The universal approximation theorem for complex-valued neural networks
We generalize the classical universal approximation theorem for neural networks to the case of complex-valued neural networks. Precisely, we consider feedforward networks with a complex activation function σ:C→C in which each neuron performs the operation CN→C,z↦σ(b+wTz) with weights w∈CN and a bias...
Uloženo v:
| Vydáno v: | Applied and computational harmonic analysis Ročník 64; s. 33 - 61 |
|---|---|
| Hlavní autor: | |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
Elsevier Inc
01.05.2023
|
| Témata: | |
| ISSN: | 1063-5203, 1096-603X |
| On-line přístup: | Získat plný text |
| Tagy: |
Přidat tag
Žádné tagy, Buďte první, kdo vytvoří štítek k tomuto záznamu!
|
| Abstract | We generalize the classical universal approximation theorem for neural networks to the case of complex-valued neural networks. Precisely, we consider feedforward networks with a complex activation function σ:C→C in which each neuron performs the operation CN→C,z↦σ(b+wTz) with weights w∈CN and a bias b∈C. We completely characterize those activation functions σ for which the associated complex networks have the universal approximation property, meaning that they can uniformly approximate any continuous function on any compact subset of Cd arbitrarily well. Unlike the classical case of real networks, the set of “good activation functions”—which give rise to networks with the universal approximation property—differs significantly depending on whether one considers deep networks or shallow networks: For deep networks with at least two hidden layers, the universal approximation property holds as long as σ is neither a polynomial, a holomorphic function, nor an antiholomorphic function. Shallow networks, on the other hand, are universal if and only if the real part or the imaginary part of σ is not a polyharmonic function. |
|---|---|
| AbstractList | We generalize the classical universal approximation theorem for neural networks to the case of complex-valued neural networks. Precisely, we consider feedforward networks with a complex activation function σ:C→C in which each neuron performs the operation CN→C,z↦σ(b+wTz) with weights w∈CN and a bias b∈C. We completely characterize those activation functions σ for which the associated complex networks have the universal approximation property, meaning that they can uniformly approximate any continuous function on any compact subset of Cd arbitrarily well. Unlike the classical case of real networks, the set of “good activation functions”—which give rise to networks with the universal approximation property—differs significantly depending on whether one considers deep networks or shallow networks: For deep networks with at least two hidden layers, the universal approximation property holds as long as σ is neither a polynomial, a holomorphic function, nor an antiholomorphic function. Shallow networks, on the other hand, are universal if and only if the real part or the imaginary part of σ is not a polyharmonic function. |
| Author | Voigtlaender, Felix |
| Author_xml | – sequence: 1 givenname: Felix orcidid: 0000-0002-5061-2756 surname: Voigtlaender fullname: Voigtlaender, Felix email: felix.voigtlaender@ku.de organization: Department of Mathematics, Technical University of Munich, 85748 Garching bei München, Germany |
| BookMark | eNp9kLFOwzAQhi1UJNrCCzDlBRLOduK6EguqKEWqxFIkNsuxL6pLGkd2Wsrbk1Amhk53w32_7v8mZNT4Bgm5p5BRoOJhl2mz1RkDxjLKMgB2RcYU5iIVwD9Gwy54WjDgN2QS4w6A0ryYj8lys8Xk0LgjhqjrRLdt8Ce3153zTdJt0QfcJ5UPifH7tsZTetT1AW3S4CH09w12Xz58xltyXek64t3fnJL35fNmsUrXby-vi6d1ajhAl3JLNSsorzSYHDWrqCiFlVaXojCsxELkdm5lPsNyxsqZKNFUrMylrHKQklM-JfKca4KPMWCljOt-n-2CdrWioAYfaqcGH2rwoShTvY8eZf_QNvRFw_dl6PEMYV_q6DCoaBw2Bq0LaDplvbuE_wDZbn1Y |
