A Universal Approximation Result for Difference of Log-Sum-Exp Neural Networks

We show that a neural network whose output is obtained as the difference of the outputs of two feedforward networks with exponential activation function in the hidden layer and logarithmic activation function in the output node, referred to as log-sum-exp (LSE) network, is a smooth universal approxi...

Celý popis

Uložené v:

Podrobná bibliografia
Vydané v:	IEEE transaction on neural networks and learning systems Ročník 31; číslo 12; s. 5603 - 5612
Hlavní autori:	Calafiore, Giuseppe C., Gaubert, Stephane, Possieri, Corrado
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	United States IEEE 01.12.2020 The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Predmet:	Algorithms Artificial neural networks Computer Science Continuity (mathematics) Convex functions Data-driven optimization Design Design optimization Diabetes mellitus difference of convex (DC) programming Feedforward neural networks feedforward neural networks (FFNs) log-sum-exp (LSE) networks Machine Learning Mathematical analysis Mathematical models Mathematics Neural networks Numerical methods Optimization Optimization and Control Subtraction subtraction-free expressions surrogate models Transforms universal approximation LSE networks DC programming Subtraction-freeexpressions Surrogate models Feedforward neural networks Data-driven optimization Universal ap-proximation
ISSN:	2162-237X, 2162-2388, 2162-2388
On-line prístup:	Získať plný text
Tagy:	Pridať tag Žiadne tagy, Buďte prvý, kto otaguje tento záznam!

Abstract	We show that a neural network whose output is obtained as the difference of the outputs of two feedforward networks with exponential activation function in the hidden layer and logarithmic activation function in the output node, referred to as log-sum-exp (LSE) network, is a smooth universal approximator of continuous functions over convex, compact sets. By using a logarithmic transform, this class of network maps to a family of subtraction-free ratios of generalized posynomials (GPOS), which we also show to be universal approximators of positive functions over log-convex, compact subsets of the positive orthant. The main advantage of difference-LSE networks with respect to classical feedforward neural networks is that, after a standard training phase, they provide surrogate models for a design that possesses a specific difference-of-convex-functions form, which makes them optimizable via relatively efficient numerical methods. In particular, by adapting an existing difference-of-convex algorithm to these models, we obtain an algorithm for performing an effective optimization-based design. We illustrate the proposed approach by applying it to the data-driven design of a diet for a patient with type-2 diabetes and to a nonconvex optimization problem.
AbstractList	We show that a neural network whose output is obtained as the difference of the outputs of two feedforward networks with exponential activation function in the hidden layer and logarithmic activation function in the output node (LSE networks) is a smooth universal approximator of continuous functions over convex, compact sets. By using a logarithmic transform, this class of networks maps to a family of subtraction-free ratios of generalized posynomials, which we also show to be universal approximators of positive functions over log-convex, compact subsets of the positive orthant. The main advantage of Difference-LSE networks with respect to classical feedforward neural networks is that, after a standard training phase, they provide surrogate models for design that possess a specific difference-of-convex-functions form, which makes them optimizable via relatively efficient numerical methods. In particular, by adapting an existing difference-of-convex algorithm to these models, we