diff --git "a/parse/train/S1ldO2EFPr/S1ldO2EFPr_middle.json" "b/parse/train/S1ldO2EFPr/S1ldO2EFPr_middle.json"
new file mode 100644--- /dev/null
+++ "b/parse/train/S1ldO2EFPr/S1ldO2EFPr_middle.json"
@@ -0,0 +1,101642 @@
+{
+ "pdf_info": [
+ {
+ "preproc_blocks": [
+ {
+ "type": "title",
+ "bbox": [
+ 107,
+ 78,
+ 502,
+ 116
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 107,
+ 78,
+ 505,
+ 97
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 78,
+ 505,
+ 97
+ ],
+ "score": 1.0,
+ "content": "GRAPH NEURAL NETWORKS EXPONENTIALLY LOSE",
+ "type": "text"
+ }
+ ],
+ "index": 0
+ },
+ {
+ "bbox": [
+ 106,
+ 99,
+ 470,
+ 118
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 99,
+ 470,
+ 118
+ ],
+ "score": 1.0,
+ "content": "EXPRESSIVE POWER FOR NODE CLASSIFICATION",
+ "type": "text"
+ }
+ ],
+ "index": 1
+ }
+ ],
+ "index": 0.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 113,
+ 133,
+ 348,
+ 181
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 111,
+ 134,
+ 246,
+ 147
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 111,
+ 134,
+ 246,
+ 147
+ ],
+ "score": 1.0,
+ "content": "Kenta Oono1, 2, Taiji Suzuki1, 3",
+ "type": "text"
+ }
+ ],
+ "index": 2
+ },
+ {
+ "bbox": [
+ 112,
+ 145,
+ 349,
+ 160
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 112,
+ 145,
+ 349,
+ 160
+ ],
+ "score": 1.0,
+ "content": "{kenta oono, taiji}@mist.i.u-tokyo.ac.jp",
+ "type": "text"
+ }
+ ],
+ "index": 3
+ },
+ {
+ "bbox": [
+ 112,
+ 156,
+ 322,
+ 170
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 112,
+ 156,
+ 322,
+ 170
+ ],
+ "score": 1.0,
+ "content": "1The University of Tokyo 2Preferred Networks, Inc.",
+ "type": "text"
+ }
+ ],
+ "index": 4
+ },
+ {
+ "bbox": [
+ 111,
+ 167,
+ 340,
+ 183
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 111,
+ 167,
+ 340,
+ 183
+ ],
+ "score": 1.0,
+ "content": "3RIKEN Center for Advanced Intelligence Project (AIP)",
+ "type": "text"
+ }
+ ],
+ "index": 5
+ }
+ ],
+ "index": 3.5
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 278,
+ 210,
+ 333,
+ 222
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 276,
+ 208,
+ 336,
+ 224
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 276,
+ 208,
+ 336,
+ 224
+ ],
+ "score": 1.0,
+ "content": "ABSTRACT",
+ "type": "text"
+ }
+ ],
+ "index": 6
+ }
+ ],
+ "index": 6
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 143,
+ 234,
+ 468,
+ 443
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 142,
+ 234,
+ 469,
+ 247
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 142,
+ 234,
+ 469,
+ 247
+ ],
+ "score": 1.0,
+ "content": "Graph Neural Networks (graph NNs) are a promising deep learning approach for",
+ "type": "text"
+ }
+ ],
+ "index": 7
+ },
+ {
+ "bbox": [
+ 142,
+ 246,
+ 469,
+ 258
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 142,
+ 246,
+ 469,
+ 258
+ ],
+ "score": 1.0,
+ "content": "analyzing graph-structured data. However, it is known that they do not improve",
+ "type": "text"
+ }
+ ],
+ "index": 8
+ },
+ {
+ "bbox": [
+ 141,
+ 255,
+ 470,
+ 270
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 141,
+ 255,
+ 470,
+ 270
+ ],
+ "score": 1.0,
+ "content": "(or sometimes worsen) their predictive performance as we pile up many layers and",
+ "type": "text"
+ }
+ ],
+ "index": 9
+ },
+ {
+ "bbox": [
+ 141,
+ 267,
+ 470,
+ 280
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 141,
+ 267,
+ 470,
+ 280
+ ],
+ "score": 1.0,
+ "content": "add non-lineality. To tackle this problem, we investigate the expressive power of",
+ "type": "text"
+ }
+ ],
+ "index": 10
+ },
+ {
+ "bbox": [
+ 141,
+ 278,
+ 470,
+ 290
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 141,
+ 278,
+ 470,
+ 290
+ ],
+ "score": 1.0,
+ "content": "graph NNs via their asymptotic behaviors as the layer size tends to infinity. Our",
+ "type": "text"
+ }
+ ],
+ "index": 11
+ },
+ {
+ "bbox": [
+ 141,
+ 290,
+ 470,
+ 301
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 141,
+ 290,
+ 470,
+ 301
+ ],
+ "score": 1.0,
+ "content": "strategy is to generalize the forward propagation of a Graph Convolutional Net-",
+ "type": "text"
+ }
+ ],
+ "index": 12
+ },
+ {
+ "bbox": [
+ 141,
+ 300,
+ 469,
+ 313
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 141,
+ 300,
+ 469,
+ 313
+ ],
+ "score": 1.0,
+ "content": "work (GCN), which is a popular graph NN variant, as a specific dynamical sys-",
+ "type": "text"
+ }
+ ],
+ "index": 13
+ },
+ {
+ "bbox": [
+ 142,
+ 312,
+ 469,
+ 322
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 142,
+ 312,
+ 469,
+ 322
+ ],
+ "score": 1.0,
+ "content": "tem. In the case of a GCN, we show that when its weights satisfy the conditions",
+ "type": "text"
+ }
+ ],
+ "index": 14
+ },
+ {
+ "bbox": [
+ 141,
+ 322,
+ 469,
+ 334
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 141,
+ 322,
+ 469,
+ 334
+ ],
+ "score": 1.0,
+ "content": "determined by the spectra of the (augmented) normalized Laplacian, its output ex-",
+ "type": "text"
+ }
+ ],
+ "index": 15
+ },
+ {
+ "bbox": [
+ 141,
+ 333,
+ 470,
+ 345
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 141,
+ 333,
+ 470,
+ 345
+ ],
+ "score": 1.0,
+ "content": "ponentially approaches the set of signals that carry information of the connected",
+ "type": "text"
+ }
+ ],
+ "index": 16
+ },
+ {
+ "bbox": [
+ 141,
+ 344,
+ 469,
+ 356
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 141,
+ 344,
+ 469,
+ 356
+ ],
+ "score": 1.0,
+ "content": "components and node degrees only for distinguishing nodes. Our theory enables",
+ "type": "text"
+ }
+ ],
+ "index": 17
+ },
+ {
+ "bbox": [
+ 141,
+ 355,
+ 469,
+ 367
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 141,
+ 355,
+ 469,
+ 367
+ ],
+ "score": 1.0,
+ "content": "us to relate the expressive power of GCNs with the topological information of the",
+ "type": "text"
+ }
+ ],
+ "index": 18
+ },
+ {
+ "bbox": [
+ 141,
+ 365,
+ 470,
+ 379
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 141,
+ 365,
+ 470,
+ 379
+ ],
+ "score": 1.0,
+ "content": "underlying graphs inherent in the graph spectra. To demonstrate this, we charac-",
+ "type": "text"
+ }
+ ],
+ "index": 19
+ },
+ {
+ "bbox": [
+ 141,
+ 376,
+ 469,
+ 388
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 141,
+ 376,
+ 469,
+ 388
+ ],
+ "score": 1.0,
+ "content": "terize the asymptotic behavior of GCNs on the Erdos – R˝ enyi graph. We show´",
+ "type": "text"
+ }
+ ],
+ "index": 20
+ },
+ {
+ "bbox": [
+ 141,
+ 388,
+ 470,
+ 400
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 141,
+ 388,
+ 470,
+ 400
+ ],
+ "score": 1.0,
+ "content": "that when the Erdos – R ˝ enyi graph is sufficiently dense and large, a broad range ´",
+ "type": "text"
+ }
+ ],
+ "index": 21
+ },
+ {
+ "bbox": [
+ 141,
+ 398,
+ 469,
+ 411
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 141,
+ 398,
+ 469,
+ 411
+ ],
+ "score": 1.0,
+ "content": "of GCNs on it suffers from the “information loss” in the limit of infinite layers",
+ "type": "text"
+ }
+ ],
+ "index": 22
+ },
+ {
+ "bbox": [
+ 142,
+ 410,
+ 469,
+ 422
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 142,
+ 410,
+ 469,
+ 422
+ ],
+ "score": 1.0,
+ "content": "with high probability. Based on the theory, we provide a principled guideline for",
+ "type": "text"
+ }
+ ],
+ "index": 23
+ },
+ {
+ "bbox": [
+ 142,
+ 420,
+ 470,
+ 434
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 142,
+ 420,
+ 470,
+ 434
+ ],
+ "score": 1.0,
+ "content": "weight normalization of graph NNs. We experimentally confirm that the proposed",
+ "type": "text"
+ }
+ ],
+ "index": 24
+ },
+ {
+ "bbox": [
+ 142,
+ 432,
+ 442,
+ 443
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 142,
+ 432,
+ 442,
+ 443
+ ],
+ "score": 1.0,
+ "content": "weight scaling enhances the predictive performance of GCNs in real data1.",
+ "type": "text"
+ }
+ ],
+ "index": 25
+ }
+ ],
+ "index": 16
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 108,
+ 463,
+ 206,
+ 475
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 462,
+ 208,
+ 478
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 462,
+ 208,
+ 478
+ ],
+ "score": 1.0,
+ "content": "1 INTRODUCTION",
+ "type": "text"
+ }
+ ],
+ "index": 26
+ }
+ ],
+ "index": 26
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 488,
+ 505,
+ 565
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 488,
+ 505,
+ 500
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 488,
+ 505,
+ 500
+ ],
+ "score": 1.0,
+ "content": "Motivated by the success of Deep Learning (DL), several attempts have been made to apply DL mod-",
+ "type": "text"
+ }
+ ],
+ "index": 27
+ },
+ {
+ "bbox": [
+ 105,
+ 499,
+ 505,
+ 512
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 499,
+ 505,
+ 512
+ ],
+ "score": 1.0,
+ "content": "els to non-Euclidean data, particularly, graph-structured data such as chemical compounds, social",
+ "type": "text"
+ }
+ ],
+ "index": 28
+ },
+ {
+ "bbox": [
+ 105,
+ 510,
+ 505,
+ 522
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 510,
+ 505,
+ 522
+ ],
+ "score": 1.0,
+ "content": "networks, and polygons. Recently, Graph Neural Networks (graph NNs) (Duvenaud et al., 2015; Li",
+ "type": "text"
+ }
+ ],
+ "index": 29
+ },
+ {
+ "bbox": [
+ 105,
+ 520,
+ 505,
+ 533
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 520,
+ 505,
+ 533
+ ],
+ "score": 1.0,
+ "content": "et al., 2016; Gilmer et al., 2017; Hamilton et al., 2017; Kipf & Welling, 2017; Nguyen et al., 2017;",
+ "type": "text"
+ }
+ ],
+ "index": 30
+ },
+ {
+ "bbox": [
+ 105,
+ 531,
+ 506,
+ 545
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 531,
+ 506,
+ 545
+ ],
+ "score": 1.0,
+ "content": "Schlichtkrull et al., 2018; Battaglia et al., 2018; Xu et al., 2019; Wu et al., 2019a) have emerged as a",
+ "type": "text"
+ }
+ ],
+ "index": 31
+ },
+ {
+ "bbox": [
+ 105,
+ 542,
+ 506,
+ 555
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 542,
+ 506,
+ 555
+ ],
+ "score": 1.0,
+ "content": "promising approach. However, despite their practical popularity, theoretical research of graph NNs",
+ "type": "text"
+ }
+ ],
+ "index": 32
+ },
+ {
+ "bbox": [
+ 105,
+ 552,
+ 245,
+ 567
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 552,
+ 245,
+ 567
+ ],
+ "score": 1.0,
+ "content": "has not been explored extensively.",
+ "type": "text"
+ }
+ ],
+ "index": 33
+ }
+ ],
+ "index": 30
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 570,
+ 505,
+ 713
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 570,
+ 505,
+ 584
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 570,
+ 505,
+ 584
+ ],
+ "score": 1.0,
+ "content": "The characterization of DL model expressive power, i.e., to identify what function classes DL mod-",
+ "type": "text"
+ }
+ ],
+ "index": 34
+ },
+ {
+ "bbox": [
+ 105,
+ 581,
+ 505,
+ 594
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 581,
+ 505,
+ 594
+ ],
+ "score": 1.0,
+ "content": "els can (approximately) represent, is a fundamental question in theoretical research of DL. Many",
+ "type": "text"
+ }
+ ],
+ "index": 35
+ },
+ {
+ "bbox": [
+ 106,
+ 593,
+ 504,
+ 605
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 593,
+ 504,
+ 605
+ ],
+ "score": 1.0,
+ "content": "studies have been conducted for Fully Connected Neural Networks (FNNs) (Cybenko, 1989; Hornik,",
+ "type": "text"
+ }
+ ],
+ "index": 36
+ },
+ {
+ "bbox": [
+ 106,
+ 603,
+ 505,
+ 616
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 603,
+ 505,
+ 616
+ ],
+ "score": 1.0,
+ "content": "1991; Hornik et al., 1989; Barron, 1993; Mhaskar, 1993; Sonoda & Murata, 2017; Yarotsky, 2017)",
+ "type": "text"
+ }
+ ],
+ "index": 37
+ },
+ {
+ "bbox": [
+ 105,
+ 614,
+ 505,
+ 627
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 614,
+ 505,
+ 627
+ ],
+ "score": 1.0,
+ "content": "and Convolutional Neural Networks (CNNs) (Petersen & Voigtlaender, 2018; Zhou, 2018; Oono &",
+ "type": "text"
+ }
+ ],
+ "index": 38
+ },
+ {
+ "bbox": [
+ 105,
+ 624,
+ 505,
+ 639
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 624,
+ 505,
+ 639
+ ],
+ "score": 1.0,
+ "content": "Suzuki, 2019). For such models, we have theoretical and empirical justification that deep and non-",
+ "type": "text"
+ }
+ ],
+ "index": 39
+ },
+ {
+ "bbox": [
+ 105,
+ 635,
+ 506,
+ 650
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 635,
+ 506,
+ 650
+ ],
+ "score": 1.0,
+ "content": "linear architectures can enhance representation power (Telgarsky, 2016; Chen et al., 2018b; Zhou",
+ "type": "text"
+ }
+ ],
+ "index": 40
+ },
+ {
+ "bbox": [
+ 105,
+ 646,
+ 506,
+ 662
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 646,
+ 506,
+ 662
+ ],
+ "score": 1.0,
+ "content": "& Feng, 2018). However, for graph NNs, several papers have reported that node representations go",
+ "type": "text"
+ }
+ ],
+ "index": 41
+ },
+ {
+ "bbox": [
+ 105,
+ 657,
+ 506,
+ 672
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 657,
+ 506,
+ 672
+ ],
+ "score": 1.0,
+ "content": "indistinguishable (known as over-smoothing) and prediction performances severely degrade when",
+ "type": "text"
+ }
+ ],
+ "index": 42
+ },
+ {
+ "bbox": [
+ 105,
+ 668,
+ 505,
+ 682
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 668,
+ 505,
+ 682
+ ],
+ "score": 1.0,
+ "content": "we stack many layers (Kipf & Welling, 2017; Wu et al., 2019b; Li et al., 2018). Besides, Wu et al.",
+ "type": "text"
+ }
+ ],
+ "index": 43
+ },
+ {
+ "bbox": [
+ 105,
+ 680,
+ 505,
+ 693
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 680,
+ 505,
+ 693
+ ],
+ "score": 1.0,
+ "content": "(2019a) reported that graph NNs achieved comparable performance even if they removed interme-",
+ "type": "text"
+ }
+ ],
+ "index": 44
+ },
+ {
+ "bbox": [
+ 105,
+ 691,
+ 505,
+ 704
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 691,
+ 505,
+ 704
+ ],
+ "score": 1.0,
+ "content": "diate non-linear functions. These studies posed a question about the current architecture and made",
+ "type": "text"
+ }
+ ],
+ "index": 45
+ },
+ {
+ "bbox": [
+ 105,
+ 701,
+ 439,
+ 716
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 701,
+ 439,
+ 716
+ ],
+ "score": 1.0,
+ "content": "us aware of the need for the theoretical analysis of the graph NN expressive power.",
+ "type": "text"
+ }
+ ],
+ "index": 46
+ }
+ ],
+ "index": 40
+ }
+ ],
+ "page_idx": 0,
+ "page_size": [
+ 612,
+ 792
+ ],
+ "discarded_blocks": [
+ {
+ "type": "discarded",
+ "bbox": [
+ 118,
+ 722,
+ 435,
+ 732
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 119,
+ 720,
+ 437,
+ 734
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 119,
+ 720,
+ 437,
+ 734
+ ],
+ "score": 1.0,
+ "content": "1Code is available at https://github.com/delta2323/gnn-asymptotics.",
+ "type": "text"
+ }
+ ]
+ }
+ ]
+ },
+ {
+ "type": "discarded",
+ "bbox": [
+ 107,
+ 27,
+ 294,
+ 37
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 25,
+ 294,
+ 38
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 25,
+ 294,
+ 38
+ ],
+ "score": 1.0,
+ "content": "Published as a conference paper at ICLR 2020",
+ "type": "text"
+ }
+ ]
+ }
+ ]
+ },
+ {
+ "type": "discarded",
+ "bbox": [
+ 302,
+ 752,
+ 308,
+ 760
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 302,
+ 751,
+ 309,
+ 762
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 302,
+ 751,
+ 309,
+ 762
+ ],
+ "score": 1.0,
+ "content": "1",
+ "type": "text"
+ }
+ ]
+ }
+ ]
+ }
+ ],
+ "para_blocks": [
+ {
+ "type": "title",
+ "bbox": [
+ 107,
+ 78,
+ 502,
+ 116
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 107,
+ 78,
+ 505,
+ 97
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 78,
+ 505,
+ 97
+ ],
+ "score": 1.0,
+ "content": "GRAPH NEURAL NETWORKS EXPONENTIALLY LOSE",
+ "type": "text"
+ }
+ ],
+ "index": 0
+ },
+ {
+ "bbox": [
+ 106,
+ 99,
+ 470,
+ 118
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 99,
+ 470,
+ 118
+ ],
+ "score": 1.0,
+ "content": "EXPRESSIVE POWER FOR NODE CLASSIFICATION",
+ "type": "text"
+ }
+ ],
+ "index": 1
+ }
+ ],
+ "index": 0.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 113,
+ 133,
+ 348,
+ 181
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 111,
+ 134,
+ 246,
+ 147
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 111,
+ 134,
+ 246,
+ 147
+ ],
+ "score": 1.0,
+ "content": "Kenta Oono1, 2, Taiji Suzuki1, 3",
+ "type": "text"
+ }
+ ],
+ "index": 2
+ },
+ {
+ "bbox": [
+ 112,
+ 145,
+ 349,
+ 160
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 112,
+ 145,
+ 349,
+ 160
+ ],
+ "score": 1.0,
+ "content": "{kenta oono, taiji}@mist.i.u-tokyo.ac.jp",
+ "type": "text"
+ }
+ ],
+ "index": 3
+ },
+ {
+ "bbox": [
+ 112,
+ 156,
+ 322,
+ 170
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 112,
+ 156,
+ 322,
+ 170
+ ],
+ "score": 1.0,
+ "content": "1The University of Tokyo 2Preferred Networks, Inc.",
+ "type": "text"
+ }
+ ],
+ "index": 4
+ },
+ {
+ "bbox": [
+ 111,
+ 167,
+ 340,
+ 183
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 111,
+ 167,
+ 340,
+ 183
+ ],
+ "score": 1.0,
+ "content": "3RIKEN Center for Advanced Intelligence Project (AIP)",
+ "type": "text"
+ }
+ ],
+ "index": 5
+ }
+ ],
+ "index": 3.5,
+ "bbox_fs": [
+ 111,
+ 134,
+ 349,
+ 183
+ ]
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 278,
+ 210,
+ 333,
+ 222
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 276,
+ 208,
+ 336,
+ 224
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 276,
+ 208,
+ 336,
+ 224
+ ],
+ "score": 1.0,
+ "content": "ABSTRACT",
+ "type": "text"
+ }
+ ],
+ "index": 6
+ }
+ ],
+ "index": 6
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 143,
+ 234,
+ 468,
+ 443
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 142,
+ 234,
+ 469,
+ 247
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 142,
+ 234,
+ 469,
+ 247
+ ],
+ "score": 1.0,
+ "content": "Graph Neural Networks (graph NNs) are a promising deep learning approach for",
+ "type": "text"
+ }
+ ],
+ "index": 7
+ },
+ {
+ "bbox": [
+ 142,
+ 246,
+ 469,
+ 258
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 142,
+ 246,
+ 469,
+ 258
+ ],
+ "score": 1.0,
+ "content": "analyzing graph-structured data. However, it is known that they do not improve",
+ "type": "text"
+ }
+ ],
+ "index": 8
+ },
+ {
+ "bbox": [
+ 141,
+ 255,
+ 470,
+ 270
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 141,
+ 255,
+ 470,
+ 270
+ ],
+ "score": 1.0,
+ "content": "(or sometimes worsen) their predictive performance as we pile up many layers and",
+ "type": "text"
+ }
+ ],
+ "index": 9
+ },
+ {
+ "bbox": [
+ 141,
+ 267,
+ 470,
+ 280
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 141,
+ 267,
+ 470,
+ 280
+ ],
+ "score": 1.0,
+ "content": "add non-lineality. To tackle this problem, we investigate the expressive power of",
+ "type": "text"
+ }
+ ],
+ "index": 10
+ },
+ {
+ "bbox": [
+ 141,
+ 278,
+ 470,
+ 290
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 141,
+ 278,
+ 470,
+ 290
+ ],
+ "score": 1.0,
+ "content": "graph NNs via their asymptotic behaviors as the layer size tends to infinity. Our",
+ "type": "text"
+ }
+ ],
+ "index": 11
+ },
+ {
+ "bbox": [
+ 141,
+ 290,
+ 470,
+ 301
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 141,
+ 290,
+ 470,
+ 301
+ ],
+ "score": 1.0,
+ "content": "strategy is to generalize the forward propagation of a Graph Convolutional Net-",
+ "type": "text"
+ }
+ ],
+ "index": 12
+ },
+ {
+ "bbox": [
+ 141,
+ 300,
+ 469,
+ 313
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 141,
+ 300,
+ 469,
+ 313
+ ],
+ "score": 1.0,
+ "content": "work (GCN), which is a popular graph NN variant, as a specific dynamical sys-",
+ "type": "text"
+ }
+ ],
+ "index": 13
+ },
+ {
+ "bbox": [
+ 142,
+ 312,
+ 469,
+ 322
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 142,
+ 312,
+ 469,
+ 322
+ ],
+ "score": 1.0,
+ "content": "tem. In the case of a GCN, we show that when its weights satisfy the conditions",
+ "type": "text"
+ }
+ ],
+ "index": 14
+ },
+ {
+ "bbox": [
+ 141,
+ 322,
+ 469,
+ 334
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 141,
+ 322,
+ 469,
+ 334
+ ],
+ "score": 1.0,
+ "content": "determined by the spectra of the (augmented) normalized Laplacian, its output ex-",
+ "type": "text"
+ }
+ ],
+ "index": 15
+ },
+ {
+ "bbox": [
+ 141,
+ 333,
+ 470,
+ 345
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 141,
+ 333,
+ 470,
+ 345
+ ],
+ "score": 1.0,
+ "content": "ponentially approaches the set of signals that carry information of the connected",
+ "type": "text"
+ }
+ ],
+ "index": 16
+ },
+ {
+ "bbox": [
+ 141,
+ 344,
+ 469,
+ 356
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 141,
+ 344,
+ 469,
+ 356
+ ],
+ "score": 1.0,
+ "content": "components and node degrees only for distinguishing nodes. Our theory enables",
+ "type": "text"
+ }
+ ],
+ "index": 17
+ },
+ {
+ "bbox": [
+ 141,
+ 355,
+ 469,
+ 367
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 141,
+ 355,
+ 469,
+ 367
+ ],
+ "score": 1.0,
+ "content": "us to relate the expressive power of GCNs with the topological information of the",
+ "type": "text"
+ }
+ ],
+ "index": 18
+ },
+ {
+ "bbox": [
+ 141,
+ 365,
+ 470,
+ 379
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 141,
+ 365,
+ 470,
+ 379
+ ],
+ "score": 1.0,
+ "content": "underlying graphs inherent in the graph spectra. To demonstrate this, we charac-",
+ "type": "text"
+ }
+ ],
+ "index": 19
+ },
+ {
+ "bbox": [
+ 141,
+ 376,
+ 469,
+ 388
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 141,
+ 376,
+ 469,
+ 388
+ ],
+ "score": 1.0,
+ "content": "terize the asymptotic behavior of GCNs on the Erdos – R˝ enyi graph. We show´",
+ "type": "text"
+ }
+ ],
+ "index": 20
+ },
+ {
+ "bbox": [
+ 141,
+ 388,
+ 470,
+ 400
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 141,
+ 388,
+ 470,
+ 400
+ ],
+ "score": 1.0,
+ "content": "that when the Erdos – R ˝ enyi graph is sufficiently dense and large, a broad range ´",
+ "type": "text"
+ }
+ ],
+ "index": 21
+ },
+ {
+ "bbox": [
+ 141,
+ 398,
+ 469,
+ 411
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 141,
+ 398,
+ 469,
+ 411
+ ],
+ "score": 1.0,
+ "content": "of GCNs on it suffers from the “information loss” in the limit of infinite layers",
+ "type": "text"
+ }
+ ],
+ "index": 22
+ },
+ {
+ "bbox": [
+ 142,
+ 410,
+ 469,
+ 422
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 142,
+ 410,
+ 469,
+ 422
+ ],
+ "score": 1.0,
+ "content": "with high probability. Based on the theory, we provide a principled guideline for",
+ "type": "text"
+ }
+ ],
+ "index": 23
+ },
+ {
+ "bbox": [
+ 142,
+ 420,
+ 470,
+ 434
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 142,
+ 420,
+ 470,
+ 434
+ ],
+ "score": 1.0,
+ "content": "weight normalization of graph NNs. We experimentally confirm that the proposed",
+ "type": "text"
+ }
+ ],
+ "index": 24
+ },
+ {
+ "bbox": [
+ 142,
+ 432,
+ 442,
+ 443
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 142,
+ 432,
+ 442,
+ 443
+ ],
+ "score": 1.0,
+ "content": "weight scaling enhances the predictive performance of GCNs in real data1.",
+ "type": "text"
+ }
+ ],
+ "index": 25
+ }
+ ],
+ "index": 16,
+ "bbox_fs": [
+ 141,
+ 234,
+ 470,
+ 443
+ ]
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 108,
+ 463,
+ 206,
+ 475
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 462,
+ 208,
+ 478
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 462,
+ 208,
+ 478
+ ],
+ "score": 1.0,
+ "content": "1 INTRODUCTION",
+ "type": "text"
+ }
+ ],
+ "index": 26
+ }
+ ],
+ "index": 26
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 488,
+ 505,
+ 565
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 488,
+ 505,
+ 500
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 488,
+ 505,
+ 500
+ ],
+ "score": 1.0,
+ "content": "Motivated by the success of Deep Learning (DL), several attempts have been made to apply DL mod-",
+ "type": "text"
+ }
+ ],
+ "index": 27
+ },
+ {
+ "bbox": [
+ 105,
+ 499,
+ 505,
+ 512
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 499,
+ 505,
+ 512
+ ],
+ "score": 1.0,
+ "content": "els to non-Euclidean data, particularly, graph-structured data such as chemical compounds, social",
+ "type": "text"
+ }
+ ],
+ "index": 28
+ },
+ {
+ "bbox": [
+ 105,
+ 510,
+ 505,
+ 522
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 510,
+ 505,
+ 522
+ ],
+ "score": 1.0,
+ "content": "networks, and polygons. Recently, Graph Neural Networks (graph NNs) (Duvenaud et al., 2015; Li",
+ "type": "text"
+ }
+ ],
+ "index": 29
+ },
+ {
+ "bbox": [
+ 105,
+ 520,
+ 505,
+ 533
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 520,
+ 505,
+ 533
+ ],
+ "score": 1.0,
+ "content": "et al., 2016; Gilmer et al., 2017; Hamilton et al., 2017; Kipf & Welling, 2017; Nguyen et al., 2017;",
+ "type": "text"
+ }
+ ],
+ "index": 30
+ },
+ {
+ "bbox": [
+ 105,
+ 531,
+ 506,
+ 545
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 531,
+ 506,
+ 545
+ ],
+ "score": 1.0,
+ "content": "Schlichtkrull et al., 2018; Battaglia et al., 2018; Xu et al., 2019; Wu et al., 2019a) have emerged as a",
+ "type": "text"
+ }
+ ],
+ "index": 31
+ },
+ {
+ "bbox": [
+ 105,
+ 542,
+ 506,
+ 555
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 542,
+ 506,
+ 555
+ ],
+ "score": 1.0,
+ "content": "promising approach. However, despite their practical popularity, theoretical research of graph NNs",
+ "type": "text"
+ }
+ ],
+ "index": 32
+ },
+ {
+ "bbox": [
+ 105,
+ 552,
+ 245,
+ 567
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 552,
+ 245,
+ 567
+ ],
+ "score": 1.0,
+ "content": "has not been explored extensively.",
+ "type": "text"
+ }
+ ],
+ "index": 33
+ }
+ ],
+ "index": 30,
+ "bbox_fs": [
+ 105,
+ 488,
+ 506,
+ 567
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 570,
+ 505,
+ 713
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 570,
+ 505,
+ 584
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 570,
+ 505,
+ 584
+ ],
+ "score": 1.0,
+ "content": "The characterization of DL model expressive power, i.e., to identify what function classes DL mod-",
+ "type": "text"
+ }
+ ],
+ "index": 34
+ },
+ {
+ "bbox": [
+ 105,
+ 581,
+ 505,
+ 594
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 581,
+ 505,
+ 594
+ ],
+ "score": 1.0,
+ "content": "els can (approximately) represent, is a fundamental question in theoretical research of DL. Many",
+ "type": "text"
+ }
+ ],
+ "index": 35
+ },
+ {
+ "bbox": [
+ 106,
+ 593,
+ 504,
+ 605
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 593,
+ 504,
+ 605
+ ],
+ "score": 1.0,
+ "content": "studies have been conducted for Fully Connected Neural Networks (FNNs) (Cybenko, 1989; Hornik,",
+ "type": "text"
+ }
+ ],
+ "index": 36
+ },
+ {
+ "bbox": [
+ 106,
+ 603,
+ 505,
+ 616
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 603,
+ 505,
+ 616
+ ],
+ "score": 1.0,
+ "content": "1991; Hornik et al., 1989; Barron, 1993; Mhaskar, 1993; Sonoda & Murata, 2017; Yarotsky, 2017)",
+ "type": "text"
+ }
+ ],
+ "index": 37
+ },
+ {
+ "bbox": [
+ 105,
+ 614,
+ 505,
+ 627
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 614,
+ 505,
+ 627
+ ],
+ "score": 1.0,
+ "content": "and Convolutional Neural Networks (CNNs) (Petersen & Voigtlaender, 2018; Zhou, 2018; Oono &",
+ "type": "text"
+ }
+ ],
+ "index": 38
+ },
+ {
+ "bbox": [
+ 105,
+ 624,
+ 505,
+ 639
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 624,
+ 505,
+ 639
+ ],
+ "score": 1.0,
+ "content": "Suzuki, 2019). For such models, we have theoretical and empirical justification that deep and non-",
+ "type": "text"
+ }
+ ],
+ "index": 39
+ },
+ {
+ "bbox": [
+ 105,
+ 635,
+ 506,
+ 650
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 635,
+ 506,
+ 650
+ ],
+ "score": 1.0,
+ "content": "linear architectures can enhance representation power (Telgarsky, 2016; Chen et al., 2018b; Zhou",
+ "type": "text"
+ }
+ ],
+ "index": 40
+ },
+ {
+ "bbox": [
+ 105,
+ 646,
+ 506,
+ 662
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 646,
+ 506,
+ 662
+ ],
+ "score": 1.0,
+ "content": "& Feng, 2018). However, for graph NNs, several papers have reported that node representations go",
+ "type": "text"
+ }
+ ],
+ "index": 41
+ },
+ {
+ "bbox": [
+ 105,
+ 657,
+ 506,
+ 672
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 657,
+ 506,
+ 672
+ ],
+ "score": 1.0,
+ "content": "indistinguishable (known as over-smoothing) and prediction performances severely degrade when",
+ "type": "text"
+ }
+ ],
+ "index": 42
+ },
+ {
+ "bbox": [
+ 105,
+ 668,
+ 505,
+ 682
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 668,
+ 505,
+ 682
+ ],
+ "score": 1.0,
+ "content": "we stack many layers (Kipf & Welling, 2017; Wu et al., 2019b; Li et al., 2018). Besides, Wu et al.",
+ "type": "text"
+ }
+ ],
+ "index": 43
+ },
+ {
+ "bbox": [
+ 105,
+ 680,
+ 505,
+ 693
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 680,
+ 505,
+ 693
+ ],
+ "score": 1.0,
+ "content": "(2019a) reported that graph NNs achieved comparable performance even if they removed interme-",
+ "type": "text"
+ }
+ ],
+ "index": 44
+ },
+ {
+ "bbox": [
+ 105,
+ 691,
+ 505,
+ 704
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 691,
+ 505,
+ 704
+ ],
+ "score": 1.0,
+ "content": "diate non-linear functions. These studies posed a question about the current architecture and made",
+ "type": "text"
+ }
+ ],
+ "index": 45
+ },
+ {
+ "bbox": [
+ 105,
+ 701,
+ 439,
+ 716
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 701,
+ 439,
+ 716
+ ],
+ "score": 1.0,
+ "content": "us aware of the need for the theoretical analysis of the graph NN expressive power.",
+ "type": "text"
+ }
+ ],
+ "index": 46
+ }
+ ],
+ "index": 40,
+ "bbox_fs": [
+ 105,
+ 570,
+ 506,
+ 716
+ ]
+ }
+ ]
+ },
+ {
+ "preproc_blocks": [
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 82,
+ 505,
+ 225
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 83,
+ 504,
+ 95
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 83,
+ 504,
+ 95
+ ],
+ "score": 1.0,
+ "content": "In this paper, we investigate the expressive power of graph NNs by analyzing their asymptotic be-",
+ "type": "text"
+ }
+ ],
+ "index": 0
+ },
+ {
+ "bbox": [
+ 105,
+ 94,
+ 505,
+ 107
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 94,
+ 505,
+ 107
+ ],
+ "score": 1.0,
+ "content": "haviors as the layer size goes to infinity. Our theory gives new theoretical conditions under which",
+ "type": "text"
+ }
+ ],
+ "index": 1
+ },
+ {
+ "bbox": [
+ 105,
+ 104,
+ 506,
+ 118
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 104,
+ 506,
+ 118
+ ],
+ "score": 1.0,
+ "content": "neither layer stacking nor non-linearity contributes to improving expressive power. We consider a",
+ "type": "text"
+ }
+ ],
+ "index": 2
+ },
+ {
+ "bbox": [
+ 105,
+ 115,
+ 505,
+ 129
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 115,
+ 505,
+ 129
+ ],
+ "score": 1.0,
+ "content": "specific dynamics that includes a transition defining a Markov process and the forward propaga-",
+ "type": "text"
+ }
+ ],
+ "index": 3
+ },
+ {
+ "bbox": [
+ 105,
+ 126,
+ 506,
+ 139
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 126,
+ 506,
+ 139
+ ],
+ "score": 1.0,
+ "content": "tion of a Graph Convolutional Network (GCN) (Kipf & Welling, 2017), which is one of the most",
+ "type": "text"
+ }
+ ],
+ "index": 4
+ },
+ {
+ "bbox": [
+ 105,
+ 137,
+ 506,
+ 150
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 137,
+ 506,
+ 150
+ ],
+ "score": 1.0,
+ "content": "popular graph NN variants, as special cases. We prove that under certain conditions, the dynamics",
+ "type": "text"
+ }
+ ],
+ "index": 5
+ },
+ {
+ "bbox": [
+ 105,
+ 149,
+ 505,
+ 161
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 149,
+ 505,
+ 161
+ ],
+ "score": 1.0,
+ "content": "exponentially approaches a subspace that is invariant under the dynamics. In the case of GCN, the",
+ "type": "text"
+ }
+ ],
+ "index": 6
+ },
+ {
+ "bbox": [
+ 105,
+ 159,
+ 506,
+ 172
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 159,
+ 506,
+ 172
+ ],
+ "score": 1.0,
+ "content": "invariant space is a set of signals that correspond to the lowest frequency of graph spectra and that",
+ "type": "text"
+ }
+ ],
+ "index": 7
+ },
+ {
+ "bbox": [
+ 105,
+ 170,
+ 506,
+ 183
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 170,
+ 506,
+ 183
+ ],
+ "score": 1.0,
+ "content": "have “no information” other than connected components and node degrees for a node classification",
+ "type": "text"
+ }
+ ],
+ "index": 8
+ },
+ {
+ "bbox": [
+ 105,
+ 181,
+ 505,
+ 194
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 181,
+ 505,
+ 194
+ ],
+ "score": 1.0,
+ "content": "task whose goal is to predict the nodes’ properties in a graph. The rate of the distance between the",
+ "type": "text"
+ }
+ ],
+ "index": 9
+ },
+ {
+ "bbox": [
+ 105,
+ 191,
+ 504,
+ 205
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 191,
+ 240,
+ 205
+ ],
+ "score": 1.0,
+ "content": "output and the invariant space is",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 240,
+ 191,
+ 280,
+ 204
+ ],
+ "score": 0.93,
+ "content": "{ \\cal O } ( ( s \\lambda ) ^ { \\widehat { L } } )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 281,
+ 191,
+ 309,
+ 205
+ ],
+ "score": 1.0,
+ "content": "where",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 310,
+ 194,
+ 316,
+ 202
+ ],
+ "score": 0.71,
+ "content": "s",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 316,
+ 191,
+ 496,
+ 205
+ ],
+ "score": 1.0,
+ "content": "is the maximum singular values of weights,",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 497,
+ 192,
+ 504,
+ 202
+ ],
+ "score": 0.72,
+ "content": "\\lambda",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 10
+ },
+ {
+ "bbox": [
+ 105,
+ 202,
+ 506,
+ 216
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 202,
+ 486,
+ 216
+ ],
+ "score": 1.0,
+ "content": "is typically a quantity determined by the spectra of the (augmented) normalized Laplacian, and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 487,
+ 204,
+ 495,
+ 213
+ ],
+ "score": 0.8,
+ "content": "L",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 495,
+ 202,
+ 506,
+ 216
+ ],
+ "score": 1.0,
+ "content": "is",
+ "type": "text"
+ }
+ ],
+ "index": 11
+ },
+ {
+ "bbox": [
+ 105,
+ 214,
+ 457,
+ 227
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 214,
+ 457,
+ 227
+ ],
+ "score": 1.0,
+ "content": "the layer size. See Sections 3.3 (general case) and 4 (GCN case) for precise statements.",
+ "type": "text"
+ }
+ ],
+ "index": 12
+ }
+ ],
+ "index": 6
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 231,
+ 505,
+ 308
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 230,
+ 505,
+ 243
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 230,
+ 505,
+ 243
+ ],
+ "score": 1.0,
+ "content": "We can interpret our theorem as the generalization of the well-known property that if a finite and dis-",
+ "type": "text"
+ }
+ ],
+ "index": 13
+ },
+ {
+ "bbox": [
+ 105,
+ 241,
+ 505,
+ 255
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 241,
+ 505,
+ 255
+ ],
+ "score": 1.0,
+ "content": "crete Markov process is irreducible and aperiodic, it exponentially converges to a unique equilibrium",
+ "type": "text"
+ }
+ ],
+ "index": 14
+ },
+ {
+ "bbox": [
+ 105,
+ 252,
+ 505,
+ 266
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 252,
+ 505,
+ 266
+ ],
+ "score": 1.0,
+ "content": "and the eigenvalues of its transition matrix determine the convergence rate (see, e.g., Chung & Gra-",
+ "type": "text"
+ }
+ ],
+ "index": 15
+ },
+ {
+ "bbox": [
+ 105,
+ 264,
+ 505,
+ 276
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 264,
+ 505,
+ 276
+ ],
+ "score": 1.0,
+ "content": "ham (1997)). Different from the Markov process case, which is linear, the existence of intermediate",
+ "type": "text"
+ }
+ ],
+ "index": 16
+ },
+ {
+ "bbox": [
+ 105,
+ 275,
+ 505,
+ 287
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 275,
+ 505,
+ 287
+ ],
+ "score": 1.0,
+ "content": "non-linear functions complicates the analysis. We overcame this problem by leveraging the combi-",
+ "type": "text"
+ }
+ ],
+ "index": 17
+ },
+ {
+ "bbox": [
+ 105,
+ 285,
+ 505,
+ 299
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 285,
+ 505,
+ 299
+ ],
+ "score": 1.0,
+ "content": "nation of the ReLU activation function (Krizhevsky et al., 2012) and the positivity of eigenvectors",
+ "type": "text"
+ }
+ ],
+ "index": 18
+ },
+ {
+ "bbox": [
+ 105,
+ 296,
+ 372,
+ 309
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 296,
+ 372,
+ 309
+ ],
+ "score": 1.0,
+ "content": "of the Laplacian associated with the smallest positive eigenvalues.",
+ "type": "text"
+ }
+ ],
+ "index": 19
+ }
+ ],
+ "index": 16
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 313,
+ 505,
+ 414
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 314,
+ 505,
+ 326
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 314,
+ 505,
+ 326
+ ],
+ "score": 1.0,
+ "content": "Our theory enables us to investigate asymptotic behaviors of graph NNs via the spectral distribution",
+ "type": "text"
+ }
+ ],
+ "index": 20
+ },
+ {
+ "bbox": [
+ 104,
+ 323,
+ 505,
+ 338
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 323,
+ 505,
+ 338
+ ],
+ "score": 1.0,
+ "content": "of the underlying graphs. To demonstrate this, we take GCNs defined on the Erdos – R ˝ enyi graph ´",
+ "type": "text"
+ }
+ ],
+ "index": 21
+ },
+ {
+ "bbox": [
+ 107,
+ 335,
+ 505,
+ 349
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 336,
+ 129,
+ 348
+ ],
+ "score": 0.9,
+ "content": "G _ { N , p }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 129,
+ 335,
+ 175,
+ 349
+ ],
+ "score": 1.0,
+ "content": ", which has",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 176,
+ 336,
+ 186,
+ 345
+ ],
+ "score": 0.82,
+ "content": "N",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 186,
+ 335,
+ 430,
+ 349
+ ],
+ "score": 1.0,
+ "content": "nodes and each edge appears independently with probability",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 430,
+ 338,
+ 437,
+ 347
+ ],
+ "score": 0.74,
+ "content": "p",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 437,
+ 335,
+ 505,
+ 349
+ ],
+ "score": 1.0,
+ "content": ", for an example.",
+ "type": "text"
+ }
+ ],
+ "index": 22
+ },
+ {
+ "bbox": [
+ 104,
+ 345,
+ 506,
+ 363
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 345,
+ 198,
+ 361
+ ],
+ "score": 1.0,
+ "content": "We prove that if log N",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 173,
+ 347,
+ 226,
+ 363
+ ],
+ "score": 0.93,
+ "content": "\\frac { \\log N } { p N } = o ( 1 )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 226,
+ 348,
+ 291,
+ 362
+ ],
+ "score": 1.0,
+ "content": "as a function of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 291,
+ 349,
+ 301,
+ 358
+ ],
+ "score": 0.8,
+ "content": "N",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 302,
+ 348,
+ 506,
+ 362
+ ],
+ "score": 1.0,
+ "content": ", any GCN whose weights have maximum singular",
+ "type": "text"
+ }
+ ],
+ "index": 23
+ },
+ {
+ "bbox": [
+ 106,
+ 363,
+ 507,
+ 382
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 366,
+ 168,
+ 379
+ ],
+ "score": 1.0,
+ "content": "values at most",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 169,
+ 363,
+ 221,
+ 382
+ ],
+ "score": 0.93,
+ "content": "\\begin{array} { r } { C \\sqrt { \\frac { N p } { \\log ( N / \\varepsilon ) } } } \\end{array}",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 221,
+ 363,
+ 507,
+ 382
+ ],
+ "score": 1.0,
+ "content": "approaches the “information-less” invariant space with probability at",
+ "type": "text"
+ }
+ ],
+ "index": 24
+ },
+ {
+ "bbox": [
+ 106,
+ 381,
+ 505,
+ 393
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 381,
+ 127,
+ 393
+ ],
+ "score": 1.0,
+ "content": "least",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 127,
+ 381,
+ 150,
+ 391
+ ],
+ "score": 0.85,
+ "content": "1 - \\varepsilon",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 151,
+ 381,
+ 181,
+ 393
+ ],
+ "score": 1.0,
+ "content": ", where",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 181,
+ 382,
+ 190,
+ 390
+ ],
+ "score": 0.82,
+ "content": "C",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 190,
+ 381,
+ 505,
+ 393
+ ],
+ "score": 1.0,
+ "content": "is a universal constant. Intuitively, if the graph on which we define graph NNs",
+ "type": "text"
+ }
+ ],
+ "index": 25
+ },
+ {
+ "bbox": [
+ 105,
+ 392,
+ 504,
+ 404
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 392,
+ 504,
+ 404
+ ],
+ "score": 1.0,
+ "content": "is sufficiently dense, graph-convolution operations mix signals on nodes fast and hence the feature",
+ "type": "text"
+ }
+ ],
+ "index": 26
+ },
+ {
+ "bbox": [
+ 106,
+ 402,
+ 330,
+ 416
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 402,
+ 330,
+ 416
+ ],
+ "score": 1.0,
+ "content": "maps lose information for distinguishing nodes quickly.",
+ "type": "text"
+ }
+ ],
+ "index": 27
+ }
+ ],
+ "index": 23.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 419,
+ 239,
+ 430
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 418,
+ 240,
+ 432
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 418,
+ 240,
+ 432
+ ],
+ "score": 1.0,
+ "content": "Our contributions are as follows:",
+ "type": "text"
+ }
+ ],
+ "index": 28
+ }
+ ],
+ "index": 28
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 132,
+ 441,
+ 505,
+ 553
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 132,
+ 440,
+ 504,
+ 453
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 132,
+ 440,
+ 504,
+ 453
+ ],
+ "score": 1.0,
+ "content": "• We relate asymptotic behaviors of graph NNs with the topological information of underly-",
+ "type": "text"
+ }
+ ],
+ "index": 29
+ },
+ {
+ "bbox": [
+ 141,
+ 452,
+ 467,
+ 464
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 141,
+ 452,
+ 467,
+ 464
+ ],
+ "score": 1.0,
+ "content": "ing graphs via the spectral distribution of the (augmented) normalized Laplacian.",
+ "type": "text"
+ }
+ ],
+ "index": 30
+ },
+ {
+ "bbox": [
+ 136,
+ 466,
+ 505,
+ 480
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 136,
+ 466,
+ 505,
+ 480
+ ],
+ "score": 1.0,
+ "content": "• We prove that if the weights of a GCN satisfy conditions determined by the graph spectra,",
+ "type": "text"
+ }
+ ],
+ "index": 31
+ },
+ {
+ "bbox": [
+ 142,
+ 478,
+ 505,
+ 490
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 142,
+ 478,
+ 505,
+ 490
+ ],
+ "score": 1.0,
+ "content": "the output of the GCN carries no information other than the node degrees and connected",
+ "type": "text"
+ }
+ ],
+ "index": 32
+ },
+ {
+ "bbox": [
+ 142,
+ 489,
+ 504,
+ 501
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 142,
+ 489,
+ 504,
+ 501
+ ],
+ "score": 1.0,
+ "content": "components for discriminating nodes when the layer size goes to infinity (Theorems 1, 2).",
+ "type": "text"
+ }
+ ],
+ "index": 33
+ },
+ {
+ "bbox": [
+ 135,
+ 504,
+ 505,
+ 516
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 135,
+ 504,
+ 505,
+ 516
+ ],
+ "score": 1.0,
+ "content": "• We apply our theory to Erdos – R ˝ enyi graphs as an example and show that when the graph ´",
+ "type": "text"
+ }
+ ],
+ "index": 34
+ },
+ {
+ "bbox": [
+ 141,
+ 515,
+ 502,
+ 527
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 141,
+ 515,
+ 502,
+ 527
+ ],
+ "score": 1.0,
+ "content": "is sufficiently dense and large, many GCNs suffer from the information loss (Theorem 3).",
+ "type": "text"
+ }
+ ],
+ "index": 35
+ },
+ {
+ "bbox": [
+ 132,
+ 530,
+ 504,
+ 543
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 132,
+ 530,
+ 504,
+ 543
+ ],
+ "score": 1.0,
+ "content": "• We propose a principled guideline for weight normalization of graph NNs and empirically",
+ "type": "text"
+ }
+ ],
+ "index": 36
+ },
+ {
+ "bbox": [
+ 142,
+ 542,
+ 246,
+ 554
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 142,
+ 542,
+ 246,
+ 554
+ ],
+ "score": 1.0,
+ "content": "confirm it using real data.",
+ "type": "text"
+ }
+ ],
+ "index": 37
+ }
+ ],
+ "index": 33
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 108,
+ 568,
+ 211,
+ 582
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 568,
+ 213,
+ 583
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 568,
+ 213,
+ 583
+ ],
+ "score": 1.0,
+ "content": "2 RELATED WORK",
+ "type": "text"
+ }
+ ],
+ "index": 38
+ }
+ ],
+ "index": 38
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 594,
+ 505,
+ 704
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 594,
+ 506,
+ 606
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 594,
+ 506,
+ 606
+ ],
+ "score": 1.0,
+ "content": "MPNN-type Graph NNs. Since many graph NN variants have been proposed, there are several",
+ "type": "text"
+ }
+ ],
+ "index": 39
+ },
+ {
+ "bbox": [
+ 106,
+ 605,
+ 505,
+ 617
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 605,
+ 505,
+ 617
+ ],
+ "score": 1.0,
+ "content": "unified formulations of graph NNs (Gilmer et al., 2017; Battaglia et al., 2018). Our approach is",
+ "type": "text"
+ }
+ ],
+ "index": 40
+ },
+ {
+ "bbox": [
+ 105,
+ 614,
+ 506,
+ 630
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 614,
+ 506,
+ 630
+ ],
+ "score": 1.0,
+ "content": "the closest to the formulation of Message Passing Neural Network (MPNN) (Gilmer et al., 2017),",
+ "type": "text"
+ }
+ ],
+ "index": 41
+ },
+ {
+ "bbox": [
+ 105,
+ 627,
+ 505,
+ 640
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 627,
+ 505,
+ 640
+ ],
+ "score": 1.0,
+ "content": "which unified graph NNs in terms of the update and readout operations. Many graph NNs fall into",
+ "type": "text"
+ }
+ ],
+ "index": 42
+ },
+ {
+ "bbox": [
+ 105,
+ 638,
+ 506,
+ 651
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 638,
+ 506,
+ 651
+ ],
+ "score": 1.0,
+ "content": "this formulation such as Duvenaud et al. (2015), Li et al. (2016), and Velickovi ˇ c et al. (2018). ´",
+ "type": "text"
+ }
+ ],
+ "index": 43
+ },
+ {
+ "bbox": [
+ 105,
+ 649,
+ 506,
+ 662
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 649,
+ 506,
+ 662
+ ],
+ "score": 1.0,
+ "content": "Among others, GCN (Kipf & Welling, 2017) is an important application of our theory because it",
+ "type": "text"
+ }
+ ],
+ "index": 44
+ },
+ {
+ "bbox": [
+ 105,
+ 660,
+ 505,
+ 673
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 660,
+ 505,
+ 673
+ ],
+ "score": 1.0,
+ "content": "is one of the most widely used graph NNs. In addition, GCNs are interesting from a theoretical",
+ "type": "text"
+ }
+ ],
+ "index": 45
+ },
+ {
+ "bbox": [
+ 105,
+ 671,
+ 506,
+ 684
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 671,
+ 506,
+ 684
+ ],
+ "score": 1.0,
+ "content": "research perspective because, in addition to an MPNN-type graph NN, we can interpret GCNs as a",
+ "type": "text"
+ }
+ ],
+ "index": 46
+ },
+ {
+ "bbox": [
+ 105,
+ 681,
+ 506,
+ 695
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 681,
+ 506,
+ 695
+ ],
+ "score": 1.0,
+ "content": "simplification of spectral-type graph NNs (Henaff et al., 2015; Defferrard et al., 2016), that make",
+ "type": "text"
+ }
+ ],
+ "index": 47
+ },
+ {
+ "bbox": [
+ 105,
+ 693,
+ 217,
+ 705
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 693,
+ 217,
+ 705
+ ],
+ "score": 1.0,
+ "content": "use of the graph Laplacian.",
+ "type": "text"
+ }
+ ],
+ "index": 48
+ }
+ ],
+ "index": 43.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 709,
+ 504,
+ 732
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 709,
+ 505,
+ 722
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 709,
+ 505,
+ 722
+ ],
+ "score": 1.0,
+ "content": "Our approach, which considers the asymptotic behaviors graph NNs as the layer size goes to infinity,",
+ "type": "text"
+ }
+ ],
+ "index": 49
+ },
+ {
+ "bbox": [
+ 105,
+ 720,
+ 505,
+ 733
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 720,
+ 505,
+ 733
+ ],
+ "score": 1.0,
+ "content": "is similar to Scarselli et al. (2009), one of the earliest works about graph NNs. They obtained node",
+ "type": "text"
+ }
+ ],
+ "index": 50
+ }
+ ],
+ "index": 49.5
+ }
+ ],
+ "page_idx": 1,
+ "page_size": [
+ 612,
+ 792
+ ],
+ "discarded_blocks": [
+ {
+ "type": "discarded",
+ "bbox": [
+ 108,
+ 27,
+ 293,
+ 37
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 26,
+ 294,
+ 38
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 26,
+ 294,
+ 38
+ ],
+ "score": 1.0,
+ "content": "Published as a conference paper at ICLR 2020",
+ "type": "text"
+ }
+ ]
+ }
+ ]
+ },
+ {
+ "type": "discarded",
+ "bbox": [
+ 302,
+ 751,
+ 309,
+ 760
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 301,
+ 750,
+ 310,
+ 763
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 301,
+ 750,
+ 310,
+ 763
+ ],
+ "score": 1.0,
+ "content": "2",
+ "type": "text"
+ }
+ ]
+ }
+ ]
+ }
+ ],
+ "para_blocks": [
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 82,
+ 505,
+ 225
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 83,
+ 504,
+ 95
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 83,
+ 504,
+ 95
+ ],
+ "score": 1.0,
+ "content": "In this paper, we investigate the expressive power of graph NNs by analyzing their asymptotic be-",
+ "type": "text"
+ }
+ ],
+ "index": 0
+ },
+ {
+ "bbox": [
+ 105,
+ 94,
+ 505,
+ 107
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 94,
+ 505,
+ 107
+ ],
+ "score": 1.0,
+ "content": "haviors as the layer size goes to infinity. Our theory gives new theoretical conditions under which",
+ "type": "text"
+ }
+ ],
+ "index": 1
+ },
+ {
+ "bbox": [
+ 105,
+ 104,
+ 506,
+ 118
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 104,
+ 506,
+ 118
+ ],
+ "score": 1.0,
+ "content": "neither layer stacking nor non-linearity contributes to improving expressive power. We consider a",
+ "type": "text"
+ }
+ ],
+ "index": 2
+ },
+ {
+ "bbox": [
+ 105,
+ 115,
+ 505,
+ 129
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 115,
+ 505,
+ 129
+ ],
+ "score": 1.0,
+ "content": "specific dynamics that includes a transition defining a Markov process and the forward propaga-",
+ "type": "text"
+ }
+ ],
+ "index": 3
+ },
+ {
+ "bbox": [
+ 105,
+ 126,
+ 506,
+ 139
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 126,
+ 506,
+ 139
+ ],
+ "score": 1.0,
+ "content": "tion of a Graph Convolutional Network (GCN) (Kipf & Welling, 2017), which is one of the most",
+ "type": "text"
+ }
+ ],
+ "index": 4
+ },
+ {
+ "bbox": [
+ 105,
+ 137,
+ 506,
+ 150
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 137,
+ 506,
+ 150
+ ],
+ "score": 1.0,
+ "content": "popular graph NN variants, as special cases. We prove that under certain conditions, the dynamics",
+ "type": "text"
+ }
+ ],
+ "index": 5
+ },
+ {
+ "bbox": [
+ 105,
+ 149,
+ 505,
+ 161
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 149,
+ 505,
+ 161
+ ],
+ "score": 1.0,
+ "content": "exponentially approaches a subspace that is invariant under the dynamics. In the case of GCN, the",
+ "type": "text"
+ }
+ ],
+ "index": 6
+ },
+ {
+ "bbox": [
+ 105,
+ 159,
+ 506,
+ 172
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 159,
+ 506,
+ 172
+ ],
+ "score": 1.0,
+ "content": "invariant space is a set of signals that correspond to the lowest frequency of graph spectra and that",
+ "type": "text"
+ }
+ ],
+ "index": 7
+ },
+ {
+ "bbox": [
+ 105,
+ 170,
+ 506,
+ 183
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 170,
+ 506,
+ 183
+ ],
+ "score": 1.0,
+ "content": "have “no information” other than connected components and node degrees for a node classification",
+ "type": "text"
+ }
+ ],
+ "index": 8
+ },
+ {
+ "bbox": [
+ 105,
+ 181,
+ 505,
+ 194
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 181,
+ 505,
+ 194
+ ],
+ "score": 1.0,
+ "content": "task whose goal is to predict the nodes’ properties in a graph. The rate of the distance between the",
+ "type": "text"
+ }
+ ],
+ "index": 9
+ },
+ {
+ "bbox": [
+ 105,
+ 191,
+ 504,
+ 205
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 191,
+ 240,
+ 205
+ ],
+ "score": 1.0,
+ "content": "output and the invariant space is",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 240,
+ 191,
