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We develop gauge equivariant convolutional neural networks on arbitrary manifolds <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathcal {M}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>M<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> using principal bundles with structure group <jats:italic>K<\/jats:italic> and equivariant maps between sections of associated vector bundles. We also discuss group equivariant neural networks for homogeneous spaces <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathcal {M}=G\/K$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>M<\/mml:mi>\n                    <mml:mo>=<\/mml:mo>\n                    <mml:mi>G<\/mml:mi>\n                    <mml:mo>\/<\/mml:mo>\n                    <mml:mi>K<\/mml:mi>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, which are instead equivariant with respect to the global symmetry <jats:italic>G<\/jats:italic> on <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathcal {M}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>M<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. Group equivariant layers can be interpreted as intertwiners between induced representations of <jats:italic>G<\/jats:italic>, and we show their relation to gauge equivariant convolutional layers. We analyze several applications of this formalism, including semantic segmentation and object detection networks. We also discuss the case of spherical networks in great detail, corresponding to the case <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathcal {M}=S^2=\\textrm{SO}(3)\/\\textrm{SO}(2)$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>M<\/mml:mi>\n                    <mml:mo>=<\/mml:mo>\n                    <mml:msup>\n                      <mml:mi>S<\/mml:mi>\n                      <mml:mn>2<\/mml:mn>\n                    <\/mml:msup>\n                    <mml:mo>=<\/mml:mo>\n                    <mml:mtext>SO<\/mml:mtext>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:mn>3<\/mml:mn>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                    <mml:mo>\/<\/mml:mo>\n                    <mml:mtext>SO<\/mml:mtext>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:mn>2<\/mml:mn>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. Here we emphasize the use of Fourier analysis involving Wigner matrices, spherical harmonics and Clebsch\u2013Gordan coefficients for <jats:inline-formula><jats:alternatives><jats:tex-math>$$G=\\textrm{SO}(3)$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>G<\/mml:mi>\n                    <mml:mo>=<\/mml:mo>\n                    <mml:mtext>SO<\/mml:mtext>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:mn>3<\/mml:mn>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, illustrating the power of representation theory for deep learning.<\/jats:p>","DOI":"10.1007\/s10462-023-10502-7","type":"journal-article","created":{"date-parts":[[2023,6,4]],"date-time":"2023-06-04T04:01:15Z","timestamp":1685851275000},"page":"14605-14662","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":50,"title":["Geometric deep learning and equivariant neural networks"],"prefix":"10.1007","volume":"56","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0172-7944","authenticated-orcid":false,"given":"Jan E.","family":"Gerken","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jimmy","family":"Aronsson","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9571-3884","authenticated-orcid":false,"given":"Oscar","family":"Carlsson","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Hampus","family":"Linander","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3165-6999","authenticated-orcid":false,"given":"Fredrik","family":"Ohlsson","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Christoffer","family":"Petersson","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9803-2734","authenticated-orcid":false,"given":"Daniel","family":"Persson","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2023,6,4]]},"reference":[{"key":"10502_CR1","unstructured":"Albin P (xxx) \u201cLinear analysis on manifolds.