obtain an algorithm for performing effective optimization-based design. We illustrate the proposed approach by applying it to data-driven design of a diet for a patient with type-2 diabetes. We show that a neural network whose output is obtained as the difference of the outputs of two feedforward networks with exponential activation function in the hidden layer and logarithmic activation function in the output node, referred to as log-sum-exp (LSE) network, is a smooth universal approximator of continuous functions over convex, compact sets. By using a logarithmic transform, this class of network maps to a family of subtraction-free ratios of generalized posynomials (GPOS), which we also show to be universal approximators of positive functions over log-convex, compact subsets of the positive orthant. The main advantage of difference-LSE networks with respect to classical feedforward neural networks is that, after a standard training phase, they provide surrogate models for a design that possesses a specific difference-of-convex-functions form, which makes them optimizable via relatively efficient numerical methods. In particular, by adapting an existing difference-of-convex algorithm to these models, we obtain an algorithm for performing an effective optimization-based design. We illustrate the proposed approach by applying it to the data-driven design of a diet for a patient with type-2 diabetes and to a nonconvex optimization problem.We show that a neural network whose output is obtained as the difference of the outputs of two feedforward networks with exponential activation function in the hidden layer and logarithmic activation function in the output node, referred to as log-sum-exp (LSE) network, is a smooth universal approximator of continuous functions over convex, compact sets. By using a logarithmic transform, this class of network maps to a family of subtraction-free ratios of generalized posynomials (GPOS), which we also show to be universal approximators of positive functions over log-convex, compact subsets of the positive orthant. The main advantage of difference-LSE networks with respect to classical feedforward neural networks is that, after a standard training phase, they provide surrogate models for a design that possesses a specific difference-of-convex-functions form, which makes them optimizable via relatively efficient numerical methods. In particular, by adapting an existing difference-of-convex algorithm to these models, we obtain an algorithm for performing an effective optimization-based design. We illustrate the proposed approach by applying it to the data-driven design of a diet for a patient with type-2 diabetes and to a nonconvex optimization problem. We show that a neural network whose output is obtained as the difference of the outputs of two feedforward networks with exponential activation function in the hidden layer and logarithmic activation function in the output node, referred to as log-sum-exp (LSE) network, is a smooth universal approximator of continuous functions over convex, compact sets. By using a logarithmic transform, this class of network maps to a family of subtraction-free ratios of generalized posynomials (GPOS), which we also show to be universal approximators of positive functions over log-convex, compact subsets of the positive orthant. The main advantage of difference-LSE networks with respect to classical feedforward neural networks is that, after a standard training phase, they provide surrogate models for a design that possesses a specific difference-of-convex-functions form, which makes them optimizable via relatively efficient numerical methods. In particular, by adapting an existing difference-of-convex algorithm to these models, we obtain an algorithm for performing an effective optimization-based design. We illustrate the proposed approach by applying it to the data-driven design of a diet for a patient with type-2 diabetes and to a nonconvex optimization problem.