+ 280,
+ 204
+ ],
+ "score": 0.93,
+ "content": "{ \\cal O } ( ( s \\lambda ) ^ { \\widehat { L } } )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 281,
+ 191,
+ 309,
+ 205
+ ],
+ "score": 1.0,
+ "content": "where",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 310,
+ 194,
+ 316,
+ 202
+ ],
+ "score": 0.71,
+ "content": "s",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 316,
+ 191,
+ 496,
+ 205
+ ],
+ "score": 1.0,
+ "content": "is the maximum singular values of weights,",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 497,
+ 192,
+ 504,
+ 202
+ ],
+ "score": 0.72,
+ "content": "\\lambda",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 10
+ },
+ {
+ "bbox": [
+ 105,
+ 202,
+ 506,
+ 216
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 202,
+ 486,
+ 216
+ ],
+ "score": 1.0,
+ "content": "is typically a quantity determined by the spectra of the (augmented) normalized Laplacian, and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 487,
+ 204,
+ 495,
+ 213
+ ],
+ "score": 0.8,
+ "content": "L",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 495,
+ 202,
+ 506,
+ 216
+ ],
+ "score": 1.0,
+ "content": "is",
+ "type": "text"
+ }
+ ],
+ "index": 11
+ },
+ {
+ "bbox": [
+ 105,
+ 214,
+ 457,
+ 227
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 214,
+ 457,
+ 227
+ ],
+ "score": 1.0,
+ "content": "the layer size. See Sections 3.3 (general case) and 4 (GCN case) for precise statements.",
+ "type": "text"
+ }
+ ],
+ "index": 12
+ }
+ ],
+ "index": 6,
+ "bbox_fs": [
+ 105,
+ 83,
+ 506,
+ 227
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 231,
+ 505,
+ 308
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 230,
+ 505,
+ 243
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 230,
+ 505,
+ 243
+ ],
+ "score": 1.0,
+ "content": "We can interpret our theorem as the generalization of the well-known property that if a finite and dis-",
+ "type": "text"
+ }
+ ],
+ "index": 13
+ },
+ {
+ "bbox": [
+ 105,
+ 241,
+ 505,
+ 255
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 241,
+ 505,
+ 255
+ ],
+ "score": 1.0,
+ "content": "crete Markov process is irreducible and aperiodic, it exponentially converges to a unique equilibrium",
+ "type": "text"
+ }
+ ],
+ "index": 14
+ },
+ {
+ "bbox": [
+ 105,
+ 252,
+ 505,
+ 266
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 252,
+ 505,
+ 266
+ ],
+ "score": 1.0,
+ "content": "and the eigenvalues of its transition matrix determine the convergence rate (see, e.g., Chung & Gra-",
+ "type": "text"
+ }
+ ],
+ "index": 15
+ },
+ {
+ "bbox": [
+ 105,
+ 264,
+ 505,
+ 276
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 264,
+ 505,
+ 276
+ ],
+ "score": 1.0,
+ "content": "ham (1997)). Different from the Markov process case, which is linear, the existence of intermediate",
+ "type": "text"
+ }
+ ],
+ "index": 16
+ },
+ {
+ "bbox": [
+ 105,
+ 275,
+ 505,
+ 287
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 275,
+ 505,
+ 287
+ ],
+ "score": 1.0,
+ "content": "non-linear functions complicates the analysis. We overcame this problem by leveraging the combi-",
+ "type": "text"
+ }
+ ],
+ "index": 17
+ },
+ {
+ "bbox": [
+ 105,
+ 285,
+ 505,
+ 299
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 285,
+ 505,
+ 299
+ ],
+ "score": 1.0,
+ "content": "nation of the ReLU activation function (Krizhevsky et al., 2012) and the positivity of eigenvectors",
+ "type": "text"
+ }
+ ],
+ "index": 18
+ },
+ {
+ "bbox": [
+ 105,
+ 296,
+ 372,
+ 309
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 296,
+ 372,
+ 309
+ ],
+ "score": 1.0,
+ "content": "of the Laplacian associated with the smallest positive eigenvalues.",
+ "type": "text"
+ }
+ ],
+ "index": 19
+ }
+ ],
+ "index": 16,
+ "bbox_fs": [
+ 105,
+ 230,
+ 505,
+ 309
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 313,
+ 505,
+ 414
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 314,
+ 505,
+ 326
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 314,
+ 505,
+ 326
+ ],
+ "score": 1.0,
+ "content": "Our theory enables us to investigate asymptotic behaviors of graph NNs via the spectral distribution",
+ "type": "text"
+ }
+ ],
+ "index": 20
+ },
+ {
+ "bbox": [
+ 104,
+ 323,
+ 505,
+ 338
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 323,
+ 505,
+ 338
+ ],
+ "score": 1.0,
+ "content": "of the underlying graphs. To demonstrate this, we take GCNs defined on the Erdos – R ˝ enyi graph ´",
+ "type": "text"
+ }
+ ],
+ "index": 21
+ },
+ {
+ "bbox": [
+ 107,
+ 335,
+ 505,
+ 349
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 336,
+ 129,
+ 348
+ ],
+ "score": 0.9,
+ "content": "G _ { N , p }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 129,
+ 335,
+ 175,
+ 349
+ ],
+ "score": 1.0,
+ "content": ", which has",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 176,
+ 336,
+ 186,
+ 345
+ ],
+ "score": 0.82,
+ "content": "N",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 186,
+ 335,
+ 430,
+ 349
+ ],
+ "score": 1.0,
+ "content": "nodes and each edge appears independently with probability",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 430,
+ 338,
+ 437,
+ 347
+ ],
+ "score": 0.74,
+ "content": "p",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 437,
+ 335,
+ 505,
+ 349
+ ],
+ "score": 1.0,
+ "content": ", for an example.",
+ "type": "text"
+ }
+ ],
+ "index": 22
+ },
+ {
+ "bbox": [
+ 104,
+ 345,
+ 506,
+ 363
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 345,
+ 198,
+ 361
+ ],
+ "score": 1.0,
+ "content": "We prove that if log N",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 173,
+ 347,
+ 226,
+ 363
+ ],
+ "score": 0.93,
+ "content": "\\frac { \\log N } { p N } = o ( 1 )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 226,
+ 348,
+ 291,
+ 362
+ ],
+ "score": 1.0,
+ "content": "as a function of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 291,
+ 349,
+ 301,
+ 358
+ ],
+ "score": 0.8,
+ "content": "N",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 302,
+ 348,
+ 506,
+ 362
+ ],
+ "score": 1.0,
+ "content": ", any GCN whose weights have maximum singular",
+ "type": "text"
+ }
+ ],
+ "index": 23
+ },
+ {
+ "bbox": [
+ 106,
+ 363,
+ 507,
+ 382
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 366,
+ 168,
+ 379
+ ],
+ "score": 1.0,
+ "content": "values at most",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 169,
+ 363,
+ 221,
+ 382
+ ],
+ "score": 0.93,
+ "content": "\\begin{array} { r } { C \\sqrt { \\frac { N p } { \\log ( N / \\varepsilon ) } } } \\end{array}",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 221,
+ 363,
+ 507,
+ 382
+ ],
+ "score": 1.0,
+ "content": "approaches the “information-less” invariant space with probability at",
+ "type": "text"
+ }
+ ],
+ "index": 24
+ },
+ {
+ "bbox": [
+ 106,
+ 381,
+ 505,
+ 393
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 381,
+ 127,
+ 393
+ ],
+ "score": 1.0,
+ "content": "least",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 127,
+ 381,
+ 150,
+ 391
+ ],
+ "score": 0.85,
+ "content": "1 - \\varepsilon",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 151,
+ 381,
+ 181,
+ 393
+ ],
+ "score": 1.0,
+ "content": ", where",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 181,
+ 382,
+ 190,
+ 390
+ ],
+ "score": 0.82,
+ "content": "C",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 190,
+ 381,
+ 505,
+ 393
+ ],
+ "score": 1.0,
+ "content": "is a universal constant. Intuitively, if the graph on which we define graph NNs",
+ "type": "text"
+ }
+ ],
+ "index": 25
+ },
+ {
+ "bbox": [
+ 105,
+ 392,
+ 504,
+ 404
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 392,
+ 504,
+ 404
+ ],
+ "score": 1.0,
+ "content": "is sufficiently dense, graph-convolution operations mix signals on nodes fast and hence the feature",
+ "type": "text"
+ }
+ ],
+ "index": 26
+ },
+ {
+ "bbox": [
+ 106,
+ 402,
+ 330,
+ 416
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 402,
+ 330,
+ 416
+ ],
+ "score": 1.0,
+ "content": "maps lose information for distinguishing nodes quickly.",
+ "type": "text"
+ }
+ ],
+ "index": 27
+ }
+ ],
+ "index": 23.5,
+ "bbox_fs": [
+ 104,
+ 314,
+ 507,
+ 416
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 419,
+ 239,
+ 430
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 418,
+ 240,
+ 432
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 418,
+ 240,
+ 432
+ ],
+ "score": 1.0,
+ "content": "Our contributions are as follows:",
+ "type": "text"
+ }
+ ],
+ "index": 28
+ }
+ ],
+ "index": 28,
+ "bbox_fs": [
+ 106,
+ 418,
+ 240,
+ 432
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 132,
+ 441,
+ 505,
+ 553
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 132,
+ 440,
+ 504,
+ 453
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 132,
+ 440,
+ 504,
+ 453
+ ],
+ "score": 1.0,
+ "content": "• We relate asymptotic behaviors of graph NNs with the topological information of underly-",
+ "type": "text"
+ }
+ ],
+ "index": 29
+ },
+ {
+ "bbox": [
+ 141,
+ 452,
+ 467,
+ 464
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 141,
+ 452,
+ 467,
+ 464
+ ],
+ "score": 1.0,
+ "content": "ing graphs via the spectral distribution of the (augmented) normalized Laplacian.",
+ "type": "text"
+ }
+ ],
+ "index": 30
+ },
+ {
+ "bbox": [
+ 136,
+ 466,
+ 505,
+ 480
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 136,
+ 466,
+ 505,
+ 480
+ ],
+ "score": 1.0,
+ "content": "• We prove that if the weights of a GCN satisfy conditions determined by the graph spectra,",
+ "type": "text"
+ }
+ ],
+ "index": 31
+ },
+ {
+ "bbox": [
+ 142,
+ 478,
+ 505,
+ 490
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 142,
+ 478,
+ 505,
+ 490
+ ],
+ "score": 1.0,
+ "content": "the output of the GCN carries no information other than the node degrees and connected",
+ "type": "text"
+ }
+ ],
+ "index": 32
+ },
+ {
+ "bbox": [
+ 142,
+ 489,
+ 504,
+ 501
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 142,
+ 489,
+ 504,
+ 501
+ ],
+ "score": 1.0,
+ "content": "components for discriminating nodes when the layer size goes to infinity (Theorems 1, 2).",
+ "type": "text"
+ }
+ ],
+ "index": 33
+ },
+ {
+ "bbox": [
+ 135,
+ 504,
+ 505,
+ 516
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 135,
+ 504,
+ 505,
+ 516
+ ],
+ "score": 1.0,
+ "content": "• We apply our theory to Erdos – R ˝ enyi graphs as an example and show that when the graph ´",
+ "type": "text"
+ }
+ ],
+ "index": 34
+ },
+ {
+ "bbox": [
+ 141,
+ 515,
+ 502,
+ 527
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 141,
+ 515,
+ 502,
+ 527
+ ],
+ "score": 1.0,
+ "content": "is sufficiently dense and large, many GCNs suffer from the information loss (Theorem 3).",
+ "type": "text"
+ }
+ ],
+ "index": 35
+ },
+ {
+ "bbox": [
+ 132,
+ 530,
+ 504,
+ 543
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 132,
+ 530,
+ 504,
+ 543
+ ],
+ "score": 1.0,
+ "content": "• We propose a principled guideline for weight normalization of graph NNs and empirically",
+ "type": "text"
+ }
+ ],
+ "index": 36
+ },
+ {
+ "bbox": [
+ 142,
+ 542,
+ 246,
+ 554
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 142,
+ 542,
+ 246,
+ 554
+ ],
+ "score": 1.0,
+ "content": "confirm it using real data.",
+ "type": "text"
+ }
+ ],
+ "index": 37
+ }
+ ],
+ "index": 33,
+ "bbox_fs": [
+ 132,
+ 440,
+ 505,
+ 554
+ ]
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 108,
+ 568,
+ 211,
+ 582
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 568,
+ 213,
+ 583
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 568,
+ 213,
+ 583
+ ],
+ "score": 1.0,
+ "content": "2 RELATED WORK",
+ "type": "text"
+ }
+ ],
+ "index": 38
+ }
+ ],
+ "index": 38
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 594,
+ 505,
+ 704
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 594,
+ 506,
+ 606
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 594,
+ 506,
+ 606
+ ],
+ "score": 1.0,
+ "content": "MPNN-type Graph NNs. Since many graph NN variants have been proposed, there are several",
+ "type": "text"
+ }
+ ],
+ "index": 39
+ },
+ {
+ "bbox": [
+ 106,
+ 605,
+ 505,
+ 617
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 605,
+ 505,
+ 617
+ ],
+ "score": 1.0,
+ "content": "unified formulations of graph NNs (Gilmer et al., 2017; Battaglia et al., 2018). Our approach is",
+ "type": "text"
+ }
+ ],
+ "index": 40
+ },
+ {
+ "bbox": [
+ 105,
+ 614,
+ 506,
+ 630
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 614,
+ 506,
+ 630
+ ],
+ "score": 1.0,
+ "content": "the closest to the formulation of Message Passing Neural Network (MPNN) (Gilmer et al., 2017),",
+ "type": "text"
+ }
+ ],
+ "index": 41
+ },
+ {
+ "bbox": [
+ 105,
+ 627,
+ 505,
+ 640
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 627,
+ 505,
+ 640
+ ],
+ "score": 1.0,
+ "content": "which unified graph NNs in terms of the update and readout operations. Many graph NNs fall into",
+ "type": "text"
+ }
+ ],
+ "index": 42
+ },
+ {
+ "bbox": [
+ 105,
+ 638,
+ 506,
+ 651
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 638,
+ 506,
+ 651
+ ],
+ "score": 1.0,
+ "content": "this formulation such as Duvenaud et al. (2015), Li et al. (2016), and Velickovi ˇ c et al. (2018). ´",
+ "type": "text"
+ }
+ ],
+ "index": 43
+ },
+ {
+ "bbox": [
+ 105,
+ 649,
+ 506,
+ 662
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 649,
+ 506,
+ 662
+ ],
+ "score": 1.0,
+ "content": "Among others, GCN (Kipf & Welling, 2017) is an important application of our theory because it",
+ "type": "text"
+ }
+ ],
+ "index": 44
+ },
+ {
+ "bbox": [
+ 105,
+ 660,
+ 505,
+ 673
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 660,
+ 505,
+ 673
+ ],
+ "score": 1.0,
+ "content": "is one of the most widely used graph NNs. In addition, GCNs are interesting from a theoretical",
+ "type": "text"
+ }
+ ],
+ "index": 45
+ },
+ {
+ "bbox": [
+ 105,
+ 671,
+ 506,
+ 684
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 671,
+ 506,
+ 684
+ ],
+ "score": 1.0,
+ "content": "research perspective because, in addition to an MPNN-type graph NN, we can interpret GCNs as a",
+ "type": "text"
+ }
+ ],
+ "index": 46
+ },
+ {
+ "bbox": [
+ 105,
+ 681,
+ 506,
+ 695
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 681,
+ 506,
+ 695
+ ],
+ "score": 1.0,
+ "content": "simplification of spectral-type graph NNs (Henaff et al., 2015; Defferrard et al., 2016), that make",
+ "type": "text"
+ }
+ ],
+ "index": 47
+ },
+ {
+ "bbox": [
+ 105,
+ 693,
+ 217,
+ 705
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 693,
+ 217,
+ 705
+ ],
+ "score": 1.0,
+ "content": "use of the graph Laplacian.",
+ "type": "text"
+ }
+ ],
+ "index": 48
+ }
+ ],
+ "index": 43.5,
+ "bbox_fs": [
+ 105,
+ 594,
+ 506,
+ 705
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 709,
+ 504,
+ 732
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 709,
+ 505,
+ 722
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 709,
+ 505,
+ 722
+ ],
+ "score": 1.0,
+ "content": "Our approach, which considers the asymptotic behaviors graph NNs as the layer size goes to infinity,",
+ "type": "text"
+ }
+ ],
+ "index": 49
+ },
+ {
+ "bbox": [
+ 105,
+ 720,
+ 505,
+ 733
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 720,
+ 505,
+ 733
+ ],
+ "score": 1.0,
+ "content": "is similar to Scarselli et al. (2009), one of the earliest works about graph NNs. They obtained node",
+ "type": "text"
+ }
+ ],
+ "index": 50
+ },
+ {
+ "bbox": [
+ 105,
+ 82,
+ 506,
+ 95
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 82,
+ 506,
+ 95
+ ],
+ "score": 1.0,
+ "content": "representations by iterating message passing between nodes until convergence. Their formulation is",
+ "type": "text",
+ "cross_page": true
+ }
+ ],
+ "index": 0
+ },
+ {
+ "bbox": [
+ 105,
+ 93,
+ 506,
+ 106
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 93,
+ 506,
+ 106
+ ],
+ "score": 1.0,
+ "content": "general in that we can use any local aggregation operation as long as it is a contraction map. Our",
+ "type": "text",
+ "cross_page": true
+ }
+ ],
+ "index": 1
+ },
+ {
+ "bbox": [
+ 105,
+ 105,
+ 506,
+ 117
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 105,
+ 506,
+ 117
+ ],
+ "score": 1.0,
+ "content": "theory differs from theirs in that we proved that the output of a graph NN approaches a certain space",
+ "type": "text",
+ "cross_page": true
+ }
+ ],
+ "index": 2
+ },
+ {
+ "bbox": [
+ 105,
+ 114,
+ 405,
+ 129
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 114,
+ 405,
+ 129
+ ],
+ "score": 1.0,
+ "content": "even if the local aggregation function is not necessarily a contraction map.",
+ "type": "text",
+ "cross_page": true
+ }
+ ],
+ "index": 3
+ }
+ ],
+ "index": 49.5,
+ "bbox_fs": [
+ 105,
+ 709,
+ 505,
+ 733
+ ]
+ }
+ ]
+ },
+ {
+ "preproc_blocks": [
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 82,
+ 504,
+ 126
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 82,
+ 506,
+ 95
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 82,
+ 506,
+ 95
+ ],
+ "score": 1.0,
+ "content": "representations by iterating message passing between nodes until convergence. Their formulation is",
+ "type": "text"
+ }
+ ],
+ "index": 0
+ },
+ {
+ "bbox": [
+ 105,
+ 93,
+ 506,
+ 106
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 93,
+ 506,
+ 106
+ ],
+ "score": 1.0,
+ "content": "general in that we can use any local aggregation operation as long as it is a contraction map. Our",
+ "type": "text"
+ }
+ ],
+ "index": 1
+ },
+ {
+ "bbox": [
+ 105,
+ 105,
+ 506,
+ 117
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 105,
+ 506,
+ 117
+ ],
+ "score": 1.0,
+ "content": "theory differs from theirs in that we proved that the output of a graph NN approaches a certain space",
+ "type": "text"
+ }
+ ],
+ "index": 2
+ },
+ {
+ "bbox": [
+ 105,
+ 114,
+ 405,
+ 129
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 114,
+ 405,
+ 129
+ ],
+ "score": 1.0,
+ "content": "even if the local aggregation function is not necessarily a contraction map.",
+ "type": "text"
+ }
+ ],
+ "index": 3
+ }
+ ],
+ "index": 1.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 132,
+ 505,
+ 308
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 132,
+ 505,
+ 145
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 132,
+ 505,
+ 145
+ ],
+ "score": 1.0,
+ "content": "Expressive Power of Graph NNs. Several studies have focused on theoretical analysis and the im-",
+ "type": "text"
+ }
+ ],
+ "index": 4
+ },
+ {
+ "bbox": [
+ 105,
+ 143,
+ 505,
+ 156
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 143,
+ 505,
+ 156
+ ],
+ "score": 1.0,
+ "content": "provement of graph NN expressive power. For example, Xu et al. (2019) proved that graph NNs are",
+ "type": "text"
+ }
+ ],
+ "index": 5
+ },
+ {
+ "bbox": [
+ 104,
+ 154,
+ 505,
+ 167
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 154,
+ 505,
+ 167
+ ],
+ "score": 1.0,
+ "content": "no more powerful than the Weisfeiler – Lehman (WL) isomorphism test (Weisfeiler & A.A., 1968)",
+ "type": "text"
+ }
+ ],
+ "index": 6
+ },
+ {
+ "bbox": [
+ 105,
+ 165,
+ 505,
+ 178
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 165,
+ 505,
+ 178
+ ],
+ "score": 1.0,
+ "content": "and proposed a Graph Isomorphism Network (GIN), that is approximately as powerful as the WL",
+ "type": "text"
+ }
+ ],
+ "index": 7
+ },
+ {
+ "bbox": [
+ 105,
+ 175,
+ 506,
+ 190
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 175,
+ 506,
+ 190
+ ],
+ "score": 1.0,
+ "content": "test. Although they experimentally showed that GIN has improved accuracy in supervised learning",
+ "type": "text"
+ }
+ ],
+ "index": 8
+ },
+ {
+ "bbox": [
+ 105,
+ 187,
+ 505,
+ 200
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 187,
+ 505,
+ 200
+ ],
+ "score": 1.0,
+ "content": "tasks, their analysis was restricted to the graph isomorphism problem. Xu et al. (2018) analyzed",
+ "type": "text"
+ }
+ ],
+ "index": 9
+ },
+ {
+ "bbox": [
+ 105,
+ 199,
+ 505,
+ 211
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 199,
+ 505,
+ 211
+ ],
+ "score": 1.0,
+ "content": "the non-asymptotic properties of GCNs through the lens of random walk theory. They proved the",
+ "type": "text"
+ }
+ ],
+ "index": 10
+ },
+ {
+ "bbox": [
+ 105,
+ 209,
+ 505,
+ 222
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 209,
+ 505,
+ 222
+ ],
+ "score": 1.0,
+ "content": "limitations of GCNs in expander-like graphs and proposed a Jumping Knowledge Network (JK-",
+ "type": "text"
+ }
+ ],
+ "index": 11
+ },
+ {
+ "bbox": [
+ 105,
+ 220,
+ 505,
+ 233
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 220,
+ 505,
+ 233
+ ],
+ "score": 1.0,
+ "content": "Net) to address the issue. To handle the non-linearity, they linearized networks by a randomization",
+ "type": "text"
+ }
+ ],
+ "index": 12
+ },
+ {
+ "bbox": [
+ 105,
+ 231,
+ 505,
+ 243
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 231,
+ 505,
+ 243
+ ],
+ "score": 1.0,
+ "content": "assumption (Choromanska et al., 2015). We take a different strategy and make use of the interpre-",
+ "type": "text"
+ }
+ ],
+ "index": 13
+ },
+ {
+ "bbox": [
+ 105,
+ 242,
+ 505,
+ 254
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 242,
+ 505,
+ 254
+ ],
+ "score": 1.0,
+ "content": "tation of ReLU as a projection onto a cone. Recently, NT & Maehara (2019) showed that a GCN",
+ "type": "text"
+ }
+ ],
+ "index": 14
+ },
+ {
+ "bbox": [
+ 105,
+ 253,
+ 506,
+ 266
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 253,
+ 506,
+ 266
+ ],
+ "score": 1.0,
+ "content": "approximately works as a low-pass filter plus an MLP in a certain setting. Although they analyzed",
+ "type": "text"
+ }
+ ],
+ "index": 15
+ },
+ {
+ "bbox": [
+ 105,
+ 263,
+ 506,
+ 277
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 263,
+ 506,
+ 277
+ ],
+ "score": 1.0,
+ "content": "finite-depth GCNs, our theory has similar spirits with theirs because our “information-less” space",
+ "type": "text"
+ }
+ ],
+ "index": 16
+ },
+ {
+ "bbox": [
+ 105,
+ 275,
+ 504,
+ 287
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 275,
+ 504,
+ 287
+ ],
+ "score": 1.0,
+ "content": "corresponds to the lowest frequency of a graph Laplacian. Another point is that they imposed as-",
+ "type": "text"
+ }
+ ],
+ "index": 17
+ },
+ {
+ "bbox": [
+ 105,
+ 285,
+ 505,
+ 299
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 285,
+ 505,
+ 299
+ ],
+ "score": 1.0,
+ "content": "sumptions that input signals consist of low-frequent true signals and high-frequent noise, whereas",
+ "type": "text"
+ }
+ ],
+ "index": 18
+ },
+ {
+ "bbox": [
+ 106,
+ 297,
+ 240,
+ 310
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 297,
+ 240,
+ 310
+ ],
+ "score": 1.0,
+ "content": "we need not such an assumption.",
+ "type": "text"
+ }
+ ],
+ "index": 19
+ }
+ ],
+ "index": 11.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 313,
+ 505,
+ 445
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 313,
+ 505,
+ 326
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 313,
+ 505,
+ 326
+ ],
+ "score": 1.0,
+ "content": "Role of Deep and Non-linear Structures. For ordinal DL models such as FNNs and CNNs, we",
+ "type": "text"
+ }
+ ],
+ "index": 20
+ },
+ {
+ "bbox": [
+ 105,
+ 324,
+ 505,
+ 338
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 324,
+ 505,
+ 338
+ ],
+ "score": 1.0,
+ "content": "have both theoretical and empirical justification of deep and non-linear architectures for enhancing",
+ "type": "text"
+ }
+ ],
+ "index": 21
+ },
+ {
+ "bbox": [
+ 105,
+ 335,
+ 505,
+ 349
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 335,
+ 505,
+ 349
+ ],
+ "score": 1.0,
+ "content": "of the expressive power (e.g., Telgarsky (2016); Petersen & Voigtlaender (2018); Oono & Suzuki",
+ "type": "text"
+ }
+ ],
+ "index": 22
+ },
+ {
+ "bbox": [
+ 105,
+ 345,
+ 505,
+ 360
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 345,
+ 505,
+ 360
+ ],
+ "score": 1.0,
+ "content": "(2019)). In contrast, several studies have witnessed severe performance degradation when stacking",
+ "type": "text"
+ }
+ ],
+ "index": 23
+ },
+ {
+ "bbox": [
+ 105,
+ 358,
+ 506,
+ 369
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 358,
+ 506,
+ 369
+ ],
+ "score": 1.0,
+ "content": "many layers on graph NNs (Kipf & Welling, 2017; Wu et al., 2019b). Li et al. (2018) reported that",
+ "type": "text"
+ }
+ ],
+ "index": 24
+ },
+ {
+ "bbox": [
+ 105,
+ 369,
+ 505,
+ 381
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 369,
+ 505,
+ 381
+ ],
+ "score": 1.0,
+ "content": "feature vectors on nodes in a graph go indistinguishable as we increase layers in several tasks. They",
+ "type": "text"
+ }
+ ],
+ "index": 25
+ },
+ {
+ "bbox": [
+ 106,
+ 381,
+ 504,
+ 392
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 381,
+ 504,
+ 392
+ ],
+ "score": 1.0,
+ "content": "named this phenomenon over-smoothing. Regarding non-linearity, Wu et al. (2019a) empirically",
+ "type": "text"
+ }
+ ],
+ "index": 26
+ },
+ {
+ "bbox": [
+ 105,
+ 390,
+ 504,
+ 403
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 390,
+ 504,
+ 403
+ ],
+ "score": 1.0,
+ "content": "showed that graph NNs achieve comparable performance even if we omit intermediate non-linearity.",
+ "type": "text"
+ }
+ ],
+ "index": 27
+ },
+ {
+ "bbox": [
+ 105,
+ 401,
+ 505,
+ 415
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 401,
+ 505,
+ 415
+ ],
+ "score": 1.0,
+ "content": "These observations gave us questions about the current models of deep graph NNs in terms of their",
+ "type": "text"
+ }
+ ],
+ "index": 28
+ },
+ {
+ "bbox": [
+ 105,
+ 412,
+ 506,
+ 426
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 412,
+ 506,
+ 426
+ ],
+ "score": 1.0,
+ "content": "expressive power. Several studies gave theoretical explanations of the over-smoothing phenomena",
+ "type": "text"
+ }
+ ],
+ "index": 29
+ },
+ {
+ "bbox": [
+ 105,
+ 423,
+ 505,
+ 436
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 423,
+ 505,
+ 436
+ ],
+ "score": 1.0,
+ "content": "for linear GNNs (Li et al., 2018; Zhang, 2019; Zhao & Akoglu, 2020). We can think of our theory",
+ "type": "text"
+ }
+ ],
+ "index": 30
+ },
+ {
+ "bbox": [
+ 105,
+ 434,
+ 314,
+ 446
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 434,
+ 314,
+ 446
+ ],
+ "score": 1.0,
+ "content": "as an extension of their results to non-linear GNNs.",
+ "type": "text"
+ }
+ ],
+ "index": 31
+ }
+ ],
+ "index": 25.5
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 108,
+ 463,
+ 326,
+ 476
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 462,
+ 328,
+ 477
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 462,
+ 328,
+ 477
+ ],
+ "score": 1.0,
+ "content": "3 PROBLEM SETTING AND MAIN RESULT",
+ "type": "text"
+ }
+ ],
+ "index": 32
+ }
+ ],
+ "index": 32
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 107,
+ 488,
+ 176,
+ 500
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 487,
+ 178,
+ 502
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 487,
+ 178,
+ 502
+ ],
+ "score": 1.0,
+ "content": "3.1 NOTATION",
+ "type": "text"
+ }
+ ],
+ "index": 33
+ }
+ ],
+ "index": 33
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 509,
+ 505,
+ 624
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 508,
+ 506,
+ 523
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 508,
+ 122,
+ 523
+ ],
+ "score": 1.0,
+ "content": "Let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 123,
+ 510,
+ 138,
+ 521
+ ],
+ "score": 0.84,
+ "content": "\\mathbb { N } _ { + }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 138,
+ 508,
+ 281,
+ 523
+ ],
+ "score": 1.0,
+ "content": "be the set of positive integers. For",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 281,
+ 510,
+ 319,
+ 522
+ ],
+ "score": 0.92,
+ "content": "N \\in { \\mathbb { N } } _ { + }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 320,
+ 508,
+ 368,
+ 523
+ ],
+ "score": 1.0,
+ "content": ", we denote",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 369,
+ 510,
+ 448,
+ 522
+ ],
+ "score": 0.92,
+ "content": "[ N ] : = \\{ 1 , \\dots , N \\}",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 448,
+ 508,
+ 506,
+ 523
+ ],
+ "score": 1.0,
+ "content": ". For a vector",
+ "type": "text"
+ }
+ ],
+ "index": 34
+ },
+ {
+ "bbox": [
+ 106,
+ 517,
+ 506,
+ 534
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 521,
+ 140,
+ 531
+ ],
+ "score": 0.9,
+ "content": "v \\in \\mathbb { R } ^ { N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 140,
+ 517,
+ 181,
+ 534
+ ],
+ "score": 1.0,
+ "content": ", we write",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 182,
+ 522,
+ 207,
+ 532
+ ],
+ "score": 0.91,
+ "content": "v \\geq 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 207,
+ 517,
+ 262,
+ 534
+ ],
+ "score": 1.0,
+ "content": "if and only if",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 263,
+ 522,
+ 293,
+ 532
+ ],
+ "score": 0.91,
+ "content": "v _ { n } \\geq 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 293,
+ 517,
+ 321,
+ 534
+ ],
+ "score": 1.0,
+ "content": "for all",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 321,
+ 521,
+ 355,
+ 532
+ ],
+ "score": 0.91,
+ "content": "n \\in [ N ]",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 355,
+ 517,
+ 451,
+ 534
+ ],
+ "score": 1.0,
+ "content": ". Similarly, for a matrix",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 452,
+ 520,
+ 501,
+ 531
+ ],
+ "score": 0.91,
+ "content": "X \\in \\mathbb { R } ^ { N \\times C }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 501,
+ 517,
+ 506,
+ 534
+ ],
+ "score": 1.0,
+ "content": ",",
+ "type": "text"
+ }
+ ],
+ "index": 35
+ },
+ {
+ "bbox": [
+ 106,
+ 531,
+ 506,
+ 545
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 531,
+ 145,
+ 545
+ ],
+ "score": 1.0,
+ "content": "we write",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 145,
+ 532,
+ 176,
+ 543
+ ],
+ "score": 0.91,
+ "content": "X \\geq 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 176,
+ 531,
+ 234,
+ 545
+ ],
+ "score": 1.0,
+ "content": "if and only if",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 235,
+ 532,
+ 273,
+ 543
+ ],
+ "score": 0.92,
+ "content": "X _ { n c } \\geq 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 274,
+ 531,
+ 303,
+ 545
+ ],
+ "score": 1.0,
+ "content": "for all",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 303,
+ 532,
+ 339,
+ 544
+ ],
+ "score": 0.91,
+ "content": "n \\in [ N ]",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 340,
+ 531,
+ 359,
+ 545
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 359,
+ 532,
+ 392,
+ 544
+ ],
+ "score": 0.92,
+ "content": "c \\in [ C ]",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 393,
+ 531,
+ 506,
+ 545
+ ],
+ "score": 1.0,
+ "content": ". We say such a vector and",
+ "type": "text"
+ }
+ ],
+ "index": 36
+ },
+ {
+ "bbox": [
+ 105,
+ 542,
+ 505,
+ 555
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 542,
+ 207,
+ 555
+ ],
+ "score": 1.0,
+ "content": "matrix is non-negative.",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 207,
+ 543,
+ 226,
+ 555
+ ],
+ "score": 0.9,
+ "content": "\\langle \\cdot , \\cdot \\rangle",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 226,
+ 542,
+ 505,
+ 555
+ ],
+ "score": 1.0,
+ "content": "denotes the inner product of vectors or matrices, depending on the",
+ "type": "text"
+ }
+ ],
+ "index": 37
+ },
+ {
+ "bbox": [
+ 104,
+ 550,
+ 507,
+ 568
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 550,
+ 142,
+ 568
+ ],
+ "score": 1.0,
+ "content": "context:",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 142,
+ 553,
+ 200,
+ 566
+ ],
+ "score": 0.92,
+ "content": "\\langle u , v \\rangle : = u ^ { \\top } v",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 200,
+ 550,
+ 215,
+ 568
+ ],
+ "score": 1.0,
+ "content": "for",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 216,
+ 553,
+ 259,
+ 565
+ ],
+ "score": 0.92,
+ "content": "u , v \\in \\mathbb { R } ^ { N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 259,
+ 550,
+ 278,
+ 568
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 278,
+ 553,
+ 362,
+ 565
+ ],
+ "score": 0.9,
+ "content": "\\langle X , Y \\rangle : = \\mathrm { t r } ( X ^ { T } Y )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 362,
+ 550,
+ 378,
+ 568
+ ],
+ "score": 1.0,
+ "content": "for",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 379,
+ 553,
+ 440,
+ 564
+ ],
+ "score": 0.87,
+ "content": "X , Y \\in \\mathbb { R } ^ { N \\times C }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 440,
+ 550,
+ 443,
+ 568
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 443,
+ 554,
+ 458,
+ 565
+ ],
+ "score": 0.53,
+ "content": "{ \\bf 1 } _ { P }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 458,
+ 550,
+ 507,
+ 568
+ ],
+ "score": 1.0,
+ "content": "equals to 1",
+ "type": "text"
+ }
+ ],
+ "index": 38
+ },
+ {
+ "bbox": [
+ 104,
+ 561,
+ 507,
+ 579
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 561,
+ 178,
+ 579
+ ],
+ "score": 1.0,
+ "content": "if the proposition",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 179,
+ 566,
+ 188,
+ 575
+ ],
+ "score": 0.84,
+ "content": "P",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 188,
+ 561,
+ 295,
+ 579
+ ],
+ "score": 1.0,
+ "content": "is true else 0. For vectors",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 295,
+ 565,
+ 329,
+ 575
+ ],
+ "score": 0.87,
+ "content": "v \\in \\mathbb { R } ^ { N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 330,
+ 561,
+ 348,
+ 579
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 348,
+ 564,
+ 384,
+ 575
+ ],
+ "score": 0.9,
+ "content": "w \\in \\mathbb { R } ^ { C }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 384,
+ 561,
+ 389,
+ 579
+ ],
+ "score": 1.0,
+ "content": ",",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 389,
+ 564,
+ 455,
+ 575
+ ],
+ "score": 0.87,
+ "content": "v \\otimes w \\in \\mathbb { R } ^ { N \\times C }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 456,
+ 561,
+ 507,
+ 579
+ ],
+ "score": 1.0,
+ "content": "denotes the",
+ "type": "text"
+ }
+ ],
+ "index": 39
+ },
+ {
+ "bbox": [
+ 104,
+ 575,
+ 504,
+ 591
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 575,
+ 194,
+ 591
+ ],
+ "score": 1.0,
+ "content": "Kronecker product of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 195,
+ 579,
+ 201,
+ 587
+ ],
+ "score": 0.76,
+ "content": "v",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 201,
+ 575,
+ 218,
+ 591
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 219,
+ 579,
+ 228,
+ 587
+ ],
+ "score": 0.78,
+ "content": "w",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 228,
+ 575,
+ 273,
+ 591
+ ],
+ "score": 1.0,
+ "content": "defined by",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 273,
+ 577,
+ 353,
+ 589
+ ],
+ "score": 0.88,
+ "content": "( v \\otimes w ) _ { n c } : = v _ { n } w _ { c } .",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 353,
+ 575,
+ 372,
+ 591
+ ],
+ "score": 1.0,
+ "content": ". For",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 372,
+ 577,
+ 422,
+ 588
+ ],
+ "score": 0.86,
+ "content": "X \\in \\mathbb { R } ^ { N \\times C }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 422,
+ 575,
+ 426,
+ 591
+ ],
+ "score": 1.0,
+ "content": ",",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 426,
+ 576,
+ 504,
+ 589
+ ],
+ "score": 0.8,
+ "content": "\\| X \\| _ { \\mathrm { F } } : = \\langle X , X \\rangle ^ { \\frac { 1 } { 2 } }",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 40
+ },
+ {
+ "bbox": [
+ 102,
+ 584,
+ 504,
+ 605
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 102,
+ 584,
+ 234,
+ 605
+ ],
+ "score": 1.0,
+ "content": "denotes the Frobenius norm of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 234,
+ 589,
+ 244,
+ 598
+ ],
+ "score": 0.85,
+ "content": "X",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 244,
+ 584,
+ 303,
+ 605
+ ],
+ "score": 1.0,
+ "content": ". For a vector",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 303,
+ 588,
+ 338,
+ 599
+ ],
+ "score": 0.84,
+ "content": "v \\in \\mathbb { R } ^ { N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 339,
+ 584,
+ 342,
+ 605
+ ],
+ "score": 1.0,
+ "content": ",",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 343,
+ 588,
+ 504,
+ 601
+ ],
+ "score": 0.88,
+ "content": "\\mathrm { d i a g } ( v ) : = ( v _ { n } \\delta _ { n m } ) _ { n , m \\in [ N ] } \\in \\mathbb { R } ^ { N \\times N }",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 41
+ },
+ {
+ "bbox": [
+ 104,
+ 597,
+ 506,
+ 614
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 597,
+ 230,
+ 614
+ ],
+ "score": 1.0,
+ "content": "denotes the diagonalization of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 231,
+ 603,
+ 237,
+ 611
+ ],
+ "score": 0.67,
+ "content": "v",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 237,
+ 597,
+ 242,
+ 614
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 243,
+ 600,
+ 298,
+ 612
+ ],
+ "score": 0.89,
+ "content": "I _ { N } \\in \\mathbb { R } ^ { N \\times N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 298,
+ 597,
+ 439,
+ 614
+ ],
+ "score": 1.0,
+ "content": "denotes the identity matrix of size",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 440,
+ 602,
+ 450,
+ 611
+ ],
+ "score": 0.82,
+ "content": "N",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 450,
+ 597,
+ 506,
+ 614
+ ],
+ "score": 1.0,
+ "content": ". For a linear",
+ "type": "text"
+ }
+ ],
+ "index": 42
+ },
+ {
+ "bbox": [
+ 104,
+ 609,
+ 478,
+ 627
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 609,
+ 142,
+ 627
+ ],
+ "score": 1.0,
+ "content": "operator",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 142,
+ 612,
+ 205,
+ 622
+ ],
+ "score": 0.91,
+ "content": "P : \\mathbb { R } ^ { \\tilde { N } } \\mathbb { R } ^ { M }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 205,
+ 609,
+ 258,
+ 627
+ ],
+ "score": 1.0,
+ "content": "and a subset",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 258,
+ 612,
+ 295,
+ 623
+ ],
+ "score": 0.89,
+ "content": "V \\subset \\mathbb { R } ^ { N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 295,
+ 609,
+ 411,
+ 627
+ ],
+ "score": 1.0,
+ "content": ", we denote the restriction of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 411,
+ 613,
+ 420,
+ 622
+ ],
+ "score": 0.84,
+ "content": "P",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 420,
+ 609,
+ 431,
+ 627
+ ],
+ "score": 1.0,
+ "content": "to",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 431,
+ 613,
+ 441,
+ 622
+ ],
+ "score": 0.81,
+ "content": "V",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 441,
+ 609,
+ 454,
+ 627
+ ],
+ "score": 1.0,
+ "content": "by",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 455,
+ 613,
+ 473,
+ 624
+ ],
+ "score": 0.92,
+ "content": "P | _ { V }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 473,
+ 609,
+ 478,
+ 627
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 43
+ }
+ ],
+ "index": 38.5
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 108,
+ 638,
+ 223,
+ 650
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 637,
+ 225,
+ 651
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 637,
+ 225,
+ 651
+ ],
+ "score": 1.0,
+ "content": "3.2 DYNAMICAL SYSTEM",
+ "type": "text"
+ }
+ ],
+ "index": 44
+ }
+ ],
+ "index": 44
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 659,
+ 503,
+ 682
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 660,
+ 505,
+ 672
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 660,
+ 505,
+ 672
+ ],
+ "score": 1.0,
+ "content": "Although we are mainly interested in GCNs, we develop our theory more generally using dynamical",
+ "type": "text"
+ }
+ ],
+ "index": 45
+ },
+ {
+ "bbox": [
+ 105,
+ 671,
+ 325,
+ 683
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 671,
+ 325,
+ 683
+ ],
+ "score": 1.0,
+ "content": "systems. We will specialize to the GCNs in Section 4.",
+ "type": "text"
+ }
+ ],
+ "index": 46
+ }
+ ],
+ "index": 45.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 686,
+ 505,
+ 732
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 104,
+ 684,
+ 507,
+ 701
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 684,
+ 124,
+ 701
+ ],
+ "score": 1.0,
+ "content": "For",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 124,
+ 687,
+ 192,
+ 699
+ ],
+ "score": 0.87,
+ "content": "N , C , H _ { l } \\ \\in \\ \\mathbb { N } _ { + }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 196,
+ 687,
+ 235,
+ 699
+ ],
+ "score": 0.74,
+ "content": "( l \\in \\mathbb { N } _ { + } )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 235,
+ 684,
+ 255,
+ 701
+ ],
+ "score": 1.0,
+ "content": ", let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 255,
+ 686,
+ 309,
+ 698
+ ],
+ "score": 0.9,
+ "content": "P \\in \\mathbb { R } ^ { N \\times N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 309,
+ 684,
+ 426,
+ 701
+ ],
+ "score": 1.0,
+ "content": "be a symmetric matrix and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 427,
+ 686,
+ 488,
+ 699
+ ],
+ "score": 0.92,
+ "content": "W _ { l h } \\ \\in \\ \\mathbb { R } ^ { C \\times C }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 489,
+ 684,
+ 507,
+ 701
+ ],
+ "score": 1.0,
+ "content": "for",
+ "type": "text"
+ }
+ ],
+ "index": 47
+ },
+ {
+ "bbox": [
+ 107,
+ 694,
+ 508,
+ 713
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 699,
+ 142,
+ 710
+ ],
+ "score": 0.87,
+ "content": "l \\in \\mathbb { N } _ { + }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 143,
+ 694,
+ 163,
+ 713
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 163,
+ 699,
+ 205,
+ 709
+ ],
+ "score": 0.87,
+ "content": "h \\in \\lceil H _ { l } \\rceil",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 205,
+ 694,
+ 259,
+ 713
+ ],
+ "score": 1.0,
+ "content": ". We define",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 259,
+ 698,
+ 357,
+ 710
+ ],
+ "score": 0.89,
+ "content": "f _ { l } : \\mathbb { R } ^ { N \\times C } \\mathbb { R } ^ { \\mathbf { \\tilde { N } } \\times C }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 358,
+ 694,
+ 375,
+ 713
+ ],
+ "score": 1.0,
+ "content": "by",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 375,
+ 699,
+ 471,
+ 711
+ ],
+ "score": 0.92,
+ "content": "f _ { l } ( X ) \\ : = \\ \\mathrm { M L P } _ { l } ( P X )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 472,
+ 694,
+ 508,
+ 713
+ ],
+ "score": 1.0,
+ "content": ". Here,",
+ "type": "text"
+ }
+ ],
+ "index": 48
+ },
+ {
+ "bbox": [
+ 106,
+ 706,
+ 506,
+ 723
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 709,
+ 213,
+ 721
+ ],
+ "score": 0.88,
+ "content": "\\mathrm { M L P } _ { l } : \\mathbb { R } ^ { N \\times C } \\to \\dot { \\mathbb { R } } ^ { N \\times C }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 214,
+ 706,
+ 239,
+ 723
+ ],
+ "score": 1.0,
+ "content": "is the",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 240,
+ 710,
+ 244,
+ 720
+ ],
+ "score": 0.66,
+ "content": "l",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 244,
+ 706,
+ 438,
+ 723
+ ],
+ "score": 1.0,
+ "content": "-th multi-layer perceptron common to all nodes",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 439,
+ 711,
+ 453,
+ 719
+ ],
+ "score": 0.61,
+ "content": "\\mathrm { { . x u } }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 453,
+ 706,
+ 506,
+ 723
+ ],
+ "score": 1.0,
+ "content": "et al., 2019)",
+ "type": "text"
+ }
+ ],
+ "index": 49
+ },
+ {
+ "bbox": [
+ 103,
+ 718,
+ 503,
+ 733
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 103,
+ 718,
+ 181,
+ 733
+ ],
+ "score": 1.0,
+ "content": "and is defined by",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 182,
+ 720,
+ 378,
+ 733
+ ],
+ "score": 0.89,
+ "content": "\\mathrm { M L P } _ { l } ( X ) : = \\sigma ( \\cdots \\sigma ( \\sigma ( \\dot { X } ) \\bar { W } _ { l 1 } ) \\bar { W } _ { l 2 } \\cdot \\cdot \\cdot W _ { l H _ { l } } )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 378,
+ 718,
+ 410,
+ 733
+ ],
+ "score": 1.0,
+ "content": ", where",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 410,
+ 720,
+ 503,
+ 730
+ ],
+ "score": 0.9,
+ "content": "\\sigma : \\mathbb { R } ^ { \\dot { N } \\times C } \\mathbb { R } ^ { N \\times \\dot { C } }",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 50
+ }
+ ],
+ "index": 48.5
+ }
+ ],
+ "page_idx": 2,
+ "page_size": [
+ 612,
+ 792
+ ],
+ "discarded_blocks": [
+ {
+ "type": "discarded",
+ "bbox": [
+ 108,
+ 27,
+ 293,
+ 37
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 26,
+ 294,
+ 38
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 26,
+ 294,
+ 38
+ ],
+ "score": 1.0,
+ "content": "Published as a conference paper at ICLR 2020",
+ "type": "text"
+ }
+ ]
+ }
+ ]
+ },
+ {
+ "type": "discarded",
+ "bbox": [
+ 302,
+ 751,
+ 309,
+ 760
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 301,
+ 750,
+ 310,
+ 762
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 301,
+ 750,
+ 310,
+ 762
+ ],
+ "score": 1.0,
+ "content": "3",
+ "type": "text"
+ }
+ ]
+ }
+ ]
+ }
+ ],
+ "para_blocks": [
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 82,
+ 504,
+ 126
+ ],
+ "lines": [],
+ "index": 1.5,
+ "bbox_fs": [
+ 105,
+ 82,
+ 506,
+ 129
+ ],
+ "lines_deleted": true
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 132,
+ 505,
+ 308
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 132,
+ 505,
+ 145
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 132,
+ 505,
+ 145
+ ],
+ "score": 1.0,
+ "content": "Expressive Power of Graph NNs. Several studies have focused on theoretical analysis and the im-",
+ "type": "text"
+ }
+ ],
+ "index": 4
+ },
+ {
+ "bbox": [
+ 105,
+ 143,
+ 505,
+ 156
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 143,
+ 505,
+ 156
+ ],
+ "score": 1.0,
+ "content": "provement of graph NN expressive power. For example, Xu et al. (2019) proved that graph NNs are",
+ "type": "text"
+ }
+ ],
+ "index": 5
+ },
+ {
+ "bbox": [
+ 104,
+ 154,
+ 505,
+ 167
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 154,
+ 505,
+ 167
+ ],
+ "score": 1.0,
+ "content": "no more powerful than the Weisfeiler – Lehman (WL) isomorphism test (Weisfeiler & A.A., 1968)",
+ "type": "text"
+ }
+ ],
+ "index": 6
+ },
+ {
+ "bbox": [
+ 105,
+ 165,
+ 505,
+ 178
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 165,
+ 505,
+ 178
+ ],
+ "score": 1.0,
+ "content": "and proposed a Graph Isomorphism Network (GIN), that is approximately as powerful as the WL",
+ "type": "text"
+ }
+ ],
+ "index": 7
+ },
+ {
+ "bbox": [
+ 105,
+ 175,
+ 506,
+ 190
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 175,
+ 506,
+ 190
+ ],
+ "score": 1.0,
+ "content": "test. Although they experimentally showed that GIN has improved accuracy in supervised learning",
+ "type": "text"
+ }
+ ],
+ "index": 8
+ },
+ {
+ "bbox": [
+ 105,
+ 187,
+ 505,
+ 200
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 187,
+ 505,
+ 200
+ ],
+ "score": 1.0,
+ "content": "tasks, their analysis was restricted to the graph isomorphism problem. Xu et al. (2018) analyzed",
+ "type": "text"
+ }
+ ],
+ "index": 9
+ },
+ {
+ "bbox": [
+ 105,
+ 199,
+ 505,
+ 211
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 199,
+ 505,
+ 211
+ ],
+ "score": 1.0,
+ "content": "the non-asymptotic properties of GCNs through the lens of random walk theory. They proved the",
+ "type": "text"
+ }
+ ],
+ "index": 10
+ },
+ {
+ "bbox": [
+ 105,
+ 209,
+ 505,
+ 222
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 209,
+ 505,
+ 222
+ ],
+ "score": 1.0,
+ "content": "limitations of GCNs in expander-like graphs and proposed a Jumping Knowledge Network (JK-",
+ "type": "text"
+ }
+ ],
+ "index": 11
+ },
+ {
+ "bbox": [
+ 105,
+ 220,
+ 505,
+ 233
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 220,
+ 505,
+ 233
+ ],
+ "score": 1.0,
+ "content": "Net) to address the issue. To handle the non-linearity, they linearized networks by a randomization",
+ "type": "text"
+ }
+ ],
+ "index": 12
+ },
+ {
+ "bbox": [
+ 105,
+ 231,
+ 505,
+ 243
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 231,
+ 505,
+ 243
+ ],
+ "score": 1.0,
+ "content": "assumption (Choromanska et al., 2015). We take a different strategy and make use of the interpre-",
+ "type": "text"
+ }
+ ],
+ "index": 13
+ },
+ {
+ "bbox": [
+ 105,
+ 242,
+ 505,
+ 254
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 242,
+ 505,
+ 254
+ ],
+ "score": 1.0,
+ "content": "tation of ReLU as a projection onto a cone. Recently, NT & Maehara (2019) showed that a GCN",
+ "type": "text"
+ }
+ ],
+ "index": 14
+ },
+ {
+ "bbox": [
+ 105,
+ 253,
+ 506,
+ 266
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 253,
+ 506,
+ 266
+ ],
+ "score": 1.0,
+ "content": "approximately works as a low-pass filter plus an MLP in a certain setting. Although they analyzed",
+ "type": "text"
+ }
+ ],
+ "index": 15
+ },
+ {
+ "bbox": [
+ 105,
+ 263,
+ 506,
+ 277
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 263,
+ 506,
+ 277
+ ],
+ "score": 1.0,
+ "content": "finite-depth GCNs, our theory has similar spirits with theirs because our “information-less” space",
+ "type": "text"
+ }
+ ],
+ "index": 16
+ },
+ {
+ "bbox": [
+ 105,
+ 275,
+ 504,
+ 287
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 275,
+ 504,
+ 287
+ ],
+ "score": 1.0,
+ "content": "corresponds to the lowest frequency of a graph Laplacian. Another point is that they imposed as-",
+ "type": "text"
+ }
+ ],
+ "index": 17
+ },
+ {
+ "bbox": [
+ 105,
+ 285,
+ 505,
+ 299
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 285,
+ 505,
+ 299
+ ],
+ "score": 1.0,
+ "content": "sumptions that input signals consist of low-frequent true signals and high-frequent noise, whereas",
+ "type": "text"
+ }
+ ],
+ "index": 18
+ },
+ {
+ "bbox": [
+ 106,
+ 297,
+ 240,
+ 310
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 297,
+ 240,
+ 310
+ ],
+ "score": 1.0,
+ "content": "we need not such an assumption.",
+ "type": "text"
+ }
+ ],
+ "index": 19
+ }
+ ],
+ "index": 11.5,
+ "bbox_fs": [
+ 104,
+ 132,
+ 506,
+ 310
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 313,
+ 505,
+ 445
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 313,
+ 505,
+ 326
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 313,
+ 505,
+ 326
+ ],
+ "score": 1.0,
+ "content": "Role of Deep and Non-linear Structures. For ordinal DL models such as FNNs and CNNs, we",
+ "type": "text"
+ }
+ ],
+ "index": 20
+ },
+ {
+ "bbox": [
+ 105,
+ 324,
+ 505,
+ 338
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 324,
+ 505,
+ 338
+ ],
+ "score": 1.0,
+ "content": "have both theoretical and empirical justification of deep and non-linear architectures for enhancing",
+ "type": "text"
+ }
+ ],
+ "index": 21
+ },
+ {
+ "bbox": [
+ 105,
+ 335,
+ 505,
+ 349
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 335,
+ 505,
+ 349
+ ],
+ "score": 1.0,
+ "content": "of the expressive power (e.g., Telgarsky (2016); Petersen & Voigtlaender (2018); Oono & Suzuki",
+ "type": "text"
+ }
+ ],
+ "index": 22
+ },
+ {
+ "bbox": [
+ 105,
+ 345,
+ 505,
+ 360
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 345,
+ 505,
+ 360
+ ],
+ "score": 1.0,
+ "content": "(2019)). In contrast, several studies have witnessed severe performance degradation when stacking",
+ "type": "text"
+ }
+ ],
+ "index": 23
+ },
+ {
+ "bbox": [
+ 105,
+ 358,
+ 506,
+ 369
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 358,
+ 506,
+ 369
+ ],
+ "score": 1.0,
+ "content": "many layers on graph NNs (Kipf & Welling, 2017; Wu et al., 2019b). Li et al. (2018) reported that",
+ "type": "text"
+ }
+ ],
+ "index": 24
+ },
+ {
+ "bbox": [
+ 105,
+ 369,
+ 505,
+ 381
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 369,
+ 505,
+ 381
+ ],
+ "score": 1.0,
+ "content": "feature vectors on nodes in a graph go indistinguishable as we increase layers in several tasks. They",
+ "type": "text"
+ }
+ ],
+ "index": 25
+ },
+ {
+ "bbox": [
+ 106,
+ 381,
