{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,1]],"date-time":"2026-08-01T00:19:34Z","timestamp":1785543574299,"version":"3.56.0"},"reference-count":53,"publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"4","license":[{"start":{"date-parts":[[2023,4,1]],"date-time":"2023-04-01T00:00:00Z","timestamp":1680307200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/ieeexplore.ieee.org\/Xplorehelp\/downloads\/license-information\/IEEE.html"},{"start":{"date-parts":[[2023,4,1]],"date-time":"2023-04-01T00:00:00Z","timestamp":1680307200000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-029"},{"start":{"date-parts":[[2023,4,1]],"date-time":"2023-04-01T00:00:00Z","timestamp":1680307200000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-037"}],"funder":[{"DOI":"10.13039\/501100004052","name":"King Abdullah University of Science and Technology","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100004052","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Office of Sponsored Research","award":["OSR-CRG2019-4033"],"award-info":[{"award-number":["OSR-CRG2019-4033"]}]},{"name":"SDAIA-KAUST Center of Excellence in Data Science and Artificial Intelligence"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["IEEE Trans. Pattern Anal. Mach. Intell."],"published-print":{"date-parts":[[2023,4,1]]},"DOI":"10.1109\/tpami.2022.3201490","type":"journal-article","created":{"date-parts":[[2022,8,24]],"date-time":"2022-08-24T19:30:53Z","timestamp":1661369453000},"page":"5027-5037","source":"Crossref","is-referenced-by-count":16,"title":["On the Decision Boundaries of Neural Networks: A Tropical Geometry Perspective"],"prefix":"10.1109","volume":"45","author":[{"given":"Motasem","family":"Alfarra","sequence":"first","affiliation":[{"name":"King Abdullah University of Science and Technology, Thuwal, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6169-3918","authenticated-orcid":false,"given":"Adel","family":"Bibi","sequence":"additional","affiliation":[{"name":"King Abdullah University of Science and Technology, Thuwal, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Hasan","family":"Hammoud","sequence":"additional","affiliation":[{"name":"King Abdullah University of Science and Technology, Thuwal, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Mohamed","family":"Gaafar","sequence":"additional","affiliation":[{"name":"Zalando SE, Berlin, Germany"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5534-587X","authenticated-orcid":false,"given":"Bernard","family":"Ghanem","sequence":"additional","affiliation":[{"name":"King Abdullah University of Science and Technology, Thuwal, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"263","reference":[{"key":"ref1","first-page":"1106","article-title":"ImageNet classification with deep convolutional neural networks","volume-title":"Proc. Adv. Neural Inf. Process. Syst.","author":"Krizhevsky"},{"key":"ref2","doi-asserted-by":"publisher","DOI":"10.1109\/MSP.2012.2205597"},{"key":"ref3","doi-asserted-by":"publisher","DOI":"10.1146\/annurev.neuro.26.041002.131047"},{"key":"ref4","article-title":"BERT: Pre-training of deep bidirectional transformers for language understanding","author":"Devlin","year":"2018"},{"key":"ref5","doi-asserted-by":"publisher","DOI":"10.1038\/ncomms13890"},{"key":"ref6","doi-asserted-by":"publisher","DOI":"10.1038\/s41591-019-0447-x"},{"key":"ref7","doi-asserted-by":"publisher","DOI":"10.1038\/s41588-019-0420-0"},{"key":"ref8","doi-asserted-by":"publisher","DOI":"10.1038\/nature14539"},{"key":"ref9","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9781107298019"},{"key":"ref10","doi-asserted-by":"publisher","DOI":"10.5555\/2969033.2969153"},{"key":"ref11","article-title":"Understanding deep neural networks with rectified linear units","volume-title":"Proc. Int. Conf. Learn. Representations","author":"Arora"},{"key":"ref12","doi-asserted-by":"publisher","DOI":"10.1109\/CVPR.2019.00929"},{"key":"ref13","article-title":"Decision boundary analysis of adversarial examples","volume-title":"Proc. Int. Conf. Learn. Representations","author":"He"},{"key":"ref14","article-title":"On the decision boundary of deep neural networks","author":"Li","year":"2018"},{"key":"ref15","article-title":"Gradient descent maximizes the margin of homogeneous neural networks","volume-title":"Proc. Int. Conf. Learn. Representations","author":"Lyu"},{"key":"ref16","first-page":"10836","article-title":"Generalization bounds of stochastic gradient descent for wide and deep neural networks","volume-title":"Proc. Adv. Neural Inf. Process. Syst.","author":"Cao"},{"key":"ref17","doi-asserted-by":"publisher","DOI":"10.1016\/j.neunet.2021.02.024"},{"key":"ref18","article-title":"The tropical geometry of shortest paths","author":"Joswig","year":"2019"},{"key":"ref19","doi-asserted-by":"publisher","DOI":"10.1137\/130936464"},{"key":"ref20","doi-asserted-by":"publisher","DOI":"10.1063\/1.5089992"},{"key":"ref21","doi-asserted-by":"publisher","DOI":"10.1090\/S0894-0347-05-00477-7"},{"key":"ref22","doi-asserted-by":"publisher","DOI":"10.1142\/s0218196711006674"},{"key":"ref23","doi-asserted-by":"publisher","DOI":"10.1287\/moor.2018.0975"},{"key":"ref24","first-page":"5819","article-title":"Tropical