diff --git "a/parse/dev/zSeoDvsDCe/zSeoDvsDCe_middle.json" "b/parse/dev/zSeoDvsDCe/zSeoDvsDCe_middle.json" new file mode 100644--- /dev/null +++ "b/parse/dev/zSeoDvsDCe/zSeoDvsDCe_middle.json" @@ -0,0 +1,165230 @@ +{ + "pdf_info": [ + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 153, + 97, + 455, + 137 + ], + "lines": [ + { + "bbox": [ + 162, + 97, + 450, + 116 + ], + "spans": [ + { + "bbox": [ + 162, + 97, + 450, + 116 + ], + "score": 1.0, + "content": "Sign and Basis Invariant Networks for", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 151, + 116, + 458, + 139 + ], + "spans": [ + { + "bbox": [ + 151, + 116, + 458, + 139 + ], + "score": 1.0, + "content": "Spectral Graph Representation Learning", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 259, + 179, + 355, + 223 + ], + "lines": [ + { + "bbox": [ + 258, + 179, + 355, + 191 + ], + "spans": [ + { + "bbox": [ + 258, + 179, + 355, + 191 + ], + "score": 1.0, + "content": "Anonymous Author(s)", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 283, + 189, + 328, + 202 + ], + "spans": [ + { + "bbox": [ + 283, + 189, + 328, + 202 + ], + "score": 1.0, + "content": "Affiliation", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 286, + 200, + 324, + 213 + ], + "spans": [ + { + "bbox": [ + 286, + 200, + 324, + 213 + ], + "score": 1.0, + "content": "Address", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 290, + 213, + 320, + 222 + ], + "spans": [ + { + "bbox": [ + 290, + 213, + 320, + 222 + ], + "score": 1.0, + "content": "email", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 3.5 + }, + { + "type": "title", + "bbox": [ + 283, + 252, + 328, + 265 + ], + "lines": [ + { + "bbox": [ + 281, + 251, + 331, + 267 + ], + "spans": [ + { + "bbox": [ + 281, + 251, + 331, + 267 + ], + "score": 1.0, + "content": "Abstract", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 91, + 276, + 469, + 408 + ], + "lines": [ + { + "bbox": [ + 92, + 276, + 470, + 289 + ], + "spans": [ + { + "bbox": [ + 92, + 280, + 99, + 288 + ], + "score": 1.0, + "content": "1", + "type": "text" + }, + { + "bbox": [ + 142, + 276, + 470, + 289 + ], + "score": 1.0, + "content": "We introduce SignNet and BasisNet—new neural architectures that are invariant", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 93, + 288, + 470, + 300 + ], + "spans": [ + { + "bbox": [ + 93, + 290, + 99, + 298 + ], + "score": 1.0, + "content": "2", + "type": "text" + }, + { + "bbox": [ + 142, + 288, + 439, + 300 + ], + "score": 1.0, + "content": "to two key symmetries displayed by eigenvectors: (i) sign flips, since if", + "type": "text" + }, + { + "bbox": [ + 439, + 289, + 446, + 297 + ], + "score": 0.73, + "content": "v", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 288, + 470, + 300 + ], + "score": 1.0, + "content": "is an", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 92, + 297, + 470, + 312 + ], + "spans": [ + { + "bbox": [ + 92, + 301, + 99, + 310 + ], + "score": 1.0, + "content": "3", + "type": "text" + }, + { + "bbox": [ + 141, + 297, + 230, + 312 + ], + "score": 1.0, + "content": "eigenvector then so is", + "type": "text" + }, + { + "bbox": [ + 231, + 299, + 245, + 309 + ], + "score": 0.71, + "content": "- v", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 297, + 470, + 312 + ], + "score": 1.0, + "content": "; and (ii) more general basis symmetries, which occur in", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 93, + 309, + 471, + 322 + ], + "spans": [ + { + "bbox": [ + 93, + 312, + 99, + 321 + ], + "score": 1.0, + "content": "4", + "type": "text" + }, + { + "bbox": [ + 141, + 309, + 471, + 322 + ], + "score": 1.0, + "content": "higher dimensional eigenspaces with infinitely many choices of basis eigenvectors.", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 93, + 320, + 470, + 333 + ], + "spans": [ + { + "bbox": [ + 93, + 323, + 99, + 331 + ], + "score": 1.0, + "content": "5", + "type": "text" + }, + { + "bbox": [ + 141, + 320, + 470, + 333 + ], + "score": 1.0, + "content": "We prove that our networks are universal, i.e., they can approximate any continu-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 92, + 331, + 470, + 343 + ], + "spans": [ + { + "bbox": [ + 92, + 333, + 100, + 343 + ], + "score": 1.0, + "content": "6", + "type": "text" + }, + { + "bbox": [ + 141, + 331, + 470, + 343 + ], + "score": 1.0, + "content": "ous function of eigenvectors with the desired invariances. 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Experi-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 90, + 385, + 470, + 399 + ], + "spans": [ + { + "bbox": [ + 90, + 388, + 99, + 397 + ], + "score": 1.0, + "content": "11", + "type": "text" + }, + { + "bbox": [ + 141, + 385, + 470, + 399 + ], + "score": 1.0, + "content": "ments show the strength of our networks for molecular graph regression, learning", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 90, + 397, + 460, + 409 + ], + "spans": [ + { + "bbox": [ + 90, + 399, + 99, + 408 + ], + "score": 1.0, + "content": "12", + "type": "text" + }, + { + "bbox": [ + 142, + 397, + 460, + 409 + ], + "score": 1.0, + "content": "expressive graph representations, and learning neural fields on triangle meshes.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 12.5 + }, + { + "type": "title", + "bbox": [ + 91, + 427, + 191, + 441 + ], + "lines": [ + { + "bbox": [ + 87, + 426, + 192, + 443 + ], + "spans": [ + { + "bbox": [ + 87, + 426, + 192, + 443 + ], + "score": 1.0, + "content": "13 1 Introduction", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 90, + 452, + 505, + 551 + ], + "lines": [ + { + "bbox": [ + 89, + 451, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 89, + 454, + 100, + 463 + ], + "score": 1.0, + "content": "14", + "type": "text" + }, + { + "bbox": [ + 105, + 451, + 505, + 465 + ], + "score": 1.0, + "content": "Numerous machine learning models process eigenvectors, which arise in various scenarios including", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 89, + 463, + 506, + 475 + ], + "spans": [ + { + "bbox": [ + 89, + 465, + 100, + 475 + ], + "score": 1.0, + "content": "15", + "type": "text" + }, + { + "bbox": [ + 105, + 463, + 506, + 475 + ], + "score": 1.0, + "content": "principal component analysis, matrix factorizations, and operators associated to graphs or manifolds.", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 89, + 474, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 89, + 476, + 100, + 486 + ], + "score": 1.0, + "content": "16", + "type": "text" + }, + { + "bbox": [ + 106, + 474, + 505, + 486 + ], + "score": 1.0, + "content": "An important example is the use of Laplacian eigenvectors to encode information about the structure", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 89, + 484, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 89, + 487, + 100, + 497 + ], + "score": 1.0, + "content": "17", + "type": "text" + }, + { + "bbox": [ + 105, + 484, + 505, + 497 + ], + "score": 1.0, + "content": "of a graph or manifold [Belkin and Niyogi, 2003, Von Luxburg, 2007, Lévy, 2006]. 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Our networks are universal and can approximate any continuous function", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 41.5 + } + ], + "page_idx": 0, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 102, + 731, + 499, + 742 + ], + "lines": [ + { + "bbox": [ + 106, + 730, + 500, + 743 + ], + "spans": [ + { + "bbox": [ + 106, + 730, + 500, + 743 + ], + "score": 1.0, + "content": "Submitted to 36th Conference on Neural Information Processing Systems (NeurIPS 2022). 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Experi-", + "type": "text" + } + ], + "index": 16, + "is_list_start_line": true + }, + { + "bbox": [ + 90, + 385, + 470, + 399 + ], + "spans": [ + { + "bbox": [ + 90, + 388, + 99, + 397 + ], + "score": 1.0, + "content": "11", + "type": "text" + }, + { + "bbox": [ + 141, + 385, + 470, + 399 + ], + "score": 1.0, + "content": "ments show the strength of our networks for molecular graph regression, learning", + "type": "text" + } + ], + "index": 17, + "is_list_start_line": true + }, + { + "bbox": [ + 90, + 397, + 460, + 409 + ], + "spans": [ + { + "bbox": [ + 90, + 399, + 99, + 408 + ], + "score": 1.0, + "content": "12", + "type": "text" + }, + { + "bbox": [ + 142, + 397, + 460, + 409 + ], + "score": 1.0, + "content": "expressive graph representations, and learning neural fields on triangle meshes.", + "type": "text" + } + ], + "index": 18, + "is_list_start_line": true + } + ], + "index": 12.5, + "bbox_fs": [ + 90, + 276, + 471, + 409 + ] + }, + { + "type": "title", + "bbox": [ + 91, + 427, + 191, + 441 + ], + "lines": [ + { + "bbox": [ + 87, + 426, + 192, + 443 + ], + "spans": [ + { + "bbox": [ + 87, + 426, + 192, + 443 + ], + "score": 1.0, + "content": "13 1 Introduction", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "index", + "bbox": [ + 90, + 452, + 505, + 551 + ], + "lines": [ + { + "bbox": [ + 89, + 451, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 89, + 454, + 100, + 463 + ], + "score": 1.0, + "content": "14", + "type": "text" + }, + { + "bbox": [ + 105, + 451, + 505, + 465 + ], + "score": 1.0, + "content": "Numerous machine learning models process eigenvectors, which arise in various scenarios including", + "type": "text" + } + ], + "index": 20, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 463, + 506, + 475 + ], + "spans": [ + { + "bbox": [ + 89, + 465, + 100, + 475 + ], + "score": 1.0, + "content": "15", + "type": "text" + }, + { + "bbox": [ + 105, + 463, + 506, + 475 + ], + "score": 1.0, + "content": "principal component analysis, matrix factorizations, and operators associated to graphs or manifolds.", + "type": "text" + } + ], + "index": 21, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 474, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 89, + 476, + 100, + 486 + ], + "score": 1.0, + "content": "16", + "type": "text" + }, + { + "bbox": [ + 106, + 474, + 505, + 486 + ], + "score": 1.0, + "content": "An important example is the use of Laplacian eigenvectors to encode information about the structure", + "type": "text" + } + ], + "index": 22, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 484, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 89, + 487, + 100, + 497 + ], + "score": 1.0, + "content": "17", + "type": "text" + }, + { + "bbox": [ + 105, + 484, + 505, + 497 + ], + "score": 1.0, + "content": "of a graph or manifold [Belkin and Niyogi, 2003, Von Luxburg, 2007, Lévy, 2006]. Positional", + "type": "text" + } + ], + "index": 23, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 496, + 505, + 508 + ], + "spans": [ + { + "bbox": [ + 89, + 498, + 100, + 507 + ], + "score": 1.0, + "content": "18", + "type": "text" + }, + { + "bbox": [ + 106, + 496, + 505, + 508 + ], + "score": 1.0, + "content": "encodings that involve Laplacian eigenvectors have recently been used to generalize Transformers", + "type": "text" + } + ], + "index": 24, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 506, + 506, + 520 + ], + "spans": [ + { + "bbox": [ + 89, + 509, + 100, + 519 + ], + "score": 1.0, + "content": "19", + "type": "text" + }, + { + "bbox": [ + 105, + 506, + 506, + 520 + ], + "score": 1.0, + "content": "to graphs [Kreuzer et al., 2021, Dwivedi and Bresson, 2021], and to improve the expressive power", + "type": "text" + } + ], + "index": 25, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 517, + 506, + 531 + ], + "spans": [ + { + "bbox": [ + 89, + 520, + 100, + 529 + ], + "score": 1.0, + "content": "20", + "type": "text" + }, + { + "bbox": [ + 105, + 517, + 506, + 531 + ], + "score": 1.0, + "content": "and empirical performance of graph neural networks (GNNs) [Dwivedi et al., 2022]. Furthermore,", + "type": "text" + } + ], + "index": 26, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 529, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 89, + 531, + 99, + 540 + ], + "score": 1.0, + "content": "21", + "type": "text" + }, + { + "bbox": [ + 106, + 529, + 505, + 541 + ], + "score": 1.0, + "content": "these eigenvectors are crucial for defining spectral operations on graphs that are foundational to graph", + "type": "text" + } + ], + "index": 27, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 539, + 417, + 552 + ], + "spans": [ + { + "bbox": [ + 89, + 542, + 100, + 551 + ], + "score": 1.0, + "content": "22", + "type": "text" + }, + { + "bbox": [ + 104, + 539, + 417, + 552 + ], + "score": 1.0, + "content": "signal processing and spectral GNNs [Ortega et al., 2018, Bruna et al., 2014].", + "type": "text" + } + ], + "index": 28, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 556, + 507, + 568 + ], + "spans": [ + { + "bbox": [ + 89, + 558, + 99, + 567 + ], + "score": 1.0, + "content": "23", + "type": "text" + }, + { + "bbox": [ + 106, + 556, + 507, + 568 + ], + "score": 1.0, + "content": "However, there are nontrivial symmetries that should be accounted for when processing eigenvectors.", + "type": "text" + } + ], + "index": 29, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 566, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 88, + 568, + 100, + 579 + ], + "score": 1.0, + "content": "24", + "type": "text" + }, + { + "bbox": [ + 105, + 566, + 170, + 579 + ], + "score": 1.0, + "content": "For instance, if", + "type": "text" + }, + { + "bbox": [ + 170, + 568, + 177, + 577 + ], + "score": 0.73, + "content": "v", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 566, + 291, + 579 + ], + "score": 1.0, + "content": "is an eigenvector, then so is", + "type": "text" + }, + { + "bbox": [ + 291, + 568, + 306, + 577 + ], + "score": 0.63, + "content": "- v", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 566, + 505, + 579 + ], + "score": 1.0, + "content": ", with the same eigenvalue. 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Accordingly, prior works randomly flip eigenvector signs during training in order to", + "type": "text" + } + ], + "index": 34, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 621, + 506, + 635 + ], + "spans": [ + { + "bbox": [ + 88, + 623, + 100, + 634 + ], + "score": 1.0, + "content": "29", + "type": "text" + }, + { + "bbox": [ + 105, + 621, + 506, + 635 + ], + "score": 1.0, + "content": "approximately learn sign invariance [Kreuzer et al., 2021, Dwivedi et al., 2020]. 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Sign invariance is a special case of basis invariance when all eigenvalues are", + "type": "text" + } + ], + "index": 37, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 654, + 505, + 666 + ], + "spans": [ + { + "bbox": [ + 88, + 656, + 99, + 666 + ], + "score": 1.0, + "content": "32", + "type": "text" + }, + { + "bbox": [ + 106, + 654, + 505, + 666 + ], + "score": 1.0, + "content": "distinct, but general basis invariance is even more difficult to deal with. 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As", + "type": "text" + }, + { + "bbox": [ + 472, + 579, + 506, + 592 + ], + "score": 0.9, + "content": "{ \\cal { O } } ( 1 ) =", + "type": "inline_equation" + } + ], + "index": 57 + }, + { + "bbox": [ + 90, + 591, + 483, + 603 + ], + "spans": [ + { + "bbox": [ + 90, + 593, + 100, + 602 + ], + "score": 1.0, + "content": "76", + "type": "text" + }, + { + "bbox": [ + 107, + 591, + 140, + 603 + ], + "score": 0.89, + "content": "\\{ - 1 , 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 591, + 483, + 603 + ], + "score": 1.0, + "content": ", sign invariance is a special case of basis invariance when all eigenvalues are distinct.", + "type": "text" + } + ], + "index": 58 + } + ], + "index": 57 + }, + { + "type": "text", + "bbox": [ + 90, + 606, + 505, + 652 + ], + "lines": [ + { + "bbox": [ + 89, + 607, + 505, + 619 + ], + "spans": [ + { + "bbox": [ + 89, + 608, + 99, + 618 + ], + "score": 1.0, + "content": "77", + "type": "text" + }, + { + "bbox": [ + 105, + 607, + 505, + 619 + ], + "score": 1.0, + "content": "Permutation equivariance. 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As", + "type": "text" + }, + { + "bbox": [ + 472, + 579, + 506, + 592 + ], + "score": 0.9, + "content": "{ \\cal { O } } ( 1 ) =", + "type": "inline_equation" + } + ], + "index": 57, + "is_list_start_line": true + }, + { + "bbox": [ + 90, + 591, + 483, + 603 + ], + "spans": [ + { + "bbox": [ + 90, + 593, + 100, + 602 + ], + "score": 1.0, + "content": "76", + "type": "text" + }, + { + "bbox": [ + 107, + 591, + 140, + 603 + ], + "score": 0.89, + "content": "\\{ - 1 , 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 591, + 483, + 603 + ], + "score": 1.0, + "content": ", sign invariance is a special case of basis invariance when all eigenvalues are distinct.", + "type": "text" + } + ], + "index": 58, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 607, + 505, + 619 + ], + "spans": [ + { + "bbox": [ + 89, + 608, + 99, + 618 + ], + "score": 1.0, + "content": "77", + "type": "text" + }, + { + "bbox": [ + 105, + 607, + 505, + 619 + ], + "score": 1.0, + "content": "Permutation equivariance. For GNN models that output node features or node predictions, one", + "type": "text" + } + ], + "index": 59, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 618, + 505, + 630 + ], + "spans": [ + { + "bbox": [ + 89, + 619, + 100, + 629 + ], + "score": 1.0, + "content": "78", + "type": "text" + }, + { + "bbox": [ + 106, + 618, + 201, + 630 + ], + "score": 1.0, + "content": "typically further desires", + "type": "text" + }, + { + "bbox": [ + 201, + 619, + 208, + 630 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 618, + 505, + 630 + ], + "score": 1.0, + "content": "to be invariant or equivariant to permutations of nodes, i.e., along the entries", + "type": "text" + } + ], + "index": 60, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 624, + 505, + 644 + ], + "spans": [ + { + "bbox": [ + 89, + 631, + 100, + 640 + ], + "score": 1.0, + "content": "79", + "type": "text" + }, + { + "bbox": [ + 104, + 624, + 243, + 644 + ], + "score": 1.0, + "content": "(or rows) of each vector. Thus, for", + "type": "text" + }, + { + "bbox": [ + 243, + 630, + 319, + 641 + ], + "score": 0.91, + "content": "f : \\mathbb { R } ^ { n \\times d } \\mathbb { R } ^ { n \\times d }", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 624, + 422, + 644 + ], + "score": 1.0, + "content": ", we typically also require", + "type": "text" + }, + { + "bbox": [ + 422, + 629, + 505, + 641 + ], + "score": 0.89, + "content": "f ( P V _ { 1 } , \\dots , P V _ { l } ) =", + "type": "inline_equation" + } + ], + "index": 61, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 636, + 466, + 655 + ], + "spans": [ + { + "bbox": [ + 88, + 641, + 100, + 651 + ], + "score": 1.0, + "content": "80", + "type": "text" + }, + { + "bbox": [ + 106, + 640, + 170, + 652 + ], + "score": 0.91, + "content": "P f ( V _ { 1 } , \\ldots , V _ { l } )", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 636, + 282, + 655 + ], + "score": 1.0, + "content": "for any permutation matrix", + "type": "text" + }, + { + "bbox": [ + 282, + 641, + 326, + 650 + ], + "score": 0.9, + "content": "P \\in \\mathbb { R } ^ { n \\times n }", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 636, + 466, + 655 + ], + "score": 1.0, + "content": ". 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A continuous function", + "type": "text" + }, + { + "bbox": [ + 261, + 427, + 315, + 437 + ], + "score": 0.91, + "content": "h : \\mathbb { R } ^ { n } \\mathbb { R } ^ { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 425, + 439, + 441 + ], + "score": 1.0, + "content": "is sign invariant if and only if", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17, + "bbox_fs": [ + 105, + 425, + 439, + 441 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 259, + 439, + 351, + 453 + ], + "lines": [ + { + "bbox": [ + 259, + 439, + 351, + 453 + ], + "spans": [ + { + "bbox": [ + 259, + 439, + 351, + 453 + ], + "score": 0.91, + "content": "h ( v ) = \\phi ( v ) + \\phi ( - v )", + "type": "interline_equation", + "image_path": "c4f1adeee5fde41fd8c6f138289e169bd062c2514e42ffc266321c91de96fdd6.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 259, + 439, + 351, + 453 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 98, + 453, + 505, + 477 + ], + "lines": [ + { + "bbox": [ + 104, + 452, + 505, + 467 + ], + "spans": [ + { + "bbox": [ + 104, + 452, + 192, + 467 + ], + "score": 1.0, + "content": "for some continuous", + "type": "text" + }, + { + "bbox": [ + 192, + 454, + 247, + 465 + ], + "score": 0.9, + "content": "\\phi : \\mathbb { R } ^ { n } \\mathbb { R } ^ { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 452, + 308, + 467 + ], + "score": 1.0, + "content": ". 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Any architecture choice will ensure sign", + "type": "text" + } + ], + "index": 21, + "is_list_start_line": true + }, + { + "bbox": [ + 87, + 493, + 505, + 506 + ], + "spans": [ + { + "bbox": [ + 87, + 497, + 100, + 505 + ], + "score": 1.0, + "content": "100", + "type": "text" + }, + { + "bbox": [ + 105, + 493, + 505, + 506 + ], + "score": 1.0, + "content": "invariance, while permutation equivariance can be achieved using elementwise MLPs (Multi-Layer", + "type": "text" + } + ], + "index": 22, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 504, + 477, + 517 + ], + "spans": [ + { + "bbox": [ + 86, + 506, + 100, + 516 + ], + "score": 1.0, + "content": "101", + "type": "text" + }, + { + "bbox": [ + 105, + 504, + 477, + 517 + ], + "score": 1.0, + "content": "Perceptrons), DeepSets [Zaheer et al., 2017], Transformers [Vaswani et al., 2017], or GNNs.", + "type": "text" + } + ], + "index": 23, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 522, + 505, + 533 + ], + "spans": [ + { + "bbox": [ + 86, + 523, + 100, + 532 + ], + "score": 1.0, + "content": "102", + "type": "text" + }, + { + "bbox": [ + 105, + 522, + 291, + 533 + ], + "score": 1.0, + "content": "Next, we address basis invariance for a single", + "type": "text" + }, + { + "bbox": [ + 291, + 522, + 298, + 531 + ], + "score": 0.82, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 522, + 505, + 533 + ], + "score": 1.0, + "content": "-dimensional subspace, i.e., we aim to parameterize", + "type": "text" + } + ], + "index": 24, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 529, + 507, + 546 + ], + "spans": [ + { + "bbox": [ + 86, + 534, + 100, + 544 + ], + "score": 1.0, + "content": "103", + "type": "text" + }, + { + "bbox": [ + 104, + 529, + 130, + 546 + ], + "score": 1.0, + "content": "maps", + "type": "text" + }, + { + "bbox": [ + 130, + 532, + 196, + 542 + ], + "score": 0.92, + "content": "h : \\mathbb { R } ^ { n \\times d } \\mathbb { R } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 529, + 382, + 546 + ], + "score": 1.0, + "content": "that are (a) invariant to right multiplication by", + "type": "text" + }, + { + "bbox": [ + 382, + 532, + 424, + 544 + ], + "score": 0.92, + "content": "Q \\in O ( d )", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 529, + 507, + 546 + ], + "score": 1.0, + "content": ", and (b) equivariant", + "type": "text" + } + ], + "index": 25, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 543, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 86, + 546, + 100, + 556 + ], + "score": 1.0, + "content": "104", + "type": "text" + }, + { + "bbox": [ + 105, + 543, + 388, + 556 + ], + "score": 1.0, + "content": "to permutations along the row axis. For (a), we use the mapping", + "type": "text" + }, + { + "bbox": [ + 388, + 543, + 440, + 554 + ], + "score": 0.89, + "content": "V \\mapsto V V ^ { \\top }", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 543, + 466, + 556 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 466, + 544, + 476, + 554 + ], + "score": 0.76, + "content": "V", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 543, + 505, + 556 + ], + "score": 1.0, + "content": "to the", + "type": "text" + } + ], + "index": 26, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 554, + 506, + 567 + ], + "spans": [ + { + "bbox": [ + 86, + 557, + 100, + 567 + ], + "score": 1.0, + "content": "105", + "type": "text" + }, + { + "bbox": [ + 104, + 554, + 303, + 567 + ], + "score": 1.0, + "content": "orthogonal projector of its column space, which is", + "type": "text" + }, + { + "bbox": [ + 303, + 555, + 325, + 567 + ], + "score": 0.92, + "content": "O ( d )", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 554, + 403, + 567 + ], + "score": 1.0, + "content": "invariant. 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This is justified by the classical", + "type": "text" + } + ], + "index": 28, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 576, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 86, + 579, + 100, + 588 + ], + "score": 1.0, + "content": "107", + "type": "text" + }, + { + "bbox": [ + 105, + 576, + 226, + 590 + ], + "score": 1.0, + "content": "first fundamental theorem of", + "type": "text" + }, + { + "bbox": [ + 227, + 577, + 248, + 588 + ], + "score": 0.93, + "content": "O ( d )", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 576, + 505, + 590 + ], + "score": 1.0, + "content": "[Kraft and Procesi, 1996], which has recently been applied in", + "type": "text" + } + ], + "index": 29, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 588, + 269, + 600 + ], + "spans": [ + { + "bbox": [ + 86, + 589, + 100, + 599 + ], + "score": 1.0, + "content": "108", + "type": "text" + }, + { + "bbox": [ + 105, + 588, + 269, + 600 + ], + "score": 1.0, + "content": "machine learning by Villar et al. [2021].", + "type": "text" + } + ], + "index": 30, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 602, + 507, + 618 + ], + "spans": [ + { + "bbox": [ + 86, + 605, + 100, + 616 + ], + "score": 1.0, + "content": "109", + "type": "text" + }, + { + "bbox": [ + 104, + 602, + 259, + 618 + ], + "score": 1.0, + "content": "Regarding (b), permuting the rows of", + "type": "text" + }, + { + "bbox": [ + 259, + 604, + 268, + 614 + ], + "score": 0.77, + "content": "V", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 602, + 396, + 618 + ], + "score": 1.0, + "content": "permutes rows and columns of", + "type": "text" + }, + { + "bbox": [ + 396, + 603, + 455, + 614 + ], + "score": 0.92, + "content": "V V ^ { \\top } \\in \\mathbb { R } ^ { n \\times n }", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 602, + 507, + 618 + ], + "score": 1.0, + "content": ". Hence, we", + "type": "text" + } + ], + "index": 31, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 613, + 507, + 628 + ], + "spans": [ + { + "bbox": [ + 86, + 616, + 100, + 627 + ], + "score": 1.0, + "content": "110", + "type": "text" + }, + { + "bbox": [ + 104, + 613, + 182, + 628 + ], + "score": 1.0, + "content": "desire the function", + "type": "text" + }, + { + "bbox": [ + 183, + 615, + 249, + 626 + ], + "score": 0.89, + "content": "\\phi : \\mathbb { R } ^ { n \\times n } \\mathbb { R } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 613, + 263, + 628 + ], + "score": 1.0, + "content": "on", + "type": "text" + }, + { + "bbox": [ + 264, + 614, + 287, + 625 + ], + "score": 0.89, + "content": "\\dot { V } \\dot { V } ^ { \\top }", + "type": "inline_equation" + }, + { + "bbox": [ + 287, + 613, + 507, + 628 + ], + "score": 1.0, + "content": "to be equivariant to both row and column permutation:", + "type": "text" + } + ], + "index": 32, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 624, + 507, + 640 + ], + "spans": [ + { + "bbox": [ + 86, + 627, + 100, + 638 + ], + "score": 1.0, + "content": "111", + "type": "text" + }, + { + "bbox": [ + 107, + 625, + 224, + 638 + ], + "score": 0.9, + "content": "\\phi ( P V V ^ { \\top } P ^ { \\top } ) = \\dot { P } \\phi ( V V ^ { \\top } )", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 624, + 507, + 640 + ], + "score": 1.0, + "content": ". 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We thus parameterize", + "type": "text" + } + ], + "index": 35, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 661, + 373, + 674 + ], + "spans": [ + { + "bbox": [ + 86, + 664, + 100, + 672 + ], + "score": 1.0, + "content": "114", + "type": "text" + }, + { + "bbox": [ + 104, + 661, + 373, + 674 + ], + "score": 1.0, + "content": "a family with the requisite invariance and equivariance as follows:", + "type": "text" + } + ], + "index": 36, + "is_list_start_line": true + } + ], + "index": 22, + "bbox_fs": [ + 86, + 482, + 506, + 517 + ] + }, + { + "type": "index", + "bbox": [ + 86, + 521, + 505, + 600 + ], + "lines": [], + "index": 27, + "bbox_fs": [ + 86, + 522, + 507, + 600 + ], + "lines_deleted": true + }, + { + "type": "index", + "bbox": [ + 86, + 603, + 505, + 672 + ], + "lines": [], + "index": 33.5, + "bbox_fs": [ + 86, + 602, + 507, + 674 + ], + "lines_deleted": true + }, + { + "type": "interline_equation", + "bbox": [ + 261, + 673, + 349, + 688 + ], + "lines": [ + { + "bbox": [ + 261, + 673, + 349, + 688 + ], + "spans": [ + { + "bbox": [ + 261, + 673, + 349, + 688 + ], + "score": 0.9, + "content": "h ( V ) = \\operatorname { I G N } ( V V ^ { \\top } ) .", + "type": "interline_equation", + "image_path": "f3acd7f63a0210d09241f4eadbfd2eda969c97a4da9a43ca609dfd7e83805e05.jpg" + } + ] + } + ], + "index": 37, + "virtual_lines": [ + { + "bbox": [ + 261, + 673, + 349, + 688 + ], + "spans": [], + "index": 37 + } + ] + }, + { + "type": "index", + "bbox": [ + 87, + 689, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 86, + 688, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 86, + 690, + 100, + 700 + ], + "score": 1.0, + "content": "115", + "type": "text" + }, + { + "bbox": [ + 105, + 688, + 376, + 702 + ], + "score": 1.0, + "content": "Proposition 2 states that this architecture universally approximates", + "type": "text" + }, + { + "bbox": [ + 376, + 689, + 398, + 700 + ], + "score": 0.92, + "content": "O ( d )", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 688, + 506, + 702 + ], + "score": 1.0, + "content": "invariant and permutation", + "type": "text" + } + ], + "index": 38, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 700, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 86, + 703, + 99, + 711 + ], + "score": 1.0, + "content": "116", + "type": "text" + }, + { + "bbox": [ + 105, + 700, + 505, + 712 + ], + "score": 1.0, + "content": "equivariant functions. The full approximation power requires high order tensors to be used for the", + "type": "text" + } + ], + "index": 39, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 711, + 497, + 723 + ], + "spans": [ + { + "bbox": [ + 86, + 712, + 100, + 722 + ], + "score": 1.0, + "content": "117", + "type": "text" + }, + { + "bbox": [ + 105, + 711, + 497, + 723 + ], + "score": 1.0, + "content": "IGN; in practice, we restrict the tensor dimensions for efficiency, as discussed in the next section.", + "type": "text" + } + ], + "index": 40, + "is_list_start_line": true + } + ], + "index": 39, + "bbox_fs": [ + 86, + 688, + 506, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 103, + 72, + 505, + 109 + ], + "lines": [ + { + "bbox": [ + 105, + 70, + 506, + 87 + ], + "spans": [ + { + "bbox": [ + 105, + 70, + 235, + 87 + ], + "score": 1.0, + "content": "Proposition 2. Any continuous,", + "type": "text" + }, + { + "bbox": [ + 235, + 73, + 257, + 85 + ], + "score": 0.9, + "content": "O ( d )", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 70, + 297, + 87 + ], + "score": 1.0, + "content": "invariant", + "type": "text" + }, + { + "bbox": [ + 297, + 72, + 361, + 83 + ], + "score": 0.91, + "content": "h : \\mathbb { R } ^ { n \\times d } \\mathbb { R } ^ { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 70, + 417, + 87 + ], + "score": 1.0, + "content": "is of the form", + "type": "text" + }, + { + "bbox": [ + 418, + 72, + 489, + 85 + ], + "score": 0.91, + "content": "h ( V ) = \\phi ( V V ^ { \\top } )", + "type": "inline_equation" + }, + { + "bbox": [ + 490, + 70, + 506, + 87 + ], + "score": 1.0, + "content": "for", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 83, + 505, + 98 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 160, + 98 + ], + "score": 1.0, + "content": "a continuous", + "type": "text" + }, + { + "bbox": [ + 160, + 86, + 167, + 96 + ], + "score": 0.78, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 83, + 264, + 98 + ], + "score": 1.0, + "content": ". For a compact domain", + "type": "text" + }, + { + "bbox": [ + 264, + 84, + 309, + 96 + ], + "score": 0.91, + "content": "{ \\mathcal { Z } } \\subseteq \\mathbb { R } ^ { n \\times d }", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 83, + 383, + 98 + ], + "score": 1.0, + "content": ", maps of the form", + "type": "text" + }, + { + "bbox": [ + 383, + 85, + 457, + 97 + ], + "score": 0.9, + "content": "V \\mapsto \\operatorname { I G N } ( V V ^ { \\top } )", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 83, + 505, + 98 + ], + "score": 1.0, + "content": "universally", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 104, + 94, + 507, + 111 + ], + "spans": [ + { + "bbox": [ + 104, + 94, + 204, + 111 + ], + "score": 1.0, + "content": "approximate continuous", + "type": "text" + }, + { + "bbox": [ + 205, + 96, + 292, + 108 + ], + "score": 0.92, + "content": "h : \\mathcal { Z } \\subseteq \\mathbb { R } ^ { n \\times d } \\to \\mathbb { R } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 292, + 94, + 326, + 111 + ], + "score": 1.0, + "content": "that are", + "type": "text" + }, + { + "bbox": [ + 327, + 96, + 348, + 109 + ], + "score": 0.92, + "content": "O ( d )", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 94, + 507, + 111 + ], + "score": 1.0, + "content": "invariant and permutation equivariant.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "title", + "bbox": [ + 106, + 120, + 309, + 132 + ], + "lines": [ + { + "bbox": [ + 105, + 118, + 311, + 135 + ], + "spans": [ + { + "bbox": [ + 105, + 118, + 311, + 135 + ], + "score": 1.0, + "content": "2.2 Neural Networks on Multiple Eigenspaces", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 106, + 140, + 505, + 196 + ], + "lines": [ + { + "bbox": [ + 105, + 140, + 505, + 153 + ], + "spans": [ + { + "bbox": [ + 105, + 140, + 291, + 153 + ], + "score": 1.0, + "content": "Next, we use the single-eigenspace models", + "type": "text" + }, + { + "bbox": [ + 291, + 141, + 318, + 153 + ], + "score": 0.9, + "content": "\\phi _ { l } ( V _ { l } )", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 140, + 505, + 153 + ], + "score": 1.0, + "content": "as building blocks for a model on multiple", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 151, + 505, + 164 + ], + "spans": [ + { + "bbox": [ + 106, + 151, + 312, + 164 + ], + "score": 1.0, + "content": "eigenspaces. 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This approach is grounded in a general decomposition theorem for product spaces that", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 185, + 199, + 196 + ], + "spans": [ + { + "bbox": [ + 105, + 185, + 199, + 196 + ], + "score": 1.0, + "content": "we prove in Section A.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 93, + 199, + 502, + 212 + ], + "lines": [ + { + "bbox": [ + 88, + 196, + 491, + 217 + ], + "spans": [ + { + "bbox": [ + 88, + 196, + 318, + 217 + ], + "score": 1.0, + "content": "SignNet. 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The form", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 86, + 247, + 505, + 259 + ], + "spans": [ + { + "bbox": [ + 86, + 248, + 100, + 258 + ], + "score": 1.0, + "content": "129", + "type": "text" + }, + { + "bbox": [ + 107, + 247, + 169, + 259 + ], + "score": 0.92, + "content": "\\phi ( v _ { i } ) + \\phi ( - v _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 247, + 505, + 259 + ], + "score": 1.0, + "content": "induces sign invariance for each eigenvector. Since we do not yet impose permutation", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 86, + 258, + 360, + 270 + ], + "spans": [ + { + "bbox": [ + 86, + 260, + 100, + 269 + ], + "score": 1.0, + "content": "130", + "type": "text" + }, + { + "bbox": [ + 104, + 258, + 360, + 270 + ], + "score": 1.0, + "content": "equivariance here, we term this model Unconstrained-SignNet.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 87, + 273, + 505, + 329 + ], + "lines": [ + { + "bbox": [ + 86, + 272, + 504, + 287 + ], + "spans": [ + { + "bbox": [ + 86, + 275, + 100, + 286 + ], + "score": 1.0, + "content": "131", + "type": "text" + }, + { + "bbox": [ + 105, + 272, + 325, + 287 + ], + "score": 1.0, + "content": "To obtain a sign invariant and permutation equivariant", + "type": "text" + }, + { + "bbox": [ + 325, + 275, + 333, + 286 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 272, + 424, + 287 + ], + "score": 1.0, + "content": "that outputs vectors in", + "type": "text" + }, + { + "bbox": [ + 425, + 274, + 448, + 284 + ], + "score": 0.9, + "content": "\\mathbb { R } ^ { n \\times s }", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 272, + 496, + 287 + ], + "score": 1.0, + "content": ", we restrict", + "type": "text" + }, + { + "bbox": [ + 497, + 275, + 504, + 286 + ], + "score": 0.83, + "content": "\\phi", + "type": "inline_equation" + } + ], + "index": 14 + }, + { + "bbox": [ + 86, + 284, + 507, + 298 + ], + "spans": [ + { + "bbox": [ + 86, + 286, + 100, + 297 + ], + "score": 1.0, + "content": "132", + "type": "text" + }, + { + "bbox": [ + 106, + 284, + 123, + 298 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 287, + 130, + 297 + ], + "score": 0.79, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 130, + 284, + 507, + 298 + ], + "score": 1.0, + "content": "to be permutation equivariant networks from vectors to vectors, such as elementwise MLPs,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 86, + 296, + 505, + 308 + ], + "spans": [ + { + "bbox": [ + 86, + 298, + 100, + 307 + ], + "score": 1.0, + "content": "133", + "type": "text" + }, + { + "bbox": [ + 106, + 296, + 505, + 308 + ], + "score": 1.0, + "content": "DeepSets [Zaheer et al., 2017], Transformers [Vaswani et al., 2017], or most standard GNNs. We", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 86, + 306, + 504, + 320 + ], + "spans": [ + { + "bbox": [ + 86, + 309, + 100, + 318 + ], + "score": 1.0, + "content": "134", + "type": "text" + }, + { + "bbox": [ + 105, + 306, + 493, + 320 + ], + "score": 1.0, + "content": "name this permutation equivariant version SignNet. 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For basis invariance, let 136", + "type": "text" + }, + { + "bbox": [ + 259, + 358, + 312, + 370 + ], + "score": 0.93, + "content": "V _ { i } \\ \\in \\ \\mathbb { R } ^ { n \\times d _ { i } }", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 356, + 440, + 372 + ], + "score": 1.0, + "content": "be an orthonormal basis of a", + "type": "text" + }, + { + "bbox": [ + 441, + 360, + 450, + 370 + ], + "score": 0.88, + "content": "d _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 356, + 506, + 372 + ], + "score": 1.0, + "content": "dimensional", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 84, + 370, + 384, + 383 + ], + "spans": [ + { + "bbox": [ + 84, + 370, + 363, + 383 + ], + "score": 1.0, + "content": "137 eigenspace. 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For efficiency, we will only use matrices and vectors in the IGNs (that is, no", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 86, + 471, + 395, + 486 + ], + "spans": [ + { + "bbox": [ + 86, + 475, + 100, + 485 + ], + "score": 1.0, + "content": "144", + "type": "text" + }, + { + "bbox": [ + 104, + 471, + 147, + 486 + ], + "score": 1.0, + "content": "tensors in", + "type": "text" + }, + { + "bbox": [ + 147, + 472, + 165, + 483 + ], + "score": 0.9, + "content": "\\mathbb { R } ^ { n ^ { p } }", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 471, + 181, + 486 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 181, + 473, + 206, + 484 + ], + "score": 0.88, + "content": "p > 2", + "type": "inline_equation" + }, + { + "bbox": [ + 206, + 471, + 395, + 486 + ], + "score": 1.0, + "content": "), i.e., we use 2-IGN. Our resulting BasisNet is", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 26 + }, + { + "type": "interline_equation", + "bbox": [ + 221, + 488, + 389, + 504 + ], + "lines": [ + { + "bbox": [ + 221, + 488, + 389, + 504 + ], + "spans": [ + { + "bbox": [ + 221, + 488, + 389, + 504 + ], + "score": 0.92, + "content": "f ( V _ { 1 } , \\dots , V _ { l } ) = \\rho \\left( [ \\mathrm { I G N } _ { d _ { i } } ( V _ { i } V _ { i } ^ { \\top } ) ] _ { i = 1 } ^ { l } \\right) .", + "type": "interline_equation", + "image_path": "75f36c163e73caac170519a4b10dc9fcecc8b5ab4618f9981ed9aa5bbc273725.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 221, + 488, + 389, + 504 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 86, + 506, + 505, + 595 + ], + "lines": [ + { + "bbox": [ + 86, + 507, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 86, + 509, + 100, + 518 + ], + "score": 1.0, + "content": "145", + "type": "text" + }, + { + "bbox": [ + 105, + 507, + 505, + 519 + ], + "score": 1.0, + "content": "Expressive-BasisNet. While we restrict SignNet to only use vectors and BasisNet to only use vectors", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 86, + 517, + 506, + 531 + ], + "spans": [ + { + "bbox": [ + 86, + 519, + 100, + 530 + ], + "score": 1.0, + "content": "146", + "type": "text" + }, + { + "bbox": [ + 105, + 517, + 506, + 531 + ], + "score": 1.0, + "content": "and matrices, higher order tensors are generally required for universally approximating permutation", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 86, + 528, + 506, + 541 + ], + "spans": [ + { + "bbox": [ + 86, + 531, + 100, + 540 + ], + "score": 1.0, + "content": "147", + "type": "text" + }, + { + "bbox": [ + 105, + 528, + 506, + 541 + ], + "score": 1.0, + "content": "equivariant or invariant functions [Keriven and Peyré, 2019, Maron et al., 2019, Maehara and NT,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 86, + 538, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 86, + 542, + 100, + 551 + ], + "score": 1.0, + "content": "148", + "type": "text" + }, + { + "bbox": [ + 105, + 538, + 506, + 552 + ], + "score": 1.0, + "content": "2019]. Thus, we will consider a theoretically powerful but computationally impractical variant of", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 86, + 550, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 86, + 552, + 100, + 563 + ], + "score": 1.0, + "content": "149", + "type": "text" + }, + { + "bbox": [ + 105, + 550, + 237, + 563 + ], + "score": 1.0, + "content": "our model, in which we replace", + "type": "text" + }, + { + "bbox": [ + 237, + 552, + 244, + 562 + ], + "score": 0.8, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 550, + 262, + 563 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 262, + 551, + 290, + 563 + ], + "score": 0.9, + "content": "\\mathrm { I G N } _ { d _ { i } }", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 550, + 505, + 563 + ], + "score": 1.0, + "content": "in BasisNet with IGNs of arbitrary tensor order. We", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 86, + 561, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 86, + 564, + 100, + 573 + ], + "score": 1.0, + "content": "150", + "type": "text" + }, + { + "bbox": [ + 105, + 561, + 399, + 574 + ], + "score": 1.0, + "content": "call this variant Expressive-BasisNet. Universal approximation requires", + "type": "text" + }, + { + "bbox": [ + 400, + 562, + 426, + 574 + ], + "score": 0.92, + "content": "\\Omega ( n ^ { n } )", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 561, + 505, + 574 + ], + "score": 1.0, + "content": "sized intermediate", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 86, + 571, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 86, + 574, + 100, + 585 + ], + "score": 1.0, + "content": "151", + "type": "text" + }, + { + "bbox": [ + 105, + 571, + 505, + 585 + ], + "score": 1.0, + "content": "tensors [Ravanbakhsh, 2020]. We study Expressive-BasisNet due to its theoretical interest, and to", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 86, + 583, + 502, + 596 + ], + "spans": [ + { + "bbox": [ + 86, + 586, + 100, + 595 + ], + "score": 1.0, + "content": "152", + "type": "text" + }, + { + "bbox": [ + 105, + 583, + 502, + 596 + ], + "score": 1.0, + "content": "juxtapose with the computational efficiency and strong expressive power of SignNet and BasisNet.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 34.5 + }, + { + "type": "text", + "bbox": [ + 102, + 600, + 432, + 611 + ], + "lines": [ + { + "bbox": [ + 105, + 599, + 433, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 433, + 613 + ], + "score": 1.0, + "content": "For a summary of properties and more details about our models, see Appendix B.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 86, + 616, + 506, + 661 + ], + "lines": [ + { + "bbox": [ + 86, + 615, + 506, + 628 + ], + "spans": [ + { + "bbox": [ + 86, + 618, + 100, + 627 + ], + "score": 1.0, + "content": "154", + "type": "text" + }, + { + "bbox": [ + 105, + 615, + 506, + 628 + ], + "score": 1.0, + "content": "In the multiple subspace case, we can prove universality of our models through a general decomposi-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 85, + 626, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 85, + 628, + 101, + 639 + ], + "score": 1.0, + "content": "155", + "type": "text" + }, + { + "bbox": [ + 105, + 626, + 505, + 639 + ], + "score": 1.0, + "content": "tion theorem, which reduces the multiple subspace case to the single subspace case. See Section A", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 85, + 637, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 85, + 639, + 100, + 650 + ], + "score": 1.0, + "content": "156", + "type": "text" + }, + { + "bbox": [ + 105, + 637, + 505, + 650 + ], + "score": 1.0, + "content": "for details; we have temporarily moved this Section in the revision due to space constraints, and we", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 85, + 648, + 393, + 661 + ], + "spans": [ + { + "bbox": [ + 85, + 649, + 101, + 660 + ], + "score": 1.0, + "content": "157", + "type": "text" + }, + { + "bbox": [ + 105, + 648, + 393, + 661 + ], + "score": 1.0, + "content": "will move this Section into the main paper in the camera-ready version.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 41.5 + }, + { + "type": "title", + "bbox": [ + 90, + 675, + 405, + 689 + ], + "lines": [ + { + "bbox": [ + 86, + 673, + 406, + 693 + ], + "spans": [ + { + "bbox": [ + 86, + 673, + 406, + 693 + ], + "score": 1.0, + "content": "158 3 Theoretical Power for Graph Representation Learning", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 44 + }, + { + "type": "text", + "bbox": [ + 90, + 699, + 504, + 723 + ], + "lines": [ + { + "bbox": [ + 88, + 700, + 504, + 712 + ], + "spans": [ + { + "bbox": [ + 88, + 702, + 100, + 711 + ], + "score": 1.0, + "content": "159", + "type": "text" + }, + { + "bbox": [ + 105, + 700, + 504, + 712 + ], + "score": 1.0, + "content": "Next, we establish that our SignNet and BasisNet can compute useful basis invariant and permutation", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 88, + 710, + 504, + 723 + ], + "spans": [ + { + "bbox": [ + 88, + 710, + 504, + 723 + ], + "score": 1.0, + "content": "160 equivariant functions on Laplacian eigenvectors for graph representation learning, including: spectral", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 45.5 + } + ], + "page_idx": 3, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 741, + 308, + 750 + ], + "lines": [ + { + "bbox": [ + 301, + 740, + 310, + 752 + ], + "spans": [ + { + "bbox": [ + 301, + 740, + 310, + 752 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 12, + "width": 9 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 103, + 72, + 505, + 109 + ], + "lines": [ + { + "bbox": [ + 105, + 70, + 506, + 87 + ], + "spans": [ + { + "bbox": [ + 105, + 70, + 235, + 87 + ], + "score": 1.0, + "content": "Proposition 2. Any continuous,", + "type": "text" + }, + { + "bbox": [ + 235, + 73, + 257, + 85 + ], + "score": 0.9, + "content": "O ( d )", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 70, + 297, + 87 + ], + "score": 1.0, + "content": "invariant", + "type": "text" + }, + { + "bbox": [ + 297, + 72, + 361, + 83 + ], + "score": 0.91, + "content": "h : \\mathbb { R } ^ { n \\times d } \\mathbb { R } ^ { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 70, + 417, + 87 + ], + "score": 1.0, + "content": "is of the form", + "type": "text" + }, + { + "bbox": [ + 418, + 72, + 489, + 85 + ], + "score": 0.91, + "content": "h ( V ) = \\phi ( V V ^ { \\top } )", + "type": "inline_equation" + }, + { + "bbox": [ + 490, + 70, + 506, + 87 + ], + "score": 1.0, + "content": "for", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 83, + 505, + 98 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 160, + 98 + ], + "score": 1.0, + "content": "a continuous", + "type": "text" + }, + { + "bbox": [ + 160, + 86, + 167, + 96 + ], + "score": 0.78, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 83, + 264, + 98 + ], + "score": 1.0, + "content": ". For a compact domain", + "type": "text" + }, + { + "bbox": [ + 264, + 84, + 309, + 96 + ], + "score": 0.91, + "content": "{ \\mathcal { Z } } \\subseteq \\mathbb { R } ^ { n \\times d }", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 83, + 383, + 98 + ], + "score": 1.0, + "content": ", maps of the form", + "type": "text" + }, + { + "bbox": [ + 383, + 85, + 457, + 97 + ], + "score": 0.9, + "content": "V \\mapsto \\operatorname { I G N } ( V V ^ { \\top } )", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 83, + 505, + 98 + ], + "score": 1.0, + "content": "universally", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 104, + 94, + 507, + 111 + ], + "spans": [ + { + "bbox": [ + 104, + 94, + 204, + 111 + ], + "score": 1.0, + "content": "approximate continuous", + "type": "text" + }, + { + "bbox": [ + 205, + 96, + 292, + 108 + ], + "score": 0.92, + "content": "h : \\mathcal { Z } \\subseteq \\mathbb { R } ^ { n \\times d } \\to \\mathbb { R } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 292, + 94, + 326, + 111 + ], + "score": 1.0, + "content": "that are", + "type": "text" + }, + { + "bbox": [ + 327, + 96, + 348, + 109 + ], + "score": 0.92, + "content": "O ( d )", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 94, + 507, + 111 + ], + "score": 1.0, + "content": "invariant and permutation equivariant.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1, + "bbox_fs": [ + 104, + 70, + 507, + 111 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 120, + 309, + 132 + ], + "lines": [ + { + "bbox": [ + 105, + 118, + 311, + 135 + ], + "spans": [ + { + "bbox": [ + 105, + 118, + 311, + 135 + ], + "score": 1.0, + "content": "2.2 Neural Networks on Multiple Eigenspaces", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 106, + 140, + 505, + 196 + ], + "lines": [ + { + "bbox": [ + 105, + 140, + 505, + 153 + ], + "spans": [ + { + "bbox": [ + 105, + 140, + 291, + 153 + ], + "score": 1.0, + "content": "Next, we use the single-eigenspace models", + "type": "text" + }, + { + "bbox": [ + 291, + 141, + 318, + 153 + ], + "score": 0.9, + "content": "\\phi _ { l } ( V _ { l } )", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 140, + 505, + 153 + ], + "score": 1.0, + "content": "as building blocks for a model on multiple", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 151, + 505, + 164 + ], + "spans": [ + { + "bbox": [ + 106, + 151, + 312, + 164 + ], + "score": 1.0, + "content": "eigenspaces. 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The form", + "type": "text" + } + ], + "index": 11, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 247, + 505, + 259 + ], + "spans": [ + { + "bbox": [ + 86, + 248, + 100, + 258 + ], + "score": 1.0, + "content": "129", + "type": "text" + }, + { + "bbox": [ + 107, + 247, + 169, + 259 + ], + "score": 0.92, + "content": "\\phi ( v _ { i } ) + \\phi ( - v _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 247, + 505, + 259 + ], + "score": 1.0, + "content": "induces sign invariance for each eigenvector. 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We", + "type": "text" + } + ], + "index": 16, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 306, + 504, + 320 + ], + "spans": [ + { + "bbox": [ + 86, + 309, + 100, + 318 + ], + "score": 1.0, + "content": "134", + "type": "text" + }, + { + "bbox": [ + 105, + 306, + 493, + 320 + ], + "score": 1.0, + "content": "name this permutation equivariant version SignNet. 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Our resulting BasisNet is", + "type": "text" + } + ], + "index": 29, + "is_list_start_line": true + } + ], + "index": 26, + "bbox_fs": [ + 86, + 404, + 506, + 486 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 221, + 488, + 389, + 504 + ], + "lines": [ + { + "bbox": [ + 221, + 488, + 389, + 504 + ], + "spans": [ + { + "bbox": [ + 221, + 488, + 389, + 504 + ], + "score": 0.92, + "content": "f ( V _ { 1 } , \\dots , V _ { l } ) = \\rho \\left( [ \\mathrm { I G N } _ { d _ { i } } ( V _ { i } V _ { i } ^ { \\top } ) ] _ { i = 1 } ^ { l } \\right) .", + "type": "interline_equation", + "image_path": "75f36c163e73caac170519a4b10dc9fcecc8b5ab4618f9981ed9aa5bbc273725.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 221, + 488, + 389, + 504 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "index", + "bbox": [ + 86, + 506, + 505, + 595 + ], + "lines": [ + { + "bbox": [ + 86, + 507, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 86, + 509, + 100, + 518 + ], + "score": 1.0, + "content": "145", + "type": "text" + }, + { + "bbox": [ + 105, + 507, + 505, + 519 + ], + "score": 1.0, + "content": "Expressive-BasisNet. While we restrict SignNet to only use vectors and BasisNet to only use vectors", + "type": "text" + } + ], + "index": 31, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 517, + 506, + 531 + ], + "spans": [ + { + "bbox": [ + 86, + 519, + 100, + 530 + ], + "score": 1.0, + "content": "146", + "type": "text" + }, + { + "bbox": [ + 105, + 517, + 506, + 531 + ], + "score": 1.0, + "content": "and matrices, higher order tensors are generally required for universally approximating permutation", + "type": "text" + } + ], + "index": 32, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 528, + 506, + 541 + ], + "spans": [ + { + "bbox": [ + 86, + 531, + 100, + 540 + ], + "score": 1.0, + "content": "147", + "type": "text" + }, + { + "bbox": [ + 105, + 528, + 506, + 541 + ], + "score": 1.0, + "content": "equivariant or invariant functions [Keriven and Peyré, 2019, Maron et al., 2019, Maehara and NT,", + "type": "text" + } + ], + "index": 33, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 538, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 86, + 542, + 100, + 551 + ], + "score": 1.0, + "content": "148", + "type": "text" + }, + { + "bbox": [ + 105, + 538, + 506, + 552 + ], + "score": 1.0, + "content": "2019]. Thus, we will consider a theoretically powerful but computationally impractical variant of", + "type": "text" + } + ], + "index": 34, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 550, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 86, + 552, + 100, + 563 + ], + "score": 1.0, + "content": "149", + "type": "text" + }, + { + "bbox": [ + 105, + 550, + 237, + 563 + ], + "score": 1.0, + "content": "our model, in which we replace", + "type": "text" + }, + { + "bbox": [ + 237, + 552, + 244, + 562 + ], + "score": 0.8, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 550, + 262, + 563 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 262, + 551, + 290, + 563 + ], + "score": 0.9, + "content": "\\mathrm { I G N } _ { d _ { i } }", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 550, + 505, + 563 + ], + "score": 1.0, + "content": "in BasisNet with IGNs of arbitrary tensor order. We", + "type": "text" + } + ], + "index": 35, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 561, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 86, + 564, + 100, + 573 + ], + "score": 1.0, + "content": "150", + "type": "text" + }, + { + "bbox": [ + 105, + 561, + 399, + 574 + ], + "score": 1.0, + "content": "call this variant Expressive-BasisNet. Universal approximation requires", + "type": "text" + }, + { + "bbox": [ + 400, + 562, + 426, + 574 + ], + "score": 0.92, + "content": "\\Omega ( n ^ { n } )", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 561, + 505, + 574 + ], + "score": 1.0, + "content": "sized intermediate", + "type": "text" + } + ], + "index": 36, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 571, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 86, + 574, + 100, + 585 + ], + "score": 1.0, + "content": "151", + "type": "text" + }, + { + "bbox": [ + 105, + 571, + 505, + 585 + ], + "score": 1.0, + "content": "tensors [Ravanbakhsh, 2020]. We study Expressive-BasisNet due to its theoretical interest, and to", + "type": "text" + } + ], + "index": 37, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 583, + 502, + 596 + ], + "spans": [ + { + "bbox": [ + 86, + 586, + 100, + 595 + ], + "score": 1.0, + "content": "152", + "type": "text" + }, + { + "bbox": [ + 105, + 583, + 502, + 596 + ], + "score": 1.0, + "content": "juxtapose with the computational efficiency and strong expressive power of SignNet and BasisNet.", + "type": "text" + } + ], + "index": 38, + "is_list_start_line": true + } + ], + "index": 34.5, + "bbox_fs": [ + 86, + 507, + 506, + 596 + ] + }, + { + "type": "text", + "bbox": [ + 102, + 600, + 432, + 611 + ], + "lines": [ + { + "bbox": [ + 105, + 599, + 433, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 433, + 613 + ], + "score": 1.0, + "content": "For a summary of properties and more details about our models, see Appendix B.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39, + "bbox_fs": [ + 105, + 599, + 433, + 613 + ] + }, + { + "type": "index", + "bbox": [ + 86, + 616, + 506, + 661 + ], + "lines": [ + { + "bbox": [ + 86, + 615, + 506, + 628 + ], + "spans": [ + { + "bbox": [ + 86, + 618, + 100, + 627 + ], + "score": 1.0, + "content": "154", + "type": "text" + }, + { + "bbox": [ + 105, + 615, + 506, + 628 + ], + "score": 1.0, + "content": "In the multiple subspace case, we can prove universality of our models through a general decomposi-", + "type": "text" + } + ], + "index": 40, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 626, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 85, + 628, + 101, + 639 + ], + "score": 1.0, + "content": "155", + "type": "text" + }, + { + "bbox": [ + 105, + 626, + 505, + 639 + ], + "score": 1.0, + "content": "tion theorem, which reduces the multiple subspace case to the single subspace case. See Section A", + "type": "text" + } + ], + "index": 41, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 637, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 85, + 639, + 100, + 650 + ], + "score": 1.0, + "content": "156", + "type": "text" + }, + { + "bbox": [ + 105, + 637, + 505, + 650 + ], + "score": 1.0, + "content": "for details; we have temporarily moved this Section in the revision due to space constraints, and we", + "type": "text" + } + ], + "index": 42, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 648, + 393, + 661 + ], + "spans": [ + { + "bbox": [ + 85, + 649, + 101, + 660 + ], + "score": 1.0, + "content": "157", + "type": "text" + }, + { + "bbox": [ + 105, + 648, + 393, + 661 + ], + "score": 1.0, + "content": "will move this Section into the main paper in the camera-ready version.", + "type": "text" + } + ], + "index": 43, + "is_list_start_line": true + } + ], + "index": 41.5, + "bbox_fs": [ + 85, + 615, + 506, + 661 + ] + }, + { + "type": "title", + "bbox": [ + 90, + 675, + 405, + 689 + ], + "lines": [ + { + "bbox": [ + 86, + 673, + 406, + 693 + ], + "spans": [ + { + "bbox": [ + 86, + 673, + 406, + 693 + ], + "score": 1.0, + "content": "158 3 Theoretical Power for Graph Representation Learning", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 44 + }, + { + "type": "index", + "bbox": [ + 90, + 699, + 504, + 723 + ], + "lines": [ + { + "bbox": [ + 88, + 700, + 504, + 712 + ], + "spans": [ + { + "bbox": [ + 88, + 702, + 100, + 711 + ], + "score": 1.0, + "content": "159", + "type": "text" + }, + { + "bbox": [ + 105, + 700, + 504, + 712 + ], + "score": 1.0, + "content": "Next, we establish that our SignNet and BasisNet can compute useful basis invariant and permutation", + "type": "text" + } + ], + "index": 45, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 710, + 504, + 723 + ], + "spans": [ + { + "bbox": [ + 88, + 710, + 504, + 723 + ], + "score": 1.0, + "content": "160 equivariant functions on Laplacian eigenvectors for graph representation learning, including: spectral", + "type": "text" + } + ], + "index": 46, + "is_list_start_line": true + } + ], + "index": 45.5, + "bbox_fs": [ + 88, + 700, + 504, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 90, + 73, + 506, + 106 + ], + "lines": [ + { + "bbox": [ + 104, + 72, + 506, + 85 + ], + "spans": [ + { + "bbox": [ + 104, + 72, + 506, + 85 + ], + "score": 1.0, + "content": "graph convolutions, spectral invariants, and existing graph positional encodings. 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This family includes important functions like heat kernels and generalized PageRanks on", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 187, + 506, + 201 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 506, + 201 + ], + "score": 1.0, + "content": "graphs [Li et al., 2019]. A spectral GNN is defined as multiple layers of spectral graph convolutions", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 198, + 505, + 212 + ], + "spans": [ + { + "bbox": [ + 106, + 198, + 237, + 212 + ], + "score": 1.0, + "content": "and node-wise linear maps, e.g.", + "type": "text" + }, + { + "bbox": [ + 237, + 199, + 399, + 212 + ], + "score": 0.67, + "content": "\\begin{array} { r } { V \\mathrm { D i a g } ( \\theta _ { 2 } ) V ^ { \\top } \\sigma \\left( V \\mathrm { D i a g } ( \\bar { \\theta } _ { 1 } ) V ^ { \\top } X W _ { 1 } \\right) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 199, + 414, + 210 + ], + "score": 0.63, + "content": "W _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 198, + 505, + 212 + ], + "score": 1.0, + "content": "is a two layer spectral", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 209, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 505, + 222 + ], + "score": 1.0, + "content": "GNN. It can be seen (in Appendix H.1) that spectral graph convolutions are permutation equivariant", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 221, + 505, + 233 + ], + "spans": [ + { + "bbox": [ + 106, + 221, + 213, + 233 + ], + "score": 1.0, + "content": "and sign invariant, and if", + "type": "text" + }, + { + "bbox": [ + 213, + 221, + 260, + 233 + ], + "score": 0.93, + "content": "\\theta _ { i } = h ( \\lambda _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 221, + 505, + 233 + ], + "score": 1.0, + "content": "(i.e. the spectral graph convolution is parametric) they are", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 231, + 356, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 356, + 244 + ], + "score": 1.0, + "content": "additionally invariant to a change of bases in each eigenspace.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 87, + 248, + 505, + 327 + ], + "lines": [ + { + "bbox": [ + 105, + 247, + 506, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 506, + 261 + ], + "score": 1.0, + "content": "Our SignNet and BasisNet can be viewed as generalizations of spectral graph convolutions, as our", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 259, + 507, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 507, + 272 + ], + "score": 1.0, + "content": "networks can universally approximate all spectral graph convolutions of the above form. For instance,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 103, + 269, + 508, + 289 + ], + "spans": [ + { + "bbox": [ + 103, + 269, + 163, + 289 + ], + "score": 1.0, + "content": "SignNet with", + "type": "text" + }, + { + "bbox": [ + 164, + 270, + 268, + 285 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\rho ( a _ { 1 } , \\ldots , a _ { k } ) = \\sum _ { i = 1 } ^ { k } a _ { k } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 269, + 287, + 289 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 288, + 271, + 394, + 285 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\phi ( v _ { i } , \\lambda _ { i } , X ) = \\frac { 1 } { 2 } \\theta _ { i } v _ { i } v _ { i } ^ { \\top } X } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 269, + 508, + 289 + ], + "score": 1.0, + "content": "directly yields the spectral", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 282, + 506, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 282, + 506, + 297 + ], + "score": 1.0, + "content": "graph convolution. This is captured in Theorem 1, which we prove in Appendix H.1. In fact, we may", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 294, + 505, + 306 + ], + "spans": [ + { + "bbox": [ + 106, + 294, + 505, + 306 + ], + "score": 1.0, + "content": "expect SignNet to learn spectral graph convolutions well, according to the principle of algorithmic", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 304, + 505, + 318 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 505, + 318 + ], + "score": 1.0, + "content": "alignment [Xu et al., 2020] (see Appendix H.1); this is supported by numerical experiments in", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 316, + 503, + 328 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 503, + 328 + ], + "score": 1.0, + "content": "Appendix J.2, in which our networks outperform baselines in learning spectral graph convolutions.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 100, + 332, + 503, + 355 + ], + "lines": [ + { + "bbox": [ + 106, + 332, + 504, + 345 + ], + "spans": [ + { + "bbox": [ + 106, + 332, + 504, + 345 + ], + "score": 1.0, + "content": "Theorem 1. SignNet universally approximates all spectral graph convolutions. BasisNet universally", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 344, + 338, + 356 + ], + "spans": [ + { + "bbox": [ + 106, + 344, + 338, + 356 + ], + "score": 1.0, + "content": "approximates all parametric spectral graph convolutions.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5 + }, + { + "type": "text", + "bbox": [ + 105, + 365, + 505, + 410 + ], + "lines": [ + { + "bbox": [ + 104, + 364, + 506, + 378 + ], + "spans": [ + { + "bbox": [ + 104, + 364, + 506, + 378 + ], + "score": 1.0, + "content": "In fact, SignNet and BasisNet are strictly stronger than spectral graph convolutions; there are functions", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 376, + 506, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 506, + 388 + ], + "score": 1.0, + "content": "computable by SignNet and BasisNet that cannot be approximated by spectral graph convolutions", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 387, + 506, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 506, + 400 + ], + "score": 1.0, + "content": "or spectral GNNs. One way to see this is through graph isomorphism power, as captured in this", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 398, + 153, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 153, + 410 + ], + "score": 1.0, + "content": "next result.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23.5 + }, + { + "type": "text", + "bbox": [ + 101, + 415, + 504, + 438 + ], + "lines": [ + { + "bbox": [ + 105, + 413, + 506, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 506, + 428 + ], + "score": 1.0, + "content": "Proposition 3. There exist infinitely many pairs of non-isomorphic graphs that SignNet and BasisNet", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 425, + 451, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 451, + 438 + ], + "score": 1.0, + "content": "can distinguish, but spectral graph convolutions or spectral GNNs cannot distinguish.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26.5 + }, + { + "type": "title", + "bbox": [ + 105, + 454, + 315, + 466 + ], + "lines": [ + { + "bbox": [ + 105, + 453, + 316, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 316, + 468 + ], + "score": 1.0, + "content": "3.2 BasisNets can Compute Spectral Invariants", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 105, + 475, + 505, + 531 + ], + "lines": [ + { + "bbox": [ + 106, + 475, + 506, + 488 + ], + "spans": [ + { + "bbox": [ + 106, + 475, + 506, + 488 + ], + "score": 1.0, + "content": "Many works measure the expressive power of graph neural networks by comparing their power for", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 486, + 505, + 499 + ], + "spans": [ + { + "bbox": [ + 106, + 486, + 505, + 499 + ], + "score": 1.0, + "content": "testing graph isomorphism [Xu et al., 2019, Sato, 2020], or by comparing their ability to compute", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 496, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 505, + 510 + ], + "score": 1.0, + "content": "certain functions on graphs like subgraph counts [Chen et al., 2020, Tahmasebi et al., 2020]. These", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 509, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 106, + 509, + 419, + 520 + ], + "score": 1.0, + "content": "works often compare GNNs to combinatorial invariants on graphs, especially the", + "type": "text" + }, + { + "bbox": [ + 419, + 509, + 425, + 518 + ], + "score": 0.76, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 509, + 505, + 520 + ], + "score": 1.0, + "content": "-Weisfeiler-Lehman", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 519, + 335, + 532 + ], + "spans": [ + { + "bbox": [ + 106, + 519, + 109, + 532 + ], + "score": 0.0, + "content": "", + "type": "text" + }, + { + "bbox": [ + 110, + 520, + 116, + 530 + ], + "score": 0.59, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 519, + 335, + 532 + ], + "score": 1.0, + "content": "-WL) tests of graph isomorphism [Morris et al., 2021].", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 86, + 536, + 505, + 602 + ], + "lines": [ + { + "bbox": [ + 105, + 536, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 505, + 547 + ], + "score": 1.0, + "content": "While we may also compare with these combinatorial invariants, as other GNN works that use spectral", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 546, + 505, + 559 + ], + "spans": [ + { + "bbox": [ + 104, + 546, + 505, + 559 + ], + "score": 1.0, + "content": "information have done [Beaini et al., 2021], we argue that it is more natural to analyze our networks", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 558, + 506, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 506, + 570 + ], + "score": 1.0, + "content": "in terms of spectral invariants, which are computed from the eigenvalues and eigenvectors of graphs.", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 568, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 505, + 581 + ], + "score": 1.0, + "content": "There is a rich literature of spectral invariants from the fields of spectral graph theory and complexity", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 579, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 505, + 591 + ], + "score": 1.0, + "content": "theory [Cvetkovic et al. ´ , 1997]. A spectral invariant must be invariant to permutations and changes of", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 591, + 369, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 369, + 602 + ], + "score": 1.0, + "content": "basis in each eigenspace, a characteristic shared by our networks.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 36.5 + }, + { + "type": "text", + "bbox": [ + 87, + 606, + 505, + 684 + ], + "lines": [ + { + "bbox": [ + 106, + 607, + 506, + 619 + ], + "spans": [ + { + "bbox": [ + 106, + 607, + 506, + 619 + ], + "score": 1.0, + "content": "The simplest spectral invariant is the multiset of eigenvalues, which we give as input to our networks.", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 616, + 506, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 506, + 631 + ], + "score": 1.0, + "content": "Another widely studied, powerful spectral invariant is the collection of graph angles, which are", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 104, + 627, + 507, + 642 + ], + "spans": [ + { + "bbox": [ + 104, + 627, + 196, + 642 + ], + "score": 1.0, + "content": "defined as the values", + "type": "text" + }, + { + "bbox": [ + 196, + 628, + 274, + 641 + ], + "score": 0.93, + "content": "{ \\alpha _ { i j } } ^ { \\star } = \\| V _ { i } V _ { i } ^ { \\top } e _ { j } \\| _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 627, + 307, + 642 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 307, + 628, + 358, + 640 + ], + "score": 0.93, + "content": "V _ { i } \\in \\mathbb { R } ^ { n \\times d _ { i } }", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 627, + 507, + 642 + ], + "score": 1.0, + "content": "is an orthonormal basis for the ith", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 639, + 506, + 652 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 243, + 652 + ], + "score": 1.0, + "content": "adjacency matrix eigenspace, and", + "type": "text" + }, + { + "bbox": [ + 244, + 642, + 254, + 651 + ], + "score": 0.86, + "content": "e _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 639, + 279, + 652 + ], + "score": 1.0, + "content": "is the", + "type": "text" + }, + { + "bbox": [ + 279, + 641, + 285, + 651 + ], + "score": 0.69, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 639, + 506, + 652 + ], + "score": 1.0, + "content": "th standard basis vector, which is zero besides a one in", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 650, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 106, + 650, + 121, + 662 + ], + "score": 1.0, + "content": "the", + "type": "text" + }, + { + "bbox": [ + 121, + 651, + 127, + 662 + ], + "score": 0.75, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 127, + 650, + 505, + 662 + ], + "score": 1.0, + "content": "th component. These are easily computed by our networks (Appendix H.3), so our networks", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 660, + 505, + 674 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 505, + 674 + ], + "score": 1.0, + "content": "inherit the strength of these invariants. We capture these results in the following theorem, which also", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 672, + 381, + 685 + ], + "spans": [ + { + "bbox": [ + 106, + 672, + 381, + 685 + ], + "score": 1.0, + "content": "lists a few properties that graph angles determine [Cvetkovic´, 1991].", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 43 + }, + { + "type": "text", + "bbox": [ + 102, + 689, + 504, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 505, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 374, + 702 + ], + "score": 1.0, + "content": "Theorem 2. 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This family includes important functions like heat kernels and generalized PageRanks on", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 187, + 506, + 201 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 506, + 201 + ], + "score": 1.0, + "content": "graphs [Li et al., 2019]. A spectral GNN is defined as multiple layers of spectral graph convolutions", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 198, + 505, + 212 + ], + "spans": [ + { + "bbox": [ + 106, + 198, + 237, + 212 + ], + "score": 1.0, + "content": "and node-wise linear maps, e.g.", + "type": "text" + }, + { + "bbox": [ + 237, + 199, + 399, + 212 + ], + "score": 0.67, + "content": "\\begin{array} { r } { V \\mathrm { D i a g } ( \\theta _ { 2 } ) V ^ { \\top } \\sigma \\left( V \\mathrm { D i a g } ( \\bar { \\theta } _ { 1 } ) V ^ { \\top } X W _ { 1 } \\right) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 199, + 414, + 210 + ], + "score": 0.63, + "content": "W _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 198, + 505, + 212 + ], + "score": 1.0, + "content": "is a two layer spectral", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 209, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 505, + 222 + ], + "score": 1.0, + "content": "GNN. It can be seen (in Appendix H.1) that spectral graph convolutions are permutation equivariant", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 221, + 505, + 233 + ], + "spans": [ + { + "bbox": [ + 106, + 221, + 213, + 233 + ], + "score": 1.0, + "content": "and sign invariant, and if", + "type": "text" + }, + { + "bbox": [ + 213, + 221, + 260, + 233 + ], + "score": 0.93, + "content": "\\theta _ { i } = h ( \\lambda _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 221, + 505, + 233 + ], + "score": 1.0, + "content": "(i.e. the spectral graph convolution is parametric) they are", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 231, + 356, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 356, + 244 + ], + "score": 1.0, + "content": "additionally invariant to a change of bases in each eigenspace.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 8, + "bbox_fs": [ + 103, + 142, + 506, + 244 + ] + }, + { + "type": "text", + "bbox": [ + 87, + 248, + 505, + 327 + ], + "lines": [ + { + "bbox": [ + 105, + 247, + 506, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 506, + 261 + ], + "score": 1.0, + "content": "Our SignNet and BasisNet can be viewed as generalizations of spectral graph convolutions, as our", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 259, + 507, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 507, + 272 + ], + "score": 1.0, + "content": "networks can universally approximate all spectral graph convolutions of the above form. For instance,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 103, + 269, + 508, + 289 + ], + "spans": [ + { + "bbox": [ + 103, + 269, + 163, + 289 + ], + "score": 1.0, + "content": "SignNet with", + "type": "text" + }, + { + "bbox": [ + 164, + 270, + 268, + 285 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\rho ( a _ { 1 } , \\ldots , a _ { k } ) = \\sum _ { i = 1 } ^ { k } a _ { k } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 269, + 287, + 289 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 288, + 271, + 394, + 285 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\phi ( v _ { i } , \\lambda _ { i } , X ) = \\frac { 1 } { 2 } \\theta _ { i } v _ { i } v _ { i } ^ { \\top } X } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 269, + 508, + 289 + ], + "score": 1.0, + "content": "directly yields the spectral", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 282, + 506, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 282, + 506, + 297 + ], + "score": 1.0, + "content": "graph convolution. This is captured in Theorem 1, which we prove in Appendix H.1. In fact, we may", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 294, + 505, + 306 + ], + "spans": [ + { + "bbox": [ + 106, + 294, + 505, + 306 + ], + "score": 1.0, + "content": "expect SignNet to learn spectral graph convolutions well, according to the principle of algorithmic", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 304, + 505, + 318 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 505, + 318 + ], + "score": 1.0, + "content": "alignment [Xu et al., 2020] (see Appendix H.1); this is supported by numerical experiments in", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 316, + 503, + 328 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 503, + 328 + ], + "score": 1.0, + "content": "Appendix J.2, in which our networks outperform baselines in learning spectral graph convolutions.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 16, + "bbox_fs": [ + 103, + 247, + 508, + 328 + ] + }, + { + "type": "text", + "bbox": [ + 100, + 332, + 503, + 355 + ], + "lines": [ + { + "bbox": [ + 106, + 332, + 504, + 345 + ], + "spans": [ + { + "bbox": [ + 106, + 332, + 504, + 345 + ], + "score": 1.0, + "content": "Theorem 1. SignNet universally approximates all spectral graph convolutions. BasisNet universally", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 344, + 338, + 356 + ], + "spans": [ + { + "bbox": [ + 106, + 344, + 338, + 356 + ], + "score": 1.0, + "content": "approximates all parametric spectral graph convolutions.