+ 504,
+ 392
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 381,
+ 504,
+ 392
+ ],
+ "score": 1.0,
+ "content": "named this phenomenon over-smoothing. Regarding non-linearity, Wu et al. (2019a) empirically",
+ "type": "text"
+ }
+ ],
+ "index": 26
+ },
+ {
+ "bbox": [
+ 105,
+ 390,
+ 504,
+ 403
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 390,
+ 504,
+ 403
+ ],
+ "score": 1.0,
+ "content": "showed that graph NNs achieve comparable performance even if we omit intermediate non-linearity.",
+ "type": "text"
+ }
+ ],
+ "index": 27
+ },
+ {
+ "bbox": [
+ 105,
+ 401,
+ 505,
+ 415
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 401,
+ 505,
+ 415
+ ],
+ "score": 1.0,
+ "content": "These observations gave us questions about the current models of deep graph NNs in terms of their",
+ "type": "text"
+ }
+ ],
+ "index": 28
+ },
+ {
+ "bbox": [
+ 105,
+ 412,
+ 506,
+ 426
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 412,
+ 506,
+ 426
+ ],
+ "score": 1.0,
+ "content": "expressive power. Several studies gave theoretical explanations of the over-smoothing phenomena",
+ "type": "text"
+ }
+ ],
+ "index": 29
+ },
+ {
+ "bbox": [
+ 105,
+ 423,
+ 505,
+ 436
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 423,
+ 505,
+ 436
+ ],
+ "score": 1.0,
+ "content": "for linear GNNs (Li et al., 2018; Zhang, 2019; Zhao & Akoglu, 2020). We can think of our theory",
+ "type": "text"
+ }
+ ],
+ "index": 30
+ },
+ {
+ "bbox": [
+ 105,
+ 434,
+ 314,
+ 446
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 434,
+ 314,
+ 446
+ ],
+ "score": 1.0,
+ "content": "as an extension of their results to non-linear GNNs.",
+ "type": "text"
+ }
+ ],
+ "index": 31
+ }
+ ],
+ "index": 25.5,
+ "bbox_fs": [
+ 105,
+ 313,
+ 506,
+ 446
+ ]
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 108,
+ 463,
+ 326,
+ 476
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 462,
+ 328,
+ 477
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 462,
+ 328,
+ 477
+ ],
+ "score": 1.0,
+ "content": "3 PROBLEM SETTING AND MAIN RESULT",
+ "type": "text"
+ }
+ ],
+ "index": 32
+ }
+ ],
+ "index": 32
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 107,
+ 488,
+ 176,
+ 500
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 487,
+ 178,
+ 502
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 487,
+ 178,
+ 502
+ ],
+ "score": 1.0,
+ "content": "3.1 NOTATION",
+ "type": "text"
+ }
+ ],
+ "index": 33
+ }
+ ],
+ "index": 33
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 509,
+ 505,
+ 624
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 508,
+ 506,
+ 523
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 508,
+ 122,
+ 523
+ ],
+ "score": 1.0,
+ "content": "Let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 123,
+ 510,
+ 138,
+ 521
+ ],
+ "score": 0.84,
+ "content": "\\mathbb { N } _ { + }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 138,
+ 508,
+ 281,
+ 523
+ ],
+ "score": 1.0,
+ "content": "be the set of positive integers. For",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 281,
+ 510,
+ 319,
+ 522
+ ],
+ "score": 0.92,
+ "content": "N \\in { \\mathbb { N } } _ { + }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 320,
+ 508,
+ 368,
+ 523
+ ],
+ "score": 1.0,
+ "content": ", we denote",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 369,
+ 510,
+ 448,
+ 522
+ ],
+ "score": 0.92,
+ "content": "[ N ] : = \\{ 1 , \\dots , N \\}",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 448,
+ 508,
+ 506,
+ 523
+ ],
+ "score": 1.0,
+ "content": ". For a vector",
+ "type": "text"
+ }
+ ],
+ "index": 34
+ },
+ {
+ "bbox": [
+ 106,
+ 517,
+ 506,
+ 534
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 521,
+ 140,
+ 531
+ ],
+ "score": 0.9,
+ "content": "v \\in \\mathbb { R } ^ { N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 140,
+ 517,
+ 181,
+ 534
+ ],
+ "score": 1.0,
+ "content": ", we write",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 182,
+ 522,
+ 207,
+ 532
+ ],
+ "score": 0.91,
+ "content": "v \\geq 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 207,
+ 517,
+ 262,
+ 534
+ ],
+ "score": 1.0,
+ "content": "if and only if",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 263,
+ 522,
+ 293,
+ 532
+ ],
+ "score": 0.91,
+ "content": "v _ { n } \\geq 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 293,
+ 517,
+ 321,
+ 534
+ ],
+ "score": 1.0,
+ "content": "for all",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 321,
+ 521,
+ 355,
+ 532
+ ],
+ "score": 0.91,
+ "content": "n \\in [ N ]",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 355,
+ 517,
+ 451,
+ 534
+ ],
+ "score": 1.0,
+ "content": ". Similarly, for a matrix",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 452,
+ 520,
+ 501,
+ 531
+ ],
+ "score": 0.91,
+ "content": "X \\in \\mathbb { R } ^ { N \\times C }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 501,
+ 517,
+ 506,
+ 534
+ ],
+ "score": 1.0,
+ "content": ",",
+ "type": "text"
+ }
+ ],
+ "index": 35
+ },
+ {
+ "bbox": [
+ 106,
+ 531,
+ 506,
+ 545
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 531,
+ 145,
+ 545
+ ],
+ "score": 1.0,
+ "content": "we write",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 145,
+ 532,
+ 176,
+ 543
+ ],
+ "score": 0.91,
+ "content": "X \\geq 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 176,
+ 531,
+ 234,
+ 545
+ ],
+ "score": 1.0,
+ "content": "if and only if",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 235,
+ 532,
+ 273,
+ 543
+ ],
+ "score": 0.92,
+ "content": "X _ { n c } \\geq 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 274,
+ 531,
+ 303,
+ 545
+ ],
+ "score": 1.0,
+ "content": "for all",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 303,
+ 532,
+ 339,
+ 544
+ ],
+ "score": 0.91,
+ "content": "n \\in [ N ]",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 340,
+ 531,
+ 359,
+ 545
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 359,
+ 532,
+ 392,
+ 544
+ ],
+ "score": 0.92,
+ "content": "c \\in [ C ]",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 393,
+ 531,
+ 506,
+ 545
+ ],
+ "score": 1.0,
+ "content": ". We say such a vector and",
+ "type": "text"
+ }
+ ],
+ "index": 36
+ },
+ {
+ "bbox": [
+ 105,
+ 542,
+ 505,
+ 555
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 542,
+ 207,
+ 555
+ ],
+ "score": 1.0,
+ "content": "matrix is non-negative.",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 207,
+ 543,
+ 226,
+ 555
+ ],
+ "score": 0.9,
+ "content": "\\langle \\cdot , \\cdot \\rangle",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 226,
+ 542,
+ 505,
+ 555
+ ],
+ "score": 1.0,
+ "content": "denotes the inner product of vectors or matrices, depending on the",
+ "type": "text"
+ }
+ ],
+ "index": 37
+ },
+ {
+ "bbox": [
+ 104,
+ 550,
+ 507,
+ 568
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 550,
+ 142,
+ 568
+ ],
+ "score": 1.0,
+ "content": "context:",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 142,
+ 553,
+ 200,
+ 566
+ ],
+ "score": 0.92,
+ "content": "\\langle u , v \\rangle : = u ^ { \\top } v",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 200,
+ 550,
+ 215,
+ 568
+ ],
+ "score": 1.0,
+ "content": "for",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 216,
+ 553,
+ 259,
+ 565
+ ],
+ "score": 0.92,
+ "content": "u , v \\in \\mathbb { R } ^ { N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 259,
+ 550,
+ 278,
+ 568
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 278,
+ 553,
+ 362,
+ 565
+ ],
+ "score": 0.9,
+ "content": "\\langle X , Y \\rangle : = \\mathrm { t r } ( X ^ { T } Y )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 362,
+ 550,
+ 378,
+ 568
+ ],
+ "score": 1.0,
+ "content": "for",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 379,
+ 553,
+ 440,
+ 564
+ ],
+ "score": 0.87,
+ "content": "X , Y \\in \\mathbb { R } ^ { N \\times C }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 440,
+ 550,
+ 443,
+ 568
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 443,
+ 554,
+ 458,
+ 565
+ ],
+ "score": 0.53,
+ "content": "{ \\bf 1 } _ { P }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 458,
+ 550,
+ 507,
+ 568
+ ],
+ "score": 1.0,
+ "content": "equals to 1",
+ "type": "text"
+ }
+ ],
+ "index": 38
+ },
+ {
+ "bbox": [
+ 104,
+ 561,
+ 507,
+ 579
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 561,
+ 178,
+ 579
+ ],
+ "score": 1.0,
+ "content": "if the proposition",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 179,
+ 566,
+ 188,
+ 575
+ ],
+ "score": 0.84,
+ "content": "P",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 188,
+ 561,
+ 295,
+ 579
+ ],
+ "score": 1.0,
+ "content": "is true else 0. For vectors",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 295,
+ 565,
+ 329,
+ 575
+ ],
+ "score": 0.87,
+ "content": "v \\in \\mathbb { R } ^ { N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 330,
+ 561,
+ 348,
+ 579
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 348,
+ 564,
+ 384,
+ 575
+ ],
+ "score": 0.9,
+ "content": "w \\in \\mathbb { R } ^ { C }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 384,
+ 561,
+ 389,
+ 579
+ ],
+ "score": 1.0,
+ "content": ",",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 389,
+ 564,
+ 455,
+ 575
+ ],
+ "score": 0.87,
+ "content": "v \\otimes w \\in \\mathbb { R } ^ { N \\times C }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 456,
+ 561,
+ 507,
+ 579
+ ],
+ "score": 1.0,
+ "content": "denotes the",
+ "type": "text"
+ }
+ ],
+ "index": 39
+ },
+ {
+ "bbox": [
+ 104,
+ 575,
+ 504,
+ 591
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 575,
+ 194,
+ 591
+ ],
+ "score": 1.0,
+ "content": "Kronecker product of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 195,
+ 579,
+ 201,
+ 587
+ ],
+ "score": 0.76,
+ "content": "v",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 201,
+ 575,
+ 218,
+ 591
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 219,
+ 579,
+ 228,
+ 587
+ ],
+ "score": 0.78,
+ "content": "w",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 228,
+ 575,
+ 273,
+ 591
+ ],
+ "score": 1.0,
+ "content": "defined by",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 273,
+ 577,
+ 353,
+ 589
+ ],
+ "score": 0.88,
+ "content": "( v \\otimes w ) _ { n c } : = v _ { n } w _ { c } .",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 353,
+ 575,
+ 372,
+ 591
+ ],
+ "score": 1.0,
+ "content": ". For",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 372,
+ 577,
+ 422,
+ 588
+ ],
+ "score": 0.86,
+ "content": "X \\in \\mathbb { R } ^ { N \\times C }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 422,
+ 575,
+ 426,
+ 591
+ ],
+ "score": 1.0,
+ "content": ",",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 426,
+ 576,
+ 504,
+ 589
+ ],
+ "score": 0.8,
+ "content": "\\| X \\| _ { \\mathrm { F } } : = \\langle X , X \\rangle ^ { \\frac { 1 } { 2 } }",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 40
+ },
+ {
+ "bbox": [
+ 102,
+ 584,
+ 504,
+ 605
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 102,
+ 584,
+ 234,
+ 605
+ ],
+ "score": 1.0,
+ "content": "denotes the Frobenius norm of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 234,
+ 589,
+ 244,
+ 598
+ ],
+ "score": 0.85,
+ "content": "X",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 244,
+ 584,
+ 303,
+ 605
+ ],
+ "score": 1.0,
+ "content": ". For a vector",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 303,
+ 588,
+ 338,
+ 599
+ ],
+ "score": 0.84,
+ "content": "v \\in \\mathbb { R } ^ { N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 339,
+ 584,
+ 342,
+ 605
+ ],
+ "score": 1.0,
+ "content": ",",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 343,
+ 588,
+ 504,
+ 601
+ ],
+ "score": 0.88,
+ "content": "\\mathrm { d i a g } ( v ) : = ( v _ { n } \\delta _ { n m } ) _ { n , m \\in [ N ] } \\in \\mathbb { R } ^ { N \\times N }",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 41
+ },
+ {
+ "bbox": [
+ 104,
+ 597,
+ 506,
+ 614
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 597,
+ 230,
+ 614
+ ],
+ "score": 1.0,
+ "content": "denotes the diagonalization of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 231,
+ 603,
+ 237,
+ 611
+ ],
+ "score": 0.67,
+ "content": "v",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 237,
+ 597,
+ 242,
+ 614
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 243,
+ 600,
+ 298,
+ 612
+ ],
+ "score": 0.89,
+ "content": "I _ { N } \\in \\mathbb { R } ^ { N \\times N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 298,
+ 597,
+ 439,
+ 614
+ ],
+ "score": 1.0,
+ "content": "denotes the identity matrix of size",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 440,
+ 602,
+ 450,
+ 611
+ ],
+ "score": 0.82,
+ "content": "N",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 450,
+ 597,
+ 506,
+ 614
+ ],
+ "score": 1.0,
+ "content": ". For a linear",
+ "type": "text"
+ }
+ ],
+ "index": 42
+ },
+ {
+ "bbox": [
+ 104,
+ 609,
+ 478,
+ 627
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 609,
+ 142,
+ 627
+ ],
+ "score": 1.0,
+ "content": "operator",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 142,
+ 612,
+ 205,
+ 622
+ ],
+ "score": 0.91,
+ "content": "P : \\mathbb { R } ^ { \\tilde { N } } \\mathbb { R } ^ { M }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 205,
+ 609,
+ 258,
+ 627
+ ],
+ "score": 1.0,
+ "content": "and a subset",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 258,
+ 612,
+ 295,
+ 623
+ ],
+ "score": 0.89,
+ "content": "V \\subset \\mathbb { R } ^ { N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 295,
+ 609,
+ 411,
+ 627
+ ],
+ "score": 1.0,
+ "content": ", we denote the restriction of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 411,
+ 613,
+ 420,
+ 622
+ ],
+ "score": 0.84,
+ "content": "P",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 420,
+ 609,
+ 431,
+ 627
+ ],
+ "score": 1.0,
+ "content": "to",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 431,
+ 613,
+ 441,
+ 622
+ ],
+ "score": 0.81,
+ "content": "V",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 441,
+ 609,
+ 454,
+ 627
+ ],
+ "score": 1.0,
+ "content": "by",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 455,
+ 613,
+ 473,
+ 624
+ ],
+ "score": 0.92,
+ "content": "P | _ { V }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 473,
+ 609,
+ 478,
+ 627
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 43
+ }
+ ],
+ "index": 38.5,
+ "bbox_fs": [
+ 102,
+ 508,
+ 507,
+ 627
+ ]
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 108,
+ 638,
+ 223,
+ 650
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 637,
+ 225,
+ 651
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 637,
+ 225,
+ 651
+ ],
+ "score": 1.0,
+ "content": "3.2 DYNAMICAL SYSTEM",
+ "type": "text"
+ }
+ ],
+ "index": 44
+ }
+ ],
+ "index": 44
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 659,
+ 503,
+ 682
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 660,
+ 505,
+ 672
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 660,
+ 505,
+ 672
+ ],
+ "score": 1.0,
+ "content": "Although we are mainly interested in GCNs, we develop our theory more generally using dynamical",
+ "type": "text"
+ }
+ ],
+ "index": 45
+ },
+ {
+ "bbox": [
+ 105,
+ 671,
+ 325,
+ 683
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 671,
+ 325,
+ 683
+ ],
+ "score": 1.0,
+ "content": "systems. We will specialize to the GCNs in Section 4.",
+ "type": "text"
+ }
+ ],
+ "index": 46
+ }
+ ],
+ "index": 45.5,
+ "bbox_fs": [
+ 105,
+ 660,
+ 505,
+ 683
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 686,
+ 505,
+ 732
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 104,
+ 684,
+ 507,
+ 701
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 684,
+ 124,
+ 701
+ ],
+ "score": 1.0,
+ "content": "For",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 124,
+ 687,
+ 192,
+ 699
+ ],
+ "score": 0.87,
+ "content": "N , C , H _ { l } \\ \\in \\ \\mathbb { N } _ { + }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 196,
+ 687,
+ 235,
+ 699
+ ],
+ "score": 0.74,
+ "content": "( l \\in \\mathbb { N } _ { + } )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 235,
+ 684,
+ 255,
+ 701
+ ],
+ "score": 1.0,
+ "content": ", let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 255,
+ 686,
+ 309,
+ 698
+ ],
+ "score": 0.9,
+ "content": "P \\in \\mathbb { R } ^ { N \\times N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 309,
+ 684,
+ 426,
+ 701
+ ],
+ "score": 1.0,
+ "content": "be a symmetric matrix and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 427,
+ 686,
+ 488,
+ 699
+ ],
+ "score": 0.92,
+ "content": "W _ { l h } \\ \\in \\ \\mathbb { R } ^ { C \\times C }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 489,
+ 684,
+ 507,
+ 701
+ ],
+ "score": 1.0,
+ "content": "for",
+ "type": "text"
+ }
+ ],
+ "index": 47
+ },
+ {
+ "bbox": [
+ 107,
+ 694,
+ 508,
+ 713
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 699,
+ 142,
+ 710
+ ],
+ "score": 0.87,
+ "content": "l \\in \\mathbb { N } _ { + }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 143,
+ 694,
+ 163,
+ 713
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 163,
+ 699,
+ 205,
+ 709
+ ],
+ "score": 0.87,
+ "content": "h \\in \\lceil H _ { l } \\rceil",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 205,
+ 694,
+ 259,
+ 713
+ ],
+ "score": 1.0,
+ "content": ". We define",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 259,
+ 698,
+ 357,
+ 710
+ ],
+ "score": 0.89,
+ "content": "f _ { l } : \\mathbb { R } ^ { N \\times C } \\mathbb { R } ^ { \\mathbf { \\tilde { N } } \\times C }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 358,
+ 694,
+ 375,
+ 713
+ ],
+ "score": 1.0,
+ "content": "by",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 375,
+ 699,
+ 471,
+ 711
+ ],
+ "score": 0.92,
+ "content": "f _ { l } ( X ) \\ : = \\ \\mathrm { M L P } _ { l } ( P X )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 472,
+ 694,
+ 508,
+ 713
+ ],
+ "score": 1.0,
+ "content": ". Here,",
+ "type": "text"
+ }
+ ],
+ "index": 48
+ },
+ {
+ "bbox": [
+ 106,
+ 706,
+ 506,
+ 723
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 709,
+ 213,
+ 721
+ ],
+ "score": 0.88,
+ "content": "\\mathrm { M L P } _ { l } : \\mathbb { R } ^ { N \\times C } \\to \\dot { \\mathbb { R } } ^ { N \\times C }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 214,
+ 706,
+ 239,
+ 723
+ ],
+ "score": 1.0,
+ "content": "is the",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 240,
+ 710,
+ 244,
+ 720
+ ],
+ "score": 0.66,
+ "content": "l",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 244,
+ 706,
+ 438,
+ 723
+ ],
+ "score": 1.0,
+ "content": "-th multi-layer perceptron common to all nodes",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 439,
+ 711,
+ 453,
+ 719
+ ],
+ "score": 0.61,
+ "content": "\\mathrm { { . x u } }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 453,
+ 706,
+ 506,
+ 723
+ ],
+ "score": 1.0,
+ "content": "et al., 2019)",
+ "type": "text"
+ }
+ ],
+ "index": 49
+ },
+ {
+ "bbox": [
+ 103,
+ 718,
+ 503,
+ 733
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 103,
+ 718,
+ 181,
+ 733
+ ],
+ "score": 1.0,
+ "content": "and is defined by",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 182,
+ 720,
+ 378,
+ 733
+ ],
+ "score": 0.89,
+ "content": "\\mathrm { M L P } _ { l } ( X ) : = \\sigma ( \\cdots \\sigma ( \\sigma ( \\dot { X } ) \\bar { W } _ { l 1 } ) \\bar { W } _ { l 2 } \\cdot \\cdot \\cdot W _ { l H _ { l } } )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 378,
+ 718,
+ 410,
+ 733
+ ],
+ "score": 1.0,
+ "content": ", where",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 410,
+ 720,
+ 503,
+ 730
+ ],
+ "score": 0.9,
+ "content": "\\sigma : \\mathbb { R } ^ { \\dot { N } \\times C } \\mathbb { R } ^ { N \\times \\dot { C } }",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 50
+ },
+ {
+ "bbox": [
+ 104,
+ 80,
+ 504,
+ 96
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 80,
+ 399,
+ 96
+ ],
+ "score": 1.0,
+ "content": "is an element-wise ReLU function (Krizhevsky et al., 2012) defined by",
+ "type": "text",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 399,
+ 82,
+ 504,
+ 95
+ ],
+ "score": 0.92,
+ "content": "\\sigma ( X ) _ { n c } : = \\operatorname* { m a x } ( X _ { n c } , 0 )",
+ "type": "inline_equation",
+ "cross_page": true
+ }
+ ],
+ "index": 0
+ },
+ {
+ "bbox": [
+ 104,
+ 92,
+ 506,
+ 109
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 92,
+ 122,
+ 109
+ ],
+ "score": 1.0,
+ "content": "for",
+ "type": "text",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 122,
+ 94,
+ 200,
+ 106
+ ],
+ "score": 0.91,
+ "content": "n \\in [ N ] , c \\in [ C ]",
+ "type": "inline_equation",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 200,
+ 92,
+ 320,
+ 109
+ ],
+ "score": 1.0,
+ "content": ". We consider the dynamics",
+ "type": "text",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 321,
+ 93,
+ 405,
+ 106
+ ],
+ "score": 0.92,
+ "content": "X ^ { ( l + 1 ) } : = f _ { l } ( X ^ { ( l ) } )",
+ "type": "inline_equation",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 406,
+ 92,
+ 506,
+ 109
+ ],
+ "score": 1.0,
+ "content": "with some initial value",
+ "type": "text",
+ "cross_page": true
+ }
+ ],
+ "index": 1
+ },
+ {
+ "bbox": [
+ 106,
+ 104,
+ 428,
+ 121
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 106,
+ 167,
+ 118
+ ],
+ "score": 0.91,
+ "content": "X ^ { ( 0 ) } \\in \\mathbb { R } ^ { N \\times C }",
+ "type": "inline_equation",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 167,
+ 104,
+ 362,
+ 121
+ ],
+ "score": 1.0,
+ "content": ". We are interested in the asymptotic behavior of",
+ "type": "text",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 362,
+ 106,
+ 381,
+ 117
+ ],
+ "score": 0.9,
+ "content": "X ^ { ( l ) }",
+ "type": "inline_equation",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 381,
+ 104,
+ 393,
+ 121
+ ],
+ "score": 1.0,
+ "content": "as",
+ "type": "text",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 393,
+ 108,
+ 423,
+ 117
+ ],
+ "score": 0.89,
+ "content": "l \\infty",
+ "type": "inline_equation",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 423,
+ 104,
+ 428,
+ 121
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text",
+ "cross_page": true
+ }
+ ],
+ "index": 2
+ }
+ ],
+ "index": 48.5,
+ "bbox_fs": [
+ 103,
+ 684,
+ 508,
+ 733
+ ]
+ }
+ ]
+ },
+ {
+ "preproc_blocks": [
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 82,
+ 505,
+ 119
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 104,
+ 80,
+ 504,
+ 96
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 80,
+ 399,
+ 96
+ ],
+ "score": 1.0,
+ "content": "is an element-wise ReLU function (Krizhevsky et al., 2012) defined by",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 399,
+ 82,
+ 504,
+ 95
+ ],
+ "score": 0.92,
+ "content": "\\sigma ( X ) _ { n c } : = \\operatorname* { m a x } ( X _ { n c } , 0 )",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 0
+ },
+ {
+ "bbox": [
+ 104,
+ 92,
+ 506,
+ 109
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 92,
+ 122,
+ 109
+ ],
+ "score": 1.0,
+ "content": "for",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 122,
+ 94,
+ 200,
+ 106
+ ],
+ "score": 0.91,
+ "content": "n \\in [ N ] , c \\in [ C ]",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 200,
+ 92,
+ 320,
+ 109
+ ],
+ "score": 1.0,
+ "content": ". We consider the dynamics",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 321,
+ 93,
+ 405,
+ 106
+ ],
+ "score": 0.92,
+ "content": "X ^ { ( l + 1 ) } : = f _ { l } ( X ^ { ( l ) } )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 406,
+ 92,
+ 506,
+ 109
+ ],
+ "score": 1.0,
+ "content": "with some initial value",
+ "type": "text"
+ }
+ ],
+ "index": 1
+ },
+ {
+ "bbox": [
+ 106,
+ 104,
+ 428,
+ 121
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 106,
+ 167,
+ 118
+ ],
+ "score": 0.91,
+ "content": "X ^ { ( 0 ) } \\in \\mathbb { R } ^ { N \\times C }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 167,
+ 104,
+ 362,
+ 121
+ ],
+ "score": 1.0,
+ "content": ". We are interested in the asymptotic behavior of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 362,
+ 106,
+ 381,
+ 117
+ ],
+ "score": 0.9,
+ "content": "X ^ { ( l ) }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 381,
+ 104,
+ 393,
+ 121
+ ],
+ "score": 1.0,
+ "content": "as",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 393,
+ 108,
+ 423,
+ 117
+ ],
+ "score": 0.89,
+ "content": "l \\infty",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 423,
+ 104,
+ 428,
+ 121
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 2
+ }
+ ],
+ "index": 1
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 109,
+ 123,
+ 502,
+ 158
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 122,
+ 505,
+ 137
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 122,
+ 123,
+ 137
+ ],
+ "score": 1.0,
+ "content": "For",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 124,
+ 124,
+ 162,
+ 135
+ ],
+ "score": 0.91,
+ "content": "M \\leq N",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 162,
+ 122,
+ 180,
+ 137
+ ],
+ "score": 1.0,
+ "content": ", let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 180,
+ 124,
+ 189,
+ 134
+ ],
+ "score": 0.8,
+ "content": "U",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 190,
+ 122,
+ 212,
+ 137
+ ],
+ "score": 1.0,
+ "content": "be a",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 212,
+ 124,
+ 224,
+ 134
+ ],
+ "score": 0.81,
+ "content": "M",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 224,
+ 122,
+ 329,
+ 137
+ ],
+ "score": 1.0,
+ "content": "-dimensional subspace of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 330,
+ 123,
+ 345,
+ 135
+ ],
+ "score": 0.9,
+ "content": "\\mathbb { R } ^ { N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 345,
+ 122,
+ 420,
+ 137
+ ],
+ "score": 1.0,
+ "content": ". We assume that",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 421,
+ 124,
+ 430,
+ 134
+ ],
+ "score": 0.82,
+ "content": "U",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 430,
+ 122,
+ 450,
+ 137
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 450,
+ 125,
+ 459,
+ 134
+ ],
+ "score": 0.81,
+ "content": "P",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 459,
+ 122,
+ 505,
+ 137
+ ],
+ "score": 1.0,
+ "content": "satisfy the",
+ "type": "text"
+ }
+ ],
+ "index": 3
+ },
+ {
+ "bbox": [
+ 106,
+ 135,
+ 504,
+ 147
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 135,
+ 338,
+ 147
+ ],
+ "score": 1.0,
+ "content": "following properties that generalize the situation where",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 338,
+ 136,
+ 347,
+ 145
+ ],
+ "score": 0.82,
+ "content": "U",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 348,
+ 135,
+ 504,
+ 147
+ ],
+ "score": 1.0,
+ "content": "is the eigenspace associated with the",
+ "type": "text"
+ }
+ ],
+ "index": 4
+ },
+ {
+ "bbox": [
+ 107,
+ 146,
+ 501,
+ 158
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 146,
+ 325,
+ 158
+ ],
+ "score": 1.0,
+ "content": "smallest eigenvalue of a (normalized) graph Laplacian",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 325,
+ 146,
+ 334,
+ 156
+ ],
+ "score": 0.8,
+ "content": "\\Delta",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 335,
+ 146,
+ 407,
+ 158
+ ],
+ "score": 1.0,
+ "content": "(that is, zero) and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 407,
+ 147,
+ 416,
+ 156
+ ],
+ "score": 0.83,
+ "content": "P",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 417,
+ 146,
+ 492,
+ 158
+ ],
+ "score": 1.0,
+ "content": "is a polynomial of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 492,
+ 147,
+ 501,
+ 156
+ ],
+ "score": 0.81,
+ "content": "\\Delta",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 5
+ }
+ ],
+ "index": 4
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 108,
+ 159,
+ 485,
+ 171
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 158,
+ 486,
+ 173
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 158,
+ 172,
+ 173
+ ],
+ "score": 1.0,
+ "content": "Assumption 1.",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 172,
+ 159,
+ 181,
+ 169
+ ],
+ "score": 0.49,
+ "content": "U",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 182,
+ 158,
+ 288,
+ 173
+ ],
+ "score": 1.0,
+ "content": "has an orthonormal basis",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 288,
+ 159,
+ 334,
+ 172
+ ],
+ "score": 0.93,
+ "content": "( e _ { m } ) _ { m \\in [ M ] }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 334,
+ 158,
+ 486,
+ 173
+ ],
+ "score": 1.0,
+ "content": "that consists of non-negative vectors.",
+ "type": "text"
+ }
+ ],
+ "index": 6
+ }
+ ],
+ "index": 6
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 172,
+ 382,
+ 184
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 172,
+ 383,
+ 185
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 172,
+ 172,
+ 185
+ ],
+ "score": 1.0,
+ "content": "Assumption 2.",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 172,
+ 173,
+ 181,
+ 183
+ ],
+ "score": 0.46,
+ "content": "U",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 182,
+ 172,
+ 256,
+ 185
+ ],
+ "score": 1.0,
+ "content": "is invariant under",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 256,
+ 173,
+ 265,
+ 183
+ ],
+ "score": 0.71,
+ "content": "P",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 265,
+ 172,
+ 294,
+ 185
+ ],
+ "score": 1.0,
+ "content": ", i.e., if",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 294,
+ 173,
+ 321,
+ 184
+ ],
+ "score": 0.81,
+ "content": "u \\in U",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 321,
+ 172,
+ 344,
+ 185
+ ],
+ "score": 1.0,
+ "content": ", then",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 345,
+ 173,
+ 379,
+ 183
+ ],
+ "score": 0.9,
+ "content": "P u \\in U",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 379,
+ 172,
+ 383,
+ 185
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 7
+ }
+ ],
+ "index": 7
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 191,
+ 505,
+ 258
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 189,
+ 505,
+ 204
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 189,
+ 150,
+ 204
+ ],
+ "score": 1.0,
+ "content": "We endow",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 150,
+ 191,
+ 166,
+ 202
+ ],
+ "score": 0.88,
+ "content": "\\mathbb { R } ^ { N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 166,
+ 189,
+ 453,
+ 204
+ ],
+ "score": 1.0,
+ "content": "with the ordinal inner product and denote the orthogonal complement of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 453,
+ 192,
+ 462,
+ 202
+ ],
+ "score": 0.81,
+ "content": "U",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 462,
+ 189,
+ 474,
+ 204
+ ],
+ "score": 1.0,
+ "content": "by",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 475,
+ 191,
+ 505,
+ 203
+ ],
+ "score": 0.86,
+ "content": "U ^ { \\perp } : =",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 8
+ },
+ {
+ "bbox": [
+ 107,
+ 200,
+ 506,
+ 216
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 202,
+ 240,
+ 215
+ ],
+ "score": 0.88,
+ "content": "\\{ u \\in \\mathbb { R } ^ { N } \\mid \\langle u , v \\rangle = 0 , \\forall v \\in U \\}",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 241,
+ 200,
+ 329,
+ 216
+ ],
+ "score": 1.0,
+ "content": ". By the symmetry of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 330,
+ 203,
+ 338,
+ 213
+ ],
+ "score": 0.83,
+ "content": "P",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 339,
+ 200,
+ 415,
+ 216
+ ],
+ "score": 1.0,
+ "content": ", we can show that",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 415,
+ 202,
+ 431,
+ 213
+ ],
+ "score": 0.89,
+ "content": "U ^ { \\perp }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 432,
+ 200,
+ 506,
+ 216
+ ],
+ "score": 1.0,
+ "content": "is invariant under",
+ "type": "text"
+ }
+ ],
+ "index": 9
+ },
+ {
+ "bbox": [
+ 107,
+ 212,
+ 505,
+ 227
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 214,
+ 115,
+ 223
+ ],
+ "score": 0.8,
+ "content": "P",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 116,
+ 212,
+ 376,
+ 227
+ ],
+ "score": 1.0,
+ "content": ", too (Appendix E.1, Proposition 2). Therefore, we can regard",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 376,
+ 214,
+ 385,
+ 223
+ ],
+ "score": 0.83,
+ "content": "P",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 386,
+ 212,
+ 471,
+ 227
+ ],
+ "score": 1.0,
+ "content": "as a linear mapping",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 472,
+ 213,
+ 505,
+ 226
+ ],
+ "score": 0.72,
+ "content": "P | _ { U ^ { \\perp } } :",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 10
+ },
+ {
+ "bbox": [
+ 107,
+ 222,
+ 506,
+ 237
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 224,
+ 155,
+ 235
+ ],
+ "score": 0.91,
+ "content": "U ^ { \\perp } \\to U ^ { \\perp }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 156,
+ 222,
+ 297,
+ 237
+ ],
+ "score": 1.0,
+ "content": ". We denote the operator norm of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 297,
+ 225,
+ 321,
+ 236
+ ],
+ "score": 0.91,
+ "content": "P | _ { U ^ { \\perp } }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 321,
+ 222,
+ 336,
+ 237
+ ],
+ "score": 1.0,
+ "content": "by",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 337,
+ 225,
+ 344,
+ 235
+ ],
+ "score": 0.78,
+ "content": "\\lambda",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 344,
+ 222,
+ 378,
+ 237
+ ],
+ "score": 1.0,
+ "content": ". When",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 378,
+ 225,
+ 387,
+ 235
+ ],
+ "score": 0.81,
+ "content": "U",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 387,
+ 222,
+ 506,
+ 237
+ ],
+ "score": 1.0,
+ "content": "is the eigenspace associated",
+ "type": "text"
+ }
+ ],
+ "index": 11
+ },
+ {
+ "bbox": [
+ 105,
+ 234,
+ 506,
+ 248
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 234,
+ 235,
+ 248
+ ],
+ "score": 1.0,
+ "content": "with the smallest eigenvalue of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 235,
+ 236,
+ 245,
+ 245
+ ],
+ "score": 0.83,
+ "content": "\\Delta",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 245,
+ 234,
+ 263,
+ 248
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 264,
+ 236,
+ 273,
+ 245
+ ],
+ "score": 0.83,
+ "content": "P",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 273,
+ 234,
+ 284,
+ 248
+ ],
+ "score": 1.0,
+ "content": "is",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 284,
+ 236,
+ 306,
+ 248
+ ],
+ "score": 0.92,
+ "content": "g ( \\Delta )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 307,
+ 234,
+ 335,
+ 248
+ ],
+ "score": 1.0,
+ "content": "where",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 335,
+ 237,
+ 342,
+ 247
+ ],
+ "score": 0.81,
+ "content": "g",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 343,
+ 234,
+ 434,
+ 248
+ ],
+ "score": 1.0,
+ "content": "is a polynomial, then,",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 435,
+ 236,
+ 442,
+ 245
+ ],
+ "score": 0.76,
+ "content": "\\lambda",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 442,
+ 234,
+ 506,
+ 248
+ ],
+ "score": 1.0,
+ "content": "corresponds to",
+ "type": "text"
+ }
+ ],
+ "index": 12
+ },
+ {
+ "bbox": [
+ 107,
+ 245,
+ 427,
+ 260
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 246,
+ 173,
+ 260
+ ],
+ "score": 0.93,
+ "content": "\\begin{array} { r } { \\lambda = \\operatorname* { s u p } _ { \\mu } \\left| g ( \\mu ) \\right| } \\end{array}",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 173,
+ 245,
+ 427,
+ 260
+ ],
+ "score": 1.0,
+ "content": "where sup ranges over all eigenvalues except the smallest one.",
+ "type": "text"
+ }
+ ],
+ "index": 13
+ }
+ ],
+ "index": 10.5
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 107,
+ 270,
+ 193,
+ 282
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 270,
+ 194,
+ 283
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 270,
+ 194,
+ 283
+ ],
+ "score": 1.0,
+ "content": "3.3 MAIN RESULT",
+ "type": "text"
+ }
+ ],
+ "index": 14
+ }
+ ],
+ "index": 14
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 289,
+ 505,
+ 352
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 154,
+ 281,
+ 511,
+ 317
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 154,
+ 281,
+ 202,
+ 317
+ ],
+ "score": 1.0,
+ "content": "he subspace is the ortho",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 203,
+ 291,
+ 216,
+ 302
+ ],
+ "score": 0.81,
+ "content": "\\mathcal { M }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 216,
+ 281,
+ 228,
+ 317
+ ],
+ "score": 1.0,
+ "content": "of ma",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 228,
+ 290,
+ 256,
+ 301
+ ],
+ "score": 0.9,
+ "content": "\\mathbb { R } ^ { N \\times C }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 256,
+ 281,
+ 266,
+ 317
+ ],
+ "score": 1.0,
+ "content": "b of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 270,
+ 289,
+ 477,
+ 304
+ ],
+ "score": 0.88,
+ "content": "\\begin{array} { r } { \\mathcal { M } : = U \\otimes \\mathbb { R } ^ { C } = \\{ \\sum _ { m = 1 } ^ { M } e _ { m } \\otimes w _ { m } \\mid w _ { m } \\in \\mathbb { R } ^ { C } \\} } \\end{array}",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 478,
+ 281,
+ 511,
+ 317
+ ],
+ "score": 1.0,
+ "content": "wheredenote",
+ "type": "text"
+ }
+ ],
+ "index": 15
+ },
+ {
+ "bbox": [
+ 107,
+ 303,
+ 457,
+ 317
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 303,
+ 153,
+ 317
+ ],
+ "score": 0.92,
+ "content": "( e _ { m } ) _ { m \\in [ M ] }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 266,
+ 304,
+ 275,
+ 314
+ ],
+ "score": 0.81,
+ "content": "U",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 406,
+ 303,
+ 457,
+ 314
+ ],
+ "score": 0.9,
+ "content": "X \\in \\mathbb { R } ^ { N \\times C }",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 16
+ },
+ {
+ "bbox": [
+ 105,
+ 315,
+ 505,
+ 329
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 315,
+ 191,
+ 329
+ ],
+ "score": 1.0,
+ "content": "the distance between",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 191,
+ 316,
+ 201,
+ 326
+ ],
+ "score": 0.83,
+ "content": "X",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 202,
+ 315,
+ 219,
+ 329
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 219,
+ 316,
+ 232,
+ 326
+ ],
+ "score": 0.83,
+ "content": "\\mathcal { M }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 232,
+ 315,
+ 245,
+ 329
+ ],
+ "score": 1.0,
+ "content": "by",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 245,
+ 315,
+ 399,
+ 328
+ ],
+ "score": 0.9,
+ "content": "d _ { \\mathcal { M } } ( X ) : = \\operatorname* { i n f } \\left\\{ \\| X - Y \\| _ { \\mathrm { F } } \\mid Y \\in \\mathcal { M } \\right\\}",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 400,
+ 315,
+ 505,
+ 329
+ ],
+ "score": 1.0,
+ "content": ". We denote the maximum",
+ "type": "text"
+ }
+ ],
+ "index": 17
+ },
+ {
+ "bbox": [
+ 102,
+ 322,
+ 509,
+ 347
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 102,
+ 322,
+ 178,
+ 347
+ ],
+ "score": 1.0,
+ "content": "singular value of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 178,
+ 329,
+ 196,
+ 340
+ ],
+ "score": 0.9,
+ "content": "W _ { l h }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 197,
+ 322,
+ 211,
+ 347
+ ],
+ "score": 1.0,
+ "content": "by",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 212,
+ 331,
+ 225,
+ 340
+ ],
+ "score": 0.85,
+ "content": "s _ { l h }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 226,
+ 322,
+ 259,
+ 347
+ ],
+ "score": 1.0,
+ "content": "and set",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 259,
+ 327,
+ 326,
+ 342
+ ],
+ "score": 0.93,
+ "content": "\\begin{array} { r } { s _ { l } : = \\prod _ { h = 1 } ^ { H _ { l } } s _ { l h } } \\end{array}",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 326,
+ 322,
+ 509,
+ 347
+ ],
+ "score": 1.0,
+ "content": ". With these preparations, we introduce the",
+ "type": "text"
+ }
+ ],
+ "index": 18
+ },
+ {
+ "bbox": [
+ 105,
+ 339,
+ 216,
+ 353
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 339,
+ 216,
+ 353
+ ],
+ "score": 1.0,
+ "content": "main theorem of the paper.",
+ "type": "text"
+ }
+ ],
+ "index": 19
+ }
+ ],
+ "index": 17
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 105,
+ 353,
+ 501,
+ 366
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 104,
+ 350,
+ 500,
+ 367
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 350,
+ 311,
+ 367
+ ],
+ "score": 1.0,
+ "content": "Theorem 1. Under Assumptions 1 and 2, we have",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 311,
+ 353,
+ 418,
+ 365
+ ],
+ "score": 0.88,
+ "content": "d _ { \\mathcal { M } } ( f _ { l } ( X ) ) \\leq s _ { l } \\lambda d _ { \\mathcal { M } } ( X )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 418,
+ 350,
+ 450,
+ 367
+ ],
+ "score": 1.0,
+ "content": "for any",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 451,
+ 352,
+ 500,
+ 364
+ ],
+ "score": 0.88,
+ "content": "X \\in \\mathbb { R } ^ { N \\times C }",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 20
+ }
+ ],
+ "index": 20
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 372,
+ 505,
+ 406
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 370,
+ 505,
+ 386
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 370,
+ 294,
+ 386
+ ],
+ "score": 1.0,
+ "content": "The proof key is that the non-linear operation",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 294,
+ 375,
+ 301,
+ 382
+ ],
+ "score": 0.76,
+ "content": "\\sigma",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 302,
+ 370,
+ 394,
+ 386
+ ],
+ "score": 1.0,
+ "content": "decreases the distance",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 395,
+ 372,
+ 411,
+ 384
+ ],
+ "score": 0.89,
+ "content": "d _ { \\mathcal { M } }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 411,
+ 370,
+ 446,
+ 386
+ ],
+ "score": 1.0,
+ "content": ", that is,",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 446,
+ 372,
+ 505,
+ 384
+ ],
+ "score": 0.91,
+ "content": "d _ { \\mathcal { M } } ( \\sigma ( \\boldsymbol { X } ) ) \\leq",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 21
+ },
+ {
+ "bbox": [
+ 107,
+ 383,
+ 505,
+ 396
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 384,
+ 139,
+ 396
+ ],
+ "score": 0.92,
+ "content": "d _ { \\mathcal { M } } ( \\boldsymbol { X } )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 140,
+ 383,
+ 266,
+ 396
+ ],
+ "score": 1.0,
+ "content": ". We use the non-negativity of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 266,
+ 385,
+ 280,
+ 394
+ ],
+ "score": 0.87,
+ "content": "e _ { m }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 280,
+ 383,
+ 505,
+ 396
+ ],
+ "score": 1.0,
+ "content": "to prove this claim. See Appendix A for the complete",
+ "type": "text"
+ }
+ ],
+ "index": 22
+ },
+ {
+ "bbox": [
+ 105,
+ 394,
+ 381,
+ 407
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 394,
+ 381,
+ 407
+ ],
+ "score": 1.0,
+ "content": "proof. We also discuss the strictness of Theorem 1 in Appendix E.3.",
+ "type": "text"
+ }
+ ],
+ "index": 23
+ }
+ ],
+ "index": 22
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 410,
+ 506,
+ 447
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 410,
+ 506,
+ 424
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 410,
+ 152,
+ 424
+ ],
+ "score": 1.0,
+ "content": "By setting",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 153,
+ 411,
+ 208,
+ 423
+ ],
+ "score": 0.95,
+ "content": "d _ { \\mathcal { M } } ( \\boldsymbol { X } ) = 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 209,
+ 410,
+ 320,
+ 424
+ ],
+ "score": 1.0,
+ "content": ", this theorem implies that",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 321,
+ 412,
+ 334,
+ 421
+ ],
+ "score": 0.81,
+ "content": "\\mathcal { M }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 334,
+ 410,
+ 411,
+ 424
+ ],
+ "score": 1.0,
+ "content": "is invariant under",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 412,
+ 412,
+ 421,
+ 423
+ ],
+ "score": 0.84,
+ "content": "f _ { l }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 421,
+ 410,
+ 506,
+ 424
+ ],
+ "score": 1.0,
+ "content": ". In addition, if the",
+ "type": "text"
+ }
+ ],
+ "index": 24
+ },
+ {
+ "bbox": [
+ 105,
+ 422,
+ 506,
+ 436
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 422,
+ 294,
+ 436
+ ],
+ "score": 1.0,
+ "content": "maximum value of singular values are small,",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 295,
+ 422,
+ 314,
+ 434
+ ],
+ "score": 0.9,
+ "content": "X ^ { ( l ) }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 314,
+ 422,
+ 427,
+ 436
+ ],
+ "score": 1.0,
+ "content": "asymptotically approaches",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 427,
+ 424,
+ 440,
+ 434
+ ],
+ "score": 0.79,
+ "content": "\\mathcal { M }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 441,
+ 422,
+ 506,
+ 436
+ ],
+ "score": 1.0,
+ "content": "in the sense of",
+ "type": "text"
+ }
+ ],
+ "index": 25
+ },
+ {
+ "bbox": [
+ 105,
+ 434,
+ 503,
+ 448
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 434,
+ 251,
+ 448
+ ],
+ "score": 1.0,
+ "content": "Johnson (1973) for any initial value",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 251,
+ 434,
+ 272,
+ 446
+ ],
+ "score": 0.9,
+ "content": "X ^ { ( 0 ) }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 272,
+ 434,
+ 503,
+ 448
+ ],
+ "score": 1.0,
+ "content": ". That is, the followings hold under Assumptions 1 and 2.",
+ "type": "text"
+ }
+ ],
+ "index": 26
+ }
+ ],
+ "index": 25
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 448,
+ 505,
+ 461
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 448,
+ 504,
+ 462
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 448,
+ 162,
+ 462
+ ],
+ "score": 1.0,
+ "content": "Corollary 1.",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 162,
+ 449,
+ 175,
+ 459
+ ],
+ "score": 0.71,
+ "content": "\\mathcal { M }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 176,
+ 448,
+ 289,
+ 462
+ ],
+ "score": 1.0,
+ "content": "is invariant under fl for any",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 289,
+ 449,
+ 319,
+ 461
+ ],
+ "score": 0.91,
+ "content": "l \\in \\mathbb { N } _ { + }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 319,
+ 448,
+ 360,
+ 462
+ ],
+ "score": 1.0,
+ "content": ", that is, if",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 360,
+ 449,
+ 394,
+ 460
+ ],
+ "score": 0.89,
+ "content": "X \\in \\mathcal { M }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 394,
+ 448,
+ 451,
+ 462
+ ],
+ "score": 1.0,
+ "content": ", then we have",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 452,
+ 448,
+ 501,
+ 461
+ ],
+ "score": 0.92,
+ "content": "f _ { l } ( X ) \\in \\mathcal { M }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 501,
+ 448,
+ 504,
+ 462
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 27
+ }
+ ],
+ "index": 27
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 462,
+ 503,
+ 489
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 461,
+ 506,
+ 477
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 461,
+ 178,
+ 477
+ ],
+ "score": 1.0,
+ "content": "Corollary 2. Let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 179,
+ 465,
+ 245,
+ 477
+ ],
+ "score": 0.91,
+ "content": "s : = \\mathrm { s u p } _ { l \\in \\mathbb { N } _ { + } } s _ { l }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 245,
+ 461,
+ 287,
+ 477
+ ],
+ "score": 1.0,
+ "content": ". We have",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 287,
+ 462,
+ 380,
+ 476
+ ],
+ "score": 0.92,
+ "content": "d _ { \\mathcal { M } } ( X ^ { ( l ) } ) = O ( ( s \\lambda ) ^ { l } )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 381,
+ 461,
+ 450,
+ 477
+ ],
+ "score": 1.0,
+ "content": ". In particular, if",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 450,
+ 464,
+ 481,
+ 475
+ ],
+ "score": 0.84,
+ "content": "s \\lambda < 1",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 481,
+ 461,
+ 506,
+ 477
+ ],
+ "score": 1.0,
+ "content": ", then",
+ "type": "text"
+ }
+ ],
+ "index": 28
+ },
+ {
+ "bbox": [
+ 106,
+ 476,
+ 388,
+ 490
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 478,
+ 119,
+ 489
+ ],
+ "score": 0.85,
+ "content": "X _ { l }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 119,
+ 476,
+ 225,
+ 490
+ ],
+ "score": 1.0,
+ "content": "exponentially approaches",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 225,
+ 478,
+ 238,
+ 488
+ ],
+ "score": 0.76,
+ "content": "\\mathcal { M }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 238,
+ 476,
+ 250,
+ 490
+ ],
+ "score": 1.0,
+ "content": "as",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 251,
+ 478,
+ 280,
+ 488
+ ],
+ "score": 0.89,
+ "content": "l \\infty",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 281,
+ 476,
+ 363,
+ 490
+ ],
+ "score": 1.0,
+ "content": "for any initial value",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 363,
+ 476,
+ 383,
+ 487
+ ],
+ "score": 0.89,
+ "content": "X ^ { ( 0 ) }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 384,
+ 476,
+ 388,
+ 490
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 29
+ }
+ ],
+ "index": 28.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 496,
+ 505,
+ 564
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 496,
+ 506,
+ 509
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 496,
+ 232,
+ 509
+ ],
+ "score": 1.0,
+ "content": "Suppose the operator norm of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 232,
+ 497,
+ 296,
+ 509
+ ],
+ "score": 0.92,
+ "content": "P | _ { U } : U \\to U",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 297,
+ 496,
+ 369,
+ 509
+ ],
+ "score": 1.0,
+ "content": "is no larger than",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 369,
+ 497,
+ 376,
+ 507
+ ],
+ "score": 0.77,
+ "content": "\\lambda",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 376,
+ 496,
+ 506,
+ 509
+ ],
+ "score": 1.0,
+ "content": ", then, under the assumption of",
+ "type": "text"
+ }
+ ],
+ "index": 30
+ },
+ {
+ "bbox": [
+ 107,
+ 507,
+ 506,
+ 523
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 509,
+ 138,
+ 520
+ ],
+ "score": 0.84,
+ "content": "s \\lambda < 1",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 138,
+ 507,
+ 142,
+ 523
+ ],
+ "score": 1.0,
+ "content": ",",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 142,
+ 508,
+ 162,
+ 519
+ ],
+ "score": 0.84,
+ "content": "X ^ { ( l ) }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 162,
+ 507,
+ 506,
+ 523
+ ],
+ "score": 1.0,
+ "content": "converges to 0, the trivial fixed point (see Appendix E.2, Proposition 3). Therefore,",
+ "type": "text"
+ }
+ ],
+ "index": 31
+ },
+ {
+ "bbox": [
+ 106,
+ 520,
+ 505,
+ 533
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 520,
+ 370,
+ 533
+ ],
+ "score": 1.0,
+ "content": "we are mainly interested in the case where the operator norm of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 370,
+ 520,
+ 389,
+ 532
+ ],
+ "score": 0.92,
+ "content": "P | _ { U }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 389,
+ 520,
+ 477,
+ 533
+ ],
+ "score": 1.0,
+ "content": "is strictly larger than",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 478,
+ 520,
+ 485,
+ 530
+ ],
+ "score": 0.77,
+ "content": "\\lambda",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 485,
+ 520,
+ 505,
+ 533
+ ],
+ "score": 1.0,
+ "content": "(see",
+ "type": "text"
+ }
+ ],
+ "index": 32
+ },
+ {
+ "bbox": [
+ 106,
+ 531,
+ 505,
+ 543
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 531,
+ 426,
+ 543
+ ],
+ "score": 1.0,