| CitedBy_id | crossref_primary_10_1088_2515_7620_ad810f crossref_primary_10_1088_1361_6420_ace9d4 crossref_primary_10_1002_asjc_3631 crossref_primary_10_3390_math12132097 crossref_primary_10_1007_s00365_025_09713_8 crossref_primary_10_1038_s41467_024_45982_w crossref_primary_10_3390_s24237516 crossref_primary_10_3390_pr11051460 crossref_primary_10_1109_LWC_2023_3309479 crossref_primary_10_3390_math12233704 crossref_primary_10_1016_j_engappai_2024_108352 crossref_primary_10_1186_s43593_025_00085_x crossref_primary_10_1108_GS_06_2024_0070 crossref_primary_10_1109_ACCESS_2024_3413785 crossref_primary_10_1063_5_0254013 crossref_primary_10_1007_s44439_025_00002_7 crossref_primary_10_1016_j_neunet_2024_106632 crossref_primary_10_1140_epjs_s11734_024_01306_z crossref_primary_10_1109_TAP_2024_3476915 crossref_primary_10_3390_app14135618 crossref_primary_10_1038_s41598_025_85440_1 crossref_primary_10_1016_j_trd_2024_104533 crossref_primary_10_1016_j_softx_2024_102017 crossref_primary_10_1109_JLT_2024_3466977 crossref_primary_10_3390_a17010014 crossref_primary_10_1016_j_neunet_2024_106922 crossref_primary_10_1038_s42005_025_02005_4 crossref_primary_10_1109_MSP_2024_3401621 crossref_primary_10_1121_10_0036384 crossref_primary_10_1093_gji_ggaf348 crossref_primary_10_61383_ejam_20253188 crossref_primary_10_1016_j_cma_2024_117406 |
| Cites_doi | 10.1016/j.neunet.2020.01.018 10.1016/0893-6080(91)90009-T 10.1090/proc/14789 10.1137/20M134695X 10.1007/BF02551274 10.1016/j.acha.2019.06.004 10.1162/089976603321891846 10.1016/0893-6080(89)90020-8 10.1142/S0129065795000299 10.1016/j.neunet.2018.08.019 10.1162/NECO_a_00824 10.1007/s00365-021-09543-4 10.1007/978-1-4471-7280-2 10.1137/0120005 10.1016/j.neunet.2017.07.002 10.1016/S0893-6080(05)80131-5 10.1038/nature14539 10.1162/neco.1996.8.1.164 10.1109/TNN.2006.875977 10.1145/3065386 10.1007/s00365-021-09546-1 |
| ContentType | Journal Article |
| Copyright | 2022 Elsevier Inc. |
| Copyright_xml | – notice: 2022 Elsevier Inc. |
| DBID | AAYXX CITATION |
| DOI | 10.1016/j.acha.2022.12.002 |
| DatabaseName | CrossRef |
| DatabaseTitle | CrossRef |
| DatabaseTitleList | |
| DeliveryMethod | fulltext_linktorsrc |
| Discipline | Engineering Mathematics |
| EISSN | 1096-603X |
| EndPage | 61 |
| ExternalDocumentID | 10_1016_j_acha_2022_12_002 S1063520322001014 |
| GroupedDBID | --K --M .~1 0R~ 1B1 1RT 1~. 1~5 23M 4.4 457 4G. 5GY 5VS 6I. 7-5 71M 8P~ 9JN AACTN AAEDT AAEDW AAFTH AAIAV AAIKJ AAKOC AALRI AAOAW AAQFI AAQXK AASFE AAXUO ABAOU ABFNM ABJNI ABMAC ABVKL ABXDB ABYKQ ACAZW ACDAQ ACGFS ACRLP ADBBV ADEZE ADFGL ADMUD AEBSH AEKER AENEX AEXQZ AFKWA AFTJW AGHFR AGUBO AGYEJ AHHHB AIEXJ AIGVJ AIKHN AITUG AJBFU AJOXV ALMA_UNASSIGNED_HOLDINGS AMFUW AMRAJ ARUGR ASPBG AVWKF AXJTR AZFZN BKOJK BLXMC CAG COF CS3 DM4 EBS EFBJH EFLBG EJD EO8 EO9 EP2 EP3 F5P FDB FEDTE FGOYB FIRID FNPLU FYGXN G-2 G-Q GBLVA HVGLF HZ~ IHE IXB J1W KOM LG5 M26 M41 MCRUF MHUIS MO0 N9A NCXOZ O-L O9- OAUVE OK1 OZT P-8 P-9 P2P PC. Q38 R2- RIG ROL RPZ SDF SDG SDP SES SEW SPC SPCBC SSW SSZ T5K WUQ XPP ZMT ~G- 9DU AATTM AAXKI AAYWO AAYXX ABWVN ACLOT ACRPL ACVFH ADCNI ADNMO ADVLN AEIPS AEUPX AFJKZ AFPUW AGQPQ AIGII AIIUN AKBMS AKRWK AKYEP ANKPU APXCP CITATION EFKBS ~HD |
| ID | FETCH-LOGICAL-c300t-3d1a2513fa0c4ea2f16b6d8dab65c2be564d9d847eb72b76becf2b488f4088313 |
| ISICitedReferencesCount | 44 |
| ISICitedReferencesURI | http://www.webofscience.com/api/gateway?GWVersion=2&SrcApp=Summon&SrcAuth=ProQuest&DestLinkType=CitingArticles&DestApp=WOS_CPL&KeyUT=000918021300001&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D |