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5, + "bbox_fs": [ + 106, + 332, + 504, + 356 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 365, + 505, + 410 + ], + "lines": [ + { + "bbox": [ + 104, + 364, + 506, + 378 + ], + "spans": [ + { + "bbox": [ + 104, + 364, + 506, + 378 + ], + "score": 1.0, + "content": "In fact, SignNet and BasisNet are strictly stronger than spectral graph convolutions; there are functions", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 376, + 506, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 506, + 388 + ], + "score": 1.0, + "content": "computable by SignNet and BasisNet that cannot be approximated by spectral graph convolutions", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 387, + 506, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 506, + 400 + ], + "score": 1.0, + "content": "or spectral GNNs. One way to see this is through graph isomorphism power, as captured in this", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 398, + 153, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 153, + 410 + ], + "score": 1.0, + "content": "next result.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23.5, + "bbox_fs": [ + 104, + 364, + 506, + 410 + ] + }, + { + "type": "text", + "bbox": [ + 101, + 415, + 504, + 438 + ], + "lines": [ + { + "bbox": [ + 105, + 413, + 506, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 506, + 428 + ], + "score": 1.0, + "content": "Proposition 3. There exist infinitely many pairs of non-isomorphic graphs that SignNet and BasisNet", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 425, + 451, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 451, + 438 + ], + "score": 1.0, + "content": "can distinguish, but spectral graph convolutions or spectral GNNs cannot distinguish.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26.5, + "bbox_fs": [ + 105, + 413, + 506, + 438 + ] + }, + { + "type": "title", + "bbox": [ + 105, + 454, + 315, + 466 + ], + "lines": [ + { + "bbox": [ + 105, + 453, + 316, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 316, + 468 + ], + "score": 1.0, + "content": "3.2 BasisNets can Compute Spectral Invariants", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 105, + 475, + 505, + 531 + ], + "lines": [ + { + "bbox": [ + 106, + 475, + 506, + 488 + ], + "spans": [ + { + "bbox": [ + 106, + 475, + 506, + 488 + ], + "score": 1.0, + "content": "Many works measure the expressive power of graph neural networks by comparing their power for", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 486, + 505, + 499 + ], + "spans": [ + { + "bbox": [ + 106, + 486, + 505, + 499 + ], + "score": 1.0, + "content": "testing graph isomorphism [Xu et al., 2019, Sato, 2020], or by comparing their ability to compute", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 496, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 505, + 510 + ], + "score": 1.0, + "content": "certain functions on graphs like subgraph counts [Chen et al., 2020, Tahmasebi et al., 2020]. These", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 509, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 106, + 509, + 419, + 520 + ], + "score": 1.0, + "content": "works often compare GNNs to combinatorial invariants on graphs, especially the", + "type": "text" + }, + { + "bbox": [ + 419, + 509, + 425, + 518 + ], + "score": 0.76, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 509, + 505, + 520 + ], + "score": 1.0, + "content": "-Weisfeiler-Lehman", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 519, + 335, + 532 + ], + "spans": [ + { + "bbox": [ + 106, + 519, + 109, + 532 + ], + "score": 0.0, + "content": "", + "type": "text" + }, + { + "bbox": [ + 110, + 520, + 116, + 530 + ], + "score": 0.59, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 519, + 335, + 532 + ], + "score": 1.0, + "content": "-WL) tests of graph isomorphism [Morris et al., 2021].", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31, + "bbox_fs": [ + 105, + 475, + 506, + 532 + ] + }, + { + "type": "text", + "bbox": [ + 86, + 536, + 505, + 602 + ], + "lines": [ + { + "bbox": [ + 105, + 536, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 505, + 547 + ], + "score": 1.0, + "content": "While we may also compare with these combinatorial invariants, as other GNN works that use spectral", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 546, + 505, + 559 + ], + "spans": [ + { + "bbox": [ + 104, + 546, + 505, + 559 + ], + "score": 1.0, + "content": "information have done [Beaini et al., 2021], we argue that it is more natural to analyze our networks", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 558, + 506, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 506, + 570 + ], + "score": 1.0, + "content": "in terms of spectral invariants, which are computed from the eigenvalues and eigenvectors of graphs.", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 568, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 505, + 581 + ], + "score": 1.0, + "content": "There is a rich literature of spectral invariants from the fields of spectral graph theory and complexity", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 579, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 505, + 591 + ], + "score": 1.0, + "content": "theory [Cvetkovic et al. ´ , 1997]. A spectral invariant must be invariant to permutations and changes of", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 591, + 369, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 369, + 602 + ], + "score": 1.0, + "content": "basis in each eigenspace, a characteristic shared by our networks.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 36.5, + "bbox_fs": [ + 104, + 536, + 506, + 602 + ] + }, + { + "type": "text", + "bbox": [ + 87, + 606, + 505, + 684 + ], + "lines": [ + { + "bbox": [ + 106, + 607, + 506, + 619 + ], + "spans": [ + { + "bbox": [ + 106, + 607, + 506, + 619 + ], + "score": 1.0, + "content": "The simplest spectral invariant is the multiset of eigenvalues, which we give as input to our networks.", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 616, + 506, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 506, + 631 + ], + "score": 1.0, + "content": "Another widely studied, powerful spectral invariant is the collection of graph angles, which are", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 104, + 627, + 507, + 642 + ], + "spans": [ + { + "bbox": [ + 104, + 627, + 196, + 642 + ], + "score": 1.0, + "content": "defined as the values", + "type": "text" + }, + { + "bbox": [ + 196, + 628, + 274, + 641 + ], + "score": 0.93, + "content": "{ \\alpha _ { i j } } ^ { \\star } = \\| V _ { i } V _ { i } ^ { \\top } e _ { j } \\| _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 627, + 307, + 642 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 307, + 628, + 358, + 640 + ], + "score": 0.93, + "content": "V _ { i } \\in \\mathbb { R } ^ { n \\times d _ { i } }", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 627, + 507, + 642 + ], + "score": 1.0, + "content": "is an orthonormal basis for the ith", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 639, + 506, + 652 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 243, + 652 + ], + "score": 1.0, + "content": "adjacency matrix eigenspace, and", + "type": "text" + }, + { + "bbox": [ + 244, + 642, + 254, + 651 + ], + "score": 0.86, + "content": "e _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 639, + 279, + 652 + ], + "score": 1.0, + "content": "is the", + "type": "text" + }, + { + "bbox": [ + 279, + 641, + 285, + 651 + ], + "score": 0.69, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 639, + 506, + 652 + ], + "score": 1.0, + "content": "th standard basis vector, which is zero besides a one in", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 650, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 106, + 650, + 121, + 662 + ], + "score": 1.0, + "content": "the", + "type": "text" + }, + { + "bbox": [ + 121, + 651, + 127, + 662 + ], + "score": 0.75, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 127, + 650, + 505, + 662 + ], + "score": 1.0, + "content": "th component. These are easily computed by our networks (Appendix H.3), so our networks", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 660, + 505, + 674 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 505, + 674 + ], + "score": 1.0, + "content": "inherit the strength of these invariants. We capture these results in the following theorem, which also", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 672, + 381, + 685 + ], + "spans": [ + { + "bbox": [ + 106, + 672, + 381, + 685 + ], + "score": 1.0, + "content": "lists a few properties that graph angles determine [Cvetkovic´, 1991].", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 43, + "bbox_fs": [ + 104, + 607, + 507, + 685 + ] + }, + { + "type": "text", + "bbox": [ + 102, + 689, + 504, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 505, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 374, + 702 + ], + "score": 1.0, + "content": "Theorem 2. 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The eigenvalues and graph", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 700, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 505, + 712 + ], + "score": 1.0, + "content": "angles (and thus BasisNet) can determine the number of length 3, 4, or 5 cycles, whether a graph is", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 711, + 414, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 254, + 723 + ], + "score": 1.0, + "content": "connected, and the number of length", + "type": "text" + }, + { + "bbox": [ + 255, + 712, + 261, + 721 + ], + "score": 0.39, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 711, + 414, + 723 + ], + "score": 1.0, + "content": "closed walks from any vertex to itself.", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 48, + "bbox_fs": [ + 105, + 687, + 505, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 86, + 73, + 505, + 139 + ], + "lines": [ + { + "bbox": [ + 85, + 72, + 506, + 85 + ], + "spans": [ + { + "bbox": [ + 85, + 74, + 99, + 84 + ], + "score": 1.0, + "content": "211", + "type": "text" + }, + { + "bbox": [ + 105, + 72, + 506, + 85 + ], + "score": 1.0, + "content": "Relation to WL and message passing. In contrast to this result, message passing GNNs are not able", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 85, + 82, + 507, + 97 + ], + "spans": [ + { + "bbox": [ + 85, + 85, + 100, + 95 + ], + "score": 1.0, + "content": "212", + "type": "text" + }, + { + "bbox": [ + 104, + 82, + 507, + 97 + ], + "score": 1.0, + "content": "to express any of these properties (see [Arvind et al., 2020, Garg et al., 2020] and Appendix H.3).", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 85, + 95, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 85, + 96, + 100, + 106 + ], + "score": 1.0, + "content": "213", + "type": "text" + }, + { + "bbox": [ + 106, + 95, + 506, + 106 + ], + "score": 1.0, + "content": "Although spectral invariants are strong, Fürer [2010] shows that the eigenvalues and graph angles—as", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 85, + 105, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 85, + 107, + 100, + 117 + ], + "score": 1.0, + "content": "214", + "type": "text" + }, + { + "bbox": [ + 105, + 105, + 506, + 118 + ], + "score": 1.0, + "content": "well as some strictly stronger spectral invariants—are not stronger than the 3-WL test (or, equivalently,", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 85, + 115, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 85, + 118, + 100, + 128 + ], + "score": 1.0, + "content": "215", + "type": "text" + }, + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "score": 1.0, + "content": "the 2-Folklore-WL test). Future work could study the combination of spectral invariants or spectral", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 85, + 127, + 450, + 140 + ], + "spans": [ + { + "bbox": [ + 85, + 128, + 100, + 138 + ], + "score": 1.0, + "content": "216", + "type": "text" + }, + { + "bbox": [ + 105, + 127, + 450, + 140 + ], + "score": 1.0, + "content": "graph positional encodings with combinatorial algorithms and graph neural networks.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5 + }, + { + "type": "title", + "bbox": [ + 103, + 151, + 438, + 163 + ], + "lines": [ + { + "bbox": [ + 105, + 150, + 439, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 150, + 439, + 167 + ], + "score": 1.0, + "content": "3.3 SignNets and BasisNets Generalize Existing Graph Positional Encodings", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 107, + 171, + 505, + 226 + ], + "lines": [ + { + "bbox": [ + 106, + 172, + 505, + 183 + ], + "spans": [ + { + "bbox": [ + 106, + 172, + 505, + 183 + ], + "score": 1.0, + "content": "Many graph positional encodings have been proposed, without any clear criteria on which to choose", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 182, + 505, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 505, + 195 + ], + "score": 1.0, + "content": "for a particular task. We prove (in Appendix H.2) that our efficient SignNet and BasisNet can", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 193, + 505, + 206 + ], + "spans": [ + { + "bbox": [ + 105, + 193, + 505, + 206 + ], + "score": 1.0, + "content": "universally approximate many previously used graph positional encodings, because we unify these", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 204, + 505, + 217 + ], + "spans": [ + { + "bbox": [ + 106, + 204, + 505, + 217 + ], + "score": 1.0, + "content": "positional encodings by expressing them as either a spectral graph convolution matrix or the diagonal", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 215, + 264, + 228 + ], + "spans": [ + { + "bbox": [ + 106, + 215, + 264, + 228 + ], + "score": 1.0, + "content": "of a spectral graph convolution matrix.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 107, + 229, + 505, + 273 + ], + "lines": [ + { + "bbox": [ + 106, + 228, + 505, + 242 + ], + "spans": [ + { + "bbox": [ + 106, + 228, + 505, + 242 + ], + "score": 1.0, + "content": "Proposition 4. SignNet and BasisNet universally approximate node positional encodings based on", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 240, + 504, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 240, + 504, + 252 + ], + "score": 1.0, + "content": "heat kernels [Feldman et al., 2022] and random walks [Dwivedi et al., 2022]. BasisNet universally", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 252, + 506, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 217, + 263 + ], + "score": 1.0, + "content": "approximates diffusion and", + "type": "text" + }, + { + "bbox": [ + 218, + 253, + 224, + 263 + ], + "score": 0.79, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 224, + 252, + 506, + 263 + ], + "score": 1.0, + "content": "-step random walk relative positional encodings [Mialon et al., 2021],", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 262, + 458, + 274 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 458, + 274 + ], + "score": 1.0, + "content": "and generalized PageRank and landing probability distance encodings [Li et al., 2020].", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13.5 + }, + { + "type": "text", + "bbox": [ + 107, + 282, + 505, + 338 + ], + "lines": [ + { + "bbox": [ + 106, + 282, + 505, + 294 + ], + "spans": [ + { + "bbox": [ + 106, + 282, + 505, + 294 + ], + "score": 1.0, + "content": "We note that diagonals of spectral convolutions are used as feature descriptors in the shape analysis", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 292, + 505, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 292, + 505, + 306 + ], + "score": 1.0, + "content": "literature, such as the heat kernel signature [Sun et al., 2009] and wave kernel signature [Aubry", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 303, + 505, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 505, + 317 + ], + "score": 1.0, + "content": "et al., 2011]. In the language of recent works in graph machine learning, these are node positional", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 314, + 505, + 328 + ], + "spans": [ + { + "bbox": [ + 105, + 314, + 505, + 328 + ], + "score": 1.0, + "content": "encodings computed from a discrete Laplacian of a triangle mesh. This connection appears to be", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 325, + 336, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 336, + 339 + ], + "score": 1.0, + "content": "unnoticed in recent works on graph positional encodings.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18 + }, + { + "type": "title", + "bbox": [ + 106, + 352, + 191, + 366 + ], + "lines": [ + { + "bbox": [ + 104, + 351, + 193, + 369 + ], + "spans": [ + { + "bbox": [ + 104, + 351, + 193, + 369 + ], + "score": 1.0, + "content": "4 Experiments", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 106, + 377, + 504, + 399 + ], + "lines": [ + { + "bbox": [ + 106, + 376, + 505, + 389 + ], + "spans": [ + { + "bbox": [ + 106, + 376, + 505, + 389 + ], + "score": 1.0, + "content": "We demonstrate the strength of our networks in various experiments. 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Base modelPositional encodingk#paramTest MAE (↓)
GatedGCNNo PEN/A492k0.252±0.007
LapPE (flip)8492k0.198±0.011
LapPE (abs.)8492k0.204±0.009
LapPE (can.)8505k0.298±0.019
SignNet (𝜙(u) only)8495k0.148±0.007
SignNet8495k0.121±0.005
SignNetAll491k0.100±0.007
Sparse TransformerNo PEN/A473k0.283±0.030
LapPE (flip)16487k0.223±0.007
SignNet16479k0.115±0.008
SignNetAll486k0.102±0.005
GINENo PEN/A470k0.170±0.002
LapPE (flip)16470k0.178±0.004
SignNet16470k0.147±0.005
SignNetAll417k0.102±0.002
PNANo PEN/A474k
LapPE (flip)8474k0.133±0.011 0.132±0.010
SignNet8476k0.105±0.007
SignNetAll487k0.084±0.006
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SignNet and BasisNet universally approximate node positional encodings based on", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 240, + 504, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 240, + 504, + 252 + ], + "score": 1.0, + "content": "heat kernels [Feldman et al., 2022] and random walks [Dwivedi et al., 2022]. 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In the language of recent works in graph machine learning, these are node positional", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 314, + 505, + 328 + ], + "spans": [ + { + "bbox": [ + 105, + 314, + 505, + 328 + ], + "score": 1.0, + "content": "encodings computed from a discrete Laplacian of a triangle mesh. This connection appears to be", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 325, + 336, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 336, + 339 + ], + "score": 1.0, + "content": "unnoticed in recent works on graph positional encodings.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18, + "bbox_fs": [ + 105, + 282, + 505, + 339 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 352, + 191, + 366 + ], + "lines": [ + { + "bbox": [ + 104, + 351, + 193, + 369 + ], + "spans": [ + { + "bbox": [ + 104, + 351, + 193, + 369 + ], + "score": 1.0, + "content": "4 Experiments", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 106, + 377, + 504, + 399 + ], + "lines": [ + { + "bbox": [ + 106, + 376, + 505, + 389 + ], + "spans": [ + { + "bbox": [ + 106, + 376, + 505, + 389 + ], + "score": 1.0, + "content": "We demonstrate the strength of our networks in various experiments. Appendix B shows simple", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 388, + 455, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 455, + 402 + ], + "score": 1.0, + "content": "pseudo-code and a diagram detailing the use of SignNet as a node positional encoding.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5, + "bbox_fs": [ + 105, + 376, + 505, + 402 + ] + }, + { + "type": "title", + "bbox": [ + 93, + 412, + 207, + 424 + ], + "lines": [ + { + "bbox": [ + 88, + 410, + 209, + 426 + ], + "spans": [ + { + "bbox": [ + 88, + 410, + 209, + 426 + ], + "score": 1.0, + "content": "235 4.1 Graph Regression", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "table", + "bbox": [ + 153, + 465, + 453, + 694 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 105, + 443, + 505, + 466 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 443, + 506, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 286, + 456 + ], + "score": 1.0, + "content": "Table 1: Results on the ZINC dataset with a", + "type": "text" + }, + { + "bbox": [ + 286, + 444, + 308, + 455 + ], + "score": 0.61, + "content": "5 0 0 \\mathrm { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 443, + 506, + 456 + ], + "score": 1.0, + "content": "parameter budget. 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Base modelPositional encodingk#paramTest MAE (↓)
GatedGCNNo PEN/A492k0.252±0.007
LapPE (flip)8492k0.198±0.011
LapPE (abs.)8492k0.204±0.009
LapPE (can.)8505k0.298±0.019
SignNet (𝜙(u) only)8495k0.148±0.007
SignNet8495k0.121±0.005
SignNetAll491k0.100±0.007
Sparse TransformerNo PEN/A473k0.283±0.030
LapPE (flip)16487k0.223±0.007
SignNet16479k0.115±0.008
SignNetAll486k0.102±0.005
GINENo PEN/A470k0.170±0.002
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SignNet16470k0.147±0.005
SignNetAll417k0.102±0.002
PNANo PEN/A474k
LapPE (flip)8474k0.133±0.011 0.132±0.010
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ZINC (10K)↓ZINC-full ↓Alchemy (10k)↓
HIMP † [Fey et al., 2020].151±.006.036±.002
CIN-small † [Bodnar et al., 2021].094±.004.044±.003
CIN† [Bodnar et al., 2021].079±.006.022±.002
GIN [Xu et al., 2019].170±.002.088±.002.180±.006
δ-2-GNN[Morris et al., 2020b].374±.022.042±.003.118±.001
δ-2-LGNN[Morris etal., 2020b].306±.044.045±.006.122±.003
SpeqNet [Morris et al., 2022].115±.001
GNN-IR [Dupty and Lee, 2022].137±.010.119±.002
PF-GNN [Dupty et al., 2021].122±.01.111±.01
Recon-GNN [Cotta et al., 2021].170±.006.125±.001
SignNet (ours).084±.006.024±.003.113±.002
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The", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 85, + 389, + 474, + 401 + ], + "spans": [ + { + "bbox": [ + 85, + 391, + 100, + 401 + ], + "score": 1.0, + "content": "248", + "type": "text" + }, + { + "bbox": [ + 105, + 389, + 474, + 401 + ], + "score": 1.0, + "content": "total number of parameters of the SignNet and the base model is kept within a 500k budget.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 86, + 405, + 506, + 580 + ], + "lines": [ + { + "bbox": [ + 86, + 405, + 505, + 417 + ], + "spans": [ + { + "bbox": [ + 86, + 407, + 99, + 416 + ], + "score": 1.0, + "content": "249", + "type": "text" + }, + { + "bbox": [ + 106, + 405, + 505, + 417 + ], + "score": 1.0, + "content": "Table 1 shows the results. 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Although the resulting architecture is no", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 86, + 503, + 506, + 516 + ], + "spans": [ + { + "bbox": [ + 86, + 506, + 99, + 514 + ], + "score": 1.0, + "content": "258", + "type": "text" + }, + { + "bbox": [ + 105, + 503, + 191, + 516 + ], + "score": 1.0, + "content": "longer sign invariant,", + "type": "text" + }, + { + "bbox": [ + 192, + 504, + 199, + 515 + ], + "score": 0.85, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 503, + 506, + 516 + ], + "score": 1.0, + "content": "still processes eigenvectors independently, meaning that only two invariances", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 85, + 513, + 506, + 527 + ], + "spans": [ + { + "bbox": [ + 85, + 516, + 100, + 526 + ], + "score": 1.0, + "content": "259", + "type": "text" + }, + { + "bbox": [ + 106, + 515, + 127, + 525 + ], + "score": 0.77, + "content": "( \\pm 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 127, + 513, + 305, + 527 + ], + "score": 1.0, + "content": "need be learned, significantly fewer than the", + "type": "text" + }, + { + "bbox": [ + 305, + 514, + 316, + 524 + ], + "score": 0.86, + "content": "2 ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 513, + 506, + 527 + ], + "score": 1.0, + "content": "total sign flip configurations. Accordingly, this", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 86, + 524, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 86, + 528, + 99, + 537 + ], + "score": 1.0, + "content": "260", + "type": "text" + }, + { + "bbox": [ + 105, + 524, + 505, + 538 + ], + "score": 1.0, + "content": "non-sign invariant learned positional encoding achieves a test MAE of 0.148, improving over the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 85, + 535, + 506, + 549 + ], + "spans": [ + { + "bbox": [ + 85, + 538, + 99, + 548 + ], + "score": 1.0, + "content": "261", + "type": "text" + }, + { + "bbox": [ + 105, + 535, + 506, + 549 + ], + "score": 1.0, + "content": "Laplacian PE (0.198) but falling short of the fully sign invariant SignNet (0.121). In all cases, using", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 85, + 546, + 506, + 560 + ], + "spans": [ + { + "bbox": [ + 85, + 549, + 100, + 559 + ], + "score": 1.0, + "content": "262", + "type": "text" + }, + { + "bbox": [ + 105, + 546, + 506, + 560 + ], + "score": 1.0, + "content": "all available eigenvectors in SignNet significantly improves performance over using a fixed number", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 85, + 556, + 506, + 572 + ], + "spans": [ + { + "bbox": [ + 85, + 560, + 100, + 570 + ], + "score": 1.0, + "content": "263", + "type": "text" + }, + { + "bbox": [ + 104, + 556, + 506, + 572 + ], + "score": 1.0, + "content": "of eigenvectors. In Appendix J.1, we also show that SignNet improves performance when no edge", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 85, + 569, + 238, + 581 + ], + "spans": [ + { + "bbox": [ + 85, + 570, + 101, + 581 + ], + "score": 1.0, + "content": "264", + "type": "text" + }, + { + "bbox": [ + 105, + 569, + 238, + 581 + ], + "score": 1.0, + "content": "features are included in the data.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 23.5 + }, + { + "type": "text", + "bbox": [ + 93, + 585, + 505, + 651 + ], + "lines": [ + { + "bbox": [ + 91, + 584, + 507, + 597 + ], + "spans": [ + { + "bbox": [ + 91, + 587, + 100, + 596 + ], + "score": 1.0, + "content": "65", + "type": "text" + }, + { + "bbox": [ + 105, + 584, + 507, + 597 + ], + "score": 1.0, + "content": "These significant performance improvements from SignNet come with only a slightly higher compu-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 91, + 595, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 91, + 599, + 99, + 607 + ], + "score": 1.0, + "content": "66", + "type": "text" + }, + { + "bbox": [ + 105, + 595, + 505, + 608 + ], + "score": 1.0, + "content": "tational cost. For example, GatedGCN with no PE takes about 8.2 seconds per training iteration on", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 91, + 606, + 506, + 619 + ], + "spans": [ + { + "bbox": [ + 91, + 610, + 99, + 618 + ], + "score": 1.0, + "content": "67", + "type": "text" + }, + { + "bbox": [ + 105, + 606, + 506, + 619 + ], + "score": 1.0, + "content": "ZINC, while GatedGCN with 8 eigenvectors and SignNet takes about 10.6 seconds; this is only a", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 91, + 617, + 505, + 630 + ], + "spans": [ + { + "bbox": [ + 91, + 621, + 99, + 628 + ], + "score": 1.0, + "content": "68", + "type": "text" + }, + { + "bbox": [ + 106, + 618, + 126, + 628 + ], + "score": 0.85, + "content": "2 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 126, + 617, + 338, + 630 + ], + "score": 1.0, + "content": "increase in time, for a reduction of test MAE by over", + "type": "text" + }, + { + "bbox": [ + 338, + 618, + 357, + 628 + ], + "score": 0.85, + "content": "50 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 617, + 505, + 630 + ], + "score": 1.0, + "content": ". 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In Table 2, we compare SignNet with state-of-the-art methods on graph-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 665, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 506, + 680 + ], + "score": 1.0, + "content": "level molecular regression tasks on ZINC (10,000 training graphs), ZINC-full (about 250,000 graphs),", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 678, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 506, + 690 + ], + "score": 1.0, + "content": "and Alchemy [Chen et al., 2019a] (10,000 training graphs). We compare against both methods that", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 689, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 506, + 700 + ], + "score": 1.0, + "content": "use domain-specific knowledge about molecules, and domain-agnostic GNNs of various architectures.", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 700, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 506, + 712 + ], + "score": 1.0, + "content": "We see that SignNet outperforms all domain-agnostic methods on ZINC and ZINC-full, and is within", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 711, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 505, + 722 + ], + "score": 1.0, + "content": "a standard deviation of the best domain-specific method. 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Numbers are test MAE, so lower is better. Best models within a standard deviation are bolded.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "table_body", + "bbox": [ + 140, + 100, + 468, + 243 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 140, + 100, + 468, + 243 + ], + "spans": [ + { + "bbox": [ + 140, + 100, + 468, + 243 + ], + "score": 0.983, + "html": "
ZINC (10K)↓ZINC-full ↓Alchemy (10k)↓
HIMP † [Fey et al., 2020].151±.006.036±.002
CIN-small † [Bodnar et al., 2021].094±.004.044±.003
CIN† [Bodnar et al., 2021].079±.006.022±.002
GIN [Xu et al., 2019].170±.002.088±.002.180±.006
δ-2-GNN[Morris et al., 2020b].374±.022.042±.003.118±.001
δ-2-LGNN[Morris etal., 2020b].306±.044.045±.006.122±.003
SpeqNet [Morris et al., 2022].115±.001
GNN-IR [Dupty and Lee, 2022].137±.010.119±.002
PF-GNN [Dupty et al., 2021].122±.01.111±.01
Recon-GNN [Cotta et al., 2021].170±.006.125±.001
SignNet (ours).084±.006.024±.003.113±.002
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We", + "type": "text" + } + ], + "index": 8, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 316, + 504, + 329 + ], + "spans": [ + { + "bbox": [ + 86, + 319, + 99, + 328 + ], + "score": 1.0, + "content": "242", + "type": "text" + }, + { + "bbox": [ + 105, + 316, + 234, + 329 + ], + "score": 1.0, + "content": "parameterize SignNet by taking", + "type": "text" + }, + { + "bbox": [ + 234, + 317, + 241, + 328 + ], + "score": 0.84, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 316, + 291, + 329 + ], + "score": 1.0, + "content": "to be a GIN", + "type": "text" + }, + { + "bbox": [ + 292, + 317, + 308, + 327 + ], + "score": 0.25, + "content": "\\mathrm { [ X u }", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 316, + 375, + 329 + ], + "score": 1.0, + "content": "et al., 2019] and", + "type": "text" + }, + { + "bbox": [ + 375, + 318, + 382, + 328 + ], + "score": 0.82, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 316, + 496, + 329 + ], + "score": 1.0, + "content": "to be an MLP. 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The", + "type": "text" + } + ], + "index": 14, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 389, + 474, + 401 + ], + "spans": [ + { + "bbox": [ + 85, + 391, + 100, + 401 + ], + "score": 1.0, + "content": "248", + "type": "text" + }, + { + "bbox": [ + 105, + 389, + 474, + 401 + ], + "score": 1.0, + "content": "total number of parameters of the SignNet and the base model is kept within a 500k budget.", + "type": "text" + } + ], + "index": 15, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 405, + 505, + 417 + ], + "spans": [ + { + "bbox": [ + 86, + 407, + 99, + 416 + ], + "score": 1.0, + "content": "249", + "type": "text" + }, + { + "bbox": [ + 106, + 405, + 505, + 417 + ], + "score": 1.0, + "content": "Table 1 shows the results. For all 4 base models, the PE learned with SignNet yields the best test MAE", + "type": "text" + } + ], + "index": 16, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 416, + 506, + 429 + ], + "spans": [ + { + "bbox": [ + 85, + 418, + 100, + 428 + ], + "score": 1.0, + "content": "250", + "type": "text" + }, + { + "bbox": [ + 105, + 416, + 506, + 429 + ], + "score": 1.0, + "content": "(mean absolute error) — lower MAE is better. Notably, this includes the cases of PNA and GINE, for", + "type": "text" + } + ], + "index": 17, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 426, + 506, + 441 + ], + "spans": [ + { + "bbox": [ + 86, + 429, + 99, + 438 + ], + "score": 1.0, + "content": "251", + "type": "text" + }, + { + "bbox": [ + 105, + 426, + 506, + 441 + ], + "score": 1.0, + "content": "which Laplacian PE with simple random sign flipping was unable to improve performance over using", + "type": "text" + } + ], + "index": 18, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 437, + 506, + 451 + ], + "spans": [ + { + "bbox": [ + 86, + 440, + 100, + 449 + ], + "score": 1.0, + "content": "252", + "type": "text" + }, + { + "bbox": [ + 105, + 437, + 506, + 451 + ], + "score": 1.0, + "content": "no PE at all. Our best performing model is PNA base combined with SignNet, which achieves 0.084", + "type": "text" + } + ], + "index": 19, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 448, + 505, + 461 + ], + "spans": [ + { + "bbox": [ + 85, + 451, + 100, + 461 + ], + "score": 1.0, + "content": "253", + "type": "text" + }, + { + "bbox": [ + 105, + 448, + 505, + 461 + ], + "score": 1.0, + "content": "test MAE. Besides SignNet, we consider two non-learned approaches to resolving eigenvector sign", + "type": "text" + } + ], + "index": 20, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 460, + 506, + 471 + ], + "spans": [ + { + "bbox": [ + 85, + 461, + 100, + 471 + ], + "score": 1.0, + "content": "254", + "type": "text" + }, + { + "bbox": [ + 106, + 460, + 506, + 471 + ], + "score": 1.0, + "content": "ambiguity—canonicalization and taking element-wise absolute values (see Appendix K.2 for details).", + "type": "text" + } + ], + "index": 21, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 469, + 506, + 485 + ], + "spans": [ + { + "bbox": [ + 85, + 472, + 100, + 483 + ], + "score": 1.0, + "content": "255", + "type": "text" + }, + { + "bbox": [ + 104, + 469, + 506, + 485 + ], + "score": 1.0, + "content": "Results with GatedGCN show that these alternatives are not more effective than random sign flipping", + "type": "text" + } + ], + "index": 22, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 481, + 506, + 494 + ], + "spans": [ + { + "bbox": [ + 85, + 483, + 100, + 493 + ], + "score": 1.0, + "content": "256", + "type": "text" + }, + { + "bbox": [ + 105, + 481, + 506, + 494 + ], + "score": 1.0, + "content": "for learning positional encodings. We also consider an ablation of our SignNet architecture where we", + "type": "text" + } + ], + "index": 23, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 491, + 507, + 507 + ], + "spans": [ + { + "bbox": [ + 86, + 495, + 99, + 504 + ], + "score": 1.0, + "content": "257", + "type": "text" + }, + { + "bbox": [ + 104, + 491, + 268, + 507 + ], + "score": 1.0, + "content": "remove the sign invariance, using simply", + "type": "text" + }, + { + "bbox": [ + 269, + 492, + 340, + 505 + ], + "score": 0.91, + "content": "\\mathrm { M L P } ( [ \\phi ( v _ { i } ) ] _ { i = 1 } ^ { k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 491, + 507, + 507 + ], + "score": 1.0, + "content": ". Although the resulting architecture is no", + "type": "text" + } + ], + "index": 24, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 503, + 506, + 516 + ], + "spans": [ + { + "bbox": [ + 86, + 506, + 99, + 514 + ], + "score": 1.0, + "content": "258", + "type": "text" + }, + { + "bbox": [ + 105, + 503, + 191, + 516 + ], + "score": 1.0, + "content": "longer sign invariant,", + "type": "text" + }, + { + "bbox": [ + 192, + 504, + 199, + 515 + ], + "score": 0.85, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 503, + 506, + 516 + ], + "score": 1.0, + "content": "still processes eigenvectors independently, meaning that only two invariances", + "type": "text" + } + ], + "index": 25, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 513, + 506, + 527 + ], + "spans": [ + { + "bbox": [ + 85, + 516, + 100, + 526 + ], + "score": 1.0, + "content": "259", + "type": "text" + }, + { + "bbox": [ + 106, + 515, + 127, + 525 + ], + "score": 0.77, + "content": "( \\pm 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 127, + 513, + 305, + 527 + ], + "score": 1.0, + "content": "need be learned, significantly fewer than the", + "type": "text" + }, + { + "bbox": [ + 305, + 514, + 316, + 524 + ], + "score": 0.86, + "content": "2 ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 513, + 506, + 527 + ], + "score": 1.0, + "content": "total sign flip configurations. Accordingly, this", + "type": "text" + } + ], + "index": 26, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 524, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 86, + 528, + 99, + 537 + ], + "score": 1.0, + "content": "260", + "type": "text" + }, + { + "bbox": [ + 105, + 524, + 505, + 538 + ], + "score": 1.0, + "content": "non-sign invariant learned positional encoding achieves a test MAE of 0.148, improving over the", + "type": "text" + } + ], + "index": 27, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 535, + 506, + 549 + ], + "spans": [ + { + "bbox": [ + 85, + 538, + 99, + 548 + ], + "score": 1.0, + "content": "261", + "type": "text" + }, + { + "bbox": [ + 105, + 535, + 506, + 549 + ], + "score": 1.0, + "content": "Laplacian PE (0.198) but falling short of the fully sign invariant SignNet (0.121). In all cases, using", + "type": "text" + } + ], + "index": 28, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 546, + 506, + 560 + ], + "spans": [ + { + "bbox": [ + 85, + 549, + 100, + 559 + ], + "score": 1.0, + "content": "262", + "type": "text" + }, + { + "bbox": [ + 105, + 546, + 506, + 560 + ], + "score": 1.0, + "content": "all available eigenvectors in SignNet significantly improves performance over using a fixed number", + "type": "text" + } + ], + "index": 29, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 556, + 506, + 572 + ], + "spans": [ + { + "bbox": [ + 85, + 560, + 100, + 570 + ], + "score": 1.0, + "content": "263", + "type": "text" + }, + { + "bbox": [ + 104, + 556, + 506, + 572 + ], + "score": 1.0, + "content": "of eigenvectors. In Appendix J.1, we also show that SignNet improves performance when no edge", + "type": "text" + } + ], + "index": 30, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 569, + 238, + 581 + ], + "spans": [ + { + "bbox": [ + 85, + 570, + 101, + 581 + ], + "score": 1.0, + "content": "264", + "type": "text" + }, + { + "bbox": [ + 105, + 569, + 238, + 581 + ], + "score": 1.0, + "content": "features are included in the data.", + "type": "text" + } + ], + "index": 31, + "is_list_start_line": true + }, + { + "bbox": [ + 91, + 584, + 507, + 597 + ], + "spans": [ + { + "bbox": [ + 91, + 587, + 100, + 596 + ], + "score": 1.0, + "content": "65", + "type": "text" + }, + { + "bbox": [ + 105, + 584, + 507, + 597 + ], + "score": 1.0, + "content": "These significant performance improvements from SignNet come with only a slightly higher compu-", + "type": "text" + } + ], + "index": 32, + "is_list_start_line": true + }, + { + "bbox": [ + 91, + 595, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 91, + 599, + 99, + 607 + ], + "score": 1.0, + "content": "66", + "type": "text" + }, + { + "bbox": [ + 105, + 595, + 505, + 608 + ], + "score": 1.0, + "content": "tational cost. For example, GatedGCN with no PE takes about 8.2 seconds per training iteration on", + "type": "text" + } + ], + "index": 33, + "is_list_start_line": true + }, + { + "bbox": [ + 91, + 606, + 506, + 619 + ], + "spans": [ + { + "bbox": [ + 91, + 610, + 99, + 618 + ], + "score": 1.0, + "content": "67", + "type": "text" + }, + { + "bbox": [ + 105, + 606, + 506, + 619 + ], + "score": 1.0, + "content": "ZINC, while GatedGCN with 8 eigenvectors and SignNet takes about 10.6 seconds; this is only a", + "type": "text" + } + ], + "index": 34, + "is_list_start_line": true + }, + { + "bbox": [ + 91, + 617, + 505, + 630 + ], + "spans": [ + { + "bbox": [ + 91, + 621, + 99, + 628 + ], + "score": 1.0, + "content": "68", + "type": "text" + }, + { + "bbox": [ + 106, + 618, + 126, + 628 + ], + "score": 0.85, + "content": "2 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 126, + 617, + 338, + 630 + ], + "score": 1.0, + "content": "increase in time, for a reduction of test MAE by over", + "type": "text" + }, + { + "bbox": [ + 338, + 618, + 357, + 628 + ], + "score": 0.85, + "content": "50 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 617, + 505, + 630 + ], + "score": 1.0, + "content": ". Also, eigenvector computation time", + "type": "text" + } + ], + "index": 35, + "is_list_start_line": true + }, + { + "bbox": [ + 91, + 628, + 506, + 641 + ], + "spans": [ + { + "bbox": [ + 91, + 632, + 100, + 640 + ], + "score": 1.0, + "content": "69", + "type": "text" + }, + { + "bbox": [ + 105, + 628, + 506, + 641 + ], + "score": 1.0, + "content": "is neglible, we need only precompute and save the eigenvectors once, and it only takes 15 seconds to", + "type": "text" + } + ], + "index": 36, + "is_list_start_line": true + }, + { + "bbox": [ + 92, + 639, + 262, + 651 + ], + "spans": [ + { + "bbox": [ + 92, + 642, + 99, + 650 + ], + "score": 1.0, + "content": "70", + "type": "text" + }, + { + "bbox": [ + 105, + 639, + 262, + 651 + ], + "score": 1.0, + "content": "do this for the 12,000 graphs of ZINC.", + "type": "text" + } + ], + "index": 37, + "is_list_start_line": true + } + ], + "index": 10, + "bbox_fs": [ + 85, + 273, + 507, + 401 + ] + }, + { + "type": "index", + "bbox": [ + 86, + 405, + 506, + 580 + ], + "lines": [], + "index": 23.5, + "bbox_fs": [ + 85, + 405, + 507, + 581 + ], + "lines_deleted": true + }, + { + "type": "index", + "bbox": [ + 93, + 585, + 505, + 651 + ], + "lines": [], + "index": 34.5, + "bbox_fs": [ + 91, + 584, + 507, + 651 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 103, + 656, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 656, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 656, + 506, + 668 + ], + "score": 1.0, + "content": "Comparison with SOTA. In Table 2, we compare SignNet with state-of-the-art methods on graph-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 665, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 506, + 680 + ], + "score": 1.0, + "content": "level molecular regression tasks on ZINC (10,000 training graphs), ZINC-full (about 250,000 graphs),", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 678, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 506, + 690 + ], + "score": 1.0, + "content": "and Alchemy [Chen et al., 2019a] (10,000 training graphs). We compare against both methods that", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 689, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 506, + 700 + ], + "score": 1.0, + "content": "use domain-specific knowledge about molecules, and domain-agnostic GNNs of various architectures.", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 700, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 506, + 712 + ], + "score": 1.0, + "content": "We see that SignNet outperforms all domain-agnostic methods on ZINC and ZINC-full, and is within", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 711, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 505, + 722 + ], + "score": 1.0, + "content": "a standard deviation of the best domain-specific method. Our mean score is the second best on", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 40.5, + "bbox_fs": [ + 105, + 656, + 506, + 722 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 113, + 100, + 494, + 177 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 78, + 503, + 100 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 77, + 505, + 90 + ], + "spans": [ + { + "bbox": [ + 105, + 77, + 505, + 90 + ], + "score": 1.0, + "content": "Table 3: Test results for texture reconstruction experiment on cat and human models, following the", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 87, + 505, + 101 + ], + "spans": [ + { + "bbox": [ + 105, + 87, + 505, + 101 + ], + "score": 1.0, + "content": "experimental setting of [Koestler et al., 2022]. 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CatHuman
MethodParamsPSNR↑DSSIM↓LPIPS↓PSNR↑DSSIM↓LPIPS↓
Intrinsic NF329k34.25.099.18932.29.119.330
Absolute value329k34.67.106.25232.42.132.363
Sign flip329k23.151.282.3521.521.052.71
SignNet324k34.91.090.14732.43.125.316
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With Laplacian", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 374, + 506, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 506, + 388 + ], + "score": 1.0, + "content": "PEs, SignNet improves performance, while sign flip data augmentation (LapPE) is less consistent.", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 385, + 505, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 505, + 399 + ], + "score": 1.0, + "content": "Mean and standard deviations are reported on 3 runs. 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CatHuman
MethodParamsPSNR↑DSSIM↓LPIPS↓PSNR↑DSSIM↓LPIPS↓
Intrinsic NF329k34.25.099.18932.29.119.330
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Sign flip329k23.151.282.3521.521.052.71
SignNet324k34.91.090.14732.43.125.316
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With Laplacian", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 374, + 506, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 506, + 388 + ], + "score": 1.0, + "content": "PEs, SignNet improves performance, while sign flip data augmentation (LapPE) is less consistent.", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 385, + 505, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 505, + 399 + ], + "score": 1.0, + "content": "Mean and standard deviations are reported on 3 runs. All runs use the same 4-layer GIN base model.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13 + } + ], + "index": 10 + }, + { + "type": "index", + "bbox": [ + 86, + 411, + 505, + 510 + ], + "lines": [], + "index": 19, + "bbox_fs": [ + 85, + 410, + 507, + 511 + ], + "lines_deleted": true + }, + { + "type": "title", + "bbox": [ + 90, + 529, + 246, + 541 + ], + "lines": [ + { + "bbox": [ + 86, + 529, + 247, + 542 + ], + "spans": [ + { + "bbox": [ + 86, + 529, + 247, + 542 + ], + "score": 1.0, + "content": "290 4.3 Neural Fields on Manifolds", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 105, + 552, + 505, + 630 + ], + "lines": [ + { + "bbox": [ + 105, + 552, + 506, + 566 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 506, + 566 + ], + "score": 1.0, + "content": "Discrete approximations to the Laplace-Beltrami operator on manifolds have proven useful for", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 563, + 505, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 505, + 576 + ], + "score": 1.0, + "content": "processing data on surfaces, such as triangle meshes [Lévy, 2006]. Recently, Koestler et al. [2022]", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 574, + 506, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 506, + 587 + ], + "score": 1.0, + "content": "propose intrinsic neural fields, which use eigenfunctions of the Laplace-Beltrami operator as positional", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 584, + 506, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 584, + 448, + 599 + ], + "score": 1.0, + "content": "encodings for learning neural fields on manifolds. For generalized eigenfunctions", + "type": "text" + }, + { + "bbox": [ + 448, + 587, + 490, + 596 + ], + "score": 0.89, + "content": "v _ { 1 } , \\ldots , v _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 491, + 584, + 506, + 599 + ], + "score": 1.0, + "content": ", at", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 596, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 138, + 609 + ], + "score": 1.0, + "content": "a point", + "type": "text" + }, + { + "bbox": [ + 138, + 598, + 145, + 608 + ], + "score": 0.78, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 596, + 328, + 609 + ], + "score": 1.0, + "content": "on the surface, they parameterize functions", + "type": "text" + }, + { + "bbox": [ + 328, + 596, + 460, + 608 + ], + "score": 0.91, + "content": "f ( \\boldsymbol { p } ) = \\mathrm { M L P } ( v _ { 1 } ( \\boldsymbol { p } ) , \\dots , v _ { k } ( \\boldsymbol { p } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 596, + 505, + 609 + ], + "score": 1.0, + "content": ". As these", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 606, + 506, + 620 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 388, + 620 + ], + "score": 1.0, + "content": "eigenfunctions have sign ambiguity, we use our SignNet to parameterize", + "type": "text" + }, + { + "bbox": [ + 389, + 607, + 506, + 619 + ], + "score": 0.85, + "content": "f ( \\boldsymbol { p } ) ^ { \\prime } = \\mathrm { M L P } ( \\rho ( \\operatorname { } [ \\phi ( v _ { i } ( \\boldsymbol { p } ) ) +", + "type": "inline_equation" + } + ], + "index": 30 + }, + { + "bbox": [ + 107, + 617, + 305, + 631 + ], + "spans": [ + { + "bbox": [ + 107, + 618, + 187, + 631 + ], + "score": 0.9, + "content": "\\phi \\bar { ( - v _ { i } ( p ) ) } ] _ { i = 1 , \\ldots , k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 617, + 217, + 631 + ], + "score": 1.0, + "content": "), with", + "type": "text" + }, + { + "bbox": [ + 217, + 619, + 224, + 630 + ], + "score": 0.81, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 617, + 242, + 631 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 242, + 619, + 249, + 629 + ], + "score": 0.84, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 617, + 305, + 631 + ], + "score": 1.0, + "content": "being MLPs.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 552, + 506, + 631 + ] + }, + { + "type": "text", + "bbox": [ + 97, + 634, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 102, + 634, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 102, + 634, + 506, + 646 + ], + "score": 1.0, + "content": "Table 3 shows our results for texture reconstruction experiments on all models from Koestler et al.", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 644, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 505, + 657 + ], + "score": 1.0, + "content": "[2022]. The total number of parameters in our SignNet-based model is kept below that of the original", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 656, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 656, + 506, + 668 + ], + "score": 1.0, + "content": "model. We see that the SignNet architecture improves over the original Intrinsic NF model and over", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 666, + 506, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 506, + 679 + ], + "score": 1.0, + "content": "other baselines — especially in the LPIPS (Learned Perceptual Image Patch Similarity) metric, which", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 678, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 506, + 690 + ], + "score": 1.0, + "content": "has been shown to be a typically better perceptual metric than PSNR or DSSIM [Zhang et al., 2018a].", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 688, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 701 + ], + "score": 1.0, + "content": "While we have not yet tested this, we believe that SignNet would allow even better improvements", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 699, + 507, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 507, + 713 + ], + "score": 1.0, + "content": "when learning over eigenfunctions of different models, as it could improve transfer and generalization.", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 104, + 710, + 397, + 722 + ], + "spans": [ + { + "bbox": [ + 104, + 710, + 397, + 722 + ], + "score": 1.0, + "content": "See Appendix D.1 for visualizations and Appendix K.5 for more details.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 35.5, + "bbox_fs": [ + 102, + 634, + 507, + 722 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 127, + 68, + 491, + 149 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 127, + 68, + 491, + 149 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 127, + 68, + 491, + 149 + ], + "spans": [ + { + "bbox": [ + 127, + 68, + 491, + 149 + ], + "score": 0.717, + "type": "image", + "image_path": "2da6c2113b5fda8aff05b634b7425ce6f343ab9c2c035c184728c3df7b902400.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 127, + 68, + 491, + 95.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 127, + 95.0, + 491, + 122.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 127, + 122.0, + 491, + 149.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 158, + 506, + 181 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 157, + 505, + 171 + ], + "spans": [ + { + "bbox": [ + 105, + 157, + 505, + 171 + ], + "score": 1.0, + "content": "Figure 4: Cotangent Laplacian eigenvectors of the cat model and first principal component of", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 107, + 169, + 272, + 182 + ], + "spans": [ + { + "bbox": [ + 107, + 169, + 165, + 181 + ], + "score": 0.95, + "content": "{ \\phi ( \\bar { v } ) + \\phi ( - v ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 169, + 272, + 182 + ], + "score": 1.0, + "content": "from our trained SignNet.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "title", + "bbox": [ + 90, + 200, + 326, + 213 + ], + "lines": [ + { + "bbox": [ + 86, + 199, + 327, + 216 + ], + "spans": [ + { + "bbox": [ + 86, + 199, + 327, + 216 + ], + "score": 1.0, + "content": "306 4.4 Visualization of Learned Positional Encodings", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 106, + 221, + 505, + 265 + ], + "lines": [ + { + "bbox": [ + 104, + 220, + 505, + 234 + ], + "spans": [ + { + "bbox": [ + 104, + 220, + 408, + 234 + ], + "score": 1.0, + "content": "To better understand SignNet, we plot the first principal component of", + "type": "text" + }, + { + "bbox": [ + 408, + 221, + 469, + 233 + ], + "score": 0.93, + "content": "\\phi ( v ) + \\phi ( - v )", + "type": "inline_equation" + }, + { + "bbox": [ + 469, + 220, + 505, + 234 + ], + "score": 1.0, + "content": "for two", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 232, + 505, + 244 + ], + "spans": [ + { + "bbox": [ + 106, + 232, + 505, + 244 + ], + "score": 1.0, + "content": "eigenvectors on the cat model in Figure 4. We see that SignNet encodes bilateral symmetry and", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 243, + 506, + 256 + ], + "spans": [ + { + "bbox": [ + 105, + 243, + 506, + 256 + ], + "score": 1.0, + "content": "structural information on the cat model. See Appendix D for plots of more eigenvectors and further", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 254, + 138, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 138, + 265 + ], + "score": 1.0, + "content": "details.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7.5 + }, + { + "type": "title", + "bbox": [ + 105, + 280, + 197, + 294 + ], + "lines": [ + { + "bbox": [ + 105, + 279, + 198, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 279, + 198, + 295 + ], + "score": 1.0, + "content": "5 Related Work", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 99, + 304, + 504, + 316 + ], + "lines": [ + { + "bbox": [ + 105, + 304, + 506, + 318 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 506, + 318 + ], + "score": 1.0, + "content": "In this section, we review selected related work. A more thorough review is deferred to Appendix E.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 106, + 321, + 505, + 388 + ], + "lines": [ + { + "bbox": [ + 105, + 321, + 505, + 334 + ], + "spans": [ + { + "bbox": [ + 105, + 321, + 505, + 334 + ], + "score": 1.0, + "content": "Laplacian eigenvectors in GNNs. Various recently proposed methods in graph deep learning have", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 332, + 506, + 345 + ], + "spans": [ + { + "bbox": [ + 106, + 332, + 506, + 345 + ], + "score": 1.0, + "content": "directly used Laplacian eigenvectors as node positional encodings that are input to a neural network", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 343, + 506, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 506, + 356 + ], + "score": 1.0, + "content": "that is, e.g., a message passing GNN [Dwivedi et al., 2020, 2022], or some variant of a Transformer", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 354, + 506, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 506, + 367 + ], + "score": 1.0, + "content": "that is adapted to graphs [Dwivedi and Bresson, 2021, Kreuzer et al., 2021, Mialon et al., 2021,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 364, + 505, + 377 + ], + "spans": [ + { + "bbox": [ + 105, + 364, + 505, + 377 + ], + "score": 1.0, + "content": "Dwivedi et al., 2022]. None of these methods address basis invariance, and they only partially address", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 375, + 506, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 506, + 389 + ], + "score": 1.0, + "content": "sign invariance for node positional encodings by randomly flipping eigenvector signs during training.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 14.5 + }, + { + "type": "text", + "bbox": [ + 106, + 392, + 505, + 491 + ], + "lines": [ + { + "bbox": [ + 106, + 392, + 504, + 403 + ], + "spans": [ + { + "bbox": [ + 106, + 392, + 504, + 403 + ], + "score": 1.0, + "content": "Graph positional encodings. Other recent methods use positional encodings besides Laplacian", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 403, + 506, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 506, + 415 + ], + "score": 1.0, + "content": "eigenvectors. These include positional encodings based on random walks [Dwivedi et al., 2022,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 414, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 106, + 414, + 505, + 426 + ], + "score": 1.0, + "content": "Mialon et al., 2021, Li et al., 2020], diffusion kernels on graphs [Mialon et al., 2021, Feldman", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 423, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 104, + 423, + 505, + 439 + ], + "score": 1.0, + "content": "et al., 2022], shortest paths [Ying et al., 2021, Li et al., 2020], and unsupervised node embedding", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 435, + 505, + 449 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 505, + 449 + ], + "score": 1.0, + "content": "methods [Wang et al., 2022]. In particular, Wang et al. [2022] use Laplacian eigenvectors for relative", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 447, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 106, + 447, + 505, + 459 + ], + "score": 1.0, + "content": "positional encodings in an invariant way, but they focus on robustness, so they have stricter invariances", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 457, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 106, + 457, + 505, + 471 + ], + "score": 1.0, + "content": "that significantly reduce expressivity (see Appendix E.2 for more details). These previously used", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 468, + 506, + 482 + ], + "spans": [ + { + "bbox": [ + 106, + 468, + 506, + 482 + ], + "score": 1.0, + "content": "positional encodings are mostly ad-hoc, less general since they can be provably expressed by SignNet", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 479, + 485, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 485, + 493 + ], + "score": 1.0, + "content": "and BasisNet (see Section 3.3), and/or are expensive to compute (e.g., all pairs shortest paths).", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 22 + }, + { + "type": "title", + "bbox": [ + 100, + 506, + 262, + 519 + ], + "lines": [ + { + "bbox": [ + 105, + 504, + 263, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 263, + 522 + ], + "score": 1.0, + "content": "6 Conclusion and Discussion", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 105, + 530, + 505, + 630 + ], + "lines": [ + { + "bbox": [ + 106, + 531, + 505, + 543 + ], + "spans": [ + { + "bbox": [ + 106, + 531, + 505, + 543 + ], + "score": 1.0, + "content": "SignNet and BasisNet are novel architectures for processing eigenvectors that are invariant to sign", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 541, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 106, + 541, + 505, + 555 + ], + "score": 1.0, + "content": "flips and choices of eigenspace bases, respectively. Both architectures are provably universal: they", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 552, + 505, + 566 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 505, + 566 + ], + "score": 1.0, + "content": "can represent any continuous function with the corresponding invariances. When used with Laplacian", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 563, + 506, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 506, + 577 + ], + "score": 1.0, + "content": "eigenvectors as inputs they can provably approximate spectral graph convolutions, spectral invariants,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 574, + 506, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 506, + 587 + ], + "score": 1.0, + "content": "graph properties such as subgraph counts, and a number of other graph positional encodings. These", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 585, + 505, + 598 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 505, + 598 + ], + "score": 1.0, + "content": "theoretical results are supported by experiments showing that SignNet and BasisNet are highly", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 596, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 505, + 609 + ], + "score": 1.0, + "content": "expressive in practice, and learn effective graph positional encodings that improve the performance", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 606, + 506, + 620 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 506, + 620 + ], + "score": 1.0, + "content": "of message passing graph neural networks. Initial explorations show that SignNet and BasisNet can", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 618, + 423, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 423, + 631 + ], + "score": 1.0, + "content": "be useful beyond graph representation learning, as eigenvectors are ubiquitous.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 88, + 634, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 86, + 633, + 506, + 647 + ], + "spans": [ + { + "bbox": [ + 86, + 636, + 99, + 646 + ], + "score": 1.0, + "content": "338", + "type": "text" + }, + { + "bbox": [ + 105, + 633, + 506, + 647 + ], + "score": 1.0, + "content": "While we conduct experiments on graph machine learning tasks and a particular task on triangle", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 85, + 644, + 506, + 658 + ], + "spans": [ + { + "bbox": [ + 85, + 647, + 100, + 657 + ], + "score": 1.0, + "content": "339", + "type": "text" + }, + { + "bbox": [ + 105, + 644, + 506, + 658 + ], + "score": 1.0, + "content": "meshes, SignNet and BasisNet should also be applicable to processing eigenvectors in other settings,", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 85, + 655, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 85, + 657, + 100, + 668 + ], + "score": 1.0, + "content": "340", + "type": "text" + }, + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "score": 1.0, + "content": "such as recommender systems and tasks in shape analysis. We show significant empirical benefit in", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 85, + 667, + 506, + 679 + ], + "spans": [ + { + "bbox": [ + 85, + 668, + 99, + 679 + ], + "score": 1.0, + "content": "341", + "type": "text" + }, + { + "bbox": [ + 106, + 667, + 506, + 679 + ], + "score": 1.0, + "content": "the tasks that we consider, but we expect less benefit in cases where node features are sufficient to do", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 85, + 678, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 85, + 679, + 100, + 690 + ], + "score": 1.0, + "content": "342", + "type": "text" + }, + { + "bbox": [ + 106, + 678, + 506, + 690 + ], + "score": 1.0, + "content": "well on the task, or if the task does not require much sophisticated graph structure information to", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 85, + 689, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 85, + 690, + 100, + 701 + ], + "score": 1.0, + "content": "343", + "type": "text" + }, + { + "bbox": [ + 106, + 689, + 505, + 701 + ], + "score": 1.0, + "content": "solve. 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We see that SignNet encodes bilateral symmetry and", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 243, + 506, + 256 + ], + "spans": [ + { + "bbox": [ + 105, + 243, + 506, + 256 + ], + "score": 1.0, + "content": "structural information on the cat model. See Appendix D for plots of more eigenvectors and further", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 254, + 138, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 138, + 265 + ], + "score": 1.0, + "content": "details.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7.5, + "bbox_fs": [ + 104, + 220, + 506, + 265 + ] + }, + { + "type": "title", + "bbox": [ + 105, + 280, + 197, + 294 + ], + "lines": [ + { + "bbox": [ + 105, + 279, + 198, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 279, + 198, + 295 + ], + "score": 1.0, + "content": "5 Related Work", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 99, + 304, + 504, + 316 + ], + "lines": [ + { + "bbox": [ + 105, + 304, + 506, + 318 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 506, + 318 + ], + "score": 1.0, + "content": "In this section, we review selected related work. A more thorough review is deferred to Appendix E.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11, + "bbox_fs": [ + 105, + 304, + 506, + 318 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 321, + 505, + 388 + ], + "lines": [ + { + "bbox": [ + 105, + 321, + 505, + 334 + ], + "spans": [ + { + "bbox": [ + 105, + 321, + 505, + 334 + ], + "score": 1.0, + "content": "Laplacian eigenvectors in GNNs. Various recently proposed methods in graph deep learning have", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 332, + 506, + 345 + ], + "spans": [ + { + "bbox": [ + 106, + 332, + 506, + 345 + ], + "score": 1.0, + "content": "directly used Laplacian eigenvectors as node positional encodings that are input to a neural network", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 343, + 506, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 506, + 356 + ], + "score": 1.0, + "content": "that is, e.g., a message passing GNN [Dwivedi et al., 2020, 2022], or some variant of a Transformer", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 354, + 506, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 506, + 367 + ], + "score": 1.0, + "content": "that is adapted to graphs [Dwivedi and Bresson, 2021, Kreuzer et al., 2021, Mialon et al., 2021,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 364, + 505, + 377 + ], + "spans": [ + { + "bbox": [ + 105, + 364, + 505, + 377 + ], + "score": 1.0, + "content": "Dwivedi et al., 2022]. None of these methods address basis invariance, and they only partially address", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 375, + 506, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 506, + 389 + ], + "score": 1.0, + "content": "sign invariance for node positional encodings by randomly flipping eigenvector signs during training.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 14.5, + "bbox_fs": [ + 105, + 321, + 506, + 389 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 392, + 505, + 491 + ], + "lines": [ + { + "bbox": [ + 106, + 392, + 504, + 403 + ], + "spans": [ + { + "bbox": [ + 106, + 392, + 504, + 403 + ], + "score": 1.0, + "content": "Graph positional encodings. Other recent methods use positional encodings besides Laplacian", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 403, + 506, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 506, + 415 + ], + "score": 1.0, + "content": "eigenvectors. These include positional encodings based on random walks [Dwivedi et al., 2022,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 414, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 106, + 414, + 505, + 426 + ], + "score": 1.0, + "content": "Mialon et al., 2021, Li et al., 2020], diffusion kernels on graphs [Mialon et al., 2021, Feldman", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 423, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 104, + 423, + 505, + 439 + ], + "score": 1.0, + "content": "et al., 2022], shortest paths [Ying et al., 2021, Li et al., 2020], and unsupervised node embedding", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 435, + 505, + 449 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 505, + 449 + ], + "score": 1.0, + "content": "methods [Wang et al., 2022]. In particular, Wang et al. [2022] use Laplacian eigenvectors for relative", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 447, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 106, + 447, + 505, + 459 + ], + "score": 1.0, + "content": "positional encodings in an invariant way, but they focus on robustness, so they have stricter invariances", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 457, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 106, + 457, + 505, + 471 + ], + "score": 1.0, + "content": "that significantly reduce expressivity (see Appendix E.2 for more details). These previously used", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 468, + 506, + 482 + ], + "spans": [ + { + "bbox": [ + 106, + 468, + 506, + 482 + ], + "score": 1.0, + "content": "positional encodings are mostly ad-hoc, less general since they can be provably expressed by SignNet", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 479, + 485, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 485, + 493 + ], + "score": 1.0, + "content": "and BasisNet (see Section 3.3), and/or are expensive to compute (e.g., all pairs shortest paths).", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 22, + "bbox_fs": [ + 104, + 392, + 506, + 493 + ] + }, + { + "type": "title", + "bbox": [ + 100, + 506, + 262, + 519 + ], + "lines": [ + { + "bbox": [ + 105, + 504, + 263, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 263, + 522 + ], + "score": 1.0, + "content": "6 Conclusion and Discussion", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 105, + 530, + 505, + 630 + ], + "lines": [ + { + "bbox": [ + 106, + 531, + 505, + 543 + ], + "spans": [ + { + "bbox": [ + 106, + 531, + 505, + 543 + ], + "score": 1.0, + "content": "SignNet and BasisNet are novel architectures for processing eigenvectors that are invariant to sign", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 541, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 106, + 541, + 505, + 555 + ], + "score": 1.0, + "content": "flips and choices of eigenspace bases, respectively. Both architectures are provably universal: they", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 552, + 505, + 566 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 505, + 566 + ], + "score": 1.0, + "content": "can represent any continuous function with the corresponding invariances. When used with Laplacian", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 563, + 506, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 506, + 577 + ], + "score": 1.0, + "content": "eigenvectors as inputs they can provably approximate spectral graph convolutions, spectral invariants,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 574, + 506, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 506, + 587 + ], + "score": 1.0, + "content": "graph properties such as subgraph counts, and a number of other graph positional encodings. These", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 585, + 505, + 598 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 505, + 598 + ], + "score": 1.0, + "content": "theoretical results are supported by experiments showing that SignNet and BasisNet are highly", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 596, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 505, + 609 + ], + "score": 1.0, + "content": "expressive in practice, and learn effective graph positional encodings that improve the performance", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 606, + 506, + 620 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 506, + 620 + ], + "score": 1.0, + "content": "of message passing graph neural networks. Initial explorations show that SignNet and BasisNet can", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 618, + 423, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 423, + 631 + ], + "score": 1.0, + "content": "be useful beyond graph representation learning, as eigenvectors are ubiquitous.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 32, + "bbox_fs": [ + 105, + 531, + 506, + 631 + ] + }, + { + "type": "index", + "bbox": [ + 88, + 634, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 86, + 633, + 506, + 647 + ], + "spans": [ + { + "bbox": [ + 86, + 636, + 99, + 646 + ], + "score": 1.0, + "content": "338", + "type": "text" + }, + { + "bbox": [ + 105, + 633, + 506, + 647 + ], + "score": 1.0, + "content": "While we conduct experiments on graph machine learning tasks and a particular task on triangle", + "type": "text" + } + ], + "index": 37, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 644, + 506, + 658 + ], + "spans": [ + { + "bbox": [ + 85, + 647, + 100, + 657 + ], + "score": 1.0, + "content": "339", + "type": "text" + }, + { + "bbox": [ + 105, + 644, + 506, + 658 + ], + "score": 1.0, + "content": "meshes, SignNet and BasisNet should also be applicable to processing eigenvectors in other settings,", + "type": "text" + } + ], + "index": 38, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 655, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 85, + 657, + 100, + 668 + ], + "score": 1.0, + "content": "340", + "type": "text" + }, + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "score": 1.0, + "content": "such as recommender systems and tasks in shape analysis. 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[Yes] We support the claims with theoretical and/or empirical", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 161, + 289, + 202, + 302 + ], + "spans": [ + { + "bbox": [ + 161, + 289, + 202, + 302 + ], + "score": 1.0, + "content": "evidence.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 144, + 302, + 506, + 317 + ], + "spans": [ + { + "bbox": [ + 144, + 302, + 506, + 317 + ], + "score": 1.0, + "content": "(b) Did you describe the limitations of your work? [Yes] We discuss some limitations in", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 161, + 313, + 225, + 325 + ], + "spans": [ + { + "bbox": [ + 161, + 313, + 225, + 325 + ], + "score": 1.0, + "content": "the conclusion.", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 146, + 325, + 506, + 340 + ], + "spans": [ + { + "bbox": [ + 146, + 325, + 506, + 340 + ], + "score": 1.0, + "content": "(c) Did you discuss any potential negative societal impacts of your work? [Yes] See", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 161, + 337, + 222, + 349 + ], + "spans": [ + { + "bbox": [ + 161, + 337, + 222, + 349 + ], + "score": 1.0, + "content": "Appendix B.2.", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 144, + 348, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 144, + 348, + 505, + 363 + ], + "score": 1.0, + "content": "(d) Have you read the ethics review guidelines and ensured that your paper conforms to", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 161, + 360, + 446, + 373 + ], + "spans": [ + { + "bbox": [ + 161, + 360, + 446, + 373 + ], + "score": 1.0, + "content": "them? [Yes] We have read the guidelines; our paper conforms to them.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 132, + 375, + 302, + 387 + ], + "lines": [ + { + "bbox": [ + 129, + 374, + 304, + 388 + ], + "spans": [ + { + "bbox": [ + 129, + 374, + 304, + 388 + ], + "score": 1.0, + "content": "2. If you are including theoretical results...", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 146, + 390, + 505, + 435 + ], + "lines": [ + { + "bbox": [ + 145, + 388, + 505, + 402 + ], + "spans": [ + { + "bbox": [ + 145, + 388, + 505, + 402 + ], + "score": 1.0, + "content": "(a) Did you state the full set of assumptions of all theoretical results? [Yes] We state all", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 162, + 400, + 380, + 412 + ], + "spans": [ + { + "bbox": [ + 162, + 400, + 380, + 412 + ], + "score": 1.0, + "content": "assumptions either in the main text or in the appendix.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 145, + 412, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 145, + 412, + 505, + 425 + ], + "score": 1.0, + "content": "(b) Did you include complete proofs of all theoretical results? [Yes] We include all proofs", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 161, + 423, + 229, + 437 + ], + "spans": [ + { + "bbox": [ + 161, + 423, + 229, + 437 + ], + "score": 1.0, + "content": "in the appendix.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24.5 + }, + { + "type": "text", + "bbox": [ + 131, + 438, + 241, + 450 + ], + "lines": [ + { + "bbox": [ + 128, + 436, + 243, + 452 + ], + "spans": [ + { + "bbox": [ + 128, + 436, + 243, + 452 + ], + "score": 1.0, + "content": "3. If you ran experiments...", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 146, + 453, + 505, + 578 + ], + "lines": [ + { + "bbox": [ + 146, + 452, + 506, + 465 + ], + "spans": [ + { + "bbox": [ + 146, + 452, + 506, + 465 + ], + "score": 1.0, + "content": "(a) Did you include the code, data, and instructions needed to reproduce the main exper-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 162, + 464, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 162, + 464, + 505, + 475 + ], + "score": 1.0, + "content": "imental results (either in the supplemental material or as a URL)? [Yes] We include", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 162, + 475, + 357, + 487 + ], + "spans": [ + { + "bbox": [ + 162, + 475, + 357, + 487 + ], + "score": 1.0, + "content": "experimental code and instructions on the usage.", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 146, + 486, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 146, + 486, + 505, + 500 + ], + "score": 1.0, + "content": "(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 161, + 497, + 505, + 511 + ], + "spans": [ + { + "bbox": [ + 161, + 497, + 505, + 511 + ], + "score": 1.0, + "content": "were chosen)? [Yes] We try to give most experimental details in the main paper and", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 161, + 510, + 204, + 520 + ], + "spans": [ + { + "bbox": [ + 161, + 510, + 204, + 520 + ], + "score": 1.0, + "content": "appendix.", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 147, + 521, + 506, + 534 + ], + "spans": [ + { + "bbox": [ + 147, + 521, + 506, + 534 + ], + "score": 1.0, + "content": "(c) Did you report error bars (e.g., with respect to the random seed after running experi-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 162, + 532, + 506, + 544 + ], + "spans": [ + { + "bbox": [ + 162, + 532, + 506, + 544 + ], + "score": 1.0, + "content": "ments multiple times)? [Yes] We report standard deviations for multiple runs and/or", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 162, + 543, + 428, + 555 + ], + "spans": [ + { + "bbox": [ + 162, + 543, + 428, + 555 + ], + "score": 1.0, + "content": "seeds for graph-level tasks, but not the texture reconstruction task.", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 146, + 555, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 146, + 555, + 505, + 569 + ], + "score": 1.0, + "content": "(d) Did you include the total amount of compute and the type of resources used (e.g., type", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 162, + 567, + 447, + 578 + ], + "spans": [ + { + "bbox": [ + 162, + 567, + 447, + 578 + ], + "score": 1.0, + "content": "of GPUs, internal cluster, or cloud provider)? [Yes] See Appendix K.1.