+ "content": "Proposition 1). Finally, we restate Theorem 1 specialized to the situation where",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 426,
+ 532,
+ 435,
+ 541
+ ],
+ "score": 0.81,
+ "content": "U",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 436,
+ 531,
+ 505,
+ 543
+ ],
+ "score": 1.0,
+ "content": "is the direct sum",
+ "type": "text"
+ }
+ ],
+ "index": 33
+ },
+ {
+ "bbox": [
+ 106,
+ 542,
+ 506,
+ 555
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 542,
+ 278,
+ 555
+ ],
+ "score": 1.0,
+ "content": "of eigenspaces associated with the largest",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 279,
+ 542,
+ 290,
+ 552
+ ],
+ "score": 0.77,
+ "content": "M",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 291,
+ 542,
+ 353,
+ 555
+ ],
+ "score": 1.0,
+ "content": "eigenvalues of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 353,
+ 542,
+ 362,
+ 552
+ ],
+ "score": 0.83,
+ "content": "P",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 362,
+ 542,
+ 485,
+ 555
+ ],
+ "score": 1.0,
+ "content": ". Note that the eigenvalues of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 485,
+ 542,
+ 494,
+ 552
+ ],
+ "score": 0.84,
+ "content": "P",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 495,
+ 542,
+ 506,
+ 555
+ ],
+ "score": 1.0,
+ "content": "is",
+ "type": "text"
+ }
+ ],
+ "index": 34
+ },
+ {
+ "bbox": [
+ 105,
+ 552,
+ 213,
+ 566
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 552,
+ 147,
+ 566
+ ],
+ "score": 1.0,
+ "content": "real since",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 147,
+ 554,
+ 156,
+ 563
+ ],
+ "score": 0.82,
+ "content": "P",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 156,
+ 552,
+ 213,
+ 566
+ ],
+ "score": 1.0,
+ "content": "is symmetric.",
+ "type": "text"
+ }
+ ],
+ "index": 35
+ }
+ ],
+ "index": 32.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 566,
+ 505,
+ 614
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 565,
+ 505,
+ 579
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 565,
+ 179,
+ 579
+ ],
+ "score": 1.0,
+ "content": "Corollary 3. Let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 180,
+ 567,
+ 245,
+ 578
+ ],
+ "score": 0.89,
+ "content": "\\lambda _ { 1 } \\leq \\cdots \\leq \\lambda _ { N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 246,
+ 565,
+ 331,
+ 579
+ ],
+ "score": 1.0,
+ "content": "be the eigenvalue of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 332,
+ 567,
+ 340,
+ 576
+ ],
+ "score": 0.8,
+ "content": "P",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 340,
+ 565,
+ 505,
+ 579
+ ],
+ "score": 1.0,
+ "content": ", sorted in ascending order. Suppose the",
+ "type": "text"
+ }
+ ],
+ "index": 36
+ },
+ {
+ "bbox": [
+ 105,
+ 576,
+ 506,
+ 591
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 576,
+ 257,
+ 591
+ ],
+ "score": 1.0,
+ "content": "multiplicity of the largest eigenvalue",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 257,
+ 578,
+ 272,
+ 588
+ ],
+ "score": 0.88,
+ "content": "\\lambda _ { N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 272,
+ 576,
+ 283,
+ 591
+ ],
+ "score": 1.0,
+ "content": "is",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 284,
+ 577,
+ 323,
+ 589
+ ],
+ "score": 0.9,
+ "content": "M ( \\leq N )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 324,
+ 576,
+ 346,
+ 591
+ ],
+ "score": 1.0,
+ "content": ", i.e.,",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 346,
+ 578,
+ 484,
+ 589
+ ],
+ "score": 0.9,
+ "content": "\\lambda _ { N - M } < \\lambda _ { N - M + 1 } = \\cdot \\cdot \\cdot = \\lambda _ { N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 484,
+ 576,
+ 506,
+ 591
+ ],
+ "score": 1.0,
+ "content": ". We",
+ "type": "text"
+ }
+ ],
+ "index": 37
+ },
+ {
+ "bbox": [
+ 105,
+ 587,
+ 506,
+ 602
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 587,
+ 132,
+ 602
+ ],
+ "score": 1.0,
+ "content": "define",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 133,
+ 588,
+ 229,
+ 601
+ ],
+ "score": 0.91,
+ "content": "\\lambda : = \\operatorname* { m a x } _ { n \\in [ N - M ] } \\left| \\lambda _ { n } \\right|",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 229,
+ 587,
+ 276,
+ 602
+ ],
+ "score": 1.0,
+ "content": ". We denote",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 277,
+ 589,
+ 286,
+ 598
+ ],
+ "score": 0.76,
+ "content": "U",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 286,
+ 587,
+ 423,
+ 602
+ ],
+ "score": 1.0,
+ "content": "by the eigenspace associated with",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 423,
+ 589,
+ 437,
+ 599
+ ],
+ "score": 0.88,
+ "content": "\\lambda _ { N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 437,
+ 587,
+ 506,
+ 602
+ ],
+ "score": 1.0,
+ "content": "and assume that",
+ "type": "text"
+ }
+ ],
+ "index": 38
+ },
+ {
+ "bbox": [
+ 106,
+ 600,
+ 394,
+ 615
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 602,
+ 115,
+ 612
+ ],
+ "score": 0.66,
+ "content": "U",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 116,
+ 600,
+ 270,
+ 615
+ ],
+ "score": 1.0,
+ "content": "satisfies Assumption 1. Then, we have",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 271,
+ 600,
+ 391,
+ 614
+ ],
+ "score": 0.93,
+ "content": "d _ { \\mathcal { M } } ( X ^ { ( l + 1 ) } ) \\leq s _ { l } \\lambda d _ { \\mathcal { M } } ( X ^ { ( l ) } ) .",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 391,
+ 600,
+ 394,
+ 615
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 39
+ }
+ ],
+ "index": 37.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 614,
+ 505,
+ 660
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 614,
+ 505,
+ 627
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 614,
+ 505,
+ 627
+ ],
+ "score": 1.0,
+ "content": "Remark 1. It is known that any Markov process on finite states converges to a unique distribution",
+ "type": "text"
+ }
+ ],
+ "index": 40
+ },
+ {
+ "bbox": [
+ 105,
+ 626,
+ 506,
+ 639
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 626,
+ 443,
+ 639
+ ],
+ "score": 1.0,
+ "content": "(equilibrium) if it is irreducible and aperiodic (see e.g., Norris (1998)). Theorem",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 443,
+ 627,
+ 450,
+ 636
+ ],
+ "score": 0.41,
+ "content": "^ { l }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 450,
+ 626,
+ 506,
+ 639
+ ],
+ "score": 1.0,
+ "content": "includes this",
+ "type": "text"
+ }
+ ],
+ "index": 41
+ },
+ {
+ "bbox": [
+ 104,
+ 637,
+ 506,
+ 649
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 637,
+ 245,
+ 649
+ ],
+ "score": 1.0,
+ "content": "proposition as a special case with",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 246,
+ 637,
+ 276,
+ 647
+ ],
+ "score": 0.86,
+ "content": "M = 1",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 277,
+ 637,
+ 281,
+ 649
+ ],
+ "score": 1.0,
+ "content": ",",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 281,
+ 637,
+ 309,
+ 647
+ ],
+ "score": 0.87,
+ "content": "C = 1",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 310,
+ 637,
+ 331,
+ 649
+ ],
+ "score": 1.0,
+ "content": ", and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 332,
+ 637,
+ 365,
+ 648
+ ],
+ "score": 0.91,
+ "content": "W _ { l } = 1",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 365,
+ 637,
+ 394,
+ 649
+ ],
+ "score": 1.0,
+ "content": "for all",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 394,
+ 637,
+ 425,
+ 648
+ ],
+ "score": 0.91,
+ "content": "l \\in \\mathbb { N } _ { + }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 425,
+ 637,
+ 506,
+ 649
+ ],
+ "score": 1.0,
+ "content": ". This is essentially",
+ "type": "text"
+ }
+ ],
+ "index": 42
+ },
+ {
+ "bbox": [
+ 105,
+ 648,
+ 496,
+ 661
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 648,
+ 484,
+ 661
+ ],
+ "score": 1.0,
+ "content": "the direct consequence of Perron – Frobenius’ theorem (see e.g., Meyer (2000)). See Appendix",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 485,
+ 649,
+ 492,
+ 658
+ ],
+ "score": 0.64,
+ "content": "F",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 492,
+ 648,
+ 496,
+ 661
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 43
+ }
+ ],
+ "index": 41.5
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 108,
+ 674,
+ 243,
+ 687
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 673,
+ 244,
+ 689
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 673,
+ 244,
+ 689
+ ],
+ "score": 1.0,
+ "content": "4 APPLICATION TO GCN",
+ "type": "text"
+ }
+ ],
+ "index": 44
+ }
+ ],
+ "index": 44
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 698,
+ 505,
+ 732
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 697,
+ 505,
+ 712
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 697,
+ 505,
+ 712
+ ],
+ "score": 1.0,
+ "content": "We formulate a GCN (Kipf & Welling, 2017) without readout operations (Gilmer et al., 2017) using",
+ "type": "text"
+ }
+ ],
+ "index": 45
+ },
+ {
+ "bbox": [
+ 106,
+ 711,
+ 505,
+ 721
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 711,
+ 505,
+ 721
+ ],
+ "score": 1.0,
+ "content": "the dynamical system in the previous section and derive a sufficient condition in terms of the spectra",
+ "type": "text"
+ }
+ ],
+ "index": 46
+ },
+ {
+ "bbox": [
+ 106,
+ 721,
+ 505,
+ 733
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 721,
+ 505,
+ 733
+ ],
+ "score": 1.0,
+ "content": "of underlying graphs in which layer stacking nor non-linearity are not helpful for node classification.",
+ "type": "text"
+ }
+ ],
+ "index": 47
+ }
+ ],
+ "index": 46
+ }
+ ],
+ "page_idx": 3,
+ "page_size": [
+ 612,
+ 792
+ ],
+ "discarded_blocks": [
+ {
+ "type": "discarded",
+ "bbox": [
+ 107,
+ 27,
+ 293,
+ 37
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 26,
+ 294,
+ 38
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 26,
+ 294,
+ 38
+ ],
+ "score": 1.0,
+ "content": "Published as a conference paper at ICLR 2020",
+ "type": "text"
+ }
+ ]
+ }
+ ]
+ },
+ {
+ "type": "discarded",
+ "bbox": [
+ 302,
+ 751,
+ 308,
+ 759
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 302,
+ 750,
+ 310,
+ 762
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 302,
+ 750,
+ 310,
+ 762
+ ],
+ "score": 1.0,
+ "content": "",
+ "type": "text",
+ "height": 12,
+ "width": 8
+ }
+ ]
+ }
+ ]
+ }
+ ],
+ "para_blocks": [
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 82,
+ 505,
+ 119
+ ],
+ "lines": [],
+ "index": 1,
+ "bbox_fs": [
+ 104,
+ 80,
+ 506,
+ 121
+ ],
+ "lines_deleted": true
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 109,
+ 123,
+ 502,
+ 158
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 122,
+ 505,
+ 137
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 122,
+ 123,
+ 137
+ ],
+ "score": 1.0,
+ "content": "For",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 124,
+ 124,
+ 162,
+ 135
+ ],
+ "score": 0.91,
+ "content": "M \\leq N",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 162,
+ 122,
+ 180,
+ 137
+ ],
+ "score": 1.0,
+ "content": ", let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 180,
+ 124,
+ 189,
+ 134
+ ],
+ "score": 0.8,
+ "content": "U",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 190,
+ 122,
+ 212,
+ 137
+ ],
+ "score": 1.0,
+ "content": "be a",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 212,
+ 124,
+ 224,
+ 134
+ ],
+ "score": 0.81,
+ "content": "M",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 224,
+ 122,
+ 329,
+ 137
+ ],
+ "score": 1.0,
+ "content": "-dimensional subspace of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 330,
+ 123,
+ 345,
+ 135
+ ],
+ "score": 0.9,
+ "content": "\\mathbb { R } ^ { N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 345,
+ 122,
+ 420,
+ 137
+ ],
+ "score": 1.0,
+ "content": ". We assume that",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 421,
+ 124,
+ 430,
+ 134
+ ],
+ "score": 0.82,
+ "content": "U",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 430,
+ 122,
+ 450,
+ 137
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 450,
+ 125,
+ 459,
+ 134
+ ],
+ "score": 0.81,
+ "content": "P",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 459,
+ 122,
+ 505,
+ 137
+ ],
+ "score": 1.0,
+ "content": "satisfy the",
+ "type": "text"
+ }
+ ],
+ "index": 3
+ },
+ {
+ "bbox": [
+ 106,
+ 135,
+ 504,
+ 147
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 135,
+ 338,
+ 147
+ ],
+ "score": 1.0,
+ "content": "following properties that generalize the situation where",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 338,
+ 136,
+ 347,
+ 145
+ ],
+ "score": 0.82,
+ "content": "U",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 348,
+ 135,
+ 504,
+ 147
+ ],
+ "score": 1.0,
+ "content": "is the eigenspace associated with the",
+ "type": "text"
+ }
+ ],
+ "index": 4
+ },
+ {
+ "bbox": [
+ 107,
+ 146,
+ 501,
+ 158
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 146,
+ 325,
+ 158
+ ],
+ "score": 1.0,
+ "content": "smallest eigenvalue of a (normalized) graph Laplacian",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 325,
+ 146,
+ 334,
+ 156
+ ],
+ "score": 0.8,
+ "content": "\\Delta",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 335,
+ 146,
+ 407,
+ 158
+ ],
+ "score": 1.0,
+ "content": "(that is, zero) and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 407,
+ 147,
+ 416,
+ 156
+ ],
+ "score": 0.83,
+ "content": "P",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 417,
+ 146,
+ 492,
+ 158
+ ],
+ "score": 1.0,
+ "content": "is a polynomial of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 492,
+ 147,
+ 501,
+ 156
+ ],
+ "score": 0.81,
+ "content": "\\Delta",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 5
+ }
+ ],
+ "index": 4,
+ "bbox_fs": [
+ 105,
+ 122,
+ 505,
+ 158
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 108,
+ 159,
+ 485,
+ 171
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 158,
+ 486,
+ 173
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 158,
+ 172,
+ 173
+ ],
+ "score": 1.0,
+ "content": "Assumption 1.",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 172,
+ 159,
+ 181,
+ 169
+ ],
+ "score": 0.49,
+ "content": "U",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 182,
+ 158,
+ 288,
+ 173
+ ],
+ "score": 1.0,
+ "content": "has an orthonormal basis",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 288,
+ 159,
+ 334,
+ 172
+ ],
+ "score": 0.93,
+ "content": "( e _ { m } ) _ { m \\in [ M ] }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 334,
+ 158,
+ 486,
+ 173
+ ],
+ "score": 1.0,
+ "content": "that consists of non-negative vectors.",
+ "type": "text"
+ }
+ ],
+ "index": 6
+ }
+ ],
+ "index": 6,
+ "bbox_fs": [
+ 106,
+ 158,
+ 486,
+ 173
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 172,
+ 382,
+ 184
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 172,
+ 383,
+ 185
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 172,
+ 172,
+ 185
+ ],
+ "score": 1.0,
+ "content": "Assumption 2.",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 172,
+ 173,
+ 181,
+ 183
+ ],
+ "score": 0.46,
+ "content": "U",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 182,
+ 172,
+ 256,
+ 185
+ ],
+ "score": 1.0,
+ "content": "is invariant under",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 256,
+ 173,
+ 265,
+ 183
+ ],
+ "score": 0.71,
+ "content": "P",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 265,
+ 172,
+ 294,
+ 185
+ ],
+ "score": 1.0,
+ "content": ", i.e., if",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 294,
+ 173,
+ 321,
+ 184
+ ],
+ "score": 0.81,
+ "content": "u \\in U",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 321,
+ 172,
+ 344,
+ 185
+ ],
+ "score": 1.0,
+ "content": ", then",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 345,
+ 173,
+ 379,
+ 183
+ ],
+ "score": 0.9,
+ "content": "P u \\in U",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 379,
+ 172,
+ 383,
+ 185
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 7
+ }
+ ],
+ "index": 7,
+ "bbox_fs": [
+ 106,
+ 172,
+ 383,
+ 185
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 191,
+ 505,
+ 258
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 189,
+ 505,
+ 204
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 189,
+ 150,
+ 204
+ ],
+ "score": 1.0,
+ "content": "We endow",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 150,
+ 191,
+ 166,
+ 202
+ ],
+ "score": 0.88,
+ "content": "\\mathbb { R } ^ { N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 166,
+ 189,
+ 453,
+ 204
+ ],
+ "score": 1.0,
+ "content": "with the ordinal inner product and denote the orthogonal complement of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 453,
+ 192,
+ 462,
+ 202
+ ],
+ "score": 0.81,
+ "content": "U",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 462,
+ 189,
+ 474,
+ 204
+ ],
+ "score": 1.0,
+ "content": "by",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 475,
+ 191,
+ 505,
+ 203
+ ],
+ "score": 0.86,
+ "content": "U ^ { \\perp } : =",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 8
+ },
+ {
+ "bbox": [
+ 107,
+ 200,
+ 506,
+ 216
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 202,
+ 240,
+ 215
+ ],
+ "score": 0.88,
+ "content": "\\{ u \\in \\mathbb { R } ^ { N } \\mid \\langle u , v \\rangle = 0 , \\forall v \\in U \\}",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 241,
+ 200,
+ 329,
+ 216
+ ],
+ "score": 1.0,
+ "content": ". By the symmetry of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 330,
+ 203,
+ 338,
+ 213
+ ],
+ "score": 0.83,
+ "content": "P",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 339,
+ 200,
+ 415,
+ 216
+ ],
+ "score": 1.0,
+ "content": ", we can show that",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 415,
+ 202,
+ 431,
+ 213
+ ],
+ "score": 0.89,
+ "content": "U ^ { \\perp }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 432,
+ 200,
+ 506,
+ 216
+ ],
+ "score": 1.0,
+ "content": "is invariant under",
+ "type": "text"
+ }
+ ],
+ "index": 9
+ },
+ {
+ "bbox": [
+ 107,
+ 212,
+ 505,
+ 227
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 214,
+ 115,
+ 223
+ ],
+ "score": 0.8,
+ "content": "P",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 116,
+ 212,
+ 376,
+ 227
+ ],
+ "score": 1.0,
+ "content": ", too (Appendix E.1, Proposition 2). Therefore, we can regard",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 376,
+ 214,
+ 385,
+ 223
+ ],
+ "score": 0.83,
+ "content": "P",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 386,
+ 212,
+ 471,
+ 227
+ ],
+ "score": 1.0,
+ "content": "as a linear mapping",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 472,
+ 213,
+ 505,
+ 226
+ ],
+ "score": 0.72,
+ "content": "P | _ { U ^ { \\perp } } :",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 10
+ },
+ {
+ "bbox": [
+ 107,
+ 222,
+ 506,
+ 237
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 224,
+ 155,
+ 235
+ ],
+ "score": 0.91,
+ "content": "U ^ { \\perp } \\to U ^ { \\perp }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 156,
+ 222,
+ 297,
+ 237
+ ],
+ "score": 1.0,
+ "content": ". We denote the operator norm of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 297,
+ 225,
+ 321,
+ 236
+ ],
+ "score": 0.91,
+ "content": "P | _ { U ^ { \\perp } }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 321,
+ 222,
+ 336,
+ 237
+ ],
+ "score": 1.0,
+ "content": "by",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 337,
+ 225,
+ 344,
+ 235
+ ],
+ "score": 0.78,
+ "content": "\\lambda",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 344,
+ 222,
+ 378,
+ 237
+ ],
+ "score": 1.0,
+ "content": ". When",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 378,
+ 225,
+ 387,
+ 235
+ ],
+ "score": 0.81,
+ "content": "U",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 387,
+ 222,
+ 506,
+ 237
+ ],
+ "score": 1.0,
+ "content": "is the eigenspace associated",
+ "type": "text"
+ }
+ ],
+ "index": 11
+ },
+ {
+ "bbox": [
+ 105,
+ 234,
+ 506,
+ 248
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 234,
+ 235,
+ 248
+ ],
+ "score": 1.0,
+ "content": "with the smallest eigenvalue of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 235,
+ 236,
+ 245,
+ 245
+ ],
+ "score": 0.83,
+ "content": "\\Delta",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 245,
+ 234,
+ 263,
+ 248
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 264,
+ 236,
+ 273,
+ 245
+ ],
+ "score": 0.83,
+ "content": "P",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 273,
+ 234,
+ 284,
+ 248
+ ],
+ "score": 1.0,
+ "content": "is",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 284,
+ 236,
+ 306,
+ 248
+ ],
+ "score": 0.92,
+ "content": "g ( \\Delta )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 307,
+ 234,
+ 335,
+ 248
+ ],
+ "score": 1.0,
+ "content": "where",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 335,
+ 237,
+ 342,
+ 247
+ ],
+ "score": 0.81,
+ "content": "g",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 343,
+ 234,
+ 434,
+ 248
+ ],
+ "score": 1.0,
+ "content": "is a polynomial, then,",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 435,
+ 236,
+ 442,
+ 245
+ ],
+ "score": 0.76,
+ "content": "\\lambda",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 442,
+ 234,
+ 506,
+ 248
+ ],
+ "score": 1.0,
+ "content": "corresponds to",
+ "type": "text"
+ }
+ ],
+ "index": 12
+ },
+ {
+ "bbox": [
+ 107,
+ 245,
+ 427,
+ 260
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 246,
+ 173,
+ 260
+ ],
+ "score": 0.93,
+ "content": "\\begin{array} { r } { \\lambda = \\operatorname* { s u p } _ { \\mu } \\left| g ( \\mu ) \\right| } \\end{array}",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 173,
+ 245,
+ 427,
+ 260
+ ],
+ "score": 1.0,
+ "content": "where sup ranges over all eigenvalues except the smallest one.",
+ "type": "text"
+ }
+ ],
+ "index": 13
+ }
+ ],
+ "index": 10.5,
+ "bbox_fs": [
+ 105,
+ 189,
+ 506,
+ 260
+ ]
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 107,
+ 270,
+ 193,
+ 282
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 270,
+ 194,
+ 283
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 270,
+ 194,
+ 283
+ ],
+ "score": 1.0,
+ "content": "3.3 MAIN RESULT",
+ "type": "text"
+ }
+ ],
+ "index": 14
+ }
+ ],
+ "index": 14
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 289,
+ 505,
+ 352
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 154,
+ 281,
+ 511,
+ 317
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 154,
+ 281,
+ 202,
+ 317
+ ],
+ "score": 1.0,
+ "content": "he subspace is the ortho",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 203,
+ 291,
+ 216,
+ 302
+ ],
+ "score": 0.81,
+ "content": "\\mathcal { M }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 216,
+ 281,
+ 228,
+ 317
+ ],
+ "score": 1.0,
+ "content": "of ma",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 228,
+ 290,
+ 256,
+ 301
+ ],
+ "score": 0.9,
+ "content": "\\mathbb { R } ^ { N \\times C }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 256,
+ 281,
+ 266,
+ 317
+ ],
+ "score": 1.0,
+ "content": "b of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 270,
+ 289,
+ 477,
+ 304
+ ],
+ "score": 0.88,
+ "content": "\\begin{array} { r } { \\mathcal { M } : = U \\otimes \\mathbb { R } ^ { C } = \\{ \\sum _ { m = 1 } ^ { M } e _ { m } \\otimes w _ { m } \\mid w _ { m } \\in \\mathbb { R } ^ { C } \\} } \\end{array}",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 478,
+ 281,
+ 511,
+ 317
+ ],
+ "score": 1.0,
+ "content": "wheredenote",
+ "type": "text"
+ }
+ ],
+ "index": 15
+ },
+ {
+ "bbox": [
+ 107,
+ 303,
+ 457,
+ 317
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 303,
+ 153,
+ 317
+ ],
+ "score": 0.92,
+ "content": "( e _ { m } ) _ { m \\in [ M ] }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 266,
+ 304,
+ 275,
+ 314
+ ],
+ "score": 0.81,
+ "content": "U",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 406,
+ 303,
+ 457,
+ 314
+ ],
+ "score": 0.9,
+ "content": "X \\in \\mathbb { R } ^ { N \\times C }",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 16
+ },
+ {
+ "bbox": [
+ 105,
+ 315,
+ 505,
+ 329
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 315,
+ 191,
+ 329
+ ],
+ "score": 1.0,
+ "content": "the distance between",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 191,
+ 316,
+ 201,
+ 326
+ ],
+ "score": 0.83,
+ "content": "X",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 202,
+ 315,
+ 219,
+ 329
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 219,
+ 316,
+ 232,
+ 326
+ ],
+ "score": 0.83,
+ "content": "\\mathcal { M }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 232,
+ 315,
+ 245,
+ 329
+ ],
+ "score": 1.0,
+ "content": "by",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 245,
+ 315,
+ 399,
+ 328
+ ],
+ "score": 0.9,
+ "content": "d _ { \\mathcal { M } } ( X ) : = \\operatorname* { i n f } \\left\\{ \\| X - Y \\| _ { \\mathrm { F } } \\mid Y \\in \\mathcal { M } \\right\\}",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 400,
+ 315,
+ 505,
+ 329
+ ],
+ "score": 1.0,
+ "content": ". We denote the maximum",
+ "type": "text"
+ }
+ ],
+ "index": 17
+ },
+ {
+ "bbox": [
+ 102,
+ 322,
+ 509,
+ 347
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 102,
+ 322,
+ 178,
+ 347
+ ],
+ "score": 1.0,
+ "content": "singular value of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 178,
+ 329,
+ 196,
+ 340
+ ],
+ "score": 0.9,
+ "content": "W _ { l h }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 197,
+ 322,
+ 211,
+ 347
+ ],
+ "score": 1.0,
+ "content": "by",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 212,
+ 331,
+ 225,
+ 340
+ ],
+ "score": 0.85,
+ "content": "s _ { l h }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 226,
+ 322,
+ 259,
+ 347
+ ],
+ "score": 1.0,
+ "content": "and set",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 259,
+ 327,
+ 326,
+ 342
+ ],
+ "score": 0.93,
+ "content": "\\begin{array} { r } { s _ { l } : = \\prod _ { h = 1 } ^ { H _ { l } } s _ { l h } } \\end{array}",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 326,
+ 322,
+ 509,
+ 347
+ ],
+ "score": 1.0,
+ "content": ". With these preparations, we introduce the",
+ "type": "text"
+ }
+ ],
+ "index": 18
+ },
+ {
+ "bbox": [
+ 105,
+ 339,
+ 216,
+ 353
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 339,
+ 216,
+ 353
+ ],
+ "score": 1.0,
+ "content": "main theorem of the paper.",
+ "type": "text"
+ }
+ ],
+ "index": 19
+ }
+ ],
+ "index": 17,
+ "bbox_fs": [
+ 102,
+ 281,
+ 511,
+ 353
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 105,
+ 353,
+ 501,
+ 366
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 104,
+ 350,
+ 500,
+ 367
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 350,
+ 311,
+ 367
+ ],
+ "score": 1.0,
+ "content": "Theorem 1. Under Assumptions 1 and 2, we have",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 311,
+ 353,
+ 418,
+ 365
+ ],
+ "score": 0.88,
+ "content": "d _ { \\mathcal { M } } ( f _ { l } ( X ) ) \\leq s _ { l } \\lambda d _ { \\mathcal { M } } ( X )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 418,
+ 350,
+ 450,
+ 367
+ ],
+ "score": 1.0,
+ "content": "for any",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 451,
+ 352,
+ 500,
+ 364
+ ],
+ "score": 0.88,
+ "content": "X \\in \\mathbb { R } ^ { N \\times C }",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 20
+ }
+ ],
+ "index": 20,
+ "bbox_fs": [
+ 104,
+ 350,
+ 500,
+ 367
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 372,
+ 505,
+ 406
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 370,
+ 505,
+ 386
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 370,
+ 294,
+ 386
+ ],
+ "score": 1.0,
+ "content": "The proof key is that the non-linear operation",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 294,
+ 375,
+ 301,
+ 382
+ ],
+ "score": 0.76,
+ "content": "\\sigma",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 302,
+ 370,
+ 394,
+ 386
+ ],
+ "score": 1.0,
+ "content": "decreases the distance",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 395,
+ 372,
+ 411,
+ 384
+ ],
+ "score": 0.89,
+ "content": "d _ { \\mathcal { M } }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 411,
+ 370,
+ 446,
+ 386
+ ],
+ "score": 1.0,
+ "content": ", that is,",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 446,
+ 372,
+ 505,
+ 384
+ ],
+ "score": 0.91,
+ "content": "d _ { \\mathcal { M } } ( \\sigma ( \\boldsymbol { X } ) ) \\leq",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 21
+ },
+ {
+ "bbox": [
+ 107,
+ 383,
+ 505,
+ 396
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 384,
+ 139,
+ 396
+ ],
+ "score": 0.92,
+ "content": "d _ { \\mathcal { M } } ( \\boldsymbol { X } )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 140,
+ 383,
+ 266,
+ 396
+ ],
+ "score": 1.0,
+ "content": ". We use the non-negativity of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 266,
+ 385,
+ 280,
+ 394
+ ],
+ "score": 0.87,
+ "content": "e _ { m }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 280,
+ 383,
+ 505,
+ 396
+ ],
+ "score": 1.0,
+ "content": "to prove this claim. See Appendix A for the complete",
+ "type": "text"
+ }
+ ],
+ "index": 22
+ },
+ {
+ "bbox": [
+ 105,
+ 394,
+ 381,
+ 407
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 394,
+ 381,
+ 407
+ ],
+ "score": 1.0,
+ "content": "proof. We also discuss the strictness of Theorem 1 in Appendix E.3.",
+ "type": "text"
+ }
+ ],
+ "index": 23
+ }
+ ],
+ "index": 22,
+ "bbox_fs": [
+ 105,
+ 370,
+ 505,
+ 407
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 410,
+ 506,
+ 447
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 410,
+ 506,
+ 424
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 410,
+ 152,
+ 424
+ ],
+ "score": 1.0,
+ "content": "By setting",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 153,
+ 411,
+ 208,
+ 423
+ ],
+ "score": 0.95,
+ "content": "d _ { \\mathcal { M } } ( \\boldsymbol { X } ) = 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 209,
+ 410,
+ 320,
+ 424
+ ],
+ "score": 1.0,
+ "content": ", this theorem implies that",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 321,
+ 412,
+ 334,
+ 421
+ ],
+ "score": 0.81,
+ "content": "\\mathcal { M }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 334,
+ 410,
+ 411,
+ 424
+ ],
+ "score": 1.0,
+ "content": "is invariant under",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 412,
+ 412,
+ 421,
+ 423
+ ],
+ "score": 0.84,
+ "content": "f _ { l }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 421,
+ 410,
+ 506,
+ 424
+ ],
+ "score": 1.0,
+ "content": ". In addition, if the",
+ "type": "text"
+ }
+ ],
+ "index": 24
+ },
+ {
+ "bbox": [
+ 105,
+ 422,
+ 506,
+ 436
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 422,
+ 294,
+ 436
+ ],
+ "score": 1.0,
+ "content": "maximum value of singular values are small,",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 295,
+ 422,
+ 314,
+ 434
+ ],
+ "score": 0.9,
+ "content": "X ^ { ( l ) }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 314,
+ 422,
+ 427,
+ 436
+ ],
+ "score": 1.0,
+ "content": "asymptotically approaches",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 427,
+ 424,
+ 440,
+ 434
+ ],
+ "score": 0.79,
+ "content": "\\mathcal { M }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 441,
+ 422,
+ 506,
+ 436
+ ],
+ "score": 1.0,
+ "content": "in the sense of",
+ "type": "text"
+ }
+ ],
+ "index": 25
+ },
+ {
+ "bbox": [
+ 105,
+ 434,
+ 503,
+ 448
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 434,
+ 251,
+ 448
+ ],
+ "score": 1.0,
+ "content": "Johnson (1973) for any initial value",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 251,
+ 434,
+ 272,
+ 446
+ ],
+ "score": 0.9,
+ "content": "X ^ { ( 0 ) }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 272,
+ 434,
+ 503,
+ 448
+ ],
+ "score": 1.0,
+ "content": ". That is, the followings hold under Assumptions 1 and 2.",
+ "type": "text"
+ }
+ ],
+ "index": 26
+ }
+ ],
+ "index": 25,
+ "bbox_fs": [
+ 105,
+ 410,
+ 506,
+ 448
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 448,
+ 505,
+ 461
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 448,
+ 504,
+ 462
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 448,
+ 162,
+ 462
+ ],
+ "score": 1.0,
+ "content": "Corollary 1.",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 162,
+ 449,
+ 175,
+ 459
+ ],
+ "score": 0.71,
+ "content": "\\mathcal { M }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 176,
+ 448,
+ 289,
+ 462
+ ],
+ "score": 1.0,
+ "content": "is invariant under fl for any",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 289,
+ 449,
+ 319,
+ 461
+ ],
+ "score": 0.91,
+ "content": "l \\in \\mathbb { N } _ { + }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 319,
+ 448,
+ 360,
+ 462
+ ],
+ "score": 1.0,
+ "content": ", that is, if",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 360,
+ 449,
+ 394,
+ 460
+ ],
+ "score": 0.89,
+ "content": "X \\in \\mathcal { M }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 394,
+ 448,
+ 451,
+ 462
+ ],
+ "score": 1.0,
+ "content": ", then we have",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 452,
+ 448,
+ 501,
+ 461
+ ],
+ "score": 0.92,
+ "content": "f _ { l } ( X ) \\in \\mathcal { M }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 501,
+ 448,
+ 504,
+ 462
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 27
+ }
+ ],
+ "index": 27,
+ "bbox_fs": [
+ 106,
+ 448,
+ 504,
+ 462
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 462,
+ 503,
+ 489
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 461,
+ 506,
+ 477
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 461,
+ 178,
+ 477
+ ],
+ "score": 1.0,
+ "content": "Corollary 2. Let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 179,
+ 465,
+ 245,
+ 477
+ ],
+ "score": 0.91,
+ "content": "s : = \\mathrm { s u p } _ { l \\in \\mathbb { N } _ { + } } s _ { l }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 245,
+ 461,
+ 287,
+ 477
+ ],
+ "score": 1.0,
+ "content": ". We have",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 287,
+ 462,
+ 380,
+ 476
+ ],
+ "score": 0.92,
+ "content": "d _ { \\mathcal { M } } ( X ^ { ( l ) } ) = O ( ( s \\lambda ) ^ { l } )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 381,
+ 461,
+ 450,
+ 477
+ ],
+ "score": 1.0,
+ "content": ". In particular, if",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 450,
+ 464,
+ 481,
+ 475
+ ],
+ "score": 0.84,
+ "content": "s \\lambda < 1",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 481,
+ 461,
+ 506,
+ 477
+ ],
+ "score": 1.0,
+ "content": ", then",
+ "type": "text"
+ }
+ ],
+ "index": 28
+ },
+ {
+ "bbox": [
+ 106,
+ 476,
+ 388,
+ 490
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 478,
+ 119,
+ 489
+ ],
+ "score": 0.85,
+ "content": "X _ { l }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 119,
+ 476,
+ 225,
+ 490
+ ],
+ "score": 1.0,
+ "content": "exponentially approaches",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 225,
+ 478,
+ 238,
+ 488
+ ],
+ "score": 0.76,
+ "content": "\\mathcal { M }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 238,
+ 476,
+ 250,
+ 490
+ ],
+ "score": 1.0,
+ "content": "as",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 251,
+ 478,
+ 280,
+ 488
+ ],
+ "score": 0.89,
+ "content": "l \\infty",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 281,
+ 476,
+ 363,
+ 490
+ ],
+ "score": 1.0,
+ "content": "for any initial value",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 363,
+ 476,
+ 383,
+ 487
+ ],
+ "score": 0.89,
+ "content": "X ^ { ( 0 ) }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 384,
+ 476,
+ 388,
+ 490
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 29
+ }
+ ],
+ "index": 28.5,
+ "bbox_fs": [
+ 105,
+ 461,
+ 506,
+ 490
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 496,
+ 505,
+ 564
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 496,
+ 506,
+ 509
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 496,
+ 232,
+ 509
+ ],
+ "score": 1.0,
+ "content": "Suppose the operator norm of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 232,
+ 497,
+ 296,
+ 509
+ ],
+ "score": 0.92,
+ "content": "P | _ { U } : U \\to U",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 297,
+ 496,
+ 369,
+ 509
+ ],
+ "score": 1.0,
+ "content": "is no larger than",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 369,
+ 497,
+ 376,
+ 507
+ ],
+ "score": 0.77,
+ "content": "\\lambda",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 376,
+ 496,
+ 506,
+ 509
+ ],
+ "score": 1.0,
+ "content": ", then, under the assumption of",
+ "type": "text"
+ }
+ ],
+ "index": 30
+ },
+ {
+ "bbox": [
+ 107,
+ 507,
+ 506,
+ 523
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 509,
+ 138,
+ 520
+ ],
+ "score": 0.84,
+ "content": "s \\lambda < 1",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 138,
+ 507,
+ 142,
+ 523
+ ],
+ "score": 1.0,
+ "content": ",",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 142,
+ 508,
+ 162,
+ 519
+ ],
+ "score": 0.84,
+ "content": "X ^ { ( l ) }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 162,
+ 507,
+ 506,
+ 523
+ ],
+ "score": 1.0,
+ "content": "converges to 0, the trivial fixed point (see Appendix E.2, Proposition 3). Therefore,",
+ "type": "text"
+ }
+ ],
+ "index": 31
+ },
+ {
+ "bbox": [
+ 106,
+ 520,
+ 505,
+ 533
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 520,
+ 370,
+ 533
+ ],
+ "score": 1.0,
+ "content": "we are mainly interested in the case where the operator norm of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 370,
+ 520,
+ 389,
+ 532
+ ],
+ "score": 0.92,
+ "content": "P | _ { U }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 389,
+ 520,
+ 477,
+ 533
+ ],
+ "score": 1.0,
+ "content": "is strictly larger than",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 478,
+ 520,
+ 485,
+ 530
+ ],
+ "score": 0.77,
+ "content": "\\lambda",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 485,
+ 520,
+ 505,
+ 533
+ ],
+ "score": 1.0,
+ "content": "(see",
+ "type": "text"
+ }
+ ],
+ "index": 32
+ },
+ {
+ "bbox": [
+ 106,
+ 531,
+ 505,
+ 543
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 531,
+ 426,
+ 543
+ ],
+ "score": 1.0,
+ "content": "Proposition 1). Finally, we restate Theorem 1 specialized to the situation where",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 426,
+ 532,
+ 435,
+ 541
+ ],
+ "score": 0.81,
+ "content": "U",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 436,
+ 531,
+ 505,
+ 543
+ ],
+ "score": 1.0,
+ "content": "is the direct sum",
+ "type": "text"
+ }
+ ],
+ "index": 33
+ },
+ {
+ "bbox": [
+ 106,
+ 542,
+ 506,
+ 555
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 542,
+ 278,
+ 555
+ ],
+ "score": 1.0,
+ "content": "of eigenspaces associated with the largest",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 279,
+ 542,
+ 290,
+ 552
+ ],
+ "score": 0.77,
+ "content": "M",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 291,
+ 542,
+ 353,
+ 555
+ ],
+ "score": 1.0,
+ "content": "eigenvalues of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 353,
+ 542,
+ 362,
+ 552
+ ],
+ "score": 0.83,
+ "content": "P",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 362,
+ 542,
+ 485,
+ 555
+ ],
+ "score": 1.0,
+ "content": ". Note that the eigenvalues of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 485,
+ 542,
+ 494,
+ 552
+ ],
+ "score": 0.84,
+ "content": "P",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 495,
+ 542,
+ 506,
+ 555
+ ],
+ "score": 1.0,
+ "content": "is",
+ "type": "text"
+ }
+ ],
+ "index": 34
+ },
+ {
+ "bbox": [
+ 105,
+ 552,
+ 213,
+ 566
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 552,
+ 147,
+ 566
+ ],
+ "score": 1.0,
+ "content": "real since",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 147,
+ 554,
+ 156,
+ 563
+ ],
+ "score": 0.82,
+ "content": "P",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 156,
+ 552,
+ 213,
+ 566
+ ],
+ "score": 1.0,
+ "content": "is symmetric.",
+ "type": "text"
+ }
+ ],
+ "index": 35
+ }
+ ],
+ "index": 32.5,
+ "bbox_fs": [
+ 105,
+ 496,
+ 506,
+ 566
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 566,
+ 505,
+ 614
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 565,
+ 505,
+ 579
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 565,
+ 179,
+ 579
+ ],
+ "score": 1.0,
+ "content": "Corollary 3. Let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 180,
+ 567,
+ 245,
+ 578
+ ],
+ "score": 0.89,
+ "content": "\\lambda _ { 1 } \\leq \\cdots \\leq \\lambda _ { N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 246,
+ 565,
+ 331,
+ 579
+ ],
+ "score": 1.0,
+ "content": "be the eigenvalue of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 332,
+ 567,
+ 340,
+ 576
+ ],
+ "score": 0.8,
+ "content": "P",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 340,
+ 565,
+ 505,
+ 579
+ ],
+ "score": 1.0,
+ "content": ", sorted in ascending order. Suppose the",
+ "type": "text"
+ }
+ ],
+ "index": 36
+ },
+ {
+ "bbox": [
+ 105,
+ 576,
+ 506,
+ 591
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 576,
+ 257,
+ 591
+ ],
+ "score": 1.0,
+ "content": "multiplicity of the largest eigenvalue",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 257,
+ 578,
+ 272,
+ 588
+ ],
+ "score": 0.88,
+ "content": "\\lambda _ { N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 272,
+ 576,
+ 283,
+ 591
+ ],
+ "score": 1.0,
+ "content": "is",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 284,
+ 577,
+ 323,
+ 589
+ ],
+ "score": 0.9,
+ "content": "M ( \\leq N )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 324,
+ 576,
+ 346,
+ 591
+ ],
+ "score": 1.0,
+ "content": ", i.e.,",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 346,
+ 578,
+ 484,
+ 589
+ ],
+ "score": 0.9,
+ "content": "\\lambda _ { N - M } < \\lambda _ { N - M + 1 } = \\cdot \\cdot \\cdot = \\lambda _ { N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 484,
+ 576,
+ 506,
+ 591
+ ],
+ "score": 1.0,
+ "content": ". We",
+ "type": "text"
+ }
+ ],
+ "index": 37
+ },
+ {
+ "bbox": [
+ 105,
+ 587,
+ 506,
+ 602
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 587,
+ 132,
+ 602
+ ],
+ "score": 1.0,
+ "content": "define",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 133,
+ 588,
+ 229,
+ 601
+ ],
+ "score": 0.91,
+ "content": "\\lambda : = \\operatorname* { m a x } _ { n \\in [ N - M ] } \\left| \\lambda _ { n } \\right|",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 229,
+ 587,
+ 276,
+ 602
+ ],
+ "score": 1.0,
+ "content": ". We denote",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 277,
+ 589,
+ 286,
+ 598
+ ],
+ "score": 0.76,
+ "content": "U",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 286,
+ 587,
+ 423,
+ 602
+ ],
+ "score": 1.0,
+ "content": "by the eigenspace associated with",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 423,
+ 589,
+ 437,
+ 599
+ ],
+ "score": 0.88,
+ "content": "\\lambda _ { N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 437,
+ 587,
+ 506,
+ 602
+ ],
+ "score": 1.0,
+ "content": "and assume that",
+ "type": "text"
+ }
+ ],
+ "index": 38
+ },
+ {
+ "bbox": [
+ 106,
+ 600,
+ 394,
+ 615
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 602,
+ 115,
+ 612
+ ],
+ "score": 0.66,
+ "content": "U",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 116,
+ 600,
+ 270,
+ 615
+ ],
+ "score": 1.0,
+ "content": "satisfies Assumption 1. Then, we have",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 271,
+ 600,
+ 391,
+ 614
+ ],
+ "score": 0.93,
+ "content": "d _ { \\mathcal { M } } ( X ^ { ( l + 1 ) } ) \\leq s _ { l } \\lambda d _ { \\mathcal { M } } ( X ^ { ( l ) } ) .",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 391,
+ 600,
+ 394,
+ 615
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 39
+ }
+ ],
+ "index": 37.5,
+ "bbox_fs": [
+ 105,
+ 565,
+ 506,
+ 615
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 614,
+ 505,
+ 660
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 614,
+ 505,
+ 627
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 614,
+ 505,
+ 627
+ ],
+ "score": 1.0,
+ "content": "Remark 1. It is known that any Markov process on finite states converges to a unique distribution",
+ "type": "text"
+ }
+ ],
+ "index": 40
+ },
+ {
+ "bbox": [
+ 105,
+ 626,
+ 506,
+ 639
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 626,
+ 443,
+ 639
+ ],
+ "score": 1.0,
+ "content": "(equilibrium) if it is irreducible and aperiodic (see e.g., Norris (1998)). Theorem",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 443,
+ 627,
+ 450,
+ 636
+ ],
+ "score": 0.41,
+ "content": "^ { l }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 450,
+ 626,
+ 506,
+ 639
+ ],
+ "score": 1.0,
+ "content": "includes this",
+ "type": "text"
+ }
+ ],
+ "index": 41
+ },
+ {
+ "bbox": [
+ 104,
+ 637,
+ 506,
+ 649
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 637,
+ 245,
+ 649
+ ],
+ "score": 1.0,
+ "content": "proposition as a special case with",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 246,
+ 637,
+ 276,
+ 647
+ ],
+ "score": 0.86,
+ "content": "M = 1",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 277,
+ 637,
+ 281,
+ 649
+ ],
+ "score": 1.0,
+ "content": ",",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 281,
+ 637,
+ 309,
+ 647
+ ],
+ "score": 0.87,
+ "content": "C = 1",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 310,
+ 637,
+ 331,
+ 649
+ ],
+ "score": 1.0,
+ "content": ", and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 332,
+ 637,
+ 365,
+ 648
+ ],
+ "score": 0.91,
+ "content": "W _ { l } = 1",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 365,
+ 637,
+ 394,
+ 649
+ ],
+ "score": 1.0,
+ "content": "for all",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 394,
+ 637,
+ 425,
+ 648
+ ],
+ "score": 0.91,
+ "content": "l \\in \\mathbb { N } _ { + }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 425,
+ 637,
+ 506,
+ 649
+ ],
+ "score": 1.0,
+ "content": ". This is essentially",
+ "type": "text"
+ }
+ ],
+ "index": 42
+ },
+ {
+ "bbox": [
+ 105,
+ 648,
+ 496,
+ 661
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 648,
+ 484,
+ 661
+ ],
+ "score": 1.0,
+ "content": "the direct consequence of Perron – Frobenius’ theorem (see e.g., Meyer (2000)). See Appendix",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 485,
+ 649,
+ 492,
+ 658
+ ],
+ "score": 0.64,
+ "content": "F",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 492,
+ 648,
+ 496,
+ 661
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 43
+ }
+ ],
+ "index": 41.5,
+ "bbox_fs": [
+ 104,
+ 614,
+ 506,
+ 661
+ ]
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 108,
+ 674,
+ 243,
+ 687
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 673,
+ 244,
+ 689
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 673,
+ 244,
+ 689
+ ],
+ "score": 1.0,
+ "content": "4 APPLICATION TO GCN",
+ "type": "text"
+ }
+ ],
+ "index": 44
+ }
+ ],
+ "index": 44
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 698,
+ 505,
+ 732
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 697,
+ 505,
+ 712
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 697,
+ 505,
+ 712
+ ],
+ "score": 1.0,
+ "content": "We formulate a GCN (Kipf & Welling, 2017) without readout operations (Gilmer et al., 2017) using",
+ "type": "text"
+ }
+ ],
+ "index": 45
+ },
+ {
+ "bbox": [
+ 106,
+ 711,
+ 505,
+ 721
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 711,
+ 505,
+ 721
+ ],
+ "score": 1.0,
+ "content": "the dynamical system in the previous section and derive a sufficient condition in terms of the spectra",
+ "type": "text"
+ }
+ ],
+ "index": 46
+ },
+ {
+ "bbox": [
+ 106,
+ 721,
+ 505,
+ 733
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 721,
+ 505,
+ 733
+ ],
+ "score": 1.0,
+ "content": "of underlying graphs in which layer stacking nor non-linearity are not helpful for node classification.",
+ "type": "text"
+ }
+ ],
+ "index": 47
+ }
+ ],
+ "index": 46,
+ "bbox_fs": [
+ 105,
+ 697,
+ 505,
+ 733
+ ]
+ }
+ ]
+ },
+ {
+ "preproc_blocks": [
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 82,
+ 505,
+ 127
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 82,
+ 505,
+ 95
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 83,
+ 122,
+ 95
+ ],
+ "score": 1.0,
+ "content": "Let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 122,
+ 82,
+ 171,
+ 95
+ ],
+ "score": 0.93,
+ "content": "G = ( V , E )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 171,
+ 83,
+ 291,
+ 95
+ ],
+ "score": 1.0,
+ "content": "be an undirected graph where",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 291,
+ 83,
+ 300,
+ 92
+ ],
+ "score": 0.79,
+ "content": "V",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 301,
+ 83,
+ 383,
+ 95
+ ],
+ "score": 1.0,
+ "content": "is a set of nodes and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 383,
+ 83,
+ 392,
+ 92
+ ],
+ "score": 0.82,
+ "content": "E",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 393,
+ 83,
+ 505,
+ 95
+ ],
+ "score": 1.0,
+ "content": "is a set of edges. We denote",
+ "type": "text"
+ }
+ ],
+ "index": 0
+ },
+ {
+ "bbox": [
+ 105,
+ 93,
+ 506,
+ 106
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 93,
+ 199,
+ 106
+ ],
+ "score": 1.0,
+ "content": "the number of nodes in",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 199,
+ 94,
+ 208,
+ 104
+ ],
+ "score": 0.79,
+ "content": "G",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 209,
+ 93,
+ 221,
+ 106
+ ],
+ "score": 1.0,
+ "content": "by",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 221,
+ 94,
+ 258,
+ 105
+ ],
+ "score": 0.92,
+ "content": "N = | V |",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 258,
+ 93,
+ 308,
+ 106
+ ],
+ "score": 1.0,
+ "content": "and identify",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 309,
+ 94,
+ 318,
+ 104
+ ],
+ "score": 0.74,
+ "content": "V",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 318,
+ 93,
+ 339,
+ 106
+ ],
+ "score": 1.0,
+ "content": "with",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 339,
+ 93,
+ 354,
+ 106
+ ],
+ "score": 0.91,
+ "content": "[ N ]",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 355,
+ 93,
+ 438,
+ 106
+ ],
+ "score": 1.0,
+ "content": "by fixing an order of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 438,
+ 94,
+ 447,
+ 104
+ ],
+ "score": 0.67,
+ "content": "V",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 447,
+ 93,
+ 506,
+ 106
+ ],
+ "score": 1.0,
+ "content": ". We associate",
+ "type": "text"
+ }
+ ],
+ "index": 1
+ },
+ {
+ "bbox": [
+ 104,
+ 104,
+ 505,
+ 117
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 104,
+ 113,
+ 117
+ ],
+ "score": 1.0,
+ "content": "a",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 113,
+ 105,
+ 123,
+ 114
+ ],
+ "score": 0.81,
+ "content": "C",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 123,
+ 104,
+ 257,
+ 117
+ ],
+ "score": 1.0,
+ "content": "dimensional signal to each node.",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 257,
+ 105,
+ 267,
+ 114
+ ],
+ "score": 0.83,
+ "content": "X",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 268,
+ 104,
+ 505,
+ 117
+ ],
+ "score": 1.0,
+ "content": "in the previous section corresponds to concatenation of the",
+ "type": "text"
+ }
+ ],
+ "index": 2
+ },
+ {
+ "bbox": [
+ 105,
+ 115,
+ 479,
+ 128
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 115,
+ 281,
+ 128
+ ],
+ "score": 1.0,
+ "content": "signals. GCNs iteratively update signals on",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 281,
+ 116,
+ 290,
+ 125
+ ],
+ "score": 0.78,
+ "content": "V",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 291,
+ 115,
+ 479,
+ 128
+ ],
+ "score": 1.0,
+ "content": "using the connection information and weights.",
+ "type": "text"
+ }
+ ],
+ "index": 3
+ }
+ ],
+ "index": 1.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 131,