\u201d https:\/\/faculty.math.illinois.edu\/palbin\/analysisonmfds\/lecturenotes.pdf"},{"key":"10502_CR2","doi-asserted-by":"publisher","unstructured":"Aronsson J (July, 2022), \u201cHomogeneous vector bundles and G-equivariant convolutional neural networks,\u201dSampling Theory, Signal Processing, and Data Analysis 20, arXiv:2105.05400. https:\/\/doi.org\/10.1007\/s43670-022-00029-3","DOI":"10.1007\/s43670-022-00029-3"},{"key":"10502_CR3","doi-asserted-by":"crossref","unstructured":"Bekkers EJ, Lafarge MW, Veta M, Eppenhof KAJ, Pluim JPW, Duits R (2018) \u201cRoto-translation covariant convolutional networks for medical image analysis,\u201d in Medical Image Computing and Computer Assisted Intervention - MICCAI 2018, Frangi AF, Schnabel JA, Davatzikos C, Alberola-L\u00f3pez C, Fichtinger G, eds., Lecture Notes in Computer Science, pp.\u00a0440\u2013448. 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Curran Associates, Inc.,"},{"key":"10502_CR6","unstructured":"Bronstein MM, Bruna J, Cohen T, Veli\u010dkovi\u0107 P (2021) \u201cGeometric deep learning: Grids, groups, graphs, geodesics, and gauges,\u201d arXiv:2104.13478"},{"key":"10502_CR7","doi-asserted-by":"publisher","first-page":"18","DOI":"10.1109\/msp.2017.2693418","volume":"34","author":"MM Bronstein","year":"2017","unstructured":"Bronstein MM, Bruna J, LeCun Y, Szlam A, Vandergheynst P (2017) \u201cGeometric deep learning: Going beyond euclidean data,\u2019\u2019. 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Association for Computational Linguistics, Red Hook, NY, USA. arXiv:1706.03762. https:\/\/doi.org\/10.18653\/v1\/p18-1008","DOI":"10.18653\/v1\/P18-1008"},{"key":"10502_CR9","unstructured":"Cheng MCN, Anagiannis V, Weiler M, de Haan P, Cohen TS, Welling M (2019) \u201cCovariance in physics and convolutional neural networks,\u201d arXiv:1906.02481"},{"key":"10502_CR10","unstructured":"Cobb OJ, Wallis CGR, Mavor-Parker AN, Marignier A, Price MA, d\u2019Avezac M, McEwen JD (2020) \u201cEfficient generalized spherical cnns,\u201d arXiv:2010.11661"},{"key":"10502_CR11","unstructured":"Cohen TS, Geiger M, K\u00f6hler J, Welling M (2018) \u201cSpherical cnns,\u201d in International Conference on Learning Representations. arXiv:1801.10130"},{"key":"10502_CR12","unstructured":"Cohen TS, Geiger M, Weiler M (2018a) \u201cIntertwiners between induced representations (with applications to the theory of equivariant neural networks),\u201d arXiv:1803.10743"},{"key":"10502_CR13","unstructured":"Cohen TS, Weiler M, Kicanaoglu B, Welling M (2019) \u201cGauge equivariant convolutional networks and the icosahedral cnn,\u201d in Proceedings of the 36th International Conference on Machine Learning, Chaudhuri K, Salakhutdinov R, eds., vol.\u00a097 of Proceedings of Machine Learning Research, pp.\u00a01321\u20131330. PMLR. arXiv:1902.04615"},{"key":"10502_CR14","unstructured":"Cohen TS, Welling M (2016) \u201cGroup equivariant convolutional networks,\u201d in Proceedings of The 33rd International Conference on Machine Learning, Balcan MF, Weinberger KQ, eds., vol.\u00a048 of Proceedings of Machine Learning Research, pp.\u00a02990\u20132999. PMLR, New York, New York, USA. arXiv:1602.07576"},{"key":"10502_CR15","unstructured":"Cohen T, Geiger M, Weiler M (2019) \u201cA general theory of equivariant cnns on homogeneous spaces,\u201d in Advances in Neural Information Processing Systems, Wallach H, Larochelle H, Beygelzimer A, F.\u00a0d\u2019Alch\u00eb-Buc, E.\u00a0Fox, and R.\u00a0Garnett, eds., vol.\u00a032. 