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 131, + 581, + 504, + 592 + ], + "lines": [ + { + "bbox": [ + 129, + 579, + 507, + 595 + ], + "spans": [ + { + "bbox": [ + 129, + 579, + 507, + 595 + ], + "score": 1.0, + "content": "4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 146, + 595, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 146, + 595, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 146, + 595, + 505, + 608 + ], + "score": 1.0, + "content": "(a) If your work uses existing assets, did you cite the creators? 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[Yes]", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 162, + 642, + 505, + 654 + ], + "spans": [ + { + "bbox": [ + 162, + 642, + 505, + 654 + ], + "score": 1.0, + "content": "We provide our code. We will also link to an open source version after anonymous", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 162, + 654, + 193, + 664 + ], + "spans": [ + { + "bbox": [ + 162, + 654, + 193, + 664 + ], + "score": 1.0, + "content": "review.", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 146, + 666, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 146, + 666, + 505, + 677 + ], + "score": 1.0, + "content": "(d) Did you discuss whether and how consent was obtained from people whose data you’re", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 162, + 676, + 328, + 688 + ], + "spans": [ + { + "bbox": [ + 162, + 676, + 328, + 688 + ], + "score": 1.0, + "content": "using/curating? 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For all authors...", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12, + "bbox_fs": [ + 129, + 253, + 210, + 267 + ] + }, + { + "type": "list", + "bbox": [ + 146, + 268, + 505, + 372 + ], + "lines": [ + { + "bbox": [ + 146, + 269, + 505, + 281 + ], + "spans": [ + { + "bbox": [ + 146, + 269, + 505, + 281 + ], + "score": 1.0, + "content": "(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s", + "type": "text" + } + ], + "index": 13, + "is_list_start_line": true + }, + { + "bbox": [ + 161, + 279, + 505, + 292 + ], + "spans": [ + { + "bbox": [ + 161, + 279, + 505, + 292 + ], + "score": 1.0, + "content": "contributions and scope? [Yes] We support the claims with theoretical and/or empirical", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 161, + 289, + 202, + 302 + ], + "spans": [ + { + "bbox": [ + 161, + 289, + 202, + 302 + ], + "score": 1.0, + "content": "evidence.", + "type": "text" + } + ], + "index": 15, + "is_list_end_line": true + }, + { + "bbox": [ + 144, + 302, + 506, + 317 + ], + "spans": [ + { + "bbox": [ + 144, + 302, + 506, + 317 + ], + "score": 1.0, + "content": "(b) Did you describe the limitations of your work? [Yes] We discuss some limitations in", + "type": "text" + } + ], + "index": 16, + "is_list_start_line": true + }, + { + "bbox": [ + 161, + 313, + 225, + 325 + ], + "spans": [ + { + "bbox": [ + 161, + 313, + 225, + 325 + ], + "score": 1.0, + "content": "the conclusion.", + "type": "text" + } + ], + "index": 17, + "is_list_end_line": true + }, + { + "bbox": [ + 146, + 325, + 506, + 340 + ], + "spans": [ + { + "bbox": [ + 146, + 325, + 506, + 340 + ], + "score": 1.0, + "content": "(c) Did you discuss any potential negative societal impacts of your work? [Yes] See", + "type": "text" + } + ], + "index": 18, + "is_list_start_line": true + }, + { + "bbox": [ + 161, + 337, + 222, + 349 + ], + "spans": [ + { + "bbox": [ + 161, + 337, + 222, + 349 + ], + "score": 1.0, + "content": "Appendix B.2.", + "type": "text" + } + ], + "index": 19, + "is_list_end_line": true + }, + { + "bbox": [ + 144, + 348, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 144, + 348, + 505, + 363 + ], + "score": 1.0, + "content": "(d) Have you read the ethics review guidelines and ensured that your paper conforms to", + "type": "text" + } + ], + "index": 20, + "is_list_start_line": true + }, + { + "bbox": [ + 161, + 360, + 446, + 373 + ], + "spans": [ + { + "bbox": [ + 161, + 360, + 446, + 373 + ], + "score": 1.0, + "content": "them? [Yes] We have read the guidelines; our paper conforms to them.", + "type": "text" + } + ], + "index": 21, + "is_list_end_line": true + } + ], + "index": 17, + "bbox_fs": [ + 144, + 269, + 506, + 373 + ] + }, + { + "type": "text", + "bbox": [ + 132, + 375, + 302, + 387 + ], + "lines": [ + { + "bbox": [ + 129, + 374, + 304, + 388 + ], + "spans": [ + { + "bbox": [ + 129, + 374, + 304, + 388 + ], + "score": 1.0, + "content": "2. If you are including theoretical results...", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22, + "bbox_fs": [ + 129, + 374, + 304, + 388 + ] + }, + { + "type": "list", + "bbox": [ + 146, + 390, + 505, + 435 + ], + "lines": [ + { + "bbox": [ + 145, + 388, + 505, + 402 + ], + "spans": [ + { + "bbox": [ + 145, + 388, + 505, + 402 + ], + "score": 1.0, + "content": "(a) Did you state the full set of assumptions of all theoretical results? 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[Yes] We include all proofs", + "type": "text" + } + ], + "index": 25, + "is_list_start_line": true + }, + { + "bbox": [ + 161, + 423, + 229, + 437 + ], + "spans": [ + { + "bbox": [ + 161, + 423, + 229, + 437 + ], + "score": 1.0, + "content": "in the appendix.", + "type": "text" + } + ], + "index": 26, + "is_list_end_line": true + } + ], + "index": 24.5, + "bbox_fs": [ + 145, + 388, + 505, + 437 + ] + }, + { + "type": "text", + "bbox": [ + 131, + 438, + 241, + 450 + ], + "lines": [ + { + "bbox": [ + 128, + 436, + 243, + 452 + ], + "spans": [ + { + "bbox": [ + 128, + 436, + 243, + 452 + ], + "score": 1.0, + "content": "3. 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[Yes] We include", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 162, + 475, + 357, + 487 + ], + "spans": [ + { + "bbox": [ + 162, + 475, + 357, + 487 + ], + "score": 1.0, + "content": "experimental code and instructions on the usage.", + "type": "text" + } + ], + "index": 30, + "is_list_end_line": true + }, + { + "bbox": [ + 146, + 486, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 146, + 486, + 505, + 500 + ], + "score": 1.0, + "content": "(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they", + "type": "text" + } + ], + "index": 31, + "is_list_start_line": true + }, + { + "bbox": [ + 161, + 497, + 505, + 511 + ], + "spans": [ + { + "bbox": [ + 161, + 497, + 505, + 511 + ], + "score": 1.0, + "content": "were chosen)? [Yes] We try to give most experimental details in the main paper and", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 161, + 510, + 204, + 520 + ], + "spans": [ + { + "bbox": [ + 161, + 510, + 204, + 520 + ], + "score": 1.0, + "content": "appendix.", + "type": "text" + } + ], + "index": 33, + "is_list_end_line": true + }, + { + "bbox": [ + 147, + 521, + 506, + 534 + ], + "spans": [ + { + "bbox": [ + 147, + 521, + 506, + 534 + ], + "score": 1.0, + "content": "(c) Did you report error bars (e.g., with respect to the random seed after running experi-", + "type": "text" + } + ], + "index": 34, + "is_list_start_line": true + }, + { + "bbox": [ + 162, + 532, + 506, + 544 + ], + "spans": [ + { + "bbox": [ + 162, + 532, + 506, + 544 + ], + "score": 1.0, + "content": "ments multiple times)? 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[Yes] See Appendix K.1.", + "type": "text" + } + ], + "index": 38, + "is_list_end_line": true + } + ], + "index": 33, + "bbox_fs": [ + 146, + 452, + 506, + 578 + ] + }, + { + "type": "text", + "bbox": [ + 131, + 581, + 504, + 592 + ], + "lines": [ + { + "bbox": [ + 129, + 579, + 507, + 595 + ], + "spans": [ + { + "bbox": [ + 129, + 579, + 507, + 595 + ], + "score": 1.0, + "content": "4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39, + "bbox_fs": [ + 129, + 579, + 507, + 595 + ] + }, + { + "type": "list", + "bbox": [ + 146, + 595, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 146, + 595, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 146, + 595, + 505, + 608 + ], + "score": 1.0, + "content": "(a) If your work uses existing assets, did you cite the creators? 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If you used crowdsourcing or conducted research with human subjects...", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 145, + 87, + 506, + 169 + ], + "lines": [ + { + "bbox": [ + 145, + 87, + 506, + 101 + ], + "spans": [ + { + "bbox": [ + 145, + 87, + 506, + 101 + ], + "score": 1.0, + "content": "(a) Did you include the full text of instructions given to participants and screenshots, if", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 161, + 99, + 411, + 111 + ], + "spans": [ + { + "bbox": [ + 161, + 99, + 411, + 111 + ], + "score": 1.0, + "content": "applicable? 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If you used crowdsourcing or conducted research with human subjects...", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0, + "bbox_fs": [ + 128, + 71, + 433, + 87 + ] + }, + { + "type": "list", + "bbox": [ + 145, + 87, + 506, + 169 + ], + "lines": [ + { + "bbox": [ + 145, + 87, + 506, + 101 + ], + "spans": [ + { + "bbox": [ + 145, + 87, + 506, + 101 + ], + "score": 1.0, + "content": "(a) Did you include the full text of instructions given to participants and screenshots, if", + "type": "text" + } + ], + "index": 1, + "is_list_start_line": true + }, + { + "bbox": [ + 161, + 99, + 411, + 111 + ], + "spans": [ + { + "bbox": [ + 161, + 99, + 411, + 111 + ], + "score": 1.0, + "content": "applicable? 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The", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 145, + 506, + 157 + ], + "spans": [ + { + "bbox": [ + 106, + 145, + 506, + 157 + ], + "score": 1.0, + "content": "universality of our architectures follows as a corollary of the following general decomposition result,", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 156, + 453, + 167 + ], + "spans": [ + { + "bbox": [ + 106, + 156, + 453, + 167 + ], + "score": 1.0, + "content": "which may enable construction of universal architectures for other invariances as well.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 106, + 170, + 505, + 215 + ], + "lines": [ + { + "bbox": [ + 106, + 171, + 504, + 183 + ], + "spans": [ + { + "bbox": [ + 106, + 171, + 291, + 183 + ], + "score": 1.0, + "content": "Theorem 3 (Decomposition Theorem). Let", + "type": "text" + }, + { + "bbox": [ + 292, + 172, + 338, + 182 + ], + "score": 0.88, + "content": "\\mathcal { X } _ { 1 } , \\ldots , \\mathcal { X } _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 171, + 469, + 183 + ], + "score": 1.0, + "content": "be topological spaces, and let", + "type": "text" + }, + { + "bbox": [ + 470, + 171, + 482, + 182 + ], + "score": 0.86, + "content": "G _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 171, + 497, + 183 + ], + "score": 1.0, + "content": "be", + "type": "text" + }, + { + "bbox": [ + 498, + 173, + 504, + 181 + ], + "score": 0.39, + "content": "a", + "type": "inline_equation" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 181, + 505, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 173, + 195 + ], + "score": 1.0, + "content": "group acting on", + "type": "text" + }, + { + "bbox": [ + 173, + 182, + 184, + 193 + ], + "score": 0.85, + "content": "\\mathcal { X } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 181, + 222, + 195 + ], + "score": 1.0, + "content": "for each", + "type": "text" + }, + { + "bbox": [ + 222, + 183, + 226, + 192 + ], + "score": 0.6, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 181, + 403, + 195 + ], + "score": 1.0, + "content": ". 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Along with the proof of Theorem 3, the mild topological assumptions are explained in", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 86, + 304, + 506, + 317 + ], + "spans": [ + { + "bbox": [ + 86, + 306, + 100, + 317 + ], + "score": 1.0, + "content": "698", + "type": "text" + }, + { + "bbox": [ + 105, + 304, + 506, + 317 + ], + "score": 1.0, + "content": "Appendix G.1. The assumptions hold for sign invariance and basis invariance. By applying this", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 86, + 315, + 316, + 327 + ], + "spans": [ + { + "bbox": [ + 86, + 317, + 100, + 327 + ], + "score": 1.0, + "content": "699", + "type": "text" + }, + { + "bbox": [ + 105, + 315, + 316, + 327 + ], + "score": 1.0, + "content": "theorem, we can prove universality of our networks:", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 100, + 330, + 504, + 364 + ], + "lines": [ + { + "bbox": [ + 106, + 330, + 506, + 343 + ], + "spans": [ + { + "bbox": [ + 106, + 330, + 506, + 343 + ], + "score": 1.0, + "content": "Corollary 1. Unconstrained-SignNet can represent any sign invariant function and Unconstrained-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 340, + 505, + 355 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 505, + 355 + ], + "score": 1.0, + "content": "BasisNet can represent any basis invariant function. Expressive-BasisNet is a universal approximator", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 353, + 390, + 365 + ], + "spans": [ + { + "bbox": [ + 106, + 353, + 390, + 365 + ], + "score": 1.0, + "content": "of functions that are both basis invariant and permutation equivariant.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 86, + 372, + 506, + 450 + ], + "lines": [ + { + "bbox": [ + 85, + 372, + 506, + 385 + ], + "spans": [ + { + "bbox": [ + 85, + 374, + 100, + 385 + ], + "score": 1.0, + "content": "703", + "type": "text" + }, + { + "bbox": [ + 105, + 372, + 506, + 385 + ], + "score": 1.0, + "content": "This result shows that Unconstrained-SignNet, Unconstrained-BasisNet, and Expressive-BasisNet", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 85, + 383, + 506, + 396 + ], + "spans": [ + { + "bbox": [ + 85, + 384, + 101, + 395 + ], + "score": 1.0, + "content": "704", + "type": "text" + }, + { + "bbox": [ + 106, + 383, + 506, + 396 + ], + "score": 1.0, + "content": "take the correct functional form for their respective invariances (proofs in Appendix G.2). Note", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 86, + 393, + 506, + 408 + ], + "spans": [ + { + "bbox": [ + 86, + 396, + 100, + 406 + ], + "score": 1.0, + "content": "705", + "type": "text" + }, + { + "bbox": [ + 105, + 393, + 506, + 408 + ], + "score": 1.0, + "content": "that Expressive-BasisNet approximates all sign invariant functions as a special case, by treating", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 85, + 406, + 506, + 417 + ], + "spans": [ + { + "bbox": [ + 85, + 406, + 100, + 417 + ], + "score": 1.0, + "content": "706", + "type": "text" + }, + { + "bbox": [ + 106, + 406, + 506, + 417 + ], + "score": 1.0, + "content": "all inputs as one dimensional eigenspaces. Accompanying the decomposition result, we show a", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 85, + 416, + 506, + 429 + ], + "spans": [ + { + "bbox": [ + 85, + 417, + 100, + 428 + ], + "score": 1.0, + "content": "707", + "type": "text" + }, + { + "bbox": [ + 106, + 416, + 506, + 429 + ], + "score": 1.0, + "content": "corresponding universal approximation result (proof in Appendix G.3). Similarly to Theorem 3,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 85, + 426, + 506, + 441 + ], + "spans": [ + { + "bbox": [ + 85, + 428, + 100, + 439 + ], + "score": 1.0, + "content": "708", + "type": "text" + }, + { + "bbox": [ + 105, + 426, + 229, + 441 + ], + "score": 1.0, + "content": "the problem of approximating", + "type": "text" + }, + { + "bbox": [ + 229, + 428, + 312, + 439 + ], + "score": 0.91, + "content": "G = G _ { 1 } \\times \\ldots \\times G _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 426, + 506, + 441 + ], + "score": 1.0, + "content": "invariant functions is reduced to approximating", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 86, + 437, + 231, + 451 + ], + "spans": [ + { + "bbox": [ + 86, + 440, + 100, + 450 + ], + "score": 1.0, + "content": "709", + "type": "text" + }, + { + "bbox": [ + 105, + 437, + 137, + 451 + ], + "score": 1.0, + "content": "several", + "type": "text" + }, + { + "bbox": [ + 137, + 439, + 149, + 449 + ], + "score": 0.88, + "content": "G _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 437, + 231, + 451 + ], + "score": 1.0, + "content": "-invariant functions.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24 + }, + { + "type": "title", + "bbox": [ + 86, + 465, + 323, + 479 + ], + "lines": [ + { + "bbox": [ + 84, + 464, + 325, + 482 + ], + "spans": [ + { + "bbox": [ + 84, + 464, + 325, + 482 + ], + "score": 1.0, + "content": "710 B More Details on SignNet and BasisNet", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "table", + "bbox": [ + 107, + 536, + 502, + 589 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 502, + 505, + 536 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 503, + 506, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 503, + 506, + 515 + ], + "score": 1.0, + "content": "Table 4: Properties of our architectures: Unconstrained-SignNet, SignNet, Unconstrained-BasisNet,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 513, + 505, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 505, + 528 + ], + "score": 1.0, + "content": "and Expressive-BasisNet. The properties are: permutation equivariance, universality (for the proper", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 525, + 392, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 392, + 538 + ], + "score": 1.0, + "content": "class of continuous invariant functions), and computational tractability.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30 + }, + { + "type": "table_body", + "bbox": [ + 107, + 536, + 502, + 589 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 536, + 502, + 589 + ], + "spans": [ + { + "bbox": [ + 107, + 536, + 502, + 589 + ], + "score": 0.969, + "html": "
Unconstr.-SignNetSignNetUnconstr.-BasisNetBasisNetExpr.-BasisNet
Permutation equiv.×√√×√x√√
Universal×√
Tractable√×
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In Table 4, we compare and contrast properties of the neural", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 86, + 621, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 86, + 624, + 100, + 633 + ], + "score": 1.0, + "content": "713", + "type": "text" + }, + { + "bbox": [ + 105, + 621, + 505, + 635 + ], + "score": 1.0, + "content": "architectures that we introduce. In Figure 5, we give pseudo-code of SignNet for learning node", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 86, + 633, + 316, + 645 + ], + "spans": [ + { + "bbox": [ + 86, + 635, + 100, + 644 + ], + "score": 1.0, + "content": "714", + "type": "text" + }, + { + "bbox": [ + 105, + 633, + 316, + 645 + ], + "score": 1.0, + "content": "positional encodings with a GNN prediction model.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 36.5 + }, + { + "type": "title", + "bbox": [ + 87, + 657, + 319, + 669 + ], + "lines": [ + { + "bbox": [ + 84, + 656, + 320, + 672 + ], + "spans": [ + { + "bbox": [ + 84, + 656, + 320, + 672 + ], + "score": 1.0, + "content": "715 B.1 Generalization Beyond Symmetric Matrices", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 87, + 678, + 506, + 722 + ], + "lines": [ + { + "bbox": [ + 86, + 677, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 86, + 680, + 100, + 690 + ], + "score": 1.0, + "content": "716", + "type": "text" + }, + { + "bbox": [ + 105, + 677, + 506, + 691 + ], + "score": 1.0, + "content": "In the main paper, we assume that the eigenspaces come from a symmetric matrix. This holds for many", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 86, + 687, + 507, + 702 + ], + "spans": [ + { + "bbox": [ + 86, + 690, + 100, + 700 + ], + "score": 1.0, + "content": "717", + "type": "text" + }, + { + "bbox": [ + 105, + 687, + 507, + 702 + ], + "score": 1.0, + "content": "cases of practical interest, as e.g. the Laplacian matrix of an undirected graph is symmetric. However,", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 85, + 699, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 85, + 701, + 100, + 712 + ], + "score": 1.0, + "content": "718", + "type": "text" + }, + { + "bbox": [ + 105, + 699, + 506, + 713 + ], + "score": 1.0, + "content": "we may also want to process directed graphs, or other data that have associated nonsymmetric matrices.", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 86, + 711, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 86, + 713, + 100, + 722 + ], + "score": 1.0, + "content": "719", + "type": "text" + }, + { + "bbox": [ + 106, + 711, + 505, + 723 + ], + "score": 1.0, + "content": "Our SignNet and BasisNet generalize in a straightforward way to handle nonsymmetric diagonalizable", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 41.5 + } + ], + "page_idx": 17, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 300, + 741, + 311, + 750 + ], + "lines": [ + { + "bbox": [ + 299, + 740, + 313, + 754 + ], + "spans": [ + { + "bbox": [ + 299, + 740, + 313, + 754 + ], + "score": 1.0, + "content": "18", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 92, + 71, + 294, + 85 + ], + "lines": [ + { + "bbox": [ + 87, + 69, + 295, + 88 + ], + "spans": [ + { + "bbox": [ + 87, + 69, + 295, + 88 + ], + "score": 1.0, + "content": "682 A Universality for Multiple Spaces", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 101, + 96, + 504, + 119 + ], + "lines": [ + { + "bbox": [ + 105, + 94, + 506, + 108 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 506, + 108 + ], + "score": 1.0, + "content": "We have temporarily moved this section to the Appendix due to space constraints, we will move it", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 106, + 316, + 120 + ], + "spans": [ + { + "bbox": [ + 105, + 106, + 316, + 120 + ], + "score": 1.0, + "content": "back to the main paper in the camera-ready version.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5, + "bbox_fs": [ + 105, + 94, + 506, + 120 + ] + }, + { + "type": "text", + "bbox": [ + 102, + 123, + 505, + 167 + ], + "lines": [ + { + "bbox": [ + 105, + 122, + 505, + 135 + ], + "spans": [ + { + "bbox": [ + 105, + 122, + 505, + 135 + ], + "score": 1.0, + "content": "While the networks introduced in the Section 2.2 possess the desired invariances, it is not immediately", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 135, + 505, + 146 + ], + "spans": [ + { + "bbox": [ + 106, + 135, + 505, + 146 + ], + "score": 1.0, + "content": "obvious whether they are powerful enough to express all functions with these invariances. The", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 145, + 506, + 157 + ], + "spans": [ + { + "bbox": [ + 106, + 145, + 506, + 157 + ], + "score": 1.0, + "content": "universality of our architectures follows as a corollary of the following general decomposition result,", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 156, + 453, + 167 + ], + "spans": [ + { + "bbox": [ + 106, + 156, + 453, + 167 + ], + "score": 1.0, + "content": "which may enable construction of universal architectures for other invariances as well.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 3.5, + "bbox_fs": [ + 105, + 122, + 506, + 167 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 170, + 505, + 215 + ], + "lines": [ + { + "bbox": [ + 106, + 171, + 504, + 183 + ], + "spans": [ + { + "bbox": [ + 106, + 171, + 291, + 183 + ], + "score": 1.0, + "content": "Theorem 3 (Decomposition Theorem). Let", + "type": "text" + }, + { + "bbox": [ + 292, + 172, + 338, + 182 + ], + "score": 0.88, + "content": "\\mathcal { X } _ { 1 } , \\ldots , \\mathcal { X } _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 171, + 469, + 183 + ], + "score": 1.0, + "content": "be topological spaces, and let", + "type": "text" + }, + { + "bbox": [ + 470, + 171, + 482, + 182 + ], + "score": 0.86, + "content": "G _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 171, + 497, + 183 + ], + "score": 1.0, + "content": "be", + "type": "text" + }, + { + "bbox": [ + 498, + 173, + 504, + 181 + ], + "score": 0.39, + "content": "a", + "type": "inline_equation" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 181, + 505, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 173, + 195 + ], + "score": 1.0, + "content": "group acting on", + "type": "text" + }, + { + "bbox": [ + 173, + 182, + 184, + 193 + ], + "score": 0.85, + "content": "\\mathcal { X } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 181, + 222, + 195 + ], + "score": 1.0, + "content": "for each", + "type": "text" + }, + { + "bbox": [ + 222, + 183, + 226, + 192 + ], + "score": 0.6, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 181, + 403, + 195 + ], + "score": 1.0, + "content": ". 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Along with the proof of Theorem 3, the mild topological assumptions are explained in", + "type": "text" + } + ], + "index": 15, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 304, + 506, + 317 + ], + "spans": [ + { + "bbox": [ + 86, + 306, + 100, + 317 + ], + "score": 1.0, + "content": "698", + "type": "text" + }, + { + "bbox": [ + 105, + 304, + 506, + 317 + ], + "score": 1.0, + "content": "Appendix G.1. The assumptions hold for sign invariance and basis invariance. By applying this", + "type": "text" + } + ], + "index": 16, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 315, + 316, + 327 + ], + "spans": [ + { + "bbox": [ + 86, + 317, + 100, + 327 + ], + "score": 1.0, + "content": "699", + "type": "text" + }, + { + "bbox": [ + 105, + 315, + 316, + 327 + ], + "score": 1.0, + "content": "theorem, we can prove universality of our networks:", + "type": "text" + } + ], + "index": 17, + "is_list_start_line": true + } + ], + "index": 15, + "bbox_fs": [ + 86, + 271, + 506, + 327 + ] + }, + { + "type": "text", + "bbox": [ + 100, + 330, + 504, + 364 + ], + "lines": [ + { + "bbox": [ + 106, + 330, + 506, + 343 + ], + "spans": [ + { + "bbox": [ + 106, + 330, + 506, + 343 + ], + "score": 1.0, + "content": "Corollary 1. Unconstrained-SignNet can represent any sign invariant function and Unconstrained-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 340, + 505, + 355 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 505, + 355 + ], + "score": 1.0, + "content": "BasisNet can represent any basis invariant function. Expressive-BasisNet is a universal approximator", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 353, + 390, + 365 + ], + "spans": [ + { + "bbox": [ + 106, + 353, + 390, + 365 + ], + "score": 1.0, + "content": "of functions that are both basis invariant and permutation equivariant.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 330, + 506, + 365 + ] + }, + { + "type": "index", + "bbox": [ + 86, + 372, + 506, + 450 + ], + "lines": [ + { + "bbox": [ + 85, + 372, + 506, + 385 + ], + "spans": [ + { + "bbox": [ + 85, + 374, + 100, + 385 + ], + "score": 1.0, + "content": "703", + "type": "text" + }, + { + "bbox": [ + 105, + 372, + 506, + 385 + ], + "score": 1.0, + "content": "This result shows that Unconstrained-SignNet, Unconstrained-BasisNet, and Expressive-BasisNet", + "type": "text" + } + ], + "index": 21, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 383, + 506, + 396 + ], + "spans": [ + { + "bbox": [ + 85, + 384, + 101, + 395 + ], + "score": 1.0, + "content": "704", + "type": "text" + }, + { + "bbox": [ + 106, + 383, + 506, + 396 + ], + "score": 1.0, + "content": "take the correct functional form for their respective invariances (proofs in Appendix G.2). Note", + "type": "text" + } + ], + "index": 22, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 393, + 506, + 408 + ], + "spans": [ + { + "bbox": [ + 86, + 396, + 100, + 406 + ], + "score": 1.0, + "content": "705", + "type": "text" + }, + { + "bbox": [ + 105, + 393, + 506, + 408 + ], + "score": 1.0, + "content": "that Expressive-BasisNet approximates all sign invariant functions as a special case, by treating", + "type": "text" + } + ], + "index": 23, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 406, + 506, + 417 + ], + "spans": [ + { + "bbox": [ + 85, + 406, + 100, + 417 + ], + "score": 1.0, + "content": "706", + "type": "text" + }, + { + "bbox": [ + 106, + 406, + 506, + 417 + ], + "score": 1.0, + "content": "all inputs as one dimensional eigenspaces. Accompanying the decomposition result, we show a", + "type": "text" + } + ], + "index": 24, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 416, + 506, + 429 + ], + "spans": [ + { + "bbox": [ + 85, + 417, + 100, + 428 + ], + "score": 1.0, + "content": "707", + "type": "text" + }, + { + "bbox": [ + 106, + 416, + 506, + 429 + ], + "score": 1.0, + "content": "corresponding universal approximation result (proof in Appendix G.3). Similarly to Theorem 3,", + "type": "text" + } + ], + "index": 25, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 426, + 506, + 441 + ], + "spans": [ + { + "bbox": [ + 85, + 428, + 100, + 439 + ], + "score": 1.0, + "content": "708", + "type": "text" + }, + { + "bbox": [ + 105, + 426, + 229, + 441 + ], + "score": 1.0, + "content": "the problem of approximating", + "type": "text" + }, + { + "bbox": [ + 229, + 428, + 312, + 439 + ], + "score": 0.91, + "content": "G = G _ { 1 } \\times \\ldots \\times G _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 426, + 506, + 441 + ], + "score": 1.0, + "content": "invariant functions is reduced to approximating", + "type": "text" + } + ], + "index": 26, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 437, + 231, + 451 + ], + "spans": [ + { + "bbox": [ + 86, + 440, + 100, + 450 + ], + "score": 1.0, + "content": "709", + "type": "text" + }, + { + "bbox": [ + 105, + 437, + 137, + 451 + ], + "score": 1.0, + "content": "several", + "type": "text" + }, + { + "bbox": [ + 137, + 439, + 149, + 449 + ], + "score": 0.88, + "content": "G _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 437, + 231, + 451 + ], + "score": 1.0, + "content": "-invariant functions.", + "type": "text" + } + ], + "index": 27, + "is_list_start_line": true + } + ], + "index": 24, + "bbox_fs": [ + 85, + 372, + 506, + 451 + ] + }, + { + "type": "title", + "bbox": [ + 86, + 465, + 323, + 479 + ], + "lines": [ + { + "bbox": [ + 84, + 464, + 325, + 482 + ], + "spans": [ + { + "bbox": [ + 84, + 464, + 325, + 482 + ], + "score": 1.0, + "content": "710 B More Details on SignNet and BasisNet", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "table", + "bbox": [ + 107, + 536, + 502, + 589 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 502, + 505, + 536 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 503, + 506, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 503, + 506, + 515 + ], + "score": 1.0, + "content": "Table 4: Properties of our architectures: Unconstrained-SignNet, SignNet, Unconstrained-BasisNet,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 513, + 505, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 505, + 528 + ], + "score": 1.0, + "content": "and Expressive-BasisNet. The properties are: permutation equivariance, universality (for the proper", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 525, + 392, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 392, + 538 + ], + "score": 1.0, + "content": "class of continuous invariant functions), and computational tractability.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30 + }, + { + "type": "table_body", + "bbox": [ + 107, + 536, + 502, + 589 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 536, + 502, + 589 + ], + "spans": [ + { + "bbox": [ + 107, + 536, + 502, + 589 + ], + "score": 0.969, + "html": "
Unconstr.-SignNetSignNetUnconstr.-BasisNetBasisNetExpr.-BasisNet
Permutation equiv.×√√×√x√√
Universal×√
Tractable√×
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Let", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 234, + 336, + 279, + 347 + ], + "score": 0.92, + "content": "A \\in \\mathbb { R } ^ { n \\times n }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 279, + 335, + 419, + 349 + ], + "score": 1.0, + "content": "be a matrix with a diagonalization", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 419, + 336, + 474, + 347 + ], + "score": 0.92, + "content": "A = V \\Lambda V ^ { - 1 }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 474, + 335, + 506, + 349 + ], + "score": 1.0, + "content": ", where", + "type": "text", + "cross_page": true + } + ], + "index": 15, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 345, + 505, + 361 + ], + "spans": [ + { + "bbox": [ + 85, + 348, + 100, + 359 + ], + "score": 1.0, + "content": "721", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 347, + 202, + 360 + ], + "score": 0.89, + "content": "\\boldsymbol { \\Lambda } = \\operatorname { D i a g } ( \\lambda _ { 1 } , \\ldots , \\lambda _ { n } )", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 203, + 345, + 307, + 361 + ], + "score": 1.0, + "content": "contains the eigenvalues", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 307, + 348, + 318, + 358 + ], + "score": 0.86, + "content": "\\lambda _ { i }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 318, + 345, + 406, + 361 + ], + "score": 1.0, + "content": ", and the columns of", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 406, + 348, + 488, + 360 + ], + "score": 0.87, + "content": "V = \\left[ v _ { 1 } \\quad \\ldots \\quad v _ { n } \\right]", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 489, + 345, + 505, + 361 + ], + "score": 1.0, + "content": "are", + "type": "text", + "cross_page": true + } + ], + "index": 16, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 358, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 85, + 360, + 100, + 370 + ], + "score": 1.0, + "content": "722", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 358, + 400, + 371 + ], + "score": 1.0, + "content": "eigenvectors. 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Unlike in the", + "type": "text", + "cross_page": true + } + ], + "index": 17, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 369, + 505, + 382 + ], + "spans": [ + { + "bbox": [ + 85, + 371, + 100, + 381 + ], + "score": 1.0, + "content": "723", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 104, + 369, + 505, + 382 + ], + "score": 1.0, + "content": "symmetric matrix case, the eigenvectors are not necessarily orthonormal, and both the eigenvalues", + "type": "text", + "cross_page": true + } + ], + "index": 18, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 380, + 244, + 394 + ], + "spans": [ + { + "bbox": [ + 86, + 383, + 100, + 392 + ], + "score": 1.0, + "content": "724", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 380, + 244, + 394 + ], + "score": 1.0, + "content": "and eigenvectors can be complex.", + "type": "text", + "cross_page": true + } + ], + "index": 19, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 396, + 506, + 410 + ], + "spans": [ + { + "bbox": [ + 86, + 398, + 99, + 408 + ], + "score": 1.0, + "content": "725", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 396, + 324, + 410 + ], + "score": 1.0, + "content": "Real eigenvectors. 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Let", + "type": "text" + }, + { + "bbox": [ + 234, + 336, + 279, + 347 + ], + "score": 0.92, + "content": "A \\in \\mathbb { R } ^ { n \\times n }", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 335, + 419, + 349 + ], + "score": 1.0, + "content": "be a matrix with a diagonalization", + "type": "text" + }, + { + "bbox": [ + 419, + 336, + 474, + 347 + ], + "score": 0.92, + "content": "A = V \\Lambda V ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 335, + 506, + 349 + ], + "score": 1.0, + "content": ", where", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 85, + 345, + 505, + 361 + ], + "spans": [ + { + "bbox": [ + 85, + 348, + 100, + 359 + ], + "score": 1.0, + "content": "721", + "type": "text" + }, + { + "bbox": [ + 106, + 347, + 202, + 360 + ], + "score": 0.89, + "content": "\\boldsymbol { \\Lambda } = \\operatorname { D i a g } ( \\lambda _ { 1 } , \\ldots , \\lambda _ { n } )", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 345, + 307, + 361 + ], + "score": 1.0, + "content": "contains the eigenvalues", + "type": "text" + }, + { + "bbox": [ + 307, + 348, + 318, + 358 + ], + "score": 0.86, + "content": "\\lambda _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 345, + 406, + 361 + ], + "score": 1.0, + "content": ", and the columns of", + "type": "text" + }, + { + "bbox": [ + 406, + 348, + 488, + 360 + ], + "score": 0.87, + "content": "V = \\left[ v _ { 1 } \\quad \\ldots \\quad v _ { n } \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 345, + 505, + 361 + ], + "score": 1.0, + "content": "are", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 85, + 358, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 85, + 360, + 100, + 370 + ], + "score": 1.0, + "content": "722", + "type": "text" + }, + { + "bbox": [ + 105, + 358, + 400, + 371 + ], + "score": 1.0, + "content": "eigenvectors. 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Unlike in the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 85, + 369, + 505, + 382 + ], + "spans": [ + { + "bbox": [ + 85, + 371, + 100, + 381 + ], + "score": 1.0, + "content": "723", + "type": "text" + }, + { + "bbox": [ + 104, + 369, + 505, + 382 + ], + "score": 1.0, + "content": "symmetric matrix case, the eigenvectors are not necessarily orthonormal, and both the eigenvalues", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 86, + 380, + 244, + 394 + ], + "spans": [ + { + "bbox": [ + 86, + 383, + 100, + 392 + ], + "score": 1.0, + "content": "724", + "type": "text" + }, + { + "bbox": [ + 105, + 380, + 244, + 394 + ], + "score": 1.0, + "content": "and eigenvectors can be complex.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 86, + 396, + 505, + 474 + ], + "lines": [ + { + "bbox": [ + 86, + 396, + 506, + 410 + ], + "spans": [ + { + "bbox": [ + 86, + 398, + 99, + 408 + ], + "score": 1.0, + "content": "725", + "type": "text" + }, + { + "bbox": [ + 105, + 396, + 324, + 410 + ], + "score": 1.0, + "content": "Real eigenvectors. 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Also, suppose that we choose the real numbers", + "type": "text" + }, + { + "bbox": [ + 371, + 419, + 379, + 428 + ], + "score": 0.83, + "content": "\\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 417, + 506, + 432 + ], + "score": 1.0, + "content": "as our base field for the vector", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 86, + 428, + 502, + 443 + ], + "spans": [ + { + "bbox": [ + 86, + 431, + 99, + 441 + ], + "score": 1.0, + "content": "728", + "type": "text" + }, + { + "bbox": [ + 106, + 428, + 374, + 443 + ], + "score": 1.0, + "content": "space in which eigenvectors lie. Note that for any scaling factor", + "type": "text" + }, + { + "bbox": [ + 375, + 429, + 426, + 441 + ], + "score": 0.92, + "content": "c \\in \\mathbb { R } \\setminus \\{ 0 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 428, + 496, + 443 + ], + "score": 1.0, + "content": "and eigenvector", + "type": "text" + }, + { + "bbox": [ + 496, + 432, + 502, + 439 + ], + "score": 0.7, + "content": "v", + "type": "inline_equation" + } + ], + "index": 23 + }, + { + "bbox": [ + 86, + 441, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 86, + 442, + 99, + 452 + ], + "score": 1.0, + "content": "729", + "type": "text" + }, + { + "bbox": [ + 106, + 441, + 160, + 452 + ], + "score": 1.0, + "content": "we have that", + "type": "text" + }, + { + "bbox": [ + 160, + 442, + 171, + 451 + ], + "score": 0.73, + "content": "c v", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 441, + 505, + 452 + ], + "score": 1.0, + "content": "is an eigenvector of the same eigenvalue. 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The basis invariance is", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7.5 + }, + { + "type": "interline_equation", + "bbox": [ + 187, + 204, + 423, + 218 + ], + "lines": [ + { + "bbox": [ + 187, + 204, + 423, + 218 + ], + "spans": [ + { + "bbox": [ + 187, + 204, + 423, + 218 + ], + "score": 0.89, + "content": "f ( V _ { 1 } , \\dots , V _ { l } ) = f ( V _ { 1 } W _ { 1 } , \\dots , V _ { l } W _ { l } ) \\qquad W _ { i } \\in \\operatorname { G L } _ { \\mathbb { C } } ( d _ { i } ) .", + "type": "interline_equation", + "image_path": "65f1b32b669cf13eaf5904ce42ed81f08891695950ea82b58d321543c700d393.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 187, + 204, + 423, + 218 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 93, + 222, + 503, + 245 + ], + "lines": [ + { + "bbox": [ + 93, + 221, + 505, + 236 + ], + "spans": [ + { + "bbox": [ + 93, + 221, + 282, + 236 + ], + "score": 1.0, + "content": "The orthogonal projector of the image of 53", + "type": "text" + }, + { + "bbox": [ + 282, + 223, + 293, + 234 + ], + "score": 0.87, + "content": "V _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 221, + 306, + 236 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 306, + 222, + 368, + 235 + ], + "score": 0.93, + "content": "V _ { i } ( V _ { i } ^ { * } V _ { i } ) ^ { - 1 } V _ { i } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 221, + 505, + 236 + ], + "score": 1.0, + "content": ", where there are now conjugate", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 90, + 233, + 378, + 246 + ], + "spans": [ + { + "bbox": [ + 90, + 233, + 378, + 246 + ], + "score": 1.0, + "content": "54 transposes replacing the transposes. Thus, BasisNet takes the form:", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5 + }, + { + "type": "interline_equation", + "bbox": [ + 199, + 249, + 412, + 271 + ], + "lines": [ + { + "bbox": [ + 199, + 249, + 412, + 271 + ], + "spans": [ + { + "bbox": [ + 199, + 249, + 412, + 271 + ], + "score": 0.93, + "content": "f ( V _ { 1 } , \\dots , V _ { l } ) = \\rho \\left( \\left[ \\phi _ { d _ { i } } ( V _ { i } ( V _ { i } ^ { * } V _ { i } ) ^ { - 1 } V _ { i } ^ { * } ) \\right] _ { i = 1 , \\dots , l } \\right) .", + "type": "interline_equation", + "image_path": "0ba266579e566fe9ddd51d92bedace84d33e0f02e399db5e0a9b1881405e00cb.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 199, + 249, + 412, + 271 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "title", + "bbox": [ + 87, + 282, + 204, + 294 + ], + "lines": [ + { + "bbox": [ + 85, + 281, + 205, + 295 + ], + "spans": [ + { + "bbox": [ + 85, + 281, + 205, + 295 + ], + "score": 1.0, + "content": "755 B.2 Broader Impacts", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 87, + 302, + 505, + 423 + ], + "lines": [ + { + "bbox": [ + 86, + 302, + 506, + 315 + ], + "spans": [ + { + "bbox": [ + 86, + 304, + 99, + 313 + ], + "score": 1.0, + "content": "756", + "type": "text" + }, + { + "bbox": [ + 105, + 302, + 506, + 315 + ], + "score": 1.0, + "content": "We believe that our models and future sign invariant or basis invariant networks could be useful in a", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 86, + 313, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 86, + 315, + 100, + 325 + ], + "score": 1.0, + "content": "757", + "type": "text" + }, + { + "bbox": [ + 105, + 313, + 505, + 325 + ], + "score": 1.0, + "content": "wide variety of applications. As eigenvectors arise in many domains, it is difficult to predict the uses", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 86, + 323, + 506, + 338 + ], + "spans": [ + { + "bbox": [ + 86, + 326, + 99, + 335 + ], + "score": 1.0, + "content": "758", + "type": "text" + }, + { + "bbox": [ + 105, + 323, + 506, + 338 + ], + "score": 1.0, + "content": "of these models. We test on several molecular property prediction tasks, which have the potential", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 86, + 335, + 505, + 347 + ], + "spans": [ + { + "bbox": [ + 86, + 336, + 100, + 347 + ], + "score": 1.0, + "content": "759", + "type": "text" + }, + { + "bbox": [ + 105, + 335, + 505, + 347 + ], + "score": 1.0, + "content": "for much positive impact, such as in drug discovery [Stokes et al., 2020]. However, recent work", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 86, + 346, + 505, + 358 + ], + "spans": [ + { + "bbox": [ + 86, + 348, + 99, + 357 + ], + "score": 1.0, + "content": "760", + "type": "text" + }, + { + "bbox": [ + 106, + 346, + 505, + 358 + ], + "score": 1.0, + "content": "has found that the same models that we use for finding beneficial drugs can also be used to design", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 86, + 355, + 506, + 371 + ], + "spans": [ + { + "bbox": [ + 86, + 358, + 99, + 369 + ], + "score": 1.0, + "content": "761", + "type": "text" + }, + { + "bbox": [ + 104, + 355, + 506, + 371 + ], + "score": 1.0, + "content": "biochemical weapons [Urbina et al., 2022]. Another major application of graph machine learning", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 86, + 367, + 506, + 380 + ], + "spans": [ + { + "bbox": [ + 86, + 369, + 100, + 380 + ], + "score": 1.0, + "content": "762", + "type": "text" + }, + { + "bbox": [ + 104, + 367, + 506, + 380 + ], + "score": 1.0, + "content": "is in social network analysis, where positive (e.g. malicious node detection [Pandit et al., 2007])", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 86, + 377, + 506, + 392 + ], + "spans": [ + { + "bbox": [ + 86, + 381, + 99, + 390 + ], + "score": 1.0, + "content": "763", + "type": "text" + }, + { + "bbox": [ + 105, + 377, + 506, + 392 + ], + "score": 1.0, + "content": "and negative (e.g. deanonymization [Narayanan and Shmatikov, 2009]) uses of machine learning", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 86, + 389, + 506, + 402 + ], + "spans": [ + { + "bbox": [ + 86, + 390, + 100, + 402 + ], + "score": 1.0, + "content": "764", + "type": "text" + }, + { + "bbox": [ + 104, + 389, + 506, + 402 + ], + "score": 1.0, + "content": "are possible. Even if there is no negative intent, bias in learned models can differentially impact", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 86, + 400, + 506, + 412 + ], + "spans": [ + { + "bbox": [ + 86, + 402, + 100, + 412 + ], + "score": 1.0, + "content": "765", + "type": "text" + }, + { + "bbox": [ + 105, + 400, + 506, + 412 + ], + "score": 1.0, + "content": "particular subgroups of people. Thus, academia, industry, and policy makers must be aware of such", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 86, + 411, + 405, + 423 + ], + "spans": [ + { + "bbox": [ + 86, + 413, + 99, + 423 + ], + "score": 1.0, + "content": "766", + "type": "text" + }, + { + "bbox": [ + 105, + 411, + 405, + 423 + ], + "score": 1.0, + "content": "potential negative uses, and work towards reducing the likelihood of them.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 20 + }, + { + "type": "title", + "bbox": [ + 100, + 437, + 302, + 451 + ], + "lines": [ + { + "bbox": [ + 105, + 435, + 303, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 303, + 455 + ], + "score": 1.0, + "content": "C More on Eigenvalue Multiplicities", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 462, + 504, + 485 + ], + "lines": [ + { + "bbox": [ + 105, + 461, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 505, + 475 + ], + "score": 1.0, + "content": "In this section, we study the properties of eigenvalues and eigenvectors computed by numerical", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 473, + 230, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 230, + 486 + ], + "score": 1.0, + "content": "algorithms on real-world data.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27.5 + }, + { + "type": "title", + "bbox": [ + 90, + 497, + 363, + 509 + ], + "lines": [ + { + "bbox": [ + 86, + 495, + 365, + 513 + ], + "spans": [ + { + "bbox": [ + 86, + 495, + 365, + 513 + ], + "score": 1.0, + "content": "770 C.1 Sign and Basis Ambiguities in Numerical Eigensolvers", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 86, + 518, + 506, + 640 + ], + "lines": [ + { + "bbox": [ + 86, + 518, + 507, + 531 + ], + "spans": [ + { + "bbox": [ + 86, + 520, + 99, + 529 + ], + "score": 1.0, + "content": "771", + "type": "text" + }, + { + "bbox": [ + 105, + 518, + 507, + 531 + ], + "score": 1.0, + "content": "When processing real-world data, we use eigenvectors that are computed by numerical algorithms.", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 86, + 527, + 506, + 542 + ], + "spans": [ + { + "bbox": [ + 86, + 530, + 99, + 540 + ], + "score": 1.0, + "content": "772", + "type": "text" + }, + { + "bbox": [ + 105, + 527, + 506, + 542 + ], + "score": 1.0, + "content": "These algorithms return specific eigenvectors for each eigenspace, so there is some choice of sign", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 86, + 538, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 86, + 541, + 99, + 551 + ], + "score": 1.0, + "content": "773", + "type": "text" + }, + { + "bbox": [ + 105, + 538, + 506, + 552 + ], + "score": 1.0, + "content": "or basis of each eigenspace. The general symmetric matrix eigensolvers numpy.linalg.eigh", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 86, + 550, + 506, + 563 + ], + "spans": [ + { + "bbox": [ + 86, + 552, + 99, + 561 + ], + "score": 1.0, + "content": "774", + "type": "text" + }, + { + "bbox": [ + 105, + 550, + 506, + 563 + ], + "score": 1.0, + "content": "and scipy.linalg.eigh both call LAPACK routines. They both proceed as follows: for a", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 85, + 560, + 507, + 576 + ], + "spans": [ + { + "bbox": [ + 85, + 563, + 100, + 573 + ], + "score": 1.0, + "content": "775", + "type": "text" + }, + { + "bbox": [ + 104, + 560, + 184, + 576 + ], + "score": 1.0, + "content": "symmetric matrix", + "type": "text" + }, + { + "bbox": [ + 185, + 562, + 193, + 571 + ], + "score": 0.72, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 193, + 560, + 313, + 576 + ], + "score": 1.0, + "content": ", they first decompose it as", + "type": "text" + }, + { + "bbox": [ + 313, + 560, + 371, + 572 + ], + "score": 0.91, + "content": "A = Q T Q ^ { \\dagger }", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 560, + 438, + 576 + ], + "score": 1.0, + "content": "for orthogonal", + "type": "text" + }, + { + "bbox": [ + 438, + 561, + 448, + 573 + ], + "score": 0.83, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 560, + 507, + 576 + ], + "score": 1.0, + "content": "and tridiago-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 85, + 570, + 506, + 586 + ], + "spans": [ + { + "bbox": [ + 85, + 573, + 100, + 584 + ], + "score": 1.0, + "content": "776", + "type": "text" + }, + { + "bbox": [ + 105, + 570, + 122, + 586 + ], + "score": 1.0, + "content": "nal", + "type": "text" + }, + { + "bbox": [ + 122, + 573, + 131, + 582 + ], + "score": 0.81, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 131, + 570, + 328, + 586 + ], + "score": 1.0, + "content": ", then they compute the eigendecomposition of", + "type": "text" + }, + { + "bbox": [ + 328, + 572, + 387, + 582 + ], + "score": 0.89, + "content": "T \\doteq \\bar { W } \\Lambda W ^ { \\top }", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 570, + 506, + 586 + ], + "score": 1.0, + "content": ", so the eigendecomposition", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 86, + 582, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 86, + 585, + 99, + 594 + ], + "score": 1.0, + "content": "777", + "type": "text" + }, + { + "bbox": [ + 105, + 582, + 118, + 596 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 119, + 583, + 127, + 593 + ], + "score": 0.76, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 582, + 140, + 596 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 140, + 582, + 239, + 595 + ], + "score": 0.89, + "content": "A = ( \\mathsf { \\bar { Q } } W ) \\dot { \\Lambda } ( W ^ { \\top } Q ^ { \\top } )", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 582, + 506, + 596 + ], + "score": 1.0, + "content": ". There are multiple ambiguities here: for diagonal sign matri-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 86, + 592, + 507, + 608 + ], + "spans": [ + { + "bbox": [ + 86, + 595, + 100, + 606 + ], + "score": 1.0, + "content": "778", + "type": "text" + }, + { + "bbox": [ + 104, + 592, + 123, + 608 + ], + "score": 1.0, + "content": "ces", + "type": "text" + }, + { + "bbox": [ + 124, + 595, + 219, + 606 + ], + "score": 0.87, + "content": "S = \\mathrm { D i a g } ( s _ { 1 } , . . . , s _ { n } )", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 592, + 240, + 608 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 240, + 594, + 339, + 606 + ], + "score": 0.92, + "content": "S ^ { \\prime } = \\mathrm { D i a g } ( s _ { 1 } ^ { \\prime } , . . . , s _ { n } ^ { \\prime } )", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 592, + 372, + 608 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 373, + 594, + 443, + 606 + ], + "score": 0.93, + "content": "s _ { i } , s _ { i } ^ { \\prime } \\in \\{ - 1 , 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 592, + 507, + 608 + ], + "score": 1.0, + "content": ", we have that", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 86, + 605, + 504, + 618 + ], + "spans": [ + { + "bbox": [ + 86, + 608, + 99, + 617 + ], + "score": 1.0, + "content": "779", + "type": "text" + }, + { + "bbox": [ + 107, + 606, + 192, + 618 + ], + "score": 0.92, + "content": "A = Q S ( S T S ) S Q ^ { \\top }", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 606, + 336, + 618 + ], + "score": 1.0, + "content": "is also a valid tridiagonalization, as", + "type": "text" + }, + { + "bbox": [ + 337, + 606, + 352, + 617 + ], + "score": 0.87, + "content": "\\it Q S", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 606, + 428, + 618 + ], + "score": 1.0, + "content": "is still orthogonal,", + "type": "text" + }, + { + "bbox": [ + 428, + 606, + 461, + 616 + ], + "score": 0.91, + "content": "S S = I", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 606, + 482, + 618 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 482, + 605, + 504, + 617 + ], + "score": 0.67, + "content": "S T S", + "type": "inline_equation" + } + ], + "index": 38 + }, + { + "bbox": [ + 86, + 617, + 506, + 631 + ], + "spans": [ + { + "bbox": [ + 86, + 619, + 100, + 630 + ], + "score": 1.0, + "content": "780", + "type": "text" + }, + { + "bbox": [ + 105, + 617, + 207, + 631 + ], + "score": 1.0, + "content": "is still tridiagonal. 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However, recent work", + "type": "text" + } + ], + "index": 18, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 346, + 505, + 358 + ], + "spans": [ + { + "bbox": [ + 86, + 348, + 99, + 357 + ], + "score": 1.0, + "content": "760", + "type": "text" + }, + { + "bbox": [ + 106, + 346, + 505, + 358 + ], + "score": 1.0, + "content": "has found that the same models that we use for finding beneficial drugs can also be used to design", + "type": "text" + } + ], + "index": 19, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 355, + 506, + 371 + ], + "spans": [ + { + "bbox": [ + 86, + 358, + 99, + 369 + ], + "score": 1.0, + "content": "761", + "type": "text" + }, + { + "bbox": [ + 104, + 355, + 506, + 371 + ], + "score": 1.0, + "content": "biochemical weapons [Urbina et al., 2022]. Another major application of graph machine learning", + "type": "text" + } + ], + "index": 20, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 367, + 506, + 380 + ], + "spans": [ + { + "bbox": [ + 86, + 369, + 100, + 380 + ], + "score": 1.0, + "content": "762", + "type": "text" + }, + { + "bbox": [ + 104, + 367, + 506, + 380 + ], + "score": 1.0, + "content": "is in social network analysis, where positive (e.g. malicious node detection [Pandit et al., 2007])", + "type": "text" + } + ], + "index": 21, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 377, + 506, + 392 + ], + "spans": [ + { + "bbox": [ + 86, + 381, + 99, + 390 + ], + "score": 1.0, + "content": "763", + "type": "text" + }, + { + "bbox": [ + 105, + 377, + 506, + 392 + ], + "score": 1.0, + "content": "and negative (e.g. deanonymization [Narayanan and Shmatikov, 2009]) uses of machine learning", + "type": "text" + } + ], + "index": 22, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 389, + 506, + 402 + ], + "spans": [ + { + "bbox": [ + 86, + 390, + 100, + 402 + ], + "score": 1.0, + "content": "764", + "type": "text" + }, + { + "bbox": [ + 104, + 389, + 506, + 402 + ], + "score": 1.0, + "content": "are possible. Even if there is no negative intent, bias in learned models can differentially impact", + "type": "text" + } + ], + "index": 23, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 400, + 506, + 412 + ], + "spans": [ + { + "bbox": [ + 86, + 402, + 100, + 412 + ], + "score": 1.0, + "content": "765", + "type": "text" + }, + { + "bbox": [ + 105, + 400, + 506, + 412 + ], + "score": 1.0, + "content": "particular subgroups of people. Thus, academia, industry, and policy makers must be aware of such", + "type": "text" + } + ], + "index": 24, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 411, + 405, + 423 + ], + "spans": [ + { + "bbox": [ + 86, + 413, + 99, + 423 + ], + "score": 1.0, + "content": "766", + "type": "text" + }, + { + "bbox": [ + 105, + 411, + 405, + 423 + ], + "score": 1.0, + "content": "potential negative uses, and work towards reducing the likelihood of them.", + "type": "text" + } + ], + "index": 25, + "is_list_start_line": true + } + ], + "index": 20, + "bbox_fs": [ + 86, + 302, + 506, + 423 + ] + }, + { + "type": "title", + "bbox": [ + 100, + 437, + 302, + 451 + ], + "lines": [ + { + "bbox": [ + 105, + 435, + 303, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 303, + 455 + ], + "score": 1.0, + "content": "C More on Eigenvalue Multiplicities", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 462, + 504, + 485 + ], + "lines": [ + { + "bbox": [ + 105, + 461, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 505, + 475 + ], + "score": 1.0, + "content": "In this section, we study the properties of eigenvalues and eigenvectors computed by numerical", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 473, + 230, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 230, + 486 + ], + "score": 1.0, + "content": "algorithms on real-world data.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27.5, + "bbox_fs": [ + 105, + 461, + 505, + 486 + ] + }, + { + "type": "title", + "bbox": [ + 90, + 497, + 363, + 509 + ], + "lines": [ + { + "bbox": [ + 86, + 495, + 365, + 513 + ], + "spans": [ + { + "bbox": [ + 86, + 495, + 365, + 513 + ], + "score": 1.0, + "content": "770 C.1 Sign and Basis Ambiguities in Numerical Eigensolvers", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "index", + "bbox": [ + 86, + 518, + 506, + 640 + ], + "lines": [ + { + "bbox": [ + 86, + 518, + 507, + 531 + ], + "spans": [ + { + "bbox": [ + 86, + 520, + 99, + 529 + ], + "score": 1.0, + "content": "771", + "type": "text" + }, + { + "bbox": [ + 105, + 518, + 507, + 531 + ], + "score": 1.0, + "content": "When processing real-world data, we use eigenvectors that are computed by numerical algorithms.", + "type": "text" + } + ], + "index": 30, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 527, + 506, + 542 + ], + "spans": [ + { + "bbox": [ + 86, + 530, + 99, + 540 + ], + "score": 1.0, + "content": "772", + "type": "text" + }, + { + "bbox": [ + 105, + 527, + 506, + 542 + ], + "score": 1.0, + "content": "These algorithms return specific eigenvectors for each eigenspace, so there is some choice of sign", + "type": "text" + } + ], + "index": 31, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 538, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 86, + 541, + 99, + 551 + ], + "score": 1.0, + "content": "773", + "type": "text" + }, + { + "bbox": [ + 105, + 538, + 506, + 552 + ], + "score": 1.0, + "content": "or basis of each eigenspace. The general symmetric matrix eigensolvers numpy.linalg.eigh", + "type": "text" + } + ], + "index": 32, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 550, + 506, + 563 + ], + "spans": [ + { + "bbox": [ + 86, + 552, + 99, + 561 + ], + "score": 1.0, + "content": "774", + "type": "text" + }, + { + "bbox": [ + 105, + 550, + 506, + 563 + ], + "score": 1.0, + "content": "and scipy.linalg.eigh both call LAPACK routines. 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There are multiple ambiguities here: for diagonal sign matri-", + "type": "text" + } + ], + "index": 36, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 592, + 507, + 608 + ], + "spans": [ + { + "bbox": [ + 86, + 595, + 100, + 606 + ], + "score": 1.0, + "content": "778", + "type": "text" + }, + { + "bbox": [ + 104, + 592, + 123, + 608 + ], + "score": 1.0, + "content": "ces", + "type": "text" + }, + { + "bbox": [ + 124, + 595, + 219, + 606 + ], + "score": 0.87, + "content": "S = \\mathrm { D i a g } ( s _ { 1 } , . . . , s _ { n } )", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 592, + 240, + 608 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 240, + 594, + 339, + 606 + ], + "score": 0.92, + "content": "S ^ { \\prime } = \\mathrm { D i a g } ( s _ { 1 } ^ { \\prime } , . . . , s _ { n } ^ { \\prime } )", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 592, + 372, + 608 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 373, + 594, + 443, + 606 + ], + "score": 0.93, + "content": "s _ { i } , s _ { i } ^ { \\prime } \\in \\{ - 1 , 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 592, + 507, + 608 + ], + "score": 1.0, + "content": ", we have that", + "type": "text" + } + ], + "index": 37, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 605, + 504, + 618 + ], + "spans": [ + { + "bbox": [ + 86, + 608, + 99, + 617 + ], + "score": 1.0, + "content": "779", + "type": "text" + }, + { + "bbox": [ + 107, + 606, + 192, + 618 + ], + "score": 0.92, + "content": "A = Q S ( S T S ) S Q ^ { \\top }", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 606, + 336, + 618 + ], + "score": 1.0, + "content": "is also a valid tridiagonalization, as", + "type": "text" + }, + { + "bbox": [ + 337, + 606, + 352, + 617 + ], + "score": 0.87, + "content": "\\it Q S", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 606, + 428, + 618 + ], + "score": 1.0, + "content": "is still orthogonal,", + "type": "text" + }, + { + "bbox": [ + 428, + 606, + 461, + 616 + ], + "score": 0.91, + "content": "S S = I", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 606, + 482, + 618 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 482, + 605, + 504, + 617 + ], + "score": 0.67, + "content": "S T S", + "type": "inline_equation" + } + ], + "index": 38, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 617, + 506, + 631 + ], + "spans": [ + { + "bbox": [ + 86, + 619, + 100, + 630 + ], + "score": 1.0, + "content": "780", + "type": "text" + }, + { + "bbox": [ + 105, + 617, + 207, + 631 + ], + "score": 1.0, + "content": "is still tridiagonal. Also,", + "type": "text" + }, + { + "bbox": [ + 207, + 617, + 298, + 630 + ], + "score": 0.92, + "content": "T = ( W S ^ { \\prime } ) \\Lambda ( S ^ { \\prime } W ^ { \\top } )", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 617, + 432, + 631 + ], + "score": 1.0, + "content": "is a valid eigendecomposition of", + "type": "text" + }, + { + "bbox": [ + 433, + 618, + 441, + 628 + ], + "score": 0.8, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 617, + 455, + 631 + ], + "score": 1.0, + "content": ", as", + "type": "text" + }, + { + "bbox": [ + 456, + 618, + 477, + 628 + ], + "score": 0.88, + "content": "W S ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 617, + 506, + 631 + ], + "score": 1.0, + "content": "is still", + "type": "text" + } + ], + "index": 39, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 629, + 154, + 642 + ], + "spans": [ + { + "bbox": [ + 86, + 631, + 99, + 640 + ], + "score": 1.0, + "content": "781", + "type": "text" + }, + { + "bbox": [ + 105, + 629, + 154, + 642 + ], + "score": 1.0, + "content": "orthogonal.", + "type": "text" + } + ], + "index": 40, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 645, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 86, + 647, + 99, + 657 + ], + "score": 1.0, + "content": "782", + "type": "text" + }, + { + "bbox": [ + 106, + 645, + 505, + 658 + ], + "score": 1.0, + "content": "In practice, we find that the general symmetric matrix eigensolvers numpy.linalg.eigh and", + "type": "text" + } + ], + "index": 41, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 655, + 506, + 669 + ], + "spans": [ + { + "bbox": [ + 86, + 658, + 99, + 668 + ], + "score": 1.0, + "content": "783", + "type": "text" + }, + { + "bbox": [ + 105, + 655, + 506, + 669 + ], + "score": 1.0, + "content": "scipy.linalg.eigh differ between frameworks but are consistent with the same framework. More", + "type": "text" + } + ], + "index": 42, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 667, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 85, + 668, + 100, + 680 + ], + "score": 1.0, + "content": "784", + "type": "text" + }, + { + "bbox": [ + 105, + 667, + 257, + 680 + ], + "score": 1.0, + "content": "specifically, for a symmetric matrix", + "type": "text" + }, + { + "bbox": [ + 258, + 667, + 266, + 677 + ], + "score": 0.78, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 667, + 506, + 680 + ], + "score": 1.0, + "content": ", we find that the eigenvectors computed with the default", + "type": "text" + } + ], + "index": 43, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 678, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 85, + 679, + 100, + 690 + ], + "score": 1.0, + "content": "785", + "type": "text" + }, + { + "bbox": [ + 105, + 678, + 505, + 690 + ], + "score": 1.0, + "content": "settings in numpy tend to differ by a choice of sign or basis from those that are computed with the", + "type": "text" + } + ], + "index": 44, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 688, + 505, + 702 + ], + "spans": [ + { + "bbox": [ + 86, + 691, + 99, + 700 + ], + "score": 1.0, + "content": "786", + "type": "text" + }, + { + "bbox": [ + 105, + 688, + 505, + 702 + ], + "score": 1.0, + "content": "default settings in scipy. On the other hand, the called LAPACK routines are deterministic, so the", + "type": "text" + } + ], + "index": 45, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 699, + 505, + 713 + ], + "spans": [ + { + "bbox": [ + 86, + 702, + 99, + 712 + ], + "score": 1.0, + "content": "787", + "type": "text" + }, + { + "bbox": [ + 105, + 699, + 505, + 713 + ], + "score": 1.0, + "content": "eigenvectors returned by numpy are the same in each call, and the eigenvectors returned by scipy are", + "type": "text" + } + ], + "index": 46, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 711, + 230, + 722 + ], + "spans": [ + { + "bbox": [ + 86, + 713, + 99, + 722 + ], + "score": 1.0, + "content": "788", + "type": "text" + }, + { + "bbox": [ + 106, + 711, + 230, + 722 + ], + "score": 1.0, + "content": "likewise the same in each call.", + "type": "text" + } + ], + "index": 47, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 233, + 506, + 248 + ], + "spans": [ + { + "bbox": [ + 86, + 236, + 99, + 245 + ], + "score": 1.0, + "content": "789", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 233, + 506, + 248 + ], + "score": 1.0, + "content": "Eigensolvers for sparse symmetric matrices like scipy.linalg.eigsh are required for large scale", + "type": "text", + "cross_page": true + } + ], + "index": 6, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 244, + 505, + 258 + ], + "spans": [ + { + "bbox": [ + 85, + 246, + 100, + 257 + ], + "score": 1.0, + "content": "790", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 244, + 505, + 258 + ], + "score": 1.0, + "content": "problems. 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Still, in the worst case the signs chosen will be arbitrary, and they do", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 305, + 505, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 305, + 505, + 317 + ], + "score": 1.0, + "content": "not handle basis ambiguities in higher dimensional eigenspaces. Other works have made choices", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 315, + 505, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 505, + 329 + ], + "score": 1.0, + "content": "of sign, such as by picking the sign so that the eigenvector’s entries are in the largest lexicographic", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 327, + 505, + 339 + ], + "spans": [ + { + "bbox": [ + 106, + 327, + 505, + 339 + ], + "score": 1.0, + "content": "order [Tam and Dunson, 2022]. This choice of sign may work poorly for learning on graphs, as it is", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 338, + 506, + 350 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 506, + 350 + ], + "score": 1.0, + "content": "sensitive to permutations on nodes. For some graph regression experiments in Section 4.1, we try a", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 348, + 400, + 362 + ], + "spans": [ + { + "bbox": [ + 106, + 348, + 400, + 362 + ], + "score": 1.0, + "content": "choice of sign that is permutation invariant, but we find it to work poorly.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 13 + }, + { + "type": "title", + "bbox": [ + 101, + 376, + 339, + 388 + ], + "lines": [ + { + "bbox": [ + 105, + 374, + 341, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 341, + 391 + ], + "score": 1.0, + "content": "C.2 Higher Dimensional Eigenspaces in Real Graphs", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 397, + 505, + 442 + ], + "lines": [ + { + "bbox": [ + 106, + 398, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 106, + 398, + 505, + 410 + ], + "score": 1.0, + "content": "Here, we investigate the normalized Laplacian eigenspace statistics of real-world graph data. 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DatasetNodesDistinct 入#Mult.Max Mult.% Vecs mult. > 1