Author	Gaubert, Stephane Calafiore, Giuseppe C. Possieri, Corrado
Author_xml	– sequence: 1 givenname: Giuseppe C. orcidid: 0000-0002-6428-5653 surname: Calafiore fullname: Calafiore, Giuseppe C. email: giuseppe.calafiore@polito.it organization: Dipartimento di Elettronica e Telecomunicazioni, Politecnico di Torino, Turin, Italy – sequence: 2 givenname: Stephane surname: Gaubert fullname: Gaubert, Stephane email: stephane.gaubert@inria.fr organization: Inria, Palaiseau, France – sequence: 3 givenname: Corrado orcidid: 0000-0003-2528-3935 surname: Possieri fullname: Possieri, Corrado email: corrado.possieri@iasi.cnr.it organization: Consiglio Nazionale delle Ricerche, Istituto di Analisi dei Sistemi ed Informatica A. Ruberti, Rome, Italy
BackLink	https://www.ncbi.nlm.nih.gov/pubmed/32167912$$D View this record in MEDLINE/PubMed https://inria.hal.science/hal-02423871$$DView record in HAL
BookMark	eNp9kU9v1DAQxS1UREvpFwAJReIChyz2OP98XLWFIkWpRFuJmzVxJuCSjRc7acu3x9td9tBDLVm27N_zeN57zQ5GNxJjbwVfCMHV5-umqa8WwIEvQJU5z8ULdgSigBRkVR3s9-WPQ3YSwi2Po-B5kalX7FDGy1IJOGLNMrkZ7R35gEOyXK-9e7ArnKwbk-8U5mFKeueTM9v35Gk0lLg-qd3P9GpepecP66Sh2UdlQ9O987_DG_ayxyHQyW49Zjdfzq9PL9L68uu302WdGlnlU0pEXVegQFMYIGwxzkqhQdm3mJPgXYd5iUgVIhRt3wIYGbvmqBBk3stj9mn77i8c9NrHL_u_2qHVF8tab844ZNGHUtyJyH7csrG5PzOFSa9sMDQMOJKbg44elVJCpjbohyforZv9GDvRkBUlqKoAFan3O2puV9Tt6_93NQKwBYx3IXjq94jgepOefkxPb9LTu_SiqHoiMnZ6TGLyaIfnpe-2UhuN3ddSXILMuPwHB1amvw
CODEN	ITNNAL
CitedBy_id	crossref_primary_10_1109_LCSYS_2020_3032083 crossref_primary_10_1109_TAP_2021_3111299 crossref_primary_10_1109_TNNLS_2025_3570807 crossref_primary_10_1007_s10898_023_01272_1 crossref_primary_10_1109_TFUZZ_2022_3160614 crossref_primary_10_1109_TCSVT_2023_3247944 crossref_primary_10_1016_j_neunet_2023_11_014 crossref_primary_10_1016_j_cma_2023_116333 crossref_primary_10_3934_mfc_2024046 crossref_primary_10_1109_TNNLS_2024_3378697 crossref_primary_10_1016_j_neucom_2022_09_108 crossref_primary_10_3233_JIFS_211417 crossref_primary_10_1134_S0361768823100080 crossref_primary_10_1007_s00020_024_02769_4 crossref_primary_10_1007_s11227_021_04038_2 crossref_primary_10_1109_JPROC_2021_3065238 crossref_primary_10_1109_LCOMM_2021_3098750 crossref_primary_10_1109_TETCI_2024_3502463 crossref_primary_10_1109_TNNLS_2021_3105732 crossref_primary_10_1109_TNNLS_2022_3190198 crossref_primary_10_1002_spe_3408 crossref_primary_10_3390_ma16093430 crossref_primary_10_1109_TIA_2023_3296065
Cites_doi	10.1137/0111030 10.1007/s10479-004-5022-1 10.1109/22.390193 10.1007/s10107-018-1235-y 10.1007/978-3-642-54455-2_1 10.1109/87.221350 10.1090/S0002-9939-1980-0553381-X 10.1142/S0129065789000414 10.1007/s10208-014-9231-y 10.1016/j.jfa.2011.09.003 10.1017/CBO9780511804441 10.1016/0893-6080(89)90020-8 10.1007/978-3-319-57240-6_1 10.1109/TNNLS.2019.2910417 10.1504/IJMMNO.2013.055204 10.1007/978-3-030-20867-7_24 10.1007/s11634-008-0030-7 10.1080/02331931003770411 10.1109/TCSI.2004.834521 10.1109/TBME.2007.893506 10.1007/978-3-0348-8268-2_8 10.1007/BF02551274 10.1137/0801001
ContentType	Journal Article
Copyright	Copyright The Institute of Electrical and Electronics Engineers, Inc. (IEEE) 2020 Attribution
Copyright_xml	– notice: Copyright The Institute of Electrical and Electronics Engineers, Inc. (IEEE) 2020 – notice: Attribution
DBID	97E RIA RIE AAYXX CITATION NPM 7QF 7QO 7QP 7QQ 7QR 7SC 7SE 7SP 7SR 7TA 7TB 7TK 7U5 8BQ 8FD F28 FR3 H8D JG9 JQ2 KR7 L7M L~C L~D P64 7X8 1XC
DOI	10.1109/TNNLS.2020.2975051