| ISSN | 1063-5203 |
| IngestDate | Sat Nov 29 07:05:09 EST 2025 Tue Nov 18 21:51:21 EST 2025 Fri Feb 23 02:38:37 EST 2024 |
| IsPeerReviewed | true |
| IsScholarly | true |
| Keywords | 30E10 Deep neural networks 41A30 41A63 31A30 Complex-valued neural networks Universal approximation theorem 68T07 Holomorphic functions Polyharmonic functions |
| Language | English |
| LinkModel | OpenURL |
| MergedId | FETCHMERGED-LOGICAL-c300t-3d1a2513fa0c4ea2f16b6d8dab65c2be564d9d847eb72b76becf2b488f4088313 |
| ORCID | 0000-0002-5061-2756 |
| PageCount | 29 |
| ParticipantIDs | crossref_citationtrail_10_1016_j_acha_2022_12_002 crossref_primary_10_1016_j_acha_2022_12_002 elsevier_sciencedirect_doi_10_1016_j_acha_2022_12_002 |
| PublicationCentury | 2000 |
| PublicationDate | May 2023 2023-05-00 |
| PublicationDateYYYYMMDD | 2023-05-01 |
| PublicationDate_xml | – month: 05 year: 2023 text: May 2023 |
| PublicationDecade | 2020 |
| PublicationTitle | Applied and computational harmonic analysis |
| PublicationYear | 2023 |
| Publisher | Elsevier Inc |
| Publisher_xml | – name: Elsevier Inc |
| References | Wolter, Yao (br0380) 2018; vol. 31 Krizhevsky, Sutskever, Hinton (br0200) 2017; 60 Evans (br0070) 2010; vol. 19 Zhou (br0420) 2020; 124 Hirose (br0110) 2003; vol. 5 Huang, Chen, Siew (br0140) 2006; 17 Stein, Shakarchi (br0320) 2003; vol. 2 Yarotsky, Zhevnerchuk (br0410) 2020; 33 Kidger, Lyons (br0180) 2020; vol. 125 Petersen, Voigtlaender (br0280) 2018; 108 Yarotsky (br0400) 2022; 55 Zhou (br0430) 2020; 48 Petersen, Voigtlaender (br0290) 2020; 148 Lu, Shen, Yang, Zhang (br0240) 2021; 53 Mhaskar (br0260) 1996; 8 Hornik (br0120) 1991; 4 Huang, Li, Chen, Siew (br0150) 2008; 71 Arena, Fortuna, Re, Xibilia (br0030) 1993 Kaup, Kaup (br0170) 1983; vol. 3 Stroock (br0330) 2008; vol. 354 Sutskever, Vinyals, Le (br0340) 2014 Hornik, Stinchcombe, White (br0130) 1989; 2 Rudin (br0300) 1976 Alt (br0010) 2016 LeCun, Bengio, Hinton (br0210) 2015; 521 Arena, Fortuna, Muscato, Xibilia (br0020) 1998; vol. 234 Rudin (br0310) 1991 Lu, Pu, Wang, Hu, Wang (br0250) 2017; 30 Yarotsky (br0390) 2017; 94 Glorot, Bordes, Bengio (br0090) 2011 Tygert, Bruna, Chintala, LeCun, Piantino, Szlam (br0360) 2016; 28 Balk (br0050) 1991; vol. 63 Cybenko (br0060) 1989; 2 Folland (br0080) 1999 Munkres (br0270) 2000 Virtue, Yu, Lustig (br0370) 2017 Huilgol (br0160) 1971; 20 Kim, Adalı (br0190) 2003; 15 Lin, Jegelka (br0230) 2018; 31 Leshno, Lin, Pinkus, Schocken (br0220) 1993; 6 Gribonval, Kutyniok, Nielsen, Voigtlaender (br0100) 2022; 55 Arena, Fortuna, Re, Xibilia (br0040) 1995; 6 Trabelsi, Bilaniuk, Zhang, Serdyuk, Subramanian, Santos, Mehri, Rostamzadeh, Bengio, Pal (br0350) 2018 Wolter (10.1016/j.acha.2022.12.002_br0380) 2018; vol. 31 Alt (10.1016/j.acha.2022.12.002_br0010) 2016 Hirose (10.1016/j.acha.2022.12.002_br0110) 2003; vol. 5 Yarotsky (10.1016/j.acha.2022.12.002_br0390) 2017; 94 Petersen (10.1016/j.acha.2022.12.002_br0290) 2020; 148 LeCun (10.1016/j.acha.2022.12.002_br0210) 2015; 521 Kidger (10.1016/j.acha.2022.12.002_br0180) 2020; vol. 125 Rudin (10.1016/j.acha.2022.12.002_br0310) 1991 Lin (10.1016/j.acha.2022.12.002_br0230) 2018; 31 Mhaskar (10.1016/j.acha.2022.12.002_br0260) 1996; 8 Petersen (10.1016/j.acha.2022.12.002_br0280) 2018; 108 Hornik (10.1016/j.acha.2022.12.002_br0120) 1991; 4 Zhou (10.1016/j.acha.2022.12.002_br0420) 2020; 124 Arena (10.1016/j.acha.2022.12.002_br0020) 1998; vol. 234 Arena (10.1016/j.acha.2022.12.002_br0030) 1993 Gribonval (10.1016/j.acha.2022.12.002_br0100) 2022; 55 Krizhevsky (10.1016/j.acha.2022.12.002_br0200) 