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Note: we will put this back in the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 298, + 241, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 298, + 241, + 313 + ], + "score": 1.0, + "content": "main paper for the camera-ready.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5 + } + ], + "index": 3.0 + }, + { + "type": "text", + "bbox": [ + 90, + 333, + 505, + 356 + ], + "lines": [ + { + "bbox": [ + 86, + 333, + 506, + 346 + ], + "spans": [ + { + "bbox": [ + 86, + 333, + 506, + 346 + ], + "score": 1.0, + "content": "902 multiplicity for adjacency matrix eigenvalues [Babai et al., 1982], with a time complexity that is", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 85, + 344, + 321, + 357 + ], + "spans": [ + { + "bbox": [ + 85, + 344, + 321, + 357 + ], + "score": 1.0, + "content": "903 lower for graphs with lower maximum multiplicities.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8.5 + }, + { + "type": "text", + "bbox": [ + 86, + 336, + 100, + 496 + ], + "lines": [ + { + "bbox": [ + 85, + 363, + 100, + 374 + ], + "spans": [ + { + "bbox": [ + 85, + 363, + 100, + 374 + ], + "score": 1.0, + "content": "904", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 85, + 374, + 100, + 385 + ], + "spans": [ + { + "bbox": [ + 85, + 374, + 100, + 385 + ], + "score": 1.0, + "content": "905", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 85, + 385, + 100, + 396 + ], + "spans": [ + { + "bbox": [ + 85, + 385, + 100, + 396 + ], + "score": 1.0, + "content": "906", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 85, + 396, + 100, + 407 + ], + "spans": [ + { + "bbox": [ + 85, + 396, + 100, + 407 + ], + "score": 1.0, + "content": "907", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 85, + 406, + 100, + 417 + ], + "spans": [ + { + "bbox": [ + 85, + 406, + 100, + 417 + ], + "score": 1.0, + "content": "908", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 85, + 417, + 100, + 429 + ], + "spans": [ + { + "bbox": [ + 85, + 417, + 100, + 429 + ], + "score": 1.0, + "content": "909", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 85, + 428, + 100, + 439 + ], + "spans": [ + { + "bbox": [ + 85, + 428, + 100, + 439 + ], + "score": 1.0, + "content": "910", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 85, + 439, + 100, + 451 + ], + "spans": [ + { + "bbox": [ + 85, + 439, + 100, + 451 + ], + "score": 1.0, + "content": "911", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 85, + 451, + 100, + 462 + ], + "spans": [ + { + "bbox": [ + 85, + 451, + 100, + 462 + ], + "score": 1.0, + "content": "912", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 85, + 462, + 100, + 473 + ], + "spans": [ + { + "bbox": [ + 85, + 462, + 100, + 473 + ], + "score": 1.0, + "content": "913", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 85, + 473, + 100, + 484 + ], + "spans": [ + { + "bbox": [ + 85, + 473, + 100, + 484 + ], + "score": 1.0, + "content": "914", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 85, + 484, + 100, + 495 + ], + "spans": [ + { + "bbox": [ + 85, + 484, + 100, + 495 + ], + "score": 1.0, + "content": "915", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 21.0 + }, + { + "type": "text", + "bbox": [ + 105, + 361, + 505, + 493 + ], + "lines": [ + { + "bbox": [ + 105, + 360, + 506, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 506, + 374 + ], + "score": 1.0, + "content": "A recent work of Wang et al. [2022] proposes full orthogonal group invariance for functions that", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 369, + 507, + 385 + ], + "spans": [ + { + "bbox": [ + 104, + 369, + 383, + 385 + ], + "score": 1.0, + "content": "process positional encodings. In particular, for positional encodings", + "type": "text" + }, + { + "bbox": [ + 383, + 371, + 426, + 383 + ], + "score": 0.92, + "content": "Z \\in \\mathbb { R } ^ { n \\times k }", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 369, + 507, + 385 + ], + "score": 1.0, + "content": ", they parameterize", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 383, + 506, + 395 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 147, + 395 + ], + "score": 1.0, + "content": "functions", + "type": "text" + }, + { + "bbox": [ + 147, + 383, + 169, + 394 + ], + "score": 0.92, + "content": "f ( Z )", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 383, + 210, + 395 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 210, + 383, + 275, + 395 + ], + "score": 0.93, + "content": "{ \\bar { f } } ( Z ) { \\bar { = } } f ( Z Q )", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 383, + 304, + 395 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 304, + 383, + 346, + 394 + ], + "score": 0.92, + "content": "Q \\in O ( k )", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 383, + 506, + 395 + ], + "score": 1.0, + "content": ". This indeed makes sense for network", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 393, + 506, + 406 + ], + "spans": [ + { + "bbox": [ + 106, + 393, + 506, + 406 + ], + "score": 1.0, + "content": "embeddings like node2vec [Grover and Leskovec, 2016], as their objective functions are based", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 405, + 505, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 505, + 417 + ], + "score": 1.0, + "content": "on inner products and are thus orthogonally invariant. While they prove stability results when", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 416, + 506, + 428 + ], + "spans": [ + { + "bbox": [ + 106, + 416, + 506, + 428 + ], + "score": 1.0, + "content": "enforcing full orthogonal invariance for eigenvectors, this is a very strict constraint compared to", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 426, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 427, + 275, + 438 + ], + "score": 1.0, + "content": "our basis invariance. For instance, when", + "type": "text" + }, + { + "bbox": [ + 275, + 426, + 302, + 437 + ], + "score": 0.9, + "content": "k = n", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 427, + 435, + 438 + ], + "score": 1.0, + "content": "and all eigenvectors are used in", + "type": "text" + }, + { + "bbox": [ + 435, + 426, + 444, + 437 + ], + "score": 0.72, + "content": "V", + "type": "inline_equation" + }, + { + "bbox": [ + 445, + 427, + 505, + 438 + ], + "score": 1.0, + "content": ", the condition", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 107, + 437, + 506, + 450 + ], + "spans": [ + { + "bbox": [ + 107, + 437, + 172, + 449 + ], + "score": 0.92, + "content": "f ( V ) = f ( V Q )", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 437, + 223, + 450 + ], + "score": 1.0, + "content": "implies that", + "type": "text" + }, + { + "bbox": [ + 223, + 438, + 231, + 449 + ], + "score": 0.86, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 437, + 506, + 450 + ], + "score": 1.0, + "content": "is a constant function on orthogonal matrices, since any orthogonal", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 448, + 507, + 462 + ], + "spans": [ + { + "bbox": [ + 104, + 448, + 136, + 462 + ], + "score": 1.0, + "content": "matrix", + "type": "text" + }, + { + "bbox": [ + 136, + 450, + 149, + 460 + ], + "score": 0.58, + "content": "W", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 448, + 231, + 462 + ], + "score": 1.0, + "content": "can be obtained as", + "type": "text" + }, + { + "bbox": [ + 231, + 449, + 275, + 460 + ], + "score": 0.89, + "content": "W = V Q", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 448, + 292, + 462 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 293, + 448, + 380, + 461 + ], + "score": 0.92, + "content": "Q = V ^ { \\top } W \\in O ( n )", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 448, + 507, + 462 + ], + "score": 1.0, + "content": ". 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While the columns of", + "type": "text" + }, + { + "bbox": [ + 336, + 471, + 419, + 483 + ], + "score": 0.9, + "content": "V \\mathrm { D i a g } ( Q _ { 1 } , \\dots , Q _ { l } )", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 470, + 507, + 484 + ], + "score": 1.0, + "content": "are still eigenvectors,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 481, + 259, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 481, + 168, + 495 + ], + "score": 1.0, + "content": "the columns of", + "type": "text" + }, + { + "bbox": [ + 169, + 483, + 185, + 493 + ], + "score": 0.88, + "content": "V Q", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 481, + 259, + 495 + ], + "score": 1.0, + "content": "generally are not.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 22.0 + }, + { + "type": "title", + "bbox": [ + 87, + 507, + 303, + 519 + ], + "lines": [ + { + "bbox": [ + 84, + 506, + 304, + 522 + ], + "spans": [ + { + "bbox": [ + 84, + 506, + 304, + 522 + ], + "score": 1.0, + "content": "916 E.3 Graph Spectra and Learning on Graphs", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 105, + 528, + 505, + 627 + ], + "lines": [ + { + "bbox": [ + 105, + 527, + 506, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 506, + 541 + ], + "score": 1.0, + "content": "More generally, graph spectra are widely used in analyzing graphs, and spectral graph theory [Chung,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 537, + 506, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 506, + 551 + ], + "score": 1.0, + "content": "1997] studies the connection between graph properties and graph spectra. Different graph kernels", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 549, + 506, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 506, + 562 + ], + "score": 1.0, + "content": "have been defined based on graph spectra, which use robust and discriminative notions of generalized", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 560, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 505, + 573 + ], + "score": 1.0, + "content": "spectral distance [Verma and Zhang, 2017], the spectral density of states [Huang et al., 2021], random", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 571, + 506, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 506, + 584 + ], + "score": 1.0, + "content": "walk return probabilities [Zhang et al., 2018b], or the trace of the heat kernel [Tsitsulin et al., 2018].", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 582, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 582, + 506, + 595 + ], + "score": 1.0, + "content": "Graph signal processing relies on spectral operations to define Fourier transforms, frequencies,", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 593, + 505, + 605 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 505, + 605 + ], + "score": 1.0, + "content": "convolutions, and other useful concepts for processing data on graphs [Ortega et al., 2018]. The", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 603, + 506, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 506, + 617 + ], + "score": 1.0, + "content": "closely related spectral graph neural networks [Wu et al., 2020, Balcilar et al., 2020] parameterize", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 615, + 367, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 367, + 628 + ], + "score": 1.0, + "content": "neural architectures that are based on similar spectral operations.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 39 + }, + { + "type": "title", + "bbox": [ + 104, + 643, + 324, + 657 + ], + "lines": [ + { + "bbox": [ + 104, + 641, + 325, + 659 + ], + "spans": [ + { + "bbox": [ + 104, + 641, + 325, + 659 + ], + "score": 1.0, + "content": "F Definitions, Notation, and Background", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 44 + }, + { + "type": "title", + "bbox": [ + 102, + 668, + 298, + 680 + ], + "lines": [ + { + "bbox": [ + 105, + 668, + 299, + 682 + ], + "spans": [ + { + "bbox": [ + 105, + 668, + 299, + 682 + ], + "score": 1.0, + "content": "F.1 Basic Topology and Algebra Definitions", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 45 + }, + { + "type": "text", + "bbox": [ + 105, + 689, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 505, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 478, + 702 + ], + "score": 1.0, + "content": "We will use some basic topology and algebra for our theoretical results. A topological space", + "type": "text" + }, + { + "bbox": [ + 478, + 689, + 505, + 701 + ], + "score": 0.91, + "content": "( \\mathcal { X } , \\tau )", + "type": "inline_equation" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 699, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 137, + 713 + ], + "score": 1.0, + "content": "is a set", + "type": "text" + }, + { + "bbox": [ + 137, + 700, + 147, + 710 + ], + "score": 0.82, + "content": "\\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 699, + 273, + 713 + ], + "score": 1.0, + "content": "along with a family of subsets", + "type": "text" + }, + { + "bbox": [ + 274, + 699, + 306, + 711 + ], + "score": 0.94, + "content": "\\tau \\subseteq 2 ^ { \\mathcal { X } }", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 699, + 506, + 713 + ], + "score": 1.0, + "content": "satisfying certain properties, which gives useful", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 711, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 437, + 723 + ], + "score": 1.0, + "content": "notions like continuity and compactness. 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This indeed makes sense for network", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 393, + 506, + 406 + ], + "spans": [ + { + "bbox": [ + 106, + 393, + 506, + 406 + ], + "score": 1.0, + "content": "embeddings like node2vec [Grover and Leskovec, 2016], as their objective functions are based", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 405, + 505, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 505, + 417 + ], + "score": 1.0, + "content": "on inner products and are thus orthogonally invariant. While they prove stability results when", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 416, + 506, + 428 + ], + "spans": [ + { + "bbox": [ + 106, + 416, + 506, + 428 + ], + "score": 1.0, + "content": "enforcing full orthogonal invariance for eigenvectors, this is a very strict constraint compared to", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 426, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 427, + 275, + 438 + ], + "score": 1.0, + "content": "our basis invariance. For instance, when", + "type": "text" + }, + { + "bbox": [ + 275, + 426, + 302, + 437 + ], + "score": 0.9, + "content": "k = n", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 427, + 435, + 438 + ], + "score": 1.0, + "content": "and all eigenvectors are used in", + "type": "text" + }, + { + "bbox": [ + 435, + 426, + 444, + 437 + ], + "score": 0.72, + "content": "V", + "type": "inline_equation" + }, + { + "bbox": [ + 445, + 427, + 505, + 438 + ], + "score": 1.0, + "content": ", the condition", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 107, + 437, + 506, + 450 + ], + "spans": [ + { + "bbox": [ + 107, + 437, + 172, + 449 + ], + "score": 0.92, + "content": "f ( V ) = f ( V Q )", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 437, + 223, + 450 + ], + "score": 1.0, + "content": "implies that", + "type": "text" + }, + { + "bbox": [ + 223, + 438, + 231, + 449 + ], + "score": 0.86, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 437, + 506, + 450 + ], + "score": 1.0, + "content": "is a constant function on orthogonal matrices, since any orthogonal", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 448, + 507, + 462 + ], + "spans": [ + { + "bbox": [ + 104, + 448, + 136, + 462 + ], + "score": 1.0, + "content": "matrix", + "type": "text" + }, + { + "bbox": [ + 136, + 450, + 149, + 460 + ], + "score": 0.58, + "content": "W", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 448, + 231, + 462 + ], + "score": 1.0, + "content": "can be obtained as", + "type": "text" + }, + { + "bbox": [ + 231, + 449, + 275, + 460 + ], + "score": 0.89, + "content": "W = V Q", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 448, + 292, + 462 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 293, + 448, + 380, + 461 + ], + "score": 0.92, + "content": "Q = V ^ { \\top } W \\in O ( n )", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 448, + 507, + 462 + ], + "score": 1.0, + "content": ". 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While the columns of", + "type": "text" + }, + { + "bbox": [ + 336, + 471, + 419, + 483 + ], + "score": 0.9, + "content": "V \\mathrm { D i a g } ( Q _ { 1 } , \\dots , Q _ { l } )", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 470, + 507, + 484 + ], + "score": 1.0, + "content": "are still eigenvectors,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 481, + 259, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 481, + 168, + 495 + ], + "score": 1.0, + "content": "the columns of", + "type": "text" + }, + { + "bbox": [ + 169, + 483, + 185, + 493 + ], + "score": 0.88, + "content": "V Q", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 481, + 259, + 495 + ], + "score": 1.0, + "content": "generally are not.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 22.0, + "bbox_fs": [ + 104, + 360, + 507, + 495 + ] + }, + { + "type": "title", + "bbox": [ + 87, + 507, + 303, + 519 + ], + "lines": [ + { + "bbox": [ + 84, + 506, + 304, + 522 + ], + "spans": [ + { + "bbox": [ + 84, + 506, + 304, + 522 + ], + "score": 1.0, + "content": "916 E.3 Graph Spectra and Learning on Graphs", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 105, + 528, + 505, + 627 + ], + "lines": [ + { + "bbox": [ + 105, + 527, + 506, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 506, + 541 + ], + "score": 1.0, + "content": "More generally, graph spectra are widely used in analyzing graphs, and spectral graph theory [Chung,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 537, + 506, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 506, + 551 + ], + "score": 1.0, + "content": "1997] studies the connection between graph properties and graph spectra. 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The", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 603, + 506, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 506, + 617 + ], + "score": 1.0, + "content": "closely related spectral graph neural networks [Wu et al., 2020, Balcilar et al., 2020] parameterize", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 615, + 367, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 367, + 628 + ], + "score": 1.0, + "content": "neural architectures that are based on similar spectral operations.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 39, + "bbox_fs": [ + 105, + 527, + 506, + 628 + ] + }, + { + "type": "title", + "bbox": [ + 104, + 643, + 324, + 657 + ], + "lines": [ + { + "bbox": [ + 104, + 641, + 325, + 659 + ], + "spans": [ + { + "bbox": [ + 104, + 641, + 325, + 659 + ], + "score": 1.0, + "content": "F Definitions, Notation, and Background", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 44 + }, + { + "type": "title", + "bbox": [ + 102, + 668, + 298, + 680 + ], + "lines": [ + { + "bbox": [ + 105, + 668, + 299, + 682 + ], + "spans": [ + { + "bbox": [ + 105, + 668, + 299, + 682 + ], + "score": 1.0, + "content": "F.1 Basic Topology and Algebra Definitions", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 45 + }, + { + "type": "text", + "bbox": [ + 105, + 689, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 505, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 478, + 702 + ], + "score": 1.0, + "content": "We will use some basic topology and algebra for our theoretical results. 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We will use this relationship in our proofs of universal representation.", + "type": "text" + } + ], + "index": 35, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 546, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 86, + 547, + 100, + 557 + ], + "score": 1.0, + "content": "965", + "type": "text" + }, + { + "bbox": [ + 106, + 546, + 505, + 558 + ], + "score": 1.0, + "content": "When we consider permutation invariance or equivariance, the permutation acts on dimensions of size", + "type": "text" + } + ], + "index": 36, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 554, + 508, + 573 + ], + "spans": [ + { + "bbox": [ + 86, + 560, + 100, + 569 + ], + "score": 1.0, + "content": "966", + "type": "text" + }, + { + "bbox": [ + 106, + 561, + 114, + 568 + ], + "score": 0.67, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 554, + 175, + 573 + ], + "score": 1.0, + "content": ". Then a tensor", + "type": "text" + }, + { + "bbox": [ + 175, + 556, + 225, + 569 + ], + "score": 0.93, + "content": "\\bar { X } \\in \\mathbb { R } ^ { n ^ { k } \\times d }", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 554, + 297, + 573 + ], + "score": 1.0, + "content": "is called an order", + "type": "text" + }, + { + "bbox": [ + 297, + 559, + 304, + 568 + ], + "score": 0.82, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 554, + 508, + 573 + ], + "score": 1.0, + "content": "tensor with respect to this permutation symmetry,", + "type": "text" + } + ], + "index": 37, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 569, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 86, + 572, + 100, + 581 + ], + "score": 1.0, + "content": "967", + "type": "text" + }, + { + "bbox": [ + 105, + 569, + 505, + 582 + ], + "score": 1.0, + "content": "where order 0 are called scalars, order 1 tensors are called vectors, and order 2 tensors are called", + "type": "text" + } + ], + "index": 38, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 581, + 505, + 592 + ], + "spans": [ + { + "bbox": [ + 86, + 583, + 100, + 592 + ], + "score": 1.0, + "content": "968", + "type": "text" + }, + { + "bbox": [ + 106, + 581, + 279, + 592 + ], + "score": 1.0, + "content": "matrices. Note that this does not depend on", + "type": "text" + }, + { + "bbox": [ + 279, + 581, + 285, + 590 + ], + "score": 0.78, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 581, + 505, + 592 + ], + "score": 1.0, + "content": "; in this work, we only ever consider vectors and scalars", + "type": "text" + } + ], + "index": 39, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 591, + 235, + 603 + ], + "spans": [ + { + "bbox": [ + 86, + 594, + 100, + 602 + ], + "score": 1.0, + "content": "969", + "type": "text" + }, + { + "bbox": [ + 105, + 591, + 182, + 603 + ], + "score": 1.0, + "content": "with respect to the", + "type": "text" + }, + { + "bbox": [ + 182, + 591, + 204, + 603 + ], + "score": 0.92, + "content": "O ( d )", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 591, + 235, + 603 + ], + "score": 1.0, + "content": "action.", + "type": "text" + } + ], + "index": 40, + "is_list_start_line": true + } + ], + "index": 32.5, + "bbox_fs": [ + 85, + 472, + 507, + 541 + ] + }, + { + "type": "index", + "bbox": [ + 88, + 545, + 505, + 603 + ], + "lines": [], + "index": 38, + "bbox_fs": [ + 86, + 546, + 508, + 603 + ], + "lines_deleted": true + }, + { + "type": "title", + "bbox": [ + 97, + 619, + 240, + 633 + ], + "lines": [ + { + "bbox": [ + 93, + 617, + 241, + 637 + ], + "spans": [ + { + "bbox": [ + 93, + 617, + 241, + 637 + ], + "score": 1.0, + "content": "70 G Proofs of Universality", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 104, + 645, + 459, + 658 + ], + "lines": [ + { + "bbox": [ + 105, + 645, + 460, + 659 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 460, + 659 + ], + "score": 1.0, + "content": "We begin by proving the two propositions for the single subspace case from Section 2.1.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 42, + "bbox_fs": [ + 105, + 645, + 460, + 659 + ] + }, + { + "type": "text", + "bbox": [ + 99, + 660, + 439, + 673 + ], + "lines": [ + { + "bbox": [ + 104, + 658, + 439, + 676 + ], + "spans": [ + { + "bbox": [ + 104, + 658, + 261, + 676 + ], + "score": 1.0, + "content": "Proposition 1. A continuous function", + "type": "text" + }, + { + "bbox": [ + 261, + 661, + 315, + 671 + ], + "score": 0.89, + "content": "h : \\mathbb { R } ^ { n } \\mathbb { R } ^ { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 658, + 439, + 676 + ], + "score": 1.0, + "content": "is sign invariant if and only if", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 43, + "bbox_fs": [ + 104, + 658, + 439, + 676 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 259, + 680, + 352, + 693 + ], + "lines": [ + { + "bbox": [ + 259, + 680, + 352, + 693 + ], + "spans": [ + { + "bbox": [ + 259, + 680, + 352, + 693 + ], + "score": 0.92, + "content": "h ( v ) = \\phi ( v ) + \\phi ( - v )", + "type": "interline_equation", + "image_path": "7145e66f5f1c9e895db97c9e74b94e0c4fd232537ad908a1b48c00c760d1953c.jpg" + } + ] + } + ], + "index": 44, + "virtual_lines": [ + { + "bbox": [ + 259, + 680, + 352, + 693 + ], + "spans": [], + "index": 44 + } + ] + }, + { + "type": "text", + "bbox": [ + 85, + 699, + 508, + 723 + ], + "lines": [ + { + "bbox": [ + 84, + 698, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 84, + 698, + 192, + 713 + ], + "score": 1.0, + "content": "for some continuous 973", + "type": "text" + }, + { + "bbox": [ + 192, + 701, + 247, + 711 + ], + "score": 0.88, + "content": "\\phi : \\mathbb { R } ^ { n } \\mathbb { R } ^ { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 698, + 308, + 713 + ], + "score": 1.0, + "content": ". 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If", + "type": "text" + }, + { + "bbox": [ + 146, + 73, + 237, + 85 + ], + "score": 0.91, + "content": "h ( v ) = \\phi ( v ) + \\phi ( - v )", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 72, + 260, + 86 + ], + "score": 1.0, + "content": ", then", + "type": "text" + }, + { + "bbox": [ + 261, + 73, + 267, + 83 + ], + "score": 0.82, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 72, + 468, + 86 + ], + "score": 1.0, + "content": "is obviously sign invariant. 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Any continuous,", + "type": "text" + }, + { + "bbox": [ + 235, + 151, + 257, + 164 + ], + "score": 0.91, + "content": "O ( d )", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 149, + 297, + 165 + ], + "score": 1.0, + "content": "invariant", + "type": "text" + }, + { + "bbox": [ + 297, + 150, + 361, + 162 + ], + "score": 0.9, + "content": "h : \\mathbb { R } ^ { n \\times d } \\mathbb { R } ^ { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 149, + 417, + 165 + ], + "score": 1.0, + "content": "is of the form", + "type": "text" + }, + { + "bbox": [ + 418, + 150, + 489, + 163 + ], + "score": 0.9, + "content": "h ( V ) = \\phi ( V V ^ { \\top } )", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 149, + 506, + 165 + ], + "score": 1.0, + "content": "for", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 86, + 161, + 506, + 177 + ], + "spans": [ + { + "bbox": [ + 86, + 165, + 100, + 175 + ], + "score": 1.0, + "content": "982", + "type": "text" + }, + { + "bbox": [ + 103, + 161, + 160, + 177 + ], + "score": 1.0, + "content": "a continuous", + "type": "text" + }, + { + "bbox": [ + 160, + 164, + 167, + 175 + ], + "score": 0.81, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 161, + 264, + 177 + ], + "score": 1.0, + "content": ". For a compact domain", + "type": "text" + }, + { + "bbox": [ + 264, + 163, + 309, + 174 + ], + "score": 0.92, + "content": "{ \\mathcal { Z } } \\subseteq \\mathbb { R } ^ { n \\times d }", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 161, + 383, + 177 + ], + "score": 1.0, + "content": ", maps of the form", + "type": "text" + }, + { + "bbox": [ + 383, + 163, + 457, + 176 + ], + "score": 0.91, + "content": "V \\mapsto \\operatorname { I G N } ( V V ^ { \\top } )", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 161, + 506, + 177 + ], + "score": 1.0, + "content": "universally", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 85, + 173, + 507, + 189 + ], + "spans": [ + { + "bbox": [ + 85, + 177, + 100, + 187 + ], + "score": 1.0, + "content": "983", + "type": "text" + }, + { + "bbox": [ + 103, + 173, + 247, + 189 + ], + "score": 1.0, + "content": "approximate continuous functions", + "type": "text" + }, + { + "bbox": [ + 248, + 174, + 336, + 186 + ], + "score": 0.9, + "content": "h : \\mathcal { Z } \\subseteq \\mathbb { R } ^ { n \\times d } \\to \\mathbb { R } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 173, + 372, + 189 + ], + "score": 1.0, + "content": "that are", + "type": "text" + }, + { + "bbox": [ + 372, + 175, + 394, + 187 + ], + "score": 0.92, + "content": "O ( d )", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 173, + 507, + 189 + ], + "score": 1.0, + "content": "invariant and permutation", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 86, + 187, + 157, + 199 + ], + "spans": [ + { + "bbox": [ + 86, + 189, + 100, + 198 + ], + "score": 1.0, + "content": "984", + "type": "text" + }, + { + "bbox": [ + 105, + 187, + 157, + 199 + ], + "score": 1.0, + "content": "equivariant.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7.5 + }, + { + "type": "text", + "bbox": [ + 94, + 210, + 505, + 234 + ], + "lines": [ + { + "bbox": [ + 90, + 209, + 505, + 224 + ], + "spans": [ + { + "bbox": [ + 90, + 209, + 483, + 224 + ], + "score": 1.0, + "content": "85 Proof. The case without permutation equivariance holds by the First Fundamental Theorem of", + "type": "text" + }, + { + "bbox": [ + 483, + 210, + 505, + 223 + ], + "score": 0.91, + "content": "O ( d )", + "type": "inline_equation" + } + ], + "index": 10 + }, + { + "bbox": [ + 91, + 221, + 156, + 234 + ], + "spans": [ + { + "bbox": [ + 91, + 221, + 156, + 234 + ], + "score": 1.0, + "content": "86 (Lemma 2).", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5 + }, + { + "type": "text", + "bbox": [ + 86, + 236, + 505, + 294 + ], + "lines": [ + { + "bbox": [ + 86, + 236, + 504, + 250 + ], + "spans": [ + { + "bbox": [ + 86, + 240, + 99, + 249 + ], + "score": 1.0, + "content": "987", + "type": "text" + }, + { + "bbox": [ + 105, + 236, + 278, + 249 + ], + "score": 1.0, + "content": "For the permutation equivariant case, let", + "type": "text" + }, + { + "bbox": [ + 278, + 236, + 383, + 250 + ], + "score": 0.92, + "content": "\\mathcal { Z } ^ { \\prime } = \\{ V V ^ { \\top } : V \\in \\mathcal { Z } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 236, + 416, + 249 + ], + "score": 1.0, + "content": "and let", + "type": "text" + }, + { + "bbox": [ + 416, + 238, + 443, + 248 + ], + "score": 0.88, + "content": "\\epsilon > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 236, + 492, + 249 + ], + "score": 1.0, + "content": ". Note that", + "type": "text" + }, + { + "bbox": [ + 492, + 238, + 504, + 248 + ], + "score": 0.85, + "content": "\\mathcal { Z } ^ { \\prime }", + "type": "inline_equation" + } + ], + "index": 12 + }, + { + "bbox": [ + 86, + 249, + 505, + 261 + ], + "spans": [ + { + "bbox": [ + 86, + 251, + 99, + 260 + ], + "score": 1.0, + "content": "988", + "type": "text" + }, + { + "bbox": [ + 105, + 249, + 385, + 261 + ], + "score": 1.0, + "content": "is compact, as it is the continuous image of a compact set. 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Then note that Keriven", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 86, + 346, + 505, + 358 + ], + "spans": [ + { + "bbox": [ + 86, + 348, + 99, + 357 + ], + "score": 1.0, + "content": "993", + "type": "text" + }, + { + "bbox": [ + 106, + 346, + 505, + 358 + ], + "score": 1.0, + "content": "and Peyré [2019] show that invariant graph networks (of generally high tensor order in hidden layers)", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 86, + 357, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 86, + 359, + 99, + 368 + ], + "score": 1.0, + "content": "994", + "type": "text" + }, + { + "bbox": [ + 105, + 357, + 505, + 369 + ], + "score": 1.0, + "content": "universally approximate continuous permutation equivariant functions from matrices to vectors on", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 85, + 366, + 506, + 380 + ], + "spans": [ + { + "bbox": [ + 85, + 369, + 100, + 379 + ], + "score": 1.0, + "content": "995", + "type": "text" + }, + { + "bbox": [ + 104, + 366, + 293, + 380 + ], + "score": 1.0, + "content": "compact sets of matrices. 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Black arrows denote functions from", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "topological constructions, and red dashed lines denote functions that we parameterize by neural", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 604, + 257, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 147, + 617 + ], + "score": 1.0, + "content": "networks", + "type": "text" + }, + { + "bbox": [ + 147, + 605, + 226, + 616 + ], + "score": 0.9, + "content": "( \\phi = \\phi _ { 1 } \\times \\ldots \\times \\phi _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 604, + 244, + 617 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 245, + 606, + 252, + 616 + ], + "score": 0.75, + "content": "\\rho \\mathrm { \\hbar }", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 604, + 257, + 617 + ], + "score": 1.0, + "content": ").", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 84, + 628, + 505, + 695 + ], + "lines": [ + { + "bbox": [ + 85, + 628, + 506, + 642 + ], + "spans": [ + { + "bbox": [ + 85, + 631, + 100, + 641 + ], + "score": 1.0, + "content": "998", + "type": "text" + }, + { + "bbox": [ + 105, + 628, + 506, + 642 + ], + "score": 1.0, + "content": "Here, we give the formal statement of Theorem 3, which provides the necessary topological assump-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 85, + 639, + 506, + 653 + ], + "spans": [ + { + "bbox": [ + 85, + 642, + 99, + 651 + ], + "score": 1.0, + "content": "999", + "type": "text" + }, + { + "bbox": [ + 105, + 639, + 362, + 653 + ], + "score": 1.0, + "content": "tions for the theorem to hold. 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If", + "type": "text" + }, + { + "bbox": [ + 146, + 73, + 237, + 85 + ], + "score": 0.91, + "content": "h ( v ) = \\phi ( v ) + \\phi ( - v )", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 72, + 260, + 86 + ], + "score": 1.0, + "content": ", then", + "type": "text" + }, + { + "bbox": [ + 261, + 73, + 267, + 83 + ], + "score": 0.82, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 72, + 468, + 86 + ], + "score": 1.0, + "content": "is obviously sign invariant. 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Also, if the base space", + "type": "text" + }, + { + "bbox": [ + 370, + 95, + 382, + 105 + ], + "score": 0.89, + "content": "\\mathcal { X } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 93, + 482, + 107 + ], + "score": 1.0, + "content": "is a Euclidean space and", + "type": "text" + }, + { + "bbox": [ + 483, + 95, + 495, + 105 + ], + "score": 0.88, + "content": "G _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 93, + 506, + 107 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 83, + 104, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 83, + 107, + 99, + 117 + ], + "score": 1.0, + "content": "1009", + "type": "text" + }, + { + "bbox": [ + 104, + 104, + 347, + 118 + ], + "score": 1.0, + "content": "a finite or compact matrix Lie group, then a map built from", + "type": "text" + }, + { + "bbox": [ + 348, + 106, + 356, + 115 + ], + "score": 0.82, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 104, + 506, + 118 + ], + "score": 1.0, + "content": "-invariant polynomials gives such an", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 83, + 116, + 340, + 128 + ], + "spans": [ + { + "bbox": [ + 83, + 119, + 99, + 127 + ], + "score": 1.0, + "content": "1010", + "type": "text" + }, + { + "bbox": [ + 105, + 116, + 340, + 128 + ], + "score": 1.0, + "content": "embedding (González and de Salas [2003] Lemma 11.13).", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 99, + 132, + 454, + 145 + ], + "lines": [ + { + "bbox": [ + 105, + 132, + 455, + 146 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 455, + 146 + ], + "score": 1.0, + "content": "Figure 10 provides a commutative diagram representing the constructions in our proof.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 83, + 148, + 506, + 216 + ], + "lines": [ + { + "bbox": [ + 82, + 148, + 504, + 161 + ], + "spans": [ + { + "bbox": [ + 82, + 150, + 100, + 160 + ], + "score": 1.0, + "content": "1012", + "type": "text" + }, + { + "bbox": [ + 105, + 148, + 291, + 161 + ], + "score": 1.0, + "content": "Theorem 3 (Decomposition Theorem). Let", + "type": "text" + }, + { + "bbox": [ + 292, + 149, + 338, + 160 + ], + "score": 0.89, + "content": "\\mathcal { X } _ { 1 } , \\ldots , \\mathcal { X } _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 148, + 469, + 161 + ], + "score": 1.0, + "content": "be topological spaces, and let", + "type": "text" + }, + { + "bbox": [ + 470, + 149, + 482, + 159 + ], + "score": 0.86, + "content": "G _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 148, + 497, + 161 + ], + "score": 1.0, + "content": "be", + "type": "text" + }, + { + "bbox": [ + 497, + 150, + 504, + 158 + ], + "score": 0.37, + "content": "a", + "type": "inline_equation" + } + ], + "index": 6 + }, + { + "bbox": [ + 82, + 158, + 505, + 173 + ], + "spans": [ + { + "bbox": [ + 82, + 161, + 100, + 171 + ], + "score": 1.0, + "content": "1013", + "type": "text" + }, + { + "bbox": [ + 105, + 158, + 269, + 173 + ], + "score": 1.0, + "content": "topological group acting continuously on", + "type": "text" + }, + { + "bbox": [ + 270, + 160, + 281, + 171 + ], + "score": 0.83, + "content": "\\mathcal { X } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 158, + 505, + 173 + ], + "score": 1.0, + "content": "for each i. Assume that there is a topological embedding", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 82, + 169, + 507, + 184 + ], + "spans": [ + { + "bbox": [ + 82, + 172, + 100, + 182 + ], + "score": 1.0, + "content": "1014", + "type": "text" + }, + { + "bbox": [ + 107, + 170, + 190, + 182 + ], + "score": 0.91, + "content": "\\psi _ { i } : \\mathcal { X } _ { i } / G _ { i } \\to \\mathbb { R } ^ { a _ { i } }", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 169, + 389, + 184 + ], + "score": 1.0, + "content": "of each quotient space into a Euclidean space", + "type": "text" + }, + { + "bbox": [ + 389, + 171, + 405, + 181 + ], + "score": 0.8, + "content": "\\mathbb { R } ^ { a _ { i } }", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 169, + 493, + 184 + ], + "score": 1.0, + "content": "for some dimension", + "type": "text" + }, + { + "bbox": [ + 493, + 175, + 502, + 181 + ], + "score": 0.78, + "content": "a _ { i }", + "type": "inline_equation" + }, + { 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Let 1020", + "type": "text" + }, + { + "bbox": [ + 153, + 281, + 223, + 293 + ], + "score": 0.94, + "content": "\\pi _ { i } : \\mathcal { X } _ { i } \\mathcal { X } _ { i } / G _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 279, + 339, + 294 + ], + "score": 1.0, + "content": "denote the quotient map for", + "type": "text" + }, + { + "bbox": [ + 339, + 281, + 366, + 293 + ], + "score": 0.92, + "content": "\\mathcal { X } _ { i } / G _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 279, + 417, + 294 + ], + "score": 1.0, + "content": ". 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By Lemma 4, each", + "type": "text" + }, + { + "bbox": [ + 302, + 384, + 330, + 396 + ], + "score": 0.92, + "content": "\\mathcal { X } _ { i } / G _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 384, + 387, + 397 + ], + "score": 1.0, + "content": "is compact if", + "type": "text" + }, + { + "bbox": [ + 388, + 385, + 399, + 395 + ], + "score": 0.88, + "content": "\\mathcal { X } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 384, + 505, + 397 + ], + "score": 1.0, + "content": "is compact. Defining the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 84, + 395, + 437, + 408 + ], + "spans": [ + { + "bbox": [ + 84, + 397, + 100, + 407 + ], + "score": 1.0, + "content": "1025", + "type": "text" + }, + { + "bbox": [ + 105, + 395, + 133, + 408 + ], + "score": 1.0, + "content": "image", + "type": "text" + }, + { + "bbox": [ + 133, + 396, + 230, + 407 + ], + "score": 0.93, + "content": "\\mathcal { Z } _ { i } = \\psi _ { i } ( \\mathcal { X } _ { i } / G _ { i } ) \\subseteq \\mathbb { R } ^ { a _ { i } }", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 395, + 309, + 408 + ], + "score": 1.0, + "content": ", we thus know that", + "type": "text" + }, + { + "bbox": [ + 310, + 396, + 321, + 406 + ], + "score": 0.88, + "content": "\\mathcal { Z } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 395, + 376, + 408 + ], + "score": 1.0, + "content": "is compact if", + "type": "text" + }, + { + "bbox": [ + 377, + 396, + 388, + 406 + ], + "score": 0.89, + "content": "\\mathcal { X } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 395, + 437, + 408 + ], + "score": 1.0, + "content": "is compact.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 96, + 412, + 506, + 448 + ], + "lines": [ + { + "bbox": [ + 92, + 410, + 507, + 426 + ], + "spans": [ + { + "bbox": [ + 92, + 410, + 159, + 426 + ], + "score": 1.0, + "content": "Moreover, as 26", + "type": "text" + }, + { + "bbox": [ + 160, + 413, + 171, + 424 + ], + "score": 0.89, + "content": "\\psi _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 410, + 387, + 426 + ], + "score": 1.0, + "content": "is a topological embedding, it has a continuous inverse", + "type": "text" + }, + { + "bbox": [ + 387, + 412, + 406, + 425 + ], + "score": 0.92, + "content": "\\psi _ { i } ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 410, + 457, + 426 + ], + "score": 1.0, + "content": "on its image", + "type": "text" + }, + { + "bbox": [ + 457, + 413, + 468, + 424 + ], + "score": 0.87, + "content": "\\mathcal { Z } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 469, + 410, + 507, + 426 + ], + "score": 1.0, + "content": ". Further,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 93, + 421, + 507, + 438 + ], + "spans": [ + { + "bbox": [ + 93, + 421, + 245, + 438 + ], + "score": 1.0, + "content": "27 we have a topological embedding", + "type": "text" + }, + { + "bbox": [ + 246, + 424, + 382, + 435 + ], + "score": 0.89, + "content": "\\psi : \\mathcal { X } / G \\to \\mathcal { Z } = \\mathcal { Z } _ { 1 } \\times . . . \\times \\mathcal { Z } _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 421, + 421, + 438 + ], + "score": 1.0, + "content": "given by", + "type": "text" + }, + { + "bbox": [ + 422, + 424, + 502, + 435 + ], + "score": 0.9, + "content": "\\psi = \\psi _ { 1 } \\times \\ldots \\times \\psi _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 421, + 507, + 438 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 91, + 431, + 313, + 453 + ], + "spans": [ + { + "bbox": [ + 91, + 431, + 203, + 453 + ], + "score": 1.0, + "content": "with continuous inverse 28", + "type": "text" + }, + { + "bbox": [ + 204, + 435, + 306, + 448 + ], + "score": 0.91, + "content": "\\psi ^ { - 1 } = \\psi _ { 1 } ^ { - 1 } \\times \\ldots \\times \\psi _ { k } ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 431, + 313, + 453 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 106, + 452, + 146, + 463 + ], + "lines": [ + { + "bbox": [ + 106, + 451, + 146, + 464 + ], + "spans": [ + { + "bbox": [ + 106, + 451, + 146, + 464 + ], + "score": 1.0, + "content": "Note that", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "interline_equation", + "bbox": [ + 236, + 462, + 375, + 478 + ], + "lines": [ + { + "bbox": [ + 236, + 462, + 375, + 478 + ], + "spans": [ + { + "bbox": [ + 236, + 462, + 375, + 478 + ], + "score": 0.91, + "content": "f = \\tilde { f } \\circ \\pi = ( \\tilde { f } \\circ \\psi ^ { - 1 } ) \\circ ( \\psi \\circ \\pi ) .", + "type": "interline_equation", + "image_path": "2c839791d2d9da809d8401d45f8649f1ee337cc7c0154017fdef228b71597202.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 236, + 462, + 375, + 478 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "text", + "bbox": [ + 84, + 483, + 160, + 494 + ], + "lines": [ + { + "bbox": [ + 82, + 481, + 161, + 496 + ], + "spans": [ + { + "bbox": [ + 82, + 481, + 161, + 496 + ], + "score": 1.0, + "content": "1030 So we define", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "interline_equation", + "bbox": [ + 180, + 498, + 431, + 544 + ], + "lines": [ + { + "bbox": [ + 180, + 498, + 431, + 543 + ], + "spans": [ + { + "bbox": [ + 180, + 498, + 431, + 543 + ], + "score": 0.89, + "content": "\\begin{array} { r l r } & { \\rho = \\tilde { f } \\circ \\psi ^ { - 1 } } & { \\rho : \\mathcal { Z } \\to \\mathbb { R } ^ { s } } \\\\ & { \\phi _ { i } = \\psi _ { i } \\circ \\pi _ { i } } & { \\phi _ { i } : \\mathcal { X } _ { i } \\to \\mathcal { Z } _ { i } } \\\\ & { \\phi = \\psi \\circ \\pi = \\phi _ { 1 } \\times \\ldots \\times \\phi _ { k } } & { \\phi : \\mathcal { X } \\to \\mathcal { Z } } \\end{array}", + "type": "interline_equation", + "image_path": "cd5a35a1ff401f8c296b0ecfe3e8d795a0e688bcd9cd9833432aec4fbb2ab164.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 180, + 498, + 431, + 513.3333333333334 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 180, + 513.3333333333334, + 431, + 528.6666666666667 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 180, + 528.6666666666667, + 431, + 544.0000000000001 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "text", + "bbox": [ + 84, + 549, + 506, + 594 + ], + "lines": [ + { + "bbox": [ + 82, + 549, + 507, + 563 + ], + "spans": [ + { + "bbox": [ + 82, + 551, + 99, + 561 + ], + "score": 1.0, + "content": "1031", + "type": "text" + }, + { + "bbox": [ + 105, + 549, + 131, + 563 + ], + "score": 1.0, + "content": "Thus,", + "type": "text" + }, + { + "bbox": [ + 131, + 549, + 260, + 561 + ], + "score": 0.92, + "content": "f = \\rho \\circ \\phi = \\rho \\circ ( \\phi _ { 1 } \\times \\ldots \\times \\phi _ { k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 549, + 408, + 563 + ], + "score": 1.0, + "content": ", so equation (9) holds. Moreover, the", + "type": "text" + }, + { + "bbox": [ + 408, + 551, + 415, + 561 + ], + "score": 0.83, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 549, + 432, + 563 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 433, + 550, + 443, + 561 + ], + "score": 0.88, + "content": "\\phi _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 549, + 507, + 563 + ], + "score": 1.0, + "content": "are continuous,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 83, + 560, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 83, + 563, + 99, + 572 + ], + "score": 1.0, + "content": "1032", + "type": "text" + }, + { + "bbox": [ + 105, + 560, + 444, + 573 + ], + "score": 1.0, + "content": "as they are compositions of continuous functions. 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Since each", + "type": "text" + }, + { + "bbox": [ + 324, + 572, + 335, + 582 + ], + "score": 0.88, + "content": "\\mathcal { Z } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 570, + 393, + 584 + ], + "score": 1.0, + "content": "is compact if", + "type": "text" + }, + { + "bbox": [ + 393, + 572, + 405, + 582 + ], + "score": 0.89, + "content": "\\mathcal { X } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 570, + 505, + 584 + ], + "score": 1.0, + "content": "is compact, the product", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 83, + 581, + 395, + 596 + ], + "spans": [ + { + "bbox": [ + 83, + 583, + 100, + 594 + ], + "score": 1.0, + "content": "1034", + "type": "text" + }, + { + "bbox": [ + 107, + 582, + 189, + 594 + ], + "score": 0.9, + "content": "\\mathcal { Z } = \\mathcal { Z } _ { 1 } \\times \\ldots \\times \\mathcal { Z } _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 581, + 264, + 596 + ], + "score": 1.0, + "content": "is compact if each", + "type": "text" + }, + { + "bbox": [ + 265, + 583, + 276, + 593 + ], + "score": 0.88, + "content": "\\mathcal { X } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 581, + 395, + 596 + ], + "score": 1.0, + "content": "is compact, thus proving (2).", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33.5 + }, + { + "type": "text", + "bbox": [ + 102, + 597, + 505, + 633 + ], + "lines": [ + { + "bbox": [ + 105, + 596, + 506, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 303, + 612 + ], + "score": 1.0, + "content": "To show the last statement (3), note simply that if", + "type": "text" + }, + { + "bbox": [ + 303, + 599, + 339, + 611 + ], + "score": 0.92, + "content": "{ \\mathcal { X } } _ { i } = { \\mathcal { X } } _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 596, + 357, + 612 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 358, + 598, + 395, + 611 + ], + "score": 0.92, + "content": "G _ { i } = G _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 395, + 596, + 506, + 612 + ], + "score": 1.0, + "content": ", then the quotient maps are", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 609, + 506, + 622 + ], + "spans": [ + { + "bbox": [ + 106, + 609, + 149, + 621 + ], + "score": 1.0, + "content": "equal, i.e.", + "type": "text" + }, + { + "bbox": [ + 150, + 611, + 183, + 622 + ], + "score": 0.87, + "content": "\\pi _ { i } = \\pi _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 609, + 440, + 621 + ], + "score": 1.0, + "content": ". Moreover, we can choose the embeddings to be equal, so say", + "type": "text" + }, + { + "bbox": [ + 440, + 609, + 475, + 622 + ], + "score": 0.92, + "content": "\\psi _ { i } = \\psi _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 609, + 506, + 621 + ], + "score": 1.0, + "content": ". Then,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 107, + 621, + 506, + 633 + ], + "spans": [ + { + "bbox": [ + 107, + 621, + 225, + 633 + ], + "score": 0.89, + "content": "\\phi _ { i } = \\psi _ { i } \\circ \\pi _ { i } = \\bar { \\psi _ { j } } \\circ \\pi _ { j } = \\phi _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 621, + 294, + 633 + ], + "score": 1.0, + "content": ", so we are done.", + "type": "text" + }, + { + "bbox": [ + 494, + 621, + 506, + 632 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37 + }, + { + "type": "title", + "bbox": [ + 102, + 645, + 291, + 658 + ], + "lines": [ + { + "bbox": [ + 106, + 645, + 291, + 661 + ], + "spans": [ + { + "bbox": [ + 106, + 645, + 291, + 661 + ], + "score": 1.0, + "content": "G.2 Universality of SignNet and BasisNet", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 91, + 667, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 89, + 667, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 89, + 669, + 100, + 678 + ], + "score": 1.0, + "content": "039", + "type": "text" + }, + { + "bbox": [ + 106, + 667, + 506, + 680 + ], + "score": 1.0, + "content": "Here, we prove Corollary 1 on the universal representation and approximation capabilities of our", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 89, + 678, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 89, + 680, + 100, + 689 + ], + "score": 1.0, + "content": "040", + "type": "text" + }, + { + "bbox": [ + 106, + 678, + 506, + 690 + ], + "score": 1.0, + "content": "Unconstrained-SignNets, Unconstrained-BasisNets, and Expressive-BasisNets. We proceed in sev-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 89, + 689, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 89, + 691, + 99, + 700 + ], + "score": 1.0, + "content": "041", + "type": "text" + }, + { + "bbox": [ + 105, + 689, + 506, + 702 + ], + "score": 1.0, + "content": "eral steps, first proving universal representation of continuous functions when we do not require", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 89, + 700, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 89, + 702, + 100, + 712 + ], + "score": 1.0, + "content": "042", + "type": "text" + }, + { + "bbox": [ + 105, + 700, + 506, + 712 + ], + "score": 1.0, + "content": "permutation equivariance, then proving universal approximation when we do require permutation", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 89, + 711, + 161, + 723 + ], + "spans": [ + { + "bbox": [ + 89, + 713, + 100, + 722 + ], + "score": 1.0, + "content": "043", + "type": "text" + }, + { + "bbox": [ + 105, + 711, + 161, + 723 + ], + "score": 1.0, + "content": "equivariance.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 42 + } + ], + "page_idx": 28, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 300, + 741, + 311, + 750 + ], + "lines": [ + { + "bbox": [ + 298, + 740, + 313, + 754 + ], + "spans": [ + { + "bbox": [ + 298, + 740, + 313, + 754 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 14, + "width": 15 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "index", + "bbox": [ + 83, + 72, + 505, + 128 + ], + "lines": [], + "index": 2, + "bbox_fs": [ + 83, + 72, + 507, + 128 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 99, + 132, + 454, + 145 + ], + "lines": [ + { + "bbox": [ + 105, + 132, + 455, + 146 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 455, + 146 + ], + "score": 1.0, + "content": "Figure 10 provides a commutative diagram representing the constructions in our proof.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5, + "bbox_fs": [ + 105, + 132, + 455, + 146 + ] + }, + { + "type": "index", + "bbox": [ + 83, + 148, + 506, + 216 + ], + "lines": [ + { + "bbox": [ + 82, + 148, + 504, + 161 + ], + "spans": [ + { + "bbox": [ + 82, + 150, + 100, + 160 + ], + "score": 1.0, + "content": "1012", + "type": "text" + }, + { + "bbox": [ + 105, + 148, + 291, + 161 + ], + "score": 1.0, + "content": "Theorem 3 (Decomposition Theorem). Let", + "type": "text" + }, + { + "bbox": [ + 292, + 149, + 338, + 160 + ], + "score": 0.89, + "content": "\\mathcal { X } _ { 1 } , \\ldots , \\mathcal { X } _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 148, + 469, + 161 + ], + "score": 1.0, + "content": "be topological spaces, and let", + "type": "text" + }, + { + "bbox": [ + 470, + 149, + 482, + 159 + ], + "score": 0.86, + "content": "G _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 148, + 497, + 161 + ], + "score": 1.0, + "content": "be", + "type": "text" + }, + { + "bbox": [ + 497, + 150, + 504, + 158 + ], + "score": 0.37, + "content": "a", + "type": "inline_equation" + } + ], + "index": 6, + "is_list_start_line": true + }, + { + "bbox": [ + 82, + 158, + 505, + 173 + ], + "spans": [ + { + "bbox": [ + 82, + 161, + 100, + 171 + ], + "score": 1.0, + "content": "1013", + "type": "text" + }, + { + "bbox": [ + 105, + 158, + 269, + 173 + ], + "score": 1.0, + "content": "topological group acting continuously on", + "type": "text" + }, + { + "bbox": [ + 270, + 160, + 281, + 171 + ], + "score": 0.83, + "content": "\\mathcal { X } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 158, + 505, + 173 + ], + "score": 1.0, + "content": "for each i. Assume that there is a topological embedding", + "type": "text" + } + ], + "index": 7, + "is_list_start_line": true + }, + { + "bbox": [ + 82, + 169, + 507, + 184 + ], + "spans": [ + { + "bbox": [ + 82, + 172, + 100, + 182 + ], + "score": 1.0, + "content": "1014", + "type": "text" + }, + { + "bbox": [ + 107, + 170, + 190, + 182 + ], + "score": 0.91, + "content": "\\psi _ { i } : \\mathcal { X } _ { i } / G _ { i } \\to \\mathbb { R } ^ { a _ { i } }", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 169, + 389, + 184 + ], + "score": 1.0, + "content": "of each quotient space into a Euclidean space", + "type": "text" + }, + { + "bbox": [ + 389, + 171, + 405, + 181 + ], + "score": 0.8, + "content": "\\mathbb { R } ^ { a _ { i } }", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 169, + 493, + 184 + ], + "score": 1.0, + "content": "for some dimension", + "type": "text" + }, + { + "bbox": [ + 493, + 175, + 502, + 181 + ], + "score": 0.78, + "content": "a _ { i }", + 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Let 1020", + "type": "text" + }, + { + "bbox": [ + 153, + 281, + 223, + 293 + ], + "score": 0.94, + "content": "\\pi _ { i } : \\mathcal { X } _ { i } \\mathcal { X } _ { i } / G _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 279, + 339, + 294 + ], + "score": 1.0, + "content": "denote the quotient map for", + "type": "text" + }, + { + "bbox": [ + 339, + 281, + 366, + 293 + ], + "score": 0.92, + "content": "\\mathcal { X } _ { i } / G _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 279, + 417, + 294 + ], + "score": 1.0, + "content": ". 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Moreover, the", + "type": "text" + }, + { + "bbox": [ + 408, + 551, + 415, + 561 + ], + "score": 0.83, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 549, + 432, + 563 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 433, + 550, + 443, + 561 + ], + "score": 0.88, + "content": "\\phi _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 549, + 507, + 563 + ], + "score": 1.0, + "content": "are continuous,", + "type": "text" + } + ], + "index": 32, + "is_list_start_line": true + }, + { + "bbox": [ + 83, + 560, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 83, + 563, + 99, + 572 + ], + "score": 1.0, + "content": "1032", + "type": "text" + }, + { + "bbox": [ + 105, + 560, + 444, + 573 + ], + "score": 1.0, + "content": "as they are compositions of continuous functions. Furthermore, (1) holds as each", + "type": "text" + }, + { + "bbox": [ + 445, + 561, + 455, + 572 + ], + "score": 0.89, + "content": "\\phi _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 560, + 505, + 573 + ], + "score": 1.0, + "content": "is invariant", + "type": "text" + } + ], + "index": 33, + "is_list_start_line": true + }, + { + "bbox": [ + 83, + 570, + 505, + 584 + ], + "spans": [ + { + "bbox": [ + 83, + 573, + 100, + 582 + ], + "score": 1.0, + "content": "1033", + "type": "text" + }, + { + "bbox": [ + 105, + 570, + 117, + 584 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 118, + 572, + 129, + 582 + ], + "score": 0.88, + "content": "G _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 130, + 570, + 187, + 584 + ], + "score": 1.0, + "content": "because each", + "type": "text" + }, + { + "bbox": [ + 188, + 572, + 198, + 582 + ], + "score": 0.85, + "content": "\\pi _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 570, + 258, + 584 + ], + "score": 1.0, + "content": "is invariant to", + "type": "text" + }, + { + "bbox": [ + 259, + 571, + 271, + 582 + ], + "score": 0.88, + "content": "G _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 570, + 324, + 584 + ], + "score": 1.0, + "content": ". Since each", + "type": "text" + }, + { + "bbox": [ + 324, + 572, + 335, + 582 + ], + "score": 0.88, + "content": "\\mathcal { Z } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 570, + 393, + 584 + ], + "score": 1.0, + "content": "is compact if", + "type": "text" + }, + { + "bbox": [ + 393, + 572, + 405, + 582 + ], + "score": 0.89, + "content": "\\mathcal { X } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 570, + 505, + 584 + ], + "score": 1.0, + "content": "is compact, the product", + "type": "text" + } + ], + "index": 34, + "is_list_start_line": true + }, + { + "bbox": [ + 83, + 581, + 395, + 596 + ], + "spans": [ + { + "bbox": [ + 83, + 583, + 100, + 594 + ], + "score": 1.0, + "content": "1034", + "type": "text" + }, + { + "bbox": [ + 107, + 582, + 189, + 594 + ], + "score": 0.9, + "content": "\\mathcal { Z } = \\mathcal { Z } _ { 1 } \\times \\ldots \\times \\mathcal { Z } _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 581, + 264, + 596 + ], + "score": 1.0, + "content": "is compact if each", + "type": "text" + }, + { + "bbox": [ + 265, + 583, + 276, + 593 + ], + "score": 0.88, + "content": "\\mathcal { X } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 581, + 395, + 596 + ], + "score": 1.0, + "content": "is compact, thus proving (2).", + "type": "text" + } + ], + "index": 35, + "is_list_start_line": true + } + ], + "index": 33.5, + "bbox_fs": [ + 82, + 549, + 507, + 596 + ] + }, + { + "type": "text", + "bbox": [ + 102, + 597, + 505, + 633 + ], + "lines": [ + { + "bbox": [ + 105, + 596, + 506, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 303, + 612 + ], + "score": 1.0, + "content": "To show the last statement (3), note simply that if", + "type": "text" + }, + { + "bbox": [ + 303, + 599, + 339, + 611 + ], + "score": 0.92, + "content": "{ \\mathcal { X } } _ { i } = { \\mathcal { X } } _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 596, + 357, + 612 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 358, + 598, + 395, + 611 + ], + "score": 0.92, + "content": "G _ { i } = G _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 395, + 596, + 506, + 612 + ], + "score": 1.0, + "content": ", then the quotient maps are", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 609, + 506, + 622 + ], + "spans": [ + { + "bbox": [ + 106, + 609, + 149, + 621 + ], + "score": 1.0, + "content": "equal, i.e.", + "type": "text" + }, + { + "bbox": [ + 150, + 611, + 183, + 622 + ], + "score": 0.87, + "content": "\\pi _ { i } = \\pi _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 609, + 440, + 621 + ], + "score": 1.0, + "content": ". 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Then,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 107, + 621, + 506, + 633 + ], + "spans": [ + { + "bbox": [ + 107, + 621, + 225, + 633 + ], + "score": 0.89, + "content": "\\phi _ { i } = \\psi _ { i } \\circ \\pi _ { i } = \\bar { \\psi _ { j } } \\circ \\pi _ { j } = \\phi _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 621, + 294, + 633 + ], + "score": 1.0, + "content": ", so we are done.", + "type": "text" + }, + { + "bbox": [ + 494, + 621, + 506, + 632 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37, + "bbox_fs": [ + 105, + 596, + 506, + 633 + ] + }, + { + "type": "title", + "bbox": [ + 102, + 645, + 291, + 658 + ], + "lines": [ + { + "bbox": [ + 106, + 645, + 291, + 661 + ], + "spans": [ + { + "bbox": [ + 106, + 645, + 291, + 661 + ], + "score": 1.0, + "content": "G.2 Universality of SignNet and BasisNet", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "index", + "bbox": [ + 91, + 667, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 89, + 667, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 89, + 669, + 100, + 678 + ], + "score": 1.0, + "content": "039", + "type": "text" + }, + { + "bbox": [ + 106, + 667, + 506, + 680 + ], + "score": 1.0, + "content": "Here, we prove Corollary 1 on the universal representation and approximation capabilities of our", + "type": "text" + } + ], + "index": 40, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 678, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 89, + 680, + 100, + 689 + ], + "score": 1.0, + "content": "040", + "type": "text" + }, + { + "bbox": [ + 106, + 678, + 506, + 690 + ], + "score": 1.0, + "content": "Unconstrained-SignNets, Unconstrained-BasisNets, and Expressive-BasisNets. We proceed in sev-", + "type": "text" + } + ], + "index": 41, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 689, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 89, + 691, + 99, + 700 + ], + "score": 1.0, + "content": "041", + "type": "text" + }, + { + "bbox": [ + 105, + 689, + 506, + 702 + ], + "score": 1.0, + "content": "eral steps, first proving universal representation of continuous functions when we do not require", + "type": "text" + } + ], + "index": 42, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 700, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 89, + 702, + 100, + 712 + ], + "score": 1.0, + "content": "042", + "type": "text" + }, + { + "bbox": [ + 105, + 700, + 506, + 712 + ], + "score": 1.0, + "content": "permutation equivariance, then proving universal approximation when we do require permutation", + "type": "text" + } + ], + "index": 43, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 711, + 161, + 723 + ], + "spans": [ + { + "bbox": [ + 89, + 713, + 100, + 722 + ], + "score": 1.0, + "content": "043", + "type": "text" + }, + { + "bbox": [ + 105, + 711, + 161, + 723 + ], + "score": 1.0, + "content": "equivariance.", + "type": "text" + } + ], + "index": 44, + "is_list_start_line": true + } + ], + "index": 42, + "bbox_fs": [ + 89, + 667, + 506, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 106, + 72, + 312, + 85 + ], + "lines": [ + { + "bbox": [ + 105, + 70, + 314, + 88 + ], + "spans": [ + { + "bbox": [ + 105, + 70, + 314, + 88 + ], + "score": 1.0, + "content": "G.2.1 Sign Invariant Universal Representation", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 101, + 91, + 504, + 115 + ], + "lines": [ + { + "bbox": [ + 105, + 90, + 505, + 104 + ], + "spans": [ + { + "bbox": [ + 105, + 90, + 153, + 104 + ], + "score": 1.0, + "content": "Recall that", + "type": "text" + }, + { + "bbox": [ + 154, + 91, + 176, + 102 + ], + "score": 0.9, + "content": "\\mathbb { S } ^ { n - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 90, + 286, + 104 + ], + "score": 1.0, + "content": "denotes the unit sphere in", + "type": "text" + }, + { + "bbox": [ + 287, + 92, + 300, + 102 + ], + "score": 0.86, + "content": "\\mathbb { R } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 90, + 505, + 104 + ], + "score": 1.0, + "content": ". As we normalize eigenvectors to unit norm, the", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 104, + 101, + 420, + 115 + ], + "spans": [ + { + "bbox": [ + 104, + 101, + 217, + 115 + ], + "score": 1.0, + "content": "domain of our functions on", + "type": "text" + }, + { + "bbox": [ + 218, + 103, + 224, + 113 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 101, + 381, + 115 + ], + "score": 1.0, + "content": "eigenvectors are on the compact space", + "type": "text" + }, + { + "bbox": [ + 381, + 102, + 415, + 115 + ], + "score": 0.93, + "content": "( \\bar { \\mathbb { S } } ^ { n - 1 } ) ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 101, + 420, + 115 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5 + }, + { + "type": "text", + "bbox": [ + 106, + 119, + 505, + 154 + ], + "lines": [ + { + "bbox": [ + 105, + 117, + 506, + 132 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 325, + 132 + ], + "score": 1.0, + "content": "Corollary 2 (Universal Representation for SignNet).", + "type": "text" + }, + { + "bbox": [ + 325, + 120, + 334, + 129 + ], + "score": 0.34, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 117, + 418, + 132 + ], + "score": 1.0, + "content": "continuous function", + "type": "text" + }, + { + "bbox": [ + 419, + 118, + 494, + 131 + ], + "score": 0.92, + "content": "f : ( \\mathbb { S } ^ { n - 1 } ) ^ { k } \\to \\mathbb { R } ^ { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 117, + 506, + 132 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 129, + 505, + 143 + ], + "spans": [ + { + "bbox": [ + 105, + 129, + 186, + 143 + ], + "score": 1.0, + "content": "sign invariant, i.e.", + "type": "text" + }, + { + "bbox": [ + 186, + 130, + 331, + 142 + ], + "score": 0.91, + "content": "f ( s _ { 1 } v _ { 1 } , \\ldots , s _ { k } v _ { k } ) = f ( { \\bar { v _ { 1 } } } , \\ldots , v _ { k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 332, + 129, + 366, + 143 + ], + "score": 1.0, + "content": "for any", + "type": "text" + }, + { + "bbox": [ + 366, + 130, + 421, + 142 + ], + "score": 0.92, + "content": "s _ { i } \\in \\{ - 1 , 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 129, + 505, + 143 + ], + "score": 1.0, + "content": ", if and only if there", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 140, + 448, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 140, + 185, + 156 + ], + "score": 1.0, + "content": "exists a continuous", + "type": "text" + }, + { + "bbox": [ + 185, + 142, + 254, + 154 + ], + "score": 0.9, + "content": "\\phi : \\mathbb { R } ^ { n } \\to \\mathbb { R } ^ { 2 n - 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 140, + 327, + 156 + ], + "score": 1.0, + "content": "and a continuous", + "type": "text" + }, + { + "bbox": [ + 328, + 142, + 406, + 154 + ], + "score": 0.91, + "content": "\\rho : \\mathbb { R } ^ { ( 2 n - 2 ) k } \\mathbb { R } ^ { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 140, + 448, + 156 + ], + "score": 1.0, + "content": "such that", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + }, + { + "type": "interline_equation", + "bbox": [ + 219, + 159, + 392, + 175 + ], + "lines": [ + { + "bbox": [ + 219, + 159, + 392, + 175 + ], + "spans": [ + { + "bbox": [ + 219, + 159, + 392, + 175 + ], + "score": 0.91, + "content": "f ( v _ { 1 } , \\dots , v _ { k } ) = \\rho \\left( [ \\phi ( v _ { i } ) + \\phi ( - v _ { i } ) ] _ { i = 1 } ^ { k } \\right) .", + "type": "interline_equation", + "image_path": "61b97349cd51fa11150e4910c84cc8dfc1bfec138a958d6d6b21df99af98488b.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 219, + 159, + 392, + 175 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "text", + "bbox": [ + 88, + 188, + 409, + 201 + ], + "lines": [ + { + "bbox": [ + 84, + 187, + 410, + 202 + ], + "spans": [ + { + "bbox": [ + 84, + 187, + 260, + 202 + ], + "score": 1.0, + "content": "1050 Proof. It can be directly seen that any", + "type": "text" + }, + { + "bbox": [ + 261, + 189, + 267, + 200 + ], + "score": 0.86, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 187, + 410, + 202 + ], + "score": 1.0, + "content": "of the above form is sign invariant.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 106, + 204, + 505, + 261 + ], + "lines": [ + { + "bbox": [ + 106, + 205, + 506, + 217 + ], + "spans": [ + { + "bbox": [ + 106, + 205, + 264, + 217 + ], + "score": 1.0, + "content": "Thus, we show that any sign invariant", + "type": "text" + }, + { + "bbox": [ + 264, + 206, + 272, + 217 + ], + "score": 0.86, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 205, + 506, + 217 + ], + "score": 1.0, + "content": "can be expressed in the above form. First, we show that", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 216, + 505, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 216, + 303, + 228 + ], + "score": 1.0, + "content": "we can apply the general Theorem 3. The group", + "type": "text" + }, + { + "bbox": [ + 304, + 216, + 361, + 228 + ], + "score": 0.94, + "content": "\\bar { G _ { i } } = \\{ 1 , - 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 216, + 505, + 228 + ], + "score": 1.0, + "content": "acts continuously and satisfies that", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 223, + 506, + 241 + ], + "spans": [ + { + "bbox": [ + 106, + 226, + 211, + 239 + ], + "score": 0.91, + "content": "\\mathbb { S } ^ { n - 1 } / \\{ 1 , - \\bar { 1 } \\} = \\mathbf { \\bar { \\mathbb { R } } } \\mathbb { P } ^ { n - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 223, + 244, + 241 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 244, + 226, + 274, + 237 + ], + "score": 0.91, + "content": "\\mathbb { R } \\mathbb { P } ^ { n - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 223, + 447, + 241 + ], + "score": 1.0, + "content": "is the real projective space of dimension", + "type": "text" + }, + { + "bbox": [ + 447, + 227, + 473, + 237 + ], + "score": 0.88, + "content": "n - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 223, + 506, + 241 + ], + "score": 1.0, + "content": ". Since", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 235, + 506, + 252 + ], + "spans": [ + { + "bbox": [ + 106, + 238, + 136, + 248 + ], + "score": 0.84, + "content": "\\mathbb { R } \\mathbb { P } ^ { n - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 235, + 282, + 252 + ], + "score": 1.0, + "content": "is a smooth manifold of dimension", + "type": "text" + }, + { + "bbox": [ + 282, + 238, + 307, + 248 + ], + "score": 0.87, + "content": "n - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 235, + 506, + 252 + ], + "score": 1.0, + "content": ", Whitney’s embedding theorem states that there", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 245, + 414, + 263 + ], + "spans": [ + { + "bbox": [ + 104, + 245, + 271, + 263 + ], + "score": 1.0, + "content": "exists a (smooth) topological embedding", + "type": "text" + }, + { + "bbox": [ + 271, + 248, + 361, + 261 + ], + "score": 0.9, + "content": "\\psi _ { i } : \\mathbb { R P } ^ { n - 1 } \\to \\mathbb { R } ^ { \\bar { 2 } n - 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 245, + 414, + 263 + ], + "score": 1.0, + "content": "(Lemma 5).", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 107, + 265, + 505, + 315 + ], + "lines": [ + { + "bbox": [ + 104, + 265, + 506, + 279 + ], + "spans": [ + { + "bbox": [ + 104, + 265, + 311, + 279 + ], + "score": 1.0, + "content": "Thus, we can apply the general theorem to see that", + "type": "text" + }, + { + "bbox": [ + 312, + 265, + 357, + 279 + ], + "score": 0.93, + "content": "f = \\rho \\circ { \\tilde { \\phi } } ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 265, + 442, + 279 + ], + "score": 1.0, + "content": "for some continuous", + "type": "text" + }, + { + "bbox": [ + 442, + 269, + 449, + 279 + ], + "score": 0.78, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 449, + 265, + 466, + 279 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 467, + 265, + 478, + 279 + ], + "score": 0.9, + "content": "\\tilde { \\phi } ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 265, + 506, + 279 + ], + "score": 1.0, + "content": ". Note", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 277, + 506, + 292 + ], + "spans": [ + { + "bbox": [ + 104, + 277, + 145, + 292 + ], + "score": 1.0, + "content": "that each", + "type": "text" + }, + { + "bbox": [ + 145, + 277, + 175, + 291 + ], + "score": 0.93, + "content": "\\tilde { \\phi } _ { i } = \\bar { \\tilde { \\phi } }", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 277, + 259, + 292 + ], + "score": 1.0, + "content": "is the same, as each", + "type": "text" + }, + { + "bbox": [ + 259, + 278, + 305, + 290 + ], + "score": 0.92, + "content": "\\mathcal { X } _ { i } = \\mathbb { S } ^ { n - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 277, + 324, + 292 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 324, + 279, + 381, + 291 + ], + "score": 0.91, + "content": "G _ { i } = \\{ 1 , - 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 277, + 506, + 292 + ], + "score": 1.0, + "content": "is the same. Also, Theorem 3", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 290, + 506, + 305 + ], + "spans": [ + { + "bbox": [ + 104, + 291, + 226, + 305 + ], + "score": 1.0, + "content": "says that we may assume that", + "type": "text" + }, + { + "bbox": [ + 226, + 290, + 233, + 303 + ], + "score": 0.86, + "content": "\\tilde { \\phi }", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 291, + 313, + 305 + ], + "score": 1.0, + "content": "is sign invariant, so", + "type": "text" + }, + { + "bbox": [ + 313, + 291, + 374, + 304 + ], + "score": 0.92, + "content": "\\tilde { \\phi } ( x ) = \\tilde { \\phi } ( - x )", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 291, + 410, + 305 + ], + "score": 1.0, + "content": ". Letting", + "type": "text" + }, + { + "bbox": [ + 410, + 290, + 473, + 304 + ], + "score": 0.94, + "content": "\\phi ( { x } ) = \\tilde { \\phi } ( { x } ) / 2", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 291, + 506, + 305 + ], + "score": 1.0, + "content": ", we are", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 303, + 504, + 315 + ], + "spans": [ + { + "bbox": [ + 106, + 303, + 189, + 315 + ], + "score": 1.0, + "content": "done with the proof.", + "type": "text" + }, + { + "bbox": [ + 495, + 304, + 504, + 312 + ], + "score": 1.0, + "content": "□", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14.5 + }, + { + "type": "title", + "bbox": [ + 108, + 329, + 400, + 342 + ], + "lines": [ + { + "bbox": [ + 106, + 329, + 401, + 343 + ], + "spans": [ + { + "bbox": [ + 106, + 329, + 401, + 343 + ], + "score": 1.0, + "content": "G.2.2 Sign Invariant Universal Representation with Extra Features", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 348, + 506, + 393 + ], + "lines": [ + { + "bbox": [ + 106, + 350, + 505, + 361 + ], + "spans": [ + { + "bbox": [ + 106, + 350, + 505, + 361 + ], + "score": 1.0, + "content": "Recall that we may want our sign invariant functions to process other data besides eigenvectors, such", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 361, + 506, + 372 + ], + "spans": [ + { + "bbox": [ + 106, + 361, + 506, + 372 + ], + "score": 1.0, + "content": "as eigenvalues or node features associated to a graph. Here, we show universal representation for", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 370, + 506, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 506, + 384 + ], + "score": 1.0, + "content": "when we have this other data that does not possess sign symmetry. The proof is a simple extension of", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 381, + 373, + 394 + ], + "spans": [ + { + "bbox": [ + 106, + 381, + 373, + 394 + ], + "score": 1.0, + "content": "Corollary 2, but we provide the technical details for completeness.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 19.5 + }, + { + "type": "text", + "bbox": [ + 102, + 396, + 506, + 420 + ], + "lines": [ + { + "bbox": [ + 106, + 396, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 106, + 396, + 505, + 410 + ], + "score": 1.0, + "content": "Corollary 3 (Universal Representation for SignNet with features). For a compact space of features", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 406, + 462, + 422 + ], + "spans": [ + { + "bbox": [ + 106, + 408, + 140, + 419 + ], + "score": 0.9, + "content": "\\Omega \\subseteq \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 406, + 156, + 422 + ], + "score": 1.0, + "content": ", let", + "type": "text" + }, + { + "bbox": [ + 157, + 408, + 259, + 421 + ], + "score": 0.89, + "content": "f ( v _ { 1 } , \\ldots , v _ { k } , x _ { 1 } , \\ldots , x _ { k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 406, + 362, + 422 + ], + "score": 1.0, + "content": "be a continuous function", + "type": "text" + }, + { + "bbox": [ + 362, + 408, + 457, + 420 + ], + "score": 0.92, + "content": "f : ( \\mathbb { S } ^ { n - 1 } \\times \\bar { \\Omega } ) ^ { k } \\overset { \\cdot } { } \\mathbb { R } ^ { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 406, + 462, + 422 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5 + }, + { + "type": "text", + "bbox": [ + 84, + 424, + 331, + 436 + ], + "lines": [ + { + "bbox": [ + 82, + 424, + 331, + 438 + ], + "spans": [ + { + "bbox": [ + 82, + 424, + 128, + 438 + ], + "score": 1.0, + "content": "1067 Then", + "type": "text" + }, + { + "bbox": [ + 128, + 425, + 136, + 436 + ], + "score": 0.84, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 136, + 424, + 331, + 438 + ], + "score": 1.0, + "content": "is sign invariant for the inputs on the sphere, i.e.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "interline_equation", + "bbox": [ + 148, + 442, + 461, + 456 + ], + "lines": [ + { + "bbox": [ + 148, + 442, + 461, + 456 + ], + "spans": [ + { + "bbox": [ + 148, + 442, + 461, + 456 + ], + "score": 0.88, + "content": "f ( s _ { 1 } v _ { 1 } , \\ldots , s _ { k } v _ { k } , x _ { 1 } , \\ldots , x _ { k } ) = f ( v _ { 1 } , \\ldots , v _ { k } , x _ { 1 } , \\ldots , x _ { k } ) \\qquad s _ { i } \\in \\{ 1 , - 1 \\} ,", + "type": "interline_equation", + "image_path": "5c764048aacb5b126b36754e2ae76180aaa40305912deeda39c41a92fdbb6ab1.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 148, + 442, + 461, + 456 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 84, + 462, + 506, + 486 + ], + "lines": [ + { + "bbox": [ + 81, + 461, + 503, + 477 + ], + "spans": [ + { + "bbox": [ + 81, + 461, + 254, + 477 + ], + "score": 1.0, + "content": "if and only if there exists a continuous 1068", + "type": "text" + }, + { + "bbox": [ + 254, + 462, + 345, + 475 + ], + "score": 0.91, + "content": "\\psi : \\mathbb { R } ^ { n + d } \\mathbb { R } ^ { 2 n - 2 + d }", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 461, + 414, + 477 + ], + "score": 1.0, + "content": "and a continuous", + "type": "text" + }, + { + "bbox": [ + 415, + 462, + 503, + 476 + ], + "score": 0.89, + "content": "\\rho : \\mathbb { R } ^ { ( 2 n - 2 + d ) k } \\mathbb { R } ^ { s }", + "type": "inline_equation" + } + ], + "index": 26 + }, + { + "bbox": [ + 82, + 475, + 146, + 486 + ], + "spans": [ + { + "bbox": [ + 82, + 475, + 146, + 486 + ], + "score": 1.0, + "content": "1069 such that", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26.5 + }, + { + "type": "interline_equation", + "bbox": [ + 151, + 493, + 460, + 507 + ], + "lines": [ + { + "bbox": [ + 151, + 493, + 460, + 507 + ], + "spans": [ + { + "bbox": [ + 151, + 493, + 460, + 507 + ], + "score": 0.88, + "content": "f ( v _ { 1 } , \\ldots , v _ { k } ) = \\rho \\left( \\phi ( v _ { 1 } , x _ { 1 } ) + \\phi ( - v _ { 1 } , x _ { 1 } ) , \\ldots , \\phi ( v _ { k } , x _ { k } ) + \\phi ( - v _ { k } , x _ { k } ) \\right) .", + "type": "interline_equation", + "image_path": "2b27ec961a2e7bc8bfa86b6bd6d972163bd48ec4d3113ff4894c2b26602b6d45.jpg" + } + ] + } + ], + "index": 28, + "virtual_lines": [ + { + "bbox": [ + 151, + 493, + 460, + 507 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 84, + 520, + 405, + 533 + ], + "lines": [ + { + "bbox": [ + 82, + 520, + 406, + 535 + ], + "spans": [ + { + "bbox": [ + 82, + 520, + 291, + 535 + ], + "score": 1.0, + "content": "1070 Proof. Once again, the sign invariance of any", + "type": "text" + }, + { + "bbox": [ + 291, + 522, + 298, + 533 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 520, + 406, + 535 + ], + "score": 1.0, + "content": "in the above form is clear.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 83, + 537, + 506, + 582 + ], + "lines": [ + { + "bbox": [ + 83, + 537, + 505, + 550 + ], + "spans": [ + { + "bbox": [ + 83, + 539, + 99, + 550 + ], + "score": 1.0, + "content": "1071", + "type": "text" + }, + { + "bbox": [ + 106, + 537, + 505, + 550 + ], + "score": 1.0, + "content": "We follow very similar steps to the proof of Corollary 2 to show that we may apply Theorem 3. We", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 82, + 548, + 506, + 562 + ], + "spans": [ + { + "bbox": [ + 82, + 550, + 100, + 561 + ], + "score": 1.0, + "content": "1072", + "type": "text" + }, + { + "bbox": [ + 105, + 548, + 144, + 562 + ], + "score": 1.0, + "content": "can view", + "type": "text" + }, + { + "bbox": [ + 145, + 549, + 153, + 559 + ], + "score": 0.79, + "content": "\\Omega", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 548, + 454, + 562 + ], + "score": 1.0, + "content": "as a quotient space, after quotienting by the trivial group that does nothing,", + "type": "text" + }, + { + "bbox": [ + 455, + 549, + 503, + 561 + ], + "score": 0.92, + "content": "\\Omega \\cong \\Omega / \\{ 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 548, + 506, + 562 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 83, + 559, + 506, + 573 + ], + "spans": [ + { + "bbox": [ + 83, + 561, + 100, + 572 + ], + "score": 1.0, + "content": "1073", + "type": "text" + }, + { + "bbox": [ + 106, + 559, + 248, + 573 + ], + "score": 1.0, + "content": "The corresponding quotient map is", + "type": "text" + }, + { + "bbox": [ + 248, + 560, + 264, + 570 + ], + "score": 0.9, + "content": "\\mathrm { i d } _ { \\Omega }", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 559, + 362, + 573 + ], + "score": 1.0, + "content": ", the identity map. 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It can be directly seen that any", + "type": "text" + }, + { + "bbox": [ + 261, + 189, + 267, + 200 + ], + "score": 0.86, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 187, + 410, + 202 + ], + "score": 1.0, + "content": "of the above form is sign invariant.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7, + "bbox_fs": [ + 84, + 187, + 410, + 202 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 204, + 505, + 261 + ], + "lines": [ + { + "bbox": [ + 106, + 205, + 506, + 217 + ], + "spans": [ + { + "bbox": [ + 106, + 205, + 264, + 217 + ], + "score": 1.0, + "content": "Thus, we show that any sign invariant", + "type": "text" + }, + { + "bbox": [ + 264, + 206, + 272, + 217 + ], + "score": 0.86, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 205, + 506, + 217 + ], + "score": 1.0, + "content": "can be expressed in the above form. First, we show that", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 216, + 505, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 216, + 303, + 228 + ], + "score": 1.0, + "content": "we can apply the general Theorem 3. The group", + "type": "text" + }, + { + "bbox": [ + 304, + 216, + 361, + 228 + ], + "score": 0.94, + "content": "\\bar { G _ { i } } = \\{ 1 , - 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 216, + 505, + 228 + ], + "score": 1.0, + "content": "acts continuously and satisfies that", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 223, + 506, + 241 + ], + "spans": [ + { + "bbox": [ + 106, + 226, + 211, + 239 + ], + "score": 0.91, + "content": "\\mathbb { S } ^ { n - 1 } / \\{ 1 , - \\bar { 1 } \\} = \\mathbf { \\bar { \\mathbb { R } } } \\mathbb { P } ^ { n - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 223, + 244, + 241 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 244, + 226, + 274, + 237 + ], + "score": 0.91, + "content": "\\mathbb { R } \\mathbb { P } ^ { n - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 223, + 447, + 241 + ], + "score": 1.0, + "content": "is the real projective space of dimension", + "type": "text" + }, + { + "bbox": [ + 447, + 227, + 473, + 237 + ], + "score": 0.88, + "content": "n - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 223, + 506, + 241 + ], + "score": 1.0, + "content": ". Since", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 235, + 506, + 252 + ], + "spans": [ + { + "bbox": [ + 106, + 238, + 136, + 248 + ], + "score": 0.84, + "content": "\\mathbb { R } \\mathbb { P } ^ { n - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 235, + 282, + 252 + ], + "score": 1.0, + "content": "is a smooth manifold of dimension", + "type": "text" + }, + { + "bbox": [ + 282, + 238, + 307, + 248 + ], + "score": 0.87, + "content": "n - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 235, + 506, + 252 + ], + "score": 1.0, + "content": ", Whitney’s embedding theorem states that there", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 245, + 414, + 263 + ], + "spans": [ + { + "bbox": [ + 104, + 245, + 271, + 263 + ], + "score": 1.0, + "content": "exists a (smooth) topological embedding", + "type": "text" + }, + { + "bbox": [ + 271, + 248, + 361, + 261 + ], + "score": 0.9, + "content": "\\psi _ { i } : \\mathbb { R P } ^ { n - 1 } \\to \\mathbb { R } ^ { \\bar { 2 } n - 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 245, + 414, + 263 + ], + "score": 1.0, + "content": "(Lemma 5).", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10, + "bbox_fs": [ + 104, + 205, + 506, + 263 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 265, + 505, + 315 + ], + "lines": [ + { + "bbox": [ + 104, + 265, + 506, + 279 + ], + "spans": [ + { + "bbox": [ + 104, + 265, + 311, + 279 + ], + "score": 1.0, + "content": "Thus, we can apply the general theorem to see that", + "type": "text" + }, + { + "bbox": [ + 312, + 265, + 357, + 279 + ], + "score": 0.93, + "content": "f = \\rho \\circ { \\tilde { \\phi } } ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 265, + 442, + 279 + ], + "score": 1.0, + "content": "for some continuous", + "type": "text" + }, + { + "bbox": [ + 442, + 269, + 449, + 279 + ], + "score": 0.78, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 449, + 265, + 466, + 279 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 467, + 265, + 478, + 279 + ], + "score": 0.9, + "content": "\\tilde { \\phi } ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 265, + 506, + 279 + ], + "score": 1.0, + "content": ". Note", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 277, + 506, + 292 + ], + "spans": [ + { + "bbox": [ + 104, + 277, + 145, + 292 + ], + "score": 1.0, + "content": "that each", + "type": "text" + }, + { + "bbox": [ + 145, + 277, + 175, + 291 + ], + "score": 0.93, + "content": "\\tilde { \\phi } _ { i } = \\bar { \\tilde { \\phi } }", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 277, + 259, + 292 + ], + "score": 1.0, + "content": "is the same, as each", + "type": "text" + }, + { + "bbox": [ + 259, + 278, + 305, + 290 + ], + "score": 0.92, + "content": "\\mathcal { X } _ { i } = \\mathbb { S } ^ { n - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 277, + 324, + 292 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 324, + 279, + 381, + 291 + ], + "score": 0.91, + "content": "G _ { i } = \\{ 1 , - 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 277, + 506, + 292 + ], + "score": 1.0, + "content": "is the same. Also, Theorem 3", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 290, + 506, + 305 + ], + "spans": [ + { + "bbox": [ + 104, + 291, + 226, + 305 + ], + "score": 1.0, + "content": "says that we may assume that", + "type": "text" + }, + { + "bbox": [ + 226, + 290, + 233, + 303 + ], + "score": 0.86, + "content": "\\tilde { \\phi }", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 291, + 313, + 305 + ], + "score": 1.0, + "content": "is sign invariant, so", + "type": "text" + }, + { + "bbox": [ + 313, + 291, + 374, + 304 + ], + "score": 0.92, + "content": "\\tilde { \\phi } ( x ) = \\tilde { \\phi } ( - x )", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 291, + 410, + 305 + ], + "score": 1.0, + "content": ". Letting", + "type": "text" + }, + { + "bbox": [ + 410, + 290, + 473, + 304 + ], + "score": 0.94, + "content": "\\phi ( { x } ) = \\tilde { \\phi } ( { x } ) / 2", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 291, + 506, + 305 + ], + "score": 1.0, + "content": ", we are", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 303, + 504, + 315 + ], + "spans": [ + { + "bbox": [ + 106, + 303, + 189, + 315 + ], + "score": 1.0, + "content": "done with the proof.", + "type": "text" + }, + { + "bbox": [ + 495, + 304, + 504, + 312 + ], + "score": 1.0, + "content": "□", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14.5, + "bbox_fs": [ + 104, + 265, + 506, + 315 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 329, + 400, + 342 + ], + "lines": [ + { + "bbox": [ + 106, + 329, + 401, + 343 + ], + "spans": [ + { + "bbox": [ + 106, + 329, + 401, + 343 + ], + "score": 1.0, + "content": "G.2.2 Sign Invariant Universal Representation with Extra Features", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 348, + 506, + 393 + ], + "lines": [ + { + "bbox": [ + 106, + 350, + 505, + 361 + ], + "spans": [ + { + "bbox": [ + 106, + 350, + 505, + 361 + ], + "score": 1.0, + "content": "Recall that we may want our sign invariant functions to process other data besides eigenvectors, such", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 361, + 506, + 372 + ], + "spans": [ + { + "bbox": [ + 106, + 361, + 506, + 372 + ], + "score": 1.0, + "content": "as eigenvalues or node features associated to a graph. Here, we show universal representation for", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 370, + 506, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 506, + 384 + ], + "score": 1.0, + "content": "when we have this other data that does not possess sign symmetry. The proof is a simple extension of", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 381, + 373, + 394 + ], + "spans": [ + { + "bbox": [ + 106, + 381, + 373, + 394 + ], + "score": 1.0, + "content": "Corollary 2, but we provide the technical details for completeness.