DatabaseName	IEEE Xplore (IEEE) IEEE All-Society Periodicals Package (ASPP) 1998–Present IEEE/IET Electronic Library CrossRef PubMed Aluminium Industry Abstracts Biotechnology Research Abstracts Calcium & Calcified Tissue Abstracts Ceramic Abstracts Chemoreception Abstracts Computer and Information Systems Abstracts Corrosion Abstracts Electronics & Communications Abstracts Engineered Materials Abstracts Materials Business File Mechanical & Transportation Engineering Abstracts Neurosciences Abstracts Solid State and Superconductivity Abstracts METADEX Technology Research Database ANTE: Abstracts in New Technology & Engineering Engineering Research Database Aerospace Database Materials Research Database ProQuest Computer Science Collection Civil Engineering Abstracts Advanced Technologies Database with Aerospace Computer and Information Systems Abstracts Academic Computer and Information Systems Abstracts Professional Biotechnology and BioEngineering Abstracts MEDLINE - Academic Hyper Article en Ligne (HAL)
DatabaseTitle	CrossRef PubMed Materials Research Database Technology Research Database Computer and Information Systems Abstracts – Academic Mechanical & Transportation Engineering Abstracts ProQuest Computer Science Collection Computer and Information Systems Abstracts Materials Business File Aerospace Database Engineered Materials Abstracts Biotechnology Research Abstracts Chemoreception Abstracts Advanced Technologies Database with Aerospace ANTE: Abstracts in New Technology & Engineering Civil Engineering Abstracts Aluminium Industry Abstracts Electronics & Communications Abstracts Ceramic Abstracts Neurosciences Abstracts METADEX Biotechnology and BioEngineering Abstracts Computer and Information Systems Abstracts Professional Solid State and Superconductivity Abstracts Engineering Research Database Calcium & Calcified Tissue Abstracts Corrosion Abstracts MEDLINE - Academic
DatabaseTitleList	MEDLINE - Academic Materials Research Database PubMed
Database_xml	– sequence: 1 dbid: NPM name: PubMed url: http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?db=PubMed sourceTypes: Index Database – sequence: 2 dbid: RIE name: IEEE Electronic Library (IEL) url: https://ieeexplore.ieee.org/ sourceTypes: Publisher – sequence: 3 dbid: 7X8 name: MEDLINE - Academic url: https://search.proquest.com/medline sourceTypes: Aggregation Database
DeliveryMethod	fulltext_linktorsrc
Discipline	Computer Science Mathematics
EISSN	2162-2388
EndPage	5612
ExternalDocumentID	oai:HAL:hal-02423871v1 32167912 10_1109_TNNLS_2020_2975051 9032340
Genre	orig-research Research Support, Non-U.S. Gov't Journal Article
GrantInformation_xml	– fundername: PGMO program of EDF and Fondation Mathématique Jacques Hadamard funderid: 10.13039/501100007493 – fundername: LabEx LMH through the “Investissement d’avenir” grantid: ANR-11-LABX-0056-LMH funderid: 10.13039/501100004100
GroupedDBID	0R~ 4.4 5VS 6IK 97E AAJGR AARMG AASAJ AAWTH ABAZT ABQJQ ABVLG ACIWK ACPRK AENEX AFRAH AGQYO AGSQL AHBIQ AKJIK AKQYR ALMA_UNASSIGNED_HOLDINGS ATWAV BEFXN BFFAM BGNUA BKEBE BPEOZ EBS EJD IFIPE IPLJI JAVBF M43 MS~ O9- OCL PQQKQ RIA RIE RNS AAYXX CITATION NPM RIG 7QF 7QO 7QP 7QQ 7QR 7SC 7SE 7SP 7SR 7TA 7TB 7TK 7U5 8BQ 8FD F28 FR3 H8D JG9 JQ2 KR7 L7M L~C L~D P64 7X8 1XC
ID	FETCH-LOGICAL-c385t-eeedd6a1ac6c2eabaeab89aca3fba5e10dda57aae8aa26bfb22c31090a9a235f3
IEDL.DBID	RIE
ISICitedReferencesCount	34
ISICitedReferencesURI	http://www.webofscience.com/api/gateway?GWVersion=2&SrcApp=Summon&SrcAuth=ProQuest&DestLinkType=CitingArticles&DestApp=WOS_CPL&KeyUT=000595533300044&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D
ISSN	2162-237X 2162-2388
IngestDate	Tue Oct 14 21:00:21 EDT 2025 Thu Oct 02 11:37:01 EDT 2025 Mon Jun 30 03:09:32 EDT 2025 Thu Jan 02 22:57:52 EST 2025 Sat Nov 29 01:40:05 EST 2025 Tue Nov 18 22:33:42 EST 2025 Wed Aug 27 02:33:58 EDT 2025
IsPeerReviewed	false
IsScholarly	true
Issue	12
Keywords	LSE networks DC programming Subtraction-freeexpressions Surrogate models Feedforward neural networks Data-driven optimization Universal ap-proximation
Language	English