2017; 60 Arena (10.1016/j.acha.2022.12.002_br0040) 1995; 6 Stein (10.1016/j.acha.2022.12.002_br0320) 2003; vol. 2 Glorot (10.1016/j.acha.2022.12.002_br0090) 2011 Kim (10.1016/j.acha.2022.12.002_br0190) 2003; 15 Sutskever (10.1016/j.acha.2022.12.002_br0340) 2014 Trabelsi (10.1016/j.acha.2022.12.002_br0350) 2018 Yarotsky (10.1016/j.acha.2022.12.002_br0400) 2022; 55 Lu (10.1016/j.acha.2022.12.002_br0250) 2017; 30 Huang (10.1016/j.acha.2022.12.002_br0140) 2006; 17 Balk (10.1016/j.acha.2022.12.002_br0050) 1991; vol. 63 Munkres (10.1016/j.acha.2022.12.002_br0270) 2000 Huang (10.1016/j.acha.2022.12.002_br0150) 2008; 71 Tygert (10.1016/j.acha.2022.12.002_br0360) 2016; 28 Evans (10.1016/j.acha.2022.12.002_br0070) 2010; vol. 19 Rudin (10.1016/j.acha.2022.12.002_br0300) 1976 Leshno (10.1016/j.acha.2022.12.002_br0220) 1993; 6 Stroock (10.1016/j.acha.2022.12.002_br0330) 2008; vol. 354 Folland (10.1016/j.acha.2022.12.002_br0080) 1999 Huilgol (10.1016/j.acha.2022.12.002_br0160) 1971; 20 Lu (10.1016/j.acha.2022.12.002_br0240) 2021; 53 Zhou (10.1016/j.acha.2022.12.002_br0430) 2020; 48 Kaup (10.1016/j.acha.2022.12.002_br0170) 1983; vol. 3 Yarotsky (10.1016/j.acha.2022.12.002_br0410) 2020; 33 Cybenko (10.1016/j.acha.2022.12.002_br0060) 1989; 2 Virtue (10.1016/j.acha.2022.12.002_br0370) 2017 Hornik (10.1016/j.acha.2022.12.002_br0130) 1989; 2 |
| References_xml | – year: 2016 ident: br0010 article-title: Linear Functional Analysis publication-title: Universitext – year: 1991 ident: br0310 article-title: Functional Analysis publication-title: International Series in Pure and Applied Mathematics – volume: 20 start-page: 37 year: 1971 end-page: 39 ident: br0160 article-title: On Liouville's theorem for biharmonic functions publication-title: SIAM J. Appl. Math. – volume: vol. 234 year: 1998 ident: br0020 article-title: Neural Networks in Multidimensional Domains: Fundamentals and New Trends in Modelling and Control – volume: vol. 5 year: 2003 ident: br0110 article-title: Complex-Valued Neural Networks: Theories and Applications – year: 1999 ident: br0080 article-title: Real Analysis publication-title: Pure and Applied Mathematics (New York) – volume: 2 start-page: 303 year: 1989 end-page: 314 ident: br0060 article-title: Approximation by superpositions of a sigmoidal function publication-title: Math. Control Signals Syst. – volume: 33 year: 2020 ident: br0410 article-title: The phase diagram of approximation rates for deep neural networks publication-title: Adv. Neural Inf. Process. Syst. – volume: 55 start-page: 407 year: 2022 end-page: 474 ident: br0400 article-title: Universal approximations of invariant maps by neural networks publication-title: Constr. Approx. – volume: vol. 3 year: 1983 ident: br0170 article-title: Holomorphic Functions of Several Variables publication-title: De Gruyter Studies in Mathematics – volume: 2 start-page: 359 year: 1989 end-page: 366 ident: br0130 article-title: Multilayer feedforward networks are universal approximators publication-title: Neural Netw. – volume: vol. 354 start-page: 164 year: 2008 end-page: 173 ident: br0330 article-title: Weyl's lemma, one of many publication-title: Groups and Analysis – volume: 6 year: 1995 ident: br0040 article-title: Multilayer perceptrons to approximate complex valued functions publication-title: Int. J. Neural Syst. – year: 1993 ident: br0030 article-title: On the capability of neural networks with complex neurons in complex valued functions approximation publication-title: 1993 IEEE International Symposium on Circuits and Systems – volume: 15 start-page: 1641 year: 2003 end-page: 1666 ident: br0190 article-title: Approximation by fully complex multilayer perceptrons publication-title: Neural Comput. – volume: vol. 2 year: 2003 ident: br0320 article-title: Complex Analysis publication-title: Princeton Lectures in Analysis – volume: 8 year: 1996 ident: br0260 article-title: Neural networks for optimal approximation of smooth and analytic functions publication-title: Neural Comput. – volume: 108 year: 2018 ident: br0280 article-title: Optimal approximation of piecewise smooth functions using deep ReLU neural networks publication-title: Neural Netw. – volume: 71 year: 2008 ident: br0150 article-title: Incremental extreme learning machine with fully complex hidden nodes publication-title: Neurocomputing – year: 2000 ident: br0270 article-title: Topology – year: 2017 ident: br0370 article-title: Better than real: complex-valued neural nets for MRI fingerprinting publication-title: 2017 IEEE International Conference on Image Processing (ICIP) – volume: 17 start-page: 879 year: 2006 end-page: 892 ident: br0140 article-title: Universal approximation using incremental constructive feedforward networks with random hidden nodes publication-title: IEEE Trans. Neural Netw. – volume: 55 start-page: 259 year: 2022 end-page: 367 ident: br0100 article-title: Approximation spaces of deep neural networks publication-title: Constr. Approx. – volume: 48 start-page: 787 year: 2020 end-page: 794 ident: br0430 article-title: Universality of deep convolutional neural networks publication-title: Appl. Comput. Harmon. Anal. – volume: 124 start-page: 319 year: 2020 end-page: 327 ident: br0420 article-title: Theory of deep convolutional neural networks: downsampling publication-title: Neural Netw. – volume: 148 year: 2020 ident: br0290 article-title: Equivalence of approximation by convolutional neural networks and fully-connected networks publication-title: Proc. Am. Math. Soc. – volume: 28 year: 2016 ident: br0360 article-title: A mathematical motivation for complex-valued convolutional networks publication-title: Neural Comput. – volume: vol. 19 year: 2010 ident: br0070 article-title: Partial Differential Equations publication-title: Graduate Studies in Mathematics – volume: 30 start-page: 6231 year: 2017 end-page: 6239 ident: br0250 article-title: The expressive power of neural networks: a view from the width publication-title: Adv. Neural Inf. Process. Syst. – year: 2018 ident: br0350 article-title: Deep complex networks publication-title: ICLR – volume: 4 start-page: 251 year: 1991 end-page: 257 ident: br0120 article-title: Approximation capabilities of multilayer feedforward networks publication-title: Neural Netw. – volume: vol. 125 start-page: 2306 year: 2020 end-page: 2327 ident: br0180 article-title: Universal approximation with deep narrow networks publication-title: Proceedings of Thirty Third Conference on Learning Theory – volume: 60 year: 2017 ident: br0200 article-title: ImageNet classification with deep convolutional neural networks publication-title: Commun. ACM – volume: 94 year: 2017 ident: br0390 article-title: Error bounds for approximations with deep ReLU networks publication-title: Neural Netw. – volume: 521 year: 2015 ident: br0210 article-title: Deep learning publication-title: Nature – year: 2014 ident: br0340 article-title: Sequence to sequence learning with neural networks publication-title: Advances in Neural Information Processing Systems – volume: 6 start-page: 861 year: 1993 end-page: 867 ident: br0220 article-title: Multilayer feedforward networks with a nonpolynomial activation function can approximate any function publication-title: Neural Netw. – year: 2011 ident: br0090 article-title: Deep sparse rectifier neural networks publication-title: Proceedings of the Fourteenth International Conference on Artificial Intelligence and Statistics – volume: vol. 31 year: 2018 ident: br0380 article-title: Complex gated recurrent neural networks publication-title: Advances in Neural Information Processing Systems – volume: vol. 63 year: 1991 ident: br0050 article-title: Polyanalytic Functions publication-title: Mathematical Research – volume: 31 start-page: 6169 year: 2018 end-page: 6178 ident: br0230 article-title: ResNet with one-neuron hidden layers is a universal approximator publication-title: Adv. Neural Inf. Process. Syst. – volume: 53 start-page: 5465 year: 2021 end-page: 5506 ident: br0240 article-title: Deep network approximation for smooth functions publication-title: SIAM J. Math. Anal. – year: 1976 ident: br0300 article-title: Principles of Mathematical Analysis publication-title: International Series in Pure and Applied Mathematics – volume: 124 start-page: 319 year: 2020 ident: 10.1016/j.acha.2022.12.002_br0420 article-title: Theory of deep convolutional neural networks: downsampling publication-title: Neural Netw. doi: 10.1016/j.neunet.2020.01.018 – volume: vol. 234 year: 1998 ident: 10.1016/j.acha.2022.12.002_br0020 – volume: vol. 19 year: 2010 ident: 10.1016/j.acha.2022.12.002_br0070 article-title: Partial Differential Equations – volume: 4 start-page: 251 issue: 2 year: 1991 ident: 10.1016/j.acha.2022.12.002_br0120 article-title: Approximation capabilities of multilayer feedforward networks publication-title: Neural Netw. doi: 10.1016/0893-6080(91)90009-T – volume: 148 issue: 4 year: 2020 ident: 10.1016/j.acha.2022.12.002_br0290 article-title: Equivalence of approximation by convolutional neural networks and fully-connected networks publication-title: Proc. Am. Math. Soc. doi: 10.1090/proc/14789 – volume: 30 start-page: 6231 year: 2017 ident: 10.1016/j.acha.2022.12.002_br0250 article-title: The expressive power of neural networks: a view from the width publication-title: Adv. Neural Inf. Process. Syst. – volume: 53 start-page: 5465 issue: 5 year: 2021 ident: 10.1016/j.acha.2022.12.002_br0240 article-title: Deep network approximation for smooth functions publication-title: SIAM J. Math. Anal. doi: 10.1137/20M134695X – year: 1993 ident: 10.1016/j.acha.2022.12.002_br0030 article-title: On the capability of neural networks with complex neurons in complex valued functions approximation – volume: 2 start-page: 303 issue: 4 year: 1989 ident: 10.1016/j.acha.2022.12.002_br0060 article-title: Approximation by superpositions of a sigmoidal function publication-title: Math. Control Signals Syst. doi: 10.1007/BF02551274 – year: 2014 ident: 10.1016/j.acha.2022.12.002_br0340 article-title: Sequence to sequence learning with neural networks – volume: 48 start-page: 787 issue: 2 year: 2020 ident: 10.1016/j.acha.2022.12.002_br0430 article-title: Universality of deep convolutional neural networks publication-title: Appl. Comput. Harmon. Anal. doi: 10.1016/j.acha.2019.06.004 – year: 1976 ident: 10.1016/j.acha.2022.12.002_br0300 article-title: Principles of Mathematical Analysis – volume: 15 start-page: 1641 issue: 7 year: 2003 ident: 10.1016/j.acha.2022.12.002_br0190 article-title: Approximation by fully complex multilayer perceptrons publication-title: Neural Comput. doi: 10.1162/089976603321891846 – year: 1991 ident: 10.1016/j.acha.2022.12.002_br0310 article-title: Functional Analysis – volume: vol. 2 year: 2003 ident: 10.1016/j.acha.2022.12.002_br0320 article-title: Complex Analysis – year: 2000 ident: 10.1016/j.acha.2022.12.002_br0270 – volume: vol. 31 year: 2018 ident: 10.1016/j.acha.2022.12.002_br0380 article-title: Complex gated recurrent neural networks – volume: vol. 5 year: 2003 ident: 10.1016/j.acha.2022.12.002_br0110 – volume: 2 start-page: 359 issue: 5 year: 1989 ident: 10.1016/j.acha.2022.12.002_br0130 article-title: Multilayer feedforward networks are universal approximators publication-title: Neural Netw. doi: 10.1016/0893-6080(89)90020-8 – year: 1999 ident: 10.1016/j.acha.2022.12.002_br0080 article-title: Real Analysis – volume: 71 issue: 4–6 year: 2008 ident: 10.1016/j.acha.2022.12.002_br0150 article-title: Incremental extreme learning machine with fully complex hidden nodes publication-title: Neurocomputing – volume: 6 issue: 04 year: 1995 ident: 10.1016/j.acha.2022.12.002_br0040 article-title: Multilayer perceptrons to approximate complex valued functions publication-title: Int. J. Neural Syst. doi: 10.1142/S0129065795000299 – year: 2017 ident: 10.1016/j.acha.2022.12.002_br0370 article-title: Better than real: complex-valued neural nets for MRI fingerprinting – volume: 108 year: 2018 ident: 10.1016/j.acha.2022.12.002_br0280 article-title: Optimal approximation of piecewise smooth functions using deep ReLU neural networks publication-title: Neural Netw. doi: 10.1016/j.neunet.2018.08.019 – volume: 28 issue: 5 year: 2016 ident: 10.1016/j.acha.2022.12.002_br0360 article-title: A mathematical motivation for complex-valued convolutional networks publication-title: Neural Comput. doi: 10.1162/NECO_a_00824 – volume: 55 start-page: 259 issue: 1 year: 2022 ident: 10.1016/j.acha.2022.12.002_br0100 article-title: Approximation spaces of deep neural networks publication-title: Constr. Approx. doi: 10.1007/s00365-021-09543-4 – year: 2016 ident: 10.1016/j.acha.2022.12.002_br0010 article-title: Linear Functional Analysis doi: 10.1007/978-1-4471-7280-2 – year: 2011 ident: 10.1016/j.acha.2022.12.002_br0090 article-title: Deep sparse rectifier neural networks – volume: 20 start-page: 37 year: 1971 ident: 10.1016/j.acha.2022.12.002_br0160 article-title: On Liouville's theorem for biharmonic functions publication-title: SIAM J. Appl. Math. doi: 10.1137/0120005 – volume: vol. 63 year: 1991 ident: 10.1016/j.acha.2022.12.002_br0050 article-title: Polyanalytic Functions – volume: vol. 354 start-page: 164 year: 2008 ident: 10.1016/j.acha.2022.12.002_br0330 article-title: Weyl's lemma, one of many – volume: 33 year: 2020 ident: 10.1016/j.acha.2022.12.002_br0410 article-title: The phase diagram of approximation rates for deep neural networks publication-title: Adv. Neural Inf. Process. Syst. – volume: vol. 3 year: 1983 ident: 10.1016/j.acha.2022.12.002_br0170 article-title: Holomorphic Functions of Several Variables – volume: 94 year: 2017 ident: 10.1016/j.acha.2022.12.002_br0390 article-title: Error bounds for approximations with deep ReLU networks publication-title: Neural Netw. doi: 10.1016/j.neunet.2017.07.002 – year: 2018 ident: 10.1016/j.acha.2022.12.002_br0350 article-title: Deep complex networks – volume: 6 start-page: 861 issue: 6 year: 1993 ident: 10.1016/j.acha.2022.12.002_br0220 article-title: Multilayer feedforward networks with a nonpolynomial activation function can approximate any function publication-title: Neural Netw. doi: 10.1016/S0893-6080(05)80131-5 – volume: 521 issue: 7553 year: 2015 ident: 10.1016/j.acha.2022.12.002_br0210 article-title: Deep learning publication-title: Nature doi: 10.1038/nature14539 – volume: 31 start-page: 6169 year: 2018 ident: 10.1016/j.acha.2022.12.002_br0230 article-title: ResNet with one-neuron hidden layers is a universal approximator publication-title: Adv. Neural