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 19.5, + "bbox_fs": [ + 105, + 350, + 506, + 394 + ] + }, + { + "type": "text", + "bbox": [ + 102, + 396, + 506, + 420 + ], + "lines": [ + { + "bbox": [ + 106, + 396, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 106, + 396, + 505, + 410 + ], + "score": 1.0, + "content": "Corollary 3 (Universal Representation for SignNet with features). For a compact space of features", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 406, + 462, + 422 + ], + "spans": [ + { + "bbox": [ + 106, + 408, + 140, + 419 + ], + "score": 0.9, + "content": "\\Omega \\subseteq \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 406, + 156, + 422 + ], + "score": 1.0, + "content": ", let", + "type": "text" + }, + { + "bbox": [ + 157, + 408, + 259, + 421 + ], + "score": 0.89, + "content": "f ( v _ { 1 } , \\ldots , v _ { k } , x _ { 1 } , \\ldots , x _ { k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 406, + 362, + 422 + ], + "score": 1.0, + "content": "be a continuous function", + "type": "text" + }, + { + "bbox": [ + 362, + 408, + 457, + 420 + ], + "score": 0.92, + "content": "f : ( \\mathbb { S } ^ { n - 1 } \\times \\bar { \\Omega } ) ^ { k } \\overset { \\cdot } { } \\mathbb { R } ^ { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 406, + 462, + 422 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5, + "bbox_fs": [ + 106, + 396, + 505, + 422 + ] + }, + { + "type": "text", + "bbox": [ + 84, + 424, + 331, + 436 + ], + "lines": [ + { + "bbox": [ + 82, + 424, + 331, + 438 + ], + "spans": [ + { + "bbox": [ + 82, + 424, + 128, + 438 + ], + "score": 1.0, + "content": "1067 Then", + "type": "text" + }, + { + "bbox": [ + 128, + 425, + 136, + 436 + ], + "score": 0.84, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 136, + 424, + 331, + 438 + ], + "score": 1.0, + "content": "is sign invariant for the inputs on the sphere, i.e.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24, + "bbox_fs": [ + 82, + 424, + 331, + 438 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 148, + 442, + 461, + 456 + ], + "lines": [ + { + "bbox": [ + 148, + 442, + 461, + 456 + ], + "spans": [ + { + "bbox": [ + 148, + 442, + 461, + 456 + ], + "score": 0.88, + "content": "f ( s _ { 1 } v _ { 1 } , \\ldots , s _ { k } v _ { k } , x _ { 1 } , \\ldots , x _ { k } ) = f ( v _ { 1 } , \\ldots , v _ { k } , x _ { 1 } , \\ldots , x _ { k } ) \\qquad s _ { i } \\in \\{ 1 , - 1 \\} ,", + "type": "interline_equation", + "image_path": "5c764048aacb5b126b36754e2ae76180aaa40305912deeda39c41a92fdbb6ab1.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 148, + 442, + 461, + 456 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 84, + 462, + 506, + 486 + ], + "lines": [ + { + "bbox": [ + 81, + 461, + 503, + 477 + ], + "spans": [ + { + "bbox": [ + 81, + 461, + 254, + 477 + ], + "score": 1.0, + "content": "if and only if there exists a continuous 1068", + "type": "text" + }, + { + "bbox": [ + 254, + 462, + 345, + 475 + ], + "score": 0.91, + "content": "\\psi : \\mathbb { R } ^ { n + d } \\mathbb { R } ^ { 2 n - 2 + d }", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 461, + 414, + 477 + ], + "score": 1.0, + "content": "and a continuous", + "type": "text" + }, + { + "bbox": [ + 415, + 462, + 503, + 476 + ], + "score": 0.89, + "content": "\\rho : \\mathbb { R } ^ { ( 2 n - 2 + d ) k } \\mathbb { R } ^ { s }", + "type": "inline_equation" + } + ], + "index": 26 + }, + { + "bbox": [ + 82, + 475, + 146, + 486 + ], + "spans": [ + { + "bbox": [ + 82, + 475, + 146, + 486 + ], + "score": 1.0, + "content": "1069 such that", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26.5, + "bbox_fs": [ + 81, + 461, + 503, + 486 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 151, + 493, + 460, + 507 + ], + "lines": [ + { + "bbox": [ + 151, + 493, + 460, + 507 + ], + "spans": [ + { + "bbox": [ + 151, + 493, + 460, + 507 + ], + "score": 0.88, + "content": "f ( v _ { 1 } , \\ldots , v _ { k } ) = \\rho \\left( \\phi ( v _ { 1 } , x _ { 1 } ) + \\phi ( - v _ { 1 } , x _ { 1 } ) , \\ldots , \\phi ( v _ { k } , x _ { k } ) + \\phi ( - v _ { k } , x _ { k } ) \\right) .", + "type": "interline_equation", + "image_path": "2b27ec961a2e7bc8bfa86b6bd6d972163bd48ec4d3113ff4894c2b26602b6d45.jpg" + } + ] + } + ], + "index": 28, + "virtual_lines": [ + { + "bbox": [ + 151, + 493, + 460, + 507 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 84, + 520, + 405, + 533 + ], + "lines": [ + { + "bbox": [ + 82, + 520, + 406, + 535 + ], + "spans": [ + { + "bbox": [ + 82, + 520, + 291, + 535 + ], + "score": 1.0, + "content": "1070 Proof. Once again, the sign invariance of any", + "type": "text" + }, + { + "bbox": [ + 291, + 522, + 298, + 533 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 520, + 406, + 535 + ], + "score": 1.0, + "content": "in the above form is clear.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29, + "bbox_fs": [ + 82, + 520, + 406, + 535 + ] + }, + { + "type": "index", + "bbox": [ + 83, + 537, + 506, + 582 + ], + "lines": [ + { + "bbox": [ + 83, + 537, + 505, + 550 + ], + "spans": [ + { + "bbox": [ + 83, + 539, + 99, + 550 + ], + "score": 1.0, + "content": "1071", + "type": "text" + }, + { + "bbox": [ + 106, + 537, + 505, + 550 + ], + "score": 1.0, + "content": "We follow very similar steps to the proof of Corollary 2 to show that we may apply Theorem 3. We", + "type": "text" + } + ], + "index": 30, + "is_list_start_line": true + }, + { + "bbox": [ + 82, + 548, + 506, + 562 + ], + "spans": [ + { + "bbox": [ + 82, + 550, + 100, + 561 + ], + "score": 1.0, + "content": "1072", + "type": "text" + }, + { + "bbox": [ + 105, + 548, + 144, + 562 + ], + "score": 1.0, + "content": "can view", + "type": "text" + }, + { + "bbox": [ + 145, + 549, + 153, + 559 + ], + "score": 0.79, + "content": "\\Omega", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 548, + 454, + 562 + ], + "score": 1.0, + "content": "as a quotient space, after quotienting by the trivial group that does nothing,", + "type": "text" + }, + { + "bbox": [ + 455, + 549, + 503, + 561 + ], + "score": 0.92, + "content": "\\Omega \\cong \\Omega / \\{ 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 548, + 506, + 562 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 31, + "is_list_start_line": true + }, + { + "bbox": [ + 83, + 559, + 506, + 573 + ], + "spans": [ + { + "bbox": [ + 83, + 561, + 100, + 572 + ], + "score": 1.0, + "content": "1073", + "type": "text" + }, + { + "bbox": [ + 106, + 559, + 248, + 573 + ], + "score": 1.0, + "content": "The corresponding quotient map is", + "type": "text" + }, + { + "bbox": [ + 248, + 560, + 264, + 570 + ], + "score": 0.9, + "content": "\\mathrm { i d } _ { \\Omega }", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 559, + 362, + 573 + ], + "score": 1.0, + "content": ", the identity map. Also,", + "type": "text" + }, + { + "bbox": [ + 363, + 560, + 371, + 569 + ], + "score": 0.79, + "content": "\\Omega", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 559, + 506, + 573 + ], + "score": 1.0, + "content": "trivially topologically embeds in", + "type": "text" + } + ], + "index": 32, + "is_list_start_line": true + }, + { + "bbox": [ + 83, + 568, + 209, + 584 + ], + "spans": [ + { + "bbox": [ + 83, + 572, + 100, + 581 + ], + "score": 1.0, + "content": "1074", + "type": "text" + }, + { + "bbox": [ + 106, + 570, + 119, + 580 + ], + "score": 0.87, + "content": "\\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 120, + 568, + 209, + 584 + ], + "score": 1.0, + "content": "by the inclusion map.", + "type": "text" + } + ], + "index": 33, + "is_list_start_line": true + } + ], + "index": 31.5, + "bbox_fs": [ + 82, + 537, + 506, + 584 + ] + }, + { + "type": "text", + "bbox": [ + 87, + 586, + 385, + 599 + ], + "lines": [ + { + "bbox": [ + 85, + 586, + 385, + 600 + ], + "spans": [ + { + "bbox": [ + 85, + 586, + 119, + 600 + ], + "score": 1.0, + "content": "1075 As", + "type": "text" + }, + { + "bbox": [ + 120, + 586, + 204, + 599 + ], + "score": 0.93, + "content": "G _ { i } = \\{ - 1 , 1 \\} \\times \\{ 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 586, + 385, + 600 + ], + "score": 1.0, + "content": "acts continuously, by Lemma 3 we have that", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34, + "bbox_fs": [ + 85, + 586, + 385, + 600 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 148, + 605, + 462, + 620 + ], + "lines": [ + { + "bbox": [ + 148, + 605, + 462, + 620 + ], + "spans": [ + { + "bbox": [ + 148, + 605, + 462, + 620 + ], + "score": 0.88, + "content": "( \\mathbb { S } ^ { n - 1 } \\times \\Omega ) / ( \\{ 1 , - 1 \\} \\times \\{ 1 \\} ) \\cong ( \\mathbb { S } ^ { n - 1 } / \\{ 1 , - 1 \\} ) \\times ( \\Omega / \\{ 1 \\} ) \\cong \\mathbb { R } \\mathbb { P } ^ { n - 1 } \\times \\Omega ,", + "type": "interline_equation", + "image_path": "81d19892c20a2aebcea2d5e5ee610ccf937a58355b63b58ed3e8d351f1a4ed67.jpg" + } + ] + } + ], + "index": 35, + "virtual_lines": [ + { + "bbox": [ + 148, + 605, + 462, + 620 + ], + "spans": [], + "index": 35 + } + ] + }, + { + "type": "text", + "bbox": [ + 83, + 626, + 438, + 638 + ], + "lines": [ + { + "bbox": [ + 81, + 624, + 436, + 640 + ], + "spans": [ + { + "bbox": [ + 81, + 624, + 240, + 640 + ], + "score": 1.0, + "content": "with corresponding quotient map 1076", + "type": "text" + }, + { + "bbox": [ + 240, + 626, + 274, + 637 + ], + "score": 0.91, + "content": "\\pi \\times \\mathrm { i d } _ { \\Omega }", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 624, + 305, + 640 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 305, + 628, + 312, + 636 + ], + "score": 0.78, + "content": "\\pi", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 624, + 402, + 640 + ], + "score": 1.0, + "content": "is the quotient map to", + "type": "text" + }, + { + "bbox": [ + 403, + 626, + 432, + 636 + ], + "score": 0.91, + "content": "\\mathbb { R } \\mathbb { P } ^ { n - 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Also, if", + "type": "text" + }, + { + "bbox": [ + 450, + 322, + 482, + 334 + ], + "score": 0.92, + "content": "d _ { i } = d _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 317, + 507, + 337 + ], + "score": 1.0, + "content": ", then", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 331, + 301, + 347 + ], + "spans": [ + { + "bbox": [ + 106, + 334, + 142, + 345 + ], + "score": 0.92, + "content": "{ \\mathcal { X } } _ { i } = { \\mathcal { X } } _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 143, + 331, + 161, + 347 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 161, + 334, + 198, + 345 + ], + "score": 0.91, + "content": "G _ { i } = G _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 331, + 263, + 347 + ], + "score": 1.0, + "content": ", so we can take", + "type": "text" + }, + { + "bbox": [ + 263, + 334, + 297, + 345 + ], + "score": 0.92, + "content": "\\phi _ { i } = \\phi _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 331, + 301, + 347 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19 + }, + { + "type": "title", + "bbox": [ + 108, + 371, + 444, + 384 + ], + "lines": [ + { + "bbox": [ + 105, + 371, + 445, + 386 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 445, + 386 + ], + "score": 1.0, + "content": "G.2.4 Basis Invariant and Permutation Equivariant Universal Approximation", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 103, + 389, + 506, + 490 + ], + "lines": [ + { + "bbox": [ + 105, + 389, + 507, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 201, + 403 + ], + "score": 1.0, + "content": "With the restriction that", + "type": "text" + }, + { + "bbox": [ + 201, + 389, + 333, + 403 + ], + "score": 0.91, + "content": "f ( V _ { 1 } , \\dots , V _ { l } ) : \\mathbb { R } ^ { n \\times \\sum _ { i } d _ { i } } \\to \\mathbb { R } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 389, + 507, + 403 + ], + "score": 1.0, + "content": "be permutation equivariant and basis invari-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 401, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 442, + 415 + ], + "score": 1.0, + "content": "ant, we need to use the impractically expensive Expressive-BasisNet to approximate", + "type": "text" + }, + { + "bbox": [ + 443, + 402, + 450, + 414 + ], + "score": 0.83, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 401, + 505, + 415 + ], + "score": 1.0, + "content": ". Universality", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 414, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 106, + 414, + 505, + 424 + ], + "score": 1.0, + "content": "of permutation invariant or equivariant functions from matrices to scalars or matrices to vectors is", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 422, + 506, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 422, + 506, + 437 + ], + "score": 1.0, + "content": "difficult to achieve in a computationally tractable manner [Maron et al., 2019, Keriven and Peyré,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 434, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 106, + 434, + 505, + 447 + ], + "score": 1.0, + "content": "2019, Maehara and NT, 2019]. 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Let", + "type": "text" + }, + { + "bbox": [ + 393, + 491, + 505, + 503 + ], + "score": 0.89, + "content": "f ( V _ { 1 } , \\dots , V _ { l } ) : \\mathrm { S t } ( d _ { 1 } , n ) \\times", + "type": "inline_equation" + } + ], + "index": 31 + }, + { + "bbox": [ + 84, + 502, + 506, + 515 + ], + "spans": [ + { + "bbox": [ + 84, + 505, + 99, + 513 + ], + "score": 1.0, + "content": "1113", + "type": "text" + }, + { + "bbox": [ + 107, + 502, + 195, + 514 + ], + "score": 0.91, + "content": "\\dots \\times \\operatorname { S t } ( d _ { l } , n ) \\to \\mathbb { R } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 502, + 258, + 515 + ], + "score": 1.0, + "content": "be continuous,", + "type": "text" + }, + { + "bbox": [ + 258, + 502, + 343, + 514 + ], + "score": 0.92, + "content": "O ( d _ { 1 } ) \\times \\ldots \\times O ( d _ { l } )", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 502, + 506, + 515 + ], + "score": 1.0, + "content": "invariant, and permutation equivariant.", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 84, + 513, + 342, + 526 + ], + "spans": [ + { + "bbox": [ + 84, + 513, + 128, + 526 + ], + "score": 1.0, + "content": "1114 Then", + "type": "text" + }, + { + "bbox": [ + 128, + 514, + 136, + 524 + ], + "score": 0.82, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 136, + 513, + 342, + 526 + ], + "score": 1.0, + "content": "can be ϵ-approximated by an Expressive-BasisNet.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 89, + 535, + 488, + 548 + ], + "lines": [ + { + "bbox": [ + 85, + 534, + 490, + 550 + ], + "spans": [ + { + "bbox": [ + 85, + 534, + 411, + 550 + ], + "score": 1.0, + "content": "1115 Proof. 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Define 119", + "type": "text" + }, + { + "bbox": [ + 174, + 615, + 264, + 628 + ], + "score": 0.92, + "content": "h : \\mathcal { Z } \\subseteq \\mathbb { R } ^ { n ^ { 2 } \\times l } \\to \\mathbb { R } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 612, + 280, + 632 + ], + "score": 1.0, + "content": "by", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39.5 + }, + { + "type": "interline_equation", + "bbox": [ + 187, + 629, + 423, + 645 + ], + "lines": [ + { + "bbox": [ + 187, + 629, + 423, + 645 + ], + "spans": [ + { + "bbox": [ + 187, + 629, + 423, + 645 + ], + "score": 0.89, + "content": "\\begin{array} { r } { h ( V _ { 1 } V _ { 1 } ^ { \\top } , \\ldots , V _ { l } V _ { l } ^ { \\top } ) = \\rho \\left( \\phi _ { d _ { 1 } } ( V _ { 1 } V _ { 1 } ^ { \\top } ) , \\ldots , \\phi _ { d _ { l } } ( V _ { l } V _ { l } ^ { \\top } ) \\right) . } \\end{array}", + "type": "interline_equation", + "image_path": "cddf73eba3d761b1fa61ef79a063fb5b5ac415e473513ed703b244528b509afc.jpg" + } + ] + } + ], + "index": 41, + "virtual_lines": [ + { + "bbox": [ + 187, + 629, + 423, + 645 + ], + "spans": [], + "index": 41 + } + ] + }, + { + "type": "text", + "bbox": [ + 83, + 645, + 505, + 694 + ], + "lines": [ + { + "bbox": [ + 83, + 645, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 83, + 646, + 100, + 656 + ], + "score": 1.0, + "content": "1120", + "type": "text" + }, + { + "bbox": [ + 104, + 645, + 168, + 657 + ], + "score": 1.0, + "content": "Then note that", + "type": "text" + }, + { + "bbox": [ + 168, + 645, + 175, + 655 + ], + "score": 0.83, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 645, + 506, + 657 + ], + "score": 1.0, + "content": "is continuous and permutation equivariant from matrices to vectors, so it can be", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 83, + 655, + 506, + 669 + ], + "spans": [ + { + "bbox": [ + 83, + 660, + 100, + 669 + ], + "score": 1.0, + "content": "1121", + "type": "text" + }, + { + "bbox": [ + 107, + 660, + 112, + 668 + ], + "score": 0.6, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 112, + 658, + 430, + 669 + ], + "score": 1.0, + "content": "-approximated by an invariant graph network [Keriven and Peyré, 2019], call it", + "type": "text" + }, + { + "bbox": [ + 430, + 655, + 451, + 668 + ], + "score": 0.73, + "content": "\\widetilde { \\mathrm { I G N } }", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 658, + 506, + 669 + ], + "score": 1.0, + "content": ". 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Universality", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 414, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 106, + 414, + 505, + 424 + ], + "score": 1.0, + "content": "of permutation invariant or equivariant functions from matrices to scalars or matrices to vectors is", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 422, + 506, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 422, + 506, + 437 + ], + "score": 1.0, + "content": "difficult to achieve in a computationally tractable manner [Maron et al., 2019, Keriven and Peyré,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 434, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 106, + 434, + 505, + 447 + ], + "score": 1.0, + "content": "2019, Maehara and NT, 2019]. One intuitive reason to expect this is that universally approximating", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 446, + 505, + 457 + ], + "spans": [ + { + "bbox": [ + 106, + 446, + 505, + 457 + ], + "score": 1.0, + "content": "such functions allows solution of the graph isomorphism problem [Chen et al., 2019b], which is a", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 456, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 505, + 469 + ], + "score": 1.0, + "content": "computationally difficult problem. While we have exact representation of basis invariant functions", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 468, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 106, + 468, + 165, + 480 + ], + "score": 1.0, + "content": "by continuous", + "type": "text" + }, + { + "bbox": [ + 166, + 469, + 173, + 479 + ], + "score": 0.81, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 468, + 190, + 480 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 191, + 468, + 201, + 479 + ], + "score": 0.88, + "content": "\\phi _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 468, + 505, + 480 + ], + "score": 1.0, + "content": "when there is no permutation equivariance constraint, we can only achieve", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 478, + 438, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 238, + 491 + ], + "score": 1.0, + "content": "approximation up to an arbitrary", + "type": "text" + }, + { + "bbox": [ + 239, + 479, + 262, + 489 + ], + "score": 0.89, + "content": "\\epsilon > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 478, + 438, + 491 + ], + "score": 1.0, + "content": "when we require permutation equivariance.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 26, + "bbox_fs": [ + 105, + 389, + 507, + 491 + ] + }, + { + "type": "index", + "bbox": [ + 84, + 491, + 505, + 525 + ], + "lines": [ + { + "bbox": [ + 83, + 490, + 505, + 504 + ], + "spans": [ + { + "bbox": [ + 83, + 493, + 100, + 502 + ], + "score": 1.0, + "content": "1112", + "type": "text" + }, + { + "bbox": [ + 105, + 490, + 392, + 504 + ], + "score": 1.0, + "content": "Corollary 5 (Universal Approximation for Expressive-BasisNets). Let", + "type": "text" + }, + { + "bbox": [ + 393, + 491, + 505, + 503 + ], + "score": 0.89, + "content": "f ( V _ { 1 } , \\dots , V _ { l } ) : \\mathrm { S t } ( d _ { 1 } , n ) \\times", + "type": "inline_equation" + } + ], + "index": 31, + "is_list_start_line": true + }, + { + "bbox": [ + 84, + 502, + 506, + 515 + ], + "spans": [ + { + "bbox": [ + 84, + 505, + 99, + 513 + ], + "score": 1.0, + "content": "1113", + "type": "text" + }, + { + "bbox": [ + 107, + 502, + 195, + 514 + ], + "score": 0.91, + "content": "\\dots \\times \\operatorname { S t } ( d _ { l } , n ) \\to \\mathbb { R } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 502, + 258, + 515 + ], + "score": 1.0, + "content": "be continuous,", + "type": "text" + }, + { + "bbox": [ + 258, + 502, + 343, + 514 + ], + "score": 0.92, + "content": "O ( d _ { 1 } ) \\times \\ldots \\times O ( d _ { l } )", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 502, + 506, + 515 + ], + "score": 1.0, + "content": "invariant, and permutation equivariant.", + "type": "text" + } + ], + "index": 32, + "is_list_start_line": true + }, + { + "bbox": [ + 84, + 513, + 342, + 526 + ], + "spans": [ + { + "bbox": [ + 84, + 513, + 128, + 526 + ], + "score": 1.0, + "content": "1114 Then", + "type": "text" + }, + { + "bbox": [ + 128, + 514, + 136, + 524 + ], + "score": 0.82, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 136, + 513, + 342, + 526 + ], + "score": 1.0, + "content": "can be ϵ-approximated by an Expressive-BasisNet.", + "type": "text" + } + ], + "index": 33, + "is_list_start_line": true + } + ], + "index": 32, + "bbox_fs": [ + 83, + 490, + 506, + 526 + ] + }, + { + "type": "text", + "bbox": [ + 89, + 535, + 488, + 548 + ], + "lines": [ + { + "bbox": [ + 85, + 534, + 490, + 550 + ], + "spans": [ + { + "bbox": [ + 85, + 534, + 411, + 550 + ], + "score": 1.0, + "content": "1115 Proof. By invariance, Corollary 4 of the decomposition theorem shows that", + "type": "text" + }, + { + "bbox": [ + 411, + 537, + 418, + 547 + ], + "score": 0.86, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 534, + 490, + 550 + ], + "score": 1.0, + "content": "can be written as", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34, + "bbox_fs": [ + 85, + 534, + 490, + 550 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 219, + 548, + 391, + 561 + ], + "lines": [ + { + "bbox": [ + 219, + 548, + 391, + 561 + ], + "spans": [ + { + "bbox": [ + 219, + 548, + 391, + 561 + ], + "score": 0.84, + "content": "f ( V _ { 1 } , \\dots , V _ { l } ) = \\rho \\left( \\varphi _ { d _ { 1 } } ( V _ { 1 } ) , \\dots , \\varphi _ { d _ { l } } ( V _ { l } ) \\right)", + "type": "interline_equation", + "image_path": "86912fe25fcca61f87347d3da9c8bbcf522615f5db83ed3f4c98de9786643cc8.jpg" + } + ] + } + ], + "index": 35, + "virtual_lines": [ + { + "bbox": [ + 219, + 548, + 391, + 561 + ], + "spans": [], + "index": 35 + } + ] + }, + { + "type": "text", + "bbox": [ + 87, + 563, + 496, + 586 + ], + "lines": [ + { + "bbox": [ + 84, + 561, + 505, + 576 + ], + "spans": [ + { + "bbox": [ + 84, + 561, + 189, + 576 + ], + "score": 1.0, + "content": "for some continuous 1116", + "type": "text" + }, + { + "bbox": [ + 189, + 562, + 214, + 574 + ], + "score": 0.94, + "content": "O ( d _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 561, + 252, + 576 + ], + "score": 1.0, + "content": "invariant", + "type": "text" + }, + { + "bbox": [ + 252, + 564, + 267, + 574 + ], + "score": 0.89, + "content": "\\varphi _ { d _ { i } }", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 561, + 330, + 576 + ], + "score": 1.0, + "content": "and continuous", + "type": "text" + }, + { + "bbox": [ + 330, + 564, + 337, + 574 + ], + "score": 0.75, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 561, + 483, + 576 + ], + "score": 1.0, + "content": ". By the first fundamental theorem of", + "type": "text" + }, + { + "bbox": [ + 483, + 562, + 505, + 574 + ], + "score": 0.91, + "content": "O ( d )", + "type": "inline_equation" + } + ], + "index": 36 + }, + { + "bbox": [ + 84, + 573, + 475, + 587 + ], + "spans": [ + { + "bbox": [ + 84, + 573, + 176, + 587 + ], + "score": 1.0, + "content": "(Lemma 2), each 1117", + "type": "text" + }, + { + "bbox": [ + 177, + 576, + 191, + 586 + ], + "score": 0.88, + "content": "\\varphi _ { d _ { i } }", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 573, + 263, + 587 + ], + "score": 1.0, + "content": "can be written as", + "type": "text" + }, + { + "bbox": [ + 263, + 574, + 353, + 586 + ], + "score": 0.93, + "content": "\\varphi _ { d _ { i } } ( V _ { i } ) = \\phi _ { d _ { i } } ( V _ { i } V _ { i } ^ { \\top } )", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 573, + 439, + 587 + ], + "score": 1.0, + "content": "for some continuous", + "type": "text" + }, + { + "bbox": [ + 439, + 574, + 453, + 586 + ], + "score": 0.9, + "content": "\\phi _ { d _ { i } }", + "type": "inline_equation" + }, + { + "bbox": [ + 454, + 573, + 475, + 587 + ], + "score": 1.0, + "content": ". Let", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36.5, + "bbox_fs": [ + 84, + 561, + 505, + 587 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 196, + 588, + 415, + 603 + ], + "lines": [ + { + "bbox": [ + 196, + 588, + 415, + 603 + ], + "spans": [ + { + "bbox": [ + 196, + 588, + 415, + 603 + ], + "score": 0.85, + "content": "{ \\mathcal { Z } } = \\{ ( V _ { 1 } V _ { 1 } ^ { \\top } , \\ldots , V _ { l } V _ { l } ^ { \\top } ) : V _ { i } \\in { \\mathrm { S t } } ( d _ { i } , n ) \\} \\subseteq \\mathbb { R } ^ { n ^ { 2 } \\times l } ,", + "type": "interline_equation", + "image_path": "bf667495dbb3b2d36cad4b6156f6cb2299aba431ea0e7d0d4ebd24eaaaea6446.jpg" + } + ] + } + ], + "index": 38, + "virtual_lines": [ + { + "bbox": [ + 196, + 588, + 415, + 603 + ], + "spans": [], + "index": 38 + } + ] + }, + { + "type": "text", + "bbox": [ + 93, + 604, + 505, + 628 + ], + "lines": [ + { + "bbox": [ + 91, + 603, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 91, + 603, + 325, + 617 + ], + "score": 1.0, + "content": "which is compact as it is the image of the compact space 18", + "type": "text" + }, + { + "bbox": [ + 325, + 604, + 428, + 616 + ], + "score": 0.87, + "content": "\\mathrm { S t } ( d _ { 1 } , n ) \\times \\ldots \\times \\mathrm { S t } ( d _ { l } , n )", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 603, + 505, + 617 + ], + "score": 1.0, + "content": "under a continuous", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 89, + 612, + 280, + 632 + ], + "spans": [ + { + "bbox": [ + 89, + 612, + 174, + 632 + ], + "score": 1.0, + "content": "function. Define 119", + "type": "text" + }, + { + "bbox": [ + 174, + 615, + 264, + 628 + ], + "score": 0.92, + "content": "h : \\mathcal { Z } \\subseteq \\mathbb { R } ^ { n ^ { 2 } \\times l } \\to \\mathbb { R } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 612, + 280, + 632 + ], + "score": 1.0, + "content": "by", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39.5, + "bbox_fs": [ + 89, + 603, + 505, + 632 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 187, + 629, + 423, + 645 + ], + "lines": [ + { + "bbox": [ + 187, + 629, + 423, + 645 + ], + "spans": [ + { + "bbox": [ + 187, + 629, + 423, + 645 + ], + "score": 0.89, + "content": "\\begin{array} { r } { h ( V _ { 1 } V _ { 1 } ^ { \\top } , \\ldots , V _ { l } V _ { l } ^ { \\top } ) = \\rho \\left( \\phi _ { d _ { 1 } } ( V _ { 1 } V _ { 1 } ^ { \\top } ) , \\ldots , \\phi _ { d _ { l } } ( V _ { l } V _ { l } ^ { \\top } ) \\right) . } \\end{array}", + "type": "interline_equation", + "image_path": "cddf73eba3d761b1fa61ef79a063fb5b5ac415e473513ed703b244528b509afc.jpg" + } + ] + } + ], + "index": 41, + "virtual_lines": [ + { + "bbox": [ + 187, + 629, + 423, + 645 + ], + "spans": [], + "index": 41 + } + ] + }, + { + "type": "index", + "bbox": [ + 83, + 645, + 505, + 694 + ], + "lines": [ + { + "bbox": [ + 83, + 645, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 83, + 646, + 100, + 656 + ], + "score": 1.0, + "content": "1120", + "type": "text" + }, + { + "bbox": [ + 104, + 645, + 168, + 657 + ], + "score": 1.0, + "content": "Then note that", + "type": "text" + }, + { + "bbox": [ + 168, + 645, + 175, + 655 + ], + "score": 0.83, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 645, + 506, + 657 + ], + "score": 1.0, + "content": "is continuous and permutation equivariant from matrices to vectors, so it can be", + "type": "text" + } + ], + "index": 42, + "is_list_start_line": true + }, + { + "bbox": [ + 83, + 655, + 506, + 669 + ], + "spans": [ + { + "bbox": [ + 83, + 660, + 100, + 669 + ], + "score": 1.0, + "content": "1121", + "type": "text" + }, + { + "bbox": [ + 107, + 660, + 112, + 668 + ], + "score": 0.6, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 112, + 658, + 430, + 669 + ], + "score": 1.0, + "content": "-approximated by an invariant graph network [Keriven and Peyré, 2019], call it", + "type": "text" + }, + { + "bbox": [ + 430, + 655, + 451, + 668 + ], + "score": 0.73, + "content": "\\widetilde { \\mathrm { I G N } }", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 658, + 506, + 669 + ], + "score": 1.0, + "content": ". If we define", + "type": "text" + } + ], + "index": 43, + "is_list_start_line": true + }, + { + "bbox": [ + 83, + 669, + 507, + 684 + ], + "spans": [ + { + "bbox": [ + 83, + 673, + 100, + 682 + ], + "score": 1.0, + "content": "1122", + "type": "text" + }, + { + "bbox": [ + 106, + 669, + 145, + 682 + ], + "score": 0.9, + "content": "\\tilde { \\rho } = \\widetilde { \\mathrm { I G N } }", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 670, + 163, + 684 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 164, + 670, + 259, + 683 + ], + "score": 0.92, + "content": "\\mathrm { I G N } _ { d _ { i } } ( V _ { i } V _ { i } ^ { \\top } ) = V _ { i } V _ { i } ^ { \\top }", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 670, + 507, + 684 + ], + "score": 1.0, + "content": "(this identity operation is linear and permutation equivariant,", + "type": "text" + } + ], + "index": 44, + "is_list_start_line": true + }, + { + "bbox": [ + 83, + 681, + 432, + 695 + ], + "spans": [ + { + "bbox": [ + 83, + 684, + 101, + 693 + ], + "score": 1.0, + "content": "1123", + "type": "text" + }, + { + "bbox": [ + 104, + 681, + 332, + 695 + ], + "score": 1.0, + "content": "so it can be exactly expressed by an IGN), then we have", + "type": "text" + }, + { + "bbox": [ + 332, + 684, + 337, + 692 + ], + "score": 0.75, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 681, + 411, + 695 + ], + "score": 1.0, + "content": "-approximation of", + "type": "text" + }, + { + "bbox": [ + 411, + 682, + 418, + 694 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 418, + 681, + 432, + 695 + ], + "score": 1.0, + "content": "by", + "type": "text" + } + ], + "index": 45, + "is_list_start_line": true + } + ], + "index": 43.5, + "bbox_fs": [ + 83, + 645, + 507, + 695 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 168, + 695, + 442, + 711 + ], + "lines": [ + { + "bbox": [ + 168, + 695, + 442, + 711 + ], + "spans": [ + { + "bbox": [ + 168, + 695, + 442, + 711 + ], + "score": 0.87, + "content": "\\widetilde { \\mathrm { I G N } } ( V _ { 1 } V _ { 1 } ^ { \\top } , \\dots , V _ { l } V _ { l } ^ { \\top } ) = \\widetilde { \\rho } \\left( \\mathrm { I G N } _ { d _ { 1 } } ( V _ { 1 } V _ { 1 } ^ { \\top } ) , \\dots , \\mathrm { I G N } _ { d _ { l } } ( V _ { l } V _ { l } ^ { \\top } ) \\right) .", + "type": "interline_equation", + "image_path": "4e63104867e723be0733de902a09bbc91361b2008566f7ff91e22bf0d8e90a35.jpg" + } + ] + } + ], + "index": 46, + "virtual_lines": [ + { + "bbox": [ + 168, + 695, + 442, + 711 + ], + "spans": [], + "index": 46 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 83, + 92, + 506, + 160 + ], + "lines": [ + { + "bbox": [ + 82, + 92, + 506, + 104 + ], + "spans": [ + { + "bbox": [ + 82, + 94, + 100, + 104 + ], + "score": 1.0, + "content": "1126", + "type": "text" + }, + { + "bbox": [ + 105, + 92, + 359, + 104 + ], + "score": 1.0, + "content": "Theorem 4. Consider the same setup as Theorem 3, where", + "type": "text" + }, + { + "bbox": [ + 360, + 93, + 371, + 104 + ], + "score": 0.87, + "content": "\\mathcal { X } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 92, + 470, + 104 + ], + "score": 1.0, + "content": "are also compact. 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Then for any", + "type": "text" + }, + { + "bbox": [ + 442, + 126, + 468, + 136 + ], + "score": 0.88, + "content": "\\varepsilon > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 124, + 506, + 139 + ], + "score": 1.0, + "content": "and any", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 82, + 135, + 506, + 148 + ], + "spans": [ + { + "bbox": [ + 82, + 137, + 100, + 148 + ], + "score": 1.0, + "content": "1130", + "type": "text" + }, + { + "bbox": [ + 107, + 137, + 115, + 146 + ], + "score": 0.76, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 116, + 135, + 240, + 148 + ], + "score": 1.0, + "content": "-invariant continuous function", + "type": "text" + }, + { + "bbox": [ + 240, + 137, + 342, + 148 + ], + "score": 0.93, + "content": "f : \\mathcal { X } _ { 1 } \\times . . . \\times \\mathcal { X } _ { k } \\to \\mathbb { R } ^ { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 135, + 392, + 148 + ], + "score": 1.0, + "content": "there exists", + "type": "text" + }, + { + "bbox": [ + 392, + 137, + 418, + 147 + ], + "score": 0.91, + "content": "\\phi \\in \\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 135, + 437, + 148 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 438, + 137, + 465, + 148 + ], + "score": 0.91, + "content": "\\rho \\in \\mathcal R", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 135, + 506, + 148 + ], + "score": 1.0, + "content": "such that", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 83, + 146, + 223, + 161 + ], + "spans": [ + { + "bbox": [ + 83, + 149, + 100, + 159 + ], + "score": 1.0, + "content": "1131", + "type": "text" + }, + { + "bbox": [ + 107, + 147, + 218, + 159 + ], + "score": 0.89, + "content": "\\| f - \\rho ( \\phi _ { 1 } , \\ldots , \\phi _ { k } ) \\| _ { \\infty } < \\varepsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 146, + 223, + 161 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 83, + 169, + 505, + 205 + ], + "lines": [ + { + "bbox": [ + 81, + 169, + 505, + 183 + ], + "spans": [ + { + "bbox": [ + 81, + 169, + 228, + 183 + ], + "score": 1.0, + "content": "Proof. 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As", + "type": "text" + }, + { + "bbox": [ + 340, + 437, + 347, + 446 + ], + "score": 0.81, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 435, + 506, + 447 + ], + "score": 1.0, + "content": "is continuous on a compact domain, it", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 83, + 446, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 83, + 448, + 101, + 457 + ], + "score": 1.0, + "content": "1144", + "type": "text" + }, + { + "bbox": [ + 104, + 446, + 338, + 459 + ], + "score": 1.0, + "content": "is in fact uniformly continuous. 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We can optionally take", + "type": "text" + }, + { + "bbox": [ + 444, + 667, + 489, + 679 + ], + "score": 0.93, + "content": "\\theta _ { i } = h ( \\lambda _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 490, + 661, + 509, + 684 + ], + "score": 1.0, + "content": "for", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 213, + 690 + ], + "score": 1.0, + "content": "some continuous function", + "type": "text" + }, + { + "bbox": [ + 213, + 679, + 258, + 688 + ], + "score": 0.88, + "content": "h : \\mathbb { R } \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 677, + 505, + 690 + ], + "score": 1.0, + "content": "of the eigenvalues. This form captures most popular spectral", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 688, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 506, + 702 + ], + "score": 1.0, + "content": "graph convolutions in the literature [Bruna et al., 2014, Hamilton, 2020, Bronstein et al., 2017]; often,", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 700, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 291, + 712 + ], + "score": 1.0, + "content": "such convolutions are parameterized by taking", + "type": "text" + }, + { + "bbox": [ + 292, + 702, + 298, + 710 + ], + "score": 0.84, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 700, + 505, + 712 + ], + "score": 1.0, + "content": "to be some analytic function such as a simple affine", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 710, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 506, + 723 + ], + "score": 1.0, + "content": "function [Kipf and Welling, 2017], a linear combination in a polynomial basis [Defferrard et al.,", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 37.5 + } + ], + "page_idx": 31, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 300, + 741, + 311, + 750 + ], + "lines": [ + { + "bbox": [ + 298, + 740, + 313, + 755 + ], + "spans": [ + { + "bbox": [ + 298, + 740, + 313, + 755 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 15, + "width": 15 + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 494, + 585, + 505, + 596 + ], + "lines": [ + { + "bbox": [ + 496, + 587, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 496, + 587, + 505, + 596 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 86, + 72, + 400, + 85 + ], + "lines": [ + { + "bbox": [ + 84, + 71, + 401, + 87 + ], + "spans": [ + { + "bbox": [ + 84, + 71, + 401, + 87 + ], + "score": 1.0, + "content": "1125 G.3 Proof of Universal Approximation for General Decompositions", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "index", + "bbox": [ + 83, + 92, + 506, + 160 + ], + "lines": [ + { + "bbox": [ + 82, + 92, + 506, + 104 + ], + "spans": [ + { + "bbox": [ + 82, + 94, + 100, + 104 + ], + "score": 1.0, + "content": "1126", + "type": "text" + }, + { + "bbox": [ + 105, + 92, + 359, + 104 + ], + "score": 1.0, + "content": "Theorem 4. Consider the same setup as Theorem 3, where", + "type": "text" + }, + { + "bbox": [ + 360, + 93, + 371, + 104 + ], + "score": 0.87, + "content": "\\mathcal { X } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 92, + 470, + 104 + ], + "score": 1.0, + "content": "are also compact. 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Now, note that", + "type": "text" + }, + { + "bbox": [ + 384, + 244, + 433, + 259 + ], + "score": 0.93, + "content": "\\scriptstyle \\sum _ { l = 1 } ^ { d } v _ { i _ { l } } v _ { i _ { l } } ^ { \\top }", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 239, + 510, + 264 + ], + "score": 1.0, + "content": "is the orthogonal", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 82, + 256, + 506, + 270 + ], + "spans": [ + { + "bbox": [ + 82, + 258, + 100, + 268 + ], + "score": 1.0, + "content": "1171", + "type": "text" + }, + { + "bbox": [ + 104, + 256, + 506, + 270 + ], + "score": 1.0, + "content": "projector onto the eigenspace [Trefethen and Bau III, 1997]. 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Let", + "type": "text" + }, + { + "bbox": [ + 432, + 285, + 474, + 296 + ], + "score": 0.92, + "content": "V _ { 1 } , \\dots , V _ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 284, + 505, + 297 + ], + "score": 1.0, + "content": "be the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 84, + 293, + 506, + 309 + ], + "spans": [ + { + "bbox": [ + 84, + 298, + 100, + 306 + ], + "score": 1.0, + "content": "1174", + "type": "text" + }, + { + "bbox": [ + 103, + 293, + 216, + 309 + ], + "score": 1.0, + "content": "eigenspaces of dimension", + "type": "text" + }, + { + "bbox": [ + 216, + 296, + 257, + 307 + ], + "score": 0.9, + "content": "d _ { 1 } , \\ldots , d _ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 293, + 290, + 309 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 290, + 295, + 341, + 306 + ], + "score": 0.91, + "content": "V _ { i } \\in \\mathbb { R } ^ { n \\times d _ { i } ^ { \\star } }", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 293, + 506, + 309 + ], + "score": 1.0, + "content": ". Let the corresponding eigenvalues be", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 84, + 306, + 378, + 319 + ], + "spans": [ + { + "bbox": [ + 84, + 309, + 100, + 318 + ], + "score": 1.0, + "content": "1175", + "type": "text" + }, + { + "bbox": [ + 106, + 308, + 149, + 318 + ], + "score": 0.87, + "content": "\\mu _ { 1 } , \\ldots , \\mu _ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 306, + 289, + 319 + ], + "score": 1.0, + "content": ". Then for any orthogonal matrices", + "type": "text" + }, + { + "bbox": [ + 290, + 307, + 338, + 318 + ], + "score": 0.91, + "content": "Q _ { i } \\in O ( d _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 306, + 378, + 319 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16 + }, + { + "type": "interline_equation", + "bbox": [ + 207, + 324, + 402, + 433 + ], + "lines": [ + { + "bbox": [ + 207, + 324, + 402, + 433 + ], + "spans": [ + { + "bbox": [ + 207, + 324, + 402, + 433 + ], + "score": 0.95, + "content": "\\begin{array} { l } { { \\displaystyle \\sum _ { i = 1 } ^ { n } h ( \\lambda _ { i } ) v _ { i } v _ { i } ^ { \\top } = \\sum _ { j = 1 } ^ { l } V _ { j } h ( \\mu _ { j } ) I _ { d _ { j } } V _ { j } ^ { \\top } } } \\\\ { ~ } \\\\ { { \\displaystyle = \\sum _ { j = 1 } ^ { l } V _ { j } h ( \\mu _ { j } ) I _ { d _ { j } } Q _ { j } Q _ { j } ^ { \\top } V _ { j } ^ { \\top } } } \\\\ { { \\displaystyle ~ = \\sum _ { j = 1 } ^ { l } ( V _ { j } Q _ { j } ) h ( \\mu _ { j } ) I _ { d _ { j } } ( V _ { j } Q _ { j } ) ^ { \\top } } , } \\end{array}", + "type": "interline_equation", + "image_path": "f7186e5a54143ef94319274046c6343d0da931c5b39aebdb5444d9c1b892e9c3.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 207, + 324, + 402, + 339.57142857142856 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 207, + 339.57142857142856, + 402, + 355.1428571428571 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 207, + 355.1428571428571, + 402, + 370.71428571428567 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 207, + 370.71428571428567, + 402, + 386.2857142857142 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 207, + 386.2857142857142, + 402, + 401.8571428571428 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 207, + 401.8571428571428, + 402, + 417.42857142857133 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 207, + 417.42857142857133, + 402, + 432.9999999999999 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 84, + 437, + 399, + 450 + ], + "lines": [ + { + "bbox": [ + 81, + 435, + 400, + 453 + ], + "spans": [ + { + "bbox": [ + 81, + 435, + 346, + 453 + ], + "score": 1.0, + "content": "1176 so the spectral graph convolution is invariant to substituting", + "type": "text" + }, + { + "bbox": [ + 347, + 438, + 369, + 451 + ], + "score": 0.91, + "content": "V _ { j } Q _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 435, + 385, + 453 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 385, + 438, + 396, + 450 + ], + "score": 0.87, + "content": "V _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 435, + 400, + 453 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 96, + 454, + 506, + 477 + ], + "lines": [ + { + "bbox": [ + 92, + 452, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 92, + 452, + 505, + 468 + ], + "score": 1.0, + "content": "77 Now, we give the proof that shows SignNet and BasisNet can universally approximate spectral graph", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 92, + 465, + 162, + 477 + ], + "spans": [ + { + "bbox": [ + 92, + 465, + 162, + 477 + ], + "score": 1.0, + "content": "78 convolutions.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26.5 + }, + { + "type": "text", + "bbox": [ + 83, + 479, + 505, + 524 + ], + "lines": [ + { + "bbox": [ + 83, + 479, + 506, + 493 + ], + "spans": [ + { + "bbox": [ + 83, + 482, + 100, + 492 + ], + "score": 1.0, + "content": "1179", + "type": "text" + }, + { + "bbox": [ + 105, + 479, + 438, + 493 + ], + "score": 1.0, + "content": "Theorem 1 (Learning Spectral Graph Convolutions). Suppose the node features", + "type": "text" + }, + { + "bbox": [ + 438, + 480, + 484, + 491 + ], + "score": 0.92, + "content": "X \\in \\mathbb { R } ^ { n \\times q }", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 479, + 506, + 493 + ], + "score": 1.0, + "content": "take", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 83, + 491, + 506, + 505 + ], + "spans": [ + { + "bbox": [ + 83, + 493, + 99, + 502 + ], + "score": 1.0, + "content": "1180", + "type": "text" + }, + { + "bbox": [ + 105, + 491, + 506, + 505 + ], + "score": 1.0, + "content": "values in compact sets. Then SignNet can universally approximate any spectral graph convolution,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 83, + 501, + 506, + 516 + ], + "spans": [ + { + "bbox": [ + 83, + 504, + 99, + 514 + ], + "score": 1.0, + "content": "1181", + "type": "text" + }, + { + "bbox": [ + 105, + 501, + 506, + 516 + ], + "score": 1.0, + "content": "and both BasisNet and Expressive-BasisNet can universally approximate any parametric spectral", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 83, + 513, + 184, + 525 + ], + "spans": [ + { + "bbox": [ + 83, + 515, + 100, + 525 + ], + "score": 1.0, + "content": "1182", + "type": "text" + }, + { + "bbox": [ + 105, + 513, + 184, + 525 + ], + "score": 1.0, + "content": "graph convolution.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29.5 + }, + { + "type": "text", + "bbox": [ + 86, + 537, + 506, + 572 + ], + "lines": [ + { + "bbox": [ + 84, + 538, + 505, + 550 + ], + "spans": [ + { + "bbox": [ + 84, + 540, + 100, + 549 + ], + "score": 1.0, + "content": "1183", + "type": "text" + }, + { + "bbox": [ + 104, + 538, + 505, + 550 + ], + "score": 1.0, + "content": "Proof. Note that eigenvectors and eigenvalues of normalized Laplacian matrices take values in", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 84, + 549, + 507, + 562 + ], + "spans": [ + { + "bbox": [ + 84, + 551, + 99, + 559 + ], + "score": 1.0, + "content": "1184", + "type": "text" + }, + { + "bbox": [ + 105, + 549, + 507, + 562 + ], + "score": 1.0, + "content": "compact sets, since the eigenvalues are in [0, 2] and we take eigenvectors to have unit-norm. Thus,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 84, + 559, + 362, + 573 + ], + "spans": [ + { + "bbox": [ + 84, + 562, + 100, + 571 + ], + "score": 1.0, + "content": "1185", + "type": "text" + }, + { + "bbox": [ + 104, + 559, + 362, + 573 + ], + "score": 1.0, + "content": "the whole domain of the spectral graph convolution is compact.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 83, + 575, + 505, + 656 + ], + "lines": [ + { + "bbox": [ + 83, + 573, + 508, + 592 + ], + "spans": [ + { + "bbox": [ + 83, + 578, + 101, + 588 + ], + "score": 1.0, + "content": "1186", + "type": "text" + }, + { + "bbox": [ + 102, + 573, + 122, + 592 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 122, + 577, + 146, + 587 + ], + "score": 0.88, + "content": "\\varepsilon > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 573, + 326, + 592 + ], + "score": 1.0, + "content": ". First, consider a spectral graph convolution", + "type": "text" + }, + { + "bbox": [ + 326, + 575, + 448, + 589 + ], + "score": 0.91, + "content": "\\begin{array} { r } { f ( V , \\Lambda , X ) = \\sum _ { i = 1 } ^ { n } \\theta _ { i } v _ { i } v _ { i } ^ { \\top } X } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 573, + 508, + 592 + ], + "score": 1.0, + "content": ". For SignNet,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 83, + 588, + 506, + 602 + ], + "spans": [ + { + "bbox": [ + 83, + 591, + 100, + 601 + ], + "score": 1.0, + "content": "1187", + "type": "text" + }, + { + "bbox": [ + 104, + 589, + 120, + 602 + ], + "score": 1.0, + "content": "let", + "type": "text" + }, + { + "bbox": [ + 120, + 589, + 171, + 601 + ], + "score": 0.93, + "content": "\\phi ( v _ { i } , \\lambda _ { i } , X )", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 589, + 282, + 602 + ], + "score": 1.0, + "content": "approximate the function", + "type": "text" + }, + { + "bbox": [ + 282, + 588, + 388, + 602 + ], + "score": 0.91, + "content": "\\tilde { \\phi } ( v _ { i } , \\lambda _ { i } , X ) = \\theta _ { i } v _ { i } v _ { i } ^ { \\top } X", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 589, + 432, + 602 + ], + "score": 1.0, + "content": "to within", + "type": "text" + }, + { + "bbox": [ + 432, + 590, + 449, + 601 + ], + "score": 0.89, + "content": "\\varepsilon / n", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 589, + 506, + 602 + ], + "score": 1.0, + "content": "error, which", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 82, + 600, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 82, + 602, + 100, + 612 + ], + "score": 1.0, + "content": "1188", + "type": "text" + }, + { + "bbox": [ + 105, + 600, + 506, + 613 + ], + "score": 1.0, + "content": "DeepSets can do since this is a continuous permutation equivariant function from vectors to vectors", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 82, + 612, + 510, + 641 + ], + "spans": [ + { + "bbox": [ + 82, + 613, + 100, + 634 + ], + "score": 1.0, + "content": "1189 1190", + "type": "text" + }, + { + "bbox": [ + 101, + 612, + 235, + 641 + ], + "score": 1.0, + "content": "[Segol and Lipman, 2019] (note1 is the all ones vector). Then", + "type": "text" + }, + { + "bbox": [ + 236, + 622, + 282, + 635 + ], + "score": 0.93, + "content": "\\rho = \\textstyle \\sum _ { i = 1 } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 612, + 302, + 641 + ], + "score": 1.0, + "content": "pass is a", + "type": "text" + }, + { + "bbox": [ + 302, + 612, + 312, + 623 + ], + "score": 0.88, + "content": "\\lambda _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 612, + 368, + 641 + ], + "score": 1.0, + "content": "as a vector in ear permutati", + "type": "text" + }, + { + "bbox": [ + 369, + 612, + 382, + 621 + ], + "score": 0.87, + "content": "\\mathbb { R } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 612, + 459, + 641 + ], + "score": 1.0, + "content": "by instead passing equivariant operati", + "type": "text" + }, + { + "bbox": [ + 459, + 612, + 474, + 623 + ], + "score": 0.89, + "content": "\\lambda _ { i } \\mathbf { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 612, + 510, + 641 + ], + "score": 1.0, + "content": ", wherehat can", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 82, + 634, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 82, + 635, + 99, + 645 + ], + "score": 1.0, + "content": "1191", + "type": "text" + }, + { + "bbox": [ + 105, + 634, + 356, + 645 + ], + "score": 1.0, + "content": "be exactly expressed by DeepSets, so the total error is within", + "type": "text" + }, + { + "bbox": [ + 362, + 634, + 505, + 645 + ], + "score": 1.0, + "content": ". The same argument applies when", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 82, + 644, + 284, + 657 + ], + "spans": [ + { + "bbox": [ + 82, + 646, + 100, + 655 + ], + "score": 1.0, + "content": "1192", + "type": "text" + }, + { + "bbox": [ + 106, + 644, + 151, + 656 + ], + "score": 0.93, + "content": "\\theta _ { i } = h ( \\lambda _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 644, + 272, + 657 + ], + "score": 1.0, + "content": "for some continuous function", + "type": "text" + }, + { + "bbox": [ + 273, + 645, + 279, + 654 + ], + "score": 0.79, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 644, + 284, + 657 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 37.5 + }, + { + "type": "text", + "bbox": [ + 83, + 660, + 505, + 723 + ], + "lines": [ + { + "bbox": [ + 83, + 660, + 504, + 673 + ], + "spans": [ + { + "bbox": [ + 83, + 662, + 100, + 672 + ], + "score": 1.0, + "content": "1193", + "type": "text" + }, + { + "bbox": [ + 105, + 660, + 443, + 672 + ], + "score": 1.0, + "content": "For the basis invariant case, consider a parametric spectral graph convolution", + "type": "text" + }, + { + "bbox": [ + 444, + 660, + 504, + 673 + ], + "score": 0.9, + "content": "f ( V , \\Lambda , X ) ~ =", + "type": "inline_equation" + } + ], + "index": 41 + }, + { + "bbox": [ + 83, + 670, + 507, + 688 + ], + "spans": [ + { + "bbox": [ + 83, + 674, + 100, + 685 + ], + "score": 1.0, + "content": "1194", + "type": "text" + }, + { + "bbox": [ + 106, + 671, + 185, + 685 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\sum _ { i = 1 } ^ { n } h ( \\lambda _ { i } ) v _ { i } v _ { i } ^ { \\top } X } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 670, + 333, + 688 + ], + "score": 1.0, + "content": ". Note that if the eigenspace bases are", + "type": "text" + }, + { + "bbox": [ + 334, + 673, + 376, + 684 + ], + "score": 0.92, + "content": "V _ { 1 } , \\dots , V _ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 670, + 444, + 688 + ], + "score": 1.0, + "content": "with eigenvalues", + "type": "text" + }, + { + "bbox": [ + 445, + 674, + 487, + 684 + ], + "score": 0.86, + "content": "\\mu _ { 1 } , \\ldots , \\mu _ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 670, + 507, + 688 + ], + "score": 1.0, + "content": ", we", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 83, + 683, + 508, + 702 + ], + "spans": [ + { + "bbox": [ + 83, + 688, + 99, + 698 + ], + "score": 1.0, + "content": "1195", + "type": "text" + }, + { + "bbox": [ + 104, + 683, + 162, + 702 + ], + "score": 1.0, + "content": "can write the", + "type": "text" + }, + { + "bbox": [ + 162, + 684, + 304, + 700 + ], + "score": 0.93, + "content": "\\begin{array} { r } { f ( V , \\Lambda , X ) = \\sum _ { i = 1 } ^ { l } h ( \\mu _ { j } ) V _ { j } V _ { j } ^ { \\top } X } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 683, + 383, + 702 + ], + "score": 1.0, + "content": ". Again, we will let", + "type": "text" + }, + { + "bbox": [ + 384, + 684, + 427, + 699 + ], + "score": 0.93, + "content": "\\rho = \\textstyle \\sum _ { i = 1 } ^ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 683, + 508, + 702 + ], + "score": 1.0, + "content": "be a sum function,", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 83, + 699, + 504, + 713 + ], + "spans": [ + { + "bbox": [ + 83, + 701, + 100, + 712 + ], + "score": 1.0, + "content": "1196", + "type": "text" + }, + { + "bbox": [ + 105, + 699, + 400, + 713 + ], + "score": 1.0, + "content": "which can be expressed exactly by DeepSets. 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For", + "type": "text" + } + ], + "index": 6, + "is_list_start_line": true + }, + { + "bbox": [ + 82, + 183, + 505, + 196 + ], + "spans": [ + { + "bbox": [ + 82, + 185, + 100, + 196 + ], + "score": 1.0, + "content": "1165", + "type": "text" + }, + { + "bbox": [ + 106, + 183, + 154, + 196 + ], + "score": 1.0, + "content": "instance, if", + "type": "text" + }, + { + "bbox": [ + 154, + 185, + 164, + 195 + ], + "score": 0.85, + "content": "v _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 183, + 183, + 196 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 183, + 185, + 194, + 194 + ], + "score": 0.84, + "content": "v _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 183, + 453, + 196 + ], + "score": 1.0, + "content": "are in the same eigenspace, and we change basis by permuting", + "type": "text" + }, + { + "bbox": [ + 453, + 184, + 486, + 196 + ], + "score": 0.92, + "content": "v _ { 1 } ^ { \\prime } = v _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 183, + 505, + 196 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 7, + "is_list_start_line": true + }, + { + "bbox": [ + 82, + 194, + 450, + 207 + ], + "spans": [ + { + "bbox": [ + 82, + 196, + 100, + 207 + ], + "score": 1.0, + "content": "1166", + "type": "text" + }, + { + "bbox": [ + 106, + 194, + 139, + 207 + ], + "score": 0.93, + "content": "v _ { 2 } ^ { \\prime } = v _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 195, + 171, + 207 + ], + "score": 1.0, + "content": ", then if", + "type": "text" + }, + { + "bbox": [ + 172, + 195, + 204, + 206 + ], + "score": 0.93, + "content": "\\theta _ { 1 } \\neq \\theta _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 195, + 450, + 207 + ], + "score": 1.0, + "content": "the spectral graph convolution will generally change as well.", + "type": "text" + } + ], + "index": 8, + "is_list_start_line": true + }, + { + "bbox": [ + 82, + 210, + 506, + 223 + ], + "spans": [ + { + "bbox": [ + 82, + 213, + 100, + 222 + ], + "score": 1.0, + "content": "1167", + "type": "text" + }, + { + "bbox": [ + 106, + 210, + 192, + 222 + ], + "score": 1.0, + "content": "On the other hand, if", + "type": "text" + }, + { + "bbox": [ + 192, + 211, + 237, + 223 + ], + "score": 0.92, + "content": "\\theta _ { i } = h ( \\lambda _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 210, + 312, + 222 + ], + "score": 1.0, + "content": "for some function", + "type": "text" + }, + { + "bbox": [ + 313, + 211, + 357, + 221 + ], + "score": 0.9, + "content": "h : \\mathbb { R } \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 210, + 506, + 222 + ], + "score": 1.0, + "content": ", then the spectral graph convolution", + "type": "text" + } + ], + "index": 9, + "is_list_start_line": true + }, + { + "bbox": [ + 82, + 221, + 506, + 234 + ], + "spans": [ + { + "bbox": [ + 82, + 223, + 100, + 234 + ], + "score": 1.0, + "content": "1168", + "type": "text" + }, + { + "bbox": [ + 104, + 221, + 259, + 234 + ], + "score": 1.0, + "content": "is basis invariant. This is because if", + "type": "text" + }, + { + "bbox": [ + 260, + 223, + 269, + 233 + ], + "score": 0.85, + "content": "v _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 221, + 289, + 234 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 289, + 223, + 299, + 234 + ], + "score": 0.86, + "content": "v _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 221, + 455, + 234 + ], + "score": 1.0, + "content": "belong to the same eigenspace, then", + "type": "text" + }, + { + "bbox": [ + 455, + 222, + 491, + 234 + ], + "score": 0.91, + "content": "\\lambda _ { i } = \\lambda _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 492, + 221, + 506, + 234 + ], + "score": 1.0, + "content": "so", + "type": "text" + } + ], + "index": 10, + "is_list_start_line": true + }, + { + "bbox": [ + 82, + 231, + 507, + 246 + ], + "spans": [ + { + "bbox": [ + 82, + 234, + 100, + 244 + ], + "score": 1.0, + "content": "1169", + "type": "text" + }, + { + "bbox": [ + 106, + 232, + 169, + 245 + ], + "score": 0.92, + "content": "h ( \\lambda _ { i } ) = h ( \\lambda _ { j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 231, + 210, + 246 + ], + "score": 1.0, + "content": ". Thus, if", + "type": "text" + }, + { + "bbox": [ + 211, + 234, + 258, + 245 + ], + "score": 0.88, + "content": "v _ { i _ { 1 } } , \\ldots , v _ { i _ { d } }", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 231, + 495, + 246 + ], + "score": 1.0, + "content": "are eigenvectors of the same eigenspace with eigenvalue", + "type": "text" + }, + { + "bbox": [ + 496, + 234, + 502, + 243 + ], + "score": 0.79, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 231, + 507, + 246 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 11, + "is_list_start_line": true + }, + { + "bbox": [ + 82, + 239, + 510, + 264 + ], + "spans": [ + { + "bbox": [ + 82, + 248, + 100, + 258 + ], + "score": 1.0, + "content": "1170", + "type": "text" + }, + { + "bbox": [ + 101, + 239, + 159, + 264 + ], + "score": 1.0, + "content": "we have that", + "type": "text" + }, + { + "bbox": [ + 159, + 244, + 318, + 260 + ], + "score": 0.89, + "content": "\\begin{array} { r } { \\sum _ { l = 1 } ^ { d } \\underline { h } ( \\lambda _ { i _ { l } } ) v _ { i _ { l } } v _ { i _ { l } } ^ { \\top } = h ( \\lambda ) \\sum _ { l = 1 } ^ { d } v _ { i _ { l } } v _ { i _ { l } } ^ { \\top } . } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 239, + 383, + 264 + ], + "score": 1.0, + "content": ". Now, note that", + "type": "text" + }, + { + "bbox": [ + 384, + 244, + 433, + 259 + ], + "score": 0.93, + "content": "\\scriptstyle \\sum _ { l = 1 } ^ { d } v _ { i _ { l } } v _ { i _ { l } } ^ { \\top }", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 239, + 510, + 264 + ], + "score": 1.0, + "content": "is the orthogonal", + "type": "text" + } + ], + "index": 12, + "is_list_start_line": true + }, + { + "bbox": [ + 82, + 256, + 506, + 270 + ], + "spans": [ + { + "bbox": [ + 82, + 258, + 100, + 268 + ], + "score": 1.0, + "content": "1171", + "type": "text" + }, + { + "bbox": [ + 104, + 256, + 506, + 270 + ], + "score": 1.0, + "content": "projector onto the eigenspace [Trefethen and Bau III, 1997]. A change of basis does not change this", + "type": "text" + } + ], + "index": 13, + "is_list_start_line": true + }, + { + "bbox": [ + 83, + 268, + 414, + 281 + ], + "spans": [ + { + "bbox": [ + 83, + 270, + 101, + 280 + ], + "score": 1.0, + "content": "1172", + "type": "text" + }, + { + "bbox": [ + 105, + 268, + 414, + 281 + ], + "score": 1.0, + "content": "orthogonal projector, so such spectral graph convolutions are basis invariant.", + "type": "text" + } + ], + "index": 14, + "is_list_start_line": true + }, + { + "bbox": [ + 81, + 284, + 505, + 297 + ], + "spans": [ + { + "bbox": [ + 81, + 284, + 431, + 297 + ], + "score": 1.0, + "content": "1173 Another way to see this basis invariance is with a simple computation. Let", + "type": "text" + }, + { + "bbox": [ + 432, + 285, + 474, + 296 + ], + "score": 0.92, + "content": "V _ { 1 } , \\dots , V _ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 284, + 505, + 297 + ], + "score": 1.0, + "content": "be the", + "type": "text" + } + ], + "index": 15, + "is_list_start_line": true + }, + { + "bbox": [ + 84, + 293, + 506, + 309 + ], + "spans": [ + { + "bbox": [ + 84, + 298, + 100, + 306 + ], + "score": 1.0, + "content": "1174", + "type": "text" + }, + { + "bbox": [ + 103, + 293, + 216, + 309 + ], + "score": 1.0, + "content": "eigenspaces of dimension", + "type": "text" + }, + { + "bbox": [ + 216, + 296, + 257, + 307 + ], + "score": 0.9, + "content": "d _ { 1 } , \\ldots , d _ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 293, + 290, + 309 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 290, + 295, + 341, + 306 + ], + "score": 0.91, + "content": "V _ { i } \\in \\mathbb { R } ^ { n \\times d _ { i } ^ { \\star } }", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 293, + 506, + 309 + ], + "score": 1.0, + "content": ". Let the corresponding eigenvalues be", + "type": "text" + } + ], + "index": 16, + "is_list_start_line": true + }, + { + "bbox": [ + 84, + 306, + 378, + 319 + ], + "spans": [ + { + "bbox": [ + 84, + 309, + 100, + 318 + ], + "score": 1.0, + "content": "1175", + "type": "text" + }, + { + "bbox": [ + 106, + 308, + 149, + 318 + ], + "score": 0.87, + "content": "\\mu _ { 1 } , \\ldots , \\mu _ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 306, + 289, + 319 + ], + "score": 1.0, + "content": ". Then for any orthogonal matrices", + "type": "text" + }, + { + "bbox": [ + 290, + 307, + 338, + 318 + ], + "score": 0.91, + "content": "Q _ { i } \\in O ( d _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 306, + 378, + 319 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + } + ], + "index": 17, + "is_list_start_line": true + } + ], + "index": 6.5, + "bbox_fs": [ + 82, + 160, + 506, + 207 + ] + }, + { + "type": "index", + "bbox": [ + 83, + 210, + 506, + 280 + ], + "lines": [], + "index": 11.5, + "bbox_fs": [ + 82, + 210, + 510, + 281 + ], + "lines_deleted": true + }, + { + "type": "index", + "bbox": [ + 84, + 284, + 504, + 318 + ], + "lines": [], + "index": 16, + "bbox_fs": [ + 81, + 284, + 506, + 319 + ], + "lines_deleted": true + }, + { + "type": "interline_equation", + "bbox": [ + 207, + 324, + 402, + 433 + ], + "lines": [ + { + "bbox": [ + 207, + 324, + 402, + 433 + ], + "spans": [ + { + "bbox": [ + 207, + 324, + 402, + 433 + ], + "score": 0.95, + "content": "\\begin{array} { l } { { \\displaystyle \\sum _ { i = 1 } ^ { n } h ( \\lambda _ { i } ) v _ { i } v _ { i } ^ { \\top } = \\sum _ { j = 1 } ^ { l } V _ { j } h ( \\mu _ { j } ) I _ { d _ { j } } V _ { j } ^ { \\top } } } \\\\ { ~ } \\\\ { { \\displaystyle = \\sum _ { j = 1 } ^ { l } V _ { j } h ( \\mu _ { j } ) I _ { d _ { j } } Q _ { j } Q _ { j } ^ { \\top } V _ { j } ^ { \\top } } } \\\\ { { \\displaystyle ~ = \\sum _ { j = 1 } ^ { l } ( V _ { j } Q _ { j } ) h ( \\mu _ { j } ) I _ { d _ { j } } ( V _ { j } Q _ { j } ) ^ { \\top } } , } \\end{array}", + "type": "interline_equation", + "image_path": "f7186e5a54143ef94319274046c6343d0da931c5b39aebdb5444d9c1b892e9c3.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 207, + 324, + 402, + 339.57142857142856 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 207, + 339.57142857142856, + 402, + 355.1428571428571 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 207, + 355.1428571428571, + 402, + 370.71428571428567 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 207, + 370.71428571428567, + 402, + 386.2857142857142 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 207, + 386.2857142857142, + 402, + 401.8571428571428 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 207, + 401.8571428571428, + 402, + 417.42857142857133 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 207, + 417.42857142857133, + 402, + 432.9999999999999 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 84, + 437, + 399, + 450 + ], + "lines": [ + { + "bbox": [ + 81, + 435, + 400, + 453 + ], + "spans": [ + { + "bbox": [ + 81, + 435, + 346, + 453 + ], + "score": 1.0, + "content": "1176 so the spectral graph convolution is invariant to substituting", + "type": "text" + }, + { + "bbox": [ + 347, + 438, + 369, + 451 + ], + "score": 0.91, + "content": "V _ { j } Q _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 435, + 385, + 453 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 385, + 438, + 396, + 450 + ], + "score": 0.87, + "content": "V _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 435, + 400, + 453 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25, + "bbox_fs": [ + 81, + 435, + 400, + 453 + ] + }, + { + "type": "index", + "bbox": [ + 96, + 454, + 506, + 477 + ], + "lines": [ + { + "bbox": [ + 92, + 452, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 92, + 452, + 505, + 468 + ], + "score": 1.0, + "content": "77 Now, we give the proof that shows SignNet and BasisNet can universally approximate spectral graph", + "type": "text" + } + ], + "index": 26, + "is_list_start_line": true + }, + { + "bbox": [ + 92, + 465, + 162, + 477 + ], + "spans": [ + { + "bbox": [ + 92, + 465, + 162, + 477 + ], + "score": 1.0, + "content": "78 convolutions.", + "type": "text" + } + ], + "index": 27, + "is_list_start_line": true + }, + { + "bbox": [ + 83, + 479, + 506, + 493 + ], + "spans": [ + { + "bbox": [ + 83, + 482, + 100, + 492 + ], + "score": 1.0, + "content": "1179", + "type": "text" + }, + { + "bbox": [ + 105, + 479, + 438, + 493 + ], + "score": 1.0, + "content": "Theorem 1 (Learning Spectral Graph Convolutions). Suppose the node features", + "type": "text" + }, + { + "bbox": [ + 438, + 480, + 484, + 491 + ], + "score": 0.92, + "content": "X \\in \\mathbb { R } ^ { n \\times q }", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 479, + 506, + 493 + ], + "score": 1.0, + "content": "take", + "type": "text" + } + ], + "index": 28, + "is_list_start_line": true + }, + { + "bbox": [ + 83, + 491, + 506, + 505 + ], + "spans": [ + { + "bbox": [ + 83, + 493, + 99, + 502 + ], + "score": 1.0, + "content": "1180", + "type": "text" + }, + { + "bbox": [ + 105, + 491, + 506, + 505 + ], + "score": 1.0, + "content": "values in compact sets. Then SignNet can universally approximate any spectral graph convolution,", + "type": "text" + } + ], + "index": 29, + "is_list_start_line": true + }, + { + "bbox": [ + 83, + 501, + 506, + 516 + ], + "spans": [ + { + "bbox": [ + 83, + 504, + 99, + 514 + ], + "score": 1.0, + "content": "1181", + "type": "text" + }, + { + "bbox": [ + 105, + 501, + 506, + 516 + ], + "score": 1.0, + "content": "and both BasisNet and Expressive-BasisNet can universally approximate any parametric spectral", + "type": "text" + } + ], + "index": 30, + "is_list_start_line": true + }, + { + "bbox": [ + 83, + 513, + 184, + 525 + ], + "spans": [ + { + "bbox": [ + 83, + 515, + 100, + 525 + ], + "score": 1.0, + "content": "1182", + "type": "text" + }, + { + "bbox": [ + 105, + 513, + 184, + 525 + ], + "score": 1.0, + "content": "graph convolution.", + "type": "text" + } + ], + "index": 31, + "is_list_start_line": true + }, + { + "bbox": [ + 84, + 538, + 505, + 550 + ], + "spans": [ + { + "bbox": [ + 84, + 540, + 100, + 549 + ], + "score": 1.0, + "content": "1183", + "type": "text" + }, + { + "bbox": [ + 104, + 538, + 505, + 550 + ], + "score": 1.0, + "content": "Proof. Note that eigenvectors and eigenvalues of normalized Laplacian matrices take values in", + "type": "text" + } + ], + "index": 32, + "is_list_start_line": true + }, + { + "bbox": [ + 84, + 549, + 507, + 562 + ], + "spans": [ + { + "bbox": [ + 84, + 551, + 99, + 559 + ], + "score": 1.0, + "content": "1184", + "type": "text" + }, + { + "bbox": [ + 105, + 549, + 507, + 562 + ], + "score": 1.0, + "content": "compact sets, since the eigenvalues are in [0, 2] and we take eigenvectors to have unit-norm. Thus,", + "type": "text" + } + ], + "index": 33, + "is_list_start_line": true + }, + { + "bbox": [ + 84, + 559, + 362, + 573 + ], + "spans": [ + { + "bbox": [ + 84, + 562, + 100, + 571 + ], + "score": 1.0, + "content": "1185", + "type": "text" + }, + { + "bbox": [ + 104, + 559, + 362, + 573 + ], + "score": 1.0, + "content": "the whole domain of the spectral graph convolution is compact.", + "type": "text" + } + ], + "index": 34, + "is_list_start_line": true + }, + { + "bbox": [ + 83, + 573, + 508, + 592 + ], + "spans": [ + { + "bbox": [ + 83, + 578, + 101, + 588 + ], + "score": 1.0, + "content": "1186", + "type": "text" + }, + { + "bbox": [ + 102, + 573, + 122, + 592 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 122, + 577, + 146, + 587 + ], + "score": 0.88, + "content": "\\varepsilon > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 573, + 326, + 592 + ], + "score": 1.0, + "content": ". First, consider a spectral graph convolution", + "type": "text" + }, + { + "bbox": [ + 326, + 575, + 448, + 589 + ], + "score": 0.91, + "content": "\\begin{array} { r } { f ( V , \\Lambda , X ) = \\sum _ { i = 1 } ^ { n } \\theta _ { i } v _ { i } v _ { i } ^ { \\top } X } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 573, + 508, + 592 + ], + "score": 1.0, + "content": ". For SignNet,", + "type": "text" + } + ], + "index": 35, + "is_list_start_line": true + }, + { + "bbox": [ + 83, + 588, + 506, + 602 + ], + "spans": [ + { + "bbox": [ + 83, + 591, + 100, + 601 + ], + "score": 1.0, + "content": "1187", + "type": "text" + }, + { + "bbox": [ + 104, + 589, + 120, + 602 + ], + "score": 1.0, + "content": "let", + "type": "text" + }, + { + "bbox": [ + 120, + 589, + 171, + 601 + ], + "score": 0.93, + "content": "\\phi ( v _ { i } , \\lambda _ { i } , X )", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 589, + 282, + 602 + ], + "score": 1.0, + "content": "approximate the function", + "type": "text" + }, + { + "bbox": [ + 282, + 588, + 388, + 602 + ], + "score": 0.91, + "content": "\\tilde { \\phi } ( v _ { i } , \\lambda _ { i } , X ) = \\theta _ { i } v _ { i } v _ { i } ^ { \\top } X", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 589, + 432, + 602 + ], + "score": 1.0, + "content": "to within", + "type": "text" + }, + { + "bbox": [ + 432, + 590, + 449, + 601 + ], + "score": 0.89, + "content": "\\varepsilon / n", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 589, + 506, + 602 + ], + "score": 1.0, + "content": "error, which", + "type": "text" + } + ], + "index": 36, + "is_list_start_line": true + }, + { + "bbox": [ + 82, + 600, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 82, + 602, + 100, + 612 + ], + "score": 1.0, + "content": "1188", + "type": "text" + }, + { + "bbox": [ + 105, + 600, + 506, + 613 + ], + "score": 1.0, + "content": "DeepSets can do since this is a continuous permutation equivariant function from vectors to vectors", + "type": "text" + } + ], + "index": 37, + "is_list_start_line": true + }, + { + "bbox": [ + 82, + 612, + 510, + 641 + ], + "spans": [ + { + "bbox": [ + 82, + 613, + 100, + 634 + ], + "score": 1.0, + "content": "1189 1190", + "type": "text" + }, + { + "bbox": [ + 101, + 612, + 235, + 641 + ], + "score": 1.0, + "content": "[Segol and Lipman, 2019] (note1 is the all ones vector). Then", + "type": "text" + }, + { + "bbox": [ + 236, + 622, + 282, + 635 + ], + "score": 0.93, + "content": "\\rho = \\textstyle \\sum _ { i = 1 } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 612, + 302, + 641 + ], + "score": 1.0, + "content": "pass is a", + "type": "text" + }, + { + "bbox": [ + 302, + 612, + 312, + 623 + ], + "score": 0.88, + "content": "\\lambda _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 612, + 368, + 641 + ], + "score": 1.0, + "content": "as a vector in ear permutati", + "type": "text" + }, + { + "bbox": [ + 369, + 612, + 382, + 621 + ], + "score": 0.87, + "content": "\\mathbb { R } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 612, + 459, + 641 + ], + "score": 1.0, + "content": "by instead passing equivariant operati", + "type": "text" + }, + { + "bbox": [ + 459, + 612, + 474, + 623 + ], + "score": 0.89, + "content": "\\lambda _ { i } \\mathbf { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 612, + 510, + 641 + ], + "score": 1.0, + "content": ", wherehat can", + "type": "text" + } + ], + "index": 38, + "is_list_start_line": true + }, + { + "bbox": [ + 82, + 634, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 82, + 635, + 99, + 645 + ], + "score": 1.0, + "content": "1191", + "type": "text" + }, + { + "bbox": [ + 105, + 634, + 356, + 645 + ], + "score": 1.0, + "content": "be exactly expressed by DeepSets, so the total error is within", + "type": "text" + }, + { + "bbox": [ + 362, + 634, + 505, + 645 + ], + "score": 1.0, + "content": ". The same argument applies when", + "type": "text" + } + ], + "index": 39, + "is_list_start_line": true + }, + { + "bbox": [ + 82, + 644, + 284, + 657 + ], + "spans": [ + { + "bbox": [ + 82, + 646, + 100, + 655 + ], + "score": 1.0, + "content": "1192", + "type": "text" + }, + { + "bbox": [ + 106, + 644, + 151, + 656 + ], + "score": 0.93, + "content": "\\theta _ { i } = h ( \\lambda _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 644, + 272, + 657 + ], + "score": 1.0, + "content": "for some continuous function", + "type": "text" + }, + { + "bbox": [ + 273, + 645, + 279, + 654 + ], + "score": 0.79, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 644, + 284, + 657 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 40, + "is_list_start_line": true + }, + { + "bbox": [ + 83, + 660, + 504, + 673 + ], + "spans": [ + { + "bbox": [ + 83, + 662, + 100, + 672 + ], + "score": 1.0, + "content": "1193", + "type": "text" + }, + { + "bbox": [ + 105, + 660, + 443, + 672 + ], + "score": 1.0, + "content": "For the basis invariant case, consider a parametric spectral graph convolution", + "type": "text" + }, + { + "bbox": [ + 444, + 660, + 504, + 673 + ], + "score": 0.9, + "content": "f ( V , \\Lambda , X ) ~ =", + "type": "inline_equation" + } + ], + "index": 41, + "is_list_start_line": true + }, + { + "bbox": [ + 83, + 670, + 507, + 688 + ], + "spans": [ + { + "bbox": [ + 83, + 674, + 100, + 685 + ], + "score": 1.0, + "content": "1194", + "type": "text" + }, + { + "bbox": [ + 106, + 671, + 185, + 685 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\sum _ { i = 1 } ^ { n } h ( \\lambda _ { i } ) v _ { i } v _ { i } ^ { \\top } X } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 670, + 333, + 688 + ], + "score": 1.0, + "content": ". Note that if the eigenspace bases are", + "type": "text" + }, + { + "bbox": [ + 334, + 673, + 376, + 684 + ], + "score": 0.92, + "content": "V _ { 1 } , \\dots , V _ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 670, + 444, + 688 + ], + "score": 1.0, + "content": "with eigenvalues", + "type": "text" + }, + { + "bbox": [ + 445, + 674, + 487, + 684 + ], + "score": 0.86, + "content": "\\mu _ { 1 } , \\ldots , \\mu _ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 670, + 507, + 688 + ], + "score": 1.0, + "content": ", we", + "type": "text" + } + ], + "index": 42, + "is_list_start_line": true + }, + { + "bbox": [ + 83, + 683, + 508, + 702 + ], + "spans": [ + { + "bbox": [ + 83, + 688, + 99, + 698 + ], + "score": 1.0, + "content": "1195", + "type": "text" + }, + { + "bbox": [ + 104, + 683, + 162, + 702 + ], + "score": 1.0, + "content": "can write the", + "type": "text" + }, + { + "bbox": [ + 162, + 684, + 304, + 700 + ], + "score": 0.93, + "content": "\\begin{array} { r } { f ( V , \\Lambda , X ) = \\sum _ { i = 1 } ^ { l } h ( \\mu _ { j } ) V _ { j } V _ { j } ^ { \\top } X } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 683, + 383, + 702 + ], + "score": 1.0, + "content": ". Again, we will let", + "type": "text" + }, + { + "bbox": [ + 384, + 684, + 427, + 699 + ], + "score": 0.93, + "content": "\\rho = \\textstyle \\sum _ { i = 1 } ^ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 683, + 508, + 702 + ], + "score": 1.0, + "content": "be a sum function,", + "type": "text" + } + ], + "index": 43, + "is_list_start_line": true + }, + { + "bbox": [ + 83, + 699, + 504, + 713 + ], + "spans": [ + { + "bbox": [ + 83, + 701, + 100, + 712 + ], + "score": 1.0, + "content": "1196", + "type": "text" + }, + { + "bbox": [ + 105, + 699, + 400, + 713 + ], + "score": 1.0, + "content": "which can be expressed exactly by DeepSets. 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Also, since a 2-IGN can learn matrix vector", + "type": "text", + "cross_page": true + } + ], + "index": 2, + "is_list_start_line": true + }, + { + "bbox": [ + 83, + 105, + 506, + 122 + ], + "spans": [ + { + "bbox": [ + 83, + 109, + 99, + 119 + ], + "score": 1.0, + "content": "1201", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 104, + 105, + 396, + 122 + ], + "score": 1.0, + "content": "multiplication (Cai and Wang [2022] Lemma 10), we can approximate", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 397, + 106, + 506, + 120 + ], + "score": 0.92, + "content": "f _ { 2 } ( h ( \\mu ) { \\bf 1 1 } ^ { \\top } , V V ^ { \\top } , X ) =", + "type": "inline_equation", + "cross_page": true + } + ], + "index": 3, + "is_list_start_line": true + }, + { + "bbox": [ + 83, + 118, + 507, + 135 + ], + "spans": [ + { + "bbox": [ + 83, + 122, + 100, + 133 + ], + "score": 1.0, + "content": "1202", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 119, + 188, + 133 + ], + "score": 0.9, + "content": "( h ( \\mu ) \\mathbf { 1 1 } ^ { \\top } , V V ^ { \\top } X )", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 189, + 118, + 203, + 135 + ], + "score": 1.0, + "content": ", as", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 203, + 119, + 255, + 133 + ], + "score": 0.93, + "content": "V _ { i } V _ { i } ^ { \\top } \\in \\mathbb { R } ^ { n ^ { 2 } }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 256, + 118, + 318, + 135 + ], + "score": 1.0, + "content": "is a matrix and", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 319, + 120, + 364, + 131 + ], + "score": 0.92, + "content": "\\ b X \\in \\mathbb { R } ^ { n \\times q }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 364, + 118, + 507, + 135 + ], + "score": 1.0, + "content": "is a vector with respect to permuta-", + "type": "text", + "cross_page": true + } + ], + "index": 4, + "is_list_start_line": true + }, + { + "bbox": [ + 82, + 132, + 506, + 144 + ], + "spans": [ + { + "bbox": [ + 82, + 134, + 100, + 144 + ], + "score": 1.0, + "content": "1203", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 132, + 506, + 144 + ], + "score": 1.0, + "content": "tion symmetries. 