License	https://ieeexplore.ieee.org/Xplorehelp/downloads/license-information/IEEE.html https://doi.org/10.15223/policy-029 https://doi.org/10.15223/policy-037 Attribution: http://creativecommons.org/licenses/by
LinkModel	DirectLink
MergedId	FETCHMERGED-LOGICAL-c385t-eeedd6a1ac6c2eabaeab89aca3fba5e10dda57aae8aa26bfb22c31090a9a235f3
Notes	ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 content type line 23
ORCID	0000-0003-2528-3935 0000-0002-6428-5653
PMID	32167912
PQID	2467298629
PQPubID	85436
PageCount	10
ParticipantIDs	pubmed_primary_32167912 proquest_miscellaneous_2377332491 proquest_journals_2467298629 crossref_primary_10_1109_TNNLS_2020_2975051 crossref_citationtrail_10_1109_TNNLS_2020_2975051 hal_primary_oai_HAL_hal_02423871v1 ieee_primary_9032340
PublicationCentury	2000
PublicationDate	2020-12-01
PublicationDateYYYYMMDD	2020-12-01
PublicationDate_xml	– month: 12 year: 2020 text: 2020-12-01 day: 01
PublicationDecade	2020
PublicationPlace	United States
PublicationPlace_xml	– name: United States – name: Piscataway
PublicationTitle	IEEE transaction on neural networks and learning systems
PublicationTitleAbbrev	TNNLS
PublicationTitleAlternate	IEEE Trans Neural Netw Learn Syst
PublicationYear	2020
Publisher	IEEE The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Publisher_xml	– name: IEEE – name: The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
References	zhang (ref12) 2018; 80 ref14 ref30 ref11 ref2 ref1 ref17 ref16 ref18 ref24 ref26 ref25 ref20 ref22 ref21 ref27 ref29 ref8 ref7 ovchinnikov (ref13) 2002; 43 akian (ref19) 2018 ref9 ref4 ref3 itenberg (ref10) 2007 ref6 ref5 goodfellow (ref15) 2013; 28 bertsekas (ref23) 1999 ripsin (ref28) 2009; 79
References_xml	– ident: ref21 doi: 10.1137/0111030 – year: 1999 ident: ref23 publication-title: Nonlinear Programming – ident: ref24 doi: 10.1007/s10479-004-5022-1 – volume: 80 start-page: 5824 year: 2018 ident: ref12 article-title: Tropical geometry of deep neural networks publication-title: Proc 35th Int Conf Mach Learn – ident: ref3 doi: 10.1109/22.390193 – ident: ref27 doi: 10.1007/s10107-018-1235-y – volume: 79 start-page: 29 year: 2009 ident: ref28 article-title: Management of blood glucose in type 2 diabetes mellitus publication-title: Amer Family Phys – ident: ref26 doi: 10.1007/978-3-642-54455-2_1 – ident: ref4 doi: 10.1109/87.221350 – ident: ref18 doi: 10.1090/S0002-9939-1980-0553381-X – ident: ref5 doi: 10.1142/S0129065789000414 – ident: ref8 doi: 10.1007/s10208-014-9231-y – volume: 28 start-page: iii-1319 year: 2013 ident: ref15 article-title: Maxout networks publication-title: Proc 30th Int Conf Int Conf Mach Learn (ICML) – ident: ref17 doi: 10.1016/j.jfa.2011.09.003 – ident: ref6 doi: 10.1017/CBO9780511804441 – ident: ref2 doi: 10.1016/0893-6080(89)90020-8 – volume: 43 start-page: 297 year: 2002 ident: ref13 article-title: Max-min representations of piecewise linear functions publication-title: Contrib Algebra Geometry – start-page: 225 year: 2018 ident: ref19 article-title: Minimax representation of nonexpansive functions and application to zero-sum recursive games publication-title: J Convex Anal – ident: ref11 doi: 10.1007/978-3-319-57240-6_1 – ident: ref7 doi: 10.1109/TNNLS.2019.2910417 – ident: ref30 doi: 10.1504/IJMMNO.2013.055204 – ident: ref16 doi: 10.1007/978-3-030-20867-7_24 – year: 2007 ident: ref10 publication-title: Tropical algebraic geometry ser Oberwolfach seminars – ident: ref25 doi: 10.1007/s11634-008-0030-7 – ident: ref20 doi: 10.1080/02331931003770411 – ident: ref14 doi: 10.1109/TCSI.2004.834521 – ident: ref29 doi: 10.1109/TBME.2007.893506 – ident: ref9 doi: 10.1007/978-3-0348-8268-2_8 – ident: ref1 doi: 10.1007/BF02551274 – ident: ref22 doi: 10.1137/0801001
SSID	ssj0000605649
Score	2.5448906
Snippet	We show that a neural network whose output is obtained as the difference of the outputs of two feedforward networks with exponential activation function in the...