geometry of deep neural networks","volume-title":"Proc. Int. Conf. Mach. Learn.","author":"Zhang"},{"key":"ref25","article-title":"Tropical polynomial division and neural networks","author":"Smyrnis","year":"2019"},{"key":"ref26","article-title":"The lottery ticket hypothesis: Finding sparse, trainable neural networks","volume-title":"Proc. Int. Conf. Learn. Representations","author":"Frankle"},{"key":"ref27","article-title":"Very deep convolutional networks for large-scale image recognition","author":"Simonyan","year":"2014"},{"key":"ref28","article-title":"Reading digits in natural images with unsupervised feature learning","volume-title":"Proc. Adv. Neural Inf. Process. Syst.","author":"Netzer"},{"key":"ref29","article-title":"Learning multiple layers of features from tiny images","author":"Krizhevsky","year":"2009"},{"key":"ref30","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-0346-0048-4"},{"key":"ref31","doi-asserted-by":"crossref","DOI":"10.1090\/gsm\/161","volume-title":"Introduction to Tropical Geometry, Ser. Graduate Studies in Mathematics","author":"Maclagan","year":"2015"},{"key":"ref32","doi-asserted-by":"publisher","DOI":"10.4169\/amer.math.monthly.121.07.563"},{"key":"ref33","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0121142"},{"key":"ref34","doi-asserted-by":"publisher","DOI":"10.1137\/0406019"},{"key":"ref35","article-title":"A randomized algorithm for enumerating zonotope vertices","author":"Stinson","year":"2016"},{"key":"ref36","article-title":"Picking winning tickets before training by preserving gradient flow","volume-title":"Proc. Int. Conf. Learn. Representations","author":"Wang"},{"key":"ref37","first-page":"4932","article-title":"One ticket to win them all: Generalizing lottery ticket initializations across datasets and optimizers","volume-title":"Proc. Adv. Neural Inf. Process. Syst.","author":"Morcos"},{"key":"ref38","first-page":"6377","article-title":"Pruning neural networks without any data by iteratively conserving synaptic flow","author":"Tanaka","year":"2020","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"ref39","first-page":"249","article-title":"Understanding the difficulty of training deep feedforward neural networks","volume-title":"Proc. Int. Conf. Artif. Intell. Statist.","author":"Glorot"},{"key":"ref40","first-page":"598","article-title":"Optimal brain damage","volume-title":"Proc. Adv. Neural Inf. Process. Syst.","author":"LeCun"},{"key":"ref41","first-page":"164","article-title":"Second order derivatives for network pruning: Optimal brain surgeon","volume-title":"Proc. Adv. Neural Inf. Process. Syst.","author":"Hassibi"},{"key":"ref42","doi-asserted-by":"publisher","DOI":"10.1109\/CVPR.2018.00948"},{"key":"ref43","article-title":"Learning both weights and connections for efficient neural networks","volume":"28","author":"Han","year":"2015","journal-title":"Adv. Neural Inform. Process. Syst."},{"key":"ref44","doi-asserted-by":"publisher","DOI":"10.18653\/v1\/K16-1029"},{"key":"ref45","doi-asserted-by":"publisher","DOI":"10.1109\/CVPR.2009.5206848"},{"key":"ref46","first-page":"9068","article-title":"Multiclass neural network minimization via tropical newton polytope approximation","volume-title":"Proc. Int. Conf. Mach. Learn.","author":"Smyrnis"},{"key":"ref47","article-title":"Intriguing properties of neural networks","volume-title":"Proc. Int. Conf. Learn. Representations","author":"Szegedy"},{"key":"ref48","doi-asserted-by":"publisher","DOI":"10.1109\/CVPR.2018.00099"},{"key":"ref49","doi-asserted-by":"publisher","DOI":"10.1145\/3319535.3354259"},{"key":"ref50","article-title":"On the geometry of adversarial examples","author":"Khoury","year":"2018"},{"key":"ref51","doi-asserted-by":"publisher","DOI":"10.1109\/SP.2017.49"},{"key":"ref52","doi-asserted-by":"publisher","DOI":"10.1561\/2200000016"},{"key":"ref53","article-title":"Structured adversarial attack: Towards general implementation and better interpretability","volume-title":"Proc. Int. Conf. Learn. Representations","author":"Xu"}],"container-title":["IEEE Transactions on Pattern Analysis and Machine Intelligence"],"original-title":[],"link":[{"URL":"http:\/\/xplorestaging.ieee.org\/ielx7\/34\/10061515\/09866576.pdf?arnumber=9866576","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,6,25]],"date-time":"2024-06-25T19:52:50Z","timestamp":1719345170000},"score":1,"resource":{"primary":{"URL":"https:\/\/ieeexplore.ieee.org\/document\/9866576\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,4,1]]},"references-count":53,"journal-issue":{"issue":"4"},"URL":"https:\/\/doi.org\/10.1109\/tpami.2022.3201490","relation":{},"ISSN":["0162-8828","2160-9292","1939-3539"],"issn-type":[{"value":"0162-8828","type":"print"},{"value":"2160-9292","type":"electronic"},{"value":"1939-3539","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,4,1]]}}}