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Since", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 298, + 143, + 484, + 156 + ], + "score": 0.92, + "content": "f _ { 3 } \\circ f _ { 2 } \\circ \\bar { f } _ { 1 } ( \\mu { \\bf 1 1 } ^ { \\top } , V V ^ { \\top } , X ) = h ( \\mu ) V V ^ { \\top } X", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 484, + 142, + 506, + 157 + ], + "score": 1.0, + "content": ", and", + "type": "text", + "cross_page": true + } + ], + "index": 6, + "is_list_start_line": true + }, + { + "bbox": [ + 82, + 154, + 506, + 168 + ], + "spans": [ + { + "bbox": [ + 82, + 156, + 100, + 167 + ], + "score": 1.0, + "content": "1205", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 154, + 283, + 168 + ], + "score": 1.0, + "content": "since 2-IGNs universally approximate each", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 284, + 155, + 293, + 166 + ], + "score": 0.87, + "content": "f _ { i }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 293, + 154, + 506, + 168 + ], + "score": 1.0, + "content": ", applying Lemma 6 shows that a 2-IGN can approx-", + "type": "text", + "cross_page": true + } + ], + "index": 7, + "is_list_start_line": true + }, + { + "bbox": [ + 82, + 165, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 82, + 167, + 100, + 177 + ], + "score": 1.0, + "content": "1206", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 165, + 132, + 178 + ], + "score": 1.0, + "content": "imate", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 132, + 165, + 185, + 178 + ], + "score": 0.93, + "content": "h ( \\mu ) V V ^ { \\top } X", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 185, + 165, + 197, + 178 + ], + "score": 1.0, + "content": "to", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 198, + 166, + 214, + 178 + ], + "score": 0.91, + "content": "\\epsilon / n", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 214, + 165, + 506, + 178 + ], + "score": 1.0, + "content": "accuracy, so we are done. 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Also, since a 2-IGN can learn matrix vector", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 83, + 105, + 506, + 122 + ], + "spans": [ + { + "bbox": [ + 83, + 109, + 99, + 119 + ], + "score": 1.0, + "content": "1201", + "type": "text" + }, + { + "bbox": [ + 104, + 105, + 396, + 122 + ], + "score": 1.0, + "content": "multiplication (Cai and Wang [2022] Lemma 10), we can approximate", + "type": "text" + }, + { + "bbox": [ + 397, + 106, + 506, + 120 + ], + "score": 0.92, + "content": "f _ { 2 } ( h ( \\mu ) { \\bf 1 1 } ^ { \\top } , V V ^ { \\top } , X ) =", + "type": "inline_equation" + } + ], + "index": 3 + }, + { + "bbox": [ + 83, + 118, + 507, + 135 + ], + "spans": [ + { + "bbox": [ + 83, + 122, + 100, + 133 + ], + "score": 1.0, + "content": "1202", + "type": "text" + }, + { + "bbox": [ + 106, + 119, + 188, + 133 + ], + "score": 0.9, + "content": "( h ( \\mu ) \\mathbf { 1 1 } ^ { \\top } , V V ^ { \\top } X )", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 118, + 203, + 135 + ], + "score": 1.0, + "content": ", as", + "type": "text" + }, + { + "bbox": [ + 203, + 119, + 255, + 133 + ], + "score": 0.93, + "content": "V _ { i } V _ { i } ^ { \\top } \\in \\mathbb { R } ^ { n ^ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 118, + 318, + 135 + ], + "score": 1.0, + "content": "is a matrix and", + "type": "text" + }, + { + "bbox": [ + 319, + 120, + 364, + 131 + ], + "score": 0.92, + "content": "\\ b X \\in \\mathbb { R } ^ { n \\times q }", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 118, + 507, + 135 + ], + "score": 1.0, + "content": "is a vector with respect to permuta-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 82, + 132, + 506, + 144 + ], + "spans": [ + { + "bbox": [ + 82, + 134, + 100, + 144 + ], + "score": 1.0, + "content": "1203", + "type": "text" + }, + { + "bbox": [ + 106, + 132, + 506, + 144 + ], + "score": 1.0, + "content": "tion symmetries. Finally, we use an elementwise MLP to approximate the scalar-vector multiplication", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 83, + 142, + 506, + 157 + ], + "spans": [ + { + "bbox": [ + 83, + 145, + 100, + 155 + ], + "score": 1.0, + "content": "1204", + "type": "text" + }, + { + "bbox": [ + 107, + 143, + 268, + 156 + ], + "score": 0.93, + "content": "f _ { 3 } ( h ( \\mu ) { \\bf 1 1 } ^ { \\top } , V V ^ { \\top } , X ) = h ( \\mu ) V V ^ { \\top } X", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 142, + 298, + 157 + ], + "score": 1.0, + "content": ". Since", + "type": "text" + }, + { + "bbox": [ + 298, + 143, + 484, + 156 + ], + "score": 0.92, + "content": "f _ { 3 } \\circ f _ { 2 } \\circ \\bar { f } _ { 1 } ( \\mu { \\bf 1 1 } ^ { \\top } , V V ^ { \\top } , X ) = h ( \\mu ) V V ^ { \\top } X", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 142, + 506, + 157 + ], + "score": 1.0, + "content": ", and", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 82, + 154, + 506, + 168 + ], + "spans": [ + { + "bbox": [ + 82, + 156, + 100, + 167 + ], + "score": 1.0, + "content": "1205", + "type": "text" + }, + { + "bbox": [ + 105, + 154, + 283, + 168 + ], + "score": 1.0, + "content": "since 2-IGNs universally approximate each", + "type": "text" + }, + { + "bbox": [ + 284, + 155, + 293, + 166 + ], + "score": 0.87, + "content": "f _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 154, + 506, + 168 + ], + "score": 1.0, + "content": ", applying Lemma 6 shows that a 2-IGN can approx-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 82, + 165, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 82, + 167, + 100, + 177 + ], + "score": 1.0, + "content": "1206", + "type": "text" + }, + { + "bbox": [ + 105, + 165, + 132, + 178 + ], + "score": 1.0, + "content": "imate", + "type": "text" + }, + { + "bbox": [ + 132, + 165, + 185, + 178 + ], + "score": 0.93, + "content": "h ( \\mu ) V V ^ { \\top } X", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 165, + 197, + 178 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 198, + 166, + 214, + 178 + ], + "score": 0.91, + "content": "\\epsilon / n", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 165, + 506, + 178 + ], + "score": 1.0, + "content": "accuracy, so we are done. Since Expressive-BasisNet is stronger than", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 83, + 176, + 506, + 190 + ], + "spans": [ + { + "bbox": [ + 83, + 178, + 100, + 189 + ], + "score": 1.0, + "content": "1207", + "type": "text" + }, + { + "bbox": [ + 105, + 176, + 354, + 190 + ], + "score": 1.0, + "content": "BasisNet, it can also universally approximate these functions.", + "type": "text" + }, + { + "bbox": [ + 494, + 177, + 506, + 189 + ], + "score": 0.998, + "content": "□", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 4.5 + }, + { + "type": "text", + "bbox": [ + 83, + 199, + 505, + 300 + ], + "lines": [ + { + "bbox": [ + 83, + 200, + 505, + 212 + ], + "spans": [ + { + "bbox": [ + 83, + 201, + 100, + 211 + ], + "score": 1.0, + "content": "1208", + "type": "text" + }, + { + "bbox": [ + 105, + 200, + 480, + 212 + ], + "score": 1.0, + "content": "From the proof, we can see that SignNet and BasisNet need only learn simple functions for the", + "type": "text" + }, + { + "bbox": [ + 480, + 201, + 487, + 211 + ], + "score": 0.8, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 200, + 505, + 212 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 83, + 210, + 505, + 223 + ], + "spans": [ + { + "bbox": [ + 83, + 213, + 100, + 222 + ], + "score": 1.0, + "content": "1209", + "type": "text" + }, + { + "bbox": [ + 106, + 211, + 114, + 222 + ], + "score": 0.83, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 210, + 138, + 223 + ], + "score": 1.0, + "content": "when", + "type": "text" + }, + { + "bbox": [ + 139, + 211, + 146, + 221 + ], + "score": 0.8, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 210, + 425, + 223 + ], + "score": 1.0, + "content": "is simple, or when the filter is non-parametric and we need only learn", + "type": "text" + }, + { + "bbox": [ + 426, + 211, + 435, + 222 + ], + "score": 0.87, + "content": "\\theta _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 210, + 505, + 223 + ], + "score": 1.0, + "content": ". Xu et al. [2020]", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 83, + 222, + 506, + 234 + ], + "spans": [ + { + "bbox": [ + 83, + 223, + 100, + 234 + ], + "score": 1.0, + "content": "1210", + "type": "text" + }, + { + "bbox": [ + 105, + 222, + 506, + 234 + ], + "score": 1.0, + "content": "propose the principle of algorithmic alignment, and show that if separate modules of a neural network", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 83, + 233, + 505, + 245 + ], + "spans": [ + { + "bbox": [ + 83, + 234, + 99, + 244 + ], + "score": 1.0, + "content": "1211", + "type": "text" + }, + { + "bbox": [ + 106, + 233, + 505, + 245 + ], + "score": 1.0, + "content": "each need only learn simple functions (that is, functions that are well-approximated by low-order", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 82, + 243, + 506, + 256 + ], + "spans": [ + { + "bbox": [ + 82, + 245, + 100, + 255 + ], + "score": 1.0, + "content": "1212", + "type": "text" + }, + { + "bbox": [ + 105, + 243, + 506, + 256 + ], + "score": 1.0, + "content": "polynomials with small coefficients), then the network may be more sample efficient. If we do not", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 82, + 254, + 506, + 267 + ], + "spans": [ + { + "bbox": [ + 82, + 257, + 100, + 267 + ], + "score": 1.0, + "content": "1213", + "type": "text" + }, + { + "bbox": [ + 104, + 254, + 506, + 267 + ], + "score": 1.0, + "content": "require permutation equivariance, and parameterize SignNet and BasisNet with simple MLPs, then", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 83, + 264, + 506, + 279 + ], + "spans": [ + { + "bbox": [ + 83, + 268, + 99, + 277 + ], + "score": 1.0, + "content": "1214", + "type": "text" + }, + { + "bbox": [ + 105, + 264, + 428, + 279 + ], + "score": 1.0, + "content": "algorithmic alignment may suggest that our models are sample efficient. Indeed,", + "type": "text" + }, + { + "bbox": [ + 429, + 265, + 459, + 277 + ], + "score": 0.92, + "content": "\\rho \\overset { \\cdot } { = } \\sum", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 264, + 506, + 279 + ], + "score": 1.0, + "content": "is a simple", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 83, + 276, + 502, + 290 + ], + "spans": [ + { + "bbox": [ + 83, + 279, + 100, + 289 + ], + "score": 1.0, + "content": "1215", + "type": "text" + }, + { + "bbox": [ + 104, + 276, + 261, + 290 + ], + "score": 1.0, + "content": "linear function with coefficients 1, and", + "type": "text" + }, + { + "bbox": [ + 261, + 277, + 370, + 289 + ], + "score": 0.92, + "content": "\\phi ( V , \\lambda , X ) = h ( \\lambda ) V V ^ { \\top } X", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 276, + 430, + 290 + ], + "score": 1.0, + "content": "is quadratic in", + "type": "text" + }, + { + "bbox": [ + 430, + 278, + 439, + 287 + ], + "score": 0.76, + "content": "V", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 276, + 492, + 290 + ], + "score": 1.0, + "content": "and linear in", + "type": "text" + }, + { + "bbox": [ + 492, + 278, + 502, + 287 + ], + "score": 0.79, + "content": "X", + "type": "inline_equation" + } + ], + "index": 17 + }, + { + "bbox": [ + 83, + 288, + 221, + 300 + ], + "spans": [ + { + "bbox": [ + 83, + 290, + 100, + 300 + ], + "score": 1.0, + "content": "1216", + "type": "text" + }, + { + "bbox": [ + 105, + 288, + 173, + 300 + ], + "score": 1.0, + "content": "so it is simple if", + "type": "text" + }, + { + "bbox": [ + 173, + 289, + 180, + 298 + ], + "score": 0.82, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 288, + 221, + 300 + ], + "score": 1.0, + "content": "is simple.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 97, + 302, + 504, + 325 + ], + "lines": [ + { + "bbox": [ + 103, + 301, + 506, + 316 + ], + "spans": [ + { + "bbox": [ + 103, + 301, + 506, + 316 + ], + "score": 1.0, + "content": "Proposition 3. There exist infinitely many pairs of non-isomorphic graphs that SignNet and BasisNet", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 312, + 450, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 450, + 325 + ], + "score": 1.0, + "content": "can distinguish, but spectral graph convolutions or spectral GNNs cannot distinguish.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19.5 + }, + { + "type": "text", + "bbox": [ + 82, + 336, + 505, + 438 + ], + "lines": [ + { + "bbox": [ + 83, + 336, + 506, + 348 + ], + "spans": [ + { + "bbox": [ + 83, + 338, + 100, + 348 + ], + "score": 1.0, + "content": "1219", + "type": "text" + }, + { + "bbox": [ + 105, + 336, + 311, + 348 + ], + "score": 1.0, + "content": "Proof. The idea is as follows: we will take graphs", + "type": "text" + }, + { + "bbox": [ + 312, + 339, + 320, + 347 + ], + "score": 0.88, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 336, + 477, + 348 + ], + "score": 1.0, + "content": "and give them the node feature matrix", + "type": "text" + }, + { + "bbox": [ + 478, + 339, + 506, + 348 + ], + "score": 0.86, + "content": "X _ { G } =", + "type": "inline_equation" + } + ], + "index": 21 + }, + { + "bbox": [ + 82, + 345, + 507, + 363 + ], + "spans": [ + { + "bbox": [ + 82, + 350, + 100, + 360 + ], + "score": 1.0, + "content": "1220", + "type": "text" + }, + { + "bbox": [ + 106, + 348, + 134, + 359 + ], + "score": 0.87, + "content": "D ^ { 1 / 2 } \\mathbf { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 345, + 507, + 363 + ], + "score": 1.0, + "content": ", i.e. each node has as feature the square root of its degree. Then any spectral graph convolution", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 82, + 358, + 506, + 372 + ], + "spans": [ + { + "bbox": [ + 82, + 361, + 99, + 371 + ], + "score": 1.0, + "content": "1221", + "type": "text" + }, + { + "bbox": [ + 105, + 358, + 300, + 372 + ], + "score": 1.0, + "content": "(or, the first layer of any spectral GNN) will map", + "type": "text" + }, + { + "bbox": [ + 300, + 361, + 366, + 372 + ], + "score": 0.92, + "content": "V \\mathrm { D i a g } ( \\theta ) V ^ { \\mathrm { ~ l ~ } } X", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 358, + 506, + 372 + ], + "score": 1.0, + "content": "to something that only depends on", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 82, + 369, + 506, + 383 + ], + "spans": [ + { + "bbox": [ + 82, + 372, + 100, + 382 + ], + "score": 1.0, + "content": "1222", + "type": "text" + }, + { + "bbox": [ + 105, + 369, + 506, + 383 + ], + "score": 1.0, + "content": "the degree sequence and number of nodes. Thus, any spectral graph convolution or spectral GNN", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 82, + 380, + 506, + 394 + ], + "spans": [ + { + "bbox": [ + 82, + 383, + 100, + 393 + ], + "score": 1.0, + "content": "1223", + "type": "text" + }, + { + "bbox": [ + 105, + 380, + 370, + 394 + ], + "score": 1.0, + "content": "will have the same output (up to permutation) for any such graphs", + "type": "text" + }, + { + "bbox": [ + 371, + 384, + 379, + 391 + ], + "score": 0.9, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 380, + 456, + 394 + ], + "score": 1.0, + "content": "with node features", + "type": "text" + }, + { + "bbox": [ + 457, + 384, + 472, + 393 + ], + "score": 0.91, + "content": "X _ { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 472, + 380, + 506, + 394 + ], + "score": 1.0, + "content": "and the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 82, + 392, + 505, + 404 + ], + "spans": [ + { + "bbox": [ + 82, + 394, + 100, + 404 + ], + "score": 1.0, + "content": "1224", + "type": "text" + }, + { + "bbox": [ + 105, + 392, + 505, + 404 + ], + "score": 1.0, + "content": "same number of nodes and same degree sequence. On the other hand, SignNet and BasisNet can", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 82, + 401, + 504, + 418 + ], + "spans": [ + { + "bbox": [ + 82, + 406, + 100, + 416 + ], + "score": 1.0, + "content": "1225", + "type": "text" + }, + { + "bbox": [ + 104, + 401, + 315, + 418 + ], + "score": 1.0, + "content": "distinguish between infinitely many pairs of graphs", + "type": "text" + }, + { + "bbox": [ + 316, + 405, + 365, + 417 + ], + "score": 0.92, + "content": "\\left( G ^ { ( 1 ) } , G ^ { ( 2 ) } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 401, + 443, + 418 + ], + "score": 1.0, + "content": "with node features", + "type": "text" + }, + { + "bbox": [ + 444, + 406, + 504, + 416 + ], + "score": 0.88, + "content": "( X _ { G ^ { ( 1 ) } } , X _ { G ^ { ( 2 ) } } )", + "type": "inline_equation" + } + ], + "index": 27 + }, + { + "bbox": [ + 82, + 414, + 506, + 428 + ], + "spans": [ + { + "bbox": [ + 82, + 417, + 100, + 427 + ], + "score": 1.0, + "content": "1226", + "type": "text" + }, + { + "bbox": [ + 105, + 414, + 506, + 428 + ], + "score": 1.0, + "content": "and the same number of nodes and degree sequence; this is because SignNet and BasisNet can tell", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 82, + 425, + 209, + 439 + ], + "spans": [ + { + "bbox": [ + 82, + 427, + 100, + 438 + ], + "score": 1.0, + "content": "1227", + "type": "text" + }, + { + "bbox": [ + 105, + 425, + 209, + 439 + ], + "score": 1.0, + "content": "when a graph is bipartite.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 82, + 442, + 505, + 528 + ], + "lines": [ + { + "bbox": [ + 83, + 441, + 507, + 457 + ], + "spans": [ + { + "bbox": [ + 83, + 445, + 100, + 455 + ], + "score": 1.0, + "content": "1228", + "type": "text" + }, + { + "bbox": [ + 104, + 441, + 145, + 457 + ], + "score": 1.0, + "content": "For each", + "type": "text" + }, + { + "bbox": [ + 145, + 444, + 172, + 455 + ], + "score": 0.86, + "content": "n \\geq 5", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 441, + 239, + 457 + ], + "score": 1.0, + "content": ", we will define", + "type": "text" + }, + { + "bbox": [ + 239, + 444, + 258, + 453 + ], + "score": 0.91, + "content": "G ^ { ( 1 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 441, + 279, + 457 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 279, + 444, + 297, + 453 + ], + "score": 0.92, + "content": "G ^ { ( 2 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 441, + 407, + 457 + ], + "score": 1.0, + "content": "as connected graphs with", + "type": "text" + }, + { + "bbox": [ + 407, + 448, + 414, + 453 + ], + "score": 0.87, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 441, + 507, + 457 + ], + "score": 1.0, + "content": "nodes, with the same", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 83, + 455, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 83, + 461, + 100, + 471 + ], + "score": 1.0, + "content": "1229", + "type": "text" + }, + { + "bbox": [ + 102, + 457, + 240, + 474 + ], + "score": 1.0, + "content": "degree sequence. Also, we define", + "type": "text" + }, + { + "bbox": [ + 240, + 461, + 258, + 470 + ], + "score": 0.92, + "content": "G ^ { ( 1 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 457, + 346, + 474 + ], + "score": 1.0, + "content": "to have node features", + "type": "text" + }, + { + "bbox": [ + 346, + 456, + 405, + 475 + ], + "score": 0.95, + "content": "X _ { i } ^ { ( 1 ) } = \\sqrt { d _ { i } ^ { ( 1 ) } }", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 455, + 435, + 474 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 436, + 459, + 452, + 473 + ], + "score": 0.95, + "content": "d _ { i } ^ { ( 1 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 455, + 505, + 474 + ], + "score": 1.0, + "content": "is the degree", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 82, + 473, + 507, + 494 + ], + "spans": [ + { + "bbox": [ + 82, + 481, + 100, + 491 + ], + "score": 1.0, + "content": "1230", + "type": "text" + }, + { + "bbox": [ + 104, + 473, + 140, + 494 + ], + "score": 1.0, + "content": "of node", + "type": "text" + }, + { + "bbox": [ + 141, + 482, + 145, + 489 + ], + "score": 0.82, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 473, + 157, + 494 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 158, + 480, + 176, + 489 + ], + "score": 0.9, + "content": "G ^ { ( 1 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 473, + 237, + 494 + ], + "score": 1.0, + "content": ", and similarly", + "type": "text" + }, + { + "bbox": [ + 238, + 480, + 257, + 489 + ], + "score": 0.92, + "content": "G ^ { ( 2 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 473, + 333, + 494 + ], + "score": 1.0, + "content": "has node features", + "type": "text" + }, + { + "bbox": [ + 333, + 475, + 393, + 494 + ], + "score": 0.93, + "content": "X _ { i } ^ { ( 2 ) } = \\sqrt { d _ { i } ^ { ( 2 ) } }", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 473, + 461, + 494 + ], + "score": 1.0, + "content": ". Now, note that", + "type": "text" + }, + { + "bbox": [ + 462, + 480, + 481, + 489 + ], + "score": 0.94, + "content": "X ^ { ( 1 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 473, + 507, + 494 + ], + "score": 1.0, + "content": "is an", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 82, + 491, + 506, + 507 + ], + "spans": [ + { + "bbox": [ + 82, + 495, + 99, + 506 + ], + "score": 1.0, + "content": "1231", + "type": "text" + }, + { + "bbox": [ + 104, + 491, + 277, + 507 + ], + "score": 1.0, + "content": "eigenvector of the normalized Laplacian of", + "type": "text" + }, + { + "bbox": [ + 277, + 495, + 295, + 504 + ], + "score": 0.91, + "content": "G ^ { ( 1 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 491, + 384, + 507 + ], + "score": 1.0, + "content": ", and it has eigenvalue", + "type": "text" + }, + { + "bbox": [ + 384, + 497, + 389, + 504 + ], + "score": 0.39, + "content": "0", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 491, + 506, + 507 + ], + "score": 1.0, + "content": ". As we take the eigenvectors", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 82, + 503, + 506, + 518 + ], + "spans": [ + { + "bbox": [ + 82, + 507, + 100, + 517 + ], + "score": 1.0, + "content": "1232", + "type": "text" + }, + { + "bbox": [ + 105, + 503, + 506, + 518 + ], + "score": 1.0, + "content": "to be orthonormal (since the normalized Laplacian is symmetric), for any spectral graph convolution", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 82, + 516, + 159, + 528 + ], + "spans": [ + { + "bbox": [ + 82, + 518, + 100, + 528 + ], + "score": 1.0, + "content": "1233", + "type": "text" + }, + { + "bbox": [ + 105, + 516, + 159, + 528 + ], + "score": 1.0, + "content": "we have that", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 32.5 + }, + { + "type": "interline_equation", + "bbox": [ + 126, + 529, + 466, + 563 + ], + "lines": [ + { + "bbox": [ + 126, + 529, + 466, + 563 + ], + "spans": [ + { + "bbox": [ + 126, + 529, + 466, + 563 + ], + "score": 0.93, + "content": "\\sum _ { i = 1 } ^ { n } \\theta _ { i } v _ { i } v _ { i } ^ { \\top } X ^ { ( 1 ) } = \\theta _ { 1 } v _ { 1 } v _ { 1 } ^ { \\top } X ^ { ( 1 ) } = \\theta _ { 1 } D _ { 1 } ^ { 1 / 2 } \\mathbf { 1 } ( D _ { 1 } ^ { 1 / 2 } \\mathbf { 1 } ) ^ { \\top } D _ { 1 } ^ { 1 / 2 } \\mathbf { 1 } = \\theta _ { 1 } \\sum _ { j = 1 } ^ { n } ( d _ { j } ^ { ( 1 ) } ) D _ { 1 } ^ { 1 / 2 } \\mathbf { 1 } .", + "type": "interline_equation", + "image_path": "73b0411817ae20399bac710f759fa286521b3064c1b5c321715b8814e56837f8.jpg" + } + ] + } + ], + "index": 37, + "virtual_lines": [ + { + "bbox": [ + 126, + 529, + 466, + 540.3333333333334 + ], + "spans": [], + "index": 36 + }, + { + "bbox": [ + 126, + 540.3333333333334, + 466, + 551.6666666666667 + ], + "spans": [], + "index": 37 + }, + { + "bbox": [ + 126, + 551.6666666666667, + 466, + 563.0000000000001 + ], + "spans": [], + "index": 38 + } + ] + }, + { + "type": "text", + "bbox": [ + 82, + 567, + 505, + 631 + ], + "lines": [ + { + "bbox": [ + 83, + 565, + 507, + 582 + ], + "spans": [ + { + "bbox": [ + 83, + 570, + 100, + 579 + ], + "score": 1.0, + "content": "1234", + "type": "text" + }, + { + "bbox": [ + 104, + 565, + 136, + 582 + ], + "score": 1.0, + "content": "Where", + "type": "text" + }, + { + "bbox": [ + 136, + 570, + 149, + 579 + ], + "score": 0.89, + "content": "D _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 565, + 281, + 582 + ], + "score": 1.0, + "content": "is the diagonal degree matrix of", + "type": "text" + }, + { + "bbox": [ + 281, + 569, + 300, + 578 + ], + "score": 0.9, + "content": "G ^ { ( 1 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 565, + 507, + 582 + ], + "score": 1.0, + "content": ". Likewise, any spectral graph convolution outputs", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 83, + 576, + 507, + 598 + ], + "spans": [ + { + "bbox": [ + 83, + 583, + 100, + 594 + ], + "score": 1.0, + "content": "1235", + "type": "text" + }, + { + "bbox": [ + 107, + 580, + 183, + 596 + ], + "score": 0.93, + "content": "\\theta _ { 1 } \\sum _ { j } ( d _ { j } ^ { ( 2 ) } ) D _ { 2 } ^ { 1 / 2 } { \\bf 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 576, + 200, + 598 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 200, + 582, + 219, + 591 + ], + "score": 0.92, + "content": "G ^ { ( 2 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 576, + 249, + 598 + ], + "score": 1.0, + "content": ". Since", + "type": "text" + }, + { + "bbox": [ + 249, + 584, + 263, + 593 + ], + "score": 0.92, + "content": "D _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 576, + 282, + 598 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 282, + 584, + 295, + 593 + ], + "score": 0.91, + "content": "D _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 576, + 507, + 598 + ], + "score": 1.0, + "content": "are the same up to a permutation, we have that any", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 82, + 593, + 506, + 609 + ], + "spans": [ + { + "bbox": [ + 82, + 597, + 100, + 608 + ], + "score": 1.0, + "content": "1236", + "type": "text" + }, + { + "bbox": [ + 104, + 593, + 312, + 609 + ], + "score": 1.0, + "content": "spectral graph convolution has the same output for", + "type": "text" + }, + { + "bbox": [ + 312, + 596, + 331, + 606 + ], + "score": 0.92, + "content": "G ^ { ( 1 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 593, + 350, + 609 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 351, + 596, + 369, + 606 + ], + "score": 0.91, + "content": "G ^ { ( 2 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 593, + 506, + 609 + ], + "score": 1.0, + "content": ", up to a permutation. In fact, this", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 82, + 604, + 506, + 620 + ], + "spans": [ + { + "bbox": [ + 82, + 608, + 100, + 618 + ], + "score": 1.0, + "content": "1237", + "type": "text" + }, + { + "bbox": [ + 105, + 604, + 506, + 620 + ], + "score": 1.0, + "content": "also holds for spectral GNNs, as the first layer will always have the same output (up to a permutation)", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 82, + 616, + 462, + 631 + ], + "spans": [ + { + "bbox": [ + 82, + 619, + 100, + 631 + ], + "score": 1.0, + "content": "1238", + "type": "text" + }, + { + "bbox": [ + 104, + 616, + 119, + 631 + ], + "score": 1.0, + "content": "on", + "type": "text" + }, + { + "bbox": [ + 120, + 619, + 138, + 628 + ], + "score": 0.92, + "content": "G ^ { ( 1 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 616, + 157, + 631 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 158, + 619, + 176, + 628 + ], + "score": 0.91, + "content": "G ^ { ( 2 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 616, + 462, + 631 + ], + "score": 1.0, + "content": ", so the latter layers will also have the same output up to a permutation.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 83, + 635, + 505, + 694 + ], + "lines": [ + { + "bbox": [ + 83, + 633, + 507, + 649 + ], + "spans": [ + { + "bbox": [ + 83, + 638, + 100, + 647 + ], + "score": 1.0, + "content": "1239", + "type": "text" + }, + { + "bbox": [ + 104, + 633, + 215, + 649 + ], + "score": 1.0, + "content": "Now, we concretely define", + "type": "text" + }, + { + "bbox": [ + 216, + 637, + 234, + 646 + ], + "score": 0.92, + "content": "G ^ { ( 1 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 633, + 253, + 649 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 254, + 637, + 272, + 646 + ], + "score": 0.91, + "content": "G ^ { ( 2 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 633, + 477, + 649 + ], + "score": 1.0, + "content": ". This is illustrated in Figure 11 and Figure 12. For", + "type": "text" + }, + { + "bbox": [ + 478, + 639, + 502, + 646 + ], + "score": 0.89, + "content": "n = 5", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 633, + 507, + 649 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 83, + 645, + 507, + 662 + ], + "spans": [ + { + "bbox": [ + 83, + 649, + 100, + 659 + ], + "score": 1.0, + "content": "1240", + "type": "text" + }, + { + "bbox": [ + 104, + 645, + 119, + 662 + ], + "score": 1.0, + "content": "let", + "type": "text" + }, + { + "bbox": [ + 120, + 649, + 138, + 658 + ], + "score": 0.9, + "content": "G ^ { ( 1 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 645, + 259, + 662 + ], + "score": 1.0, + "content": "contain a triangle with nodes", + "type": "text" + }, + { + "bbox": [ + 259, + 653, + 303, + 660 + ], + "score": 0.89, + "content": "w _ { 1 } , w _ { 2 } , w _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 645, + 507, + 662 + ], + "score": 1.0, + "content": ", and have a path of length 2 coming out of one of", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 83, + 659, + 506, + 671 + ], + "spans": [ + { + "bbox": [ + 83, + 661, + 99, + 670 + ], + "score": 1.0, + "content": "1241", + "type": "text" + }, + { + "bbox": [ + 106, + 659, + 221, + 671 + ], + "score": 1.0, + "content": "the nodes in the triangle, say", + "type": "text" + }, + { + "bbox": [ + 221, + 664, + 233, + 670 + ], + "score": 0.89, + "content": "w _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 659, + 281, + 671 + ], + "score": 1.0, + "content": "connects to", + "type": "text" + }, + { + "bbox": [ + 281, + 664, + 293, + 670 + ], + "score": 0.88, + "content": "w _ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 659, + 314, + 671 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 315, + 664, + 326, + 671 + ], + "score": 0.89, + "content": "w _ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 659, + 375, + 671 + ], + "score": 1.0, + "content": "connects to", + "type": "text" + }, + { + "bbox": [ + 375, + 664, + 387, + 671 + ], + "score": 0.87, + "content": "w _ { 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 659, + 506, + 671 + ], + "score": 1.0, + "content": ". This is not bipartite, as there", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 83, + 668, + 507, + 684 + ], + "spans": [ + { + "bbox": [ + 83, + 672, + 100, + 683 + ], + "score": 1.0, + "content": "1242", + "type": "text" + }, + { + "bbox": [ + 104, + 668, + 177, + 684 + ], + "score": 1.0, + "content": "is a triangle. Let", + "type": "text" + }, + { + "bbox": [ + 177, + 671, + 196, + 681 + ], + "score": 0.91, + "content": "G ^ { ( 2 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 668, + 393, + 684 + ], + "score": 1.0, + "content": "be a bipartite graph that has 2 nodes on the left", + "type": "text" + }, + { + "bbox": [ + 393, + 673, + 424, + 684 + ], + "score": 0.93, + "content": "( v _ { 1 } , v _ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 668, + 507, + 684 + ], + "score": 1.0, + "content": "and 3 nodes on the", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 83, + 680, + 461, + 696 + ], + "spans": [ + { + "bbox": [ + 83, + 684, + 100, + 693 + ], + "score": 1.0, + "content": "1243", + "type": "text" + }, + { + "bbox": [ + 104, + 680, + 128, + 696 + ], + "score": 1.0, + "content": "right", + "type": "text" + }, + { + "bbox": [ + 128, + 684, + 173, + 694 + ], + "score": 0.89, + "content": "( v _ { 3 } , v _ { 4 } , v _ { 5 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 680, + 214, + 696 + ], + "score": 1.0, + "content": ". Connect", + "type": "text" + }, + { + "bbox": [ + 214, + 687, + 223, + 693 + ], + "score": 0.88, + "content": "v _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 224, + 680, + 385, + 696 + ], + "score": 1.0, + "content": "with all nodes on the right, and connect", + "type": "text" + }, + { + "bbox": [ + 386, + 687, + 395, + 693 + ], + "score": 0.88, + "content": "v _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 680, + 417, + 696 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 417, + 687, + 427, + 693 + ], + "score": 0.88, + "content": "v _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 680, + 446, + 696 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 446, + 687, + 456, + 693 + ], + "score": 0.88, + "content": "\\boldsymbol { v } _ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 680, + 461, + 696 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 46 + }, + { + "type": "text", + "bbox": [ + 86, + 699, + 505, + 723 + ], + "lines": [ + { + "bbox": [ + 83, + 697, + 507, + 713 + ], + "spans": [ + { + "bbox": [ + 83, + 702, + 100, + 711 + ], + "score": 1.0, + "content": "1244", + "type": "text" + }, + { + "bbox": [ + 104, + 697, + 172, + 713 + ], + "score": 1.0, + "content": "Note that both", + "type": "text" + }, + { + "bbox": [ + 172, + 700, + 191, + 710 + ], + "score": 0.91, + "content": "G ^ { ( 1 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 697, + 213, + 713 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 213, + 700, + 232, + 710 + ], + "score": 0.92, + "content": "G ^ { ( 2 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 697, + 507, + 713 + ], + "score": 1.0, + "content": "have the same number of nodes and the same degree sequence", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 83, + 710, + 507, + 723 + ], + "spans": [ + { + "bbox": [ + 83, + 712, + 100, + 722 + ], + "score": 1.0, + "content": "1245", + "type": "text" + }, + { + "bbox": [ + 107, + 712, + 160, + 723 + ], + "score": 0.88, + "content": "\\{ 3 , 2 , 2 , 2 , 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 710, + 507, + 723 + ], + "score": 1.0, + "content": ". Thus, spectral graph convolutions or spectral GNNs cannot distinguish them. How-", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 49.5 + } + ], + "page_idx": 33, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 300, + 741, + 311, + 750 + ], + "lines": [ + { + "bbox": [ + 298, + 740, + 313, + 754 + ], + "spans": [ + { + "bbox": [ + 298, + 740, + 313, + 754 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 14, + "width": 15 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "index", + "bbox": [ + 83, + 72, + 506, + 189 + ], + "lines": [], + "index": 4.5, + "bbox_fs": [ + 82, + 72, + 507, + 190 + ], + "lines_deleted": true + }, + { + "type": "index", + "bbox": [ + 83, + 199, + 505, + 300 + ], + "lines": [], + "index": 14, + "bbox_fs": [ + 82, + 200, + 506, + 300 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 97, + 302, + 504, + 325 + ], + "lines": [ + { + "bbox": [ + 103, + 301, + 506, + 316 + ], + "spans": [ + { + "bbox": [ + 103, + 301, + 506, + 316 + ], + "score": 1.0, + "content": "Proposition 3. There exist infinitely many pairs of non-isomorphic graphs that SignNet and BasisNet", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 312, + 450, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 450, + 325 + ], + "score": 1.0, + "content": "can distinguish, but spectral graph convolutions or spectral GNNs cannot distinguish.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19.5, + "bbox_fs": [ + 103, + 301, + 506, + 325 + ] + }, + { + "type": "index", + "bbox": [ + 82, + 336, + 505, + 438 + ], + "lines": [ + { + "bbox": [ + 83, + 336, + 506, + 348 + ], + "spans": [ + { + "bbox": [ + 83, + 338, + 100, + 348 + ], + "score": 1.0, + "content": "1219", + "type": "text" + }, + { + "bbox": [ + 105, + 336, + 311, + 348 + ], + "score": 1.0, + "content": "Proof. The idea is as follows: we will take graphs", + "type": "text" + }, + { + "bbox": [ + 312, + 339, + 320, + 347 + ], + "score": 0.88, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 336, + 477, + 348 + ], + "score": 1.0, + "content": "and give them the node feature matrix", + "type": "text" + }, + { + "bbox": [ + 478, + 339, + 506, + 348 + ], + "score": 0.86, + "content": "X _ { G } =", + "type": "inline_equation" + } + ], + "index": 21, + "is_list_start_line": true + }, + { + "bbox": [ + 82, + 345, + 507, + 363 + ], + "spans": [ + { + "bbox": [ + 82, + 350, + 100, + 360 + ], + "score": 1.0, + "content": "1220", + "type": "text" + }, + { + "bbox": [ + 106, + 348, + 134, + 359 + ], + "score": 0.87, + "content": "D ^ { 1 / 2 } \\mathbf { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 345, + 507, + 363 + ], + "score": 1.0, + "content": ", i.e. each node has as feature the square root of its degree. Then any spectral graph convolution", + "type": "text" + } + ], + "index": 22, + "is_list_start_line": true + }, + { + "bbox": [ + 82, + 358, + 506, + 372 + ], + "spans": [ + { + "bbox": [ + 82, + 361, + 99, + 371 + ], + "score": 1.0, + "content": "1221", + "type": "text" + }, + { + "bbox": [ + 105, + 358, + 300, + 372 + ], + "score": 1.0, + "content": "(or, the first layer of any spectral GNN) will map", + "type": "text" + }, + { + "bbox": [ + 300, + 361, + 366, + 372 + ], + "score": 0.92, + "content": "V \\mathrm { D i a g } ( \\theta ) V ^ { \\mathrm { ~ l ~ } } X", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 358, + 506, + 372 + ], + "score": 1.0, + "content": "to something that only depends on", + "type": "text" + } + ], + "index": 23, + "is_list_start_line": true + }, + { + "bbox": [ + 82, + 369, + 506, + 383 + ], + "spans": [ + { + "bbox": [ + 82, + 372, + 100, + 382 + ], + "score": 1.0, + "content": "1222", + "type": "text" + }, + { + "bbox": [ + 105, + 369, + 506, + 383 + ], + "score": 1.0, + "content": "the degree sequence and number of nodes. 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Also, we define", + "type": "text" + }, + { + "bbox": [ + 240, + 461, + 258, + 470 + ], + "score": 0.92, + "content": "G ^ { ( 1 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 457, + 346, + 474 + ], + "score": 1.0, + "content": "to have node features", + "type": "text" + }, + { + "bbox": [ + 346, + 456, + 405, + 475 + ], + "score": 0.95, + "content": "X _ { i } ^ { ( 1 ) } = \\sqrt { d _ { i } ^ { ( 1 ) } }", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 455, + 435, + 474 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 436, + 459, + 452, + 473 + ], + "score": 0.95, + "content": "d _ { i } ^ { ( 1 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 455, + 505, + 474 + ], + "score": 1.0, + "content": "is the degree", + "type": "text" + } + ], + "index": 31, + "is_list_start_line": true + }, + { + "bbox": [ + 82, + 473, + 507, + 494 + ], + "spans": [ + { + "bbox": [ + 82, + 481, + 100, + 491 + ], + "score": 1.0, + "content": "1230", + "type": "text" + }, + { + "bbox": [ + 104, + 473, + 140, + 494 + ], + "score": 1.0, + "content": "of node", + "type": "text" + }, + { + "bbox": [ + 141, + 482, + 145, + 489 + ], + "score": 0.82, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 473, + 157, + 494 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 158, + 480, + 176, + 489 + ], + "score": 0.9, + "content": "G ^ { ( 1 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 473, + 237, + 494 + ], + "score": 1.0, + "content": ", and similarly", + "type": "text" + }, + { + "bbox": [ + 238, + 480, + 257, + 489 + ], + "score": 0.92, + "content": "G ^ { ( 2 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 473, + 333, + 494 + ], + "score": 1.0, + "content": "has node features", + "type": "text" + }, + { + "bbox": [ + 333, + 475, + 393, + 494 + ], + "score": 0.93, + "content": "X _ { i } ^ { ( 2 ) } = \\sqrt { d _ { i } ^ { ( 2 ) } }", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 473, + 461, + 494 + ], + "score": 1.0, + "content": ". Now, note that", + "type": "text" + }, + { + "bbox": [ + 462, + 480, + 481, + 489 + ], + "score": 0.94, + "content": "X ^ { ( 1 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 473, + 507, + 494 + ], + "score": 1.0, + "content": "is an", + "type": "text" + } + ], + "index": 32, + "is_list_start_line": true + }, + { + "bbox": [ + 82, + 491, + 506, + 507 + ], + "spans": [ + { + "bbox": [ + 82, + 495, + 99, + 506 + ], + "score": 1.0, + "content": "1231", + "type": "text" + }, + { + "bbox": [ + 104, + 491, + 277, + 507 + ], + "score": 1.0, + "content": "eigenvector of the normalized Laplacian of", + "type": "text" + }, + { + "bbox": [ + 277, + 495, + 295, + 504 + ], + "score": 0.91, + "content": "G ^ { ( 1 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 491, + 384, + 507 + ], + "score": 1.0, + "content": ", and it has eigenvalue", + "type": "text" + }, + { + "bbox": [ + 384, + 497, + 389, + 504 + ], + "score": 0.39, + "content": "0", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 491, + 506, + 507 + ], + "score": 1.0, + "content": ". As we take the eigenvectors", + "type": "text" + } + ], + "index": 33, + "is_list_start_line": true + }, + { + "bbox": [ + 82, + 503, + 506, + 518 + ], + "spans": [ + { + "bbox": [ + 82, + 507, + 100, + 517 + ], + "score": 1.0, + "content": "1232", + "type": "text" + }, + { + "bbox": [ + 105, + 503, + 506, + 518 + ], + "score": 1.0, + "content": "to be orthonormal (since the normalized Laplacian is symmetric), for any spectral graph convolution", + "type": "text" + } + ], + "index": 34, + "is_list_start_line": true + }, + { + "bbox": [ + 82, + 516, + 159, + 528 + ], + "spans": [ + { + "bbox": [ + 82, + 518, + 100, + 528 + ], + "score": 1.0, + "content": "1233", + "type": "text" + }, + { + "bbox": [ + 105, + 516, + 159, + 528 + ], + "score": 1.0, + "content": "we have that", + "type": "text" + } + ], + "index": 35, + "is_list_start_line": true + } + ], + "index": 25, + "bbox_fs": [ + 82, + 336, + 507, + 439 + ] + }, + { + "type": "index", + "bbox": [ + 82, + 442, + 505, + 528 + ], + "lines": [], + "index": 32.5, + "bbox_fs": [ + 82, + 441, + 507, + 528 + ], + "lines_deleted": true + }, + { + "type": "interline_equation", + "bbox": [ + 126, + 529, + 466, + 563 + ], + "lines": [ + { + "bbox": [ + 126, + 529, + 466, + 563 + ], + "spans": [ + { + "bbox": [ + 126, + 529, + 466, + 563 + ], + "score": 0.93, + "content": "\\sum _ { i = 1 } ^ { n } \\theta _ { i } v _ { i } v _ { i } ^ { \\top } X ^ { ( 1 ) } = \\theta _ { 1 } v _ { 1 } v _ { 1 } ^ { \\top } X ^ { ( 1 ) } = \\theta _ { 1 } D _ { 1 } ^ { 1 / 2 } \\mathbf { 1 } ( D _ { 1 } ^ { 1 / 2 } \\mathbf { 1 } ) ^ { \\top } D _ { 1 } ^ { 1 / 2 } \\mathbf { 1 } = \\theta _ { 1 } \\sum _ { j = 1 } ^ { n } ( d _ { j } ^ { ( 1 ) } ) D _ { 1 } ^ { 1 / 2 } \\mathbf { 1 } .", + "type": "interline_equation", + "image_path": "73b0411817ae20399bac710f759fa286521b3064c1b5c321715b8814e56837f8.jpg" + } + ] + } + ], + "index": 37, + "virtual_lines": [ + { + "bbox": [ + 126, + 529, + 466, + 540.3333333333334 + ], + "spans": [], + "index": 36 + }, + { + "bbox": [ + 126, + 540.3333333333334, + 466, + 551.6666666666667 + ], + "spans": [], + "index": 37 + }, + { + "bbox": [ + 126, + 551.6666666666667, + 466, + 563.0000000000001 + ], + "spans": [], + "index": 38 + } + ] + }, + { + "type": "index", + "bbox": [ + 82, + 567, + 505, + 631 + ], + "lines": [ + { + "bbox": [ + 83, + 565, + 507, + 582 + ], + "spans": [ + { + "bbox": [ + 83, + 570, + 100, + 579 + ], + "score": 1.0, + "content": "1234", + "type": "text" + }, + { + "bbox": [ + 104, + 565, + 136, + 582 + ], + "score": 1.0, + "content": "Where", + "type": "text" + }, + { + "bbox": [ + 136, + 570, + 149, + 579 + ], + "score": 0.89, + "content": "D _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 565, + 281, + 582 + ], + "score": 1.0, + "content": "is the diagonal degree matrix of", + "type": "text" + }, + { + "bbox": [ + 281, + 569, + 300, + 578 + ], + "score": 0.9, + "content": "G ^ { ( 1 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 565, + 507, + 582 + ], + "score": 1.0, + "content": ". Likewise, any spectral graph convolution outputs", + "type": "text" + } + ], + "index": 39, + "is_list_start_line": true + }, + { + "bbox": [ + 83, + 576, + 507, + 598 + ], + "spans": [ + { + "bbox": [ + 83, + 583, + 100, + 594 + ], + "score": 1.0, + "content": "1235", + "type": "text" + }, + { + "bbox": [ + 107, + 580, + 183, + 596 + ], + "score": 0.93, + "content": "\\theta _ { 1 } \\sum _ { j } ( d _ { j } ^ { ( 2 ) } ) D _ { 2 } ^ { 1 / 2 } { \\bf 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 576, + 200, + 598 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 200, + 582, + 219, + 591 + ], + "score": 0.92, + "content": "G ^ { ( 2 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 576, + 249, + 598 + ], + "score": 1.0, + "content": ". Since", + "type": "text" + }, + { + "bbox": [ + 249, + 584, + 263, + 593 + ], + "score": 0.92, + "content": "D _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 576, + 282, + 598 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 282, + 584, + 295, + 593 + ], + "score": 0.91, + "content": "D _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 576, + 507, + 598 + ], + "score": 1.0, + "content": "are the same up to a permutation, we have that any", + "type": "text" + } + ], + "index": 40, + "is_list_start_line": true + }, + { + "bbox": [ + 82, + 593, + 506, + 609 + ], + "spans": [ + { + "bbox": [ + 82, + 597, + 100, + 608 + ], + "score": 1.0, + "content": "1236", + "type": "text" + }, + { + "bbox": [ + 104, + 593, + 312, + 609 + ], + "score": 1.0, + "content": "spectral graph convolution has the same output for", + "type": "text" + }, + { + "bbox": [ + 312, + 596, + 331, + 606 + ], + "score": 0.92, + "content": "G ^ { ( 1 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 593, + 350, + 609 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 351, + 596, + 369, + 606 + ], + "score": 0.91, + "content": "G ^ { ( 2 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 593, + 506, + 609 + ], + "score": 1.0, + "content": ", up to a permutation. In fact, this", + "type": "text" + } + ], + "index": 41, + "is_list_start_line": true + }, + { + "bbox": [ + 82, + 604, + 506, + 620 + ], + "spans": [ + { + "bbox": [ + 82, + 608, + 100, + 618 + ], + "score": 1.0, + "content": "1237", + "type": "text" + }, + { + "bbox": [ + 105, + 604, + 506, + 620 + ], + "score": 1.0, + "content": "also holds for spectral GNNs, as the first layer will always have the same output (up to a permutation)", + "type": "text" + } + ], + "index": 42, + "is_list_start_line": true + }, + { + "bbox": [ + 82, + 616, + 462, + 631 + ], + "spans": [ + { + "bbox": [ + 82, + 619, + 100, + 631 + ], + "score": 1.0, + "content": "1238", + "type": "text" + }, + { + "bbox": [ + 104, + 616, + 119, + 631 + ], + "score": 1.0, + "content": "on", + "type": "text" + }, + { + "bbox": [ + 120, + 619, + 138, + 628 + ], + "score": 0.92, + "content": "G ^ { ( 1 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 616, + 157, + 631 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 158, + 619, + 176, + 628 + ], + "score": 0.91, + "content": "G ^ { ( 2 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 616, + 462, + 631 + ], + "score": 1.0, + "content": ", so the latter layers will also have the same output up to a permutation.", + "type": "text" + } + ], + "index": 43, + "is_list_start_line": true + }, + { + "bbox": [ + 83, + 633, + 507, + 649 + ], + "spans": [ + { + "bbox": [ + 83, + 638, + 100, + 647 + ], + "score": 1.0, + "content": "1239", + "type": "text" + }, + { + "bbox": [ + 104, + 633, + 215, + 649 + ], + "score": 1.0, + "content": "Now, we concretely define", + "type": "text" + }, + { + "bbox": [ + 216, + 637, + 234, + 646 + ], + "score": 0.92, + "content": "G ^ { ( 1 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 633, + 253, + 649 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 254, + 637, + 272, + 646 + ], + "score": 0.91, + "content": "G ^ { ( 2 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 633, + 477, + 649 + ], + "score": 1.0, + "content": ". This is illustrated in Figure 11 and Figure 12. For", + "type": "text" + }, + { + "bbox": [ + 478, + 639, + 502, + 646 + ], + "score": 0.89, + "content": "n = 5", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 633, + 507, + 649 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 44, + "is_list_start_line": true + }, + { + "bbox": [ + 83, + 645, + 507, + 662 + ], + "spans": [ + { + "bbox": [ + 83, + 649, + 100, + 659 + ], + "score": 1.0, + "content": "1240", + "type": "text" + }, + { + "bbox": [ + 104, + 645, + 119, + 662 + ], + "score": 1.0, + "content": "let", + "type": "text" + }, + { + "bbox": [ + 120, + 649, + 138, + 658 + ], + "score": 0.9, + "content": "G ^ { ( 1 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 645, + 259, + 662 + ], + "score": 1.0, + "content": "contain a triangle with nodes", + "type": "text" + }, + { + "bbox": [ + 259, + 653, + 303, + 660 + ], + "score": 0.89, + "content": "w _ { 1 } , w _ { 2 } , w _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 645, + 507, + 662 + ], + "score": 1.0, + "content": ", and have a path of length 2 coming out of one of", + "type": "text" + } + ], + "index": 45, + "is_list_start_line": true + }, + { + "bbox": [ + 83, + 659, + 506, + 671 + ], + "spans": [ + { + "bbox": [ + 83, + 661, + 99, + 670 + ], + "score": 1.0, + "content": "1241", + "type": "text" + }, + { + "bbox": [ + 106, + 659, + 221, + 671 + ], + "score": 1.0, + "content": "the nodes in the triangle, say", + "type": "text" + }, + { + "bbox": [ + 221, + 664, + 233, + 670 + ], + "score": 0.89, + "content": "w _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 659, + 281, + 671 + ], + "score": 1.0, + "content": "connects to", + "type": "text" + }, + { + "bbox": [ + 281, + 664, + 293, + 670 + ], + "score": 0.88, + "content": "w _ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 659, + 314, + 671 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 315, + 664, + 326, + 671 + ], + "score": 0.89, + "content": "w _ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 659, + 375, + 671 + ], + "score": 1.0, + "content": "connects to", + "type": "text" + }, + { + "bbox": [ + 375, + 664, + 387, + 671 + ], + "score": 0.87, + "content": "w _ { 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 659, + 506, + 671 + ], + "score": 1.0, + "content": ". This is not bipartite, as there", + "type": "text" + } + ], + "index": 46, + "is_list_start_line": true + }, + { + "bbox": [ + 83, + 668, + 507, + 684 + ], + "spans": [ + { + "bbox": [ + 83, + 672, + 100, + 683 + ], + "score": 1.0, + "content": "1242", + "type": "text" + }, + { + "bbox": [ + 104, + 668, + 177, + 684 + ], + "score": 1.0, + "content": "is a triangle. 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Connect", + "type": "text" + }, + { + "bbox": [ + 214, + 687, + 223, + 693 + ], + "score": 0.88, + "content": "v _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 224, + 680, + 385, + 696 + ], + "score": 1.0, + "content": "with all nodes on the right, and connect", + "type": "text" + }, + { + "bbox": [ + 386, + 687, + 395, + 693 + ], + "score": 0.88, + "content": "v _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 680, + 417, + 696 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 417, + 687, + 427, + 693 + ], + "score": 0.88, + "content": "v _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 680, + 446, + 696 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 446, + 687, + 456, + 693 + ], + "score": 0.88, + "content": "\\boldsymbol { v } _ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 680, + 461, + 696 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 48, + "is_list_start_line": true + }, + { + "bbox": [ + 83, + 697, + 507, + 713 + ], + "spans": [ + { + "bbox": [ + 83, + 702, + 100, + 711 + ], + "score": 1.0, + "content": "1244", + "type": "text" + }, + { + "bbox": [ + 104, + 697, + 172, + 713 + ], + "score": 1.0, + "content": "Note that both", + "type": "text" + }, + { + "bbox": [ + 172, + 700, + 191, + 710 + ], + "score": 0.91, + "content": "G ^ { ( 1 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 697, + 213, + 713 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 213, + 700, + 232, + 710 + ], + "score": 0.92, + "content": "G ^ { ( 2 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 697, + 507, + 713 + ], + "score": 1.0, + "content": "have the same number of nodes and the same degree sequence", + "type": "text" + } + ], + "index": 49, + "is_list_start_line": true + }, + { + "bbox": [ + 83, + 710, + 507, + 723 + ], + "spans": [ + { + "bbox": [ + 83, + 712, + 100, + 722 + ], + "score": 1.0, + "content": "1245", + "type": "text" + }, + { + "bbox": [ + 107, + 712, + 160, + 723 + ], + "score": 0.88, + "content": "\\{ 3 , 2 , 2 , 2 , 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 710, + 507, + 723 + ], + "score": 1.0, + "content": ". 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This is because the multiplicity of", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 410, + 505, + 422 + ], + "spans": [ + { + "bbox": [ + 106, + 410, + 505, + 422 + ], + "score": 1.0, + "content": "the eigenvalue 2 is the number of bipartite components. In particular, SignNet can approximate", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 418, + 506, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 158, + 433 + ], + "score": 1.0, + "content": "the function", + "type": "text" + }, + { + "bbox": [ + 158, + 422, + 229, + 433 + ], + "score": 0.92, + "content": "\\phi ( v _ { i } , \\lambda _ { i } , X ) = \\lambda _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 418, + 248, + 433 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 249, + 423, + 300, + 433 + ], + "score": 0.9, + "content": "\\rho \\approx \\mathrm { m a x } _ { i = 1 } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 418, + 506, + 433 + ], + "score": 1.0, + "content": ". Likewise, BasisNet can approximate the function", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 107, + 432, + 263, + 447 + ], + "spans": [ + { + "bbox": [ + 107, + 433, + 189, + 445 + ], + "score": 0.92, + "content": "\\phi _ { d _ { i } } ( V _ { i } V _ { i } ^ { \\top } , \\lambda _ { i } ) = \\lambda _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 432, + 207, + 447 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 208, + 433, + 258, + 445 + ], + "score": 0.93, + "content": "\\rho \\approx \\mathrm { m a x } _ { i = 1 } ^ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 432, + 263, + 447 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 106, + 448, + 505, + 531 + ], + "lines": [ + { + "bbox": [ + 106, + 448, + 506, + 461 + ], + "spans": [ + { + "bbox": [ + 106, + 448, + 506, + 461 + ], + "score": 1.0, + "content": "This in fact gives an infinite family of graphs that SignNet / BasisNet can distinguish, but spectral", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 459, + 506, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 433, + 473 + ], + "score": 1.0, + "content": "graph convolutions or spectral graph GNNs cannot. To see why, suppose we have", + "type": "text" + }, + { + "bbox": [ + 433, + 461, + 451, + 470 + ], + "score": 0.92, + "content": "G ^ { ( 1 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 459, + 470, + 473 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 471, + 461, + 489, + 470 + ], + "score": 0.91, + "content": "G ^ { ( 2 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 490, + 459, + 506, + 473 + ], + "score": 1.0, + "content": "for", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 471, + 505, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 130, + 484 + ], + "score": 1.0, + "content": "some", + "type": "text" + }, + { + "bbox": [ + 131, + 475, + 155, + 483 + ], + "score": 0.89, + "content": "n \\geq 5", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 471, + 313, + 484 + ], + "score": 1.0, + "content": ". Then we construct a pair of graphs on", + "type": "text" + }, + { + "bbox": [ + 313, + 474, + 336, + 482 + ], + "score": 0.9, + "content": "n + 1", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 471, + 505, + 484 + ], + "score": 1.0, + "content": "nodes with the same degree sequence. To", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 480, + 506, + 497 + ], + "spans": [ + { + "bbox": [ + 104, + 480, + 278, + 497 + ], + "score": 1.0, + "content": "do this, we add another node to the path of", + "type": "text" + }, + { + "bbox": [ + 278, + 484, + 297, + 493 + ], + "score": 0.91, + "content": "G ^ { ( 1 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 480, + 423, + 497 + ], + "score": 1.0, + "content": ", thus giving it degree sequence", + "type": "text" + }, + { + "bbox": [ + 423, + 486, + 484, + 496 + ], + "score": 0.92, + "content": "\\{ 3 , 2 , \\ldots , 2 , 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 480, + 506, + 497 + ], + "score": 1.0, + "content": ". For", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 107, + 492, + 506, + 510 + ], + "spans": [ + { + "bbox": [ + 107, + 496, + 126, + 506 + ], + "score": 0.89, + "content": "G ^ { ( 2 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 126, + 492, + 189, + 510 + ], + "score": 1.0, + "content": ", we add a node", + "type": "text" + }, + { + "bbox": [ + 189, + 501, + 209, + 508 + ], + "score": 0.89, + "content": "v _ { n + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 492, + 271, + 510 + ], + "score": 1.0, + "content": "to the side that", + "type": "text" + }, + { + "bbox": [ + 271, + 501, + 282, + 507 + ], + "score": 0.88, + "content": "v _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 492, + 394, + 510 + ], + "score": 1.0, + "content": "is not contained on (e.g. for", + "type": "text" + }, + { + "bbox": [ + 395, + 498, + 419, + 506 + ], + "score": 0.89, + "content": "n = 5", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 492, + 454, + 510 + ], + "score": 1.0, + "content": ", we add", + "type": "text" + }, + { + "bbox": [ + 454, + 501, + 464, + 507 + ], + "score": 0.87, + "content": "\\boldsymbol { v } _ { 6 }", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 492, + 506, + 510 + ], + "score": 1.0, + "content": "to the left", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 506, + 506, + 520 + ], + "spans": [ + { + "bbox": [ + 106, + 506, + 138, + 520 + ], + "score": 1.0, + "content": "side, as", + "type": "text" + }, + { + "bbox": [ + 138, + 512, + 148, + 518 + ], + "score": 0.84, + "content": "\\boldsymbol { v } _ { 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 506, + 274, + 520 + ], + "score": 1.0, + "content": "was on the right), then connect", + "type": "text" + }, + { + "bbox": [ + 275, + 512, + 285, + 518 + ], + "score": 0.89, + "content": "v _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 506, + 297, + 520 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 297, + 512, + 318, + 519 + ], + "score": 0.9, + "content": "v _ { n + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 506, + 441, + 520 + ], + "score": 1.0, + "content": "to also give a degree sequence", + "type": "text" + }, + { + "bbox": [ + 442, + 509, + 503, + 519 + ], + "score": 0.92, + "content": "\\{ 3 , 2 , \\ldots , 2 , 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 506, + 506, + 520 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 516, + 424, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 243, + 532 + ], + "score": 1.0, + "content": "Note that the non-bipartiteness of", + "type": "text" + }, + { + "bbox": [ + 243, + 520, + 262, + 529 + ], + "score": 0.92, + "content": "G ^ { ( 1 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 516, + 344, + 532 + ], + "score": 1.0, + "content": "and bipartiteness of", + "type": "text" + }, + { + "bbox": [ + 345, + 520, + 363, + 529 + ], + "score": 0.92, + "content": "G ^ { ( 2 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 516, + 424, + 532 + ], + "score": 1.0, + "content": "are preserved.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 23 + }, + { + "type": "title", + "bbox": [ + 106, + 560, + 259, + 572 + ], + "lines": [ + { + "bbox": [ + 105, + 558, + 261, + 575 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 261, + 575 + ], + "score": 1.0, + "content": "H.2 Existing Positional Encodings", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 106, + 580, + 505, + 624 + ], + "lines": [ + { + "bbox": [ + 105, + 580, + 506, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 580, + 506, + 594 + ], + "score": 1.0, + "content": "Here, we show that our SignNets and BasisNets universally approximate various types of existing", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 591, + 506, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 506, + 604 + ], + "score": 1.0, + "content": "graph positional encodings. The key is to show that these positional encodings are related to spectral", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 603, + 506, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 506, + 614 + ], + "score": 1.0, + "content": "graph convolution matrices and the diagonals of these matrices, and to show that our networks can", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 614, + 278, + 625 + ], + "spans": [ + { + "bbox": [ + 106, + 614, + 278, + 625 + ], + "score": 1.0, + "content": "approximate these matrices and diagonals.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29.5 + }, + { + "type": "text", + "bbox": [ + 106, + 628, + 505, + 676 + ], + "lines": [ + { + "bbox": [ + 105, + 627, + 506, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 506, + 641 + ], + "score": 1.0, + "content": "Proposition 5. If the eigenvalues take values in a compact set, SignNets and BasisNets universally ap-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 103, + 636, + 502, + 655 + ], + "spans": [ + { + "bbox": [ + 103, + 636, + 360, + 655 + ], + "score": 1.0, + "content": "proximate the diagonal of any spectral graph convolution matrix", + "type": "text" + }, + { + "bbox": [ + 360, + 638, + 502, + 652 + ], + "score": 0.85, + "content": "\\begin{array} { r } { \\pmb { f } ( V , \\Lambda ) = \\mathrm { d i a g } \\left( \\sum _ { i = 1 } ^ { n } h ( \\lambda _ { i } ) \\hat { v _ { i } v _ { i } ^ { \\top } } \\right) } \\end{array}", + "type": "inline_equation" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 650, + 506, + 664 + ], + "spans": [ + { + "bbox": [ + 105, + 650, + 461, + 664 + ], + "score": 1.0, + "content": "BasisNets can additionally universally approximate any spectral graph convolution matrix", + "type": "text" + }, + { + "bbox": [ + 461, + 652, + 506, + 663 + ], + "score": 0.91, + "content": "f ( V , \\Lambda ) =", + "type": "inline_equation" + } + ], + "index": 34 + }, + { + "bbox": [ + 107, + 658, + 181, + 680 + ], + "spans": [ + { + "bbox": [ + 107, + 662, + 175, + 676 + ], + "score": 0.92, + "content": "\\textstyle \\sum _ { i = 1 } ^ { n } h ( \\lambda _ { i } ) v _ { i } v _ { i } ^ { \\top }", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 658, + 181, + 680 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33.5 + }, + { + "type": "text", + "bbox": [ + 85, + 688, + 505, + 723 + ], + "lines": [ + { + "bbox": [ + 104, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 104, + 687, + 189, + 700 + ], + "score": 1.0, + "content": "Proof. Note that the", + "type": "text" + }, + { + "bbox": [ + 189, + 690, + 198, + 699 + ], + "score": 0.84, + "content": "v _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 687, + 415, + 700 + ], + "score": 1.0, + "content": "come from a compact set as they are of unit norm. The", + "type": "text" + }, + { + "bbox": [ + 416, + 689, + 426, + 699 + ], + "score": 0.88, + "content": "\\lambda _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "are from a compact", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 103, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 103, + 698, + 402, + 712 + ], + "score": 1.0, + "content": "set by assumption; this assumption holds for the normalized Laplacian, as", + "type": "text" + }, + { + "bbox": [ + 403, + 699, + 445, + 710 + ], + "score": 0.9, + "content": "\\lambda _ { i } \\in [ 0 , 2 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 445, + 698, + 506, + 712 + ], + "score": 1.0, + "content": ". 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This is because the multiplicity of", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 410, + 505, + 422 + ], + "spans": [ + { + "bbox": [ + 106, + 410, + 505, + 422 + ], + "score": 1.0, + "content": "the eigenvalue 2 is the number of bipartite components. In particular, SignNet can approximate", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 418, + 506, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 158, + 433 + ], + "score": 1.0, + "content": "the function", + "type": "text" + }, + { + "bbox": [ + 158, + 422, + 229, + 433 + ], + "score": 0.92, + "content": "\\phi ( v _ { i } , \\lambda _ { i } , X ) = \\lambda _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 418, + 248, + 433 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 249, + 423, + 300, + 433 + ], + "score": 0.9, + "content": "\\rho \\approx \\mathrm { m a x } _ { i = 1 } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 418, + 506, + 433 + ], + "score": 1.0, + "content": ". Likewise, BasisNet can approximate the function", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 107, + 432, + 263, + 447 + ], + "spans": [ + { + "bbox": [ + 107, + 433, + 189, + 445 + ], + "score": 0.92, + "content": "\\phi _ { d _ { i } } ( V _ { i } V _ { i } ^ { \\top } , \\lambda _ { i } ) = \\lambda _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 432, + 207, + 447 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 208, + 433, + 258, + 445 + ], + "score": 0.93, + "content": "\\rho \\approx \\mathrm { m a x } _ { i = 1 } ^ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 432, + 263, + 447 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17, + "bbox_fs": [ + 105, + 387, + 506, + 447 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 448, + 505, + 531 + ], + "lines": [ + { + "bbox": [ + 106, + 448, + 506, + 461 + ], + "spans": [ + { + "bbox": [ + 106, + 448, + 506, + 461 + ], + "score": 1.0, + "content": "This in fact gives an infinite family of graphs that SignNet / BasisNet can distinguish, but spectral", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 459, + 506, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 433, + 473 + ], + "score": 1.0, + "content": "graph convolutions or spectral graph GNNs cannot. To see why, suppose we have", + "type": "text" + }, + { + "bbox": [ + 433, + 461, + 451, + 470 + ], + "score": 0.92, + "content": "G ^ { ( 1 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 459, + 470, + 473 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 471, + 461, + 489, + 470 + ], + "score": 0.91, + "content": "G ^ { ( 2 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 490, + 459, + 506, + 473 + ], + "score": 1.0, + "content": "for", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 471, + 505, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 130, + 484 + ], + "score": 1.0, + "content": "some", + "type": "text" + }, + { + "bbox": [ + 131, + 475, + 155, + 483 + ], + "score": 0.89, + "content": "n \\geq 5", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 471, + 313, + 484 + ], + "score": 1.0, + "content": ". Then we construct a pair of graphs on", + "type": "text" + }, + { + "bbox": [ + 313, + 474, + 336, + 482 + ], + "score": 0.9, + "content": "n + 1", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 471, + 505, + 484 + ], + "score": 1.0, + "content": "nodes with the same degree sequence. To", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 480, + 506, + 497 + ], + "spans": [ + { + "bbox": [ + 104, + 480, + 278, + 497 + ], + "score": 1.0, + "content": "do this, we add another node to the path of", + "type": "text" + }, + { + "bbox": [ + 278, + 484, + 297, + 493 + ], + "score": 0.91, + "content": "G ^ { ( 1 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 480, + 423, + 497 + ], + "score": 1.0, + "content": ", thus giving it degree sequence", + "type": "text" + }, + { + "bbox": [ + 423, + 486, + 484, + 496 + ], + "score": 0.92, + "content": "\\{ 3 , 2 , \\ldots , 2 , 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 480, + 506, + 497 + ], + "score": 1.0, + "content": ". For", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 107, + 492, + 506, + 510 + ], + "spans": [ + { + "bbox": [ + 107, + 496, + 126, + 506 + ], + "score": 0.89, + "content": "G ^ { ( 2 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 126, + 492, + 189, + 510 + ], + "score": 1.0, + "content": ", we add a node", + "type": "text" + }, + { + "bbox": [ + 189, + 501, + 209, + 508 + ], + "score": 0.89, + "content": "v _ { n + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 492, + 271, + 510 + ], + "score": 1.0, + "content": "to the side that", + "type": "text" + }, + { + "bbox": [ + 271, + 501, + 282, + 507 + ], + "score": 0.88, + "content": "v _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 492, + 394, + 510 + ], + "score": 1.0, + "content": "is not contained on (e.g. for", + "type": "text" + }, + { + "bbox": [ + 395, + 498, + 419, + 506 + ], + "score": 0.89, + "content": "n = 5", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 492, + 454, + 510 + ], + "score": 1.0, + "content": ", we add", + "type": "text" + }, + { + "bbox": [ + 454, + 501, + 464, + 507 + ], + "score": 0.87, + "content": "\\boldsymbol { v } _ { 6 }", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 492, + 506, + 510 + ], + "score": 1.0, + "content": "to the left", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 506, + 506, + 520 + ], + "spans": [ + { + "bbox": [ + 106, + 506, + 138, + 520 + ], + "score": 1.0, + "content": "side, as", + "type": "text" + }, + { + "bbox": [ + 138, + 512, + 148, + 518 + ], + "score": 0.84, + "content": "\\boldsymbol { v } _ { 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 506, + 274, + 520 + ], + "score": 1.0, + "content": "was on the right), then connect", + "type": "text" + }, + { + "bbox": [ + 275, + 512, + 285, + 518 + ], + "score": 0.89, + "content": "v _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 506, + 297, + 520 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 297, + 512, + 318, + 519 + ], + "score": 0.9, + "content": "v _ { n + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 506, + 441, + 520 + ], + "score": 1.0, + "content": "to also give a degree sequence", + "type": "text" + }, + { + "bbox": [ + 442, + 509, + 503, + 519 + ], + "score": 0.92, + "content": "\\{ 3 , 2 , \\ldots , 2 , 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 506, + 506, + 520 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 516, + 424, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 243, + 532 + ], + "score": 1.0, + "content": "Note that the non-bipartiteness of", + "type": "text" + }, + { + "bbox": [ + 243, + 520, + 262, + 529 + ], + "score": 0.92, + "content": "G ^ { ( 1 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 516, + 344, + 532 + ], + "score": 1.0, + "content": "and bipartiteness of", + "type": "text" + }, + { + "bbox": [ + 345, + 520, + 363, + 529 + ], + "score": 0.92, + "content": "G ^ { ( 2 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 516, + 424, + 532 + ], + "score": 1.0, + "content": "are preserved.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 23, + "bbox_fs": [ + 104, + 448, + 506, + 532 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 560, + 259, + 572 + ], + "lines": [ + { + "bbox": [ + 105, + 558, + 261, + 575 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 261, + 575 + ], + "score": 1.0, + "content": "H.2 Existing Positional Encodings", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 106, + 580, + 505, + 624 + ], + "lines": [ + { + "bbox": [ + 105, + 580, + 506, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 580, + 506, + 594 + ], + "score": 1.0, + "content": "Here, we show that our SignNets and BasisNets universally approximate various types of existing", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 591, + 506, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 506, + 604 + ], + "score": 1.0, + "content": "graph positional encodings. The key is to show that these positional encodings are related to spectral", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 603, + 506, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 506, + 614 + ], + "score": 1.0, + "content": "graph convolution matrices and the diagonals of these matrices, and to show that our networks can", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 614, + 278, + 625 + ], + "spans": [ + { + "bbox": [ + 106, + 614, + 278, + 625 + ], + "score": 1.0, + "content": "approximate these matrices and diagonals.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29.5, + "bbox_fs": [ + 105, + 580, + 506, + 625 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 628, + 505, + 676 + ], + "lines": [ + { + "bbox": [ + 105, + 627, + 506, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 506, + 641 + ], + "score": 1.0, + "content": "Proposition 5. If the eigenvalues take values in a compact set, SignNets and BasisNets universally ap-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 103, + 636, + 502, + 655 + ], + "spans": [ + { + "bbox": [ + 103, + 636, + 360, + 655 + ], + "score": 1.0, + "content": "proximate the diagonal of any spectral graph convolution matrix", + "type": "text" + }, + { + "bbox": [ + 360, + 638, + 502, + 652 + ], + "score": 0.85, + "content": "\\begin{array} { r } { \\pmb { f } ( V , \\Lambda ) = \\mathrm { d i a g } \\left( \\sum _ { i = 1 } ^ { n } h ( \\lambda _ { i } ) \\hat { v _ { i } v _ { i } ^ { \\top } } \\right) } \\end{array}", + "type": "inline_equation" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 650, + 506, + 664 + ], + "spans": [ + { + "bbox": [ + 105, + 650, + 461, + 664 + ], + "score": 1.0, + "content": "BasisNets can additionally universally approximate any spectral graph convolution matrix", + "type": "text" + }, + { + "bbox": [ + 461, + 652, + 506, + 663 + ], + "score": 0.91, + "content": "f ( V , \\Lambda ) =", + "type": "inline_equation" + } + ], + "index": 34 + }, + { + "bbox": [ + 107, + 658, + 181, + 680 + ], + "spans": [ + { + "bbox": [ + 107, + 662, + 175, + 676 + ], + "score": 0.92, + "content": "\\textstyle \\sum _ { i = 1 } ^ { n } h ( \\lambda _ { i } ) v _ { i } v _ { i } ^ { \\top }", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 658, + 181, + 680 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33.5, + "bbox_fs": [ + 103, + 627, + 506, + 680 + ] + }, + { + "type": "text", + "bbox": [ + 85, + 688, + 505, + 723 + ], + "lines": [ + { + "bbox": [ + 104, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 104, + 687, + 189, + 700 + ], + "score": 1.0, + "content": "Proof. Note that the", + "type": "text" + }, + { + "bbox": [ + 189, + 690, + 198, + 699 + ], + "score": 0.84, + "content": "v _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 687, + 415, + 700 + ], + "score": 1.0, + "content": "come from a compact set as they are of unit norm. The", + "type": "text" + }, + { + "bbox": [ + 416, + 689, + 426, + 699 + ], + "score": 0.88, + "content": "\\lambda _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "are from a compact", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 103, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 103, + 698, + 402, + 712 + ], + "score": 1.0, + "content": "set by assumption; this assumption holds for the normalized Laplacian, as", + "type": "text" + }, + { + "bbox": [ + 403, + 699, + 445, + 710 + ], + "score": 0.9, + "content": "\\lambda _ { i } \\in [ 0 , 2 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 445, + 698, + 506, + 712 + ], + "score": 1.0, + "content": ". Also, as diag", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 102, + 709, + 461, + 725 + ], + "spans": [ + { + "bbox": [ + 102, + 709, + 361, + 725 + ], + "score": 1.0, + "content": "is linear, the spectral graph convolution diagonal can be written", + "type": "text" + }, + { + "bbox": [ + 362, + 710, + 456, + 723 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\sum _ { i = 1 } ^ { n } h ( \\lambda _ { i } ) \\mathrm { d i a g } ( v _ { i } v _ { i } ^ { \\top } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 709, + 461, + 725 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37, + "bbox_fs": [ + 102, + 687, + 506, + 725 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 83, + 71, + 505, + 129 + ], + "lines": [ + { + "bbox": [ + 83, + 68, + 508, + 90 + ], + "spans": [ + { + "bbox": [ + 83, + 74, + 99, + 84 + ], + "score": 1.0, + "content": "1271", + "type": "text" + }, + { + "bbox": [ + 102, + 68, + 123, + 90 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 123, + 73, + 150, + 83 + ], + "score": 0.91, + "content": "\\epsilon > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 68, + 228, + 90 + ], + "score": 1.0, + "content": ". For SignNet, let", + "type": "text" + }, + { + "bbox": [ + 228, + 72, + 275, + 85 + ], + "score": 0.93, + "content": "\\rho = \\textstyle \\sum _ { i = 1 } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 68, + 508, + 90 + ], + "score": 1.0, + "content": ", which can be exactly expressed as it is a permutation", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 83, + 83, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 83, + 85, + 100, + 95 + ], + "score": 1.0, + "content": "1272", + "type": "text" + }, + { + "bbox": [ + 105, + 83, + 345, + 96 + ], + "score": 1.0, + "content": "equivariant linear operation from vectors to vectors. Then", + "type": "text" + }, + { + "bbox": [ + 345, + 84, + 381, + 96 + ], + "score": 0.94, + "content": "\\phi ( v _ { i } , \\lambda _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 83, + 506, + 96 + ], + "score": 1.0, + "content": "can approximate the function", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 83, + 95, + 506, + 109 + ], + "spans": [ + { + "bbox": [ + 83, + 97, + 100, + 107 + ], + "score": 1.0, + "content": "1273", + "type": "text" + }, + { + "bbox": [ + 107, + 95, + 163, + 108 + ], + "score": 0.92, + "content": "\\lambda _ { i } \\mathrm { d i a g } ( v _ { i } v _ { i } ^ { \\top } )", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 95, + 506, + 109 + ], + "score": 1.0, + "content": "to arbitrary precision, as it is a permutation equivariant function from vectors to", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 83, + 105, + 505, + 119 + ], + "spans": [ + { + "bbox": [ + 83, + 108, + 100, + 118 + ], + "score": 1.0, + "content": "1274", + "type": "text" + }, + { + "bbox": [ + 105, + 105, + 295, + 119 + ], + "score": 1.0, + "content": "vectors [Segol and Lipman, 2019]. Thus, letting", + "type": "text" + }, + { + "bbox": [ + 296, + 107, + 303, + 118 + ], + "score": 0.84, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 105, + 414, + 119 + ], + "score": 1.0, + "content": "approximate the function to", + "type": "text" + }, + { + "bbox": [ + 415, + 106, + 431, + 118 + ], + "score": 0.9, + "content": "\\epsilon / n", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 105, + 505, + 119 + ], + "score": 1.0, + "content": "accuracy, SignNet", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 83, + 118, + 239, + 131 + ], + "spans": [ + { + "bbox": [ + 83, + 119, + 100, + 129 + ], + "score": 1.0, + "content": "1275", + "type": "text" + }, + { + "bbox": [ + 105, + 118, + 175, + 131 + ], + "score": 1.0, + "content": "can approximate", + "type": "text" + }, + { + "bbox": [ + 175, + 118, + 182, + 129 + ], + "score": 0.86, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 118, + 194, + 131 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 194, + 119, + 199, + 127 + ], + "score": 0.7, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 118, + 239, + 131 + ], + "score": 1.0, + "content": "accuracy.