SourceID	hal proquest pubmed crossref ieee
SourceType	Open Access Repository Aggregation Database Index Database Enrichment Source Publisher
StartPage	5603
SubjectTerms	Algorithms Artificial neural networks Computer Science Continuity (mathematics) Convex functions Data-driven optimization Design Design optimization Diabetes mellitus difference of convex (DC) programming Feedforward neural networks feedforward neural networks (FFNs) log-sum-exp (LSE) networks Machine Learning Mathematical analysis Mathematical models Mathematics Neural networks Numerical methods Optimization Optimization and Control Subtraction subtraction-free expressions surrogate models Transforms universal approximation
Title	A Universal Approximation Result for Difference of Log-Sum-Exp Neural Networks
URI	https://ieeexplore.ieee.org/document/9032340 https://www.ncbi.nlm.nih.gov/pubmed/32167912 https://www.proquest.com/docview/2467298629 https://www.proquest.com/docview/2377332491 https://inria.hal.science/hal-02423871
Volume	31
WOSCitedRecordID	wos000595533300044&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D
hasFullText	1
inHoldings	1
isFullTextHit
isPrint
journalDatabaseRights	– providerCode: PRVIEE databaseName: IEEE Electronic Library (IEL) customDbUrl: eissn: 2162-2388 dateEnd: 99991231 omitProxy: false ssIdentifier: ssj0000605649 issn: 2162-237X databaseCode: RIE dateStart: 20120101 isFulltext: true titleUrlDefault: https://ieeexplore.ieee.org/ providerName: IEEE
link	http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwlV1La9wwEB6SUEouTdv0sW0S1NJbq0SW1pZ1XPIgh8WUJi17M3o5DaTrkF2H_PyOZNlQaAs5GIw9kgZ_I2tGmgfAJ-ly56zRdJobQaeSa1oqXdJGSq1tIayM3oQ_5rKqysVCfd2AL2MsjPc-Op_5w3Abz_Jda7uwVXakmOBiigb6ppSyj9Ua91MY6uVF1HZ5VnDKhVwMMTJMHV1W1fwCrUHODkMoKUriNjwVPJxBZPyPJWnzZ3CIjJVW_q10xsXnbOdxbD-HZ0nJJLNeKl7Ahl--hJ2hgANJ83kXqhlJjhmBOmQXf7juQxnJN7_qbtYEVVpykmqoWE_ahszbK3rR_aKnD7ckZPbAllXvSr56Bd_PTi-Pz2kqsECtKPM1RU6dK3SGoFjutdF4IVJWi8bo3GfMOZ0jZr7UmhemMZzbkEmUaaW5yBvxGraW7dK_BZJ7YabM2iyk9HNOmcw4hsS8dPgrVnYC2fCNa5uyj4ciGDd1tEKYqiNEdYCoThBN4PPY5rbPvfFf6o8I3UgY0mafz-Z1eBb0EIGW4T0S7QZ8RqoEzQT2BqTrNIdXNUfGuUKLT03gw_gaZ184UtFL33ZII6QUqJMq7PlNLyFj34N4vfv7mO9hO7Dfu8bswdb6rvP78MTer69Xdwco4ovyIIr4b9lJ9Pg
linkProvider	IEEE
linkToHtml	http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwlV1Lb9QwEB6VgqAXCpTHlgIGcaNuEzuJ4-MKWi0iRIguaG-WYztQqWyq7qbqz-_YeUiVChKHSFEydib5xvGMPQ-A98Km1ppK0yStOE0E0zSXOqe1EFqbjBsRvAl_FqIs88VCftuA_TEWxjkXnM_cgT8Ne_m2Ma1fKjuUEWc8QQP9bpokLO6itcYVlQg18yzouyzOGGVcLIYomUgezsuyOEF7kEUHPpgUZXEL7nPmdyFidmNSuvPbu0SGWit_VzvD9HO8_X-MP4KHvZpJpp1cPIYNt3wC20MJB9KP6B0op6R3zfDUPr_41WkXzEi-u1V7tiao1JJPfRUV40hTk6L5RU_aP_To6pz43B7YsuycyVdP4cfx0fzjjPYlFqjhebqmyKm1mY4RFsOcrjQeiJXRvK506uLIWp0iai7XmmVVXTFmfC7RSEvNeFrzZ7C5bJbuBZDU8SqJjIl9Uj9rZRVXNkJillv8GUszgXj4xsr0-cd9GYwzFeyQSKoAkfIQqR6iCXwY25x32Tf-Sf0OoRsJfeLs2bRQ_prXRDjahpdItOPxGal6aCawNyCt-lG8UgwZZxJtPjmBt-NtHH9-U0UvXdMiDReCo1YqsefnnYSMfQ_itXv7M9_Ag9n8a6GKz-WXl7DlX6VzlNmDzfVF617BPXO5Pl1dvA6Cfg311PdX
openUrl	ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=A+Universal+Approximation+Result+for+Difference+of+Log-Sum-Exp+Neural+Networks&rft.jtitle=IEEE+transaction+on+neural+networks+and+learning+systems&rft.au=Calafiore%2C+Giuseppe+C&rft.au=Gaubert%2C+Stephane&rft.au=Possieri%2C+Corrado&rft.date=2020-12-01&rft.eissn=2162-2388&rft.volume=31&rft.issue=12&rft.spage=5603&rft_id=info:doi/10.1109%2FTNNLS.2020.2975051&rft_id=info%3Apmid%2F32167912&rft.externalDocID=32167912
thumbnail_l	http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=2162-237X&client=summon
thumbnail_m	http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=2162-237X&client=summon
thumbnail_s	http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=2162-237X&client=summon