+ 505,
+ 228
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 104,
+ 128,
+ 505,
+ 147
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 128,
+ 123,
+ 147
+ ],
+ "score": 1.0,
+ "content": "Let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 123,
+ 131,
+ 270,
+ 145
+ ],
+ "score": 0.93,
+ "content": "A : = ( \\mathbf { 1 } _ { \\{ ( i , j ) \\in E \\} } ) _ { i , j \\in [ N ] } \\ \\in \\ \\mathbb { R } ^ { N \\times N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 270,
+ 128,
+ 391,
+ 147
+ ],
+ "score": 1.0,
+ "content": "be the adjacency matrix and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 391,
+ 132,
+ 505,
+ 145
+ ],
+ "score": 0.91,
+ "content": "D : = \\mathrm { d i a g } ( \\deg ( i ) _ { i \\in [ N ] } ) \\ \\in",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 4
+ },
+ {
+ "bbox": [
+ 106,
+ 141,
+ 506,
+ 160
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 144,
+ 135,
+ 155
+ ],
+ "score": 0.89,
+ "content": "\\mathbb { R } ^ { N \\times N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 136,
+ 141,
+ 235,
+ 160
+ ],
+ "score": 1.0,
+ "content": "be the degree matrix of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 236,
+ 146,
+ 245,
+ 155
+ ],
+ "score": 0.82,
+ "content": "G",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 245,
+ 141,
+ 273,
+ 160
+ ],
+ "score": 1.0,
+ "content": "where",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 274,
+ 144,
+ 415,
+ 157
+ ],
+ "score": 0.93,
+ "content": "\\deg ( i ) : = | \\{ j \\in V \\mid ( i , j ) \\in E \\} |",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 416,
+ 141,
+ 506,
+ 160
+ ],
+ "score": 1.0,
+ "content": "is the degree of node",
+ "type": "text"
+ }
+ ],
+ "index": 5
+ },
+ {
+ "bbox": [
+ 106,
+ 156,
+ 505,
+ 171
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 158,
+ 111,
+ 168
+ ],
+ "score": 0.61,
+ "content": "i",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 111,
+ 156,
+ 135,
+ 171
+ ],
+ "score": 1.0,
+ "content": ". Let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 135,
+ 156,
+ 196,
+ 169
+ ],
+ "score": 0.88,
+ "content": "{ \\tilde { A } } : = A + I _ { N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 197,
+ 156,
+ 201,
+ 171
+ ],
+ "score": 1.0,
+ "content": ",",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 202,
+ 156,
+ 265,
+ 169
+ ],
+ "score": 0.9,
+ "content": "{ \\tilde { D } } : = D + I _ { N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 265,
+ 156,
+ 447,
+ 171
+ ],
+ "score": 1.0,
+ "content": "be the adjacent and degree matrix of graph",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 448,
+ 158,
+ 457,
+ 168
+ ],
+ "score": 0.8,
+ "content": "G",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 457,
+ 156,
+ 505,
+ 171
+ ],
+ "score": 1.0,
+ "content": "augmented",
+ "type": "text"
+ }
+ ],
+ "index": 6
+ },
+ {
+ "bbox": [
+ 106,
+ 169,
+ 505,
+ 183
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 169,
+ 456,
+ 183
+ ],
+ "score": 1.0,
+ "content": "with self-loops. We define the augmented normalized Laplacian (Wu et al., 2019a) of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 456,
+ 170,
+ 466,
+ 181
+ ],
+ "score": 0.81,
+ "content": "G",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 466,
+ 169,
+ 480,
+ 183
+ ],
+ "score": 1.0,
+ "content": "by",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 480,
+ 169,
+ 505,
+ 182
+ ],
+ "score": 0.87,
+ "content": "\\tilde { \\Delta } : =",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 7
+ },
+ {
+ "bbox": [
+ 107,
+ 181,
+ 505,
+ 196
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 181,
+ 178,
+ 194
+ ],
+ "score": 0.92,
+ "content": "I _ { N } - \\tilde { D } ^ { - \\frac { 1 } { 2 } } \\tilde { A } \\tilde { D } ^ { - \\frac { 1 } { 2 } }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 178,
+ 181,
+ 209,
+ 196
+ ],
+ "score": 1.0,
+ "content": "and set",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 210,
+ 181,
+ 264,
+ 194
+ ],
+ "score": 0.92,
+ "content": "P : = I _ { N } - \\tilde { \\Delta }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 265,
+ 181,
+ 284,
+ 196
+ ],
+ "score": 1.0,
+ "content": ". Let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 284,
+ 182,
+ 330,
+ 194
+ ],
+ "score": 0.88,
+ "content": "L , C \\in \\mathbb { N } _ { + }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 330,
+ 181,
+ 505,
+ 196
+ ],
+ "score": 1.0,
+ "content": "be the layer and channel sizes, respectively.",
+ "type": "text"
+ }
+ ],
+ "index": 8
+ },
+ {
+ "bbox": [
+ 104,
+ 191,
+ 506,
+ 208
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 191,
+ 156,
+ 208
+ ],
+ "score": 1.0,
+ "content": "For weights",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 156,
+ 193,
+ 207,
+ 204
+ ],
+ "score": 0.8,
+ "content": "W _ { l } \\in \\mathbb { R } ^ { C \\times C }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 211,
+ 194,
+ 243,
+ 205
+ ],
+ "score": 0.6,
+ "content": "( l \\in [ L ] )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 244,
+ 191,
+ 295,
+ 208
+ ],
+ "score": 1.0,
+ "content": ", we define a",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 295,
+ 194,
+ 322,
+ 204
+ ],
+ "score": 0.81,
+ "content": "\\mathrm { G C N } ^ { 2 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 322,
+ 191,
+ 386,
+ 208
+ ],
+ "score": 1.0,
+ "content": "associated with",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 386,
+ 194,
+ 396,
+ 204
+ ],
+ "score": 0.81,
+ "content": "G",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 396,
+ 191,
+ 408,
+ 208
+ ],
+ "score": 1.0,
+ "content": "by",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 409,
+ 194,
+ 477,
+ 205
+ ],
+ "score": 0.93,
+ "content": "f = f _ { L } \\circ \\cdot \\cdot \\cdot \\circ f _ { 1 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 477,
+ 191,
+ 506,
+ 208
+ ],
+ "score": 1.0,
+ "content": "where",
+ "type": "text"
+ }
+ ],
+ "index": 9
+ },
+ {
+ "bbox": [
+ 107,
+ 201,
+ 506,
+ 219
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 204,
+ 201,
+ 217
+ ],
+ "score": 0.91,
+ "content": "f _ { l } : \\mathbb { R } ^ { \\breve { N } \\times C } \\mathbb { R } ^ { N \\times C }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 201,
+ 201,
+ 261,
+ 219
+ ],
+ "score": 1.0,
+ "content": "is defined by",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 261,
+ 205,
+ 349,
+ 217
+ ],
+ "score": 0.91,
+ "content": "f _ { l } ( X ) : = \\sigma ( P X W _ { l } )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 349,
+ 201,
+ 506,
+ 219
+ ],
+ "score": 1.0,
+ "content": ". We are interested in the asymptotic",
+ "type": "text"
+ }
+ ],
+ "index": 10
+ },
+ {
+ "bbox": [
+ 104,
+ 215,
+ 318,
+ 230
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 215,
+ 197,
+ 230
+ ],
+ "score": 1.0,
+ "content": "behavior of the output",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 197,
+ 216,
+ 219,
+ 227
+ ],
+ "score": 0.91,
+ "content": "X ^ { ( L ) }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 219,
+ 215,
+ 280,
+ 230
+ ],
+ "score": 1.0,
+ "content": "of the GCN as",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 281,
+ 218,
+ 314,
+ 227
+ ],
+ "score": 0.88,
+ "content": "L \\to \\infty",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 314,
+ 215,
+ 318,
+ 230
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 11
+ }
+ ],
+ "index": 7.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 234,
+ 505,
+ 279
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 234,
+ 506,
+ 246
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 234,
+ 144,
+ 246
+ ],
+ "score": 1.0,
+ "content": "Suppose",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 144,
+ 234,
+ 153,
+ 244
+ ],
+ "score": 0.8,
+ "content": "G",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 154,
+ 234,
+ 172,
+ 246
+ ],
+ "score": 1.0,
+ "content": "has",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 172,
+ 235,
+ 184,
+ 244
+ ],
+ "score": 0.81,
+ "content": "M",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 185,
+ 234,
+ 313,
+ 246
+ ],
+ "score": 1.0,
+ "content": "connected components and let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 313,
+ 234,
+ 401,
+ 245
+ ],
+ "score": 0.91,
+ "content": "V = V _ { 1 } \\sqcup \\cdots \\sqcup V _ { M }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 402,
+ 234,
+ 506,
+ 246
+ ],
+ "score": 1.0,
+ "content": "be the decomposition of",
+ "type": "text"
+ }
+ ],
+ "index": 12
+ },
+ {
+ "bbox": [
+ 106,
+ 245,
+ 505,
+ 257
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 245,
+ 158,
+ 257
+ ],
+ "score": 1.0,
+ "content": "the node set",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 159,
+ 245,
+ 168,
+ 255
+ ],
+ "score": 0.76,
+ "content": "V",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 168,
+ 245,
+ 439,
+ 257
+ ],
+ "score": 1.0,
+ "content": "into connected components. We denote an indicator vector of the",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 439,
+ 246,
+ 449,
+ 255
+ ],
+ "score": 0.79,
+ "content": "m",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 450,
+ 245,
+ 505,
+ 257
+ ],
+ "score": 1.0,
+ "content": "-th connected",
+ "type": "text"
+ }
+ ],
+ "index": 13
+ },
+ {
+ "bbox": [
+ 104,
+ 255,
+ 506,
+ 269
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 255,
+ 166,
+ 269
+ ],
+ "score": 1.0,
+ "content": "component by",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 166,
+ 256,
+ 290,
+ 269
+ ],
+ "score": 0.89,
+ "content": "u _ { m } : = ( \\mathbf { 1 } _ { \\{ n \\in V _ { m } \\} } ) _ { n \\in [ N ] } \\in \\mathbb { R } ^ { N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 291,
+ 255,
+ 506,
+ 269
+ ],
+ "score": 1.0,
+ "content": ". The following proposition shows that GCN satisfies",
+ "type": "text"
+ }
+ ],
+ "index": 14
+ },
+ {
+ "bbox": [
+ 106,
+ 267,
+ 338,
+ 279
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 267,
+ 338,
+ 279
+ ],
+ "score": 1.0,
+ "content": "the assumption of Corollay 3 (see Appendix B for proof).",
+ "type": "text"
+ }
+ ],
+ "index": 15
+ }
+ ],
+ "index": 13.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 281,
+ 505,
+ 328
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 279,
+ 506,
+ 295
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 279,
+ 187,
+ 295
+ ],
+ "score": 1.0,
+ "content": "Proposition 1. Let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 188,
+ 281,
+ 256,
+ 293
+ ],
+ "score": 0.9,
+ "content": "\\lambda _ { 1 } \\leq \\cdots \\leq \\lambda _ { N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 257,
+ 279,
+ 343,
+ 295
+ ],
+ "score": 1.0,
+ "content": "be the eigenvalue of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 344,
+ 282,
+ 352,
+ 291
+ ],
+ "score": 0.76,
+ "content": "P",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 353,
+ 279,
+ 506,
+ 295
+ ],
+ "score": 1.0,
+ "content": "sorted in ascending order. Then, we",
+ "type": "text"
+ }
+ ],
+ "index": 16
+ },
+ {
+ "bbox": [
+ 105,
+ 291,
+ 505,
+ 307
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 291,
+ 129,
+ 307
+ ],
+ "score": 1.0,
+ "content": "have",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 129,
+ 293,
+ 171,
+ 304
+ ],
+ "score": 0.81,
+ "content": "- 1 < \\lambda _ { 1 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 171,
+ 291,
+ 177,
+ 307
+ ],
+ "score": 1.0,
+ "content": ",",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 177,
+ 294,
+ 229,
+ 304
+ ],
+ "score": 0.85,
+ "content": "\\lambda _ { N - M } ~ < ~ 1",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 229,
+ 291,
+ 254,
+ 307
+ ],
+ "score": 1.0,
+ "content": ", and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 254,
+ 293,
+ 376,
+ 304
+ ],
+ "score": 0.9,
+ "content": "\\lambda _ { N - M + 1 } = \\cdot \\cdot \\cdot = \\lambda _ { N } = 1",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 377,
+ 291,
+ 480,
+ 307
+ ],
+ "score": 1.0,
+ "content": ". In particular, we have",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 481,
+ 293,
+ 505,
+ 304
+ ],
+ "score": 0.84,
+ "content": "\\lambda : =",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 17
+ },
+ {
+ "bbox": [
+ 106,
+ 304,
+ 506,
+ 320
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 306,
+ 215,
+ 318
+ ],
+ "score": 0.88,
+ "content": "\\begin{array} { r } { \\operatorname* { m a x } _ { n = 1 , \\dots , N - M } | \\lambda _ { n } | < 1 } \\end{array}",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 215,
+ 304,
+ 257,
+ 320
+ ],
+ "score": 1.0,
+ "content": ". Further,",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 257,
+ 304,
+ 317,
+ 317
+ ],
+ "score": 0.89,
+ "content": "e _ { m } : = \\tilde { D } ^ { \\frac { 1 } { 2 } } u _ { m }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 317,
+ 304,
+ 333,
+ 320
+ ],
+ "score": 1.0,
+ "content": "for",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 334,
+ 305,
+ 375,
+ 318
+ ],
+ "score": 0.9,
+ "content": "m \\in [ M ]",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 375,
+ 304,
+ 506,
+ 320
+ ],
+ "score": 1.0,
+ "content": "are the basis of the eigenspace",
+ "type": "text"
+ }
+ ],
+ "index": 18
+ },
+ {
+ "bbox": [
+ 106,
+ 317,
+ 241,
+ 329
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 317,
+ 241,
+ 329
+ ],
+ "score": 1.0,
+ "content": "associated with the eigenvalue 1.",
+ "type": "text"
+ }
+ ],
+ "index": 19
+ }
+ ],
+ "index": 17.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 108,
+ 330,
+ 504,
+ 356
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 327,
+ 505,
+ 345
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 327,
+ 256,
+ 345
+ ],
+ "score": 1.0,
+ "content": "Theorem 2. For any initial value",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 256,
+ 330,
+ 276,
+ 342
+ ],
+ "score": 0.9,
+ "content": "X ^ { ( 0 ) }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 277,
+ 327,
+ 387,
+ 345
+ ],
+ "score": 1.0,
+ "content": ", the output of l-th layer",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 388,
+ 330,
+ 407,
+ 342
+ ],
+ "score": 0.9,
+ "content": "X ^ { ( l ) }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 407,
+ 327,
+ 447,
+ 345
+ ],
+ "score": 1.0,
+ "content": "satisfies",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 447,
+ 330,
+ 505,
+ 343
+ ],
+ "score": 0.88,
+ "content": "d _ { \\mathcal { M } } ( X ^ { ( l ) } ) ~ \\leq",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 20
+ },
+ {
+ "bbox": [
+ 107,
+ 341,
+ 448,
+ 357
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 342,
+ 171,
+ 356
+ ],
+ "score": 0.92,
+ "content": "( s \\lambda ) ^ { l } d _ { \\mathcal { M } } ( X ^ { ( 0 ) } )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 171,
+ 341,
+ 231,
+ 357
+ ],
+ "score": 1.0,
+ "content": ". In particular,",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 231,
+ 342,
+ 273,
+ 356
+ ],
+ "score": 0.94,
+ "content": "d _ { \\mathcal { M } } ( { \\boldsymbol { X } } ^ { ( l ) } )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 273,
+ 341,
+ 414,
+ 357
+ ],
+ "score": 1.0,
+ "content": "exponentially converges to 0 when",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 414,
+ 344,
+ 443,
+ 354
+ ],
+ "score": 0.88,
+ "content": "s \\lambda < 1",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 444,
+ 341,
+ 448,
+ 357
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 21
+ }
+ ],
+ "index": 20.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 364,
+ 505,
+ 455
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 363,
+ 506,
+ 377
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 363,
+ 506,
+ 377
+ ],
+ "score": 1.0,
+ "content": "In the context of node classification tasks, we can interpret this corollary as the “information loss” of",
+ "type": "text"
+ }
+ ],
+ "index": 22
+ },
+ {
+ "bbox": [
+ 105,
+ 375,
+ 505,
+ 388
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 375,
+ 283,
+ 388
+ ],
+ "score": 1.0,
+ "content": "GCNs in the limit of infinite layers. For any",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 283,
+ 376,
+ 317,
+ 386
+ ],
+ "score": 0.91,
+ "content": "X \\in { \\mathcal { M } }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 318,
+ 375,
+ 372,
+ 388
+ ],
+ "score": 1.0,
+ "content": ", if two nodes",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 372,
+ 376,
+ 407,
+ 387
+ ],
+ "score": 0.91,
+ "content": "i , j \\in V",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 407,
+ 375,
+ 505,
+ 388
+ ],
+ "score": 1.0,
+ "content": "are in a same connected",
+ "type": "text"
+ }
+ ],
+ "index": 23
+ },
+ {
+ "bbox": [
+ 105,
+ 387,
+ 504,
+ 398
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 387,
+ 388,
+ 398
+ ],
+ "score": 1.0,
+ "content": "component and their degrees are identical, then, the column vectors of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 388,
+ 387,
+ 398,
+ 396
+ ],
+ "score": 0.85,
+ "content": "X",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 398,
+ 387,
+ 498,
+ 398
+ ],
+ "score": 1.0,
+ "content": "that correspond to nodes",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 499,
+ 387,
+ 504,
+ 396
+ ],
+ "score": 0.67,
+ "content": "i",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 24
+ },
+ {
+ "bbox": [
+ 105,
+ 396,
+ 505,
+ 410
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 396,
+ 123,
+ 410
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 123,
+ 398,
+ 129,
+ 408
+ ],
+ "score": 0.82,
+ "content": "j",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 130,
+ 396,
+ 401,
+ 410
+ ],
+ "score": 1.0,
+ "content": "are identical. It means that we cannot distinguish these nodes using",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 402,
+ 398,
+ 412,
+ 407
+ ],
+ "score": 0.76,
+ "content": "X",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 412,
+ 396,
+ 470,
+ 410
+ ],
+ "score": 1.0,
+ "content": ". In this sense,",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 470,
+ 397,
+ 484,
+ 407
+ ],
+ "score": 0.78,
+ "content": "\\mathcal { M }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 484,
+ 396,
+ 505,
+ 410
+ ],
+ "score": 1.0,
+ "content": "only",
+ "type": "text"
+ }
+ ],
+ "index": 25
+ },
+ {
+ "bbox": [
+ 105,
+ 408,
+ 505,
+ 421
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 408,
+ 505,
+ 421
+ ],
+ "score": 1.0,
+ "content": "has information about connected components and node degrees and we can interpret this theorem",
+ "type": "text"
+ }
+ ],
+ "index": 26
+ },
+ {
+ "bbox": [
+ 105,
+ 419,
+ 505,
+ 432
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 419,
+ 505,
+ 432
+ ],
+ "score": 1.0,
+ "content": "as the exponential information loss of GCNs in terms of the layer size. Similarly to the discussion",
+ "type": "text"
+ }
+ ],
+ "index": 27
+ },
+ {
+ "bbox": [
+ 105,
+ 430,
+ 505,
+ 444
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 430,
+ 201,
+ 444
+ ],
+ "score": 1.0,
+ "content": "in the previous section,",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 201,
+ 430,
+ 221,
+ 442
+ ],
+ "score": 0.9,
+ "content": "X ^ { ( l ) }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 221,
+ 430,
+ 392,
+ 444
+ ],
+ "score": 1.0,
+ "content": "converges to the trivial fixed point 0 when",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 393,
+ 432,
+ 417,
+ 442
+ ],
+ "score": 0.89,
+ "content": "s < 1",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 418,
+ 430,
+ 465,
+ 444
+ ],
+ "score": 1.0,
+ "content": "(remember",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 465,
+ 431,
+ 498,
+ 442
+ ],
+ "score": 0.88,
+ "content": "\\lambda _ { N } = 1",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 499,
+ 430,
+ 505,
+ 444
+ ],
+ "score": 1.0,
+ "content": ").",
+ "type": "text"
+ }
+ ],
+ "index": 28
+ },
+ {
+ "bbox": [
+ 105,
+ 442,
+ 505,
+ 455
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 442,
+ 244,
+ 455
+ ],
+ "score": 1.0,
+ "content": "An interesting point is that even if",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 245,
+ 443,
+ 269,
+ 454
+ ],
+ "score": 0.87,
+ "content": "s \\geq 1",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 269,
+ 442,
+ 272,
+ 455
+ ],
+ "score": 1.0,
+ "content": ",",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 273,
+ 442,
+ 292,
+ 453
+ ],
+ "score": 0.88,
+ "content": "X ^ { ( l ) }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 292,
+ 442,
+ 465,
+ 455
+ ],
+ "score": 1.0,
+ "content": "can suffer from this information loss when",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 466,
+ 443,
+ 501,
+ 453
+ ],
+ "score": 0.9,
+ "content": "s < \\lambda ^ { - 1 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 501,
+ 442,
+ 505,
+ 455
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 29
+ }
+ ],
+ "index": 25.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 108,
+ 460,
+ 503,
+ 493
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 460,
+ 504,
+ 472
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 460,
+ 192,
+ 472
+ ],
+ "score": 1.0,
+ "content": "We note that the rate",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 192,
+ 461,
+ 204,
+ 470
+ ],
+ "score": 0.84,
+ "content": "s \\lambda",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 205,
+ 460,
+ 504,
+ 472
+ ],
+ "score": 1.0,
+ "content": "in Theorem 2 depends on the spectra of the augmented normalized Lapla-",
+ "type": "text"
+ }
+ ],
+ "index": 30
+ },
+ {
+ "bbox": [
+ 106,
+ 471,
+ 504,
+ 483
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 471,
+ 378,
+ 483
+ ],
+ "score": 1.0,
+ "content": "cian, which is determined by the topology of the underlying graph",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 378,
+ 471,
+ 387,
+ 481
+ ],
+ "score": 0.7,
+ "content": "G",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 387,
+ 471,
+ 504,
+ 483
+ ],
+ "score": 1.0,
+ "content": ". Hence, our result explicitly",
+ "type": "text"
+ }
+ ],
+ "index": 31
+ },
+ {
+ "bbox": [
+ 106,
+ 482,
+ 450,
+ 495
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 482,
+ 450,
+ 495
+ ],
+ "score": 1.0,
+ "content": "relates the topological information of graphs and asymptotic behaviors of graph NNs.",
+ "type": "text"
+ }
+ ],
+ "index": 32
+ }
+ ],
+ "index": 31
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 496,
+ 505,
+ 552
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 497,
+ 504,
+ 509
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 497,
+ 504,
+ 509
+ ],
+ "score": 1.0,
+ "content": "Remark 2. The old preprint (version 2) of Luan et al. (2019) formulated a theorem that explains the",
+ "type": "text"
+ }
+ ],
+ "index": 33
+ },
+ {
+ "bbox": [
+ 106,
+ 508,
+ 504,
+ 520
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 508,
+ 504,
+ 520
+ ],
+ "score": 1.0,
+ "content": "over-smoothing of non-linear GNNs. Specifically, it claimed that if a graph does not have a bipartite",
+ "type": "text"
+ }
+ ],
+ "index": 34
+ },
+ {
+ "bbox": [
+ 106,
+ 518,
+ 505,
+ 532
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 518,
+ 505,
+ 532
+ ],
+ "score": 1.0,
+ "content": "component and the input distribution is continuous, the rank of the output of a GCN converges to the",
+ "type": "text"
+ }
+ ],
+ "index": 35
+ },
+ {
+ "bbox": [
+ 105,
+ 530,
+ 506,
+ 543
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 530,
+ 506,
+ 543
+ ],
+ "score": 1.0,
+ "content": "number of connected components of the underlying graph as the layer size goes to infinity almost",
+ "type": "text"
+ }
+ ],
+ "index": 36
+ },
+ {
+ "bbox": [
+ 105,
+ 541,
+ 448,
+ 554
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 541,
+ 437,
+ 554
+ ],
+ "score": 1.0,
+ "content": "surely. However, it is not true in general as we give a counterexample in Appendix",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 437,
+ 543,
+ 444,
+ 551
+ ],
+ "score": 0.53,
+ "content": "C",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 444,
+ 541,
+ 448,
+ 554
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 37
+ }
+ ],
+ "index": 35
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 108,
+ 567,
+ 442,
+ 581
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 567,
+ 443,
+ 583
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 567,
+ 443,
+ 583
+ ],
+ "score": 1.0,
+ "content": "5 ASYMPTOTIC BEHAVIOR OF GCN ON ERDOS˝ – RENYI ´ GRAPH",
+ "type": "text"
+ }
+ ],
+ "index": 38
+ }
+ ],
+ "index": 38
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 592,
+ 505,
+ 703
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 592,
+ 505,
+ 605
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 592,
+ 505,
+ 605
+ ],
+ "score": 1.0,
+ "content": "Theorem 2 gives us a way to characterize the asymptotic behaviors of GCNs via the spectral distri-",
+ "type": "text"
+ }
+ ],
+ "index": 39
+ },
+ {
+ "bbox": [
+ 104,
+ 601,
+ 504,
+ 619
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 601,
+ 481,
+ 619
+ ],
+ "score": 1.0,
+ "content": "butions of the underlying graphs. To demonstrate this, we consider an Erdos – R ˝ enyi graph ´",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 481,
+ 604,
+ 504,
+ 617
+ ],
+ "score": 0.9,
+ "content": "G _ { N , p }",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 40
+ },
+ {
+ "bbox": [
+ 105,
+ 613,
+ 506,
+ 628
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 613,
+ 397,
+ 628
+ ],
+ "score": 1.0,
+ "content": "(Erdos & R ¨ enyi, 1959; Gilbert, 1959), which is a random graph that has ´",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 397,
+ 615,
+ 407,
+ 625
+ ],
+ "score": 0.83,
+ "content": "N",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 408,
+ 613,
+ 506,
+ 628
+ ],
+ "score": 1.0,
+ "content": "nodes and whose edges",
+ "type": "text"
+ }
+ ],
+ "index": 41
+ },
+ {
+ "bbox": [
+ 105,
+ 624,
+ 506,
+ 640
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 624,
+ 374,
+ 640
+ ],
+ "score": 1.0,
+ "content": "between two distinct nodes appear independently with probability",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 374,
+ 626,
+ 414,
+ 638
+ ],
+ "score": 0.91,
+ "content": "p \\in [ 0 , 1 ]",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 414,
+ 624,
+ 506,
+ 640
+ ],
+ "score": 1.0,
+ "content": ", as an example. First,",
+ "type": "text"
+ }
+ ],
+ "index": 42
+ },
+ {
+ "bbox": [
+ 105,
+ 636,
+ 505,
+ 650
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 636,
+ 236,
+ 650
+ ],
+ "score": 1.0,
+ "content": "consider a (non-random) graph",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 236,
+ 637,
+ 245,
+ 647
+ ],
+ "score": 0.79,
+ "content": "G",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 246,
+ 636,
+ 268,
+ 650
+ ],
+ "score": 1.0,
+ "content": "with",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 268,
+ 637,
+ 281,
+ 647
+ ],
+ "score": 0.74,
+ "content": "M",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 281,
+ 636,
+ 399,
+ 650
+ ],
+ "score": 1.0,
+ "content": "connected components. Let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 399,
+ 637,
+ 505,
+ 649
+ ],
+ "score": 0.89,
+ "content": "0 = \\tilde { \\mu } _ { 1 } = \\cdot \\cdot \\cdot = \\tilde { \\mu } _ { M } <",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 43
+ },
+ {
+ "bbox": [
+ 107,
+ 648,
+ 505,
+ 661
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 648,
+ 207,
+ 660
+ ],
+ "score": 0.92,
+ "content": "\\tilde { \\mu } _ { M + 1 } \\leq \\dots \\leq \\tilde { \\mu } _ { N } < 2",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 207,
+ 648,
+ 443,
+ 661
+ ],
+ "score": 1.0,
+ "content": "be eigenvalues of the augmented normalized Laplacian of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 443,
+ 649,
+ 452,
+ 658
+ ],
+ "score": 0.8,
+ "content": "G",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 453,
+ 648,
+ 505,
+ 661
+ ],
+ "score": 1.0,
+ "content": "(see, Propo-",
+ "type": "text"
+ }
+ ],
+ "index": 44
+ },
+ {
+ "bbox": [
+ 104,
+ 658,
+ 506,
+ 672
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 658,
+ 174,
+ 672
+ ],
+ "score": 1.0,
+ "content": "sition 1) and set",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 174,
+ 659,
+ 330,
+ 671
+ ],
+ "score": 0.9,
+ "content": "\\begin{array} { r } { \\lambda : = \\operatorname* { m i n } _ { m = M + 1 , \\ldots , N } | 1 - \\tilde { \\mu } _ { m } | ( < 1 ) } \\end{array}",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 330,
+ 658,
+ 506,
+ 672
+ ],
+ "score": 1.0,
+ "content": ". By Theorem 2, the output of GCN “loses",
+ "type": "text"
+ }
+ ],
+ "index": 45
+ },
+ {
+ "bbox": [
+ 106,
+ 669,
+ 505,
+ 682
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 669,
+ 505,
+ 682
+ ],
+ "score": 1.0,
+ "content": "information” as the layer size goes to infinity when the largest singular values of weights are strictly",
+ "type": "text"
+ }
+ ],
+ "index": 46
+ },
+ {
+ "bbox": [
+ 104,
+ 679,
+ 506,
+ 694
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 679,
+ 159,
+ 694
+ ],
+ "score": 1.0,
+ "content": "smaller than",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 159,
+ 680,
+ 176,
+ 691
+ ],
+ "score": 0.91,
+ "content": "\\lambda ^ { - 1 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 177,
+ 679,
+ 367,
+ 694
+ ],
+ "score": 1.0,
+ "content": ". Therefore, the closer the positive eigenvalues",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 368,
+ 683,
+ 382,
+ 693
+ ],
+ "score": 0.86,
+ "content": "\\mu _ { m }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 383,
+ 679,
+ 506,
+ 694
+ ],
+ "score": 1.0,
+ "content": "are to 1, the broader range of",
+ "type": "text"
+ }
+ ],
+ "index": 47
+ },
+ {
+ "bbox": [
+ 106,
+ 692,
+ 290,
+ 704
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 692,
+ 290,
+ 704
+ ],
+ "score": 1.0,
+ "content": "GCNs satisfies the assumption of Theorem 2.",
+ "type": "text"
+ }
+ ],
+ "index": 48
+ }
+ ],
+ "index": 43.5
+ }
+ ],
+ "page_idx": 4,
+ "page_size": [
+ 612,
+ 792
+ ],
+ "discarded_blocks": [
+ {
+ "type": "discarded",
+ "bbox": [
+ 108,
+ 711,
+ 504,
+ 732
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 118,
+ 709,
+ 506,
+ 724
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 118,
+ 709,
+ 416,
+ 724
+ ],
+ "score": 1.0,
+ "content": "2Following the original paper (Kipf & Welling, 2017), we use one-layer MLPs (i.e.,",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 416,
+ 712,
+ 445,
+ 721
+ ],
+ "score": 0.9,
+ "content": "H _ { l } = 1",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 445,
+ 709,
+ 468,
+ 724
+ ],
+ "score": 1.0,
+ "content": "for all",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 469,
+ 712,
+ 499,
+ 722
+ ],
+ "score": 0.86,
+ "content": "l \\in \\mathbb { N } .",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 499,
+ 709,
+ 506,
+ 724
+ ],
+ "score": 1.0,
+ "content": ").",
+ "type": "text"
+ }
+ ]
+ },
+ {
+ "bbox": [
+ 106,
+ 720,
+ 288,
+ 734
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 720,
+ 288,
+ 734
+ ],
+ "score": 1.0,
+ "content": "However, our result holds for the multi-layer case",
+ "type": "text"
+ }
+ ]
+ }
+ ]
+ },
+ {
+ "type": "discarded",
+ "bbox": [
+ 108,
+ 27,
+ 293,
+ 37
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 26,
+ 294,
+ 38
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 26,
+ 294,
+ 38
+ ],
+ "score": 1.0,
+ "content": "Published as a conference paper at ICLR 2020",
+ "type": "text"
+ }
+ ]
+ }
+ ]
+ },
+ {
+ "type": "discarded",
+ "bbox": [
+ 302,
+ 751,
+ 308,
+ 760
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 302,
+ 750,
+ 309,
+ 763
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 302,
+ 750,
+ 309,
+ 763
+ ],
+ "score": 1.0,
+ "content": "5",
+ "type": "text"
+ }
+ ]
+ }
+ ]
+ }
+ ],
+ "para_blocks": [
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 82,
+ 505,
+ 127
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 82,
+ 505,
+ 95
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 83,
+ 122,
+ 95
+ ],
+ "score": 1.0,
+ "content": "Let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 122,
+ 82,
+ 171,
+ 95
+ ],
+ "score": 0.93,
+ "content": "G = ( V , E )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 171,
+ 83,
+ 291,
+ 95
+ ],
+ "score": 1.0,
+ "content": "be an undirected graph where",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 291,
+ 83,
+ 300,
+ 92
+ ],
+ "score": 0.79,
+ "content": "V",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 301,
+ 83,
+ 383,
+ 95
+ ],
+ "score": 1.0,
+ "content": "is a set of nodes and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 383,
+ 83,
+ 392,
+ 92
+ ],
+ "score": 0.82,
+ "content": "E",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 393,
+ 83,
+ 505,
+ 95
+ ],
+ "score": 1.0,
+ "content": "is a set of edges. We denote",
+ "type": "text"
+ }
+ ],
+ "index": 0
+ },
+ {
+ "bbox": [
+ 105,
+ 93,
+ 506,
+ 106
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 93,
+ 199,
+ 106
+ ],
+ "score": 1.0,
+ "content": "the number of nodes in",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 199,
+ 94,
+ 208,
+ 104
+ ],
+ "score": 0.79,
+ "content": "G",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 209,
+ 93,
+ 221,
+ 106
+ ],
+ "score": 1.0,
+ "content": "by",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 221,
+ 94,
+ 258,
+ 105
+ ],
+ "score": 0.92,
+ "content": "N = | V |",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 258,
+ 93,
+ 308,
+ 106
+ ],
+ "score": 1.0,
+ "content": "and identify",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 309,
+ 94,
+ 318,
+ 104
+ ],
+ "score": 0.74,
+ "content": "V",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 318,
+ 93,
+ 339,
+ 106
+ ],
+ "score": 1.0,
+ "content": "with",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 339,
+ 93,
+ 354,
+ 106
+ ],
+ "score": 0.91,
+ "content": "[ N ]",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 355,
+ 93,
+ 438,
+ 106
+ ],
+ "score": 1.0,
+ "content": "by fixing an order of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 438,
+ 94,
+ 447,
+ 104
+ ],
+ "score": 0.67,
+ "content": "V",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 447,
+ 93,
+ 506,
+ 106
+ ],
+ "score": 1.0,
+ "content": ". We associate",
+ "type": "text"
+ }
+ ],
+ "index": 1
+ },
+ {
+ "bbox": [
+ 104,
+ 104,
+ 505,
+ 117
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 104,
+ 113,
+ 117
+ ],
+ "score": 1.0,
+ "content": "a",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 113,
+ 105,
+ 123,
+ 114
+ ],
+ "score": 0.81,
+ "content": "C",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 123,
+ 104,
+ 257,
+ 117
+ ],
+ "score": 1.0,
+ "content": "dimensional signal to each node.",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 257,
+ 105,
+ 267,
+ 114
+ ],
+ "score": 0.83,
+ "content": "X",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 268,
+ 104,
+ 505,
+ 117
+ ],
+ "score": 1.0,
+ "content": "in the previous section corresponds to concatenation of the",
+ "type": "text"
+ }
+ ],
+ "index": 2
+ },
+ {
+ "bbox": [
+ 105,
+ 115,
+ 479,
+ 128
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 115,
+ 281,
+ 128
+ ],
+ "score": 1.0,
+ "content": "signals. GCNs iteratively update signals on",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 281,
+ 116,
+ 290,
+ 125
+ ],
+ "score": 0.78,
+ "content": "V",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 291,
+ 115,
+ 479,
+ 128
+ ],
+ "score": 1.0,
+ "content": "using the connection information and weights.",
+ "type": "text"
+ }
+ ],
+ "index": 3
+ }
+ ],
+ "index": 1.5,
+ "bbox_fs": [
+ 104,
+ 82,
+ 506,
+ 128
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 131,
+ 505,
+ 228
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 104,
+ 128,
+ 505,
+ 147
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 128,
+ 123,
+ 147
+ ],
+ "score": 1.0,
+ "content": "Let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 123,
+ 131,
+ 270,
+ 145
+ ],
+ "score": 0.93,
+ "content": "A : = ( \\mathbf { 1 } _ { \\{ ( i , j ) \\in E \\} } ) _ { i , j \\in [ N ] } \\ \\in \\ \\mathbb { R } ^ { N \\times N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 270,
+ 128,
+ 391,
+ 147
+ ],
+ "score": 1.0,
+ "content": "be the adjacency matrix and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 391,
+ 132,
+ 505,
+ 145
+ ],
+ "score": 0.91,
+ "content": "D : = \\mathrm { d i a g } ( \\deg ( i ) _ { i \\in [ N ] } ) \\ \\in",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 4
+ },
+ {
+ "bbox": [
+ 106,
+ 141,
+ 506,
+ 160
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 144,
+ 135,
+ 155
+ ],
+ "score": 0.89,
+ "content": "\\mathbb { R } ^ { N \\times N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 136,
+ 141,
+ 235,
+ 160
+ ],
+ "score": 1.0,
+ "content": "be the degree matrix of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 236,
+ 146,
+ 245,
+ 155
+ ],
+ "score": 0.82,
+ "content": "G",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 245,
+ 141,
+ 273,
+ 160
+ ],
+ "score": 1.0,
+ "content": "where",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 274,
+ 144,
+ 415,
+ 157
+ ],
+ "score": 0.93,
+ "content": "\\deg ( i ) : = | \\{ j \\in V \\mid ( i , j ) \\in E \\} |",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 416,
+ 141,
+ 506,
+ 160
+ ],
+ "score": 1.0,
+ "content": "is the degree of node",
+ "type": "text"
+ }
+ ],
+ "index": 5
+ },
+ {
+ "bbox": [
+ 106,
+ 156,
+ 505,
+ 171
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 158,
+ 111,
+ 168
+ ],
+ "score": 0.61,
+ "content": "i",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 111,
+ 156,
+ 135,
+ 171
+ ],
+ "score": 1.0,
+ "content": ". Let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 135,
+ 156,
+ 196,
+ 169
+ ],
+ "score": 0.88,
+ "content": "{ \\tilde { A } } : = A + I _ { N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 197,
+ 156,
+ 201,
+ 171
+ ],
+ "score": 1.0,
+ "content": ",",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 202,
+ 156,
+ 265,
+ 169
+ ],
+ "score": 0.9,
+ "content": "{ \\tilde { D } } : = D + I _ { N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 265,
+ 156,
+ 447,
+ 171
+ ],
+ "score": 1.0,
+ "content": "be the adjacent and degree matrix of graph",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 448,
+ 158,
+ 457,
+ 168
+ ],
+ "score": 0.8,
+ "content": "G",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 457,
+ 156,
+ 505,
+ 171
+ ],
+ "score": 1.0,
+ "content": "augmented",
+ "type": "text"
+ }
+ ],
+ "index": 6
+ },
+ {
+ "bbox": [
+ 106,
+ 169,
+ 505,
+ 183
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 169,
+ 456,
+ 183
+ ],
+ "score": 1.0,
+ "content": "with self-loops. We define the augmented normalized Laplacian (Wu et al., 2019a) of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 456,
+ 170,
+ 466,
+ 181
+ ],
+ "score": 0.81,
+ "content": "G",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 466,
+ 169,
+ 480,
+ 183
+ ],
+ "score": 1.0,
+ "content": "by",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 480,
+ 169,
+ 505,
+ 182
+ ],
+ "score": 0.87,
+ "content": "\\tilde { \\Delta } : =",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 7
+ },
+ {
+ "bbox": [
+ 107,
+ 181,
+ 505,
+ 196
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 181,
+ 178,
+ 194
+ ],
+ "score": 0.92,
+ "content": "I _ { N } - \\tilde { D } ^ { - \\frac { 1 } { 2 } } \\tilde { A } \\tilde { D } ^ { - \\frac { 1 } { 2 } }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 178,
+ 181,
+ 209,
+ 196
+ ],
+ "score": 1.0,
+ "content": "and set",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 210,
+ 181,
+ 264,
+ 194
+ ],
+ "score": 0.92,
+ "content": "P : = I _ { N } - \\tilde { \\Delta }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 265,
+ 181,
+ 284,
+ 196
+ ],
+ "score": 1.0,
+ "content": ". Let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 284,
+ 182,
+ 330,
+ 194
+ ],
+ "score": 0.88,
+ "content": "L , C \\in \\mathbb { N } _ { + }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 330,
+ 181,
+ 505,
+ 196
+ ],
+ "score": 1.0,
+ "content": "be the layer and channel sizes, respectively.",
+ "type": "text"
+ }
+ ],
+ "index": 8
+ },
+ {
+ "bbox": [
+ 104,
+ 191,
+ 506,
+ 208
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 191,
+ 156,
+ 208
+ ],
+ "score": 1.0,
+ "content": "For weights",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 156,
+ 193,
+ 207,
+ 204
+ ],
+ "score": 0.8,
+ "content": "W _ { l } \\in \\mathbb { R } ^ { C \\times C }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 211,
+ 194,
+ 243,
+ 205
+ ],
+ "score": 0.6,
+ "content": "( l \\in [ L ] )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 244,
+ 191,
+ 295,
+ 208
+ ],
+ "score": 1.0,
+ "content": ", we define a",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 295,
+ 194,
+ 322,
+ 204
+ ],
+ "score": 0.81,
+ "content": "\\mathrm { G C N } ^ { 2 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 322,
+ 191,
+ 386,
+ 208
+ ],
+ "score": 1.0,
+ "content": "associated with",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 386,
+ 194,
+ 396,
+ 204
+ ],
+ "score": 0.81,
+ "content": "G",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 396,
+ 191,
+ 408,
+ 208
+ ],
+ "score": 1.0,
+ "content": "by",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 409,
+ 194,
+ 477,
+ 205
+ ],
+ "score": 0.93,
+ "content": "f = f _ { L } \\circ \\cdot \\cdot \\cdot \\circ f _ { 1 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 477,
+ 191,
+ 506,
+ 208
+ ],
+ "score": 1.0,
+ "content": "where",
+ "type": "text"
+ }
+ ],
+ "index": 9
+ },
+ {
+ "bbox": [
+ 107,
+ 201,
+ 506,
+ 219
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 204,
+ 201,
+ 217
+ ],
+ "score": 0.91,
+ "content": "f _ { l } : \\mathbb { R } ^ { \\breve { N } \\times C } \\mathbb { R } ^ { N \\times C }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 201,
+ 201,
+ 261,
+ 219
+ ],
+ "score": 1.0,
+ "content": "is defined by",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 261,
+ 205,
+ 349,
+ 217
+ ],
+ "score": 0.91,
+ "content": "f _ { l } ( X ) : = \\sigma ( P X W _ { l } )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 349,
+ 201,
+ 506,
+ 219
+ ],
+ "score": 1.0,
+ "content": ". We are interested in the asymptotic",
+ "type": "text"
+ }
+ ],
+ "index": 10
+ },
+ {
+ "bbox": [
+ 104,
+ 215,
+ 318,
+ 230
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 215,
+ 197,
+ 230
+ ],
+ "score": 1.0,
+ "content": "behavior of the output",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 197,
+ 216,
+ 219,
+ 227
+ ],
+ "score": 0.91,
+ "content": "X ^ { ( L ) }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 219,
+ 215,
+ 280,
+ 230
+ ],
+ "score": 1.0,
+ "content": "of the GCN as",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 281,
+ 218,
+ 314,
+ 227
+ ],
+ "score": 0.88,
+ "content": "L \\to \\infty",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 314,
+ 215,
+ 318,
+ 230
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 11
+ }
+ ],
+ "index": 7.5,
+ "bbox_fs": [
+ 104,
+ 128,
+ 506,
+ 230
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 234,
+ 505,
+ 279
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 234,
+ 506,
+ 246
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 234,
+ 144,
+ 246
+ ],
+ "score": 1.0,
+ "content": "Suppose",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 144,
+ 234,
+ 153,
+ 244
+ ],
+ "score": 0.8,
+ "content": "G",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 154,
+ 234,
+ 172,
+ 246
+ ],
+ "score": 1.0,
+ "content": "has",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 172,
+ 235,
+ 184,
+ 244
+ ],
+ "score": 0.81,
+ "content": "M",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 185,
+ 234,
+ 313,
+ 246
+ ],
+ "score": 1.0,
+ "content": "connected components and let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 313,
+ 234,
+ 401,
+ 245
+ ],
+ "score": 0.91,
+ "content": "V = V _ { 1 } \\sqcup \\cdots \\sqcup V _ { M }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 402,
+ 234,
+ 506,
+ 246
+ ],
+ "score": 1.0,
+ "content": "be the decomposition of",
+ "type": "text"
+ }
+ ],
+ "index": 12
+ },
+ {
+ "bbox": [
+ 106,
+ 245,
+ 505,
+ 257
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 245,
+ 158,
+ 257
+ ],
+ "score": 1.0,
+ "content": "the node set",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 159,
+ 245,
+ 168,
+ 255
+ ],
+ "score": 0.76,
+ "content": "V",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 168,
+ 245,
+ 439,
+ 257
+ ],
+ "score": 1.0,
+ "content": "into connected components. We denote an indicator vector of the",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 439,
+ 246,
+ 449,
+ 255
+ ],
+ "score": 0.79,
+ "content": "m",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 450,
+ 245,
+ 505,
+ 257
+ ],
+ "score": 1.0,
+ "content": "-th connected",
+ "type": "text"
+ }
+ ],
+ "index": 13
+ },
+ {
+ "bbox": [
+ 104,
+ 255,
+ 506,
+ 269
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 255,
+ 166,
+ 269
+ ],
+ "score": 1.0,
+ "content": "component by",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 166,
+ 256,
+ 290,
+ 269
+ ],
+ "score": 0.89,
+ "content": "u _ { m } : = ( \\mathbf { 1 } _ { \\{ n \\in V _ { m } \\} } ) _ { n \\in [ N ] } \\in \\mathbb { R } ^ { N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 291,
+ 255,
+ 506,
+ 269
+ ],
+ "score": 1.0,
+ "content": ". The following proposition shows that GCN satisfies",
+ "type": "text"
+ }
+ ],
+ "index": 14
+ },
+ {
+ "bbox": [
+ 106,
+ 267,
+ 338,
+ 279
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 267,
+ 338,
+ 279
+ ],
+ "score": 1.0,
+ "content": "the assumption of Corollay 3 (see Appendix B for proof).",
+ "type": "text"
+ }
+ ],
+ "index": 15
+ }
+ ],
+ "index": 13.5,
+ "bbox_fs": [
+ 104,
+ 234,
+ 506,
+ 279
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 281,
+ 505,
+ 328
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 279,
+ 506,
+ 295
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 279,
+ 187,
+ 295
+ ],
+ "score": 1.0,
+ "content": "Proposition 1. Let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 188,
+ 281,
+ 256,
+ 293
+ ],
+ "score": 0.9,
+ "content": "\\lambda _ { 1 } \\leq \\cdots \\leq \\lambda _ { N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 257,
+ 279,
+ 343,
+ 295
+ ],
+ "score": 1.0,
+ "content": "be the eigenvalue of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 344,
+ 282,
+ 352,
+ 291
+ ],
+ "score": 0.76,
+ "content": "P",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 353,
+ 279,
+ 506,
+ 295
+ ],
+ "score": 1.0,
+ "content": "sorted in ascending order. Then, we",
+ "type": "text"
+ }
+ ],
+ "index": 16
+ },
+ {
+ "bbox": [
+ 105,
+ 291,
+ 505,
+ 307
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 291,
+ 129,
+ 307
+ ],
+ "score": 1.0,
+ "content": "have",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 129,
+ 293,
+ 171,
+ 304
+ ],
+ "score": 0.81,
+ "content": "- 1 < \\lambda _ { 1 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 171,
+ 291,
+ 177,
+ 307
+ ],
+ "score": 1.0,
+ "content": ",",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 177,
+ 294,
+ 229,
+ 304
+ ],
+ "score": 0.85,
+ "content": "\\lambda _ { N - M } ~ < ~ 1",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 229,
+ 291,
+ 254,
+ 307
+ ],
+ "score": 1.0,
+ "content": ", and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 254,
+ 293,
+ 376,
+ 304
+ ],
+ "score": 0.9,
+ "content": "\\lambda _ { N - M + 1 } = \\cdot \\cdot \\cdot = \\lambda _ { N } = 1",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 377,
+ 291,
+ 480,
+ 307
+ ],
+ "score": 1.0,
+ "content": ". In particular, we have",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 481,
+ 293,
+ 505,
+ 304
+ ],
+ "score": 0.84,
+ "content": "\\lambda : =",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 17
+ },
+ {
+ "bbox": [
+ 106,
+ 304,
+ 506,
+ 320
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 306,
+ 215,
+ 318
+ ],
+ "score": 0.88,
+ "content": "\\begin{array} { r } { \\operatorname* { m a x } _ { n = 1 , \\dots , N - M } | \\lambda _ { n } | < 1 } \\end{array}",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 215,
+ 304,
+ 257,
+ 320
+ ],
+ "score": 1.0,
+ "content": ". Further,",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 257,
+ 304,
+ 317,
+ 317
+ ],
+ "score": 0.89,
+ "content": "e _ { m } : = \\tilde { D } ^ { \\frac { 1 } { 2 } } u _ { m }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 317,
+ 304,
+ 333,
+ 320
+ ],
+ "score": 1.0,
+ "content": "for",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 334,
+ 305,
+ 375,
+ 318
+ ],
+ "score": 0.9,
+ "content": "m \\in [ M ]",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 375,
+ 304,
+ 506,
+ 320
+ ],
+ "score": 1.0,
+ "content": "are the basis of the eigenspace",
+ "type": "text"
+ }
+ ],
+ "index": 18
+ },
+ {
+ "bbox": [
+ 106,
+ 317,
+ 241,
+ 329
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 317,
+ 241,
+ 329
+ ],
+ "score": 1.0,
+ "content": "associated with the eigenvalue 1.",
+ "type": "text"
+ }
+ ],
+ "index": 19
+ }
+ ],
+ "index": 17.5,
+ "bbox_fs": [
+ 105,
+ 279,
+ 506,
+ 329
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 108,
+ 330,
+ 504,
+ 356
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 327,
+ 505,
+ 345
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 327,
+ 256,
+ 345
+ ],
+ "score": 1.0,
+ "content": "Theorem 2. For any initial value",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 256,
+ 330,
+ 276,
+ 342
+ ],
+ "score": 0.9,
+ "content": "X ^ { ( 0 ) }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 277,
+ 327,
+ 387,
+ 345
+ ],
+ "score": 1.0,
+ "content": ", the output of l-th layer",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 388,
+ 330,
+ 407,
+ 342
+ ],
+ "score": 0.9,
+ "content": "X ^ { ( l ) }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 407,
+ 327,
+ 447,
+ 345
+ ],
+ "score": 1.0,
+ "content": "satisfies",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 447,
+ 330,
+ 505,
+ 343
+ ],
+ "score": 0.88,
+ "content": "d _ { \\mathcal { M } } ( X ^ { ( l ) } ) ~ \\leq",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 20
+ },
+ {
+ "bbox": [
+ 107,
+ 341,
+ 448,
+ 357
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 342,
+ 171,
+ 356
+ ],
+ "score": 0.92,
+ "content": "( s \\lambda ) ^ { l } d _ { \\mathcal { M } } ( X ^ { ( 0 ) } )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 171,
+ 341,
+ 231,
+ 357
+ ],
+ "score": 1.0,
+ "content": ". In particular,",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 231,
+ 342,
+ 273,
+ 356
+ ],
+ "score": 0.94,
+ "content": "d _ { \\mathcal { M } } ( { \\boldsymbol { X } } ^ { ( l ) } )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 273,
+ 341,
+ 414,
+ 357
+ ],
+ "score": 1.0,
+ "content": "exponentially converges to 0 when",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 414,
+ 344,
+ 443,
+ 354
+ ],
+ "score": 0.88,
+ "content": "s \\lambda < 1",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 444,
+ 341,
+ 448,
+ 357
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 21
+ }
+ ],
+ "index": 20.5,
+ "bbox_fs": [
+ 105,
+ 327,
+ 505,
+ 357
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 364,
+ 505,
+ 455
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 363,
+ 506,
+ 377
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 363,
+ 506,
+ 377
+ ],
+ "score": 1.0,
+ "content": "In the context of node classification tasks, we can interpret this corollary as the “information loss” of",
+ "type": "text"
+ }
+ ],
+ "index": 22
+ },
+ {
+ "bbox": [
+ 105,
+ 375,