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 101, + 133, + 506, + 229 + ], + "lines": [ + { + "bbox": [ + 104, + 133, + 507, + 151 + ], + "spans": [ + { + "bbox": [ + 104, + 133, + 122, + 151 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 123, + 137, + 127, + 146 + ], + "score": 0.55, + "content": "l", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 133, + 251, + 151 + ], + "score": 1.0, + "content": "be the number of eigenspaces", + "type": "text" + }, + { + "bbox": [ + 251, + 136, + 294, + 147 + ], + "score": 0.91, + "content": "V _ { 1 } , \\dots , V _ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 133, + 309, + 151 + ], + "score": 1.0, + "content": ", so", + "type": "text" + }, + { + "bbox": [ + 309, + 133, + 428, + 149 + ], + "score": 0.93, + "content": "\\begin{array} { r } { f ( V , \\Lambda ) = \\sum _ { i = 1 } ^ { l } h ( \\mu _ { i } ) V _ { i } V _ { i } ^ { \\top } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 133, + 507, + 151 + ], + "score": 1.0, + "content": ". For BasisNet, we", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 146, + 506, + 159 + ], + "spans": [ + { + "bbox": [ + 105, + 146, + 432, + 159 + ], + "score": 1.0, + "content": "need only show that it can approximate the spectral graph convolution matrix to", + "type": "text" + }, + { + "bbox": [ + 432, + 147, + 446, + 159 + ], + "score": 0.89, + "content": "\\epsilon / l", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 146, + 506, + 159 + ], + "score": 1.0, + "content": "accuracy, as a", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 157, + 505, + 171 + ], + "spans": [ + { + "bbox": [ + 105, + 157, + 316, + 171 + ], + "score": 1.0, + "content": "2-IGN can exactly express the diag function in each", + "type": "text" + }, + { + "bbox": [ + 317, + 158, + 331, + 170 + ], + "score": 0.9, + "content": "\\phi _ { d _ { i } }", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 157, + 505, + 171 + ], + "score": 1.0, + "content": ", since it is a linear permutation equivariant", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 169, + 506, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 169, + 437, + 183 + ], + "score": 1.0, + "content": "function from matrices to vectors. A 2-IGN can universally approximate the function", + "type": "text" + }, + { + "bbox": [ + 438, + 169, + 506, + 182 + ], + "score": 0.92, + "content": "f _ { 1 } ( \\mu _ { i } , V _ { i } V _ { i } ^ { \\top } ) =", + "type": "inline_equation" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 181, + 505, + 194 + ], + "spans": [ + { + "bbox": [ + 106, + 181, + 165, + 194 + ], + "score": 0.92, + "content": "( h ( \\mu _ { i } ) , V _ { i } V _ { i } ^ { \\top } )", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 181, + 505, + 194 + ], + "score": 1.0, + "content": ", as it can express any elementwise MLP. Also, a 2-IGN can universally approximate", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 192, + 506, + 206 + ], + "spans": [ + { + "bbox": [ + 105, + 194, + 241, + 206 + ], + "score": 1.0, + "content": "the scalar-matrix multiplication", + "type": "text" + }, + { + "bbox": [ + 241, + 192, + 376, + 206 + ], + "score": 0.91, + "content": "f _ { 2 } ( h ( \\mu _ { i } ) , V _ { i } V _ { i } ^ { \\top } ) ~ = ~ h ( \\mu _ { i } ) V _ { i } V _ { i } ^ { \\top }", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 194, + 506, + 206 + ], + "score": 1.0, + "content": "by another elementwise MLP.", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 204, + 506, + 218 + ], + "spans": [ + { + "bbox": [ + 105, + 204, + 132, + 218 + ], + "score": 1.0, + "content": "Since", + "type": "text" + }, + { + "bbox": [ + 132, + 204, + 266, + 218 + ], + "score": 0.93, + "content": "h ( \\mu _ { i } ) V _ { i } V _ { i } ^ { \\top } = f _ { 2 } \\overset { \\cdot } { \\circ } f _ { 1 } ( \\mu _ { i } , \\bar { V _ { i } } { V _ { i } ^ { \\top } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 204, + 506, + 218 + ], + "score": 1.0, + "content": ", Lemma 6 shows that a single 2-IGN can approximate this", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 217, + 288, + 230 + ], + "spans": [ + { + "bbox": [ + 106, + 217, + 168, + 230 + ], + "score": 1.0, + "content": "composition to", + "type": "text" + }, + { + "bbox": [ + 169, + 218, + 182, + 228 + ], + "score": 0.91, + "content": "\\epsilon / l", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 217, + 288, + 230 + ], + "score": 1.0, + "content": "accuracy, so we are done.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 8.5 + }, + { + "type": "text", + "bbox": [ + 106, + 250, + 505, + 295 + ], + "lines": [ + { + "bbox": [ + 105, + 250, + 505, + 262 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 505, + 262 + ], + "score": 1.0, + "content": "Proposition 4. SignNet and BasisNet universally approximate node positional encodings based on", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 262, + 505, + 273 + ], + "spans": [ + { + "bbox": [ + 106, + 262, + 505, + 273 + ], + "score": 1.0, + "content": "heat kernels [Feldman et al., 2022] and random walks [Dwivedi et al., 2022]. BasisNet universally", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 273, + 506, + 284 + ], + "spans": [ + { + "bbox": [ + 105, + 273, + 217, + 284 + ], + "score": 1.0, + "content": "approximates diffusion and", + "type": "text" + }, + { + "bbox": [ + 218, + 274, + 224, + 284 + ], + "score": 0.75, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 224, + 273, + 506, + 284 + ], + "score": 1.0, + "content": "-step random walk relative positional encodings [Mialon et al., 2021],", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 283, + 459, + 296 + ], + "spans": [ + { + "bbox": [ + 105, + 283, + 459, + 296 + ], + "score": 1.0, + "content": "and generalized PageRank and landing probability distance encodings [Li et al., 2020].", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14.5 + }, + { + "type": "text", + "bbox": [ + 93, + 306, + 505, + 373 + ], + "lines": [ + { + "bbox": [ + 91, + 306, + 506, + 319 + ], + "spans": [ + { + "bbox": [ + 91, + 309, + 100, + 318 + ], + "score": 1.0, + "content": "89", + "type": "text" + }, + { + "bbox": [ + 105, + 306, + 506, + 319 + ], + "score": 1.0, + "content": "Proof. We will show that we can apply the above Proposition 5, by showing that all of these", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 91, + 317, + 505, + 330 + ], + "spans": [ + { + "bbox": [ + 91, + 320, + 99, + 328 + ], + "score": 1.0, + "content": "90", + "type": "text" + }, + { + "bbox": [ + 105, + 317, + 505, + 330 + ], + "score": 1.0, + "content": "positional encodings are spectral graph convolutions. The heat kernel embeddings are of the form", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 91, + 324, + 508, + 345 + ], + "spans": [ + { + "bbox": [ + 91, + 331, + 99, + 339 + ], + "score": 1.0, + "content": "91", + "type": "text" + }, + { + "bbox": [ + 102, + 324, + 127, + 345 + ], + "score": 1.0, + "content": "diag", + "type": "text" + }, + { + "bbox": [ + 127, + 327, + 226, + 341 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\big ( \\sum _ { i = 1 } ^ { n } \\exp ( - t \\lambda _ { i } ) v _ { i } v _ { i } ^ { \\top } \\big ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 324, + 366, + 345 + ], + "score": 1.0, + "content": "for some choices of the parameter", + "type": "text" + }, + { + "bbox": [ + 367, + 330, + 371, + 339 + ], + "score": 0.69, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 324, + 508, + 345 + ], + "score": 1.0, + "content": ", so they can be approximated by", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 91, + 339, + 505, + 352 + ], + "spans": [ + { + "bbox": [ + 91, + 342, + 100, + 351 + ], + "score": 1.0, + "content": "92", + "type": "text" + }, + { + "bbox": [ + 106, + 339, + 505, + 352 + ], + "score": 1.0, + "content": "SignNets or BasisNets. Also, the diffusion kernel [Mialon et al., 2021] is just the matrix of this", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 90, + 349, + 506, + 365 + ], + "spans": [ + { + "bbox": [ + 90, + 352, + 100, + 362 + ], + "score": 1.0, + "content": "93", + "type": "text" + }, + { + "bbox": [ + 104, + 349, + 189, + 365 + ], + "score": 1.0, + "content": "heat kernel, and the", + "type": "text" + }, + { + "bbox": [ + 189, + 352, + 195, + 362 + ], + "score": 0.8, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 349, + 309, + 365 + ], + "score": 1.0, + "content": "-step random walk kernel is", + "type": "text" + }, + { + "bbox": [ + 310, + 350, + 399, + 363 + ], + "score": 0.91, + "content": "{ \\textstyle \\sum _ { i = 1 } ^ { n } } ( 1 - \\gamma \\lambda _ { i } ) ^ { p } v _ { i } v _ { i } ^ { \\top }", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 349, + 482, + 365 + ], + "score": 1.0, + "content": "for some parameter", + "type": "text" + }, + { + "bbox": [ + 483, + 353, + 489, + 362 + ], + "score": 0.78, + "content": "\\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 490, + 349, + 506, + 365 + ], + "score": 1.0, + "content": ", so", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 91, + 360, + 320, + 374 + ], + "spans": [ + { + "bbox": [ + 91, + 363, + 100, + 372 + ], + "score": 1.0, + "content": "94", + "type": "text" + }, + { + "bbox": [ + 105, + 360, + 320, + 374 + ], + "score": 1.0, + "content": "BasisNets can universally approximate both of these.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 19.5 + }, + { + "type": "text", + "bbox": [ + 105, + 377, + 505, + 424 + ], + "lines": [ + { + "bbox": [ + 105, + 377, + 505, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 377, + 283, + 390 + ], + "score": 1.0, + "content": "For the other positional encodings, we let", + "type": "text" + }, + { + "bbox": [ + 283, + 379, + 293, + 389 + ], + "score": 0.83, + "content": "v _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 377, + 505, + 390 + ], + "score": 1.0, + "content": "be the eigenvectors of the random walk Laplacian", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 387, + 506, + 402 + ], + "spans": [ + { + "bbox": [ + 106, + 389, + 152, + 400 + ], + "score": 0.91, + "content": "I - D ^ { - 1 } A", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 387, + 300, + 402 + ], + "score": 1.0, + "content": "instead of the normalized Laplacian", + "type": "text" + }, + { + "bbox": [ + 300, + 388, + 381, + 400 + ], + "score": 0.92, + "content": "I - { D ^ { - 1 / 2 } A D ^ { - 1 / 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 387, + 506, + 402 + ], + "score": 1.0, + "content": ". The eigenvalues of these two", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 399, + 506, + 415 + ], + "spans": [ + { + "bbox": [ + 104, + 399, + 232, + 415 + ], + "score": 1.0, + "content": "Laplacians are the same, and if", + "type": "text" + }, + { + "bbox": [ + 232, + 402, + 241, + 413 + ], + "score": 0.87, + "content": "\\tilde { v } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 399, + 446, + 415 + ], + "score": 1.0, + "content": "is an eigenvector of the normalized Laplacian then", + "type": "text" + }, + { + "bbox": [ + 446, + 401, + 483, + 413 + ], + "score": 0.92, + "content": "D ^ { - 1 / 2 } \\tilde { v } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 399, + 506, + 415 + ], + "score": 1.0, + "content": "is an", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 413, + 468, + 425 + ], + "spans": [ + { + "bbox": [ + 106, + 413, + 468, + 425 + ], + "score": 1.0, + "content": "eigenvector of the random walk Laplacian with the same eigenvalue [Von Luxburg, 2007].", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24.5 + }, + { + "type": "text", + "bbox": [ + 90, + 428, + 504, + 451 + ], + "lines": [ + { + "bbox": [ + 88, + 426, + 505, + 442 + ], + "spans": [ + { + "bbox": [ + 88, + 431, + 100, + 440 + ], + "score": 1.0, + "content": "299", + "type": "text" + }, + { + "bbox": [ + 104, + 426, + 148, + 442 + ], + "score": 1.0, + "content": "Then with", + "type": "text" + }, + { + "bbox": [ + 148, + 431, + 157, + 440 + ], + "score": 0.85, + "content": "v _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 426, + 505, + 442 + ], + "score": 1.0, + "content": "as the eigenvectors of the random walk Laplacian, the random walk positional encodings", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 89, + 439, + 296, + 452 + ], + "spans": [ + { + "bbox": [ + 89, + 442, + 100, + 451 + ], + "score": 1.0, + "content": "300", + "type": "text" + }, + { + "bbox": [ + 104, + 439, + 296, + 452 + ], + "score": 1.0, + "content": "(RWPE) in Dwivedi et al. [2022] take the form", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27.5 + }, + { + "type": "interline_equation", + "bbox": [ + 206, + 455, + 404, + 489 + ], + "lines": [ + { + "bbox": [ + 206, + 455, + 404, + 489 + ], + "spans": [ + { + "bbox": [ + 206, + 455, + 404, + 489 + ], + "score": 0.93, + "content": "\\operatorname { d i a g } \\left( ( D ^ { - 1 } A ) ^ { k } \\right) = \\operatorname { d i a g } \\left( \\sum _ { i = 1 } ^ { n } ( 1 - \\lambda _ { i } ) ^ { k } v _ { i } v _ { i } ^ { \\top } \\right) ,", + "type": "interline_equation", + "image_path": "4839f9c714294b8ea73ccd640ec8269cab9ddc4b3d063e003a39bb28c937cc0f.jpg" + } + ] + } + ], + "index": 29.5, + "virtual_lines": [ + { + "bbox": [ + 206, + 455, + 404, + 472.0 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 206, + 472.0, + 404, + 489.0 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 85, + 492, + 220, + 504 + ], + "lines": [ + { + "bbox": [ + 82, + 491, + 221, + 505 + ], + "spans": [ + { + "bbox": [ + 82, + 491, + 210, + 505 + ], + "score": 1.0, + "content": "1301 for any choices of integer", + "type": "text" + }, + { + "bbox": [ + 210, + 493, + 217, + 502 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 491, + 221, + 505 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 87, + 508, + 369, + 520 + ], + "lines": [ + { + "bbox": [ + 84, + 507, + 370, + 522 + ], + "spans": [ + { + "bbox": [ + 84, + 507, + 370, + 522 + ], + "score": 1.0, + "content": "1302 The distance encodings proposed in Li et al. [2020] take the form", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "interline_equation", + "bbox": [ + 229, + 523, + 380, + 538 + ], + "lines": [ + { + "bbox": [ + 229, + 523, + 380, + 538 + ], + "spans": [ + { + "bbox": [ + 229, + 523, + 380, + 538 + ], + "score": 0.92, + "content": "f _ { 3 } ( A D ^ { - 1 } , ( A D ^ { - 1 } ) ^ { 2 } , ( A D ^ { - 1 } ) ^ { 3 } , \\cdot \\cdot \\cdot ) ,", + "type": "interline_equation", + "image_path": "a9184b956438d64bc72d74ece9c02c68670d711733dfcd9ff1e0d279fa1ebad4.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 229, + 523, + 380, + 538 + ], + "spans": [], + "index": 33 + } + ] + }, + { + "type": "text", + "bbox": [ + 84, + 541, + 505, + 575 + ], + "lines": [ + { + "bbox": [ + 83, + 540, + 506, + 555 + ], + "spans": [ + { + "bbox": [ + 83, + 544, + 100, + 553 + ], + "score": 1.0, + "content": "1303", + "type": "text" + }, + { + "bbox": [ + 104, + 540, + 179, + 555 + ], + "score": 1.0, + "content": "for some function", + "type": "text" + }, + { + "bbox": [ + 180, + 542, + 190, + 553 + ], + "score": 0.87, + "content": "f _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 540, + 295, + 555 + ], + "score": 1.0, + "content": ". We restrict to continuous", + "type": "text" + }, + { + "bbox": [ + 296, + 542, + 306, + 554 + ], + "score": 0.88, + "content": "f _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 540, + 506, + 555 + ], + "score": 1.0, + "content": "here; shortest path distances can be obtained by a", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 83, + 553, + 505, + 565 + ], + "spans": [ + { + "bbox": [ + 83, + 555, + 101, + 564 + ], + "score": 1.0, + "content": "1304", + "type": "text" + }, + { + "bbox": [ + 104, + 553, + 164, + 565 + ], + "score": 1.0, + "content": "discontinuous", + "type": "text" + }, + { + "bbox": [ + 165, + 554, + 175, + 565 + ], + "score": 0.89, + "content": "f _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 553, + 505, + 565 + ], + "score": 1.0, + "content": "that we discuss below. Their generalized PageRank based distance encodings can", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 82, + 564, + 168, + 577 + ], + "spans": [ + { + "bbox": [ + 82, + 564, + 168, + 577 + ], + "score": 1.0, + "content": "1305 be obtained by", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35 + }, + { + "type": "interline_equation", + "bbox": [ + 243, + 573, + 367, + 613 + ], + "lines": [ + { + "bbox": [ + 243, + 573, + 367, + 613 + ], + "spans": [ + { + "bbox": [ + 243, + 573, + 367, + 613 + ], + "score": 0.95, + "content": "\\sum _ { i = 1 } ^ { n } \\left( \\sum _ { k \\geq 1 } \\gamma _ { k } ( 1 - \\lambda _ { i } ) ^ { k } \\right) v _ { i } v _ { i } ^ { \\top }", + "type": "interline_equation", + "image_path": "d980f65cb177744157bf862be09dbb663a65ef87946aa9259afa1da86732a37d.jpg" + } + ] + } + ], + "index": 37.5, + "virtual_lines": [ + { + "bbox": [ + 243, + 573, + 367, + 593.0 + ], + "spans": [], + "index": 37 + }, + { + "bbox": [ + 243, + 593.0, + 367, + 613.0 + ], + "spans": [], + "index": 38 + } + ] + }, + { + "type": "text", + "bbox": [ + 83, + 614, + 507, + 636 + ], + "lines": [ + { + "bbox": [ + 83, + 612, + 505, + 627 + ], + "spans": [ + { + "bbox": [ + 83, + 615, + 100, + 625 + ], + "score": 1.0, + "content": "1306", + "type": "text" + }, + { + "bbox": [ + 103, + 612, + 142, + 627 + ], + "score": 1.0, + "content": "for some", + "type": "text" + }, + { + "bbox": [ + 142, + 614, + 173, + 626 + ], + "score": 0.91, + "content": "\\gamma _ { k } \\in \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 612, + 505, + 627 + ], + "score": 1.0, + "content": ", so this is a spectral graph convolution. They also define so-called landing probability", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 82, + 624, + 302, + 637 + ], + "spans": [ + { + "bbox": [ + 82, + 624, + 302, + 637 + ], + "score": 1.0, + "content": "1307 based positional encodings, which take the form", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39.5 + }, + { + "type": "interline_equation", + "bbox": [ + 265, + 640, + 344, + 673 + ], + "lines": [ + { + "bbox": [ + 265, + 640, + 344, + 673 + ], + "spans": [ + { + "bbox": [ + 265, + 640, + 344, + 673 + ], + "score": 0.94, + "content": "\\sum _ { i = 1 } ^ { n } ( 1 - \\lambda _ { i } ) ^ { k } v _ { i } v _ { i } ^ { \\top } ,", + "type": "interline_equation", + "image_path": "5e9fed7a20bb8d88af668acdaf4fa25d852ac4c75f9642d676007069023581ba.jpg" + } + ] + } + ], + "index": 41.5, + "virtual_lines": [ + { + "bbox": [ + 265, + 640, + 344, + 656.5 + ], + "spans": [], + "index": 41 + }, + { + "bbox": [ + 265, + 656.5, + 344, + 673.0 + ], + "spans": [], + "index": 42 + } + ] + }, + { + "type": "text", + "bbox": [ + 88, + 676, + 488, + 689 + ], + "lines": [ + { + "bbox": [ + 84, + 675, + 487, + 690 + ], + "spans": [ + { + "bbox": [ + 84, + 675, + 213, + 690 + ], + "score": 1.0, + "content": "1308 for some choices of integer", + "type": "text" + }, + { + "bbox": [ + 213, + 677, + 219, + 686 + ], + "score": 0.79, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 675, + 487, + 690 + ], + "score": 1.0, + "content": ". 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For BasisNet, we", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 146, + 506, + 159 + ], + "spans": [ + { + "bbox": [ + 105, + 146, + 432, + 159 + ], + "score": 1.0, + "content": "need only show that it can approximate the spectral graph convolution matrix to", + "type": "text" + }, + { + "bbox": [ + 432, + 147, + 446, + 159 + ], + "score": 0.89, + "content": "\\epsilon / l", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 146, + 506, + 159 + ], + "score": 1.0, + "content": "accuracy, as a", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 157, + 505, + 171 + ], + "spans": [ + { + "bbox": [ + 105, + 157, + 316, + 171 + ], + "score": 1.0, + "content": "2-IGN can exactly express the diag function in each", + "type": "text" + }, + { + "bbox": [ + 317, + 158, + 331, + 170 + ], + "score": 0.9, + "content": "\\phi _ { d _ { i } }", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 157, + 505, + 171 + ], + "score": 1.0, + "content": ", since it is a linear permutation equivariant", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 169, + 506, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 169, + 437, + 183 + ], + "score": 1.0, + "content": "function from matrices to vectors. A 2-IGN can universally approximate the function", + "type": "text" + }, + { + "bbox": [ + 438, + 169, + 506, + 182 + ], + "score": 0.92, + "content": "f _ { 1 } ( \\mu _ { i } , V _ { i } V _ { i } ^ { \\top } ) =", + "type": "inline_equation" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 181, + 505, + 194 + ], + "spans": [ + { + "bbox": [ + 106, + 181, + 165, + 194 + ], + "score": 0.92, + "content": "( h ( \\mu _ { i } ) , V _ { i } V _ { i } ^ { \\top } )", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 181, + 505, + 194 + ], + "score": 1.0, + "content": ", as it can express any elementwise MLP. Also, a 2-IGN can universally approximate", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 192, + 506, + 206 + ], + "spans": [ + { + "bbox": [ + 105, + 194, + 241, + 206 + ], + "score": 1.0, + "content": "the scalar-matrix multiplication", + "type": "text" + }, + { + "bbox": [ + 241, + 192, + 376, + 206 + ], + "score": 0.91, + "content": "f _ { 2 } ( h ( \\mu _ { i } ) , V _ { i } V _ { i } ^ { \\top } ) ~ = ~ h ( \\mu _ { i } ) V _ { i } V _ { i } ^ { \\top }", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 194, + 506, + 206 + ], + "score": 1.0, + "content": "by another elementwise MLP.", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 204, + 506, + 218 + ], + "spans": [ + { + "bbox": [ + 105, + 204, + 132, + 218 + ], + "score": 1.0, + "content": "Since", + "type": "text" + }, + { + "bbox": [ + 132, + 204, + 266, + 218 + ], + "score": 0.93, + "content": "h ( \\mu _ { i } ) V _ { i } V _ { i } ^ { \\top } = f _ { 2 } \\overset { \\cdot } { \\circ } f _ { 1 } ( \\mu _ { i } , \\bar { V _ { i } } { V _ { i } ^ { \\top } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 204, + 506, + 218 + ], + "score": 1.0, + "content": ", Lemma 6 shows that a single 2-IGN can approximate this", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 217, + 288, + 230 + ], + "spans": [ + { + "bbox": [ + 106, + 217, + 168, + 230 + ], + "score": 1.0, + "content": "composition to", + "type": "text" + }, + { + "bbox": [ + 169, + 218, + 182, + 228 + ], + "score": 0.91, + "content": "\\epsilon / l", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 217, + 288, + 230 + ], + "score": 1.0, + "content": "accuracy, so we are done.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 8.5, + "bbox_fs": [ + 104, + 133, + 507, + 230 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 250, + 505, + 295 + ], + "lines": [ + { + "bbox": [ + 105, + 250, + 505, + 262 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 505, + 262 + ], + "score": 1.0, + "content": "Proposition 4. SignNet and BasisNet universally approximate node positional encodings based on", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 262, + 505, + 273 + ], + "spans": [ + { + "bbox": [ + 106, + 262, + 505, + 273 + ], + "score": 1.0, + "content": "heat kernels [Feldman et al., 2022] and random walks [Dwivedi et al., 2022]. BasisNet universally", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 273, + 506, + 284 + ], + "spans": [ + { + "bbox": [ + 105, + 273, + 217, + 284 + ], + "score": 1.0, + "content": "approximates diffusion and", + "type": "text" + }, + { + "bbox": [ + 218, + 274, + 224, + 284 + ], + "score": 0.75, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 224, + 273, + 506, + 284 + ], + "score": 1.0, + "content": "-step random walk relative positional encodings [Mialon et al., 2021],", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 283, + 459, + 296 + ], + "spans": [ + { + "bbox": [ + 105, + 283, + 459, + 296 + ], + "score": 1.0, + "content": "and generalized PageRank and landing probability distance encodings [Li et al., 2020].", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14.5, + "bbox_fs": [ + 105, + 250, + 506, + 296 + ] + }, + { + "type": "index", + "bbox": [ + 93, + 306, + 505, + 373 + ], + "lines": [ + { + "bbox": [ + 91, + 306, + 506, + 319 + ], + "spans": [ + { + "bbox": [ + 91, + 309, + 100, + 318 + ], + "score": 1.0, + "content": "89", + "type": "text" + }, + { + "bbox": [ + 105, + 306, + 506, + 319 + ], + "score": 1.0, + "content": "Proof. We will show that we can apply the above Proposition 5, by showing that all of these", + "type": "text" + } + ], + "index": 17, + "is_list_start_line": true + }, + { + "bbox": [ + 91, + 317, + 505, + 330 + ], + "spans": [ + { + "bbox": [ + 91, + 320, + 99, + 328 + ], + "score": 1.0, + "content": "90", + "type": "text" + }, + { + "bbox": [ + 105, + 317, + 505, + 330 + ], + "score": 1.0, + "content": "positional encodings are spectral graph convolutions. The heat kernel embeddings are of the form", + "type": "text" + } + ], + "index": 18, + "is_list_start_line": true + }, + { + "bbox": [ + 91, + 324, + 508, + 345 + ], + "spans": [ + { + "bbox": [ + 91, + 331, + 99, + 339 + ], + "score": 1.0, + "content": "91", + "type": "text" + }, + { + "bbox": [ + 102, + 324, + 127, + 345 + ], + "score": 1.0, + "content": "diag", + "type": "text" + }, + { + "bbox": [ + 127, + 327, + 226, + 341 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\big ( \\sum _ { i = 1 } ^ { n } \\exp ( - t \\lambda _ { i } ) v _ { i } v _ { i } ^ { \\top } \\big ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 324, + 366, + 345 + ], + "score": 1.0, + "content": "for some choices of the parameter", + "type": "text" + }, + { + "bbox": [ + 367, + 330, + 371, + 339 + ], + "score": 0.69, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 324, + 508, + 345 + ], + "score": 1.0, + "content": ", so they can be approximated by", + "type": "text" + } + ], + "index": 19, + "is_list_start_line": true + }, + { + "bbox": [ + 91, + 339, + 505, + 352 + ], + "spans": [ + { + "bbox": [ + 91, + 342, + 100, + 351 + ], + "score": 1.0, + "content": "92", + "type": "text" + }, + { + "bbox": [ + 106, + 339, + 505, + 352 + ], + "score": 1.0, + "content": "SignNets or BasisNets. Also, the diffusion kernel [Mialon et al., 2021] is just the matrix of this", + "type": "text" + } + ], + "index": 20, + "is_list_start_line": true + }, + { + "bbox": [ + 90, + 349, + 506, + 365 + ], + "spans": [ + { + "bbox": [ + 90, + 352, + 100, + 362 + ], + "score": 1.0, + "content": "93", + "type": "text" + }, + { + "bbox": [ + 104, + 349, + 189, + 365 + ], + "score": 1.0, + "content": "heat kernel, and the", + "type": "text" + }, + { + "bbox": [ + 189, + 352, + 195, + 362 + ], + "score": 0.8, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 349, + 309, + 365 + ], + "score": 1.0, + "content": "-step random walk kernel is", + "type": "text" + }, + { + "bbox": [ + 310, + 350, + 399, + 363 + ], + "score": 0.91, + "content": "{ \\textstyle \\sum _ { i = 1 } ^ { n } } ( 1 - \\gamma \\lambda _ { i } ) ^ { p } v _ { i } v _ { i } ^ { \\top }", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 349, + 482, + 365 + ], + "score": 1.0, + "content": "for some parameter", + "type": "text" + }, + { + "bbox": [ + 483, + 353, + 489, + 362 + ], + "score": 0.78, + "content": "\\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 490, + 349, + 506, + 365 + ], + "score": 1.0, + "content": ", so", + "type": "text" + } + ], + "index": 21, + "is_list_start_line": true + }, + { + "bbox": [ + 91, + 360, + 320, + 374 + ], + "spans": [ + { + "bbox": [ + 91, + 363, + 100, + 372 + ], + "score": 1.0, + "content": "94", + "type": "text" + }, + { + "bbox": [ + 105, + 360, + 320, + 374 + ], + "score": 1.0, + "content": "BasisNets can universally approximate both of these.", + "type": "text" + } + ], + "index": 22, + "is_list_start_line": true + } + ], + "index": 19.5, + "bbox_fs": [ + 90, + 306, + 508, + 374 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 377, + 505, + 424 + ], + "lines": [ + { + "bbox": [ + 105, + 377, + 505, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 377, + 283, + 390 + ], + "score": 1.0, + "content": "For the other positional encodings, we let", + "type": "text" + }, + { + "bbox": [ + 283, + 379, + 293, + 389 + ], + "score": 0.83, + "content": "v _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 377, + 505, + 390 + ], + "score": 1.0, + "content": "be the eigenvectors of the random walk Laplacian", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 387, + 506, + 402 + ], + "spans": [ + { + "bbox": [ + 106, + 389, + 152, + 400 + ], + "score": 0.91, + "content": "I - D ^ { - 1 } A", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 387, + 300, + 402 + ], + "score": 1.0, + "content": "instead of the normalized Laplacian", + "type": "text" + }, + { + "bbox": [ + 300, + 388, + 381, + 400 + ], + "score": 0.92, + "content": "I - { D ^ { - 1 / 2 } A D ^ { - 1 / 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 387, + 506, + 402 + ], + "score": 1.0, + "content": ". The eigenvalues of these two", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 399, + 506, + 415 + ], + "spans": [ + { + "bbox": [ + 104, + 399, + 232, + 415 + ], + "score": 1.0, + "content": "Laplacians are the same, and if", + "type": "text" + }, + { + "bbox": [ + 232, + 402, + 241, + 413 + ], + "score": 0.87, + "content": "\\tilde { v } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 399, + 446, + 415 + ], + "score": 1.0, + "content": "is an eigenvector of the normalized Laplacian then", + "type": "text" + }, + { + "bbox": [ + 446, + 401, + 483, + 413 + ], + "score": 0.92, + "content": "D ^ { - 1 / 2 } \\tilde { v } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 399, + 506, + 415 + ], + "score": 1.0, + "content": "is an", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 413, + 468, + 425 + ], + "spans": [ + { + "bbox": [ + 106, + 413, + 468, + 425 + ], + "score": 1.0, + "content": "eigenvector of the random walk Laplacian with the same eigenvalue [Von Luxburg, 2007].", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24.5, + "bbox_fs": [ + 104, + 377, + 506, + 425 + ] + }, + { + "type": "index", + "bbox": [ + 90, + 428, + 504, + 451 + ], + "lines": [ + { + "bbox": [ + 88, + 426, + 505, + 442 + ], + "spans": [ + { + "bbox": [ + 88, + 431, + 100, + 440 + ], + "score": 1.0, + "content": "299", + "type": "text" + }, + { + "bbox": [ + 104, + 426, + 148, + 442 + ], + "score": 1.0, + "content": "Then with", + "type": "text" + }, + { + "bbox": [ + 148, + 431, + 157, + 440 + ], + "score": 0.85, + "content": "v _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 426, + 505, + 442 + ], + "score": 1.0, + "content": "as the eigenvectors of the random walk Laplacian, the random walk positional encodings", + "type": "text" + } + ], + "index": 27, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 439, + 296, + 452 + ], + "spans": [ + { + "bbox": [ + 89, + 442, + 100, + 451 + ], + "score": 1.0, + "content": "300", + "type": "text" + }, + { + "bbox": [ + 104, + 439, + 296, + 452 + ], + "score": 1.0, + "content": "(RWPE) in Dwivedi et al. [2022] take the form", + "type": "text" + } + ], + "index": 28, + "is_list_start_line": true + } + ], + "index": 27.5, + "bbox_fs": [ + 88, + 426, + 505, + 452 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 206, + 455, + 404, + 489 + ], + "lines": [ + { + "bbox": [ + 206, + 455, + 404, + 489 + ], + "spans": [ + { + "bbox": [ + 206, + 455, + 404, + 489 + ], + "score": 0.93, + "content": "\\operatorname { d i a g } \\left( ( D ^ { - 1 } A ) ^ { k } \\right) = \\operatorname { d i a g } \\left( \\sum _ { i = 1 } ^ { n } ( 1 - \\lambda _ { i } ) ^ { k } v _ { i } v _ { i } ^ { \\top } \\right) ,", + "type": "interline_equation", + "image_path": "4839f9c714294b8ea73ccd640ec8269cab9ddc4b3d063e003a39bb28c937cc0f.jpg" + } + ] + } + ], + "index": 29.5, + "virtual_lines": [ + { + "bbox": [ + 206, + 455, + 404, + 472.0 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 206, + 472.0, + 404, + 489.0 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 85, + 492, + 220, + 504 + ], + "lines": [ + { + "bbox": [ + 82, + 491, + 221, + 505 + ], + "spans": [ + { + "bbox": [ + 82, + 491, + 210, + 505 + ], + "score": 1.0, + "content": "1301 for any choices of integer", + "type": "text" + }, + { + "bbox": [ + 210, + 493, + 217, + 502 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 491, + 221, + 505 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31, + "bbox_fs": [ + 82, + 491, + 221, + 505 + ] + }, + { + "type": "text", + "bbox": [ + 87, + 508, + 369, + 520 + ], + "lines": [ + { + "bbox": [ + 84, + 507, + 370, + 522 + ], + "spans": [ + { + "bbox": [ + 84, + 507, + 370, + 522 + ], + "score": 1.0, + "content": "1302 The distance encodings proposed in Li et al. [2020] take the form", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32, + "bbox_fs": [ + 84, + 507, + 370, + 522 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 229, + 523, + 380, + 538 + ], + "lines": [ + { + "bbox": [ + 229, + 523, + 380, + 538 + ], + "spans": [ + { + "bbox": [ + 229, + 523, + 380, + 538 + ], + "score": 0.92, + "content": "f _ { 3 } ( A D ^ { - 1 } , ( A D ^ { - 1 } ) ^ { 2 } , ( A D ^ { - 1 } ) ^ { 3 } , \\cdot \\cdot \\cdot ) ,", + "type": "interline_equation", + "image_path": "a9184b956438d64bc72d74ece9c02c68670d711733dfcd9ff1e0d279fa1ebad4.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 229, + 523, + 380, + 538 + ], + "spans": [], + "index": 33 + } + ] + }, + { + "type": "index", + "bbox": [ + 84, + 541, + 505, + 575 + ], + "lines": [ + { + "bbox": [ + 83, + 540, + 506, + 555 + ], + "spans": [ + { + "bbox": [ + 83, + 544, + 100, + 553 + ], + "score": 1.0, + "content": "1303", + "type": "text" + }, + { + "bbox": [ + 104, + 540, + 179, + 555 + ], + "score": 1.0, + "content": "for some function", + "type": "text" + }, + { + "bbox": [ + 180, + 542, + 190, + 553 + ], + "score": 0.87, + "content": "f _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 540, + 295, + 555 + ], + "score": 1.0, + "content": ". We restrict to continuous", + "type": "text" + }, + { + "bbox": [ + 296, + 542, + 306, + 554 + ], + "score": 0.88, + "content": "f _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 540, + 506, + 555 + ], + "score": 1.0, + "content": "here; shortest path distances can be obtained by a", + "type": "text" + } + ], + "index": 34, + "is_list_start_line": true + }, + { + "bbox": [ + 83, + 553, + 505, + 565 + ], + "spans": [ + { + "bbox": [ + 83, + 555, + 101, + 564 + ], + "score": 1.0, + "content": "1304", + "type": "text" + }, + { + "bbox": [ + 104, + 553, + 164, + 565 + ], + "score": 1.0, + "content": "discontinuous", + "type": "text" + }, + { + "bbox": [ + 165, + 554, + 175, + 565 + ], + "score": 0.89, + "content": "f _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 553, + 505, + 565 + ], + "score": 1.0, + "content": "that we discuss below. Their generalized PageRank based distance encodings can", + "type": "text" + } + ], + "index": 35, + "is_list_start_line": true + }, + { + "bbox": [ + 82, + 564, + 168, + 577 + ], + "spans": [ + { + "bbox": [ + 82, + 564, + 168, + 577 + ], + "score": 1.0, + "content": "1305 be obtained by", + "type": "text" + } + ], + "index": 36, + "is_list_start_line": true + } + ], + "index": 35, + "bbox_fs": [ + 82, + 540, + 506, + 577 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 243, + 573, + 367, + 613 + ], + "lines": [ + { + "bbox": [ + 243, + 573, + 367, + 613 + ], + "spans": [ + { + "bbox": [ + 243, + 573, + 367, + 613 + ], + "score": 0.95, + "content": "\\sum _ { i = 1 } ^ { n } \\left( \\sum _ { k \\geq 1 } \\gamma _ { k } ( 1 - \\lambda _ { i } ) ^ { k } \\right) v _ { i } v _ { i } ^ { \\top }", + "type": "interline_equation", + "image_path": "d980f65cb177744157bf862be09dbb663a65ef87946aa9259afa1da86732a37d.jpg" + } + ] + } + ], + "index": 37.5, + "virtual_lines": [ + { + "bbox": [ + 243, + 573, + 367, + 593.0 + ], + "spans": [], + "index": 37 + }, + { + "bbox": [ + 243, + 593.0, + 367, + 613.0 + ], + "spans": [], + "index": 38 + } + ] + }, + { + "type": "index", + "bbox": [ + 83, + 614, + 507, + 636 + ], + "lines": [ + { + "bbox": [ + 83, + 612, + 505, + 627 + ], + "spans": [ + { + "bbox": [ + 83, + 615, + 100, + 625 + ], + "score": 1.0, + "content": "1306", + "type": "text" + }, + { + "bbox": [ + 103, + 612, + 142, + 627 + ], + "score": 1.0, + "content": "for some", + "type": "text" + }, + { + "bbox": [ + 142, + 614, + 173, + 626 + ], + "score": 0.91, + "content": "\\gamma _ { k } \\in \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 612, + 505, + 627 + ], + "score": 1.0, + "content": ", so this is a spectral graph convolution. They also define so-called landing probability", + "type": "text" + } + ], + "index": 39, + "is_list_start_line": true + }, + { + "bbox": [ + 82, + 624, + 302, + 637 + ], + "spans": [ + { + "bbox": [ + 82, + 624, + 302, + 637 + ], + "score": 1.0, + "content": "1307 based positional encodings, which take the form", + "type": "text" + } + ], + "index": 40, + "is_list_start_line": true + } + ], + "index": 39.5, + "bbox_fs": [ + 82, + 612, + 505, + 637 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 265, + 640, + 344, + 673 + ], + "lines": [ + { + "bbox": [ + 265, + 640, + 344, + 673 + ], + "spans": [ + { + "bbox": [ + 265, + 640, + 344, + 673 + ], + "score": 0.94, + "content": "\\sum _ { i = 1 } ^ { n } ( 1 - \\lambda _ { i } ) ^ { k } v _ { i } v _ { i } ^ { \\top } ,", + "type": "interline_equation", + "image_path": "5e9fed7a20bb8d88af668acdaf4fa25d852ac4c75f9642d676007069023581ba.jpg" + } + ] + } + ], + "index": 41.5, + "virtual_lines": [ + { + "bbox": [ + 265, + 640, + 344, + 656.5 + ], + "spans": [], + "index": 41 + }, + { + "bbox": [ + 265, + 656.5, + 344, + 673.0 + ], + "spans": [], + "index": 42 + } + ] + }, + { + "type": "text", + "bbox": [ + 88, + 676, + 488, + 689 + ], + "lines": [ + { + "bbox": [ + 84, + 675, + 487, + 690 + ], + "spans": [ + { + "bbox": [ + 84, + 675, + 213, + 690 + ], + "score": 1.0, + "content": "1308 for some choices of integer", + "type": "text" + }, + { + "bbox": [ + 213, + 677, + 219, + 686 + ], + "score": 0.79, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 675, + 487, + 690 + ], + "score": 1.0, + "content": ". Thus, BasisNets can approximate these distance encoding matrices.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 43, + "bbox_fs": [ + 84, + 675, + 487, + 690 + ] + }, + { + "type": "index", + "bbox": [ + 83, + 699, + 506, + 723 + ], + "lines": [ + { + "bbox": [ + 81, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 81, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "1309 Another powerful class of positional encodings is based on shortest path distances between nodes", + "type": "text" + } + ], + "index": 44, + "is_list_start_line": true + }, + { + "bbox": [ + 81, + 710, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 81, + 710, + 506, + 723 + ], + "score": 1.0, + "content": "1310 in the graph [Ying et al., 2021, Li et al., 2020]. Shortest path distances can be expressed in a", + "type": "text" + } + ], + "index": 45, + "is_list_start_line": true + }, + { + "bbox": [ + 83, + 73, + 505, + 85 + ], + "spans": [ + { + "bbox": [ + 83, + 74, + 99, + 84 + ], + "score": 1.0, + "content": "1311", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 73, + 505, + 85 + ], + "score": 1.0, + "content": "form similar to the spectral graph convolution, but require a highly discontinuous function. 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If we", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 83, + 83, + 505, + 97 + ], + "spans": [ + { + "bbox": [ + 83, + 85, + 100, + 95 + ], + "score": 1.0, + "content": "1312", + "type": "text" + }, + { + "bbox": [ + 105, + 83, + 134, + 97 + ], + "score": 1.0, + "content": "define", + "type": "text" + }, + { + "bbox": [ + 135, + 83, + 256, + 96 + ], + "score": 0.93, + "content": "f _ { 3 } ( x _ { 1 } , \\dots , x _ { n } ) = \\operatorname* { m i n } _ { i : x _ { i } \\neq 0 } i", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 83, + 392, + 97 + ], + "score": 1.0, + "content": "to be the lowest index such that", + "type": "text" + }, + { + "bbox": [ + 393, + 85, + 403, + 95 + ], + "score": 0.83, + "content": "x _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 83, + 505, + 97 + ], + "score": 1.0, + "content": "is nonzero, then we can", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 82, + 91, + 509, + 126 + ], + "spans": [ + { + "bbox": [ + 82, + 97, + 100, + 119 + ], + "score": 1.0, + "content": "1313 1314", + "type": "text" + }, + { + "bbox": [ + 101, + 91, + 209, + 126 + ], + "score": 1.0, + "content": "write the shortest path diselementwise to return an", + "type": "text" + }, + { + "bbox": [ + 209, + 108, + 235, + 118 + ], + "score": 0.89, + "content": "n \\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 91, + 271, + 126 + ], + "score": 1.0, + "content": "matrix as matrix.", + "type": "text" + }, + { + "bbox": [ + 272, + 95, + 421, + 107 + ], + "score": 0.88, + "content": "f _ { 3 } ( D ^ { - 1 } A , ( D ^ { - 1 } A ) ^ { 2 } , \\dots , ( D ^ { - 1 } A ) ^ { n } )", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 107, + 419, + 120 + ], + "score": 0.88, + "content": "\\textstyle ( D ^ { - 1 } A ) ^ { k } = \\sum _ { i = 1 } ^ { n } ( 1 - \\lambda _ { i } ) ^ { k } v _ { i } v _ { i } ^ { \\top }", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 91, + 452, + 126 + ], + "score": 1.0, + "content": ", where BasisN", + "type": "text" + }, + { + "bbox": [ + 452, + 96, + 463, + 107 + ], + "score": 0.88, + "content": "f _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 463, + 91, + 509, + 126 + ], + "score": 1.0, + "content": "is applied can learn", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 83, + 117, + 383, + 132 + ], + "spans": [ + { + "bbox": [ + 83, + 121, + 99, + 129 + ], + "score": 1.0, + "content": "1315", + "type": "text" + }, + { + "bbox": [ + 105, + 117, + 368, + 132 + ], + "score": 1.0, + "content": "the inside arguments, but cannot learn the discontinuous function", + "type": "text" + }, + { + "bbox": [ + 369, + 120, + 379, + 130 + ], + "score": 0.86, + "content": "f _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 117, + 383, + 132 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "title", + "bbox": [ + 87, + 143, + 216, + 155 + ], + "lines": [ + { + "bbox": [ + 84, + 142, + 216, + 157 + ], + "spans": [ + { + "bbox": [ + 84, + 142, + 216, + 157 + ], + "score": 1.0, + "content": "1316 H.3 Spectral Invariants", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 106, + 163, + 505, + 273 + ], + "lines": [ + { + "bbox": [ + 105, + 163, + 506, + 177 + ], + "spans": [ + { + "bbox": [ + 105, + 163, + 254, + 176 + ], + "score": 1.0, + "content": "Here, we consider the graph angles", + "type": "text" + }, + { + "bbox": [ + 254, + 163, + 331, + 177 + ], + "score": 0.93, + "content": "\\alpha _ { i j } = \\lVert V _ { i } V _ { i } ^ { \\top } e _ { j } \\rVert _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 163, + 350, + 176 + ], + "score": 1.0, + "content": ", for", + "type": "text" + }, + { + "bbox": [ + 350, + 164, + 400, + 176 + ], + "score": 0.92, + "content": "i = 1 , \\dots , l", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 163, + 429, + 176 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 429, + 165, + 434, + 174 + ], + "score": 0.73, + "content": "l", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 163, + 506, + 176 + ], + "score": 1.0, + "content": "is the number of", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 174, + 506, + 187 + ], + "spans": [ + { + "bbox": [ + 105, + 174, + 178, + 187 + ], + "score": 1.0, + "content": "eigenspaces, and", + "type": "text" + }, + { + "bbox": [ + 178, + 176, + 231, + 186 + ], + "score": 0.92, + "content": "j = 1 , \\dotsc , n", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 174, + 506, + 187 + ], + "score": 1.0, + "content": ". It is clear that graph angles are permutation equivariant and basis", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 184, + 505, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 184, + 505, + 199 + ], + "score": 1.0, + "content": "invariant. These graph angles have been extensively studied, so we cite a number of interesting", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 197, + 505, + 209 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 505, + 209 + ], + "score": 1.0, + "content": "properties of them. That graph angles determine the number of length 3, 4 and 5 cycles, the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 208, + 505, + 219 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 297, + 219 + ], + "score": 1.0, + "content": "connectivity of a graph, and the number of length", + "type": "text" + }, + { + "bbox": [ + 298, + 208, + 304, + 218 + ], + "score": 0.78, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 208, + 505, + 219 + ], + "score": 1.0, + "content": "closed walks is all shown in Chapter 4 of Cvetkovic´", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 218, + 506, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 218, + 506, + 231 + ], + "score": 1.0, + "content": "et al. [1997]. Other properties may be of use for graph representation learning as well. For instance,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 229, + 505, + 242 + ], + "spans": [ + { + "bbox": [ + 105, + 229, + 318, + 242 + ], + "score": 1.0, + "content": "the eigenvalues of node-deleted subgraphs of a graph", + "type": "text" + }, + { + "bbox": [ + 318, + 230, + 326, + 241 + ], + "score": 0.8, + "content": "\\mathcal { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 229, + 505, + 242 + ], + "score": 1.0, + "content": "are determined by the eigenvalues and graph", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 241, + 505, + 252 + ], + "spans": [ + { + "bbox": [ + 106, + 241, + 145, + 252 + ], + "score": 1.0, + "content": "angles of", + "type": "text" + }, + { + "bbox": [ + 145, + 241, + 153, + 251 + ], + "score": 0.76, + "content": "\\mathcal { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 241, + 505, + 252 + ], + "score": 1.0, + "content": "; this may be useful in extending recent graph neural networks that are motivated by node", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 250, + 506, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 506, + 264 + ], + "score": 1.0, + "content": "deletion and the reconstruction conjecture [Cotta et al., 2021, Bevilacqua et al., 2022, Papp et al.,", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 261, + 228, + 274 + ], + "spans": [ + { + "bbox": [ + 106, + 261, + 228, + 274 + ], + "score": 1.0, + "content": "2021, Tahmasebi et al., 2020].", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 9.5 + }, + { + "type": "text", + "bbox": [ + 107, + 278, + 505, + 334 + ], + "lines": [ + { + "bbox": [ + 106, + 279, + 505, + 291 + ], + "spans": [ + { + "bbox": [ + 106, + 279, + 505, + 291 + ], + "score": 1.0, + "content": "Now, we prove that BasisNet can universally approximate the graph angles. The graph properties we", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 290, + 505, + 302 + ], + "spans": [ + { + "bbox": [ + 106, + 290, + 505, + 302 + ], + "score": 1.0, + "content": "consider in the theorem are all integer valued (e.g. the number of cycles of length 3 in a graph is an", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 300, + 506, + 313 + ], + "spans": [ + { + "bbox": [ + 106, + 300, + 506, + 313 + ], + "score": 1.0, + "content": "integer). Thus, any two graphs that differ in these properties will differ by at least 1, so as long as", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 311, + 506, + 325 + ], + "spans": [ + { + "bbox": [ + 106, + 311, + 211, + 325 + ], + "score": 1.0, + "content": "we have approximation to", + "type": "text" + }, + { + "bbox": [ + 211, + 312, + 245, + 324 + ], + "score": 0.91, + "content": "\\varepsilon < 1 / 2", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 311, + 506, + 325 + ], + "score": 1.0, + "content": ", we can distinguish any two graphs that differ in these properties.", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 322, + 249, + 334 + ], + "spans": [ + { + "bbox": [ + 106, + 322, + 249, + 334 + ], + "score": 1.0, + "content": "Recall the statement of Theorem 2.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 106, + 337, + 505, + 371 + ], + "lines": [ + { + "bbox": [ + 105, + 337, + 505, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 337, + 378, + 351 + ], + "score": 1.0, + "content": "Theorem 2. 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The eigenvalues and graph", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 348, + 505, + 361 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 505, + 361 + ], + "score": 1.0, + "content": "angles (and thus BasisNets) can determine the number of length 3, 4, and 5 cycles, whether a graph", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 360, + 423, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 263, + 371 + ], + "score": 1.0, + "content": "is connected, and the number of length", + "type": "text" + }, + { + "bbox": [ + 264, + 360, + 270, + 369 + ], + "score": 0.33, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 360, + 423, + 371 + ], + "score": 1.0, + "content": "closed walks from any vertex to itself.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 85, + 385, + 271, + 397 + ], + "lines": [ + { + "bbox": [ + 83, + 385, + 271, + 399 + ], + "spans": [ + { + "bbox": [ + 83, + 385, + 271, + 399 + ], + "score": 1.0, + "content": "1335 Proof. 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It is clear that graph angles are permutation equivariant and basis", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 184, + 505, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 184, + 505, + 199 + ], + "score": 1.0, + "content": "invariant. These graph angles have been extensively studied, so we cite a number of interesting", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 197, + 505, + 209 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 505, + 209 + ], + "score": 1.0, + "content": "properties of them. That graph angles determine the number of length 3, 4 and 5 cycles, the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 208, + 505, + 219 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 297, + 219 + ], + "score": 1.0, + "content": "connectivity of a graph, and the number of length", + "type": "text" + }, + { + "bbox": [ + 298, + 208, + 304, + 218 + ], + "score": 0.78, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 208, + 505, + 219 + ], + "score": 1.0, + "content": "closed walks is all shown in Chapter 4 of Cvetkovic´", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 218, + 506, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 218, + 506, + 231 + ], + "score": 1.0, + "content": "et al. [1997]. Other properties may be of use for graph representation learning as well. For instance,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 229, + 505, + 242 + ], + "spans": [ + { + "bbox": [ + 105, + 229, + 318, + 242 + ], + "score": 1.0, + "content": "the eigenvalues of node-deleted subgraphs of a graph", + "type": "text" + }, + { + "bbox": [ + 318, + 230, + 326, + 241 + ], + "score": 0.8, + "content": "\\mathcal { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 229, + 505, + 242 + ], + "score": 1.0, + "content": "are determined by the eigenvalues and graph", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 241, + 505, + 252 + ], + "spans": [ + { + "bbox": [ + 106, + 241, + 145, + 252 + ], + "score": 1.0, + "content": "angles of", + "type": "text" + }, + { + "bbox": [ + 145, + 241, + 153, + 251 + ], + "score": 0.76, + "content": "\\mathcal { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 241, + 505, + 252 + ], + "score": 1.0, + "content": "; this may be useful in extending recent graph neural networks that are motivated by node", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 250, + 506, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 506, + 264 + ], + "score": 1.0, + "content": "deletion and the reconstruction conjecture [Cotta et al., 2021, Bevilacqua et al., 2022, Papp et al.,", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 261, + 228, + 274 + ], + "spans": [ + { + "bbox": [ + 106, + 261, + 228, + 274 + ], + "score": 1.0, + "content": "2021, Tahmasebi et al., 2020].", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 9.5, + "bbox_fs": [ + 105, + 163, + 506, + 274 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 278, + 505, + 334 + ], + "lines": [ + { + "bbox": [ + 106, + 279, + 505, + 291 + ], + "spans": [ + { + "bbox": [ + 106, + 279, + 505, + 291 + ], + "score": 1.0, + "content": "Now, we prove that BasisNet can universally approximate the graph angles. The graph properties we", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 290, + 505, + 302 + ], + "spans": [ + { + "bbox": [ + 106, + 290, + 505, + 302 + ], + "score": 1.0, + "content": "consider in the theorem are all integer valued (e.g. the number of cycles of length 3 in a graph is an", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 300, + 506, + 313 + ], + "spans": [ + { + "bbox": [ + 106, + 300, + 506, + 313 + ], + "score": 1.0, + "content": "integer). Thus, any two graphs that differ in these properties will differ by at least 1, so as long as", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 311, + 506, + 325 + ], + "spans": [ + { + "bbox": [ + 106, + 311, + 211, + 325 + ], + "score": 1.0, + "content": "we have approximation to", + "type": "text" + }, + { + "bbox": [ + 211, + 312, + 245, + 324 + ], + "score": 0.91, + "content": "\\varepsilon < 1 / 2", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 311, + 506, + 325 + ], + "score": 1.0, + "content": ", we can distinguish any two graphs that differ in these properties.", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 322, + 249, + 334 + ], + "spans": [ + { + "bbox": [ + 106, + 322, + 249, + 334 + ], + "score": 1.0, + "content": "Recall the statement of Theorem 2.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17, + "bbox_fs": [ + 106, + 279, + 506, + 334 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 337, + 505, + 371 + ], + "lines": [ + { + "bbox": [ + 105, + 337, + 505, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 337, + 378, + 351 + ], + "score": 1.0, + "content": "Theorem 2. 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The eigenvalues and graph", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 348, + 505, + 361 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 505, + 361 + ], + "score": 1.0, + "content": "angles (and thus BasisNets) can determine the number of length 3, 4, and 5 cycles, whether a graph", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 360, + 423, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 263, + 371 + ], + "score": 1.0, + "content": "is connected, and the number of length", + "type": "text" + }, + { + "bbox": [ + 264, + 360, + 270, + 369 + ], + "score": 0.33, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 360, + 423, + 371 + ], + "score": 1.0, + "content": "closed walks from any vertex to itself.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21, + "bbox_fs": [ + 105, + 337, + 505, + 371 + ] + }, + { + "type": "text", + "bbox": [ + 85, + 385, + 271, + 397 + ], + "lines": [ + { + "bbox": [ + 83, + 385, + 271, + 399 + ], + "spans": [ + { + "bbox": [ + 83, + 385, + 271, + 399 + ], + "score": 1.0, + "content": "1335 Proof. 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Then a 2-IGN can learn an elementwise", + "type": "text" + } + ], + "index": 28, + "is_list_start_line": true + }, + { + "bbox": [ + 83, + 481, + 506, + 496 + ], + "spans": [ + { + "bbox": [ + 83, + 484, + 100, + 494 + ], + "score": 1.0, + "content": "1340", + "type": "text" + }, + { + "bbox": [ + 105, + 482, + 313, + 496 + ], + "score": 1.0, + "content": "MLP to approximate the elementwise square root", + "type": "text" + }, + { + "bbox": [ + 313, + 481, + 455, + 495 + ], + "score": 0.92, + "content": "f _ { 2 } ( \\mathrm { d i a g } ( V _ { i } V _ { i } ^ { \\top } ) ) = \\sqrt { \\mathrm { d i a g } ( V _ { i } V _ { i } ^ { \\top } ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 482, + 506, + 496 + ], + "score": 1.0, + "content": "to arbitrary", + "type": "text" + } + ], + "index": 29, + "is_list_start_line": true + }, + { + "bbox": [ + 82, + 493, + 506, + 506 + ], + "spans": [ + { + "bbox": [ + 82, + 495, + 100, + 506 + ], + "score": 1.0, + "content": "1341", + "type": "text" + }, + { + "bbox": [ + 105, + 493, + 317, + 506 + ], + "score": 1.0, + "content": "precision. 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A", + "type": "text" + } + ], + "index": 33, + "is_list_start_line": true + }, + { + "bbox": [ + 82, + 537, + 506, + 550 + ], + "spans": [ + { + "bbox": [ + 82, + 539, + 100, + 549 + ], + "score": 1.0, + "content": "1345", + "type": "text" + }, + { + "bbox": [ + 105, + 537, + 216, + 550 + ], + "score": 1.0, + "content": "DeepSets can approximate", + "type": "text" + }, + { + "bbox": [ + 216, + 538, + 226, + 549 + ], + "score": 0.88, + "content": "f _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 537, + 506, + 550 + ], + "score": 1.0, + "content": "without any higher order tensors besides vectors [Zaheer et al., 2017,", + "type": "text" + } + ], + "index": 34, + "is_list_start_line": true + }, + { + "bbox": [ + 83, + 548, + 213, + 561 + ], + "spans": [ + { + "bbox": [ + 83, + 550, + 100, + 560 + ], + "score": 1.0, + "content": "1346", + "type": "text" + }, + { + "bbox": [ + 105, + 548, + 213, + 561 + ], + "score": 1.0, + "content": "Segol and Lipman, 2019].", + "type": "text" + } + ], + "index": 35, + "is_list_start_line": true + } + ], + "index": 30, + "bbox_fs": [ + 82, + 430, + 507, + 561 + ] + }, + { + "type": "text", + "bbox": [ + 84, + 564, + 505, + 598 + ], + "lines": [ + { + "bbox": [ + 81, + 563, + 506, + 578 + ], + "spans": [ + { + "bbox": [ + 81, + 563, + 244, + 578 + ], + "score": 1.0, + "content": "As 2-IGNs can approximate each 1347", + "type": "text" + }, + { + "bbox": [ + 245, + 565, + 254, + 576 + ], + "score": 0.87, + "content": "f _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 563, + 442, + 578 + ], + "score": 1.0, + "content": "individually, a single 2-IGN can approximate", + "type": "text" + }, + { + "bbox": [ + 443, + 565, + 491, + 576 + ], + "score": 0.93, + "content": "f _ { 3 } \\circ f _ { 2 } \\circ f _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 491, + 563, + 506, + 578 + ], + "score": 1.0, + "content": "by", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 84, + 576, + 506, + 588 + ], + "spans": [ + { + "bbox": [ + 84, + 578, + 100, + 586 + ], + "score": 1.0, + "content": "1348", + "type": "text" + }, + { + "bbox": [ + 102, + 576, + 506, + 588 + ], + "score": 1.0, + "content": "Lemma 6. Also, since the graph properties considered in the theorem are integer-valued, BasisNet", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 83, + 587, + 506, + 599 + ], + "spans": [ + { + "bbox": [ + 83, + 587, + 379, + 599 + ], + "score": 1.0, + "content": "1349 can distinguish any two graphs that differ in one of these properties.", + "type": "text" + }, + { + "bbox": [ + 494, + 587, + 506, + 598 + ], + "score": 0.998, + "content": "□", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37, + "bbox_fs": [ + 81, + 563, + 506, + 599 + ] + }, + { + "type": "index", + "bbox": [ + 83, + 612, + 505, + 723 + ], + "lines": [ + { + "bbox": [ + 83, + 611, + 506, + 625 + ], + "spans": [ + { + "bbox": [ + 83, + 614, + 99, + 624 + ], + "score": 1.0, + "content": "1350", + "type": "text" + }, + { + "bbox": [ + 104, + 611, + 506, + 625 + ], + "score": 1.0, + "content": "To see that message passing graph neural networks (MPNNs) cannot determine these quantities, we", + "type": "text" + } + ], + "index": 39, + "is_list_start_line": true + }, + { + "bbox": [ + 82, + 623, + 505, + 636 + ], + "spans": [ + { + "bbox": [ + 82, + 625, + 99, + 635 + ], + "score": 1.0, + "content": "1351", + "type": "text" + }, + { + "bbox": [ + 105, + 623, + 505, + 636 + ], + "score": 1.0, + "content": "use the fact that MPNNs cannot distinguish between two graphs that have the same number of nodes", + "type": "text" + } + ], + "index": 40, + "is_list_start_line": true + }, + { + "bbox": [ + 83, + 633, + 506, + 647 + ], + "spans": [ + { + "bbox": [ + 83, + 636, + 99, + 646 + ], + "score": 1.0, + "content": "1352", + "type": "text" + }, + { + "bbox": [ + 105, + 633, + 358, + 647 + ], + "score": 1.0, + "content": "and where each node (in both graphs) has the same degree. 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These lemmas generally only require basic", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 105, + 391, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 391, + 119 + ], + "score": 1.0, + "content": "tools to prove. Our first lemma is a crucial property of quotient spaces.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 105, + 119, + 505, + 155 + ], + "lines": [ + { + "bbox": [ + 105, + 119, + 506, + 133 + ], + "spans": [ + { + "bbox": [ + 105, + 119, + 268, + 133 + ], + "score": 1.0, + "content": "Lemma 1 (Passing to the quotient). Let", + "type": "text" + }, + { + "bbox": [ + 268, + 120, + 278, + 129 + ], + "score": 0.8, + "content": "\\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 278, + 119, + 296, + 133 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 297, + 120, + 306, + 131 + ], + "score": 0.76, + "content": "\\mathcal { V }", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 119, + 427, + 133 + ], + "score": 1.0, + "content": "be topological spaces, and let", + "type": "text" + }, + { + "bbox": [ + 427, + 120, + 449, + 131 + ], + "score": 0.91, + "content": "\\mathcal { X } / G", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 119, + 506, + 133 + ], + "score": 1.0, + "content": "be a quotient", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 130, + 505, + 144 + ], + "spans": [ + { + "bbox": [ + 105, + 130, + 266, + 144 + ], + "score": 1.0, + "content": "space, with corresponding quotient map", + "type": "text" + }, + { + "bbox": [ + 267, + 133, + 273, + 140 + ], + "score": 0.66, + "content": "\\pi", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 130, + 383, + 144 + ], + "score": 1.0, + "content": ". 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Let", + "type": "text" + }, + { + "bbox": [ + 492, + 403, + 506, + 414 + ], + "score": 0.79, + "content": "p :", + "type": "inline_equation" + } + ], + "index": 22 + }, + { + "bbox": [ + 83, + 412, + 506, + 428 + ], + "spans": [ + { + "bbox": [ + 83, + 416, + 99, + 426 + ], + "score": 1.0, + "content": "1383", + "type": "text" + }, + { + "bbox": [ + 106, + 413, + 170, + 424 + ], + "score": 0.91, + "content": "\\mathbb { R } ^ { n \\times d } \\to \\mathbb { R } ^ { n \\times n }", + "type": "inline_equation" + }, + { + "bbox": [ + 170, + 412, + 217, + 428 + ], + "score": 1.0, + "content": "be the map", + "type": "text" + }, + { + "bbox": [ + 217, + 414, + 275, + 426 + ], + "score": 0.92, + "content": "p ( V ) = V V ^ { \\top }", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 412, + 506, + 428 + ], + "score": 1.0, + "content": ". Then González and de Salas [2003] Lemma 11.13 shows", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 83, + 424, + 507, + 439 + ], + "spans": [ + { + "bbox": [ + 83, + 428, + 100, + 438 + ], + "score": 1.0, + "content": "1384", + "type": "text" + }, + { + "bbox": [ + 104, + 424, + 202, + 439 + ], + "score": 1.0, + "content": "that the quotient space", + "type": "text" + }, + { + "bbox": [ + 202, + 426, + 252, + 438 + ], + "score": 0.92, + "content": "\\mathbb { R } ^ { n \\times d } / O ( d )", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 424, + 404, + 439 + ], + "score": 1.0, + "content": "is homeomorphic to a closed subset", + "type": "text" + }, + { + "bbox": [ + 404, + 425, + 503, + 438 + ], + "score": 0.93, + "content": "p ( \\mathbb { R } ^ { n \\times d } ) = \\mathcal { Z } \\subseteq \\mathbb { R } ^ { n \\times n }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 424, + 507, + 439 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 82, + 437, + 506, + 450 + ], + "spans": [ + { + "bbox": [ + 82, + 439, + 100, + 449 + ], + "score": 1.0, + "content": "1385", + "type": "text" + }, + { + "bbox": [ + 105, + 437, + 122, + 450 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 122, + 438, + 129, + 449 + ], + "score": 0.84, + "content": "\\tilde { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 129, + 437, + 308, + 450 + ], + "score": 1.0, + "content": "refer to this homeomorphism, and note that", + "type": "text" + }, + { + "bbox": [ + 308, + 438, + 349, + 449 + ], + "score": 0.92, + "content": "{ \\tilde { p } } \\circ \\pi = p", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 437, + 506, + 450 + ], + "score": 1.0, + "content": "by passing to the quotient (Lemma 1).", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 82, + 447, + 504, + 463 + ], + "spans": [ + { + "bbox": [ + 82, + 451, + 100, + 462 + ], + "score": 1.0, + "content": "1386", + "type": "text" + }, + { + "bbox": [ + 105, + 447, + 198, + 463 + ], + "score": 1.0, + "content": "Then any continuous", + "type": "text" + }, + { + "bbox": [ + 198, + 449, + 219, + 461 + ], + "score": 0.89, + "content": "O ( d )", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 447, + 262, + 463 + ], + "score": 1.0, + "content": "invariant", + "type": "text" + }, + { + "bbox": [ + 262, + 450, + 270, + 461 + ], + "score": 0.84, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 447, + 402, + 463 + ], + "score": 1.0, + "content": "passes to a unique continuous", + "type": "text" + }, + { + "bbox": [ + 402, + 448, + 504, + 461 + ], + "score": 0.9, + "content": "\\widetilde { f } \\ : \\ \\mathbb { R } ^ { n \\times d } / O ( d ) \\ \\to \\ \\mathbb { R } ^ { s }", + "type": "inline_equation" + } + ], + "index": 26 + }, + { + "bbox": [ + 82, + 461, + 506, + 475 + ], + "spans": [ + { + "bbox": [ + 82, + 464, + 100, + 474 + ], + "score": 1.0, + "content": "1387", + "type": "text" + }, + { + "bbox": [ + 105, + 461, + 169, + 475 + ], + "score": 1.0, + "content": "(Lemma 1), so", + "type": "text" + }, + { + "bbox": [ + 169, + 461, + 212, + 474 + ], + "score": 0.93, + "content": "f = \\tilde { f } \\circ \\pi", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 461, + 241, + 475 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 241, + 464, + 249, + 473 + ], + "score": 0.75, + "content": "\\pi", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 461, + 364, + 475 + ], + "score": 1.0, + "content": "is the quotient map. Define", + "type": "text" + }, + { + "bbox": [ + 365, + 462, + 416, + 473 + ], + "score": 0.9, + "content": "h : { \\mathcal { Z } } \\to \\mathbb { R } ^ { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 461, + 431, + 475 + ], + "score": 1.0, + "content": "by", + "type": "text" + }, + { + "bbox": [ + 432, + 461, + 483, + 474 + ], + "score": 0.92, + "content": "h = \\tilde { f } \\circ \\tilde { p } ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 461, + 506, + 475 + ], + "score": 1.0, + "content": ", and", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 82, + 473, + 506, + 486 + ], + "spans": [ + { + "bbox": [ + 82, + 475, + 100, + 486 + ], + "score": 1.0, + "content": "1388", + "type": "text" + }, + { + "bbox": [ + 105, + 473, + 145, + 486 + ], + "score": 1.0, + "content": "note that", + "type": "text" + }, + { + "bbox": [ + 145, + 474, + 152, + 483 + ], + "score": 0.81, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 473, + 506, + 486 + ], + "score": 1.0, + "content": "is a composition of continuous functions and hence continuous. Finally, we have that", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 83, + 484, + 506, + 499 + ], + "spans": [ + { + "bbox": [ + 83, + 488, + 99, + 497 + ], + "score": 1.0, + "content": "1389", + "type": "text" + }, + { + "bbox": [ + 106, + 484, + 292, + 498 + ], + "score": 0.91, + "content": "\\begin{array} { r } { h ( V V ^ { \\top } ) = h ( \\tilde { p } \\circ \\pi ( V ) ) = \\tilde { f } \\circ \\pi ( V ) = f ( V ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 485, + 361, + 499 + ], + "score": 1.0, + "content": ", so we are done.", + "type": "text" + }, + { + "bbox": [ + 494, + 486, + 506, + 498 + ], + "score": 0.995, + "content": "□", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 91, + 508, + 505, + 532 + ], + "lines": [ + { + "bbox": [ + 89, + 507, + 506, + 522 + ], + "spans": [ + { + "bbox": [ + 89, + 510, + 100, + 520 + ], + "score": 1.0, + "content": "390", + "type": "text" + }, + { + "bbox": [ + 103, + 507, + 506, + 522 + ], + "score": 1.0, + "content": "The next lemma allows us to decompose a quotient of a product space into a product of smaller", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 88, + 520, + 172, + 532 + ], + "spans": [ + { + "bbox": [ + 88, + 520, + 172, + 532 + ], + "score": 1.0, + "content": "391 quotient spaces.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30.5 + }, + { + "type": "text", + "bbox": [ + 93, + 533, + 505, + 567 + ], + "lines": [ + { + "bbox": [ + 93, + 532, + 506, + 547 + ], + "spans": [ + { + "bbox": [ + 93, + 532, + 170, + 547 + ], + "score": 1.0, + "content": "92 Lemma 3. Let", + "type": "text" + }, + { + "bbox": [ + 170, + 533, + 217, + 545 + ], + "score": 0.92, + "content": "\\mathcal { X } _ { 1 } , \\ldots , \\mathcal { X } _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 532, + 326, + 547 + ], + "score": 1.0, + "content": "be topological spaces and", + "type": "text" + }, + { + "bbox": [ + 326, + 533, + 374, + 545 + ], + "score": 0.93, + "content": "G _ { 1 } , \\ldots , G _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 532, + 506, + 547 + ], + "score": 1.0, + "content": "be topological groups such that", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 93, + 544, + 506, + 557 + ], + "spans": [ + { + "bbox": [ + 93, + 544, + 127, + 557 + ], + "score": 1.0, + "content": "each 93", + "type": "text" + }, + { + "bbox": [ + 128, + 545, + 140, + 555 + ], + "score": 0.88, + "content": "G _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 544, + 225, + 557 + ], + "score": 1.0, + "content": "acts continuously on", + "type": "text" + }, + { + "bbox": [ + 225, + 545, + 236, + 555 + ], + "score": 0.88, + "content": "\\mathcal { X } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 544, + 357, + 557 + ], + "score": 1.0, + "content": ". Denote the quotient maps by", + "type": "text" + }, + { + "bbox": [ + 358, + 545, + 428, + 556 + ], + "score": 0.93, + "content": "\\pi _ { i } : \\mathcal { X } _ { i } \\mathcal { X } _ { i } / G _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 544, + 506, + 557 + ], + "score": 1.0, + "content": ". Then the quotient", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 91, + 555, + 298, + 568 + ], + "spans": [ + { + "bbox": [ + 91, + 555, + 298, + 568 + ], + "score": 1.0, + "content": "94 of the product is the product of the quotient, i.e.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33 + }, + { + "type": "interline_equation", + "bbox": [ + 169, + 568, + 439, + 582 + ], + "lines": [ + { + "bbox": [ + 169, + 568, + 439, + 582 + ], + "spans": [ + { + "bbox": [ + 169, + 568, + 439, + 582 + ], + "score": 0.73, + "content": "( { \\mathcal { X } } _ { 1 } \\times \\ldots \\times { \\mathcal { X } } _ { k } ) / ( G _ { 1 } \\times \\ldots \\times G _ { k } ) \\cong ( { \\mathcal { X } } _ { 1 } / G _ { 1 } ) \\times \\ldots \\times ( { \\mathcal { X } } _ { k } / G _ { k } ) ,", + "type": "interline_equation", + "image_path": "d676959cc7db9abfd041355e41887bce3911a4d1b84dd02fdd59dc4e9683bf19.jpg" + } + ] + } + ], + "index": 35, + "virtual_lines": [ + { + "bbox": [ + 169, + 568, + 439, + 582 + ], + "spans": [], + "index": 35 + } + ] + }, + { + "type": "text", + "bbox": [ + 86, + 583, + 438, + 597 + ], + "lines": [ + { + "bbox": [ + 82, + 582, + 432, + 598 + ], + "spans": [ + { + "bbox": [ + 82, + 582, + 161, + 598 + ], + "score": 1.0, + "content": "1395 and π1 × . . .", + "type": "text" + }, + { + "bbox": [ + 162, + 583, + 362, + 596 + ], + "score": 0.38, + "content": "\\times \\pi _ { k } : \\mathcal { X } _ { 1 } \\times . . . \\mathcal { X } _ { k } \\to ( \\mathcal { X } _ { 1 } / G _ { 1 } ) \\times . . . \\times ( \\mathcal { X } _ { k } / G _ { k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 582, + 432, + 598 + ], + "score": 1.0, + "content": ") is quotient map.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 84, + 606, + 505, + 652 + ], + "lines": [ + { + "bbox": [ + 83, + 605, + 506, + 621 + ], + "spans": [ + { + "bbox": [ + 83, + 609, + 99, + 618 + ], + "score": 1.0, + "content": "1396", + "type": "text" + }, + { + "bbox": [ + 104, + 605, + 217, + 621 + ], + "score": 1.0, + "content": "Proof. First, we show that", + "type": "text" + }, + { + "bbox": [ + 218, + 608, + 276, + 618 + ], + "score": 0.91, + "content": "\\pi _ { 1 } \\times \\ldots \\times \\pi _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 605, + 506, + 621 + ], + "score": 1.0, + "content": "is a quotient map. This is because 1. the quotient map", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 82, + 617, + 506, + 632 + ], + "spans": [ + { + "bbox": [ + 82, + 619, + 100, + 630 + ], + "score": 1.0, + "content": "1397", + "type": "text" + }, + { + "bbox": [ + 105, + 617, + 334, + 632 + ], + "score": 1.0, + "content": "of any continuous group action is an open map, so each", + "type": "text" + }, + { + "bbox": [ + 334, + 619, + 344, + 629 + ], + "score": 0.86, + "content": "\\pi _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 617, + 506, + 632 + ], + "score": 1.0, + "content": "is an open map, 2. the product of open", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 82, + 628, + 506, + 643 + ], + "spans": [ + { + "bbox": [ + 82, + 631, + 100, + 641 + ], + "score": 1.0, + "content": "1398", + "type": "text" + }, + { + "bbox": [ + 105, + 628, + 205, + 643 + ], + "score": 1.0, + "content": "maps is an open map, so", + "type": "text" + }, + { + "bbox": [ + 206, + 630, + 264, + 640 + ], + "score": 0.91, + "content": "\\pi _ { 1 } \\times \\ldots \\times \\pi _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 628, + 506, + 643 + ], + "score": 1.0, + "content": "is an open map and 3. a continuous surjective open map is a", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 83, + 639, + 452, + 654 + ], + "spans": [ + { + "bbox": [ + 83, + 642, + 99, + 651 + ], + "score": 1.0, + "content": "1399", + "type": "text" + }, + { + "bbox": [ + 105, + 639, + 175, + 654 + ], + "score": 1.0, + "content": "quotient map, so", + "type": "text" + }, + { + "bbox": [ + 175, + 641, + 233, + 651 + ], + "score": 0.91, + "content": "\\pi _ { 1 } \\times \\ldots \\times \\pi _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 639, + 452, + 654 + ], + "score": 1.0, + "content": ", which is continuous and surjective, is a quotient map.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 38.5 + }, + { + "type": "text", + "bbox": [ + 106, + 655, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "score": 1.0, + "content": "Now, we need only apply the theorem of uniqueness of quotient spaces to show (51) (see e.g. Lee", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 667, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 106, + 667, + 240, + 680 + ], + "score": 1.0, + "content": "[2013], Theorem A.31). Letting", + "type": "text" + }, + { + "bbox": [ + 240, + 667, + 475, + 678 + ], + "score": 0.85, + "content": "q : { \\mathcal { X } } _ { 1 } \\times \\ldots \\times { \\mathcal { X } } _ { k } \\to ( { \\mathcal { X } } _ { 1 } \\times \\ldots \\times { \\mathcal { X } } _ { k } ) / ( G _ { 1 } \\times \\ldots \\times G _ { k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 667, + 506, + 680 + ], + "score": 1.0, + "content": "denote", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 677, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 324, + 691 + ], + "score": 1.0, + "content": "the quotient map for this space, it is easily seen that", + "type": "text" + }, + { + "bbox": [ + 325, + 679, + 446, + 690 + ], + "score": 0.87, + "content": "q ( x _ { 1 } , \\ldots , x _ { k } ) = q ( y _ { 1 } \\ldots , y _ { k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 677, + 506, + 691 + ], + "score": 1.0, + "content": "if and only if", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 689, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 335, + 700 + ], + "score": 0.88, + "content": "\\pi _ { 1 } \\stackrel { \\textstyle \\ldots } { \\times } \\ldots \\times \\pi _ { k } ( \\bar { x _ { 1 } } , \\ldots , x _ { k } ) \\stackrel { \\textstyle \\ldots } { = } \\pi _ { 1 } \\times \\ldots \\times \\bar { \\pi } _ { k } ( y _ { 1 } , \\ldots , y _ { k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 689, + 506, + 701 + ], + "score": 1.0, + "content": ", since either of these is true if and only if", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 699, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 150, + 713 + ], + "score": 1.0, + "content": "there exist", + "type": "text" + }, + { + "bbox": [ + 150, + 701, + 183, + 712 + ], + "score": 0.91, + "content": "g _ { i } \\in G _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 699, + 222, + 713 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 223, + 702, + 262, + 712 + ], + "score": 0.9, + "content": "x _ { i } = g _ { i } y _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 699, + 299, + 713 + ], + "score": 1.0, + "content": "for each", + "type": "text" + }, + { + "bbox": [ + 299, + 702, + 303, + 710 + ], + "score": 0.75, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 699, + 506, + 713 + ], + "score": 1.0, + "content": ". 