z-Plane Neural Networks: Bounded Geometric Activation Replaces ReLU and LayerNorm","display_name":"Z-Plane Neural Networks: Bounded Geometric Activation Replaces ReLU and LayerNorm","publication_year":2026,"publication_date":"2026-06-14","ids":{"openalex":"https://openalex.org/W7164919563","doi":"https://doi.org/10.48550/arxiv.2606.15669"},"language":null,"primary_location":{"id":"doi:10.48550/arxiv.2606.15669","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2606.15669","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":"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.2606.15669","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5020854888","display_name":"Sungwoo Goo","orcid":"https://orcid.org/0000-0003-2809-5921"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Goo, Sungwoo","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5026070944","display_name":"Hwi\u2010yeol Yun","orcid":"https://orcid.org/0000-0001-8793-2449"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Yun, Hwi-yeol","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5138744831","display_name":"Sangkeun Jung","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Jung, Sangkeun","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/T12611","display_name":"Neural Networks and Reservoir Computing","score":0.8044999837875366,"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/T12611","display_name":"Neural Networks and Reservoir Computing","score":0.8044999837875366,"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":0.02590000070631504,"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"}},{"id":"https://openalex.org/T12808","display_name":"Ferroelectric and Negative Capacitance Devices","score":0.023800000548362732,"subfield":{"id":"https://openalex.org/subfields/2208","display_name":"Electrical and Electronic Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/bounded-function","display_name":"Bounded function","score":0.6708999872207642},{"id":"https://openalex.org/keywords/artificial-neural-network","display_name":"Artificial neural network","score":0.5687000155448914},{"id":"https://openalex.org/keywords/mnist-database","display_name":"MNIST database","score":0.5685999989509583},{"id":"https://openalex.org/keywords/gradient-descent","display_name":"Gradient descent","score":0.49970000982284546},{"id":"https://openalex.org/keywords/perceptron","display_name":"Perceptron","score":0.45989999175071716},{"id":"https://openalex.org/keywords/euclidean-geometry","display_name":"Euclidean geometry","score":0.45910000801086426},{"id":"https://openalex.org/keywords/phasor","display_name":"Phasor","score":0.4440999925136566},{"id":"https://openalex.org/keywords/isotropy","display_name":"Isotropy","score":0.4325999915599823},{"id":"https://openalex.org/keywords/normalization","display_name":"Normalization (sociology)","score":0.4244999885559082}],"concepts":[{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.6708999872207642},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.5687000155448914},{"id":"https://openalex.org/C190502265","wikidata":"https://www.wikidata.org/wiki/Q17069496","display_name":"MNIST database","level":3,"score":0.5685999989509583},{"id":"https://openalex.org/C153258448","wikidata":"https://www.wikidata.org/wiki/Q1199743","display_name":"Gradient descent","level":3,"score":0.49970000982284546},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.4884999990463257},{"id":"https://openalex.org/C60908668","wikidata":"https://www.wikidata.org/wiki/Q690207","display_name":"Perceptron","level":3,"score":0.45989999175071716},{"id":"https://openalex.org/C129782007","wikidata":"https://www.wikidata.org/wiki/Q162886","display_name":"Euclidean geometry","level":2,"score":0.45910000801086426},{"id":"https://openalex.org/C176605952","wikidata":"https://www.wikidata.org/wiki/Q827674","display_name":"Phasor","level":4,"score":0.4440999925136566},{"id":"https://openalex.org/C184050105","wikidata":"https://www.wikidata.org/wiki/Q273163","display_name":"Isotropy","level":2,"score":0.4325999915599823},{"id":"https://openalex.org/C136886441","wikidata":"https://www.wikidata.org/wiki/Q926129","display_name":"Normalization (sociology)","level":2,"score":0.4244999885559082},{"id":"https://openalex.org/C190470478","wikidata":"https://www.wikidata.org/wiki/Q2370229","display_name":"Invariant (physics)","level":2,"score":0.4178999960422516},{"id":"https://openalex.org/C38365724","wikidata":"https://www.wikidata.org/wiki/Q4677469","display_name":"Activation function","level":3,"score":0.39809998869895935},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.3873000144958496},{"id":"https://openalex.org/C41608201","wikidata":"https://www.wikidata.org/wiki/Q980509","display_name":"Embedding","level":2,"score":0.3781000077724457},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.376800000667572},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.375900000333786},{"id":"https://openalex.org/C184720557","wikidata":"https://www.wikidata.org/wiki/Q7825049","display_name":"Topology (electrical circuits)","level":2,"score":0.36160001158714294},{"id":"https://openalex.org/C17137986","wikidata":"https://www.wikidata.org/wiki/Q215067","display_name":"Orthogonality","level":2,"score":0.34439998865127563},{"id":"https://openalex.org/C57691317","wikidata":"https://www.wikidata.org/wiki/Q1289248","display_name":"Scalar (mathematics)","level":2,"score":0.33390000462532043},{"id":"https://openalex.org/C7305733","wikidata":"https://www.wikidata.org/wiki/Q207961","display_name":"Geometric shape","level":2,"score":0.3280999958515167},{"id":"https://openalex.org/C112972136","wikidata":"https://www.wikidata.org/wiki/Q7595718","display_name":"Stability (learning theory)","level":2,"score":0.3174000084400177},{"id":"https://openalex.org/C31836371","wikidata":"https://www.wikidata.org/wiki/Q1856609","display_name":"Scalar potential","level":2,"score":0.3149000108242035},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.31450000405311584},{"id":"https://openalex.org/C207821765","wikidata":"https://www.wikidata.org/wiki/Q405372","display_name":"Instability","level":2,"score":0.3095000088214874},{"id":"https://openalex.org/C70836080","wikidata":"https://www.wikidata.org/wiki/Q837940","display_name":"Impulse (physics)","level":2,"score":0.29249998927116394},{"id":"https://openalex.org/C81363708","wikidata":"https://www.wikidata.org/wiki/Q17084460","display_name":"Convolutional neural network","level":2,"score":0.2854999899864197},{"id":"https://openalex.org/C108583219","wikidata":"https://www.wikidata.org/wiki/Q197536","display_name":"Deep learning","level":2,"score":0.2806999981403351},{"id":"https://openalex.org/C120174047","wikidata":"https://www.wikidata.org/wiki/Q847073","display_name":"Euclidean distance","level":2,"score":0.27900001406669617},{"id":"https://openalex.org/C2984842247","wikidata":"https://www.wikidata.org/wiki/Q197536","display_name":"Deep neural networks","level":3,"score":0.27000001072883606},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.2596000134944916},{"id":"https://openalex.org/C56435381","wikidata":"https://www.wikidata.org/wiki/Q1196371","display_name":"Geometric transformation","level":3,"score":0.25589999556541443},{"id":"https://openalex.org/C191640071","wikidata":"https://www.wikidata.org/wiki/Q5377056","display_name":"Energy functional","level":2,"score":0.2535000145435333}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.48550/arxiv.2606.15669","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2606.15669","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":"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.2606.15669","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2606.15669","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":"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":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":[],"abstract_inverted_index":{"Modern":[0],"deep":[1,22,141],"neural":[2],"networks":[3],"rely":[4],"on":[5,64,120],"Euclidean":[6],"scalar":[7],"activations":[8],"(e.g.,":[9,15],"ReLU)":[10],"and":[11,35,99,117,127],"global":[12],"normalization":[13],"techniques":[14],"LayerNorm)":[16],"to":[17],"prevent":[18],"gradient":[19,101],"instability":[20],"in":[21],"architectures.":[23],"However,":[24],"these":[25],"mechanisms":[26],"inherently":[27],"cause":[28],"dead":[29],"neurons,":[30],"discard":[31],"critical":[32],"directional":[33],"information,":[34],"destroy":[36],"the":[37,44,52,81,86,121],"orthogonality":[38],"of":[39,47,115],"feature":[40],"representations.":[41],"Inspired":[42],"by":[43,103],"frequency-modulation":[45],"transmission":[46],"biological":[48],"axons,":[49],"we":[50],"propose":[51],"Z-Plane":[53,110],"Neural":[54],"Network,":[55],"which":[56,79],"maps":[57],"hidden":[58],"states":[59],"into":[60],"2D":[61],"phasor":[62],"bundles":[63],"a":[65,69,108],"hypersphere.":[66],"We":[67,89],"introduce":[68],"novel":[70],"geometric":[71,134],"activation":[72,95,135],"function,":[73],"Radial":[74],"Bounding($\\mathbf{x}":[75],"/":[76],"\\max(1,":[77],"\\|\\mathbf{x}\\|_2)$),":[78],"limits":[80],"energy":[82],"magnitude":[83],"while":[84],"preserving":[85,104],"phase":[87],"(direction).":[88],"demonstrate":[90],"mathematically":[91],"that":[92,132],"this":[93],"isotropic":[94],"maintains":[96],"1-Lipschitz":[97],"continuity":[98],"prevents":[100],"vanishing":[102],"tangential":[105],"gradients.":[106],"Empirically,":[107],"100-layer":[109],"Multi-Layer":[111],"Perceptron":[112],"(MLP)-entirely":[113],"devoid":[114],"ReLU":[116],"LayerNorm-successfully":[118],"converges":[119],"MNIST":[122],"dataset":[123],"with":[124],"98.34%":[125],"accuracy":[126],"absolute":[128],"numerical":[129],"stability,":[130],"proving":[131],"bounded":[133],"alone":[136],"is":[137],"sufficient":[138],"for":[139],"stable":[140],"learning.":[142]},"counts_by_year":[],"updated_date":"2026-07-01T06:00:48.157686","created_date":"2026-06-17T00:00:00"}