+ 505,
+ 388
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 375,
+ 283,
+ 388
+ ],
+ "score": 1.0,
+ "content": "GCNs in the limit of infinite layers. For any",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 283,
+ 376,
+ 317,
+ 386
+ ],
+ "score": 0.91,
+ "content": "X \\in { \\mathcal { M } }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 318,
+ 375,
+ 372,
+ 388
+ ],
+ "score": 1.0,
+ "content": ", if two nodes",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 372,
+ 376,
+ 407,
+ 387
+ ],
+ "score": 0.91,
+ "content": "i , j \\in V",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 407,
+ 375,
+ 505,
+ 388
+ ],
+ "score": 1.0,
+ "content": "are in a same connected",
+ "type": "text"
+ }
+ ],
+ "index": 23
+ },
+ {
+ "bbox": [
+ 105,
+ 387,
+ 504,
+ 398
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 387,
+ 388,
+ 398
+ ],
+ "score": 1.0,
+ "content": "component and their degrees are identical, then, the column vectors of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 388,
+ 387,
+ 398,
+ 396
+ ],
+ "score": 0.85,
+ "content": "X",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 398,
+ 387,
+ 498,
+ 398
+ ],
+ "score": 1.0,
+ "content": "that correspond to nodes",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 499,
+ 387,
+ 504,
+ 396
+ ],
+ "score": 0.67,
+ "content": "i",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 24
+ },
+ {
+ "bbox": [
+ 105,
+ 396,
+ 505,
+ 410
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 396,
+ 123,
+ 410
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 123,
+ 398,
+ 129,
+ 408
+ ],
+ "score": 0.82,
+ "content": "j",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 130,
+ 396,
+ 401,
+ 410
+ ],
+ "score": 1.0,
+ "content": "are identical. It means that we cannot distinguish these nodes using",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 402,
+ 398,
+ 412,
+ 407
+ ],
+ "score": 0.76,
+ "content": "X",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 412,
+ 396,
+ 470,
+ 410
+ ],
+ "score": 1.0,
+ "content": ". In this sense,",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 470,
+ 397,
+ 484,
+ 407
+ ],
+ "score": 0.78,
+ "content": "\\mathcal { M }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 484,
+ 396,
+ 505,
+ 410
+ ],
+ "score": 1.0,
+ "content": "only",
+ "type": "text"
+ }
+ ],
+ "index": 25
+ },
+ {
+ "bbox": [
+ 105,
+ 408,
+ 505,
+ 421
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 408,
+ 505,
+ 421
+ ],
+ "score": 1.0,
+ "content": "has information about connected components and node degrees and we can interpret this theorem",
+ "type": "text"
+ }
+ ],
+ "index": 26
+ },
+ {
+ "bbox": [
+ 105,
+ 419,
+ 505,
+ 432
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 419,
+ 505,
+ 432
+ ],
+ "score": 1.0,
+ "content": "as the exponential information loss of GCNs in terms of the layer size. Similarly to the discussion",
+ "type": "text"
+ }
+ ],
+ "index": 27
+ },
+ {
+ "bbox": [
+ 105,
+ 430,
+ 505,
+ 444
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 430,
+ 201,
+ 444
+ ],
+ "score": 1.0,
+ "content": "in the previous section,",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 201,
+ 430,
+ 221,
+ 442
+ ],
+ "score": 0.9,
+ "content": "X ^ { ( l ) }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 221,
+ 430,
+ 392,
+ 444
+ ],
+ "score": 1.0,
+ "content": "converges to the trivial fixed point 0 when",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 393,
+ 432,
+ 417,
+ 442
+ ],
+ "score": 0.89,
+ "content": "s < 1",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 418,
+ 430,
+ 465,
+ 444
+ ],
+ "score": 1.0,
+ "content": "(remember",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 465,
+ 431,
+ 498,
+ 442
+ ],
+ "score": 0.88,
+ "content": "\\lambda _ { N } = 1",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 499,
+ 430,
+ 505,
+ 444
+ ],
+ "score": 1.0,
+ "content": ").",
+ "type": "text"
+ }
+ ],
+ "index": 28
+ },
+ {
+ "bbox": [
+ 105,
+ 442,
+ 505,
+ 455
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 442,
+ 244,
+ 455
+ ],
+ "score": 1.0,
+ "content": "An interesting point is that even if",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 245,
+ 443,
+ 269,
+ 454
+ ],
+ "score": 0.87,
+ "content": "s \\geq 1",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 269,
+ 442,
+ 272,
+ 455
+ ],
+ "score": 1.0,
+ "content": ",",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 273,
+ 442,
+ 292,
+ 453
+ ],
+ "score": 0.88,
+ "content": "X ^ { ( l ) }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 292,
+ 442,
+ 465,
+ 455
+ ],
+ "score": 1.0,
+ "content": "can suffer from this information loss when",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 466,
+ 443,
+ 501,
+ 453
+ ],
+ "score": 0.9,
+ "content": "s < \\lambda ^ { - 1 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 501,
+ 442,
+ 505,
+ 455
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 29
+ }
+ ],
+ "index": 25.5,
+ "bbox_fs": [
+ 105,
+ 363,
+ 506,
+ 455
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 108,
+ 460,
+ 503,
+ 493
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 460,
+ 504,
+ 472
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 460,
+ 192,
+ 472
+ ],
+ "score": 1.0,
+ "content": "We note that the rate",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 192,
+ 461,
+ 204,
+ 470
+ ],
+ "score": 0.84,
+ "content": "s \\lambda",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 205,
+ 460,
+ 504,
+ 472
+ ],
+ "score": 1.0,
+ "content": "in Theorem 2 depends on the spectra of the augmented normalized Lapla-",
+ "type": "text"
+ }
+ ],
+ "index": 30
+ },
+ {
+ "bbox": [
+ 106,
+ 471,
+ 504,
+ 483
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 471,
+ 378,
+ 483
+ ],
+ "score": 1.0,
+ "content": "cian, which is determined by the topology of the underlying graph",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 378,
+ 471,
+ 387,
+ 481
+ ],
+ "score": 0.7,
+ "content": "G",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 387,
+ 471,
+ 504,
+ 483
+ ],
+ "score": 1.0,
+ "content": ". Hence, our result explicitly",
+ "type": "text"
+ }
+ ],
+ "index": 31
+ },
+ {
+ "bbox": [
+ 106,
+ 482,
+ 450,
+ 495
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 482,
+ 450,
+ 495
+ ],
+ "score": 1.0,
+ "content": "relates the topological information of graphs and asymptotic behaviors of graph NNs.",
+ "type": "text"
+ }
+ ],
+ "index": 32
+ }
+ ],
+ "index": 31,
+ "bbox_fs": [
+ 106,
+ 460,
+ 504,
+ 495
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 496,
+ 505,
+ 552
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 497,
+ 504,
+ 509
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 497,
+ 504,
+ 509
+ ],
+ "score": 1.0,
+ "content": "Remark 2. The old preprint (version 2) of Luan et al. (2019) formulated a theorem that explains the",
+ "type": "text"
+ }
+ ],
+ "index": 33
+ },
+ {
+ "bbox": [
+ 106,
+ 508,
+ 504,
+ 520
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 508,
+ 504,
+ 520
+ ],
+ "score": 1.0,
+ "content": "over-smoothing of non-linear GNNs. Specifically, it claimed that if a graph does not have a bipartite",
+ "type": "text"
+ }
+ ],
+ "index": 34
+ },
+ {
+ "bbox": [
+ 106,
+ 518,
+ 505,
+ 532
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 518,
+ 505,
+ 532
+ ],
+ "score": 1.0,
+ "content": "component and the input distribution is continuous, the rank of the output of a GCN converges to the",
+ "type": "text"
+ }
+ ],
+ "index": 35
+ },
+ {
+ "bbox": [
+ 105,
+ 530,
+ 506,
+ 543
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 530,
+ 506,
+ 543
+ ],
+ "score": 1.0,
+ "content": "number of connected components of the underlying graph as the layer size goes to infinity almost",
+ "type": "text"
+ }
+ ],
+ "index": 36
+ },
+ {
+ "bbox": [
+ 105,
+ 541,
+ 448,
+ 554
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 541,
+ 437,
+ 554
+ ],
+ "score": 1.0,
+ "content": "surely. However, it is not true in general as we give a counterexample in Appendix",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 437,
+ 543,
+ 444,
+ 551
+ ],
+ "score": 0.53,
+ "content": "C",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 444,
+ 541,
+ 448,
+ 554
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 37
+ }
+ ],
+ "index": 35,
+ "bbox_fs": [
+ 105,
+ 497,
+ 506,
+ 554
+ ]
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 108,
+ 567,
+ 442,
+ 581
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 567,
+ 443,
+ 583
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 567,
+ 443,
+ 583
+ ],
+ "score": 1.0,
+ "content": "5 ASYMPTOTIC BEHAVIOR OF GCN ON ERDOS˝ – RENYI ´ GRAPH",
+ "type": "text"
+ }
+ ],
+ "index": 38
+ }
+ ],
+ "index": 38
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 592,
+ 505,
+ 703
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 592,
+ 505,
+ 605
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 592,
+ 505,
+ 605
+ ],
+ "score": 1.0,
+ "content": "Theorem 2 gives us a way to characterize the asymptotic behaviors of GCNs via the spectral distri-",
+ "type": "text"
+ }
+ ],
+ "index": 39
+ },
+ {
+ "bbox": [
+ 104,
+ 601,
+ 504,
+ 619
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 601,
+ 481,
+ 619
+ ],
+ "score": 1.0,
+ "content": "butions of the underlying graphs. To demonstrate this, we consider an Erdos – R ˝ enyi graph ´",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 481,
+ 604,
+ 504,
+ 617
+ ],
+ "score": 0.9,
+ "content": "G _ { N , p }",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 40
+ },
+ {
+ "bbox": [
+ 105,
+ 613,
+ 506,
+ 628
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 613,
+ 397,
+ 628
+ ],
+ "score": 1.0,
+ "content": "(Erdos & R ¨ enyi, 1959; Gilbert, 1959), which is a random graph that has ´",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 397,
+ 615,
+ 407,
+ 625
+ ],
+ "score": 0.83,
+ "content": "N",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 408,
+ 613,
+ 506,
+ 628
+ ],
+ "score": 1.0,
+ "content": "nodes and whose edges",
+ "type": "text"
+ }
+ ],
+ "index": 41
+ },
+ {
+ "bbox": [
+ 105,
+ 624,
+ 506,
+ 640
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 624,
+ 374,
+ 640
+ ],
+ "score": 1.0,
+ "content": "between two distinct nodes appear independently with probability",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 374,
+ 626,
+ 414,
+ 638
+ ],
+ "score": 0.91,
+ "content": "p \\in [ 0 , 1 ]",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 414,
+ 624,
+ 506,
+ 640
+ ],
+ "score": 1.0,
+ "content": ", as an example. First,",
+ "type": "text"
+ }
+ ],
+ "index": 42
+ },
+ {
+ "bbox": [
+ 105,
+ 636,
+ 505,
+ 650
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 636,
+ 236,
+ 650
+ ],
+ "score": 1.0,
+ "content": "consider a (non-random) graph",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 236,
+ 637,
+ 245,
+ 647
+ ],
+ "score": 0.79,
+ "content": "G",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 246,
+ 636,
+ 268,
+ 650
+ ],
+ "score": 1.0,
+ "content": "with",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 268,
+ 637,
+ 281,
+ 647
+ ],
+ "score": 0.74,
+ "content": "M",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 281,
+ 636,
+ 399,
+ 650
+ ],
+ "score": 1.0,
+ "content": "connected components. Let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 399,
+ 637,
+ 505,
+ 649
+ ],
+ "score": 0.89,
+ "content": "0 = \\tilde { \\mu } _ { 1 } = \\cdot \\cdot \\cdot = \\tilde { \\mu } _ { M } <",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 43
+ },
+ {
+ "bbox": [
+ 107,
+ 648,
+ 505,
+ 661
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 648,
+ 207,
+ 660
+ ],
+ "score": 0.92,
+ "content": "\\tilde { \\mu } _ { M + 1 } \\leq \\dots \\leq \\tilde { \\mu } _ { N } < 2",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 207,
+ 648,
+ 443,
+ 661
+ ],
+ "score": 1.0,
+ "content": "be eigenvalues of the augmented normalized Laplacian of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 443,
+ 649,
+ 452,
+ 658
+ ],
+ "score": 0.8,
+ "content": "G",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 453,
+ 648,
+ 505,
+ 661
+ ],
+ "score": 1.0,
+ "content": "(see, Propo-",
+ "type": "text"
+ }
+ ],
+ "index": 44
+ },
+ {
+ "bbox": [
+ 104,
+ 658,
+ 506,
+ 672
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 658,
+ 174,
+ 672
+ ],
+ "score": 1.0,
+ "content": "sition 1) and set",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 174,
+ 659,
+ 330,
+ 671
+ ],
+ "score": 0.9,
+ "content": "\\begin{array} { r } { \\lambda : = \\operatorname* { m i n } _ { m = M + 1 , \\ldots , N } | 1 - \\tilde { \\mu } _ { m } | ( < 1 ) } \\end{array}",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 330,
+ 658,
+ 506,
+ 672
+ ],
+ "score": 1.0,
+ "content": ". By Theorem 2, the output of GCN “loses",
+ "type": "text"
+ }
+ ],
+ "index": 45
+ },
+ {
+ "bbox": [
+ 106,
+ 669,
+ 505,
+ 682
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 669,
+ 505,
+ 682
+ ],
+ "score": 1.0,
+ "content": "information” as the layer size goes to infinity when the largest singular values of weights are strictly",
+ "type": "text"
+ }
+ ],
+ "index": 46
+ },
+ {
+ "bbox": [
+ 104,
+ 679,
+ 506,
+ 694
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 679,
+ 159,
+ 694
+ ],
+ "score": 1.0,
+ "content": "smaller than",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 159,
+ 680,
+ 176,
+ 691
+ ],
+ "score": 0.91,
+ "content": "\\lambda ^ { - 1 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 177,
+ 679,
+ 367,
+ 694
+ ],
+ "score": 1.0,
+ "content": ". Therefore, the closer the positive eigenvalues",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 368,
+ 683,
+ 382,
+ 693
+ ],
+ "score": 0.86,
+ "content": "\\mu _ { m }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 383,
+ 679,
+ 506,
+ 694
+ ],
+ "score": 1.0,
+ "content": "are to 1, the broader range of",
+ "type": "text"
+ }
+ ],
+ "index": 47
+ },
+ {
+ "bbox": [
+ 106,
+ 692,
+ 290,
+ 704
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 692,
+ 290,
+ 704
+ ],
+ "score": 1.0,
+ "content": "GCNs satisfies the assumption of Theorem 2.",
+ "type": "text"
+ }
+ ],
+ "index": 48
+ }
+ ],
+ "index": 43.5,
+ "bbox_fs": [
+ 104,
+ 592,
+ 506,
+ 704
+ ]
+ }
+ ]
+ },
+ {
+ "preproc_blocks": [
+ {
+ "type": "image",
+ "bbox": [
+ 150,
+ 88,
+ 448,
+ 207
+ ],
+ "blocks": [
+ {
+ "type": "image_body",
+ "bbox": [
+ 150,
+ 88,
+ 448,
+ 207
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 150,
+ 88,
+ 448,
+ 207
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 150,
+ 88,
+ 448,
+ 207
+ ],
+ "score": 0.966,
+ "type": "image",
+ "image_path": "52ed0a06decf73aeea33aba4d223d5646b9ee3b007a61062524feaa9a9984ba1.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 1,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 150,
+ 88,
+ 448,
+ 127.66666666666666
+ ],
+ "spans": [],
+ "index": 0
+ },
+ {
+ "bbox": [
+ 150,
+ 127.66666666666666,
+ 448,
+ 167.33333333333331
+ ],
+ "spans": [],
+ "index": 1
+ },
+ {
+ "bbox": [
+ 150,
+ 167.33333333333331,
+ 448,
+ 206.99999999999997
+ ],
+ "spans": [],
+ "index": 2
+ }
+ ]
+ },
+ {
+ "type": "image_caption",
+ "bbox": [
+ 107,
+ 225,
+ 505,
+ 259
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 224,
+ 505,
+ 238
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 224,
+ 257,
+ 238
+ ],
+ "score": 1.0,
+ "content": "Figure 1: Visualization of vector field",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 257,
+ 225,
+ 340,
+ 237
+ ],
+ "score": 0.93,
+ "content": "V ( X ) : = f ( X ) - X",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 340,
+ 224,
+ 505,
+ 238
+ ],
+ "score": 1.0,
+ "content": "induced by the one-step transition. Color",
+ "type": "text"
+ }
+ ],
+ "index": 3
+ },
+ {
+ "bbox": [
+ 106,
+ 236,
+ 504,
+ 249
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 236,
+ 242,
+ 249
+ ],
+ "score": 1.0,
+ "content": "maps indicate the absolute value",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 242,
+ 237,
+ 273,
+ 249
+ ],
+ "score": 0.93,
+ "content": "| V ( X ) |",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 273,
+ 236,
+ 324,
+ 249
+ ],
+ "score": 1.0,
+ "content": "at the point",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 325,
+ 237,
+ 335,
+ 246
+ ],
+ "score": 0.79,
+ "content": "X",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 335,
+ 236,
+ 464,
+ 249
+ ],
+ "score": 1.0,
+ "content": ". Dotted lines are the subspace",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 464,
+ 237,
+ 477,
+ 247
+ ],
+ "score": 0.7,
+ "content": "\\mathcal { M }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 477,
+ 236,
+ 504,
+ 249
+ ],
+ "score": 1.0,
+ "content": ". Left:",
+ "type": "text"
+ }
+ ],
+ "index": 4
+ },
+ {
+ "bbox": [
+ 107,
+ 248,
+ 278,
+ 259
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 248,
+ 278,
+ 259
+ ],
+ "score": 1.0,
+ "content": "Case 1. Right: Case 2. Best view in color.",
+ "type": "text"
+ }
+ ],
+ "index": 5
+ }
+ ],
+ "index": 4
+ }
+ ],
+ "index": 2.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 278,
+ 505,
+ 359
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 104,
+ 277,
+ 506,
+ 295
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 277,
+ 221,
+ 295
+ ],
+ "score": 1.0,
+ "content": "For an Erdos – R ˝ enyi graph ´",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 221,
+ 281,
+ 243,
+ 293
+ ],
+ "score": 0.91,
+ "content": "G _ { N , p }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 244,
+ 277,
+ 432,
+ 295
+ ],
+ "score": 1.0,
+ "content": ", Chung & Radcliffe (2011) showed that when",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 433,
+ 279,
+ 486,
+ 294
+ ],
+ "score": 0.92,
+ "content": "\\begin{array} { r } { \\frac { \\log N } { N p } = o ( 1 ) } \\end{array}",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 486,
+ 277,
+ 506,
+ 295
+ ],
+ "score": 1.0,
+ "content": ", the",
+ "type": "text"
+ }
+ ],
+ "index": 6
+ },
+ {
+ "bbox": [
+ 106,
+ 292,
+ 505,
+ 305
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 292,
+ 505,
+ 305
+ ],
+ "score": 1.0,
+ "content": "eigenvalues of the (usual) normalized Laplacian except for the smallest one converge to 1 with high",
+ "type": "text"
+ }
+ ],
+ "index": 7
+ },
+ {
+ "bbox": [
+ 104,
+ 303,
+ 505,
+ 317
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 303,
+ 505,
+ 317
+ ],
+ "score": 1.0,
+ "content": "probability (see Theorem 2 therein)3. We can interpret this theorem as the convergence of Erdos- ˝",
+ "type": "text"
+ }
+ ],
+ "index": 8
+ },
+ {
+ "bbox": [
+ 106,
+ 315,
+ 505,
+ 326
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 315,
+ 505,
+ 326
+ ],
+ "score": 1.0,
+ "content": "Renyi graphs to the complete graph in terms of graph spectra. We can prove that the augmented ´",
+ "type": "text"
+ }
+ ],
+ "index": 9
+ },
+ {
+ "bbox": [
+ 105,
+ 325,
+ 505,
+ 338
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 325,
+ 505,
+ 338
+ ],
+ "score": 1.0,
+ "content": "normalized Laplacian behaves similarly (Lemma 6). By combining this fact with the discussion in",
+ "type": "text"
+ }
+ ],
+ "index": 10
+ },
+ {
+ "bbox": [
+ 106,
+ 337,
+ 505,
+ 349
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 337,
+ 505,
+ 349
+ ],
+ "score": 1.0,
+ "content": "the previous paragraph, we obtain the asymptotic behavior of GCNs on the Erdos – R ˝ enyi graph. ´",
+ "type": "text"
+ }
+ ],
+ "index": 11
+ },
+ {
+ "bbox": [
+ 106,
+ 347,
+ 270,
+ 360
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 347,
+ 270,
+ 360
+ ],
+ "score": 1.0,
+ "content": "See Appendix D for the complete proof.",
+ "type": "text"
+ }
+ ],
+ "index": 12
+ }
+ ],
+ "index": 9
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 362,
+ 505,
+ 418
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 360,
+ 505,
+ 380
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 360,
+ 334,
+ 380
+ ],
+ "score": 1.0,
+ "content": "Theorem 3. Consider a GCN on the Erdos-R ˝ enyi graph ´",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 334,
+ 365,
+ 357,
+ 377
+ ],
+ "score": 0.91,
+ "content": "G _ { N , p }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 357,
+ 360,
+ 397,
+ 380
+ ],
+ "score": 1.0,
+ "content": "such that",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 397,
+ 362,
+ 450,
+ 379
+ ],
+ "score": 0.93,
+ "content": "\\begin{array} { r } { \\frac { \\log N } { N p } = o ( 1 ) } \\end{array}",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 450,
+ 360,
+ 505,
+ 380
+ ],
+ "score": 1.0,
+ "content": "as a function",
+ "type": "text"
+ }
+ ],
+ "index": 13
+ },
+ {
+ "bbox": [
+ 105,
+ 376,
+ 506,
+ 389
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 376,
+ 117,
+ 389
+ ],
+ "score": 1.0,
+ "content": "of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 118,
+ 377,
+ 128,
+ 387
+ ],
+ "score": 0.68,
+ "content": "N",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 128,
+ 376,
+ 169,
+ 389
+ ],
+ "score": 1.0,
+ "content": ". For any",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 169,
+ 378,
+ 196,
+ 388
+ ],
+ "score": 0.86,
+ "content": "\\varepsilon > 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 196,
+ 376,
+ 506,
+ 389
+ ],
+ "score": 1.0,
+ "content": ", if the supremum s of the maximum singular values of weights in the GCN",
+ "type": "text"
+ }
+ ],
+ "index": 14
+ },
+ {
+ "bbox": [
+ 105,
+ 388,
+ 506,
+ 407
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 390,
+ 205,
+ 406
+ ],
+ "score": 1.0,
+ "content": "satisfies s < s0 := 17 q",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 141,
+ 388,
+ 241,
+ 407
+ ],
+ "score": 0.93,
+ "content": "\\begin{array} { r } { s < s _ { 0 } : = \\frac { 1 } { 7 } \\sqrt { \\frac { N p - p + 1 } { \\log ( 4 N / \\varepsilon ) } } } \\end{array}",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 243,
+ 391,
+ 352,
+ 405
+ ],
+ "score": 1.0,
+ "content": "then, for sufficiently large",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 352,
+ 392,
+ 362,
+ 402
+ ],
+ "score": 0.63,
+ "content": "N",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 362,
+ 391,
+ 506,
+ 405
+ ],
+ "score": 1.0,
+ "content": ", the GCN satisfies the condition of",
+ "type": "text"
+ }
+ ],
+ "index": 15
+ },
+ {
+ "bbox": [
+ 241,
+ 396,
+ 246,
+ 408
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 241,
+ 396,
+ 246,
+ 408
+ ],
+ "score": 1.0,
+ "content": ",",
+ "type": "text"
+ }
+ ],
+ "index": 16
+ },
+ {
+ "bbox": [
+ 106,
+ 405,
+ 276,
+ 420
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 405,
+ 249,
+ 420
+ ],
+ "score": 1.0,
+ "content": "Theorem 2 with probability at least",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 249,
+ 407,
+ 272,
+ 417
+ ],
+ "score": 0.87,
+ "content": "1 - \\varepsilon",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 272,
+ 405,
+ 276,
+ 420
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 17
+ }
+ ],
+ "index": 15
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 427,
+ 505,
+ 471
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 427,
+ 505,
+ 440
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 427,
+ 505,
+ 440
+ ],
+ "score": 1.0,
+ "content": "Theorem 3 requires that an underlying graph is not extremely sparse. For example, suppose the",
+ "type": "text"
+ }
+ ],
+ "index": 18
+ },
+ {
+ "bbox": [
+ 105,
+ 437,
+ 505,
+ 451
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 437,
+ 155,
+ 451
+ ],
+ "score": 1.0,
+ "content": "node size is",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 156,
+ 438,
+ 208,
+ 449
+ ],
+ "score": 0.77,
+ "content": "N = 2 0 , 0 0 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 209,
+ 437,
+ 505,
+ 451
+ ],
+ "score": 1.0,
+ "content": ", which is the approximately the maximum node size of datasets we use in",
+ "type": "text"
+ }
+ ],
+ "index": 19
+ },
+ {
+ "bbox": [
+ 105,
+ 449,
+ 504,
+ 461
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 449,
+ 268,
+ 461
+ ],
+ "score": 1.0,
+ "content": "experiments, and the edge probability is",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 269,
+ 449,
+ 325,
+ 461
+ ],
+ "score": 0.93,
+ "content": "p = \\log N / N",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 325,
+ 449,
+ 462,
+ 461
+ ],
+ "score": 1.0,
+ "content": ". Then, each node has the order of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 462,
+ 450,
+ 504,
+ 461
+ ],
+ "score": 0.91,
+ "content": "N p \\approx 4 . 3",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 20
+ },
+ {
+ "bbox": [
+ 106,
+ 461,
+ 171,
+ 472
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 461,
+ 171,
+ 472
+ ],
+ "score": 1.0,
+ "content": "adjacent nodes.",
+ "type": "text"
+ }
+ ],
+ "index": 21
+ }
+ ],
+ "index": 19.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 477,
+ 505,
+ 532
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 477,
+ 504,
+ 489
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 477,
+ 315,
+ 489
+ ],
+ "score": 1.0,
+ "content": "Under the condition of Theorem 3, the upper bound",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 315,
+ 479,
+ 351,
+ 488
+ ],
+ "score": 0.89,
+ "content": "s _ { 0 } \\to \\infty",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 351,
+ 477,
+ 362,
+ 489
+ ],
+ "score": 1.0,
+ "content": "as",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 363,
+ 478,
+ 398,
+ 487
+ ],
+ "score": 0.91,
+ "content": "N \\to \\infty",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 399,
+ 477,
+ 504,
+ 489
+ ],
+ "score": 1.0,
+ "content": ". It means that if the graph",
+ "type": "text"
+ }
+ ],
+ "index": 22
+ },
+ {
+ "bbox": [
+ 105,
+ 488,
+ 505,
+ 500
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 488,
+ 505,
+ 500
+ ],
+ "score": 1.0,
+ "content": "is sufficiently large and not extremely sparse, most GCNs suffer from the information loss. For the",
+ "type": "text"
+ }
+ ],
+ "index": 23
+ },
+ {
+ "bbox": [
+ 105,
+ 498,
+ 506,
+ 511
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 498,
+ 252,
+ 511
+ ],
+ "score": 1.0,
+ "content": "dependence on the edge probability",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 253,
+ 501,
+ 259,
+ 511
+ ],
+ "score": 0.4,
+ "content": "p",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 259,
+ 498,
+ 263,
+ 511
+ ],
+ "score": 1.0,
+ "content": ",",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 263,
+ 501,
+ 273,
+ 510
+ ],
+ "score": 0.63,
+ "content": "s _ { 0 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 274,
+ 498,
+ 388,
+ 511
+ ],
+ "score": 1.0,
+ "content": "is an increasing function of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 388,
+ 501,
+ 394,
+ 510
+ ],
+ "score": 0.78,
+ "content": "p",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 395,
+ 498,
+ 506,
+ 511
+ ],
+ "score": 1.0,
+ "content": ", which means the denser a",
+ "type": "text"
+ }
+ ],
+ "index": 24
+ },
+ {
+ "bbox": [
+ 105,
+ 510,
+ 506,
+ 523
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 510,
+ 506,
+ 523
+ ],
+ "score": 1.0,
+ "content": "graph is, the more quickly graph convolution operations mix signals on nodes and move them close",
+ "type": "text"
+ }
+ ],
+ "index": 25
+ },
+ {
+ "bbox": [
+ 105,
+ 521,
+ 162,
+ 532
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 521,
+ 162,
+ 532
+ ],
+ "score": 1.0,
+ "content": "to each other.",
+ "type": "text"
+ }
+ ],
+ "index": 26
+ }
+ ],
+ "index": 24
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 537,
+ 505,
+ 593
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 537,
+ 505,
+ 551
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 537,
+ 505,
+ 551
+ ],
+ "score": 1.0,
+ "content": "Theorem 3 implies that graph NNs perform poorly on dense NNs. More aggressively, we can hy-",
+ "type": "text"
+ }
+ ],
+ "index": 27
+ },
+ {
+ "bbox": [
+ 105,
+ 549,
+ 505,
+ 561
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 549,
+ 505,
+ 561
+ ],
+ "score": 1.0,
+ "content": "pothesize that the sparsity of practically available graphs is one of the reasons for the success of",
+ "type": "text"
+ }
+ ],
+ "index": 28
+ },
+ {
+ "bbox": [
+ 105,
+ 559,
+ 505,
+ 573
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 559,
+ 505,
+ 573
+ ],
+ "score": 1.0,
+ "content": "graph NNs in node classification tasks. To confirm this hypothesis, we artificially add edges to ci-",
+ "type": "text"
+ }
+ ],
+ "index": 29
+ },
+ {
+ "bbox": [
+ 105,
+ 569,
+ 506,
+ 585
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 569,
+ 506,
+ 585
+ ],
+ "score": 1.0,
+ "content": "tation networks to make them dense in the experiments and observe the failure of graph NNs as",
+ "type": "text"
+ }
+ ],
+ "index": 30
+ },
+ {
+ "bbox": [
+ 105,
+ 582,
+ 216,
+ 595
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 582,
+ 216,
+ 595
+ ],
+ "score": 1.0,
+ "content": "expected (see Section 6.3).",
+ "type": "text"
+ }
+ ],
+ "index": 31
+ }
+ ],
+ "index": 29
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 108,
+ 610,
+ 194,
+ 622
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 609,
+ 196,
+ 624
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 609,
+ 196,
+ 624
+ ],
+ "score": 1.0,
+ "content": "6 EXPERIMENT",
+ "type": "text"
+ }
+ ],
+ "index": 32
+ }
+ ],
+ "index": 32
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 107,
+ 635,
+ 313,
+ 647
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 634,
+ 315,
+ 648
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 634,
+ 315,
+ 648
+ ],
+ "score": 1.0,
+ "content": "6.1 SYNTHESIS DATA: ONE-STEP TRANSITION",
+ "type": "text"
+ }
+ ],
+ "index": 33
+ }
+ ],
+ "index": 33
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 655,
+ 505,
+ 701
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 655,
+ 506,
+ 669
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 655,
+ 293,
+ 669
+ ],
+ "score": 1.0,
+ "content": "We numerically investigate how the transition",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 293,
+ 656,
+ 375,
+ 668
+ ],
+ "score": 0.92,
+ "content": "f ( X ) : = \\sigma ( P X W )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 375,
+ 655,
+ 506,
+ 669
+ ],
+ "score": 1.0,
+ "content": "changes inputs using the vector",
+ "type": "text"
+ }
+ ],
+ "index": 34
+ },
+ {
+ "bbox": [
+ 105,
+ 667,
+ 504,
+ 680
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 667,
+ 127,
+ 680
+ ],
+ "score": 1.0,
+ "content": "field",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 127,
+ 667,
+ 219,
+ 680
+ ],
+ "score": 0.93,
+ "content": "V ( X ) : = { f ( X ) } - { X ^ { 4 } }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 220,
+ 667,
+ 326,
+ 680
+ ],
+ "score": 1.0,
+ "content": ". For this purpose, we set",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 326,
+ 668,
+ 357,
+ 677
+ ],
+ "score": 0.85,
+ "content": "N = 2",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 357,
+ 667,
+ 361,
+ 680
+ ],
+ "score": 1.0,
+ "content": ",",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 361,
+ 668,
+ 393,
+ 678
+ ],
+ "score": 0.87,
+ "content": "M = 1",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 393,
+ 667,
+ 415,
+ 680
+ ],
+ "score": 1.0,
+ "content": ", and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 415,
+ 668,
+ 444,
+ 678
+ ],
+ "score": 0.9,
+ "content": "C = 1",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 445,
+ 667,
+ 466,
+ 680
+ ],
+ "score": 1.0,
+ "content": ". Let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 467,
+ 667,
+ 504,
+ 679
+ ],
+ "score": 0.92,
+ "content": "\\lambda _ { 1 } \\leq \\lambda _ { 2 }",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 35
+ },
+ {
+ "bbox": [
+ 105,
+ 678,
+ 505,
+ 691
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 678,
+ 194,
+ 691
+ ],
+ "score": 1.0,
+ "content": "be the eigenvalues of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 194,
+ 680,
+ 203,
+ 688
+ ],
+ "score": 0.81,
+ "content": "P",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 203,
+ 678,
+ 255,
+ 691
+ ],
+ "score": 1.0,
+ "content": ". We choose",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 255,
+ 679,
+ 267,
+ 689
+ ],
+ "score": 0.73,
+ "content": "W",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 268,
+ 678,
+ 280,
+ 691
+ ],
+ "score": 1.0,
+ "content": "as",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 280,
+ 678,
+ 374,
+ 690
+ ],
+ "score": 0.9,
+ "content": "| \\lambda _ { 2 } | ^ { - 1 } \\le W < | \\lambda _ { 1 } | ^ { - 1 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 374,
+ 678,
+ 505,
+ 691
+ ],
+ "score": 1.0,
+ "content": "so that Theorem 1 is applicable",
+ "type": "text"
+ }
+ ],
+ "index": 36
+ },
+ {
+ "bbox": [
+ 104,
+ 688,
+ 504,
+ 702
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 688,
+ 471,
+ 702
+ ],
+ "score": 1.0,
+ "content": "but is not reduced to the trivial situation (see, Appendix E.2). We choose the eigenvector",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 472,
+ 689,
+ 504,
+ 700
+ ],
+ "score": 0.92,
+ "content": "\\bar { \\boldsymbol { e } } \\in \\mathbb { R } ^ { 2 }",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 37
+ }
+ ],
+ "index": 35.5
+ }
+ ],
+ "page_idx": 5,
+ "page_size": [
+ 612,
+ 792
+ ],
+ "discarded_blocks": [
+ {
+ "type": "discarded",
+ "bbox": [
+ 117,
+ 710,
+ 454,
+ 732
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 118,
+ 709,
+ 376,
+ 722
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 118,
+ 709,
+ 376,
+ 722
+ ],
+ "score": 1.0,
+ "content": "3Chung et al. (2004) and Coja-Oghlan (2007) proved similar theorems.",
+ "type": "text"
+ }
+ ]
+ },
+ {
+ "bbox": [
+ 119,
+ 720,
+ 455,
+ 733
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 119,
+ 720,
+ 371,
+ 733
+ ],
+ "score": 1.0,
+ "content": "4Since we consider the one-step transition only, we omit the subscript",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 371,
+ 722,
+ 375,
+ 730
+ ],
+ "score": 0.68,
+ "content": "l",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 376,
+ 720,
+ 396,
+ 733
+ ],
+ "score": 1.0,
+ "content": "from",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 396,
+ 721,
+ 419,
+ 732
+ ],
+ "score": 0.76,
+ "content": "f _ { l } , X _ { l }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 420,
+ 720,
+ 438,
+ 733
+ ],
+ "score": 1.0,
+ "content": ", and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 438,
+ 721,
+ 451,
+ 731
+ ],
+ "score": 0.86,
+ "content": "W _ { l }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 451,
+ 720,
+ 455,
+ 733
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ]
+ }
+ ]
+ },
+ {
+ "type": "discarded",
+ "bbox": [
+ 107,
+ 27,
+ 293,
+ 37
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 26,
+ 293,
+ 38
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 26,
+ 293,
+ 38
+ ],
+ "score": 1.0,
+ "content": "Published as a conference paper at ICLR 2020",
+ "type": "text"
+ }
+ ]
+ }
+ ]
+ },
+ {
+ "type": "discarded",
+ "bbox": [
+ 302,
+ 752,
+ 309,
+ 760
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 302,
+ 751,
+ 310,
+ 762
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 302,
+ 751,
+ 310,
+ 762
+ ],
+ "score": 1.0,
+ "content": "6",
+ "type": "text"
+ }
+ ]
+ }
+ ]
+ }
+ ],
+ "para_blocks": [
+ {
+ "type": "image",
+ "bbox": [
+ 150,
+ 88,
+ 448,
+ 207
+ ],
+ "blocks": [
+ {
+ "type": "image_body",
+ "bbox": [
+ 150,
+ 88,
+ 448,
+ 207
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 150,
+ 88,
+ 448,
+ 207
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 150,
+ 88,
+ 448,
+ 207
+ ],
+ "score": 0.966,
+ "type": "image",
+ "image_path": "52ed0a06decf73aeea33aba4d223d5646b9ee3b007a61062524feaa9a9984ba1.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 1,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 150,
+ 88,
+ 448,
+ 127.66666666666666
+ ],
+ "spans": [],
+ "index": 0
+ },
+ {
+ "bbox": [
+ 150,
+ 127.66666666666666,
+ 448,
+ 167.33333333333331
+ ],
+ "spans": [],
+ "index": 1
+ },
+ {
+ "bbox": [
+ 150,
+ 167.33333333333331,
+ 448,
+ 206.99999999999997
+ ],
+ "spans": [],
+ "index": 2
+ }
+ ]
+ },
+ {
+ "type": "image_caption",
+ "bbox": [
+ 107,
+ 225,
+ 505,
+ 259
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 224,
+ 505,
+ 238
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 224,
+ 257,
+ 238
+ ],
+ "score": 1.0,
+ "content": "Figure 1: Visualization of vector field",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 257,
+ 225,
+ 340,
+ 237
+ ],
+ "score": 0.93,
+ "content": "V ( X ) : = f ( X ) - X",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 340,
+ 224,
+ 505,
+ 238
+ ],
+ "score": 1.0,
+ "content": "induced by the one-step transition. Color",
+ "type": "text"
+ }
+ ],
+ "index": 3
+ },
+ {
+ "bbox": [
+ 106,
+ 236,
+ 504,
+ 249
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 236,
+ 242,
+ 249
+ ],
+ "score": 1.0,
+ "content": "maps indicate the absolute value",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 242,
+ 237,
+ 273,
+ 249
+ ],
+ "score": 0.93,
+ "content": "| V ( X ) |",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 273,
+ 236,
+ 324,
+ 249
+ ],
+ "score": 1.0,
+ "content": "at the point",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 325,
+ 237,
+ 335,
+ 246
+ ],
+ "score": 0.79,
+ "content": "X",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 335,
+ 236,
+ 464,
+ 249
+ ],
+ "score": 1.0,
+ "content": ". Dotted lines are the subspace",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 464,
+ 237,
+ 477,
+ 247
+ ],
+ "score": 0.7,
+ "content": "\\mathcal { M }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 477,
+ 236,
+ 504,
+ 249
+ ],
+ "score": 1.0,
+ "content": ". Left:",
+ "type": "text"
+ }
+ ],
+ "index": 4
+ },
+ {
+ "bbox": [
+ 107,
+ 248,
+ 278,
+ 259
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 248,
+ 278,
+ 259
+ ],
+ "score": 1.0,
+ "content": "Case 1. Right: Case 2. Best view in color.",
+ "type": "text"
+ }
+ ],
+ "index": 5
+ }
+ ],
+ "index": 4
+ }
+ ],
+ "index": 2.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 278,
+ 505,
+ 359
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 104,
+ 277,
+ 506,
+ 295
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 277,
+ 221,
+ 295
+ ],
+ "score": 1.0,
+ "content": "For an Erdos – R ˝ enyi graph ´",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 221,
+ 281,
+ 243,
+ 293
+ ],
+ "score": 0.91,
+ "content": "G _ { N , p }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 244,
+ 277,
+ 432,
+ 295
+ ],
+ "score": 1.0,
+ "content": ", Chung & Radcliffe (2011) showed that when",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 433,
+ 279,
+ 486,
+ 294
+ ],
+ "score": 0.92,
+ "content": "\\begin{array} { r } { \\frac { \\log N } { N p } = o ( 1 ) } \\end{array}",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 486,
+ 277,
+ 506,
+ 295
+ ],
+ "score": 1.0,
+ "content": ", the",
+ "type": "text"
+ }
+ ],
+ "index": 6
+ },
+ {
+ "bbox": [
+ 106,
+ 292,
+ 505,
+ 305
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 292,
+ 505,
+ 305
+ ],
+ "score": 1.0,
+ "content": "eigenvalues of the (usual) normalized Laplacian except for the smallest one converge to 1 with high",
+ "type": "text"
+ }
+ ],
+ "index": 7
+ },
+ {
+ "bbox": [
+ 104,
+ 303,
+ 505,
+ 317
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 303,
+ 505,
+ 317
+ ],
+ "score": 1.0,
+ "content": "probability (see Theorem 2 therein)3. We can interpret this theorem as the convergence of Erdos- ˝",
+ "type": "text"
+ }
+ ],
+ "index": 8
+ },
+ {
+ "bbox": [
+ 106,
+ 315,
+ 505,
+ 326
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 315,
+ 505,
+ 326
+ ],
+ "score": 1.0,
+ "content": "Renyi graphs to the complete graph in terms of graph spectra. We can prove that the augmented ´",
+ "type": "text"
+ }
+ ],
+ "index": 9
+ },
+ {
+ "bbox": [
+ 105,
+ 325,
+ 505,
+ 338
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 325,
+ 505,
+ 338
+ ],
+ "score": 1.0,
+ "content": "normalized Laplacian behaves similarly (Lemma 6). By combining this fact with the discussion in",
+ "type": "text"
+ }
+ ],
+ "index": 10
+ },
+ {
+ "bbox": [
+ 106,
+ 337,
+ 505,
+ 349
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 337,
+ 505,
+ 349
+ ],
+ "score": 1.0,
+ "content": "the previous paragraph, we obtain the asymptotic behavior of GCNs on the Erdos – R ˝ enyi graph. ´",
+ "type": "text"
+ }
+ ],
+ "index": 11
+ },
+ {
+ "bbox": [
+ 106,
+ 347,
+ 270,
+ 360
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 347,
+ 270,
+ 360
+ ],
+ "score": 1.0,
+ "content": "See Appendix D for the complete proof.",
+ "type": "text"
+ }
+ ],
+ "index": 12
+ }
+ ],
+ "index": 9,
+ "bbox_fs": [
+ 104,
+ 277,
+ 506,
+ 360
+ ]
+ },
+ {
+ "type": "list",
+ "bbox": [
+ 107,
+ 362,
+ 505,
+ 418
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 360,
+ 505,
+ 380
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 360,
+ 334,
+ 380
+ ],
+ "score": 1.0,
+ "content": "Theorem 3. Consider a GCN on the Erdos-R ˝ enyi graph ´",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 334,
+ 365,
+ 357,
+ 377
+ ],
+ "score": 0.91,
+ "content": "G _ { N , p }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 357,
+ 360,
+ 397,
+ 380
+ ],
+ "score": 1.0,
+ "content": "such that",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 397,
+ 362,
+ 450,
+ 379
+ ],
+ "score": 0.93,
+ "content": "\\begin{array} { r } { \\frac { \\log N } { N p } = o ( 1 ) } \\end{array}",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 450,
+ 360,
+ 505,
+ 380
+ ],
+ "score": 1.0,
+ "content": "as a function",
+ "type": "text"
+ }
+ ],
+ "index": 13,
+ "is_list_start_line": true
+ },
+ {
+ "bbox": [
+ 105,
+ 376,
+ 506,
+ 389
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 376,
+ 117,
+ 389
+ ],
+ "score": 1.0,
+ "content": "of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 118,
+ 377,
+ 128,
+ 387
+ ],
+ "score": 0.68,
+ "content": "N",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 128,
+ 376,
+ 169,
+ 389
+ ],
+ "score": 1.0,
+ "content": ". For any",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 169,
+ 378,
+ 196,
+ 388
+ ],
+ "score": 0.86,
+ "content": "\\varepsilon > 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 196,
+ 376,
+ 506,
+ 389
+ ],
+ "score": 1.0,
+ "content": ", if the supremum s of the maximum singular values of weights in the GCN",
+ "type": "text"
+ }
+ ],
+ "index": 14,
+ "is_list_start_line": true
+ },
+ {
+ "bbox": [
+ 105,
+ 388,
+ 506,
+ 407
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 390,
+ 205,
+ 406
+ ],
+ "score": 1.0,
+ "content": "satisfies s < s0 := 17 q",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 141,
+ 388,
+ 241,
+ 407
+ ],
+ "score": 0.93,
+ "content": "\\begin{array} { r } { s < s _ { 0 } : = \\frac { 1 } { 7 } \\sqrt { \\frac { N p - p + 1 } { \\log ( 4 N / \\varepsilon ) } } } \\end{array}",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 243,
+ 391,
+ 352,
+ 405
+ ],
+ "score": 1.0,
+ "content": "then, for sufficiently large",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 352,
+ 392,
+ 362,
+ 402
+ ],
+ "score": 0.63,
+ "content": "N",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 362,
+ 391,
+ 506,
+ 405
+ ],
+ "score": 1.0,
+ "content": ", the GCN satisfies the condition of",
+ "type": "text"
+ }
+ ],
+ "index": 15,
+ "is_list_start_line": true
+ },
+ {
+ "bbox": [
+ 241,
+ 396,
+ 246,
+ 408
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 241,
+ 396,
+ 246,
+ 408
+ ],
+ "score": 1.0,
+ "content": ",",
+ "type": "text"
+ }
+ ],
+ "index": 16,
+ "is_list_end_line": true
+ },
+ {
+ "bbox": [
+ 106,
+ 405,
+ 276,
+ 420
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 405,
+ 249,
+ 420
+ ],
+ "score": 1.0,
+ "content": "Theorem 2 with probability at least",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 249,
+ 407,
+ 272,
+ 417
+ ],
+ "score": 0.87,
+ "content": "1 - \\varepsilon",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 272,
+ 405,
+ 276,
+ 420
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 17,
+ "is_list_start_line": true,
+ "is_list_end_line": true
+ }
+ ],
+ "index": 15,
+ "bbox_fs": [
+ 105,
+ 360,
+ 506,
+ 420
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 427,
+ 505,
+ 471
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 427,
+ 505,
+ 440
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 427,
+ 505,
+ 440
+ ],
+ "score": 1.0,
+ "content": "Theorem 3 requires that an underlying graph is not extremely sparse. For example, suppose the",
+ "type": "text"
+ }
+ ],
+ "index": 18
+ },
+ {
+ "bbox": [
+ 105,
+ 437,
+ 505,
+ 451
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 437,
+ 155,
+ 451
+ ],
+ "score": 1.0,
+ "content": "node size is",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 156,
+ 438,
+ 208,
+ 449
+ ],
+ "score": 0.77,
+ "content": "N = 2 0 , 0 0 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 209,
+ 437,
+ 505,
+ 451
+ ],
+ "score": 1.0,
+ "content": ", which is the approximately the maximum node size of datasets we use in",
+ "type": "text"
+ }
+ ],
+ "index": 19
+ },
+ {
+ "bbox": [
+ 105,
+ 449,
+ 504,
+ 461
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 449,
+ 268,
+ 461
+ ],
+ "score": 1.0,
+ "content": "experiments, and the edge probability is",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 269,
+ 449,
+ 325,
+ 461
+ ],
+ "score": 0.93,
+ "content": "p = \\log N / N",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 325,
+ 449,
+ 462,
+ 461
+ ],
+ "score": 1.0,
+ "content": ". Then, each node has the order of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 462,
+ 450,
+ 504,
+ 461
+ ],
+ "score": 0.91,
+ "content": "N p \\approx 4 . 3",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 20
+ },
+ {
+ "bbox": [
+ 106,
+ 461,
+ 171,
+ 472
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 461,
+ 171,
+ 472
+ ],
+ "score": 1.0,
+ "content": "adjacent nodes.",
+ "type": "text"
+ }
+ ],
+ "index": 21
+ }
+ ],
+ "index": 19.5,
+ "bbox_fs": [
+ 105,
+ 427,
+ 505,
+ 472
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 477,
+ 505,
+ 532
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 477,
+ 504,
+ 489
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 477,
+ 315,
+ 489
+ ],
+ "score": 1.0,
+ "content": "Under the condition of Theorem 3, the upper bound",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 315,
+ 479,
+ 351,
+ 488
+ ],
+ "score": 0.89,
+ "content": "s _ { 0 } \\to \\infty",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 351,
+ 477,
+ 362,
+ 489
+ ],
+ "score": 1.0,
+ "content": "as",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 363,
+ 478,
+ 398,
+ 487
+ ],
+ "score": 0.91,
+ "content": "N \\to \\infty",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 399,
+ 477,
+ 504,
+ 489
+ ],
+ "score": 1.0,
+ "content": ". It means that if the graph",
+ "type": "text"
+ }
+ ],
+ "index": 22
+ },
+ {
+ "bbox": [
+ 105,
+ 488,
+ 505,
+ 500
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 488,
+ 505,
+ 500
+ ],
+ "score": 1.0,
+ "content": "is sufficiently large and not extremely sparse, most GCNs suffer from the information loss. For the",
+ "type": "text"
+ }
+ ],
+ "index": 23
+ },
+ {
+ "bbox": [
+ 105,
+ 498,
+ 506,
+ 511
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 498,
+ 252,
+ 511
+ ],
+ "score": 1.0,
+ "content": "dependence on the edge probability",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 253,
+ 501,
+ 259,
+ 511
+ ],
+ "score": 0.4,
+ "content": "p",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 259,
+ 498,
+ 263,
+ 511
+ ],
+ "score": 1.0,
+ "content": ",",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 263,
+ 501,
+ 273,
+ 510
+ ],
+ "score": 0.63,
+ "content": "s _ { 0 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 274,
+ 498,
+ 388,
+ 511
+ ],
+ "score": 1.0,
+ "content": "is an increasing function of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 388,
+ 501,
+ 394,
+ 510
+ ],
+ "score": 0.78,
+ "content": "p",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 395,
+ 498,
+ 506,
+ 511
+ ],
+ "score": 1.0,
+ "content": ", which means the denser a",
+ "type": "text"
+ }
+ ],
+ "index": 24
+ },
+ {
+ "bbox": [
+ 105,
+ 510,
+ 506,
+ 523
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 510,
+ 506,
+ 523
+ ],
+ "score": 1.0,
+ "content": "graph is, the more quickly graph convolution operations mix signals on nodes and move them close",
+ "type": "text"
+ }
+ ],
+ "index": 25
+ },
+ {
+ "bbox": [
+ 105,
+ 521,
+ 162,
+ 532
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 521,
+ 162,
+ 532
+ ],
+ "score": 1.0,
+ "content": "to each other.",
+ "type": "text"
+ }
+ ],
+ "index": 26
+ }
+ ],
+ "index": 24,
+ "bbox_fs": [
+ 105,
+ 477,
+ 506,
+ 532
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 537,
+ 505,
+ 593
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 537,
+ 505,
+ 551
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 537,
+ 505,
+ 551
+ ],
+ "score": 1.0,
+ "content": "Theorem 3 implies that graph NNs perform poorly on dense NNs. More aggressively, we can hy-",
+ "type": "text"
+ }
+ ],
+ "index": 27
+ },
+ {
+ "bbox": [
+ 105,
+ 549,
+ 505,
+ 561
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 549,
+ 505,
+ 561
+ ],
+ "score": 1.0,
+ "content": "pothesize that the sparsity of practically available graphs is one of the reasons for the success of",
+ "type": "text"
+ }
+ ],
+ "index": 28
+ },
+ {
+ "bbox": [
+ 105,
+ 559,
+ 505,
+ 573
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 559,
+ 505,
+ 573
+ ],
+ "score": 1.0,
+ "content": "graph NNs in node classification tasks. To confirm this hypothesis, we artificially add edges to ci-",
+ "type": "text"
+ }
+ ],
+ "index": 29
+ },
+ {
+ "bbox": [
+ 105,
+ 569,
+ 506,
+ 585
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 569,
+ 506,
+ 585
+ ],
+ "score": 1.0,
+ "content": "tation networks to make them dense in the experiments and observe the failure of graph NNs as",
+ "type": "text"
+ }
+ ],
+ "index": 30
+ },
+ {
+ "bbox": [
+ 105,
+ 582,
+ 216,
+ 595
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 582,
+ 216,
+ 595
+ ],
+ "score": 1.0,
+ "content": "expected (see Section 6.3).",
+ "type": "text"
+ }
+ ],
+ "index": 31
+ }
+ ],
+ "index": 29,
+ "bbox_fs": [
+ 105,
+ 537,
+ 506,
+ 595
+ ]
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 108,
+ 610,
+ 194,
+ 622
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 609,
+ 196,
+ 624
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 609,
+ 196,
+ 624
+ ],
+ "score": 1.0,
+ "content": "6 EXPERIMENT",
+ "type": "text"
+ }
+ ],
+ "index": 32
+ }
+ ],
+ "index": 32
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 107,
+ 635,
+ 313,
+ 647
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 634,
+ 315,
+ 648
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 634,
+ 315,
+ 648
+ ],
+ "score": 1.0,
+ "content": "6.1 SYNTHESIS DATA: ONE-STEP TRANSITION",
+ "type": "text"
+ }
+ ],
+ "index": 33
+ }
+ ],
+ "index": 33
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 655,
+ 505,
+ 701
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 655,
+ 506,
+ 669
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 655,
+ 293,
+ 669
+ ],
+ "score": 1.0,
+ "content": "We numerically investigate how the transition",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 293,
+ 656,
+ 375,
+ 668
+ ],
+ "score": 0.92,
+ "content": "f ( X ) : = \\sigma ( P X W )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 375,
+ 655,
+ 506,
+ 669
+ ],
+ "score": 1.0,
+ "content": "changes inputs using the vector",
+ "type": "text"
+ }
+ ],
+ "index": 34
+ },
+ {
+ "bbox": [
+ 105,
+ 667,
+ 504,
+ 680
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 667,