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Let", + "type": "text" + }, + { + "bbox": [ + 492, + 403, + 506, + 414 + ], + "score": 0.79, + "content": "p :", + "type": "inline_equation" + } + ], + "index": 22, + "is_list_start_line": true + }, + { + "bbox": [ + 83, + 412, + 506, + 428 + ], + "spans": [ + { + "bbox": [ + 83, + 416, + 99, + 426 + ], + "score": 1.0, + "content": "1383", + "type": "text" + }, + { + "bbox": [ + 106, + 413, + 170, + 424 + ], + "score": 0.91, + "content": "\\mathbb { R } ^ { n \\times d } \\to \\mathbb { R } ^ { n \\times n }", + "type": "inline_equation" + }, + { + "bbox": [ + 170, + 412, + 217, + 428 + ], + "score": 1.0, + "content": "be the map", + "type": "text" + }, + { + "bbox": [ + 217, + 414, + 275, + 426 + ], + "score": 0.92, + "content": "p ( V ) = V V ^ { \\top }", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 412, + 506, + 428 + ], + "score": 1.0, + "content": ". Then González and de Salas [2003] Lemma 11.13 shows", + "type": "text" + } + ], + "index": 23, + "is_list_start_line": true + }, + { + "bbox": [ + 83, + 424, + 507, + 439 + ], + "spans": [ + { + "bbox": [ + 83, + 428, + 100, + 438 + ], + "score": 1.0, + "content": "1384", + "type": "text" + }, + { + "bbox": [ + 104, + 424, + 202, + 439 + ], + "score": 1.0, + "content": "that the quotient space", + "type": "text" + }, + { + "bbox": [ + 202, + 426, + 252, + 438 + ], + "score": 0.92, + "content": "\\mathbb { R } ^ { n \\times d } / O ( d )", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 424, + 404, + 439 + ], + "score": 1.0, + "content": "is homeomorphic to a closed subset", + "type": "text" + }, + { + "bbox": [ + 404, + 425, + 503, + 438 + ], + "score": 0.93, + "content": "p ( \\mathbb { R } ^ { n \\times d } ) = \\mathcal { Z } \\subseteq \\mathbb { R } ^ { n \\times n }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 424, + 507, + 439 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 24, + "is_list_start_line": true + }, + { + "bbox": [ + 82, + 437, + 506, + 450 + ], + "spans": [ + { + "bbox": [ + 82, + 439, + 100, + 449 + ], + "score": 1.0, + "content": "1385", + "type": "text" + }, + { + "bbox": [ + 105, + 437, + 122, + 450 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 122, + 438, + 129, + 449 + ], + "score": 0.84, + "content": "\\tilde { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 129, + 437, + 308, + 450 + ], + "score": 1.0, + "content": "refer to this homeomorphism, and note that", + "type": "text" + }, + { + "bbox": [ + 308, + 438, + 349, + 449 + ], + "score": 0.92, + "content": "{ \\tilde { p } } \\circ \\pi = p", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 437, + 506, + 450 + ], + "score": 1.0, + "content": "by passing to the quotient (Lemma 1).", + "type": "text" + } + ], + "index": 25, + "is_list_start_line": true + }, + { + "bbox": [ + 82, + 447, + 504, + 463 + ], + "spans": [ + { + "bbox": [ + 82, + 451, + 100, + 462 + ], + "score": 1.0, + "content": "1386", + "type": "text" + }, + { + "bbox": [ + 105, + 447, + 198, + 463 + ], + "score": 1.0, + "content": "Then any continuous", + "type": "text" + }, + { + "bbox": [ + 198, + 449, + 219, + 461 + ], + "score": 0.89, + "content": "O ( d )", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 447, + 262, + 463 + ], + "score": 1.0, + "content": "invariant", + "type": "text" + }, + { + "bbox": [ + 262, + 450, + 270, + 461 + ], + "score": 0.84, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 447, + 402, + 463 + ], + "score": 1.0, + "content": "passes to a unique continuous", + "type": "text" + }, + { + "bbox": [ + 402, + 448, + 504, + 461 + ], + "score": 0.9, + "content": "\\widetilde { f } \\ : \\ \\mathbb { R } ^ { n \\times d } / O ( d ) \\ \\to \\ \\mathbb { R } ^ { s }", + "type": "inline_equation" + } + ], + "index": 26, + "is_list_start_line": true + }, + { + "bbox": [ + 82, + 461, + 506, + 475 + ], + "spans": [ + { + "bbox": [ + 82, + 464, + 100, + 474 + ], + "score": 1.0, + "content": "1387", + "type": "text" + }, + { + "bbox": [ + 105, + 461, + 169, + 475 + ], + "score": 1.0, + "content": "(Lemma 1), so", + "type": "text" + }, + { + "bbox": [ + 169, + 461, + 212, + 474 + ], + "score": 0.93, + "content": "f = \\tilde { f } \\circ \\pi", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 461, + 241, + 475 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 241, + 464, + 249, + 473 + ], + "score": 0.75, + "content": "\\pi", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 461, + 364, + 475 + ], + "score": 1.0, + "content": "is the quotient map. Define", + "type": "text" + }, + { + "bbox": [ + 365, + 462, + 416, + 473 + ], + "score": 0.9, + "content": "h : { \\mathcal { Z } } \\to \\mathbb { R } ^ { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 461, + 431, + 475 + ], + "score": 1.0, + "content": "by", + "type": "text" + }, + { + "bbox": [ + 432, + 461, + 483, + 474 + ], + "score": 0.92, + "content": "h = \\tilde { f } \\circ \\tilde { p } ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 461, + 506, + 475 + ], + "score": 1.0, + "content": ", and", + "type": "text" + } + ], + "index": 27, + "is_list_start_line": true + }, + { + "bbox": [ + 82, + 473, + 506, + 486 + ], + "spans": [ + { + "bbox": [ + 82, + 475, + 100, + 486 + ], + "score": 1.0, + "content": "1388", + "type": "text" + }, + { + "bbox": [ + 105, + 473, + 145, + 486 + ], + "score": 1.0, + "content": "note that", + "type": "text" + }, + { + "bbox": [ + 145, + 474, + 152, + 483 + ], + "score": 0.81, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 473, + 506, + 486 + ], + "score": 1.0, + "content": "is a composition of continuous functions and hence continuous. Finally, we have that", + "type": "text" + } + ], + "index": 28, + "is_list_start_line": true + }, + { + "bbox": [ + 83, + 484, + 506, + 499 + ], + "spans": [ + { + "bbox": [ + 83, + 488, + 99, + 497 + ], + "score": 1.0, + "content": "1389", + "type": "text" + }, + { + "bbox": [ + 106, + 484, + 292, + 498 + ], + "score": 0.91, + "content": "\\begin{array} { r } { h ( V V ^ { \\top } ) = h ( \\tilde { p } \\circ \\pi ( V ) ) = \\tilde { f } \\circ \\pi ( V ) = f ( V ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 485, + 361, + 499 + ], + "score": 1.0, + "content": ", so we are done.", + "type": "text" + }, + { + "bbox": [ + 494, + 486, + 506, + 498 + ], + "score": 0.995, + "content": "□", + "type": "text" + } + ], + "index": 29, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 507, + 506, + 522 + ], + "spans": [ + { + "bbox": [ + 89, + 510, + 100, + 520 + ], + "score": 1.0, + "content": "390", + "type": "text" + }, + { + "bbox": [ + 103, + 507, + 506, + 522 + ], + "score": 1.0, + "content": "The next lemma allows us to decompose a quotient of a product space into a product of smaller", + "type": "text" + } + ], + "index": 30, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 520, + 172, + 532 + ], + "spans": [ + { + "bbox": [ + 88, + 520, + 172, + 532 + ], + "score": 1.0, + "content": "391 quotient spaces.", + "type": "text" + } + ], + "index": 31, + "is_list_start_line": true + } + ], + "index": 25, + "bbox_fs": [ + 82, + 389, + 507, + 499 + ] + }, + { + "type": "index", + "bbox": [ + 91, + 508, + 505, + 532 + ], + "lines": [], + "index": 30.5, + "bbox_fs": [ + 88, + 507, + 506, + 532 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 93, + 533, + 505, + 567 + ], + "lines": [ + { + "bbox": [ + 93, + 532, + 506, + 547 + ], + "spans": [ + { + "bbox": [ + 93, + 532, + 170, + 547 + ], + "score": 1.0, + "content": "92 Lemma 3. Let", + "type": "text" + }, + { + "bbox": [ + 170, + 533, + 217, + 545 + ], + "score": 0.92, + "content": "\\mathcal { X } _ { 1 } , \\ldots , \\mathcal { X } _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 532, + 326, + 547 + ], + "score": 1.0, + "content": "be topological spaces and", + "type": "text" + }, + { + "bbox": [ + 326, + 533, + 374, + 545 + ], + "score": 0.93, + "content": "G _ { 1 } , \\ldots , G _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 532, + 506, + 547 + ], + "score": 1.0, + "content": "be topological groups such that", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 93, + 544, + 506, + 557 + ], + "spans": [ + { + "bbox": [ + 93, + 544, + 127, + 557 + ], + "score": 1.0, + "content": "each 93", + "type": "text" + }, + { + "bbox": [ + 128, + 545, + 140, + 555 + ], + "score": 0.88, + "content": "G _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 544, + 225, + 557 + ], + "score": 1.0, + "content": "acts continuously on", + "type": "text" + }, + { + "bbox": [ + 225, + 545, + 236, + 555 + ], + "score": 0.88, + "content": "\\mathcal { X } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 544, + 357, + 557 + ], + "score": 1.0, + "content": ". Denote the quotient maps by", + "type": "text" + }, + { + "bbox": [ + 358, + 545, + 428, + 556 + ], + "score": 0.93, + "content": "\\pi _ { i } : \\mathcal { X } _ { i } \\mathcal { X } _ { i } / G _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 544, + 506, + 557 + ], + "score": 1.0, + "content": ". Then the quotient", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 91, + 555, + 298, + 568 + ], + "spans": [ + { + "bbox": [ + 91, + 555, + 298, + 568 + ], + "score": 1.0, + "content": "94 of the product is the product of the quotient, i.e.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33, + "bbox_fs": [ + 91, + 532, + 506, + 568 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 169, + 568, + 439, + 582 + ], + "lines": [ + { + "bbox": [ + 169, + 568, + 439, + 582 + ], + "spans": [ + { + "bbox": [ + 169, + 568, + 439, + 582 + ], + "score": 0.73, + "content": "( { \\mathcal { X } } _ { 1 } \\times \\ldots \\times { \\mathcal { X } } _ { k } ) / ( G _ { 1 } \\times \\ldots \\times G _ { k } ) \\cong ( { \\mathcal { X } } _ { 1 } / G _ { 1 } ) \\times \\ldots \\times ( { \\mathcal { X } } _ { k } / G _ { k } ) ,", + "type": "interline_equation", + "image_path": "d676959cc7db9abfd041355e41887bce3911a4d1b84dd02fdd59dc4e9683bf19.jpg" + } + ] + } + ], + "index": 35, + "virtual_lines": [ + { + "bbox": [ + 169, + 568, + 439, + 582 + ], + "spans": [], + "index": 35 + } + ] + }, + { + "type": "text", + "bbox": [ + 86, + 583, + 438, + 597 + ], + "lines": [ + { + "bbox": [ + 82, + 582, + 432, + 598 + ], + "spans": [ + { + "bbox": [ + 82, + 582, + 161, + 598 + ], + "score": 1.0, + "content": "1395 and π1 × . . .", + "type": "text" + }, + { + "bbox": [ + 162, + 583, + 362, + 596 + ], + "score": 0.38, + "content": "\\times \\pi _ { k } : \\mathcal { X } _ { 1 } \\times . . . \\mathcal { X } _ { k } \\to ( \\mathcal { X } _ { 1 } / G _ { 1 } ) \\times . . . \\times ( \\mathcal { X } _ { k } / G _ { k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 582, + 432, + 598 + ], + "score": 1.0, + "content": ") is quotient map.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36, + "bbox_fs": [ + 82, + 582, + 432, + 598 + ] + }, + { + "type": "index", + "bbox": [ + 84, + 606, + 505, + 652 + ], + "lines": [ + { + "bbox": [ + 83, + 605, + 506, + 621 + ], + "spans": [ + { + "bbox": [ + 83, + 609, + 99, + 618 + ], + "score": 1.0, + "content": "1396", + "type": "text" + }, + { + "bbox": [ + 104, + 605, + 217, + 621 + ], + "score": 1.0, + "content": "Proof. First, we show that", + "type": "text" + }, + { + "bbox": [ + 218, + 608, + 276, + 618 + ], + "score": 0.91, + "content": "\\pi _ { 1 } \\times \\ldots \\times \\pi _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 605, + 506, + 621 + ], + "score": 1.0, + "content": "is a quotient map. This is because 1. the quotient map", + "type": "text" + } + ], + "index": 37, + "is_list_start_line": true + }, + { + "bbox": [ + 82, + 617, + 506, + 632 + ], + "spans": [ + { + "bbox": [ + 82, + 619, + 100, + 630 + ], + "score": 1.0, + "content": "1397", + "type": "text" + }, + { + "bbox": [ + 105, + 617, + 334, + 632 + ], + "score": 1.0, + "content": "of any continuous group action is an open map, so each", + "type": "text" + }, + { + "bbox": [ + 334, + 619, + 344, + 629 + ], + "score": 0.86, + "content": "\\pi _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 617, + 506, + 632 + ], + "score": 1.0, + "content": "is an open map, 2. the product of open", + "type": "text" + } + ], + "index": 38, + "is_list_start_line": true + }, + { + "bbox": [ + 82, + 628, + 506, + 643 + ], + "spans": [ + { + "bbox": [ + 82, + 631, + 100, + 641 + ], + "score": 1.0, + "content": "1398", + "type": "text" + }, + { + "bbox": [ + 105, + 628, + 205, + 643 + ], + "score": 1.0, + "content": "maps is an open map, so", + "type": "text" + }, + { + "bbox": [ + 206, + 630, + 264, + 640 + ], + "score": 0.91, + "content": "\\pi _ { 1 } \\times \\ldots \\times \\pi _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 628, + 506, + 643 + ], + "score": 1.0, + "content": "is an open map and 3. a continuous surjective open map is a", + "type": "text" + } + ], + "index": 39, + "is_list_start_line": true + }, + { + "bbox": [ + 83, + 639, + 452, + 654 + ], + "spans": [ + { + "bbox": [ + 83, + 642, + 99, + 651 + ], + "score": 1.0, + "content": "1399", + "type": "text" + }, + { + "bbox": [ + 105, + 639, + 175, + 654 + ], + "score": 1.0, + "content": "quotient map, so", + "type": "text" + }, + { + "bbox": [ + 175, + 641, + 233, + 651 + ], + "score": 0.91, + "content": "\\pi _ { 1 } \\times \\ldots \\times \\pi _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 639, + 452, + 654 + ], + "score": 1.0, + "content": ", which is continuous and surjective, is a quotient map.", + "type": "text" + } + ], + "index": 40, + "is_list_start_line": true + } + ], + "index": 38.5, + "bbox_fs": [ + 82, + 605, + 506, + 654 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 655, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "score": 1.0, + "content": "Now, we need only apply the theorem of uniqueness of quotient spaces to show (51) (see e.g. Lee", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 667, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 106, + 667, + 240, + 680 + ], + "score": 1.0, + "content": "[2013], Theorem A.31). Letting", + "type": "text" + }, + { + "bbox": [ + 240, + 667, + 475, + 678 + ], + "score": 0.85, + "content": "q : { \\mathcal { X } } _ { 1 } \\times \\ldots \\times { \\mathcal { X } } _ { k } \\to ( { \\mathcal { X } } _ { 1 } \\times \\ldots \\times { \\mathcal { X } } _ { k } ) / ( G _ { 1 } \\times \\ldots \\times G _ { k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 667, + 506, + 680 + ], + "score": 1.0, + "content": "denote", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 677, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 324, + 691 + ], + "score": 1.0, + "content": "the quotient map for this space, it is easily seen that", + "type": "text" + }, + { + "bbox": [ + 325, + 679, + 446, + 690 + ], + "score": 0.87, + "content": "q ( x _ { 1 } , \\ldots , x _ { k } ) = q ( y _ { 1 } \\ldots , y _ { k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 677, + 506, + 691 + ], + "score": 1.0, + "content": "if and only if", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 689, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 335, + 700 + ], + "score": 0.88, + "content": "\\pi _ { 1 } \\stackrel { \\textstyle \\ldots } { \\times } \\ldots \\times \\pi _ { k } ( \\bar { x _ { 1 } } , \\ldots , x _ { k } ) \\stackrel { \\textstyle \\ldots } { = } \\pi _ { 1 } \\times \\ldots \\times \\bar { \\pi } _ { k } ( y _ { 1 } , \\ldots , y _ { k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 689, + 506, + 701 + ], + "score": 1.0, + "content": ", since either of these is true if and only if", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 699, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 150, + 713 + ], + "score": 1.0, + "content": "there exist", + "type": "text" + }, + { + "bbox": [ + 150, + 701, + 183, + 712 + ], + "score": 0.91, + "content": "g _ { i } \\in G _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 699, + 222, + 713 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 223, + 702, + 262, + 712 + ], + "score": 0.9, + "content": "x _ { i } = g _ { i } y _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 699, + 299, + 713 + ], + "score": 1.0, + "content": "for each", + "type": "text" + }, + { + "bbox": [ + 299, + 702, + 303, + 710 + ], + "score": 0.75, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 699, + 506, + 713 + ], + "score": 1.0, + "content": ". Thus, we have an isomorphism of these quotient", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 104, + 711, + 505, + 724 + ], + "spans": [ + { + "bbox": [ + 104, + 711, + 138, + 724 + ], + "score": 1.0, + "content": "spaces.", + "type": "text" + }, + { + "bbox": [ + 494, + 711, + 505, + 722 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 43.5, + "bbox_fs": [ + 104, + 655, + 506, + 724 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 92, + 72, + 505, + 95 + ], + "lines": [ + { + "bbox": [ + 89, + 72, + 506, + 86 + ], + "spans": [ + { + "bbox": [ + 89, + 75, + 100, + 84 + ], + "score": 1.0, + "content": "406", + "type": "text" + }, + { + "bbox": [ + 104, + 72, + 506, + 86 + ], + "score": 1.0, + "content": "The following lemma shows that quotients of compact spaces are also compact, which is useful for", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 88, + 84, + 284, + 97 + ], + "spans": [ + { + "bbox": [ + 88, + 84, + 284, + 97 + ], + "score": 1.0, + "content": "407 universal approximation on quotient spaces.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 92, + 99, + 506, + 123 + ], + "lines": [ + { + "bbox": [ + 90, + 98, + 505, + 113 + ], + "spans": [ + { + "bbox": [ + 90, + 98, + 366, + 113 + ], + "score": 1.0, + "content": "08 Lemma 4 (Compactness of quotients of compact spaces). Let", + "type": "text" + }, + { + "bbox": [ + 366, + 100, + 376, + 109 + ], + "score": 0.72, + "content": "\\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 377, + 98, + 505, + 113 + ], + "score": 1.0, + "content": "be a compact space. Then the", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 88, + 110, + 238, + 123 + ], + "spans": [ + { + "bbox": [ + 88, + 110, + 167, + 123 + ], + "score": 1.0, + "content": "409 quotient space", + "type": "text" + }, + { + "bbox": [ + 167, + 110, + 190, + 123 + ], + "score": 0.9, + "content": "\\mathcal { X } / G", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 110, + 238, + 123 + ], + "score": 1.0, + "content": "is compact.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 83, + 140, + 506, + 186 + ], + "lines": [ + { + "bbox": [ + 83, + 140, + 506, + 155 + ], + "spans": [ + { + "bbox": [ + 83, + 142, + 100, + 153 + ], + "score": 1.0, + "content": "1410", + "type": "text" + }, + { + "bbox": [ + 105, + 140, + 259, + 155 + ], + "score": 1.0, + "content": "Proof. Denoting the quotient map by", + "type": "text" + }, + { + "bbox": [ + 259, + 141, + 321, + 153 + ], + "score": 0.92, + "content": "\\pi : { \\mathcal { X } } \\to { \\mathcal { X } } / G", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 140, + 367, + 155 + ], + "score": 1.0, + "content": "and letting", + "type": "text" + }, + { + "bbox": [ + 368, + 141, + 397, + 153 + ], + "score": 0.92, + "content": "\\{ U _ { \\alpha } \\} _ { \\alpha }", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 140, + 480, + 155 + ], + "score": 1.0, + "content": "be an open cover of", + "type": "text" + }, + { + "bbox": [ + 480, + 141, + 502, + 153 + ], + "score": 0.9, + "content": "\\mathcal { X } / G", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 140, + 506, + 155 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 82, + 150, + 506, + 166 + ], + "spans": [ + { + "bbox": [ + 82, + 153, + 99, + 164 + ], + "score": 1.0, + "content": "1411", + "type": "text" + }, + { + "bbox": [ + 104, + 150, + 161, + 166 + ], + "score": 1.0, + "content": "we have that", + "type": "text" + }, + { + "bbox": [ + 162, + 151, + 215, + 163 + ], + "score": 0.84, + "content": "\\{ \\check { \\pi } ^ { - 1 } ( U _ { \\alpha } ) \\} _ { \\alpha }", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 150, + 299, + 166 + ], + "score": 1.0, + "content": "is an open cover of", + "type": "text" + }, + { + "bbox": [ + 299, + 153, + 309, + 162 + ], + "score": 0.83, + "content": "\\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 150, + 396, + 166 + ], + "score": 1.0, + "content": ". By compactness of", + "type": "text" + }, + { + "bbox": [ + 397, + 153, + 406, + 162 + ], + "score": 0.83, + "content": "\\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 150, + 506, + 166 + ], + "score": 1.0, + "content": ", we can choose a finite", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 83, + 160, + 507, + 178 + ], + "spans": [ + { + "bbox": [ + 83, + 164, + 100, + 174 + ], + "score": 1.0, + "content": "1412", + "type": "text" + }, + { + "bbox": [ + 104, + 160, + 146, + 178 + ], + "score": 1.0, + "content": "subcover", + "type": "text" + }, + { + "bbox": [ + 146, + 163, + 227, + 175 + ], + "score": 0.86, + "content": "\\{ \\pi ^ { - 1 } ( U _ { \\alpha _ { i } } ) \\} _ { i = 1 , \\dots , n }", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 160, + 257, + 178 + ], + "score": 1.0, + "content": ". Then", + "type": "text" + }, + { + "bbox": [ + 258, + 163, + 423, + 175 + ], + "score": 0.92, + "content": "\\{ \\pi ( \\pi ^ { - 1 } ( U _ { \\alpha _ { i } } ) ) \\} _ { i = 1 , \\dots , n } = \\{ U _ { \\alpha _ { i } } \\} _ { i = 1 , \\dots , n }", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 160, + 507, + 178 + ], + "score": 1.0, + "content": "by surjectivity, and", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 83, + 172, + 505, + 188 + ], + "spans": [ + { + "bbox": [ + 83, + 175, + 100, + 186 + ], + "score": 1.0, + "content": "1413", + "type": "text" + }, + { + "bbox": [ + 107, + 174, + 163, + 186 + ], + "score": 0.91, + "content": "\\{ U _ { \\alpha _ { i } } \\} _ { i = 1 , \\dots , n }", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 172, + 262, + 188 + ], + "score": 1.0, + "content": "is thus an open cover of", + "type": "text" + }, + { + "bbox": [ + 262, + 174, + 284, + 186 + ], + "score": 0.9, + "content": "\\mathcal { X } / G", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 172, + 289, + 188 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 495, + 174, + 505, + 185 + ], + "score": 0.998, + "content": "□", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5.5 + }, + { + "type": "text", + "bbox": [ + 84, + 204, + 506, + 249 + ], + "lines": [ + { + "bbox": [ + 83, + 204, + 506, + 218 + ], + "spans": [ + { + "bbox": [ + 83, + 206, + 99, + 216 + ], + "score": 1.0, + "content": "1414", + "type": "text" + }, + { + "bbox": [ + 105, + 204, + 506, + 218 + ], + "score": 1.0, + "content": "The Whitney embedding theorem gives a nice condition that we apply to show that the quotient", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 82, + 215, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 82, + 217, + 100, + 227 + ], + "score": 1.0, + "content": "1415", + "type": "text" + }, + { + "bbox": [ + 105, + 216, + 136, + 227 + ], + "score": 1.0, + "content": "spaces", + "type": "text" + }, + { + "bbox": [ + 136, + 215, + 159, + 227 + ], + "score": 0.91, + "content": "\\chi / \\bar { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 216, + 430, + 227 + ], + "score": 1.0, + "content": "that we deal with embed into Euclidean space. It says that when", + "type": "text" + }, + { + "bbox": [ + 431, + 215, + 453, + 227 + ], + "score": 0.93, + "content": "\\mathcal { X } / G", + "type": "inline_equation" + }, + { + "bbox": [ + 454, + 216, + 505, + 227 + ], + "score": 1.0, + "content": "is a smooth", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 82, + 226, + 506, + 238 + ], + "spans": [ + { + "bbox": [ + 82, + 227, + 100, + 238 + ], + "score": 1.0, + "content": "1416", + "type": "text" + }, + { + "bbox": [ + 105, + 226, + 506, + 237 + ], + "score": 1.0, + "content": "manifold, then it can be embedded into a Euclidean space of double the dimension of the manifold.", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 83, + 237, + 282, + 250 + ], + "spans": [ + { + "bbox": [ + 83, + 239, + 99, + 248 + ], + "score": 1.0, + "content": "1417", + "type": "text" + }, + { + "bbox": [ + 105, + 237, + 282, + 250 + ], + "score": 1.0, + "content": "The proof is outside the scope of this paper.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 9.5 + }, + { + "type": "text", + "bbox": [ + 95, + 253, + 504, + 275 + ], + "lines": [ + { + "bbox": [ + 92, + 253, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 92, + 253, + 438, + 266 + ], + "score": 1.0, + "content": "18 Lemma 5 (Whitney Embedding Theorem [Whitney, 1944]). Every smooth manifold", + "type": "text" + }, + { + "bbox": [ + 439, + 254, + 452, + 263 + ], + "score": 0.61, + "content": "\\mathcal { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 253, + 505, + 266 + ], + "score": 1.0, + "content": "of dimension", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 90, + 263, + 274, + 277 + ], + "spans": [ + { + "bbox": [ + 90, + 263, + 274, + 277 + ], + "score": 1.0, + "content": "n > 0 can be smoothly embedded in R2n 19 .", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12.5 + }, + { + "type": "text", + "bbox": [ + 83, + 285, + 505, + 341 + ], + "lines": [ + { + "bbox": [ + 83, + 285, + 506, + 299 + ], + "spans": [ + { + "bbox": [ + 83, + 288, + 99, + 296 + ], + "score": 1.0, + "content": "1420", + "type": "text" + }, + { + "bbox": [ + 105, + 285, + 506, + 299 + ], + "score": 1.0, + "content": "Finally, we give a lemma that helps prove universal approximation results. It says that if functions", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 83, + 297, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 83, + 298, + 99, + 308 + ], + "score": 1.0, + "content": "1421", + "type": "text" + }, + { + "bbox": [ + 106, + 297, + 114, + 308 + ], + "score": 0.83, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 297, + 365, + 309 + ], + "score": 1.0, + "content": "that we want to approximate can be written as compositions", + "type": "text" + }, + { + "bbox": [ + 365, + 297, + 438, + 308 + ], + "score": 0.93, + "content": "f = f _ { L } \\circ \\dots \\circ f _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 297, + 505, + 309 + ], + "score": 1.0, + "content": ", then it suffices", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 83, + 307, + 505, + 320 + ], + "spans": [ + { + "bbox": [ + 83, + 308, + 100, + 319 + ], + "score": 1.0, + "content": "1422", + "type": "text" + }, + { + "bbox": [ + 105, + 307, + 237, + 320 + ], + "score": 1.0, + "content": "to universally approximate each", + "type": "text" + }, + { + "bbox": [ + 238, + 308, + 248, + 319 + ], + "score": 0.87, + "content": "f _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 307, + 473, + 320 + ], + "score": 1.0, + "content": "and compose the results to universally approximate the", + "type": "text" + }, + { + "bbox": [ + 473, + 308, + 480, + 319 + ], + "score": 0.83, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 307, + 505, + 320 + ], + "score": 1.0, + "content": ". This", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 83, + 318, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 83, + 320, + 99, + 330 + ], + "score": 1.0, + "content": "1423", + "type": "text" + }, + { + "bbox": [ + 106, + 318, + 505, + 331 + ], + "score": 1.0, + "content": "is especially useful for proving universality of neural networks, as we may use some layers to", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 83, + 329, + 448, + 342 + ], + "spans": [ + { + "bbox": [ + 83, + 331, + 100, + 341 + ], + "score": 1.0, + "content": "1424", + "type": "text" + }, + { + "bbox": [ + 105, + 329, + 179, + 342 + ], + "score": 1.0, + "content": "approximate each", + "type": "text" + }, + { + "bbox": [ + 180, + 330, + 189, + 341 + ], + "score": 0.85, + "content": "f _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 329, + 437, + 342 + ], + "score": 1.0, + "content": ", then compose these layers to approximate the target function", + "type": "text" + }, + { + "bbox": [ + 438, + 330, + 444, + 341 + ], + "score": 0.84, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 445, + 329, + 448, + 342 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 84, + 344, + 505, + 390 + ], + "lines": [ + { + "bbox": [ + 83, + 343, + 506, + 359 + ], + "spans": [ + { + "bbox": [ + 83, + 347, + 99, + 357 + ], + "score": 1.0, + "content": "1425", + "type": "text" + }, + { + "bbox": [ + 104, + 343, + 359, + 359 + ], + "score": 1.0, + "content": "Lemma 6 (Layer-wise universality implies universality). Let", + "type": "text" + }, + { + "bbox": [ + 359, + 344, + 397, + 356 + ], + "score": 0.91, + "content": "\\mathcal { Z } \\subseteq \\mathbb { R } ^ { d _ { 0 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 343, + 506, + 359 + ], + "score": 1.0, + "content": "be a compact domain, let", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 83, + 355, + 503, + 369 + ], + "spans": [ + { + "bbox": [ + 83, + 358, + 100, + 367 + ], + "score": 1.0, + "content": "1426", + "type": "text" + }, + { + "bbox": [ + 107, + 357, + 154, + 367 + ], + "score": 0.9, + "content": "\\mathcal { F } _ { 1 } , \\ldots , \\mathcal { F } _ { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 355, + 328, + 369 + ], + "score": 1.0, + "content": "be families of continuous functions where", + "type": "text" + }, + { + "bbox": [ + 328, + 357, + 339, + 367 + ], + "score": 0.88, + "content": "{ \\mathcal { F } } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 355, + 448, + 369 + ], + "score": 1.0, + "content": "consists of functions from", + "type": "text" + }, + { + "bbox": [ + 448, + 356, + 503, + 366 + ], + "score": 0.91, + "content": "\\mathbb { R } ^ { d _ { i - 1 } } \\mathbb { R } ^ { d _ { i } }", + "type": "inline_equation" + } + ], + "index": 20 + }, + { + "bbox": [ + 82, + 365, + 506, + 380 + ], + "spans": [ + { + "bbox": [ + 82, + 368, + 100, + 379 + ], + "score": 1.0, + "content": "1427", + "type": "text" + }, + { + "bbox": [ + 104, + 365, + 145, + 380 + ], + "score": 1.0, + "content": "for some", + "type": "text" + }, + { + "bbox": [ + 145, + 367, + 188, + 378 + ], + "score": 0.91, + "content": "d _ { 1 } , \\ldots , d _ { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 365, + 210, + 380 + ], + "score": 1.0, + "content": ". Let", + "type": "text" + }, + { + "bbox": [ + 210, + 367, + 220, + 377 + ], + "score": 0.81, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 365, + 327, + 380 + ], + "score": 1.0, + "content": "be the family of functions", + "type": "text" + }, + { + "bbox": [ + 328, + 367, + 469, + 379 + ], + "score": 0.9, + "content": "\\{ f _ { L } \\circ . . . f _ { 1 } : \\mathcal { Z } \\mathbb { R } ^ { d _ { L } } , f _ { i } \\in \\mathcal { F } _ { i } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 469, + 365, + 506, + 380 + ], + "score": 1.0, + "content": "that are", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 83, + 378, + 246, + 391 + ], + "spans": [ + { + "bbox": [ + 83, + 380, + 99, + 389 + ], + "score": 1.0, + "content": "1428", + "type": "text" + }, + { + "bbox": [ + 105, + 378, + 211, + 391 + ], + "score": 1.0, + "content": "compositions of functions", + "type": "text" + }, + { + "bbox": [ + 211, + 378, + 243, + 390 + ], + "score": 0.91, + "content": "f _ { i } \\in \\mathcal { F } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 378, + 246, + 391 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20.5 + }, + { + "type": "text", + "bbox": [ + 88, + 394, + 506, + 418 + ], + "lines": [ + { + "bbox": [ + 84, + 393, + 506, + 407 + ], + "spans": [ + { + "bbox": [ + 84, + 393, + 144, + 407 + ], + "score": 1.0, + "content": "For each 1429", + "type": "text" + }, + { + "bbox": [ + 145, + 396, + 149, + 404 + ], + "score": 0.53, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 393, + 166, + 407 + ], + "score": 1.0, + "content": ", let", + "type": "text" + }, + { + "bbox": [ + 167, + 395, + 178, + 405 + ], + "score": 0.86, + "content": "\\Phi _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 393, + 450, + 407 + ], + "score": 1.0, + "content": "be a family of continuous functions that universally approximates", + "type": "text" + }, + { + "bbox": [ + 450, + 395, + 461, + 405 + ], + "score": 0.87, + "content": "{ \\mathcal { F } } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 393, + 506, + 407 + ], + "score": 1.0, + "content": ". Then the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 85, + 405, + 442, + 418 + ], + "spans": [ + { + "bbox": [ + 85, + 405, + 199, + 418 + ], + "score": 1.0, + "content": "1430 family of compositions", + "type": "text" + }, + { + "bbox": [ + 200, + 405, + 324, + 417 + ], + "score": 0.92, + "content": "\\Phi = \\left\\{ \\phi _ { L } \\circ . . . \\circ \\phi _ { 1 } : \\phi _ { i } \\in \\Phi _ { i } \\right\\}", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 405, + 428, + 418 + ], + "score": 1.0, + "content": "universally approximates", + "type": "text" + }, + { + "bbox": [ + 429, + 406, + 437, + 415 + ], + "score": 0.82, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 405, + 442, + 418 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5 + }, + { + "type": "text", + "bbox": [ + 83, + 435, + 506, + 475 + ], + "lines": [ + { + "bbox": [ + 81, + 435, + 506, + 451 + ], + "spans": [ + { + "bbox": [ + 81, + 435, + 151, + 451 + ], + "score": 1.0, + "content": "Proof. Let 1431", + "type": "text" + }, + { + "bbox": [ + 152, + 437, + 242, + 449 + ], + "score": 0.92, + "content": "f = f _ { L } \\circ . . . \\circ f _ { 1 } \\in \\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 435, + 262, + 451 + ], + "score": 1.0, + "content": ". 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Then each", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 83, + 448, + 506, + 462 + ], + "spans": [ + { + "bbox": [ + 83, + 451, + 100, + 461 + ], + "score": 1.0, + "content": "1432", + "type": "text" + }, + { + "bbox": [ + 106, + 448, + 118, + 461 + ], + "score": 0.89, + "content": "\\mathcal { \\tilde { Z } } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 118, + 450, + 241, + 462 + ], + "score": 1.0, + "content": "is compact by continuity of the", + "type": "text" + }, + { + "bbox": [ + 241, + 451, + 251, + 461 + ], + "score": 0.86, + "content": "f _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 450, + 270, + 462 + ], + "score": 1.0, + "content": ". 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Thus, we", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 82, + 623, + 351, + 636 + ], + "spans": [ + { + "bbox": [ + 82, + 623, + 153, + 636 + ], + "score": 1.0, + "content": "may define 1443", + "type": "text" + }, + { + "bbox": [ + 153, + 624, + 275, + 636 + ], + "score": 0.87, + "content": "\\phi = \\phi _ { L } \\circ . . . \\circ \\phi _ { 1 } : \\mathcal { Z } \\mathbb { R } ^ { d _ { L } }", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 623, + 351, + 636 + ], + "score": 1.0, + "content": ", and compute that", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37 + }, + { + "type": "interline_equation", + "bbox": [ + 151, + 642, + 459, + 672 + ], + "lines": [ + { + "bbox": [ + 151, + 642, + 459, + 672 + ], + "spans": [ + { + "bbox": [ + 151, + 642, + 459, + 672 + ], + "score": 0.89, + "content": "\\begin{array} { r l } & { \\| \\phi - f \\| _ { \\infty } \\leq \\| \\phi - \\phi _ { L } \\circ f _ { L - 1 } \\circ . . . \\circ f _ { 1 } \\| _ { \\infty } + \\| \\phi _ { L } \\circ f _ { L - 1 } \\circ . . . \\circ f _ { 1 } - f \\| _ { \\infty } } \\\\ & { \\qquad < \\| \\phi - \\phi _ { L } \\circ f _ { L - 1 } \\circ . . . \\circ f _ { 1 } \\| _ { \\infty } + \\epsilon / 2 , } \\end{array}", + "type": "interline_equation", + "image_path": "b7f0b67eb993843680cd8b3d392aa29e3f46e2ac017fd8811321b4017f5f8a13.jpg" + } + ] + } + ], + "index": 40, + "virtual_lines": [ + { + "bbox": [ + 151, + 642, + 459, + 652.0 + ], + "spans": [], + "index": 39 + }, + { + "bbox": [ + 151, + 652.0, + 459, + 662.0 + ], + "spans": [], + "index": 40 + }, + { + "bbox": [ + 151, + 662.0, + 459, + 672.0 + ], + "spans": [], + "index": 41 + } + ] + }, + { + "type": "text", + "bbox": [ + 83, + 677, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 83, + 677, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 83, + 679, + 100, + 690 + ], + "score": 1.0, + "content": "1444", + "type": "text" + }, + { + "bbox": [ + 105, + 677, + 131, + 691 + ], + "score": 1.0, + "content": "since", + "type": "text" + }, + { + "bbox": [ + 131, + 677, + 214, + 690 + ], + "score": 0.88, + "content": "\\| \\phi _ { L } - f _ { L } \\| _ { \\infty } < \\epsilon / 2", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 677, + 340, + 691 + ], + "score": 1.0, + "content": ". 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As this holds for all", + "type": "text" + }, + { + "bbox": [ + 235, + 704, + 241, + 710 + ], + "score": 0.73, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 699, + 280, + 713 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + }, + { + "bbox": [ + 281, + 701, + 428, + 712 + ], + "score": 0.88, + "content": "\\| \\phi - \\phi _ { L } \\circ f _ { L - 1 } \\circ . . . \\circ f _ { 1 } \\| _ { \\infty } \\leq \\epsilon / 2", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 699, + 443, + 713 + ], + "score": 1.0, + "content": ", so", + "type": "text" + }, + { + "bbox": [ + 443, + 700, + 504, + 712 + ], + "score": 0.88, + "content": "\\| \\phi - f \\| _ { \\infty } < \\epsilon", + "type": "inline_equation" + } + ], + "index": 44 + }, + { + "bbox": [ + 82, + 711, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 82, + 712, + 101, + 722 + ], + "score": 1.0, + "content": "1447", + "type": "text" + }, + { + "bbox": [ + 105, + 711, + 177, + 723 + ], + "score": 1.0, + "content": "and we are done.", + "type": "text" + }, + { + "bbox": [ + 494, + 711, + 505, + 722 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 43.5 + } + ], + "page_idx": 38, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 300, + 741, + 311, + 750 + ], + "lines": [ + { + "bbox": [ + 298, + 740, + 313, + 754 + ], + "spans": [ + { + "bbox": [ + 298, + 740, + 313, + 754 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 14, + "width": 15 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "index", + "bbox": [ + 92, + 72, + 505, + 95 + ], + "lines": [ + { + "bbox": [ + 89, + 72, + 506, + 86 + ], + "spans": [ + { + "bbox": [ + 89, + 75, + 100, + 84 + ], + "score": 1.0, + "content": "406", + "type": "text" + }, + { + "bbox": [ + 104, + 72, + 506, + 86 + ], + "score": 1.0, + "content": "The following lemma shows that quotients of compact spaces are also compact, which is useful for", + "type": "text" + } + ], + "index": 0, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 84, + 284, + 97 + ], + "spans": [ + { + "bbox": [ + 88, + 84, + 284, + 97 + ], + "score": 1.0, + "content": "407 universal approximation on quotient spaces.", + "type": "text" + } + ], + "index": 1, + "is_list_start_line": true + }, + { + "bbox": [ + 90, + 98, + 505, + 113 + ], + "spans": [ + { + "bbox": [ + 90, + 98, + 366, + 113 + ], + "score": 1.0, + "content": "08 Lemma 4 (Compactness of quotients of compact spaces). Let", + "type": "text" + }, + { + "bbox": [ + 366, + 100, + 376, + 109 + ], + "score": 0.72, + "content": "\\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 377, + 98, + 505, + 113 + ], + "score": 1.0, + "content": "be a compact space. Then the", + "type": "text" + } + ], + "index": 2, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 110, + 238, + 123 + ], + "spans": [ + { + "bbox": [ + 88, + 110, + 167, + 123 + ], + "score": 1.0, + "content": "409 quotient space", + "type": "text" + }, + { + "bbox": [ + 167, + 110, + 190, + 123 + ], + "score": 0.9, + "content": "\\mathcal { X } / G", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 110, + 238, + 123 + ], + "score": 1.0, + "content": "is compact.", + "type": "text" + } + ], + "index": 3, + "is_list_start_line": true + }, + { + "bbox": [ + 83, + 140, + 506, + 155 + ], + "spans": [ + { + "bbox": [ + 83, + 142, + 100, + 153 + ], + "score": 1.0, + "content": "1410", + "type": "text" + }, + { + "bbox": [ + 105, + 140, + 259, + 155 + ], + "score": 1.0, + "content": "Proof. Denoting the quotient map by", + "type": "text" + }, + { + "bbox": [ + 259, + 141, + 321, + 153 + ], + "score": 0.92, + "content": "\\pi : { \\mathcal { X } } \\to { \\mathcal { X } } / G", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 140, + 367, + 155 + ], + "score": 1.0, + "content": "and letting", + "type": "text" + }, + { + "bbox": [ + 368, + 141, + 397, + 153 + ], + "score": 0.92, + "content": "\\{ U _ { \\alpha } \\} _ { \\alpha }", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 140, + 480, + 155 + ], + "score": 1.0, + "content": "be an open cover of", + "type": "text" + }, + { + "bbox": [ + 480, + 141, + 502, + 153 + ], + "score": 0.9, + "content": "\\mathcal { X } / G", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 140, + 506, + 155 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 4, + "is_list_start_line": true + }, + { + "bbox": [ + 82, + 150, + 506, + 166 + ], + "spans": [ + { + "bbox": [ + 82, + 153, + 99, + 164 + ], + "score": 1.0, + "content": "1411", + "type": "text" + }, + { + "bbox": [ + 104, + 150, + 161, + 166 + ], + "score": 1.0, + "content": "we have that", + "type": "text" + }, + { + "bbox": [ + 162, + 151, + 215, + 163 + ], + "score": 0.84, + "content": "\\{ \\check { \\pi } ^ { - 1 } ( U _ { \\alpha } ) \\} _ { \\alpha }", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 150, + 299, + 166 + ], + "score": 1.0, + "content": "is an open cover of", + "type": "text" + }, + { + "bbox": [ + 299, + 153, + 309, + 162 + ], + "score": 0.83, + "content": "\\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 150, + 396, + 166 + ], + "score": 1.0, + "content": ". By compactness of", + "type": "text" + }, + { + "bbox": [ + 397, + 153, + 406, + 162 + ], + "score": 0.83, + "content": "\\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 150, + 506, + 166 + ], + "score": 1.0, + "content": ", we can choose a finite", + "type": "text" + } + ], + "index": 5, + "is_list_start_line": true + }, + { + "bbox": [ + 83, + 160, + 507, + 178 + ], + "spans": [ + { + "bbox": [ + 83, + 164, + 100, + 174 + ], + "score": 1.0, + "content": "1412", + "type": "text" + }, + { + "bbox": [ + 104, + 160, + 146, + 178 + ], + "score": 1.0, + "content": "subcover", + "type": "text" + }, + { + "bbox": [ + 146, + 163, + 227, + 175 + ], + "score": 0.86, + "content": "\\{ \\pi ^ { - 1 } ( U _ { \\alpha _ { i } } ) \\} _ { i = 1 , \\dots , n }", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 160, + 257, + 178 + ], + "score": 1.0, + "content": ". Then", + "type": "text" + }, + { + "bbox": [ + 258, + 163, + 423, + 175 + ], + "score": 0.92, + "content": "\\{ \\pi ( \\pi ^ { - 1 } ( U _ { \\alpha _ { i } } ) ) \\} _ { i = 1 , \\dots , n } = \\{ U _ { \\alpha _ { i } } \\} _ { i = 1 , \\dots , n }", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 160, + 507, + 178 + ], + "score": 1.0, + "content": "by surjectivity, and", + "type": "text" + } + ], + "index": 6, + "is_list_start_line": true + }, + { + "bbox": [ + 83, + 172, + 505, + 188 + ], + "spans": [ + { + "bbox": [ + 83, + 175, + 100, + 186 + ], + "score": 1.0, + "content": "1413", + "type": "text" + }, + { + "bbox": [ + 107, + 174, + 163, + 186 + ], + "score": 0.91, + "content": "\\{ U _ { \\alpha _ { i } } \\} _ { i = 1 , \\dots , n }", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 172, + 262, + 188 + ], + "score": 1.0, + "content": "is thus an open cover of", + "type": "text" + }, + { + "bbox": [ + 262, + 174, + 284, + 186 + ], + "score": 0.9, + "content": "\\mathcal { X } / G", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 172, + 289, + 188 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 495, + 174, + 505, + 185 + ], + "score": 0.998, + "content": "□", + "type": "text" + } + ], + "index": 7, + "is_list_start_line": true + }, + { + "bbox": [ + 83, + 204, + 506, + 218 + ], + "spans": [ + { + "bbox": [ + 83, + 206, + 99, + 216 + ], + "score": 1.0, + "content": "1414", + "type": "text" + }, + { + "bbox": [ + 105, + 204, + 506, + 218 + ], + "score": 1.0, + "content": "The Whitney embedding theorem gives a nice condition that we apply to show that the quotient", + "type": "text" + } + ], + "index": 8, + "is_list_start_line": true + }, + { + "bbox": [ + 82, + 215, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 82, + 217, + 100, + 227 + ], + "score": 1.0, + "content": "1415", + "type": "text" + }, + { + "bbox": [ + 105, + 216, + 136, + 227 + ], + "score": 1.0, + "content": "spaces", + "type": "text" + }, + { + "bbox": [ + 136, + 215, + 159, + 227 + ], + "score": 0.91, + "content": "\\chi / \\bar { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 216, + 430, + 227 + ], + "score": 1.0, + "content": "that we deal with embed into Euclidean space. It says that when", + "type": "text" + }, + { + "bbox": [ + 431, + 215, + 453, + 227 + ], + "score": 0.93, + "content": "\\mathcal { X } / G", + "type": "inline_equation" + }, + { + "bbox": [ + 454, + 216, + 505, + 227 + ], + "score": 1.0, + "content": "is a smooth", + "type": "text" + } + ], + "index": 9, + "is_list_start_line": true + }, + { + "bbox": [ + 82, + 226, + 506, + 238 + ], + "spans": [ + { + "bbox": [ + 82, + 227, + 100, + 238 + ], + "score": 1.0, + "content": "1416", + "type": "text" + }, + { + "bbox": [ + 105, + 226, + 506, + 237 + ], + "score": 1.0, + "content": "manifold, then it can be embedded into a Euclidean space of double the dimension of the manifold.", + "type": "text" + } + ], + "index": 10, + "is_list_start_line": true + }, + { + "bbox": [ + 83, + 237, + 282, + 250 + ], + "spans": [ + { + "bbox": [ + 83, + 239, + 99, + 248 + ], + "score": 1.0, + "content": "1417", + "type": "text" + }, + { + "bbox": [ + 105, + 237, + 282, + 250 + ], + "score": 1.0, + "content": "The proof is outside the scope of this paper.", + "type": "text" + } + ], + "index": 11, + "is_list_start_line": true + } + ], + "index": 0.5, + "bbox_fs": [ + 88, + 72, + 506, + 97 + ] + }, + { + "type": "index", + "bbox": [ + 92, + 99, + 506, + 123 + ], + "lines": [], + "index": 2.5, + "bbox_fs": [ + 88, + 98, + 505, + 123 + ], + "lines_deleted": true + }, + { + "type": "index", + "bbox": [ + 83, + 140, + 506, + 186 + ], + "lines": [], + "index": 5.5, + "bbox_fs": [ + 82, + 140, + 507, + 188 + ], + "lines_deleted": true + }, + { + "type": "index", + "bbox": [ + 84, + 204, + 506, + 249 + ], + "lines": [], + "index": 9.5, + "bbox_fs": [ + 82, + 204, + 506, + 250 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 95, + 253, + 504, + 275 + ], + "lines": [ + { + "bbox": [ + 92, + 253, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 92, + 253, + 438, + 266 + ], + "score": 1.0, + "content": "18 Lemma 5 (Whitney Embedding Theorem [Whitney, 1944]). Every smooth manifold", + "type": "text" + }, + { + "bbox": [ + 439, + 254, + 452, + 263 + ], + "score": 0.61, + "content": "\\mathcal { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 253, + 505, + 266 + ], + "score": 1.0, + "content": "of dimension", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 90, + 263, + 274, + 277 + ], + "spans": [ + { + "bbox": [ + 90, + 263, + 274, + 277 + ], + "score": 1.0, + "content": "n > 0 can be smoothly embedded in R2n 19 .", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12.5, + "bbox_fs": [ + 90, + 253, + 505, + 277 + ] + }, + { + "type": "index", + "bbox": [ + 83, + 285, + 505, + 341 + ], + "lines": [ + { + "bbox": [ + 83, + 285, + 506, + 299 + ], + "spans": [ + { + "bbox": [ + 83, + 288, + 99, + 296 + ], + "score": 1.0, + "content": "1420", + "type": "text" + }, + { + "bbox": [ + 105, + 285, + 506, + 299 + ], + "score": 1.0, + "content": "Finally, we give a lemma that helps prove universal approximation results. It says that if functions", + "type": "text" + } + ], + "index": 14, + "is_list_start_line": true + }, + { + "bbox": [ + 83, + 297, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 83, + 298, + 99, + 308 + ], + "score": 1.0, + "content": "1421", + "type": "text" + }, + { + "bbox": [ + 106, + 297, + 114, + 308 + ], + "score": 0.83, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 297, + 365, + 309 + ], + "score": 1.0, + "content": "that we want to approximate can be written as compositions", + "type": "text" + }, + { + "bbox": [ + 365, + 297, + 438, + 308 + ], + "score": 0.93, + "content": "f = f _ { L } \\circ \\dots \\circ f _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 297, + 505, + 309 + ], + "score": 1.0, + "content": ", then it suffices", + "type": "text" + } + ], + "index": 15, + "is_list_start_line": true + }, + { + "bbox": [ + 83, + 307, + 505, + 320 + ], + "spans": [ + { + "bbox": [ + 83, + 308, + 100, + 319 + ], + "score": 1.0, + "content": "1422", + "type": "text" + }, + { + "bbox": [ + 105, + 307, + 237, + 320 + ], + "score": 1.0, + "content": "to universally approximate each", + "type": "text" + }, + { + "bbox": [ + 238, + 308, + 248, + 319 + ], + "score": 0.87, + "content": "f _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 307, + 473, + 320 + ], + "score": 1.0, + "content": "and compose the results to universally approximate the", + "type": "text" + }, + { + "bbox": [ + 473, + 308, + 480, + 319 + ], + "score": 0.83, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 307, + 505, + 320 + ], + "score": 1.0, + "content": ". 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Let", + "type": "text" + }, + { + "bbox": [ + 359, + 344, + 397, + 356 + ], + "score": 0.91, + "content": "\\mathcal { Z } \\subseteq \\mathbb { R } ^ { d _ { 0 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 343, + 506, + 359 + ], + "score": 1.0, + "content": "be a compact domain, let", + "type": "text" + } + ], + "index": 19, + "is_list_start_line": true + }, + { + "bbox": [ + 83, + 355, + 503, + 369 + ], + "spans": [ + { + "bbox": [ + 83, + 358, + 100, + 367 + ], + "score": 1.0, + "content": "1426", + "type": "text" + }, + { + "bbox": [ + 107, + 357, + 154, + 367 + ], + "score": 0.9, + "content": "\\mathcal { F } _ { 1 } , \\ldots , \\mathcal { F } _ { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 355, + 328, + 369 + ], + "score": 1.0, + "content": "be families of continuous functions where", + "type": "text" + }, + { + "bbox": [ + 328, + 357, + 339, + 367 + ], + "score": 0.88, + "content": "{ \\mathcal { F } } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 355, + 448, + 369 + ], + "score": 1.0, + "content": "consists of functions from", + "type": "text" + }, + { + "bbox": [ + 448, + 356, + 503, + 366 + ], + "score": 0.91, + "content": "\\mathbb { R } ^ { d _ { i - 1 } } \\mathbb { R } ^ { d _ { i } }", + "type": "inline_equation" + } + ], + "index": 20, + "is_list_start_line": true + }, + { + "bbox": [ + 82, + 365, + 506, + 380 + ], + "spans": [ + { + "bbox": [ + 82, + 368, + 100, + 379 + ], + "score": 1.0, + "content": "1427", + "type": "text" + }, + { + "bbox": [ + 104, + 365, + 145, + 380 + ], + "score": 1.0, + "content": "for some", + "type": "text" + }, + { + "bbox": [ + 145, + 367, + 188, + 378 + ], + "score": 0.91, + "content": "d _ { 1 } , \\ldots , d _ { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 365, + 210, + 380 + ], + "score": 1.0, + "content": ". Let", + "type": "text" + }, + { + "bbox": [ + 210, + 367, + 220, + 377 + ], + "score": 0.81, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 365, + 327, + 380 + ], + "score": 1.0, + "content": "be the family of functions", + "type": "text" + }, + { + "bbox": [ + 328, + 367, + 469, + 379 + ], + "score": 0.9, + "content": "\\{ f _ { L } \\circ . . . f _ { 1 } : \\mathcal { Z } \\mathbb { R } ^ { d _ { L } } , f _ { i } \\in \\mathcal { F } _ { i } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 469, + 365, + 506, + 380 + ], + "score": 1.0, + "content": "that are", + "type": "text" + } + ], + "index": 21, + "is_list_start_line": true + }, + { + "bbox": [ + 83, + 378, + 246, + 391 + ], + "spans": [ + { + "bbox": [ + 83, + 380, + 99, + 389 + ], + "score": 1.0, + "content": "1428", + "type": "text" + }, + { + "bbox": [ + 105, + 378, + 211, + 391 + ], + "score": 1.0, + "content": "compositions of functions", + "type": "text" + }, + { + "bbox": [ + 211, + 378, + 243, + 390 + ], + "score": 0.91, + "content": "f _ { i } \\in \\mathcal { F } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 378, + 246, + 391 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 22, + "is_list_start_line": true + } + ], + "index": 16, + "bbox_fs": [ + 83, + 285, + 506, + 342 + ] + }, + { + "type": "index", + "bbox": [ + 84, + 344, + 505, + 390 + ], + "lines": [], + "index": 20.5, + "bbox_fs": [ + 82, + 343, + 506, + 391 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 88, + 394, + 506, + 418 + ], + "lines": [ + { + "bbox": [ + 84, + 393, + 506, + 407 + ], + "spans": [ + { + "bbox": [ + 84, + 393, + 144, + 407 + ], + "score": 1.0, + "content": "For each 1429", + "type": "text" + }, + { + "bbox": [ + 145, + 396, + 149, + 404 + ], + "score": 0.53, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 393, + 166, + 407 + ], + "score": 1.0, + "content": ", let", + "type": "text" + }, + { + "bbox": [ + 167, + 395, + 178, + 405 + ], + "score": 0.86, + "content": "\\Phi _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 393, + 450, + 407 + ], + "score": 1.0, + "content": "be a family of continuous functions that universally approximates", + "type": "text" + }, + { + "bbox": [ + 450, + 395, + 461, + 405 + ], + "score": 0.87, + "content": "{ \\mathcal { F } } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 393, + 506, + 407 + ], + "score": 1.0, + "content": ". Then the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 85, + 405, + 442, + 418 + ], + "spans": [ + { + "bbox": [ + 85, + 405, + 199, + 418 + ], + "score": 1.0, + "content": "1430 family of compositions", + "type": "text" + }, + { + "bbox": [ + 200, + 405, + 324, + 417 + ], + "score": 0.92, + "content": "\\Phi = \\left\\{ \\phi _ { L } \\circ . . . \\circ \\phi _ { 1 } : \\phi _ { i } \\in \\Phi _ { i } \\right\\}", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 405, + 428, + 418 + ], + "score": 1.0, + "content": "universally approximates", + "type": "text" + }, + { + "bbox": [ + 429, + 406, + 437, + 415 + ], + "score": 0.82, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 405, + 442, + 418 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5, + "bbox_fs": [ + 84, + 393, + 506, + 418 + ] + }, + { + "type": "text", + "bbox": [ + 83, + 435, + 506, + 475 + ], + "lines": [ + { + "bbox": [ + 81, + 435, + 506, + 451 + ], + "spans": [ + { + "bbox": [ + 81, + 435, + 151, + 451 + ], + "score": 1.0, + "content": "Proof. Let 1431", + "type": "text" + }, + { + "bbox": [ + 152, + 437, + 242, + 449 + ], + "score": 0.92, + "content": "f = f _ { L } \\circ . . . \\circ f _ { 1 } \\in \\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 435, + 262, + 451 + ], + "score": 1.0, + "content": ". Let", + "type": "text" + }, + { + "bbox": [ + 262, + 436, + 296, + 449 + ], + "score": 0.92, + "content": "\\tilde { \\mathcal { Z } } _ { 1 } = \\mathcal { Z }", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 435, + 349, + 451 + ], + "score": 1.0, + "content": ", and then for", + "type": "text" + }, + { + "bbox": [ + 349, + 437, + 373, + 448 + ], + "score": 0.89, + "content": "i \\geq 2", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 435, + 386, + 451 + ], + "score": 1.0, + "content": "let", + "type": "text" + }, + { + "bbox": [ + 386, + 436, + 457, + 450 + ], + "score": 0.93, + "content": "\\tilde { \\mathcal { Z } } _ { i } = f _ { i - 1 } ( \\tilde { \\mathcal { Z } } _ { i - 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 458, + 435, + 506, + 451 + ], + "score": 1.0, + "content": ". Then each", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 83, + 448, + 506, + 462 + ], + "spans": [ + { + "bbox": [ + 83, + 451, + 100, + 461 + ], + "score": 1.0, + "content": "1432", + "type": "text" + }, + { + "bbox": [ + 106, + 448, + 118, + 461 + ], + "score": 0.89, + "content": "\\mathcal { \\tilde { Z } } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 118, + 450, + 241, + 462 + ], + "score": 1.0, + "content": "is compact by continuity of the", + "type": "text" + }, + { + "bbox": [ + 241, + 451, + 251, + 461 + ], + "score": 0.86, + "content": "f _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 450, + 270, + 462 + ], + "score": 1.0, + "content": ". 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Base modelPositional encodingk#paramsTest MAE (↓)
GINNo PE16497k0.348±0.014
LapPE (flip)16498k0.341±0.011
SignNet16500k0.238±0.012
GATNo PE16501k0.464±0.011
LapPE (flip)16502k0.462±0.013
SignNet16499k0.243±0.008
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Low-passHigh-passBand-passBand-rejectionComb
GCN.111±.0683.092±5.111.720±3.151.418±1.031.753±1.17
GAT.113±.065.954±.6961.105±.964.543±.340.638±.446
GPR-GNN.033±.032.012±.007.137±.081.256±.197.369±.460
ARMA.053±.029.042±.024.107±.039.148±.089.202±.116
ChebNet.003±.002.001±.001.005±.003.009±.006.022±.016
BernNet.001±.002.001±.001.000±.000.048±.042.027±.019
Transformer3.662±1.973.715±1.981.531±1.301.506±1.293.178±1.93
Transformer Eig Flip4.454±2.324.425±2.381.651±1.532.567±1.733.720±1.94
Transformer Eig Abs2.727±1.403.172±1.611.264±.7881.445±.9432.607±1.32
DeepSets SignNet.004±.013.086±.405.021±.115.008±.037.003±.016
Transformer SignNet.003±.016.004±.025.001±.004.006±.023.093±.641
DeepSets BasisNet.009±.018.003±.015.008±.030.004±.011.015±.060
Transformer BasisNet.079±.471.014±.038.005±.018.006±.016.014±.051
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Numbers are", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 88, + 388, + 100 + ], + "spans": [ + { + "bbox": [ + 106, + 88, + 388, + 100 + ], + "score": 1.0, + "content": "the mean and standard deviation over 4 runs each with different seeds.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "table_body", + "bbox": [ + 173, + 104, + 438, + 192 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 173, + 104, + 438, + 192 + ], + "spans": [ + { + "bbox": [ + 173, + 104, + 438, + 192 + ], + "score": 0.982, + "html": "
Base modelPositional encodingk#paramsTest MAE (↓)
GINNo PE16497k0.348±0.014
LapPE (flip)16498k0.341±0.011
SignNet16500k0.238±0.012
GATNo PE16501k0.464±0.011
LapPE (flip)16502k0.462±0.013
SignNet16499k0.243±0.008
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Low-passHigh-passBand-passBand-rejectionComb
GCN.111±.0683.092±5.111.720±3.151.418±1.031.753±1.17
GAT.113±.065.954±.6961.105±.964.543±.340.638±.446
GPR-GNN.033±.032.012±.007.137±.081.256±.197.369±.460
ARMA.053±.029.042±.024.107±.039.148±.089.202±.116
ChebNet.003±.002.001±.001.005±.003.009±.006.022±.016
BernNet.001±.002.001±.001.000±.000.048±.042.027±.019
Transformer3.662±1.973.715±1.981.531±1.301.506±1.293.178±1.93
Transformer Eig Flip4.454±2.324.425±2.381.651±1.532.567±1.733.720±1.94
Transformer Eig Abs2.727±1.403.172±1.611.264±.7881.445±.9432.607±1.32
DeepSets SignNet.004±.013.086±.405.021±.115.008±.037.003±.016
Transformer SignNet.003±.016.004±.025.001±.004.006±.023.093±.641
DeepSets BasisNet.009±.018.003±.015.008±.030.004±.011.015±.060
Transformer BasisNet.079±.471.014±.038.005±.018.006±.016.014±.051
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[2020], He et al. [2021]. We take the dataset of 50", + "type": "text" + } + ], + "index": 19, + "is_list_start_line": true + }, + { + "bbox": [ + 82, + 574, + 505, + 586 + ], + "spans": [ + { + "bbox": [ + 82, + 576, + 100, + 586 + ], + "score": 1.0, + "content": "1458", + "type": "text" + }, + { + "bbox": [ + 106, + 574, + 505, + 586 + ], + "score": 1.0, + "content": "images in He et al. 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Then we apply the same spectral graph convolutions on them as in", + "type": "text" + } + ], + "index": 21, + "is_list_start_line": true + }, + { + "bbox": [ + 82, + 595, + 506, + 609 + ], + "spans": [ + { + "bbox": [ + 82, + 598, + 100, + 608 + ], + "score": 1.0, + "content": "1460", + "type": "text" + }, + { + "bbox": [ + 105, + 595, + 506, + 609 + ], + "score": 1.0, + "content": "He et al. [2021], and train neural networks to learn these as regression targets. 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These models are all approximately sign", + "type": "text" + } + ], + "index": 27, + "is_list_start_line": true + }, + { + "bbox": [ + 83, + 666, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 83, + 669, + 100, + 678 + ], + "score": 1.0, + "content": "1466", + "type": "text" + }, + { + "bbox": [ + 105, + 666, + 506, + 680 + ], + "score": 1.0, + "content": "invariant (they either use eigenvectors in a sign invariant way or do not use eigenvectors). 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That is, we are only given graph", + "type": "text" + } + ], + "index": 31, + "is_list_start_line": true + }, + { + "bbox": [ + 82, + 709, + 471, + 725 + ], + "spans": [ + { + "bbox": [ + 82, + 712, + 100, + 722 + ], + "score": 1.0, + "content": "1470", + "type": "text" + }, + { + "bbox": [ + 104, + 709, + 471, + 725 + ], + "score": 1.0, + "content": "information through the eigenvectors and eigenvalues, and we do not use message passing.", + "type": "text" + } + ], + "index": 32, + "is_list_start_line": true + }, + { + "bbox": [ + 82, + 71, + 506, + 87 + ], + "spans": [ + { + "bbox": [ + 82, + 74, + 99, + 85 + ], + "score": 1.0, + "content": "1471", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 104, + 71, + 506, + 87 + ], + "score": 1.0, + "content": "Table 8 displays the results, which validate our theoretical results in Section 3.1. Without any message", + "type": "text", + "cross_page": true + } + ], + "index": 0, + "is_list_start_line": true + }, + { + "bbox": [ + 83, + 83, + 504, + 96 + ], + "spans": [ + { + "bbox": [ + 83, + 86, + 100, + 95 + ], + "score": 1.0, + "content": "1472", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 83, + 504, + 96 + ], + "score": 1.0, + "content": "passing, SignNet and BasisNet allow DeepSets and Transformers to perform strongly, beating the", + "type": "text", + "cross_page": true + } + ], + "index": 1, + "is_list_start_line": true + }, + { + "bbox": [ + 82, + 94, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 82, + 96, + 100, + 107 + ], + "score": 1.0, + "content": "1473", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 94, + 506, + 107 + ], + "score": 1.0, + "content": "spectral GNNs GPR-GNN and ARMA on all tasks. 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Without any message", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 83, + 83, + 504, + 96 + ], + "spans": [ + { + "bbox": [ + 83, + 86, + 100, + 95 + ], + "score": 1.0, + "content": "1472", + "type": "text" + }, + { + "bbox": [ + 106, + 83, + 504, + 96 + ], + "score": 1.0, + "content": "passing, SignNet and BasisNet allow DeepSets and Transformers to perform strongly, beating the", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 82, + 94, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 82, + 96, + 100, + 107 + ], + "score": 1.0, + "content": "1473", + "type": "text" + }, + { + "bbox": [ + 105, + 94, + 506, + 107 + ], + "score": 1.0, + "content": "spectral GNNs GPR-GNN and ARMA on all tasks. 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Most experiments were run on a server with 8", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 187, + 505, + 201 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 505, + 201 + ], + "score": 1.0, + "content": "NVIDIA RTX 2080 Ti GPUs. We run all of our experiments in Python, using the PyTorch [Paszke", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 199, + 506, + 212 + ], + "spans": [ + { + "bbox": [ + 105, + 199, + 506, + 212 + ], + "score": 1.0, + "content": "et al., 2019] framework (license URL). We also make use of Deep Graph Library (DGL) [Wang et al.,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 209, + 506, + 224 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 506, + 224 + ], + "score": 1.0, + "content": "2019] (Apache License 2.0), and PyTorch Geometric (PyG) [Fey and Lenssen, 2019] (MIT License)", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 221, + 239, + 235 + ], + "spans": [ + { + "bbox": [ + 105, + 221, + 239, + 235 + ], + "score": 1.0, + "content": "for experiments with graph data.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 102, + 238, + 344, + 249 + ], + "lines": [ + { + "bbox": [ + 105, + 237, + 346, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 346, + 252 + ], + "score": 1.0, + "content": "We open source our code [redacted for anonymous review].", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 101, + 254, + 505, + 353 + ], + "lines": [ + { + "bbox": [ + 106, + 254, + 506, + 266 + ], + "spans": [ + { + "bbox": [ + 106, + 254, + 506, + 266 + ], + "score": 1.0, + "content": "The data we use are all freely available online. The datasets we use are ZINC [Irwin et al., 2012],", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 264, + 506, + 278 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 506, + 278 + ], + "score": 1.0, + "content": "Alchemy [Chen et al., 2019a], the synthetic counting substructures dataset [Chen et al., 2020],", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 275, + 505, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 505, + 289 + ], + "score": 1.0, + "content": "the multi-task graph property regression synthetic dataset [Corso et al., 2020] (MIT License), the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 287, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 106, + 287, + 505, + 299 + ], + "score": 1.0, + "content": "images dataset used by Balcilar et al. [2020] (GNU General Public License v3.0), the cat mesh from", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 298, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 106, + 298, + 505, + 309 + ], + "score": 1.0, + "content": "free3d.com/3d-model/cat-v1--522281.html (Personal Use License), and the human mesh", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 309, + 505, + 321 + ], + "spans": [ + { + "bbox": [ + 106, + 309, + 505, + 321 + ], + "score": 1.0, + "content": "from turbosquid.com/3d-models/water-park-slides-3d-max/1093267 (TurboSquid 3D", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 318, + 506, + 332 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 506, + 332 + ], + "score": 1.0, + "content": "Model License). If no license is listed, this means that we cannot find a license for the dataset. As", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 331, + 505, + 343 + ], + "spans": [ + { + "bbox": [ + 106, + 331, + 505, + 343 + ], + "score": 1.0, + "content": "they appear to be freely available with permissive licenses or no licenses, we do not ask for permission", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 342, + 257, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 257, + 354 + ], + "score": 1.0, + "content": "from the creators or hosts of the data.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 84, + 358, + 506, + 413 + ], + "lines": [ + { + "bbox": [ + 83, + 358, + 506, + 370 + ], + "spans": [ + { + "bbox": [ + 83, + 361, + 99, + 369 + ], + "score": 1.0, + "content": "1492", + "type": "text" + }, + { + "bbox": [ + 106, + 358, + 506, + 370 + ], + "score": 1.0, + "content": "We do not believe that any of this data contains offensive content or personally identifiable information.", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 82, + 368, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 82, + 371, + 100, + 381 + ], + "score": 1.0, + "content": "1493", + "type": "text" + }, + { + "bbox": [ + 106, + 368, + 505, + 381 + ], + "score": 1.0, + "content": "The 50 images used in the spectral graph convolution experiments are mostly images of objects, with", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 82, + 380, + 506, + 393 + ], + "spans": [ + { + "bbox": [ + 82, + 382, + 100, + 392 + ], + "score": 1.0, + "content": "1494", + "type": "text" + }, + { + "bbox": [ + 105, + 380, + 506, + 393 + ], + "score": 1.0, + "content": "a few low resolution images of humans that do not appear to have offensive content. 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To handle the", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 567, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 271, + 581 + ], + "score": 1.0, + "content": "variable sized input in this case, we take", + "type": "text" + }, + { + "bbox": [ + 271, + 569, + 278, + 579 + ], + "score": 0.81, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 567, + 450, + 581 + ], + "score": 1.0, + "content": "to be an MLP preceded by a sum over the", + "type": "text" + }, + { + "bbox": [ + 450, + 568, + 457, + 579 + ], + "score": 0.84, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 458, + 567, + 505, + 581 + ], + "score": 1.0, + "content": "outputs. 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Most experiments were run on a server with 8", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 187, + 505, + 201 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 505, + 201 + ], + "score": 1.0, + "content": "NVIDIA RTX 2080 Ti GPUs. We run all of our experiments in Python, using the PyTorch [Paszke", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 199, + 506, + 212 + ], + "spans": [ + { + "bbox": [ + 105, + 199, + 506, + 212 + ], + "score": 1.0, + "content": "et al., 2019] framework (license URL). 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The datasets we use are ZINC [Irwin et al., 2012],", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 264, + 506, + 278 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 506, + 278 + ], + "score": 1.0, + "content": "Alchemy [Chen et al., 2019a], the synthetic counting substructures dataset [Chen et al., 2020],", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 275, + 505, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 505, + 289 + ], + "score": 1.0, + "content": "the multi-task graph property regression synthetic dataset [Corso et al., 2020] (MIT License), the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 287, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 106, + 287, + 505, + 299 + ], + "score": 1.0, + "content": "images dataset used by Balcilar et al. 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To handle the", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 567, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 271, + 581 + ], + "score": 1.0, + "content": "variable sized input in this case, we take", + "type": "text" + }, + { + "bbox": [ + 271, + 569, + 278, + 579 + ], + "score": 0.81, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 567, + 450, + 581 + ], + "score": 1.0, + "content": "to be an MLP preceded by a sum over the", + "type": "text" + }, + { + "bbox": [ + 450, + 568, + 457, + 579 + ], + "score": 0.84, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 458, + 567, + 505, + 581 + ], + "score": 1.0, + "content": "outputs. 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This", + "type": "text" + } + ], + "index": 54, + "is_list_start_line": true + }, + { + "bbox": [ + 82, + 635, + 506, + 647 + ], + "spans": [ + { + "bbox": [ + 82, + 636, + 100, + 646 + ], + "score": 1.0, + "content": "1514", + "type": "text" + }, + { + "bbox": [ + 105, + 635, + 506, + 647 + ], + "score": 1.0, + "content": "is a common heuristic used in prior work on Laplacian positional encoding [Kreuzer et al., 2021,", + "type": "text" + } + ], + "index": 55, + "is_list_start_line": true + }, + { + "bbox": [ + 82, + 644, + 506, + 658 + ], + "spans": [ + { + "bbox": [ + 82, + 647, + 100, + 657 + ], + "score": 1.0, + "content": "1515", + "type": "text" + }, + { + "bbox": [ + 105, + 644, + 390, + 658 + ], + "score": 1.0, + "content": "Dwivedi et al., 2020]. Second, take the element-wise absolute value", + "type": "text" + }, + { + "bbox": [ + 390, + 645, + 405, + 657 + ], + "score": 0.91, + "content": "| v _ { i } |", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 644, + 506, + 658 + ], + "score": 1.0, + "content": ". This is a non-injective", + "type": "text" + } + ], + "index": 56, + "is_list_start_line": true + }, + { + "bbox": [ + 83, + 656, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 83, + 658, + 99, + 667 + ], + "score": 1.0, + "content": "1516", + "type": "text" + }, + { + "bbox": [ + 105, + 656, + 506, + 668 + ], + "score": 1.0, + "content": "map, creating sign invariance at the cost of destroying positional information. 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When the tie-break also fails, the sign is chosen randomly. Results for", + "type": "text" + } + ], + "index": 60, + "is_list_start_line": true + }, + { + "bbox": [ + 83, + 699, + 505, + 713 + ], + "spans": [ + { + "bbox": [ + 83, + 702, + 99, + 711 + ], + "score": 1.0, + "content": "1520", + "type": "text" + }, + { + "bbox": [ + 105, + 699, + 505, + 713 + ], + "score": 1.0, + "content": "GatedGCN base model on ZINC in Table 1 show that all three of these approaches are significantly", + "type": "text" + } + ], + "index": 61, + "is_list_start_line": true + }, + { + "bbox": [ + 83, + 711, + 309, + 723 + ], + "spans": [ + { + "bbox": [ + 83, + 712, + 99, + 723 + ], + "score": 1.0, + "content": "1521", + "type": "text" + }, + { + "bbox": [ + 105, + 711, + 309, + 723 + ], + "score": 1.0, + "content": "poorer positional encodings compared to SignNet.", + "type": "text" + } + ], + "index": 62, + "is_list_start_line": true + }, + { + "bbox": [ + 83, + 72, + 505, + 85 + ], + "spans": [ + { + "bbox": [ + 83, + 75, + 99, + 84 + ], + "score": 1.0, + "content": "1522", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 72, + 505, + 85 + ], + "score": 1.0, + "content": "Our training pipeline largely follows that of Dwivedi et al. 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The Sparse Transformer base model architecture we use, which", + "type": "text", + "cross_page": true + } + ], + "index": 2, + "is_list_start_line": true + }, + { + "bbox": [ + 83, + 105, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 83, + 108, + 99, + 117 + ], + "score": 1.0, + "content": "1525", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 105, + 506, + 118 + ], + "score": 1.0, + "content": "like GAT computes attention only across neighbouring nodes, is introduced by Kreuzer et al. 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For the state-of-the-art comparison, all baseline results are from their respective", + "type": "text", + "cross_page": true + } + ], + "index": 5, + "is_list_start_line": true + }, + { + "bbox": [ + 82, + 138, + 262, + 150 + ], + "spans": [ + { + "bbox": [ + 82, + 139, + 100, + 150 + ], + "score": 1.0, + "content": "1528", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 138, + 262, + 150 + ], + "score": 1.0, + "content": "papers, except for GIN, which we run.", + "type": "text", + "cross_page": true + } + ], + "index": 6, + "is_list_start_line": true + }, + { + "bbox": [ + 83, + 153, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 83, + 156, + 99, + 165 + ], + "score": 1.0, + "content": "1529", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 153, + 505, + 167 + ], + "score": 1.0, + "content": "ZINC-full. We also run our method on the full ZINC dataset, termed ZINC-full. The result we", + "type": "text", + "cross_page": true + } + ], + "index": 7, + "is_list_start_line": true + }, + { + "bbox": [ + 83, + 164, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 83, + 167, + 99, + 177 + ], + "score": 1.0, + "content": "1530", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 104, + 164, + 505, + 178 + ], + "score": 1.0, + "content": "report for SignNet is a larger version of the GatedGCN base model with a SignNet that takes in", + "type": "text", + "cross_page": true + } + ], + "index": 8, + "is_list_start_line": true + }, + { + "bbox": [ + 82, + 176, + 506, + 189 + ], + "spans": [ + { + "bbox": [ + 82, + 178, + 99, + 189 + ], + "score": 1.0, + "content": "1531", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 176, + 506, + 189 + ], + "score": 1.0, + "content": "all eigenvectors. This model has 994,113 parameters in total. All baseline results are from their", + "type": "text", + "cross_page": true + } + ], + "index": 9, + "is_list_start_line": true + }, + { + "bbox": [ + 83, + 186, + 391, + 200 + ], + "spans": [ + { + "bbox": [ + 83, + 190, + 99, + 199 + ], + "score": 1.0, + "content": "1532", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 186, + 391, + 200 + ], + "score": 1.0, + "content": "respective papers, except for GIN, which is from [Bodnar et al., 2021].", + "type": "text", + "cross_page": true + } + ], + "index": 10, + "is_list_start_line": true + } + ], + "index": 57, + "bbox_fs": [ + 82, + 600, + 507, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 83, + 73, + 505, + 149 + ], + "lines": [ + { + "bbox": [ + 83, + 72, + 505, + 85 + ], + "spans": [ + { + "bbox": [ + 83, + 75, + 99, + 84 + ], + "score": 1.0, + "content": "1522", + "type": "text" + }, + { + "bbox": [ + 106, + 72, + 505, + 85 + ], + "score": 1.0, + "content": "Our training pipeline largely follows that of Dwivedi et al. 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The Sparse Transformer base model architecture we use, which", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 83, + 105, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 83, + 108, + 99, + 117 + ], + "score": 1.0, + "content": "1525", + "type": "text" + }, + { + "bbox": [ + 106, + 105, + 506, + 118 + ], + "score": 1.0, + "content": "like GAT computes attention only across neighbouring nodes, is introduced by Kreuzer et al. 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The result we", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 83, + 164, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 83, + 167, + 99, + 177 + ], + "score": 1.0, + "content": "1530", + "type": "text" + }, + { + "bbox": [ + 104, + 164, + 505, + 178 + ], + "score": 1.0, + "content": "report for SignNet is a larger version of the GatedGCN base model with a SignNet that takes in", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 82, + 176, + 506, + 189 + ], + "spans": [ + { + "bbox": [ + 82, + 178, + 99, + 189 + ], + "score": 1.0, + "content": "1531", + "type": "text" + }, + { + "bbox": [ + 106, + 176, + 506, + 189 + ], + "score": 1.0, + "content": "all eigenvectors. This model has 994,113 parameters in total. 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We use the same data split as Morris et al. [2020b]. Our base model is a GIN that takes", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 226, + 505, + 237 + ], + "spans": [ + { + "bbox": [ + 106, + 226, + 371, + 237 + ], + "score": 1.0, + "content": "in edge features (i.e. a GINE). The SignNet consists of GIN for", + "type": "text" + }, + { + "bbox": [ + 371, + 226, + 379, + 237 + ], + "score": 0.85, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 226, + 472, + 237 + ], + "score": 1.0, + "content": "and a Transformer for", + "type": "text" + }, + { + "bbox": [ + 473, + 227, + 479, + 237 + ], + "score": 0.79, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 226, + 505, + 237 + ], + "score": 1.0, + "content": ", as in", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 236, + 506, + 248 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 506, + 248 + ], + "score": 1.0, + "content": "the counting substructures and graph property regression experiments in Section 4.2. The model", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 247, + 506, + 260 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 506, + 260 + ], + "score": 1.0, + "content": "has 907,371 parameters in total. Our training setting is very similar to that of Morris et al. [2022],", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 258, + 506, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 506, + 271 + ], + "score": 1.0, + "content": "as we build off of their code. We train with an Adam optimizer [Kingma and Ba, 2014] with a", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 269, + 506, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 269, + 506, + 281 + ], + "score": 1.0, + "content": "starting learning rate of .001, and a minimum learning rate of .000001. The learning rate schedule", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 280, + 506, + 292 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 506, + 292 + ], + "score": 1.0, + "content": "cuts the learning rate in half with a patience of 20 epochs, and training ends when we reach the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 291, + 506, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 506, + 303 + ], + "score": 1.0, + "content": "minimum learning rate. All baseline results are from their respective papers, except for GIN, which", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 302, + 223, + 313 + ], + "spans": [ + { + "bbox": [ + 106, + 302, + 223, + 313 + ], + "score": 1.0, + "content": "is from [Morris et al., 2022].", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 15.5 + }, + { + "type": "title", + "bbox": [ + 84, + 328, + 286, + 340 + ], + "lines": [ + { + "bbox": [ + 81, + 327, + 287, + 342 + ], + "spans": [ + { + "bbox": [ + 81, + 327, + 287, + 342 + ], + "score": 1.0, + "content": "1543 K.3 Spectral Graph Convolution Details", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 103, + 349, + 505, + 470 + ], + "lines": [ + { + "bbox": [ + 105, + 349, + 506, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 506, + 362 + ], + "score": 1.0, + "content": "In Appendix J.2, we conduct node regression experiments for learning spectral graph convolutions.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 360, + 504, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 460, + 374 + ], + "score": 1.0, + "content": "The experimental setup is mostly taken from He et al. [2021]. However, we resize the", + "type": "text" + }, + { + "bbox": [ + 461, + 361, + 504, + 371 + ], + "score": 0.88, + "content": "1 0 0 \\times 1 0 0", + "type": "inline_equation" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 371, + 504, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 148, + 383 + ], + "score": 1.0, + "content": "images to", + "type": "text" + }, + { + "bbox": [ + 148, + 372, + 182, + 382 + ], + "score": 0.9, + "content": "3 2 \\times 3 2", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 371, + 468, + 383 + ], + "score": 1.0, + "content": ". Thus, each image is viewed as a 1024-node graph. The node features", + "type": "text" + }, + { + "bbox": [ + 469, + 372, + 504, + 382 + ], + "score": 0.88, + "content": "X \\in \\mathbb { R } ^ { n }", + "type": "inline_equation" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 382, + 505, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 382, + 505, + 394 + ], + "score": 1.0, + "content": "are the grayscale pixel intensities of each node. Just as in He et al. [2021], we only train and", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 393, + 505, + 404 + ], + "spans": [ + { + "bbox": [ + 106, + 393, + 505, + 404 + ], + "score": 1.0, + "content": "evaluate on nodes that are not connected to the boundary of the grid (that is, we only evaluate on the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 403, + 506, + 416 + ], + "spans": [ + { + "bbox": [ + 106, + 405, + 140, + 415 + ], + "score": 0.89, + "content": "2 8 \\times 2 8", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 403, + 506, + 416 + ], + "score": 1.0, + "content": "middle section). For all experiments we limit each model to 50,000 parameters. We use the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 415, + 506, + 427 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 506, + 427 + ], + "score": 1.0, + "content": "Adam [Kingma and Ba, 2014] optimizer for all experiments. 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This takes in the output of", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 83, + 486, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 83, + 488, + 100, + 498 + ], + "score": 1.0, + "content": "1556", + "type": "text" + }, + { + "bbox": [ + 106, + 486, + 505, + 498 + ], + "score": 1.0, + "content": "SignNet or BasisNet and concatenates it with the node features, then outputs a scalar prediction for", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 83, + 496, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 83, + 499, + 99, + 508 + ], + "score": 1.0, + "content": "1557", + "type": "text" + }, + { + "bbox": [ + 105, + 496, + 505, + 510 + ], + "score": 1.0, + "content": "each node. 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