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Curran Associates, Inc., arXiv:2006.10731"},{"key":"10502_CR29","doi-asserted-by":"crossref","unstructured":"Favoni M, Ipp A, M\u00fcller DI, Schuh D (Jan., 2022) \u201cLattice gauge equivariant convolutional neural networks,\u201dPhys. Rev. Lett. 128 arXiv:2012.12901. https:\/\/doi.org\/10.1103\/physrevlett.128.032003","DOI":"10.1103\/PhysRevLett.128.032003"},{"key":"10502_CR30","unstructured":"Finzi M, Welling M, Wilson AG (2021) \u201cA practical method for constructing equivariant multilayer perceptrons for arbitrary matrix groups,\u201d arXiv:2104.09459"},{"key":"10502_CR31","doi-asserted-by":"crossref","unstructured":"Folland GB (Feb., 2016) A Course in Abstract Harmonic Analysis. Chapman and Hall\/CRC, https:\/\/doi.org\/10.1201\/b19172","DOI":"10.1201\/b19172"},{"key":"10502_CR32","doi-asserted-by":"crossref","unstructured":"Fox J, Zhao B, Rajamanickam S, Ramprasad R, Song L (2021) \u201cConcentric spherical gnn for 3d representation learning,\u201d arXiv:2103.10484","DOI":"10.2172\/1772205"},{"key":"10502_CR33","doi-asserted-by":"publisher","first-page":"373","DOI":"10.1007\/s00454-016-9761-y","volume":"55","author":"P Frosini","year":"2016","unstructured":"Frosini P, Jab\u0142o\u0144ski G (2016) \u201cCombining persistent homology and invariance groups for shape comparison,\u2019\u2019. 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Curran Associates, Inc"},{"key":"10502_CR36","unstructured":"Gerken J, Carlsson O, Linander H, Ohlsson F, Petersson C, Persson D (2022) \u201cEquivariance versus augmentation for spherical images,\u201d in Proceedings of The 39th International Conference on Machine Learning, Balcan MF, Weinberger KQ, eds., vol.\u00a0162 of Proceedings of Machine Learning Research, pp.\u00a07404\u20137421. PMLR, New York, New York, USA. arXiv:2202.03990"},{"key":"10502_CR37","unstructured":"G\u0142uch G, Urbanke R (2021) \u201cNoether: The more things change, the more stay the same,\u201d arXiv:2104.05508"},{"issue":"1","key":"10502_CR38","doi-asserted-by":"publisher","first-page":"113","DOI":"10.1007\/s13137-011-0018-x","volume":"2","author":"EW Grafarend","year":"2011","unstructured":"Grafarend EW, K\u00fchnel W (2011) A minimal atlas for the rotation group SO(3). 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Accessed May 27, 2021"},{"key":"10502_CR43","unstructured":"Jiang CM, Huang J, Kashinath K, Prabhat, Marcus P, Niessner M (Sept., 2018) \u201cSpherical cnns on unstructured grids,\u201d in International Conference on Learning Representations"},{"key":"10502_CR44","doi-asserted-by":"publisher","unstructured":"Kaniuth E, Taylor KF (2012) Induced Representations of Locally Compact Groups. Cambridge University Press. https:\/\/doi.org\/10.1017\/cbo9781139045391","DOI":"10.1017\/cbo9781139045391"},{"key":"10502_CR45","doi-asserted-by":"crossref","unstructured":"Kol\u00e1\u0159 I, Slov\u00e1k J, Michor PW (1993) Natural Operations in Differential Geometry. 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Curran Associates, Inc., arXiv:1806.09231"},{"key":"10502_CR48","unstructured":"Kondor R, Trivedi S (2018) \u201cOn the generalization of equivariance and convolution in neural networks to the action of compact groups,\u201d in Proceedings of the 35th International Conference on Machine Learning, Dy J, Krause A, eds., vol.\u00a080 of Proceedings of Machine Learning Research, pp.\u00a02747\u20132755. PMLR. arXiv:1802.03690"},{"key":"10502_CR49","unstructured":"Lang L, Weiler M (2020) \u201cA Wigner-Eckart Theorem for Group Equivariant Convolution Kernels,\u201d arXiv:2010.10952"},{"key":"10502_CR50","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4419-9982-5","volume-title":"Introduction to Smooth Manifolds","author":"JM Lee","year":"2012","unstructured":"Lee JM (2012) Introduction to Smooth Manifolds. 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