+ 127,
+ 680
+ ],
+ "score": 1.0,
+ "content": "field",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 127,
+ 667,
+ 219,
+ 680
+ ],
+ "score": 0.93,
+ "content": "V ( X ) : = { f ( X ) } - { X ^ { 4 } }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 220,
+ 667,
+ 326,
+ 680
+ ],
+ "score": 1.0,
+ "content": ". For this purpose, we set",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 326,
+ 668,
+ 357,
+ 677
+ ],
+ "score": 0.85,
+ "content": "N = 2",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 357,
+ 667,
+ 361,
+ 680
+ ],
+ "score": 1.0,
+ "content": ",",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 361,
+ 668,
+ 393,
+ 678
+ ],
+ "score": 0.87,
+ "content": "M = 1",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 393,
+ 667,
+ 415,
+ 680
+ ],
+ "score": 1.0,
+ "content": ", and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 415,
+ 668,
+ 444,
+ 678
+ ],
+ "score": 0.9,
+ "content": "C = 1",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 445,
+ 667,
+ 466,
+ 680
+ ],
+ "score": 1.0,
+ "content": ". Let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 467,
+ 667,
+ 504,
+ 679
+ ],
+ "score": 0.92,
+ "content": "\\lambda _ { 1 } \\leq \\lambda _ { 2 }",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 35
+ },
+ {
+ "bbox": [
+ 105,
+ 678,
+ 505,
+ 691
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 678,
+ 194,
+ 691
+ ],
+ "score": 1.0,
+ "content": "be the eigenvalues of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 194,
+ 680,
+ 203,
+ 688
+ ],
+ "score": 0.81,
+ "content": "P",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 203,
+ 678,
+ 255,
+ 691
+ ],
+ "score": 1.0,
+ "content": ". We choose",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 255,
+ 679,
+ 267,
+ 689
+ ],
+ "score": 0.73,
+ "content": "W",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 268,
+ 678,
+ 280,
+ 691
+ ],
+ "score": 1.0,
+ "content": "as",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 280,
+ 678,
+ 374,
+ 690
+ ],
+ "score": 0.9,
+ "content": "| \\lambda _ { 2 } | ^ { - 1 } \\le W < | \\lambda _ { 1 } | ^ { - 1 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 374,
+ 678,
+ 505,
+ 691
+ ],
+ "score": 1.0,
+ "content": "so that Theorem 1 is applicable",
+ "type": "text"
+ }
+ ],
+ "index": 36
+ },
+ {
+ "bbox": [
+ 104,
+ 688,
+ 504,
+ 702
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 688,
+ 471,
+ 702
+ ],
+ "score": 1.0,
+ "content": "but is not reduced to the trivial situation (see, Appendix E.2). We choose the eigenvector",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 472,
+ 689,
+ 504,
+ 700
+ ],
+ "score": 0.92,
+ "content": "\\bar { \\boldsymbol { e } } \\in \\mathbb { R } ^ { 2 }",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 37
+ },
+ {
+ "bbox": [
+ 105,
+ 464,
+ 505,
+ 477
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 464,
+ 170,
+ 477
+ ],
+ "score": 1.0,
+ "content": "associated with",
+ "type": "text",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 170,
+ 465,
+ 181,
+ 476
+ ],
+ "score": 0.87,
+ "content": "\\lambda _ { 2 }",
+ "type": "inline_equation",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 182,
+ 464,
+ 492,
+ 477
+ ],
+ "score": 1.0,
+ "content": "in two ways as described below. See Appendix H.1 for the concrete values of",
+ "type": "text",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 492,
+ 465,
+ 501,
+ 475
+ ],
+ "score": 0.8,
+ "content": "P",
+ "type": "inline_equation",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 501,
+ 464,
+ 505,
+ 477
+ ],
+ "score": 1.0,
+ "content": ",",
+ "type": "text",
+ "cross_page": true
+ }
+ ],
+ "index": 13
+ },
+ {
+ "bbox": [
+ 106,
+ 476,
+ 504,
+ 488
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 478,
+ 112,
+ 486
+ ],
+ "score": 0.61,
+ "content": "e",
+ "type": "inline_equation",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 113,
+ 476,
+ 133,
+ 488
+ ],
+ "score": 1.0,
+ "content": ", and",
+ "type": "text",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 134,
+ 476,
+ 146,
+ 486
+ ],
+ "score": 0.72,
+ "content": "W",
+ "type": "inline_equation",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 146,
+ 476,
+ 296,
+ 488
+ ],
+ "score": 1.0,
+ "content": ". Figure 1 shows the visualization of",
+ "type": "text",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 296,
+ 476,
+ 305,
+ 486
+ ],
+ "score": 0.63,
+ "content": "V",
+ "type": "inline_equation",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 306,
+ 476,
+ 498,
+ 488
+ ],
+ "score": 1.0,
+ "content": ". First, we choose the non-negative eigenvector",
+ "type": "text",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 498,
+ 478,
+ 504,
+ 486
+ ],
+ "score": 0.67,
+ "content": "e",
+ "type": "inline_equation",
+ "cross_page": true
+ }
+ ],
+ "index": 14
+ },
+ {
+ "bbox": [
+ 105,
+ 485,
+ 506,
+ 500
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 485,
+ 413,
+ 500
+ ],
+ "score": 1.0,
+ "content": "so that it satisfies Assumption 1 (Case 1). We see that the transition function",
+ "type": "text",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 414,
+ 487,
+ 421,
+ 498
+ ],
+ "score": 0.85,
+ "content": "f",
+ "type": "inline_equation",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 421,
+ 485,
+ 506,
+ 500
+ ],
+ "score": 1.0,
+ "content": "uniformly decreases",
+ "type": "text",
+ "cross_page": true
+ }
+ ],
+ "index": 15
+ },
+ {
+ "bbox": [
+ 105,
+ 497,
+ 505,
+ 511
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 497,
+ 181,
+ 511
+ ],
+ "score": 1.0,
+ "content": "the distance from",
+ "type": "text",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 182,
+ 498,
+ 195,
+ 508
+ ],
+ "score": 0.8,
+ "content": "\\mathcal { M }",
+ "type": "inline_equation",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 195,
+ 497,
+ 505,
+ 511
+ ],
+ "score": 1.0,
+ "content": ". This is consistent with the consequence of Theorem 1. Next, we choose",
+ "type": "text",
+ "cross_page": true
+ }
+ ],
+ "index": 16
+ },
+ {
+ "bbox": [
+ 105,
+ 508,
+ 506,
+ 525
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 510,
+ 172,
+ 525
+ ],
+ "score": 1.0,
+ "content": "the eigenvector",
+ "type": "text",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 172,
+ 508,
+ 236,
+ 523
+ ],
+ "score": 0.93,
+ "content": "\\boldsymbol { e } = \\left[ e _ { 1 } \\quad e _ { 2 } \\right] ^ { \\intercal }",
+ "type": "inline_equation",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 236,
+ 510,
+ 331,
+ 525
+ ],
+ "score": 1.0,
+ "content": "such that the signs of",
+ "type": "text",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 331,
+ 513,
+ 342,
+ 522
+ ],
+ "score": 0.84,
+ "content": "e _ { 1 }",
+ "type": "inline_equation",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 342,
+ 510,
+ 362,
+ 525
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 362,
+ 512,
+ 373,
+ 522
+ ],
+ "score": 0.84,
+ "content": "e _ { 2 }",
+ "type": "inline_equation",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 374,
+ 510,
+ 506,
+ 525
+ ],
+ "score": 1.0,
+ "content": "differ (Case 2), which violates",
+ "type": "text",
+ "cross_page": true
+ }
+ ],
+ "index": 17
+ },
+ {
+ "bbox": [
+ 106,
+ 522,
+ 505,
+ 534
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 522,
+ 225,
+ 534
+ ],
+ "score": 1.0,
+ "content": "Assumption 1. We see that",
+ "type": "text",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 225,
+ 523,
+ 238,
+ 532
+ ],
+ "score": 0.81,
+ "content": "\\mathcal { M }",
+ "type": "inline_equation",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 239,
+ 522,
+ 332,
+ 534
+ ],
+ "score": 1.0,
+ "content": "is not invariant under",
+ "type": "text",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 333,
+ 523,
+ 340,
+ 534
+ ],
+ "score": 0.85,
+ "content": "f",
+ "type": "inline_equation",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 340,
+ 522,
+ 360,
+ 534
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 360,
+ 523,
+ 368,
+ 533
+ ],
+ "score": 0.85,
+ "content": "f",
+ "type": "inline_equation",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 368,
+ 522,
+ 505,
+ 534
+ ],
+ "score": 1.0,
+ "content": "does not uniformly decrease the",
+ "type": "text",
+ "cross_page": true
+ }
+ ],
+ "index": 18
+ },
+ {
+ "bbox": [
+ 105,
+ 532,
+ 494,
+ 545
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 532,
+ 163,
+ 545
+ ],
+ "score": 1.0,
+ "content": "distance from",
+ "type": "text",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 164,
+ 533,
+ 177,
+ 543
+ ],
+ "score": 0.79,
+ "content": "\\mathcal { M }",
+ "type": "inline_equation",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 177,
+ 532,
+ 494,
+ 545
+ ],
+ "score": 1.0,
+ "content": ". Therefore, we cannot remove the non-negativity assumption from Theorem 1.",
+ "type": "text",
+ "cross_page": true
+ }
+ ],
+ "index": 19
+ }
+ ],
+ "index": 35.5,
+ "bbox_fs": [
+ 104,
+ 655,
+ 506,
+ 702
+ ]
+ }
+ ]
+ },
+ {
+ "preproc_blocks": [
+ {
+ "type": "image",
+ "bbox": [
+ 151,
+ 85,
+ 459,
+ 207
+ ],
+ "blocks": [
+ {
+ "type": "image_body",
+ "bbox": [
+ 151,
+ 85,
+ 459,
+ 207
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 151,
+ 85,
+ 459,
+ 207
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 151,
+ 85,
+ 459,
+ 207
+ ],
+ "score": 0.97,
+ "type": "image",
+ "image_path": "2aeca4d04f65609a1fa1584f41938b2120f8df09b7dfae3a4549ab38b22e533d.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 1,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 151,
+ 85,
+ 459,
+ 125.66666666666666
+ ],
+ "spans": [],
+ "index": 0
+ },
+ {
+ "bbox": [
+ 151,
+ 125.66666666666666,
+ 459,
+ 166.33333333333331
+ ],
+ "spans": [],
+ "index": 1
+ },
+ {
+ "bbox": [
+ 151,
+ 166.33333333333331,
+ 459,
+ 206.99999999999997
+ ],
+ "spans": [],
+ "index": 2
+ }
+ ]
+ },
+ {
+ "type": "image_caption",
+ "bbox": [
+ 106,
+ 225,
+ 505,
+ 262
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 225,
+ 505,
+ 237
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 225,
+ 317,
+ 237
+ ],
+ "score": 1.0,
+ "content": "Figure 2: The actual distances to the invariant space",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 318,
+ 226,
+ 331,
+ 235
+ ],
+ "score": 0.79,
+ "content": "\\mathcal { M }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 331,
+ 225,
+ 505,
+ 237
+ ],
+ "score": 1.0,
+ "content": "and their upper bounds. Solid lines are the",
+ "type": "text"
+ }
+ ],
+ "index": 3
+ },
+ {
+ "bbox": [
+ 104,
+ 236,
+ 506,
+ 251
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 236,
+ 232,
+ 251
+ ],
+ "score": 1.0,
+ "content": "log relative distance defined by",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 232,
+ 236,
+ 372,
+ 250
+ ],
+ "score": 0.91,
+ "content": "y ( l ) = \\log ( d _ { \\mathcal { M } } ( \\mathbf { \\bar { { X } } } ^ { ( l ) } ) / d _ { \\mathcal { M } } ( X ^ { ( 0 ) } ) )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 372,
+ 236,
+ 506,
+ 251
+ ],
+ "score": 1.0,
+ "content": "and dotted lines are upper bound",
+ "type": "text"
+ }
+ ],
+ "index": 4
+ },
+ {
+ "bbox": [
+ 106,
+ 248,
+ 456,
+ 264
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 249,
+ 173,
+ 262
+ ],
+ "score": 0.92,
+ "content": "y ( l ) = l \\log ( s \\lambda )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 173,
+ 248,
+ 203,
+ 264
+ ],
+ "score": 1.0,
+ "content": ", where",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 204,
+ 249,
+ 224,
+ 260
+ ],
+ "score": 0.91,
+ "content": "X ^ { ( 0 ) }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 225,
+ 248,
+ 316,
+ 264
+ ],
+ "score": 1.0,
+ "content": "is the input signal and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 316,
+ 249,
+ 335,
+ 260
+ ],
+ "score": 0.89,
+ "content": "X ^ { ( l ) }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 335,
+ 248,
+ 414,
+ 264
+ ],
+ "score": 1.0,
+ "content": "is the output of the",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 414,
+ 250,
+ 418,
+ 260
+ ],
+ "score": 0.44,
+ "content": "l",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 419,
+ 248,
+ 456,
+ 264
+ ],
+ "score": 1.0,
+ "content": "-th layer.",
+ "type": "text"
+ }
+ ],
+ "index": 5
+ }
+ ],
+ "index": 4
+ }
+ ],
+ "index": 2.5
+ },
+ {
+ "type": "image",
+ "bbox": [
+ 109,
+ 276,
+ 497,
+ 385
+ ],
+ "blocks": [
+ {
+ "type": "image_body",
+ "bbox": [
+ 109,
+ 276,
+ 497,
+ 385
+ ],
+ "group_id": 1,
+ "lines": [
+ {
+ "bbox": [
+ 109,
+ 276,
+ 497,
+ 385
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 109,
+ 276,
+ 497,
+ 385
+ ],
+ "score": 0.963,
+ "type": "image",
+ "image_path": "8900fa8d6997b22156aeea03d2008a13643ccde54ceb13a5e4d7e973496948c4.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 7,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 109,
+ 276,
+ 497,
+ 312.3333333333333
+ ],
+ "spans": [],
+ "index": 6
+ },
+ {
+ "bbox": [
+ 109,
+ 312.3333333333333,
+ 497,
+ 348.66666666666663
+ ],
+ "spans": [],
+ "index": 7
+ },
+ {
+ "bbox": [
+ 109,
+ 348.66666666666663,
+ 497,
+ 384.99999999999994
+ ],
+ "spans": [],
+ "index": 8
+ }
+ ]
+ },
+ {
+ "type": "image_caption",
+ "bbox": [
+ 106,
+ 399,
+ 505,
+ 444
+ ],
+ "group_id": 1,
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 399,
+ 505,
+ 412
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 399,
+ 505,
+ 412
+ ],
+ "score": 1.0,
+ "content": "Figure 3: Node prediction results on Noisy Cora. Left: Effect of the maximum singular values on",
+ "type": "text"
+ }
+ ],
+ "index": 9
+ },
+ {
+ "bbox": [
+ 106,
+ 411,
+ 505,
+ 423
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 411,
+ 448,
+ 423
+ ],
+ "score": 1.0,
+ "content": "weights on model performance. The horizontal dotted line indicates the chance rate",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 448,
+ 411,
+ 481,
+ 422
+ ],
+ "score": 0.88,
+ "content": "( 3 0 . 2 \\% )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 481,
+ 411,
+ 505,
+ 423
+ ],
+ "score": 1.0,
+ "content": ". The",
+ "type": "text"
+ }
+ ],
+ "index": 10
+ },
+ {
+ "bbox": [
+ 105,
+ 421,
+ 505,
+ 435
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 421,
+ 505,
+ 435
+ ],
+ "score": 1.0,
+ "content": "error bar is the standard deviation of 3 trials. Right: Transition of maximum singular values during",
+ "type": "text"
+ }
+ ],
+ "index": 11
+ },
+ {
+ "bbox": [
+ 105,
+ 433,
+ 419,
+ 444
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 433,
+ 419,
+ 444
+ ],
+ "score": 1.0,
+ "content": "training. See Appendix I.3 for results using other datasets. Best view in color.",
+ "type": "text"
+ }
+ ],
+ "index": 12
+ }
+ ],
+ "index": 10.5
+ }
+ ],
+ "index": 8.75
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 464,
+ 505,
+ 544
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 464,
+ 505,
+ 477
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 464,
+ 170,
+ 477
+ ],
+ "score": 1.0,
+ "content": "associated with",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 170,
+ 465,
+ 181,
+ 476
+ ],
+ "score": 0.87,
+ "content": "\\lambda _ { 2 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 182,
+ 464,
+ 492,
+ 477
+ ],
+ "score": 1.0,
+ "content": "in two ways as described below. See Appendix H.1 for the concrete values of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 492,
+ 465,
+ 501,
+ 475
+ ],
+ "score": 0.8,
+ "content": "P",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 501,
+ 464,
+ 505,
+ 477
+ ],
+ "score": 1.0,
+ "content": ",",
+ "type": "text"
+ }
+ ],
+ "index": 13
+ },
+ {
+ "bbox": [
+ 106,
+ 476,
+ 504,
+ 488
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 478,
+ 112,
+ 486
+ ],
+ "score": 0.61,
+ "content": "e",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 113,
+ 476,
+ 133,
+ 488
+ ],
+ "score": 1.0,
+ "content": ", and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 134,
+ 476,
+ 146,
+ 486
+ ],
+ "score": 0.72,
+ "content": "W",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 146,
+ 476,
+ 296,
+ 488
+ ],
+ "score": 1.0,
+ "content": ". Figure 1 shows the visualization of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 296,
+ 476,
+ 305,
+ 486
+ ],
+ "score": 0.63,
+ "content": "V",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 306,
+ 476,
+ 498,
+ 488
+ ],
+ "score": 1.0,
+ "content": ". First, we choose the non-negative eigenvector",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 498,
+ 478,
+ 504,
+ 486
+ ],
+ "score": 0.67,
+ "content": "e",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 14
+ },
+ {
+ "bbox": [
+ 105,
+ 485,
+ 506,
+ 500
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 485,
+ 413,
+ 500
+ ],
+ "score": 1.0,
+ "content": "so that it satisfies Assumption 1 (Case 1). We see that the transition function",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 414,
+ 487,
+ 421,
+ 498
+ ],
+ "score": 0.85,
+ "content": "f",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 421,
+ 485,
+ 506,
+ 500
+ ],
+ "score": 1.0,
+ "content": "uniformly decreases",
+ "type": "text"
+ }
+ ],
+ "index": 15
+ },
+ {
+ "bbox": [
+ 105,
+ 497,
+ 505,
+ 511
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 497,
+ 181,
+ 511
+ ],
+ "score": 1.0,
+ "content": "the distance from",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 182,
+ 498,
+ 195,
+ 508
+ ],
+ "score": 0.8,
+ "content": "\\mathcal { M }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 195,
+ 497,
+ 505,
+ 511
+ ],
+ "score": 1.0,
+ "content": ". This is consistent with the consequence of Theorem 1. Next, we choose",
+ "type": "text"
+ }
+ ],
+ "index": 16
+ },
+ {
+ "bbox": [
+ 105,
+ 508,
+ 506,
+ 525
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 510,
+ 172,
+ 525
+ ],
+ "score": 1.0,
+ "content": "the eigenvector",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 172,
+ 508,
+ 236,
+ 523
+ ],
+ "score": 0.93,
+ "content": "\\boldsymbol { e } = \\left[ e _ { 1 } \\quad e _ { 2 } \\right] ^ { \\intercal }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 236,
+ 510,
+ 331,
+ 525
+ ],
+ "score": 1.0,
+ "content": "such that the signs of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 331,
+ 513,
+ 342,
+ 522
+ ],
+ "score": 0.84,
+ "content": "e _ { 1 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 342,
+ 510,
+ 362,
+ 525
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 362,
+ 512,
+ 373,
+ 522
+ ],
+ "score": 0.84,
+ "content": "e _ { 2 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 374,
+ 510,
+ 506,
+ 525
+ ],
+ "score": 1.0,
+ "content": "differ (Case 2), which violates",
+ "type": "text"
+ }
+ ],
+ "index": 17
+ },
+ {
+ "bbox": [
+ 106,
+ 522,
+ 505,
+ 534
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 522,
+ 225,
+ 534
+ ],
+ "score": 1.0,
+ "content": "Assumption 1. We see that",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 225,
+ 523,
+ 238,
+ 532
+ ],
+ "score": 0.81,
+ "content": "\\mathcal { M }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 239,
+ 522,
+ 332,
+ 534
+ ],
+ "score": 1.0,
+ "content": "is not invariant under",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 333,
+ 523,
+ 340,
+ 534
+ ],
+ "score": 0.85,
+ "content": "f",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 340,
+ 522,
+ 360,
+ 534
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 360,
+ 523,
+ 368,
+ 533
+ ],
+ "score": 0.85,
+ "content": "f",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 368,
+ 522,
+ 505,
+ 534
+ ],
+ "score": 1.0,
+ "content": "does not uniformly decrease the",
+ "type": "text"
+ }
+ ],
+ "index": 18
+ },
+ {
+ "bbox": [
+ 105,
+ 532,
+ 494,
+ 545
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 532,
+ 163,
+ 545
+ ],
+ "score": 1.0,
+ "content": "distance from",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 164,
+ 533,
+ 177,
+ 543
+ ],
+ "score": 0.79,
+ "content": "\\mathcal { M }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 177,
+ 532,
+ 494,
+ 545
+ ],
+ "score": 1.0,
+ "content": ". Therefore, we cannot remove the non-negativity assumption from Theorem 1.",
+ "type": "text"
+ }
+ ],
+ "index": 19
+ }
+ ],
+ "index": 16
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 105,
+ 557,
+ 352,
+ 569
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 557,
+ 352,
+ 570
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 557,
+ 352,
+ 570
+ ],
+ "score": 1.0,
+ "content": "6.2 SYNTHESIS DATA: DISTANCE TO INVARIANT SPACE",
+ "type": "text"
+ }
+ ],
+ "index": 20
+ }
+ ],
+ "index": 20
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 578,
+ 505,
+ 665
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 578,
+ 505,
+ 591
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 578,
+ 297,
+ 591
+ ],
+ "score": 1.0,
+ "content": "We evaluate the distance to the invariant space",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 297,
+ 579,
+ 311,
+ 589
+ ],
+ "score": 0.78,
+ "content": "\\mathcal { M }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 311,
+ 578,
+ 505,
+ 591
+ ],
+ "score": 1.0,
+ "content": "using synthesis data. We randomly generate an",
+ "type": "text"
+ }
+ ],
+ "index": 21
+ },
+ {
+ "bbox": [
+ 106,
+ 589,
+ 505,
+ 603
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 589,
+ 321,
+ 603
+ ],
+ "score": 1.0,
+ "content": "Erdos – R ˝ enyi graph, a GCN on it, and an input signal ´",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 321,
+ 590,
+ 341,
+ 601
+ ],
+ "score": 0.91,
+ "content": "X ^ { ( 0 ) }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 342,
+ 589,
+ 496,
+ 603
+ ],
+ "score": 1.0,
+ "content": ". We compute the distance between the",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 496,
+ 591,
+ 500,
+ 600
+ ],
+ "score": 0.64,
+ "content": "l",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 501,
+ 589,
+ 505,
+ 603
+ ],
+ "score": 1.0,
+ "content": "-",
+ "type": "text"
+ }
+ ],
+ "index": 22
+ },
+ {
+ "bbox": [
+ 105,
+ 601,
+ 506,
+ 615
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 601,
+ 197,
+ 615
+ ],
+ "score": 1.0,
+ "content": "th intermediate output",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 197,
+ 601,
+ 216,
+ 612
+ ],
+ "score": 0.9,
+ "content": "X ^ { ( l ) }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 216,
+ 601,
+ 311,
+ 615
+ ],
+ "score": 1.0,
+ "content": "and the invariant space",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 311,
+ 603,
+ 324,
+ 613
+ ],
+ "score": 0.76,
+ "content": "\\mathcal { M }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 324,
+ 601,
+ 438,
+ 615
+ ],
+ "score": 1.0,
+ "content": "for various edge probability",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 438,
+ 604,
+ 444,
+ 614
+ ],
+ "score": 0.77,
+ "content": "p",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 445,
+ 601,
+ 506,
+ 615
+ ],
+ "score": 1.0,
+ "content": "and maximum",
+ "type": "text"
+ }
+ ],
+ "index": 23
+ },
+ {
+ "bbox": [
+ 102,
+ 610,
+ 506,
+ 635
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 102,
+ 610,
+ 487,
+ 635
+ ],
+ "score": 1.0,
+ "content": "singular value s. Figure 2 plots the logarithm of the relative distance y(l) = log dM(X(l))dM(X(0))",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 483,
+ 616,
+ 506,
+ 630
+ ],
+ "score": 1.0,
+ "content": "with",
+ "type": "text"
+ }
+ ],
+ "index": 24
+ },
+ {
+ "bbox": [
+ 105,
+ 631,
+ 505,
+ 644
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 631,
+ 208,
+ 644
+ ],
+ "score": 1.0,
+ "content": "respect to the layer index",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 208,
+ 631,
+ 213,
+ 641
+ ],
+ "score": 0.31,
+ "content": "l",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 213,
+ 631,
+ 435,
+ 644
+ ],
+ "score": 1.0,
+ "content": ". From Theorem 1, we know that it is upper bounded by",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 435,
+ 632,
+ 501,
+ 643
+ ],
+ "score": 0.9,
+ "content": "y ( l ) = l \\log ( s \\lambda )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 502,
+ 631,
+ 505,
+ 644
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 25
+ },
+ {
+ "bbox": [
+ 106,
+ 641,
+ 505,
+ 654
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 641,
+ 362,
+ 654
+ ],
+ "score": 1.0,
+ "content": "We see that this bound well approximates the actual value when",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 363,
+ 643,
+ 375,
+ 652
+ ],
+ "score": 0.85,
+ "content": "s \\lambda",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 375,
+ 641,
+ 505,
+ 654
+ ],
+ "score": 1.0,
+ "content": "is small. On the other hand, it is",
+ "type": "text"
+ }
+ ],
+ "index": 26
+ },
+ {
+ "bbox": [
+ 106,
+ 653,
+ 448,
+ 665
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 653,
+ 166,
+ 665
+ ],
+ "score": 1.0,
+ "content": "loose for large",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 167,
+ 654,
+ 178,
+ 663
+ ],
+ "score": 0.76,
+ "content": "s \\lambda",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 178,
+ 653,
+ 295,
+ 665
+ ],
+ "score": 1.0,
+ "content": ". We leave tighter bounds for",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 295,
+ 654,
+ 311,
+ 664
+ ],
+ "score": 0.9,
+ "content": "d _ { \\mathcal { M } }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 311,
+ 653,
+ 448,
+ 665
+ ],
+ "score": 1.0,
+ "content": "in such a case for future research.",
+ "type": "text"
+ }
+ ],
+ "index": 27
+ }
+ ],
+ "index": 24
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 108,
+ 678,
+ 451,
+ 689
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 677,
+ 453,
+ 691
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 677,
+ 453,
+ 691
+ ],
+ "score": 1.0,
+ "content": "6.3 REAL DATA: EFFECT OF MAXIMUM SINGULAR VALUES ON PERFORMANCE",
+ "type": "text"
+ }
+ ],
+ "index": 28
+ }
+ ],
+ "index": 28
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 108,
+ 698,
+ 504,
+ 732
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 698,
+ 505,
+ 711
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 698,
+ 208,
+ 711
+ ],
+ "score": 1.0,
+ "content": "Theorem 2 implies that if",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 208,
+ 701,
+ 214,
+ 709
+ ],
+ "score": 0.77,
+ "content": "s",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 214,
+ 698,
+ 327,
+ 711
+ ],
+ "score": 1.0,
+ "content": "is smaller than the threshold",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 328,
+ 699,
+ 345,
+ 709
+ ],
+ "score": 0.89,
+ "content": "\\lambda ^ { - 1 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 345,
+ 698,
+ 505,
+ 711
+ ],
+ "score": 1.0,
+ "content": ", we cannot expect deep GCN to achieve",
+ "type": "text"
+ }
+ ],
+ "index": 29
+ },
+ {
+ "bbox": [
+ 105,
+ 709,
+ 506,
+ 722
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 709,
+ 427,
+ 722
+ ],
+ "score": 1.0,
+ "content": "good prediction accuracy. Conversely, if we can successfully train the model,",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 428,
+ 712,
+ 434,
+ 720
+ ],
+ "score": 0.69,
+ "content": "s",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 434,
+ 709,
+ 506,
+ 722
+ ],
+ "score": 1.0,
+ "content": "should avoid the",
+ "type": "text"
+ }
+ ],
+ "index": 30
+ },
+ {
+ "bbox": [
+ 105,
+ 720,
+ 419,
+ 734
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 720,
+ 134,
+ 734
+ ],
+ "score": 1.0,
+ "content": "region",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 135,
+ 720,
+ 169,
+ 731
+ ],
+ "score": 0.92,
+ "content": "s \\leq \\lambda ^ { - 1 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 170,
+ 720,
+ 419,
+ 734
+ ],
+ "score": 1.0,
+ "content": ". We empirically confirm these hypotheses using real datasets.",
+ "type": "text"
+ }
+ ],
+ "index": 31
+ }
+ ],
+ "index": 30
+ }
+ ],
+ "page_idx": 6,
+ "page_size": [
+ 612,
+ 792
+ ],
+ "discarded_blocks": [
+ {
+ "type": "discarded",
+ "bbox": [
+ 107,
+ 27,
+ 293,
+ 37
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 26,
+ 293,
+ 38
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 26,
+ 293,
+ 38
+ ],
+ "score": 1.0,
+ "content": "Published as a conference paper at ICLR 2020",
+ "type": "text"
+ }
+ ]
+ }
+ ]
+ },
+ {
+ "type": "discarded",
+ "bbox": [
+ 302,
+ 751,
+ 309,
+ 759
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 302,
+ 750,
+ 309,
+ 762
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 302,
+ 750,
+ 309,
+ 762
+ ],
+ "score": 1.0,
+ "content": "7",
+ "type": "text"
+ }
+ ]
+ }
+ ]
+ }
+ ],
+ "para_blocks": [
+ {
+ "type": "image",
+ "bbox": [
+ 151,
+ 85,
+ 459,
+ 207
+ ],
+ "blocks": [
+ {
+ "type": "image_body",
+ "bbox": [
+ 151,
+ 85,
+ 459,
+ 207
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 151,
+ 85,
+ 459,
+ 207
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 151,
+ 85,
+ 459,
+ 207
+ ],
+ "score": 0.97,
+ "type": "image",
+ "image_path": "2aeca4d04f65609a1fa1584f41938b2120f8df09b7dfae3a4549ab38b22e533d.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 1,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 151,
+ 85,
+ 459,
+ 125.66666666666666
+ ],
+ "spans": [],
+ "index": 0
+ },
+ {
+ "bbox": [
+ 151,
+ 125.66666666666666,
+ 459,
+ 166.33333333333331
+ ],
+ "spans": [],
+ "index": 1
+ },
+ {
+ "bbox": [
+ 151,
+ 166.33333333333331,
+ 459,
+ 206.99999999999997
+ ],
+ "spans": [],
+ "index": 2
+ }
+ ]
+ },
+ {
+ "type": "image_caption",
+ "bbox": [
+ 106,
+ 225,
+ 505,
+ 262
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 225,
+ 505,
+ 237
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 225,
+ 317,
+ 237
+ ],
+ "score": 1.0,
+ "content": "Figure 2: The actual distances to the invariant space",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 318,
+ 226,
+ 331,
+ 235
+ ],
+ "score": 0.79,
+ "content": "\\mathcal { M }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 331,
+ 225,
+ 505,
+ 237
+ ],
+ "score": 1.0,
+ "content": "and their upper bounds. Solid lines are the",
+ "type": "text"
+ }
+ ],
+ "index": 3
+ },
+ {
+ "bbox": [
+ 104,
+ 236,
+ 506,
+ 251
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 236,
+ 232,
+ 251
+ ],
+ "score": 1.0,
+ "content": "log relative distance defined by",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 232,
+ 236,
+ 372,
+ 250
+ ],
+ "score": 0.91,
+ "content": "y ( l ) = \\log ( d _ { \\mathcal { M } } ( \\mathbf { \\bar { { X } } } ^ { ( l ) } ) / d _ { \\mathcal { M } } ( X ^ { ( 0 ) } ) )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 372,
+ 236,
+ 506,
+ 251
+ ],
+ "score": 1.0,
+ "content": "and dotted lines are upper bound",
+ "type": "text"
+ }
+ ],
+ "index": 4
+ },
+ {
+ "bbox": [
+ 106,
+ 248,
+ 456,
+ 264
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 249,
+ 173,
+ 262
+ ],
+ "score": 0.92,
+ "content": "y ( l ) = l \\log ( s \\lambda )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 173,
+ 248,
+ 203,
+ 264
+ ],
+ "score": 1.0,
+ "content": ", where",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 204,
+ 249,
+ 224,
+ 260
+ ],
+ "score": 0.91,
+ "content": "X ^ { ( 0 ) }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 225,
+ 248,
+ 316,
+ 264
+ ],
+ "score": 1.0,
+ "content": "is the input signal and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 316,
+ 249,
+ 335,
+ 260
+ ],
+ "score": 0.89,
+ "content": "X ^ { ( l ) }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 335,
+ 248,
+ 414,
+ 264
+ ],
+ "score": 1.0,
+ "content": "is the output of the",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 414,
+ 250,
+ 418,
+ 260
+ ],
+ "score": 0.44,
+ "content": "l",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 419,
+ 248,
+ 456,
+ 264
+ ],
+ "score": 1.0,
+ "content": "-th layer.",
+ "type": "text"
+ }
+ ],
+ "index": 5
+ }
+ ],
+ "index": 4
+ }
+ ],
+ "index": 2.5
+ },
+ {
+ "type": "image",
+ "bbox": [
+ 109,
+ 276,
+ 497,
+ 385
+ ],
+ "blocks": [
+ {
+ "type": "image_body",
+ "bbox": [
+ 109,
+ 276,
+ 497,
+ 385
+ ],
+ "group_id": 1,
+ "lines": [
+ {
+ "bbox": [
+ 109,
+ 276,
+ 497,
+ 385
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 109,
+ 276,
+ 497,
+ 385
+ ],
+ "score": 0.963,
+ "type": "image",
+ "image_path": "8900fa8d6997b22156aeea03d2008a13643ccde54ceb13a5e4d7e973496948c4.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 7,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 109,
+ 276,
+ 497,
+ 312.3333333333333
+ ],
+ "spans": [],
+ "index": 6
+ },
+ {
+ "bbox": [
+ 109,
+ 312.3333333333333,
+ 497,
+ 348.66666666666663
+ ],
+ "spans": [],
+ "index": 7
+ },
+ {
+ "bbox": [
+ 109,
+ 348.66666666666663,
+ 497,
+ 384.99999999999994
+ ],
+ "spans": [],
+ "index": 8
+ }
+ ]
+ },
+ {
+ "type": "image_caption",
+ "bbox": [
+ 106,
+ 399,
+ 505,
+ 444
+ ],
+ "group_id": 1,
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 399,
+ 505,
+ 412
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 399,
+ 505,
+ 412
+ ],
+ "score": 1.0,
+ "content": "Figure 3: Node prediction results on Noisy Cora. Left: Effect of the maximum singular values on",
+ "type": "text"
+ }
+ ],
+ "index": 9
+ },
+ {
+ "bbox": [
+ 106,
+ 411,
+ 505,
+ 423
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 411,
+ 448,
+ 423
+ ],
+ "score": 1.0,
+ "content": "weights on model performance. The horizontal dotted line indicates the chance rate",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 448,
+ 411,
+ 481,
+ 422
+ ],
+ "score": 0.88,
+ "content": "( 3 0 . 2 \\% )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 481,
+ 411,
+ 505,
+ 423
+ ],
+ "score": 1.0,
+ "content": ". The",
+ "type": "text"
+ }
+ ],
+ "index": 10
+ },
+ {
+ "bbox": [
+ 105,
+ 421,
+ 505,
+ 435
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 421,
+ 505,
+ 435
+ ],
+ "score": 1.0,
+ "content": "error bar is the standard deviation of 3 trials. Right: Transition of maximum singular values during",
+ "type": "text"
+ }
+ ],
+ "index": 11
+ },
+ {
+ "bbox": [
+ 105,
+ 433,
+ 419,
+ 444
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 433,
+ 419,
+ 444
+ ],
+ "score": 1.0,
+ "content": "training. See Appendix I.3 for results using other datasets. Best view in color.",
+ "type": "text"
+ }
+ ],
+ "index": 12
+ }
+ ],
+ "index": 10.5
+ }
+ ],
+ "index": 8.75
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 464,
+ 505,
+ 544
+ ],
+ "lines": [],
+ "index": 16,
+ "bbox_fs": [
+ 105,
+ 464,
+ 506,
+ 545
+ ],
+ "lines_deleted": true
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 105,
+ 557,
+ 352,
+ 569
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 557,
+ 352,
+ 570
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 557,
+ 352,
+ 570
+ ],
+ "score": 1.0,
+ "content": "6.2 SYNTHESIS DATA: DISTANCE TO INVARIANT SPACE",
+ "type": "text"
+ }
+ ],
+ "index": 20
+ }
+ ],
+ "index": 20
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 578,
+ 505,
+ 665
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 578,
+ 505,
+ 591
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 578,
+ 297,
+ 591
+ ],
+ "score": 1.0,
+ "content": "We evaluate the distance to the invariant space",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 297,
+ 579,
+ 311,
+ 589
+ ],
+ "score": 0.78,
+ "content": "\\mathcal { M }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 311,
+ 578,
+ 505,
+ 591
+ ],
+ "score": 1.0,
+ "content": "using synthesis data. We randomly generate an",
+ "type": "text"
+ }
+ ],
+ "index": 21
+ },
+ {
+ "bbox": [
+ 106,
+ 589,
+ 505,
+ 603
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 589,
+ 321,
+ 603
+ ],
+ "score": 1.0,
+ "content": "Erdos – R ˝ enyi graph, a GCN on it, and an input signal ´",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 321,
+ 590,
+ 341,
+ 601
+ ],
+ "score": 0.91,
+ "content": "X ^ { ( 0 ) }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 342,
+ 589,
+ 496,
+ 603
+ ],
+ "score": 1.0,
+ "content": ". We compute the distance between the",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 496,
+ 591,
+ 500,
+ 600
+ ],
+ "score": 0.64,
+ "content": "l",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 501,
+ 589,
+ 505,
+ 603
+ ],
+ "score": 1.0,
+ "content": "-",
+ "type": "text"
+ }
+ ],
+ "index": 22
+ },
+ {
+ "bbox": [
+ 105,
+ 601,
+ 506,
+ 615
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 601,
+ 197,
+ 615
+ ],
+ "score": 1.0,
+ "content": "th intermediate output",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 197,
+ 601,
+ 216,
+ 612
+ ],
+ "score": 0.9,
+ "content": "X ^ { ( l ) }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 216,
+ 601,
+ 311,
+ 615
+ ],
+ "score": 1.0,
+ "content": "and the invariant space",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 311,
+ 603,
+ 324,
+ 613
+ ],
+ "score": 0.76,
+ "content": "\\mathcal { M }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 324,
+ 601,
+ 438,
+ 615
+ ],
+ "score": 1.0,
+ "content": "for various edge probability",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 438,
+ 604,
+ 444,
+ 614
+ ],
+ "score": 0.77,
+ "content": "p",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 445,
+ 601,
+ 506,
+ 615
+ ],
+ "score": 1.0,
+ "content": "and maximum",
+ "type": "text"
+ }
+ ],
+ "index": 23
+ },
+ {
+ "bbox": [
+ 102,
+ 610,
+ 506,
+ 635
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 102,
+ 610,
+ 487,
+ 635
+ ],
+ "score": 1.0,
+ "content": "singular value s. Figure 2 plots the logarithm of the relative distance y(l) = log dM(X(l))dM(X(0))",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 483,
+ 616,
+ 506,
+ 630
+ ],
+ "score": 1.0,
+ "content": "with",
+ "type": "text"
+ }
+ ],
+ "index": 24
+ },
+ {
+ "bbox": [
+ 105,
+ 631,
+ 505,
+ 644
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 631,
+ 208,
+ 644
+ ],
+ "score": 1.0,
+ "content": "respect to the layer index",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 208,
+ 631,
+ 213,
+ 641
+ ],
+ "score": 0.31,
+ "content": "l",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 213,
+ 631,
+ 435,
+ 644
+ ],
+ "score": 1.0,
+ "content": ". From Theorem 1, we know that it is upper bounded by",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 435,
+ 632,
+ 501,
+ 643
+ ],
+ "score": 0.9,
+ "content": "y ( l ) = l \\log ( s \\lambda )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 502,
+ 631,
+ 505,
+ 644
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 25
+ },
+ {
+ "bbox": [
+ 106,
+ 641,
+ 505,
+ 654
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 641,
+ 362,
+ 654
+ ],
+ "score": 1.0,
+ "content": "We see that this bound well approximates the actual value when",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 363,
+ 643,
+ 375,
+ 652
+ ],
+ "score": 0.85,
+ "content": "s \\lambda",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 375,
+ 641,
+ 505,
+ 654
+ ],
+ "score": 1.0,
+ "content": "is small. On the other hand, it is",
+ "type": "text"
+ }
+ ],
+ "index": 26
+ },
+ {
+ "bbox": [
+ 106,
+ 653,
+ 448,
+ 665
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 653,
+ 166,
+ 665
+ ],
+ "score": 1.0,
+ "content": "loose for large",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 167,
+ 654,
+ 178,
+ 663
+ ],
+ "score": 0.76,
+ "content": "s \\lambda",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 178,
+ 653,
+ 295,
+ 665
+ ],
+ "score": 1.0,
+ "content": ". We leave tighter bounds for",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 295,
+ 654,
+ 311,
+ 664
+ ],
+ "score": 0.9,
+ "content": "d _ { \\mathcal { M } }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 311,
+ 653,
+ 448,
+ 665
+ ],
+ "score": 1.0,
+ "content": "in such a case for future research.",
+ "type": "text"
+ }
+ ],
+ "index": 27
+ }
+ ],
+ "index": 24,
+ "bbox_fs": [
+ 102,
+ 578,
+ 506,
+ 665
+ ]
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 108,
+ 678,
+ 451,
+ 689
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 677,
+ 453,
+ 691
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 677,
+ 453,
+ 691
+ ],
+ "score": 1.0,
+ "content": "6.3 REAL DATA: EFFECT OF MAXIMUM SINGULAR VALUES ON PERFORMANCE",
+ "type": "text"
+ }
+ ],
+ "index": 28
+ }
+ ],
+ "index": 28
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 108,
+ 698,
+ 504,
+ 732
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 698,
+ 505,
+ 711
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 698,
+ 208,
+ 711
+ ],
+ "score": 1.0,
+ "content": "Theorem 2 implies that if",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 208,
+ 701,
+ 214,
+ 709
+ ],
+ "score": 0.77,
+ "content": "s",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 214,
+ 698,
+ 327,
+ 711
+ ],
+ "score": 1.0,
+ "content": "is smaller than the threshold",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 328,
+ 699,
+ 345,
+ 709
+ ],
+ "score": 0.89,
+ "content": "\\lambda ^ { - 1 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 345,
+ 698,
+ 505,
+ 711
+ ],
+ "score": 1.0,
+ "content": ", we cannot expect deep GCN to achieve",
+ "type": "text"
+ }
+ ],
+ "index": 29
+ },
+ {
+ "bbox": [
+ 105,
+ 709,
+ 506,
+ 722
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 709,
+ 427,
+ 722
+ ],
+ "score": 1.0,
+ "content": "good prediction accuracy. Conversely, if we can successfully train the model,",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 428,
+ 712,
+ 434,
+ 720
+ ],
+ "score": 0.69,
+ "content": "s",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 434,
+ 709,
+ 506,
+ 722
+ ],
+ "score": 1.0,
+ "content": "should avoid the",
+ "type": "text"
+ }
+ ],
+ "index": 30
+ },
+ {
+ "bbox": [
+ 105,
+ 720,
+ 419,
+ 734
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 720,
+ 134,
+ 734
+ ],
+ "score": 1.0,
+ "content": "region",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 135,
+ 720,
+ 169,
+ 731
+ ],
+ "score": 0.92,
+ "content": "s \\leq \\lambda ^ { - 1 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 170,
+ 720,
+ 419,
+ 734
+ ],
+ "score": 1.0,
+ "content": ". We empirically confirm these hypotheses using real datasets.",
+ "type": "text"
+ }
+ ],
+ "index": 31
+ }
+ ],
+ "index": 30,
+ "bbox_fs": [
+ 105,
+ 698,
+ 506,
+ 734
+ ]
+ }
+ ]
+ },
+ {
+ "preproc_blocks": [
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 82,
+ 505,
+ 181
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 83,
+ 505,
+ 94
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 83,
+ 505,
+ 94
+ ],
+ "score": 1.0,
+ "content": "We use Cora, CiteSeer, and PubMed (Sen et al., 2008), which are standard citation network datasets.",
+ "type": "text"
+ }
+ ],
+ "index": 0
+ },
+ {
+ "bbox": [
+ 106,
+ 93,
+ 506,
+ 106
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 93,
+ 506,
+ 106
+ ],
+ "score": 1.0,
+ "content": "The task is to classify the genre of papers using word occurrences and citation relationships. We",
+ "type": "text"
+ }
+ ],
+ "index": 1
+ },
+ {
+ "bbox": [
+ 105,
+ 105,
+ 506,
+ 117
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 105,
+ 506,
+ 117
+ ],
+ "score": 1.0,
+ "content": "regard each paper as a node and citation relationship as an edge. Due to space constraints, we focus",
+ "type": "text"
+ }
+ ],
+ "index": 2
+ },
+ {
+ "bbox": [
+ 105,
+ 115,
+ 505,
+ 128
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 115,
+ 505,
+ 128
+ ],
+ "score": 1.0,
+ "content": "on Cora in the main article. See Appendix H.3 and I.3 for the other datasets. The discussion in",
+ "type": "text"
+ }
+ ],
+ "index": 3
+ },
+ {
+ "bbox": [
+ 105,
+ 126,
+ 505,
+ 139
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 126,
+ 505,
+ 139
+ ],
+ "score": 1.0,
+ "content": "Section 5 implies that Theorem 2 can support a wide range of GCNs when the underlying graph",
+ "type": "text"
+ }
+ ],
+ "index": 4
+ },
+ {
+ "bbox": [
+ 105,
+ 137,
+ 505,
+ 149
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 137,
+ 505,
+ 149
+ ],
+ "score": 1.0,
+ "content": "is relatively dense. However, the citation networks are too sparse to examine the aforementioned",
+ "type": "text"
+ }
+ ],
+ "index": 5
+ },
+ {
+ "bbox": [
+ 105,
+ 147,
+ 505,
+ 160
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 148,
+ 354,
+ 160
+ ],
+ "score": 1.0,
+ "content": "hypotheses — Theorem 2 gives a non-trivial result only when",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 354,
+ 147,
+ 488,
+ 159
+ ],
+ "score": 0.91,
+ "content": "\\stackrel { \\cdot } { 1 } \\leq s < \\lambda ^ { - 1 } \\approx 1 + 3 . 6 2 \\times 1 0 ^ { - 3 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 488,
+ 148,
+ 505,
+ 160
+ ],
+ "score": 1.0,
+ "content": ". To",
+ "type": "text"
+ }
+ ],
+ "index": 6
+ },
+ {
+ "bbox": [
+ 105,
+ 159,
+ 505,
+ 172
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 159,
+ 505,
+ 172
+ ],
+ "score": 1.0,
+ "content": "circumvent this, we make noisy versions of citation networks by randomly adding edges to graphs.",
+ "type": "text"
+ }
+ ],
+ "index": 7
+ },
+ {
+ "bbox": [
+ 105,
+ 170,
+ 385,
+ 182
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 170,
+ 333,
+ 182
+ ],
+ "score": 1.0,
+ "content": "Through this manipulation, we can increase the value of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 333,
+ 170,
+ 350,
+ 180
+ ],
+ "score": 0.9,
+ "content": "\\lambda ^ { - 1 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 351,
+ 170,
+ 385,
+ 182
+ ],
+ "score": 1.0,
+ "content": "to 1.11.",
+ "type": "text"
+ }
+ ],
+ "index": 8
+ }
+ ],
+ "index": 4
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 187,
+ 505,
+ 286
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 186,
+ 505,
+ 200
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 186,
+ 505,
+ 200
+ ],
+ "score": 1.0,
+ "content": "Figure 3 (left) shows the accuracy for the test dataset in terms of the maximum singular values",
+ "type": "text"
+ }
+ ],
+ "index": 9
+ },
+ {
+ "bbox": [
+ 106,
+ 199,
+ 505,
+ 210
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 199,
+ 505,
+ 210
+ ],
+ "score": 1.0,
+ "content": "and the number of graph convolution layers. We can observe that when GCNs whose maximum",
+ "type": "text"
+ }
+ ],
+ "index": 10
+ },
+ {
+ "bbox": [
+ 105,
+ 208,
+ 506,
+ 222
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 208,
+ 167,
+ 222
+ ],
+ "score": 1.0,
+ "content": "singular value",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 168,
+ 212,
+ 174,
+ 219
+ ],
+ "score": 0.62,
+ "content": "s",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 174,
+ 208,
+ 260,
+ 222
+ ],
+ "score": 1.0,
+ "content": "is out of the region",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 261,
+ 208,
+ 301,
+ 219
+ ],
+ "score": 0.91,
+ "content": "s \\ < \\ \\lambda ^ { - 1 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 302,
+ 208,
+ 506,
+ 222
+ ],
+ "score": 1.0,
+ "content": "outperform those inside the region in almost all",
+ "type": "text"
+ }
+ ],
+ "index": 11
+ },
+ {
+ "bbox": [
+ 105,
+ 219,
+ 505,
+ 233
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 219,
+ 347,
+ 233
+ ],
+ "score": 1.0,
+ "content": "configurations. Furthermore, the accuracy of GCNs with",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 347,
+ 220,
+ 381,
+ 230
+ ],
+ "score": 0.9,
+ "content": "s ~ = ~ 1 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 382,
+ 219,
+ 505,
+ 233
+ ],
+ "score": 1.0,
+ "content": "are better than those without",
+ "type": "text"
+ }
+ ],
+ "index": 12
+ },
+ {
+ "bbox": [
+ 105,
+ 231,
+ 505,
+ 243
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 231,
+ 505,
+ 243
+ ],
+ "score": 1.0,
+ "content": "normalization (unnormalized). Figure 3 (right) shows the transition of the maximum singular values",
+ "type": "text"
+ }
+ ],
+ "index": 13
+ },
+ {
+ "bbox": [
+ 105,
+ 241,
+ 505,
+ 255
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 241,
+ 505,
+ 255
+ ],
+ "score": 1.0,
+ "content": "of the weights during training when we use a three-layered GCN. We can observe that the maximum",
+ "type": "text"
+ }
+ ],
+ "index": 14
+ },
+ {
+ "bbox": [
+ 105,
+ 252,
+ 505,
+ 266
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 252,
+ 164,
+ 266
+ ],
+ "score": 1.0,
+ "content": "singular value",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 165,
+ 255,
+ 171,
+ 263
+ ],
+ "score": 0.63,
+ "content": "s",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 171,
+ 252,
+ 288,
+ 266
+ ],
+ "score": 1.0,
+ "content": "does not shrink to the region",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 288,
+ 252,
+ 323,
+ 264
+ ],
+ "score": 0.92,
+ "content": "s \\leq \\lambda ^ { - 1 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 323,
+ 252,
+ 505,
+ 266
+ ],
+ "score": 1.0,
+ "content": ". In addition, when the layer size is small and",
+ "type": "text"
+ }
+ ],
+ "index": 15
+ },
+ {
+ "bbox": [
+ 105,
+ 263,
+ 505,
+ 277
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 263,
+ 323,
+ 277
+ ],
+ "score": 1.0,
+ "content": "predictive accuracy is high, GCNs gradually increase",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 323,
+ 266,
+ 329,
+ 274
+ ],
+ "score": 0.62,
+ "content": "s",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 330,
+ 263,
+ 505,
+ 277
+ ],
+ "score": 1.0,
+ "content": "from the initial value and avoid the region.",
+ "type": "text"
+ }
+ ],
+ "index": 16
+ },
+ {
+ "bbox": [
+ 106,
+ 275,
+ 389,
+ 287
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 275,
+ 389,
+ 287
+ ],
+ "score": 1.0,
+ "content": "In conclusion, the experiment results are consistent with the theorems.",
+ "type": "text"
+ }
+ ],
+ "index": 17
+ }
+ ],
+ "index": 13
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 108,
+ 303,
+ 497,
+ 314
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 303,
+ 498,
+ 315
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 303,
+ 498,
+ 315
+ ],
+ "score": 1.0,
+ "content": "6.4 REAL DATA: EFFECT OF SIGNAL COMPONENT PERPENDICULAR TO INVARIANT SPACE",
+ "type": "text"
+ }
+ ],
+ "index": 18
+ }
+ ],
+ "index": 18
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 325,
+ 355,
+ 468
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 324,
+ 355,
+ 338
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 324,
+ 228,
+ 338
+ ],
+ "score": 1.0,
+ "content": "We can decompose the output",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 228,
+ 326,
+ 238,
+ 335
+ ],
+ "score": 0.82,
+ "content": "X",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 238,
+ 324,
+ 294,
+ 338
+ ],
+ "score": 1.0,
+ "content": "of a model as",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 295,
+ 325,
+ 355,
+ 336
+ ],
+ "score": 0.92,
+ "content": "X = X _ { 0 } + X _ { 1 }",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 19
+ },
+ {
+ "bbox": [
+ 107,
+ 336,
+ 356,
+ 348
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 336,
+ 210,
+ 348
+ ],
+ "score": 0.72,
+ "content": "( X _ { 0 } \\in \\mathcal { M } , \\bar { X _ { 1 } } \\in \\mathcal { M } ^ { \\perp } )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 210,
+ 336,
+ 324,
+ 348
+ ],
+ "score": 1.0,
+ "content": ". According to the theory,",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 324,
+ 337,
+ 338,
+ 347
+ ],
+ "score": 0.87,
+ "content": "X _ { 0 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 339,
+ 336,
+ 356,
+ 348
+ ],
+ "score": 1.0,
+ "content": "has",
+ "type": "text"
+ }
+ ],
+ "index": 20
+ },
+ {
+ "bbox": [
+ 105,
+ 347,
+ 357,
+ 359
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 347,
+ 357,
+ 359
+ ],
+ "score": 1.0,
+ "content": "limited information for node classification. We hypothesize",
+ "type": "text"
+ }
+ ],
+ "index": 21
+ },
+ {
+ "bbox": [
+ 105,
+ 358,
+ 355,
+ 371
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 358,
+ 341,
+ 371
+ ],
+ "score": 1.0,
+ "content": "that the model emphasizes the perpendicular component",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 341,
+ 358,
+ 355,
+ 369
+ ],
+ "score": 0.87,