Inf. Process. Syst. – volume: 8 issue: 1 year: 1996 ident: 10.1016/j.acha.2022.12.002_br0260 article-title: Neural networks for optimal approximation of smooth and analytic functions publication-title: Neural Comput. doi: 10.1162/neco.1996.8.1.164 – volume: 17 start-page: 879 issue: 4 year: 2006 ident: 10.1016/j.acha.2022.12.002_br0140 article-title: Universal approximation using incremental constructive feedforward networks with random hidden nodes publication-title: IEEE Trans. Neural Netw. doi: 10.1109/TNN.2006.875977 – volume: 60 issue: 6 year: 2017 ident: 10.1016/j.acha.2022.12.002_br0200 article-title: ImageNet classification with deep convolutional neural networks publication-title: Commun. ACM doi: 10.1145/3065386 – volume: 55 start-page: 407 issue: 1 year: 2022 ident: 10.1016/j.acha.2022.12.002_br0400 article-title: Universal approximations of invariant maps by neural networks publication-title: Constr. Approx. doi: 10.1007/s00365-021-09546-1 – volume: vol. 125 start-page: 2306 year: 2020 ident: 10.1016/j.acha.2022.12.002_br0180 article-title: Universal approximation with deep narrow networks |
| SSID | ssj0011459 |
| Score | 2.572248 |
| Snippet | We generalize the classical universal approximation theorem for neural networks to the case of complex-valued neural networks. Precisely, we consider... |
| SourceID | crossref elsevier |
| SourceType | Enrichment Source Index Database Publisher |
| StartPage | 33 |
| SubjectTerms | Complex-valued neural networks Deep neural networks Holomorphic functions Polyharmonic functions Universal approximation theorem |
| Title | The universal approximation theorem for complex-valued neural networks |
| URI | https://dx.doi.org/10.1016/j.acha.2022.12.002 |
| Volume | 64 |
| WOSCitedRecordID | wos000918021300001&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: PRVESC databaseName: Elsevier SD Freedom Collection Journals 2021 customDbUrl: eissn: 1096-603X dateEnd: 99991231 omitProxy: false ssIdentifier: ssj0011459 issn: 1063-5203 databaseCode: AIEXJ dateStart: 20211209 isFulltext: true titleUrlDefault: https://www.sciencedirect.com providerName: Elsevier |
| link | http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwtV1LS8QwEA6LetCD-MQ3OXiTQpv0kV4EEUUFxYPK3krTJrqydhe3K_35Tl61PtGDl9ItbXbbLzvzTTrzDUL7AeXgCGTilb5UAUopPRblzBPKGDJWSsp93Wwiubpi_X563esdulqYl2FSVaxp0vG_Qg3HAGxVOvsHuNtB4QDsA-iwBdhh-2vgpybdQukAKM3wZmAKFG3Z4pPOLdTJ5KLxlNy3ygEQWoCjMmnhky5pdUzVlsCNp7VbQVS617qFTm61TRx-d6PBfT3MdZ86zY_FcNB0VxhIJ5_PGkWgMRCw-rRrNeOwY_aMloV1oEZc_ZNpNqsEjzBNHpTeEyF6GdYnb47IvXz_4J_arEGXkPaYqTEyNUYWkExric6SJErBqs0enZ_0L9r3SEGo2-W1d2DLpkyG38df8jU16dCNmyW0aOMEfGTwXUY9Ua2ghY56JHy6bCV3J6voFHDHLe74He7Y4o4Bd_wed2xwxw73NXR7enJzfObZJhleQX2_9mgZ5MBRqcz9IhQ5kUHM45KVOY-jgnARxWGZlsBBBE8IT2L4z0rCwWzLEBwMDeg6mqlGldhAmEeBgPA3DwWcL4uI-TzkTKRxyiEwL-gmCtzzyQqrIK8amQyz75HZRAftNWOjn_Lj2ZF77JllgIbZZTCLfrhu60_fso3m36b5Dpqpn6diF80VL_Vg8rxnp9ArtxN_bQ |
| linkProvider | Elsevier |
| 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=The+universal+approximation+theorem+for+complex-valued+neural+networks&rft.jtitle=Applied+and+computational+harmonic+analysis&rft.au=Voigtlaender%2C+Felix&rft.date=2023-05-01&rft.issn=1063-5203&rft.volume=64&rft.spage=33&rft.epage=61&rft_id=info:doi/10.1016%2Fj.acha.2022.12.002&rft.externalDBID=n%2Fa&rft.externalDocID=10_1016_j_acha_2022_12_002 |
| thumbnail_l | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=1063-5203&client=summon |
| thumbnail_m | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=1063-5203&client=summon |
| thumbnail_s | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=1063-5203&client=summon |