+ "content": "X _ { 1 }",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 22
+ },
+ {
+ "bbox": [
+ 105,
+ 369,
+ 357,
+ 381
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 369,
+ 357,
+ 381
+ ],
+ "score": 1.0,
+ "content": "to perform good predictions. To quantitatively evaluate it, we",
+ "type": "text"
+ }
+ ],
+ "index": 23
+ },
+ {
+ "bbox": [
+ 105,
+ 379,
+ 357,
+ 393
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 379,
+ 357,
+ 393
+ ],
+ "score": 1.0,
+ "content": "define the relative magnitude of the perpendicular component",
+ "type": "text"
+ }
+ ],
+ "index": 24
+ },
+ {
+ "bbox": [
+ 106,
+ 391,
+ 357,
+ 403
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 391,
+ 161,
+ 403
+ ],
+ "score": 1.0,
+ "content": "of the output",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 162,
+ 392,
+ 172,
+ 401
+ ],
+ "score": 0.8,
+ "content": "X",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 172,
+ 391,
+ 186,
+ 403
+ ],
+ "score": 1.0,
+ "content": "by",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 186,
+ 391,
+ 255,
+ 403
+ ],
+ "score": 0.93,
+ "content": "t ( \\mathbf { \\bar { \\boldsymbol { X } } } ) : = \\boldsymbol { X } _ { 1 } / \\boldsymbol { X } _ { 0 } ^ { \\mathbf { \\bar { \\alpha } } }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 256,
+ 391,
+ 357,
+ 403
+ ],
+ "score": 1.0,
+ "content": ". Figure 4 compares this",
+ "type": "text"
+ }
+ ],
+ "index": 25
+ },
+ {
+ "bbox": [
+ 105,
+ 402,
+ 358,
+ 414
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 402,
+ 358,
+ 414
+ ],
+ "score": 1.0,
+ "content": "quantity and the prediction accuracy on the noisy version of",
+ "type": "text"
+ }
+ ],
+ "index": 26
+ },
+ {
+ "bbox": [
+ 105,
+ 413,
+ 357,
+ 425
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 413,
+ 357,
+ 425
+ ],
+ "score": 1.0,
+ "content": "Cora (see Appendix I.4 for other datasets). We observe that",
+ "type": "text"
+ }
+ ],
+ "index": 27
+ },
+ {
+ "bbox": [
+ 105,
+ 423,
+ 357,
+ 436
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 423,
+ 248,
+ 436
+ ],
+ "score": 1.0,
+ "content": "these two quantities are correlated",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 248,
+ 424,
+ 295,
+ 435
+ ],
+ "score": 0.85,
+ "content": "R = 0 . 5 4 5 )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 295,
+ 423,
+ 357,
+ 436
+ ],
+ "score": 1.0,
+ "content": "). If we remove",
+ "type": "text"
+ }
+ ],
+ "index": 28
+ },
+ {
+ "bbox": [
+ 105,
+ 434,
+ 357,
+ 448
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 434,
+ 357,
+ 448
+ ],
+ "score": 1.0,
+ "content": "GCNs have only one layer (corresponding to right points in the",
+ "type": "text"
+ }
+ ],
+ "index": 29
+ },
+ {
+ "bbox": [
+ 105,
+ 446,
+ 357,
+ 457
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 446,
+ 357,
+ 457
+ ],
+ "score": 1.0,
+ "content": "figure), the correlation coefficient is 0.827. This result does not",
+ "type": "text"
+ }
+ ],
+ "index": 30
+ },
+ {
+ "bbox": [
+ 105,
+ 456,
+ 254,
+ 468
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 456,
+ 254,
+ 468
+ ],
+ "score": 1.0,
+ "content": "contradict to the hypothesis above 5.",
+ "type": "text"
+ }
+ ],
+ "index": 31
+ }
+ ],
+ "index": 25
+ },
+ {
+ "type": "image",
+ "bbox": [
+ 372,
+ 325,
+ 495,
+ 450
+ ],
+ "blocks": [
+ {
+ "type": "image_body",
+ "bbox": [
+ 372,
+ 325,
+ 495,
+ 450
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 372,
+ 325,
+ 495,
+ 450
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 372,
+ 325,
+ 495,
+ 450
+ ],
+ "score": 0.957,
+ "type": "image",
+ "image_path": "ca029b92d1bfb0988aea15f768e174e2b67f0ad352d4ddba46a86c712446b225.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 32.5,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 372,
+ 325,
+ 495,
+ 387.5
+ ],
+ "spans": [],
+ "index": 32
+ },
+ {
+ "bbox": [
+ 372,
+ 387.5,
+ 495,
+ 450.0
+ ],
+ "spans": [],
+ "index": 33
+ }
+ ]
+ },
+ {
+ "type": "image_caption",
+ "bbox": [
+ 364,
+ 465,
+ 505,
+ 488
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 363,
+ 465,
+ 505,
+ 478
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 363,
+ 465,
+ 405,
+ 478
+ ],
+ "score": 1.0,
+ "content": "Figure 4:",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 406,
+ 465,
+ 442,
+ 477
+ ],
+ "score": 0.9,
+ "content": "\\log t ( X )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 442,
+ 465,
+ 505,
+ 478
+ ],
+ "score": 1.0,
+ "content": "and prediction",
+ "type": "text"
+ }
+ ],
+ "index": 34
+ },
+ {
+ "bbox": [
+ 363,
+ 475,
+ 466,
+ 490
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 363,
+ 475,
+ 466,
+ 490
+ ],
+ "score": 1.0,
+ "content": "accuracy on Noisy Cora.",
+ "type": "text"
+ }
+ ],
+ "index": 35
+ }
+ ],
+ "index": 34.5
+ }
+ ],
+ "index": 33.5
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 108,
+ 487,
+ 190,
+ 501
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 486,
+ 192,
+ 504
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 486,
+ 192,
+ 504
+ ],
+ "score": 1.0,
+ "content": "7 DISCUSSION",
+ "type": "text"
+ }
+ ],
+ "index": 36
+ }
+ ],
+ "index": 36
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 514,
+ 504,
+ 592
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 514,
+ 506,
+ 528
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 514,
+ 506,
+ 528
+ ],
+ "score": 1.0,
+ "content": "Applicability to Graph NNs on Sparse Graphs. We have theoretically and empirically shown that",
+ "type": "text"
+ }
+ ],
+ "index": 37
+ },
+ {
+ "bbox": [
+ 105,
+ 525,
+ 506,
+ 538
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 525,
+ 402,
+ 538
+ ],
+ "score": 1.0,
+ "content": "when the underlying graph is sufficiently dense and large, the threshold",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 402,
+ 525,
+ 420,
+ 536
+ ],
+ "score": 0.9,
+ "content": "\\lambda ^ { - 1 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 420,
+ 525,
+ 506,
+ 538
+ ],
+ "score": 1.0,
+ "content": "is large (Theorem 2",
+ "type": "text"
+ }
+ ],
+ "index": 38
+ },
+ {
+ "bbox": [
+ 105,
+ 536,
+ 506,
+ 550
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 536,
+ 506,
+ 550
+ ],
+ "score": 1.0,
+ "content": "and Section 6.3), which means many graph CNNs are eligible. However, real-world graphs are not",
+ "type": "text"
+ }
+ ],
+ "index": 39
+ },
+ {
+ "bbox": [
+ 105,
+ 548,
+ 505,
+ 560
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 548,
+ 505,
+ 560
+ ],
+ "score": 1.0,
+ "content": "often dense, which means that Theorem 2 is applicable to very limited GCNs. In addition, Coja-",
+ "type": "text"
+ }
+ ],
+ "index": 40
+ },
+ {
+ "bbox": [
+ 105,
+ 559,
+ 506,
+ 572
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 559,
+ 414,
+ 572
+ ],
+ "score": 1.0,
+ "content": "Oghlan (2007) theoretically proved that if the expected average degree of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 414,
+ 559,
+ 437,
+ 571
+ ],
+ "score": 0.92,
+ "content": "G _ { N , p }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 437,
+ 559,
+ 506,
+ 572
+ ],
+ "score": 1.0,
+ "content": "is bounded, the",
+ "type": "text"
+ }
+ ],
+ "index": 41
+ },
+ {
+ "bbox": [
+ 105,
+ 569,
+ 506,
+ 583
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 569,
+ 345,
+ 583
+ ],
+ "score": 1.0,
+ "content": "smallest positive eigenvalue of the normalized Laplacian of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 345,
+ 570,
+ 367,
+ 582
+ ],
+ "score": 0.91,
+ "content": "G _ { N , p }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 368,
+ 569,
+ 378,
+ 583
+ ],
+ "score": 1.0,
+ "content": "is",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 378,
+ 570,
+ 397,
+ 582
+ ],
+ "score": 0.9,
+ "content": "o ( 1 )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 397,
+ 569,
+ 506,
+ 583
+ ],
+ "score": 1.0,
+ "content": "with high probability. The",
+ "type": "text"
+ }
+ ],
+ "index": 42
+ },
+ {
+ "bbox": [
+ 105,
+ 581,
+ 428,
+ 593
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 581,
+ 428,
+ 593
+ ],
+ "score": 1.0,
+ "content": "asymptotic behaviors of graph NNs on sparse graphs are left for future research.",
+ "type": "text"
+ }
+ ],
+ "index": 43
+ }
+ ],
+ "index": 40
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 597,
+ 505,
+ 697
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 597,
+ 505,
+ 610
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 597,
+ 505,
+ 610
+ ],
+ "score": 1.0,
+ "content": "Remedy for Over-smoothing. Based on our theory, we can propose several techniques for mitigat-",
+ "type": "text"
+ }
+ ],
+ "index": 44
+ },
+ {
+ "bbox": [
+ 105,
+ 608,
+ 504,
+ 622
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 608,
+ 504,
+ 622
+ ],
+ "score": 1.0,
+ "content": "ing the over-smoothing phenomena. One idea is to (randomly) sample edges in an underlying graph.",
+ "type": "text"
+ }
+ ],
+ "index": 45
+ },
+ {
+ "bbox": [
+ 105,
+ 619,
+ 505,
+ 633
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 619,
+ 505,
+ 633
+ ],
+ "score": 1.0,
+ "content": "The sparsity of practically available graphs could be a factor in the success of graph NNs. Assum-",
+ "type": "text"
+ }
+ ],
+ "index": 46
+ },
+ {
+ "bbox": [
+ 105,
+ 630,
+ 505,
+ 645
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 630,
+ 505,
+ 645
+ ],
+ "score": 1.0,
+ "content": "ing this hypothesis is correct, there is a possibility that we can relive the effect of over-smoothing",
+ "type": "text"
+ }
+ ],
+ "index": 47
+ },
+ {
+ "bbox": [
+ 105,
+ 642,
+ 506,
+ 655
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 642,
+ 506,
+ 655
+ ],
+ "score": 1.0,
+ "content": "by sparsification. Since we can never restore the information in pruned edges if we remove them",
+ "type": "text"
+ }
+ ],
+ "index": 48
+ },
+ {
+ "bbox": [
+ 105,
+ 653,
+ 505,
+ 665
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 653,
+ 505,
+ 665
+ ],
+ "score": 1.0,
+ "content": "permanently, random edge sampling could work better as FastGCN (Chen et al., 2018a) and Graph-",
+ "type": "text"
+ }
+ ],
+ "index": 49
+ },
+ {
+ "bbox": [
+ 106,
+ 664,
+ 505,
+ 676
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 664,
+ 505,
+ 676
+ ],
+ "score": 1.0,
+ "content": "SAGE (Hamilton et al., 2017) do. Another idea is to scale node representations (i.e., intermediate",
+ "type": "text"
+ }
+ ],
+ "index": 50
+ },
+ {
+ "bbox": [
+ 105,
+ 674,
+ 505,
+ 687
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 674,
+ 469,
+ 687
+ ],
+ "score": 1.0,
+ "content": "or final output of graph NNs) appropriately so that they keep away from the invariant space",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 470,
+ 675,
+ 483,
+ 685
+ ],
+ "score": 0.57,
+ "content": "\\mathcal { M }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 483,
+ 674,
+ 505,
+ 687
+ ],
+ "score": 1.0,
+ "content": ". Our",
+ "type": "text"
+ }
+ ],
+ "index": 51
+ },
+ {
+ "bbox": [
+ 105,
+ 685,
+ 505,
+ 699
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 685,
+ 505,
+ 699
+ ],
+ "score": 1.0,
+ "content": "proposed weight scaling mechanism takes this strategy. Recently, Zhao & Akoglu (2020) has pro-",
+ "type": "text"
+ }
+ ],
+ "index": 52
+ }
+ ],
+ "index": 48
+ }
+ ],
+ "page_idx": 7,
+ "page_size": [
+ 612,
+ 792
+ ],
+ "discarded_blocks": [
+ {
+ "type": "discarded",
+ "bbox": [
+ 107,
+ 712,
+ 504,
+ 732
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 118,
+ 709,
+ 506,
+ 724
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 118,
+ 709,
+ 506,
+ 724
+ ],
+ "score": 1.0,
+ "content": "5We cannot conclude that large perpendicular components are essential for good performance, since the",
+ "type": "text"
+ }
+ ]
+ },
+ {
+ "bbox": [
+ 106,
+ 720,
+ 324,
+ 733
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 720,
+ 198,
+ 733
+ ],
+ "score": 1.0,
+ "content": "maximum singular value",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 198,
+ 723,
+ 203,
+ 730
+ ],
+ "score": 0.54,
+ "content": "s",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 204,
+ 720,
+ 324,
+ 733
+ ],
+ "score": 1.0,
+ "content": "is correlated to the accuracy, too.",
+ "type": "text"
+ }
+ ]
+ }
+ ]
+ },
+ {
+ "type": "discarded",
+ "bbox": [
+ 108,
+ 27,
+ 293,
+ 37
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 26,
+ 294,
+ 38
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 26,
+ 294,
+ 38
+ ],
+ "score": 1.0,
+ "content": "Published as a conference paper at ICLR 2020",
+ "type": "text"
+ }
+ ]
+ }
+ ]
+ },
+ {
+ "type": "discarded",
+ "bbox": [
+ 302,
+ 752,
+ 308,
+ 760
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 302,
+ 750,
+ 309,
+ 761
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 302,
+ 750,
+ 309,
+ 761
+ ],
+ "score": 1.0,
+ "content": "8",
+ "type": "text"
+ }
+ ]
+ }
+ ]
+ }
+ ],
+ "para_blocks": [
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 82,
+ 505,
+ 181
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 83,
+ 505,
+ 94
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 83,
+ 505,
+ 94
+ ],
+ "score": 1.0,
+ "content": "We use Cora, CiteSeer, and PubMed (Sen et al., 2008), which are standard citation network datasets.",
+ "type": "text"
+ }
+ ],
+ "index": 0
+ },
+ {
+ "bbox": [
+ 106,
+ 93,
+ 506,
+ 106
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 93,
+ 506,
+ 106
+ ],
+ "score": 1.0,
+ "content": "The task is to classify the genre of papers using word occurrences and citation relationships. We",
+ "type": "text"
+ }
+ ],
+ "index": 1
+ },
+ {
+ "bbox": [
+ 105,
+ 105,
+ 506,
+ 117
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 105,
+ 506,
+ 117
+ ],
+ "score": 1.0,
+ "content": "regard each paper as a node and citation relationship as an edge. Due to space constraints, we focus",
+ "type": "text"
+ }
+ ],
+ "index": 2
+ },
+ {
+ "bbox": [
+ 105,
+ 115,
+ 505,
+ 128
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 115,
+ 505,
+ 128
+ ],
+ "score": 1.0,
+ "content": "on Cora in the main article. See Appendix H.3 and I.3 for the other datasets. The discussion in",
+ "type": "text"
+ }
+ ],
+ "index": 3
+ },
+ {
+ "bbox": [
+ 105,
+ 126,
+ 505,
+ 139
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 126,
+ 505,
+ 139
+ ],
+ "score": 1.0,
+ "content": "Section 5 implies that Theorem 2 can support a wide range of GCNs when the underlying graph",
+ "type": "text"
+ }
+ ],
+ "index": 4
+ },
+ {
+ "bbox": [
+ 105,
+ 137,
+ 505,
+ 149
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 137,
+ 505,
+ 149
+ ],
+ "score": 1.0,
+ "content": "is relatively dense. However, the citation networks are too sparse to examine the aforementioned",
+ "type": "text"
+ }
+ ],
+ "index": 5
+ },
+ {
+ "bbox": [
+ 105,
+ 147,
+ 505,
+ 160
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 148,
+ 354,
+ 160
+ ],
+ "score": 1.0,
+ "content": "hypotheses — Theorem 2 gives a non-trivial result only when",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 354,
+ 147,
+ 488,
+ 159
+ ],
+ "score": 0.91,
+ "content": "\\stackrel { \\cdot } { 1 } \\leq s < \\lambda ^ { - 1 } \\approx 1 + 3 . 6 2 \\times 1 0 ^ { - 3 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 488,
+ 148,
+ 505,
+ 160
+ ],
+ "score": 1.0,
+ "content": ". To",
+ "type": "text"
+ }
+ ],
+ "index": 6
+ },
+ {
+ "bbox": [
+ 105,
+ 159,
+ 505,
+ 172
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 159,
+ 505,
+ 172
+ ],
+ "score": 1.0,
+ "content": "circumvent this, we make noisy versions of citation networks by randomly adding edges to graphs.",
+ "type": "text"
+ }
+ ],
+ "index": 7
+ },
+ {
+ "bbox": [
+ 105,
+ 170,
+ 385,
+ 182
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 170,
+ 333,
+ 182
+ ],
+ "score": 1.0,
+ "content": "Through this manipulation, we can increase the value of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 333,
+ 170,
+ 350,
+ 180
+ ],
+ "score": 0.9,
+ "content": "\\lambda ^ { - 1 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 351,
+ 170,
+ 385,
+ 182
+ ],
+ "score": 1.0,
+ "content": "to 1.11.",
+ "type": "text"
+ }
+ ],
+ "index": 8
+ }
+ ],
+ "index": 4,
+ "bbox_fs": [
+ 105,
+ 83,
+ 506,
+ 182
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 187,
+ 505,
+ 286
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 186,
+ 505,
+ 200
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 186,
+ 505,
+ 200
+ ],
+ "score": 1.0,
+ "content": "Figure 3 (left) shows the accuracy for the test dataset in terms of the maximum singular values",
+ "type": "text"
+ }
+ ],
+ "index": 9
+ },
+ {
+ "bbox": [
+ 106,
+ 199,
+ 505,
+ 210
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 199,
+ 505,
+ 210
+ ],
+ "score": 1.0,
+ "content": "and the number of graph convolution layers. We can observe that when GCNs whose maximum",
+ "type": "text"
+ }
+ ],
+ "index": 10
+ },
+ {
+ "bbox": [
+ 105,
+ 208,
+ 506,
+ 222
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 208,
+ 167,
+ 222
+ ],
+ "score": 1.0,
+ "content": "singular value",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 168,
+ 212,
+ 174,
+ 219
+ ],
+ "score": 0.62,
+ "content": "s",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 174,
+ 208,
+ 260,
+ 222
+ ],
+ "score": 1.0,
+ "content": "is out of the region",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 261,
+ 208,
+ 301,
+ 219
+ ],
+ "score": 0.91,
+ "content": "s \\ < \\ \\lambda ^ { - 1 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 302,
+ 208,
+ 506,
+ 222
+ ],
+ "score": 1.0,
+ "content": "outperform those inside the region in almost all",
+ "type": "text"
+ }
+ ],
+ "index": 11
+ },
+ {
+ "bbox": [
+ 105,
+ 219,
+ 505,
+ 233
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 219,
+ 347,
+ 233
+ ],
+ "score": 1.0,
+ "content": "configurations. Furthermore, the accuracy of GCNs with",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 347,
+ 220,
+ 381,
+ 230
+ ],
+ "score": 0.9,
+ "content": "s ~ = ~ 1 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 382,
+ 219,
+ 505,
+ 233
+ ],
+ "score": 1.0,
+ "content": "are better than those without",
+ "type": "text"
+ }
+ ],
+ "index": 12
+ },
+ {
+ "bbox": [
+ 105,
+ 231,
+ 505,
+ 243
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 231,
+ 505,
+ 243
+ ],
+ "score": 1.0,
+ "content": "normalization (unnormalized). Figure 3 (right) shows the transition of the maximum singular values",
+ "type": "text"
+ }
+ ],
+ "index": 13
+ },
+ {
+ "bbox": [
+ 105,
+ 241,
+ 505,
+ 255
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 241,
+ 505,
+ 255
+ ],
+ "score": 1.0,
+ "content": "of the weights during training when we use a three-layered GCN. We can observe that the maximum",
+ "type": "text"
+ }
+ ],
+ "index": 14
+ },
+ {
+ "bbox": [
+ 105,
+ 252,
+ 505,
+ 266
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 252,
+ 164,
+ 266
+ ],
+ "score": 1.0,
+ "content": "singular value",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 165,
+ 255,
+ 171,
+ 263
+ ],
+ "score": 0.63,
+ "content": "s",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 171,
+ 252,
+ 288,
+ 266
+ ],
+ "score": 1.0,
+ "content": "does not shrink to the region",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 288,
+ 252,
+ 323,
+ 264
+ ],
+ "score": 0.92,
+ "content": "s \\leq \\lambda ^ { - 1 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 323,
+ 252,
+ 505,
+ 266
+ ],
+ "score": 1.0,
+ "content": ". In addition, when the layer size is small and",
+ "type": "text"
+ }
+ ],
+ "index": 15
+ },
+ {
+ "bbox": [
+ 105,
+ 263,
+ 505,
+ 277
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 263,
+ 323,
+ 277
+ ],
+ "score": 1.0,
+ "content": "predictive accuracy is high, GCNs gradually increase",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 323,
+ 266,
+ 329,
+ 274
+ ],
+ "score": 0.62,
+ "content": "s",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 330,
+ 263,
+ 505,
+ 277
+ ],
+ "score": 1.0,
+ "content": "from the initial value and avoid the region.",
+ "type": "text"
+ }
+ ],
+ "index": 16
+ },
+ {
+ "bbox": [
+ 106,
+ 275,
+ 389,
+ 287
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 275,
+ 389,
+ 287
+ ],
+ "score": 1.0,
+ "content": "In conclusion, the experiment results are consistent with the theorems.",
+ "type": "text"
+ }
+ ],
+ "index": 17
+ }
+ ],
+ "index": 13,
+ "bbox_fs": [
+ 105,
+ 186,
+ 506,
+ 287
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 108,
+ 303,
+ 497,
+ 314
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 303,
+ 498,
+ 315
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 303,
+ 498,
+ 315
+ ],
+ "score": 1.0,
+ "content": "6.4 REAL DATA: EFFECT OF SIGNAL COMPONENT PERPENDICULAR TO INVARIANT SPACE",
+ "type": "text"
+ }
+ ],
+ "index": 18
+ }
+ ],
+ "index": 18,
+ "bbox_fs": [
+ 106,
+ 303,
+ 498,
+ 315
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 325,
+ 355,
+ 468
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 324,
+ 355,
+ 338
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 324,
+ 228,
+ 338
+ ],
+ "score": 1.0,
+ "content": "We can decompose the output",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 228,
+ 326,
+ 238,
+ 335
+ ],
+ "score": 0.82,
+ "content": "X",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 238,
+ 324,
+ 294,
+ 338
+ ],
+ "score": 1.0,
+ "content": "of a model as",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 295,
+ 325,
+ 355,
+ 336
+ ],
+ "score": 0.92,
+ "content": "X = X _ { 0 } + X _ { 1 }",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 19
+ },
+ {
+ "bbox": [
+ 107,
+ 336,
+ 356,
+ 348
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 336,
+ 210,
+ 348
+ ],
+ "score": 0.72,
+ "content": "( X _ { 0 } \\in \\mathcal { M } , \\bar { X _ { 1 } } \\in \\mathcal { M } ^ { \\perp } )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 210,
+ 336,
+ 324,
+ 348
+ ],
+ "score": 1.0,
+ "content": ". According to the theory,",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 324,
+ 337,
+ 338,
+ 347
+ ],
+ "score": 0.87,
+ "content": "X _ { 0 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 339,
+ 336,
+ 356,
+ 348
+ ],
+ "score": 1.0,
+ "content": "has",
+ "type": "text"
+ }
+ ],
+ "index": 20
+ },
+ {
+ "bbox": [
+ 105,
+ 347,
+ 357,
+ 359
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 347,
+ 357,
+ 359
+ ],
+ "score": 1.0,
+ "content": "limited information for node classification. We hypothesize",
+ "type": "text"
+ }
+ ],
+ "index": 21
+ },
+ {
+ "bbox": [
+ 105,
+ 358,
+ 355,
+ 371
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 358,
+ 341,
+ 371
+ ],
+ "score": 1.0,
+ "content": "that the model emphasizes the perpendicular component",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 341,
+ 358,
+ 355,
+ 369
+ ],
+ "score": 0.87,
+ "content": "X _ { 1 }",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 22
+ },
+ {
+ "bbox": [
+ 105,
+ 369,
+ 357,
+ 381
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 369,
+ 357,
+ 381
+ ],
+ "score": 1.0,
+ "content": "to perform good predictions. To quantitatively evaluate it, we",
+ "type": "text"
+ }
+ ],
+ "index": 23
+ },
+ {
+ "bbox": [
+ 105,
+ 379,
+ 357,
+ 393
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 379,
+ 357,
+ 393
+ ],
+ "score": 1.0,
+ "content": "define the relative magnitude of the perpendicular component",
+ "type": "text"
+ }
+ ],
+ "index": 24
+ },
+ {
+ "bbox": [
+ 106,
+ 391,
+ 357,
+ 403
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 391,
+ 161,
+ 403
+ ],
+ "score": 1.0,
+ "content": "of the output",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 162,
+ 392,
+ 172,
+ 401
+ ],
+ "score": 0.8,
+ "content": "X",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 172,
+ 391,
+ 186,
+ 403
+ ],
+ "score": 1.0,
+ "content": "by",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 186,
+ 391,
+ 255,
+ 403
+ ],
+ "score": 0.93,
+ "content": "t ( \\mathbf { \\bar { \\boldsymbol { X } } } ) : = \\boldsymbol { X } _ { 1 } / \\boldsymbol { X } _ { 0 } ^ { \\mathbf { \\bar { \\alpha } } }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 256,
+ 391,
+ 357,
+ 403
+ ],
+ "score": 1.0,
+ "content": ". Figure 4 compares this",
+ "type": "text"
+ }
+ ],
+ "index": 25
+ },
+ {
+ "bbox": [
+ 105,
+ 402,
+ 358,
+ 414
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 402,
+ 358,
+ 414
+ ],
+ "score": 1.0,
+ "content": "quantity and the prediction accuracy on the noisy version of",
+ "type": "text"
+ }
+ ],
+ "index": 26
+ },
+ {
+ "bbox": [
+ 105,
+ 413,
+ 357,
+ 425
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 413,
+ 357,
+ 425
+ ],
+ "score": 1.0,
+ "content": "Cora (see Appendix I.4 for other datasets). We observe that",
+ "type": "text"
+ }
+ ],
+ "index": 27
+ },
+ {
+ "bbox": [
+ 105,
+ 423,
+ 357,
+ 436
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 423,
+ 248,
+ 436
+ ],
+ "score": 1.0,
+ "content": "these two quantities are correlated",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 248,
+ 424,
+ 295,
+ 435
+ ],
+ "score": 0.85,
+ "content": "R = 0 . 5 4 5 )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 295,
+ 423,
+ 357,
+ 436
+ ],
+ "score": 1.0,
+ "content": "). If we remove",
+ "type": "text"
+ }
+ ],
+ "index": 28
+ },
+ {
+ "bbox": [
+ 105,
+ 434,
+ 357,
+ 448
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 434,
+ 357,
+ 448
+ ],
+ "score": 1.0,
+ "content": "GCNs have only one layer (corresponding to right points in the",
+ "type": "text"
+ }
+ ],
+ "index": 29
+ },
+ {
+ "bbox": [
+ 105,
+ 446,
+ 357,
+ 457
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 446,
+ 357,
+ 457
+ ],
+ "score": 1.0,
+ "content": "figure), the correlation coefficient is 0.827. This result does not",
+ "type": "text"
+ }
+ ],
+ "index": 30
+ },
+ {
+ "bbox": [
+ 105,
+ 456,
+ 254,
+ 468
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 456,
+ 254,
+ 468
+ ],
+ "score": 1.0,
+ "content": "contradict to the hypothesis above 5.",
+ "type": "text"
+ }
+ ],
+ "index": 31
+ }
+ ],
+ "index": 25,
+ "bbox_fs": [
+ 105,
+ 324,
+ 358,
+ 468
+ ]
+ },
+ {
+ "type": "image",
+ "bbox": [
+ 372,
+ 325,
+ 495,
+ 450
+ ],
+ "blocks": [
+ {
+ "type": "image_body",
+ "bbox": [
+ 372,
+ 325,
+ 495,
+ 450
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 372,
+ 325,
+ 495,
+ 450
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 372,
+ 325,
+ 495,
+ 450
+ ],
+ "score": 0.957,
+ "type": "image",
+ "image_path": "ca029b92d1bfb0988aea15f768e174e2b67f0ad352d4ddba46a86c712446b225.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 32.5,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 372,
+ 325,
+ 495,
+ 387.5
+ ],
+ "spans": [],
+ "index": 32
+ },
+ {
+ "bbox": [
+ 372,
+ 387.5,
+ 495,
+ 450.0
+ ],
+ "spans": [],
+ "index": 33
+ }
+ ]
+ },
+ {
+ "type": "image_caption",
+ "bbox": [
+ 364,
+ 465,
+ 505,
+ 488
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 363,
+ 465,
+ 505,
+ 478
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 363,
+ 465,
+ 405,
+ 478
+ ],
+ "score": 1.0,
+ "content": "Figure 4:",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 406,
+ 465,
+ 442,
+ 477
+ ],
+ "score": 0.9,
+ "content": "\\log t ( X )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 442,
+ 465,
+ 505,
+ 478
+ ],
+ "score": 1.0,
+ "content": "and prediction",
+ "type": "text"
+ }
+ ],
+ "index": 34
+ },
+ {
+ "bbox": [
+ 363,
+ 475,
+ 466,
+ 490
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 363,
+ 475,
+ 466,
+ 490
+ ],
+ "score": 1.0,
+ "content": "accuracy on Noisy Cora.",
+ "type": "text"
+ }
+ ],
+ "index": 35
+ }
+ ],
+ "index": 34.5
+ }
+ ],
+ "index": 33.5
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 108,
+ 487,
+ 190,
+ 501
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 486,
+ 192,
+ 504
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 486,
+ 192,
+ 504
+ ],
+ "score": 1.0,
+ "content": "7 DISCUSSION",
+ "type": "text"
+ }
+ ],
+ "index": 36
+ }
+ ],
+ "index": 36
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 514,
+ 504,
+ 592
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 514,
+ 506,
+ 528
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 514,
+ 506,
+ 528
+ ],
+ "score": 1.0,
+ "content": "Applicability to Graph NNs on Sparse Graphs. We have theoretically and empirically shown that",
+ "type": "text"
+ }
+ ],
+ "index": 37
+ },
+ {
+ "bbox": [
+ 105,
+ 525,
+ 506,
+ 538
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 525,
+ 402,
+ 538
+ ],
+ "score": 1.0,
+ "content": "when the underlying graph is sufficiently dense and large, the threshold",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 402,
+ 525,
+ 420,
+ 536
+ ],
+ "score": 0.9,
+ "content": "\\lambda ^ { - 1 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 420,
+ 525,
+ 506,
+ 538
+ ],
+ "score": 1.0,
+ "content": "is large (Theorem 2",
+ "type": "text"
+ }
+ ],
+ "index": 38
+ },
+ {
+ "bbox": [
+ 105,
+ 536,
+ 506,
+ 550
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 536,
+ 506,
+ 550
+ ],
+ "score": 1.0,
+ "content": "and Section 6.3), which means many graph CNNs are eligible. However, real-world graphs are not",
+ "type": "text"
+ }
+ ],
+ "index": 39
+ },
+ {
+ "bbox": [
+ 105,
+ 548,
+ 505,
+ 560
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 548,
+ 505,
+ 560
+ ],
+ "score": 1.0,
+ "content": "often dense, which means that Theorem 2 is applicable to very limited GCNs. In addition, Coja-",
+ "type": "text"
+ }
+ ],
+ "index": 40
+ },
+ {
+ "bbox": [
+ 105,
+ 559,
+ 506,
+ 572
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 559,
+ 414,
+ 572
+ ],
+ "score": 1.0,
+ "content": "Oghlan (2007) theoretically proved that if the expected average degree of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 414,
+ 559,
+ 437,
+ 571
+ ],
+ "score": 0.92,
+ "content": "G _ { N , p }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 437,
+ 559,
+ 506,
+ 572
+ ],
+ "score": 1.0,
+ "content": "is bounded, the",
+ "type": "text"
+ }
+ ],
+ "index": 41
+ },
+ {
+ "bbox": [
+ 105,
+ 569,
+ 506,
+ 583
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 569,
+ 345,
+ 583
+ ],
+ "score": 1.0,
+ "content": "smallest positive eigenvalue of the normalized Laplacian of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 345,
+ 570,
+ 367,
+ 582
+ ],
+ "score": 0.91,
+ "content": "G _ { N , p }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 368,
+ 569,
+ 378,
+ 583
+ ],
+ "score": 1.0,
+ "content": "is",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 378,
+ 570,
+ 397,
+ 582
+ ],
+ "score": 0.9,
+ "content": "o ( 1 )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 397,
+ 569,
+ 506,
+ 583
+ ],
+ "score": 1.0,
+ "content": "with high probability. The",
+ "type": "text"
+ }
+ ],
+ "index": 42
+ },
+ {
+ "bbox": [
+ 105,
+ 581,
+ 428,
+ 593
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 581,
+ 428,
+ 593
+ ],
+ "score": 1.0,
+ "content": "asymptotic behaviors of graph NNs on sparse graphs are left for future research.",
+ "type": "text"
+ }
+ ],
+ "index": 43
+ }
+ ],
+ "index": 40,
+ "bbox_fs": [
+ 105,
+ 514,
+ 506,
+ 593
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 597,
+ 505,
+ 697
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 597,
+ 505,
+ 610
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 597,
+ 505,
+ 610
+ ],
+ "score": 1.0,
+ "content": "Remedy for Over-smoothing. Based on our theory, we can propose several techniques for mitigat-",
+ "type": "text"
+ }
+ ],
+ "index": 44
+ },
+ {
+ "bbox": [
+ 105,
+ 608,
+ 504,
+ 622
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 608,
+ 504,
+ 622
+ ],
+ "score": 1.0,
+ "content": "ing the over-smoothing phenomena. One idea is to (randomly) sample edges in an underlying graph.",
+ "type": "text"
+ }
+ ],
+ "index": 45
+ },
+ {
+ "bbox": [
+ 105,
+ 619,
+ 505,
+ 633
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 619,
+ 505,
+ 633
+ ],
+ "score": 1.0,
+ "content": "The sparsity of practically available graphs could be a factor in the success of graph NNs. Assum-",
+ "type": "text"
+ }
+ ],
+ "index": 46
+ },
+ {
+ "bbox": [
+ 105,
+ 630,
+ 505,
+ 645
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 630,
+ 505,
+ 645
+ ],
+ "score": 1.0,
+ "content": "ing this hypothesis is correct, there is a possibility that we can relive the effect of over-smoothing",
+ "type": "text"
+ }
+ ],
+ "index": 47
+ },
+ {
+ "bbox": [
+ 105,
+ 642,
+ 506,
+ 655
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 642,
+ 506,
+ 655
+ ],
+ "score": 1.0,
+ "content": "by sparsification. Since we can never restore the information in pruned edges if we remove them",
+ "type": "text"
+ }
+ ],
+ "index": 48
+ },
+ {
+ "bbox": [
+ 105,
+ 653,
+ 505,
+ 665
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 653,
+ 505,
+ 665
+ ],
+ "score": 1.0,
+ "content": "permanently, random edge sampling could work better as FastGCN (Chen et al., 2018a) and Graph-",
+ "type": "text"
+ }
+ ],
+ "index": 49
+ },
+ {
+ "bbox": [
+ 106,
+ 664,
+ 505,
+ 676
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 664,
+ 505,
+ 676
+ ],
+ "score": 1.0,
+ "content": "SAGE (Hamilton et al., 2017) do. Another idea is to scale node representations (i.e., intermediate",
+ "type": "text"
+ }
+ ],
+ "index": 50
+ },
+ {
+ "bbox": [
+ 105,
+ 674,
+ 505,
+ 687
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 674,
+ 469,
+ 687
+ ],
+ "score": 1.0,
+ "content": "or final output of graph NNs) appropriately so that they keep away from the invariant space",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 470,
+ 675,
+ 483,
+ 685
+ ],
+ "score": 0.57,
+ "content": "\\mathcal { M }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 483,
+ 674,
+ 505,
+ 687
+ ],
+ "score": 1.0,
+ "content": ". Our",
+ "type": "text"
+ }
+ ],
+ "index": 51
+ },
+ {
+ "bbox": [
+ 105,
+ 685,
+ 505,
+ 699
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 685,
+ 505,
+ 699
+ ],
+ "score": 1.0,
+ "content": "proposed weight scaling mechanism takes this strategy. Recently, Zhao & Akoglu (2020) has pro-",
+ "type": "text"
+ }
+ ],
+ "index": 52
+ },
+ {
+ "bbox": [
+ 105,
+ 82,
+ 506,
+ 95
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 82,
+ 506,
+ 95
+ ],
+ "score": 1.0,
+ "content": "posed PairNorm to alleviate the over-smoothing phenomena. Although the scaling target is different",
+ "type": "text",
+ "cross_page": true
+ }
+ ],
+ "index": 0
+ },
+ {
+ "bbox": [
+ 106,
+ 93,
+ 478,
+ 106
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 93,
+ 478,
+ 106
+ ],
+ "score": 1.0,
+ "content": "– they rescaled signals whereas we normalized weights – theirs and ours have similar spirits.",
+ "type": "text",
+ "cross_page": true
+ }
+ ],
+ "index": 1
+ }
+ ],
+ "index": 48,
+ "bbox_fs": [
+ 105,
+ 597,
+ 506,
+ 699
+ ]
+ }
+ ]
+ },
+ {
+ "preproc_blocks": [
+ {
+ "type": "text",
+ "bbox": [
+ 105,
+ 82,
+ 503,
+ 105
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 82,
+ 506,
+ 95
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 82,
+ 506,
+ 95
+ ],
+ "score": 1.0,
+ "content": "posed PairNorm to alleviate the over-smoothing phenomena. Although the scaling target is different",
+ "type": "text"
+ }
+ ],
+ "index": 0
+ },
+ {
+ "bbox": [
+ 106,
+ 93,
+ 478,
+ 106
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 93,
+ 478,
+ 106
+ ],
+ "score": 1.0,
+ "content": "– they rescaled signals whereas we normalized weights – theirs and ours have similar spirits.",
+ "type": "text"
+ }
+ ],
+ "index": 1
+ }
+ ],
+ "index": 0.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 110,
+ 504,
+ 187
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 110,
+ 506,
+ 123
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 110,
+ 506,
+ 123
+ ],
+ "score": 1.0,
+ "content": "Graph NNs with Large Weights. Our theory suggests that the maximum singular values of weights",
+ "type": "text"
+ }
+ ],
+ "index": 2
+ },
+ {
+ "bbox": [
+ 105,
+ 120,
+ 505,
+ 133
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 120,
+ 307,
+ 133
+ ],
+ "score": 1.0,
+ "content": "in a GCN should not be smaller than a threshold",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 307,
+ 121,
+ 325,
+ 132
+ ],
+ "score": 0.9,
+ "content": "\\bar { \\lambda } ^ { - 1 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 325,
+ 120,
+ 505,
+ 133
+ ],
+ "score": 1.0,
+ "content": "because it suffers from information loss for",
+ "type": "text"
+ }
+ ],
+ "index": 3
+ },
+ {
+ "bbox": [
+ 105,
+ 131,
+ 505,
+ 146
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 131,
+ 505,
+ 146
+ ],
+ "score": 1.0,
+ "content": "node classification. On the other hand, if the scale of weights are very large, the model complexity",
+ "type": "text"
+ }
+ ],
+ "index": 4
+ },
+ {
+ "bbox": [
+ 105,
+ 143,
+ 506,
+ 156
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 143,
+ 506,
+ 156
+ ],
+ "score": 1.0,
+ "content": "of the function class represented by graph NNs increases, which may cause large generalization",
+ "type": "text"
+ }
+ ],
+ "index": 5
+ },
+ {
+ "bbox": [
+ 105,
+ 154,
+ 506,
+ 167
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 154,
+ 506,
+ 167
+ ],
+ "score": 1.0,
+ "content": "errors. Therefore, from a statistical learning theory perspective, we conjecture that the graph NNs",
+ "type": "text"
+ }
+ ],
+ "index": 6
+ },
+ {
+ "bbox": [
+ 105,
+ 165,
+ 506,
+ 179
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 165,
+ 506,
+ 179
+ ],
+ "score": 1.0,
+ "content": "with too-large weights perform poorly, too. A trade-off should exist between the expressive power",
+ "type": "text"
+ }
+ ],
+ "index": 7
+ },
+ {
+ "bbox": [
+ 106,
+ 177,
+ 505,
+ 189
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 177,
+ 505,
+ 189
+ ],
+ "score": 1.0,
+ "content": "and model complexity and there should be a “sweet spot” on the weight scale that balances the two.",
+ "type": "text"
+ }
+ ],
+ "index": 8
+ }
+ ],
+ "index": 5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 193,
+ 505,
+ 325
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 192,
+ 505,
+ 206
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 192,
+ 505,
+ 206
+ ],
+ "score": 1.0,
+ "content": "Relation to Double Descent Phenomena. Belkin et al. (2019) pointed out that modern deep mod-",
+ "type": "text"
+ }
+ ],
+ "index": 9
+ },
+ {
+ "bbox": [
+ 105,
+ 204,
+ 505,
+ 216
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 204,
+ 505,
+ 216
+ ],
+ "score": 1.0,
+ "content": "els often have double descent risk curves: when a model is under-parameterized, a classical bias-",
+ "type": "text"
+ }
+ ],
+ "index": 10
+ },
+ {
+ "bbox": [
+ 106,
+ 214,
+ 505,
+ 228
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 214,
+ 505,
+ 228
+ ],
+ "score": 1.0,
+ "content": "variance trade-off occurs. However, once the model has a large capacity and perfectly fits the training",
+ "type": "text"
+ }
+ ],
+ "index": 11
+ },
+ {
+ "bbox": [
+ 105,
+ 226,
+ 505,
+ 239
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 226,
+ 505,
+ 239
+ ],
+ "score": 1.0,
+ "content": "data, the test error decreases as we increase the number of parameters. To the best of our knowl-",
+ "type": "text"
+ }
+ ],
+ "index": 12
+ },
+ {
+ "bbox": [
+ 106,
+ 238,
+ 505,
+ 249
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 238,
+ 505,
+ 249
+ ],
+ "score": 1.0,
+ "content": "edge, no literature reported the double descent phenomena for graph NNs (it is consistent with the",
+ "type": "text"
+ }
+ ],
+ "index": 13
+ },
+ {
+ "bbox": [
+ 105,
+ 248,
+ 505,
+ 260
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 248,
+ 505,
+ 260
+ ],
+ "score": 1.0,
+ "content": "picture of the classical U-shaped risk curve in the previous paragraph). It is known that double",
+ "type": "text"
+ }
+ ],
+ "index": 14
+ },
+ {
+ "bbox": [
+ 105,
+ 259,
+ 505,
+ 272
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 259,
+ 505,
+ 272
+ ],
+ "score": 1.0,
+ "content": "descent phenomena do not occur in some situations, especially depending on regularization types.",
+ "type": "text"
+ }
+ ],
+ "index": 15
+ },
+ {
+ "bbox": [
+ 105,
+ 270,
+ 505,
+ 282
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 270,
+ 505,
+ 282
+ ],
+ "score": 1.0,
+ "content": "For example, while Belkin et al. (2019) employed the interpolating hypothesis with the minimum",
+ "type": "text"
+ }
+ ],
+ "index": 16
+ },
+ {
+ "bbox": [
+ 105,
+ 281,
+ 505,
+ 293
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 281,
+ 505,
+ 293
+ ],
+ "score": 1.0,
+ "content": "norm, Mei & Montanari (2019) found that the double descent was alleviated or disappeared when",
+ "type": "text"
+ }
+ ],
+ "index": 17
+ },
+ {
+ "bbox": [
+ 105,
+ 291,
+ 505,
+ 305
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 291,
+ 505,
+ 305
+ ],
+ "score": 1.0,
+ "content": "they used Ridge-type regularization techniques. Therefore, one can hypothesize the over-smoothing",
+ "type": "text"
+ }
+ ],
+ "index": 18
+ },
+ {
+ "bbox": [
+ 105,
+ 303,
+ 505,
+ 316
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 303,
+ 505,
+ 316
+ ],
+ "score": 1.0,
+ "content": "is a cause or consequence of regularization that is more like a Ridge-type rather than minimum-norm",
+ "type": "text"
+ }
+ ],
+ "index": 19
+ },
+ {
+ "bbox": [
+ 106,
+ 313,
+ 167,
+ 325
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 313,
+ 167,
+ 325
+ ],
+ "score": 1.0,
+ "content": "inductive bias.",
+ "type": "text"
+ }
+ ],
+ "index": 20
+ }
+ ],
+ "index": 14.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 330,
+ 505,
+ 419
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 330,
+ 505,
+ 343
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 330,
+ 505,
+ 343
+ ],
+ "score": 1.0,
+ "content": "Limitations in Graph NN Architectures. Our analysis is limited to graph NNs with the ReLU",
+ "type": "text"
+ }
+ ],
+ "index": 21
+ },
+ {
+ "bbox": [
+ 106,
+ 342,
+ 505,
+ 354
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 342,
+ 505,
+ 354
+ ],
+ "score": 1.0,
+ "content": "activation function because we implicitly use the property that ReLU is a projection onto the cone",
+ "type": "text"
+ }
+ ],
+ "index": 22
+ },
+ {
+ "bbox": [
+ 107,
+ 352,
+ 505,
+ 365
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 352,
+ 145,
+ 365
+ ],
+ "score": 0.92,
+ "content": "\\{ X \\geq 0 \\}",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 146,
+ 352,
+ 505,
+ 365
+ ],
+ "score": 1.0,
+ "content": "(Appendix A, Lemma 3). This fact enables the ReLU function to get along with the non-",
+ "type": "text"
+ }
+ ],
+ "index": 23
+ },
+ {
+ "bbox": [
+ 105,
+ 363,
+ 505,
+ 376
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 363,
+ 505,
+ 376
+ ],
+ "score": 1.0,
+ "content": "negativity of eigenvectors associated with the largest eigenvalues. Therefore, it is far from trivial to",
+ "type": "text"
+ }
+ ],
+ "index": 24
+ },
+ {
+ "bbox": [
+ 105,
+ 375,
+ 505,
+ 387
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 375,
+ 505,
+ 387
+ ],
+ "score": 1.0,
+ "content": "extend our results to other activation functions such as the sigmoid function or Leaky ReLU (Maas",
+ "type": "text"
+ }
+ ],
+ "index": 25
+ },
+ {
+ "bbox": [
+ 105,
+ 385,
+ 505,
+ 398
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 385,
+ 505,
+ 398
+ ],
+ "score": 1.0,
+ "content": "et al., 2013). Another point is that our formulation considers the update operation (Gilmer et al.,",
+ "type": "text"
+ }
+ ],
+ "index": 26
+ },
+ {
+ "bbox": [
+ 105,
+ 396,
+ 506,
+ 409
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 396,
+ 506,
+ 409
+ ],
+ "score": 1.0,
+ "content": "2017) of graph NNs only and does not take readout operations into account. In particular, we cannot",
+ "type": "text"
+ }
+ ],
+ "index": 27
+ },
+ {
+ "bbox": [
+ 105,
+ 406,
+ 452,
+ 421
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 406,
+ 452,
+ 421
+ ],
+ "score": 1.0,
+ "content": "directly apply our theory to graph classification tasks in which each sample is a graph.",
+ "type": "text"
+ }
+ ],
+ "index": 28
+ }
+ ],
+ "index": 24.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 424,
+ 505,
+ 523
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 424,
+ 505,
+ 436
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 424,
+ 505,
+ 436
+ ],
+ "score": 1.0,
+ "content": "Over-smoothing of Residual GNNs. Considering the correspondence of graph NNs and Markov",
+ "type": "text"
+ }
+ ],
+ "index": 29
+ },
+ {
+ "bbox": [
+ 105,
+ 435,
+ 505,
+ 448
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 435,
+ 505,
+ 448
+ ],
+ "score": 1.0,
+ "content": "processes (see Appendix F), one can imagine that residual links do not contribute to alleviating",
+ "type": "text"
+ }
+ ],
+ "index": 30
+ },
+ {
+ "bbox": [
+ 105,
+ 446,
+ 505,
+ 459
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 446,
+ 505,
+ 459
+ ],
+ "score": 1.0,
+ "content": "the over-smoothing phenomena because adding residual connections to a graph NN corresponds",
+ "type": "text"
+ }
+ ],
+ "index": 31
+ },
+ {
+ "bbox": [
+ 105,
+ 457,
+ 506,
+ 470
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 457,
+ 506,
+ 470
+ ],
+ "score": 1.0,
+ "content": "to converting a Markov process to its lazy version. When a Markov process converges to a stable",
+ "type": "text"
+ }
+ ],
+ "index": 32
+ },
+ {
+ "bbox": [
+ 105,
+ 467,
+ 506,
+ 482
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 467,
+ 506,
+ 482
+ ],
+ "score": 1.0,
+ "content": "distribution, the corresponding lazy process also converges eventually under certain conditions. It",
+ "type": "text"
+ }
+ ],
+ "index": 33
+ },
+ {
+ "bbox": [
+ 106,
+ 479,
+ 505,
+ 491
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 479,
+ 505,
+ 491
+ ],
+ "score": 1.0,
+ "content": "implies that residual links might not be helpful. However, Li et al. (2019) reported that graph NNs",
+ "type": "text"
+ }
+ ],
+ "index": 34
+ },
+ {
+ "bbox": [
+ 106,
+ 491,
+ 504,
+ 502
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 491,
+ 504,
+ 502
+ ],
+ "score": 1.0,
+ "content": "with as many as 56 layers performed well if they added residual connections. Considering that,",
+ "type": "text"
+ }
+ ],
+ "index": 35
+ },
+ {
+ "bbox": [
+ 105,
+ 500,
+ 505,
+ 513
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 500,
+ 505,
+ 513
+ ],
+ "score": 1.0,
+ "content": "the situation could be more complicated than our intuitions. The analysis of the role of residual",
+ "type": "text"
+ }
+ ],
+ "index": 36
+ },
+ {
+ "bbox": [
+ 106,
+ 513,
+ 387,
+ 524
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 513,
+ 387,
+ 524
+ ],
+ "score": 1.0,
+ "content": "connections in graph NNs is a promising direction for future research.",
+ "type": "text"
+ }
+ ],
+ "index": 37
+ }
+ ],
+ "index": 33
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 107,
+ 554,
+ 195,
+ 567
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 553,
+ 198,
+ 570
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 553,
+ 198,
+ 570
+ ],
+ "score": 1.0,
+ "content": "8 CONCLUSION",
+ "type": "text"
+ }
+ ],
+ "index": 38
+ }
+ ],
+ "index": 38
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 589,
+ 505,
+ 732
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 588,
+ 506,
+ 601
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 588,
+ 506,
+ 601
+ ],
+ "score": 1.0,
+ "content": "In this paper, to understand the empirically observed phenomena that deep non-linear graph NNs",
+ "type": "text"
+ }
+ ],
+ "index": 39
+ },
+ {
+ "bbox": [
+ 106,
+ 601,
+ 505,
+ 612
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 601,
+ 505,
+ 612
+ ],
+ "score": 1.0,
+ "content": "do not perform well, we analyzed their asymptotic behaviors by interpreting them as a dynamical",
+ "type": "text"
+ }
+ ],
+ "index": 40
+ },
+ {
+ "bbox": [
+ 105,
+ 610,
+ 506,
+ 624
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 610,
+ 506,
+ 624
+ ],
+ "score": 1.0,
+ "content": "system that includes GCN and Markov process as special cases. We gave theoretical conditions",
+ "type": "text"
+ }
+ ],
+ "index": 41
+ },
+ {
+ "bbox": [
+ 105,
+ 621,
+ 505,
+ 635
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 621,
+ 505,
+ 635
+ ],
+ "score": 1.0,
+ "content": "under which GCNs suffer from the information loss in the limit of infinite layers. Our theory directly",
+ "type": "text"
+ }
+ ],
+ "index": 42
+ },
+ {
+ "bbox": [
+ 105,
+ 632,
+ 506,
+ 647
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 632,
+ 506,
+ 647
+ ],
+ "score": 1.0,
+ "content": "related the expressive power of graph NNs and topological information of the underlying graphs via",
+ "type": "text"
+ }
+ ],
+ "index": 43
+ },
+ {
+ "bbox": [
+ 105,
+ 644,
+ 505,
+ 656
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 644,
+ 505,
+ 656
+ ],
+ "score": 1.0,
+ "content": "spectra of the Laplacian. It enabled us to leverage spectral and random graph theory to analyze the",
+ "type": "text"
+ }
+ ],
+ "index": 44
+ },
+ {
+ "bbox": [
+ 105,
+ 654,
+ 506,
+ 668
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 654,
+ 506,
+ 668
+ ],
+ "score": 1.0,
+ "content": "expressive power of graph NNs. To demonstrate this, we considered GCN on the Erdos – R ˝ enyi ´",
+ "type": "text"
+ }
+ ],
+ "index": 45
+ },
+ {
+ "bbox": [
+ 105,
+ 664,
+ 506,
+ 680
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 664,
+ 506,
+ 680
+ ],
+ "score": 1.0,
+ "content": "graph as an example and showed that when the underlying graph is sufficiently dense and large,",
+ "type": "text"
+ }
+ ],
+ "index": 46
+ },
+ {
+ "bbox": [
+ 104,
+ 676,
+ 506,
+ 691
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 676,
+ 506,
+ 691
+ ],
+ "score": 1.0,
+ "content": "a wide range of GCNs on the graph suffer from information loss. Based on the theory, we gave",
+ "type": "text"
+ }
+ ],
+ "index": 47
+ },
+ {
+ "bbox": [
+ 105,
+ 688,
+ 505,
+ 700
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 688,
+ 505,
+ 700
+ ],
+ "score": 1.0,
+ "content": "a principled guideline for how to determine the scale of weights of graph NNs and empirically",
+ "type": "text"
+ }
+ ],
+ "index": 48
+ },
+ {
+ "bbox": [
+ 106,
+ 699,
+ 505,
+ 711
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 699,
+ 505,
+ 711
+ ],
+ "score": 1.0,
+ "content": "showed that the weight normalization implied by our theory performed well in real datasets. One",
+ "type": "text"
+ }
+ ],
+ "index": 49
+ },
+ {
+ "bbox": [
+ 105,
+ 709,
+ 505,
+ 723
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 709,
+ 505,
+ 723
+ ],
+ "score": 1.0,
+ "content": "promising direction of research is to analyze the optimization and statistical properties such as the",
+ "type": "text"
+ }
+ ],
+ "index": 50
+ },
+ {
+ "bbox": [
+ 105,
+ 720,
+ 503,
+ 734
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 720,
+ 503,
+ 734
+ ],
+ "score": 1.0,
+ "content": "generalization power (Verma & Zhang, 2019) of graph NNs via spectral and random graph theories.",
+ "type": "text"
+ }
+ ],
+ "index": 51
+ }
+ ],
+ "index": 45
+ }
+ ],
+ "page_idx": 8,
+ "page_size": [
+ 612,
+ 792
+ ],
+ "discarded_blocks": [
+ {
+ "type": "discarded",
+ "bbox": [
+ 108,
+ 27,
+ 293,
+ 37
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 26,
+ 294,
+ 38
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 26,
+ 294,
+ 38
+ ],
+ "score": 1.0,
+ "content": "Published as a conference paper at ICLR 2020",
+ "type": "text"
+ }
+ ]
+ }
+ ]
+ },
+ {
+ "type": "discarded",
+ "bbox": [
+ 302,
+ 751,
+ 308,
+ 759
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 302,
+ 751,
+ 309,
+ 762
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 302,
+ 751,
+ 309,
+ 762
+ ],
+ "score": 1.0,
+ "content": "9",
+ "type": "text"
+ }
+ ]
+ }
+ ]
+ }
+ ],
+ "para_blocks": [
+ {
+ "type": "text",
+ "bbox": [
+ 105,
+ 82,
+ 503,
+ 105
+ ],
+ "lines": [],
+ "index": 0.5,
+ "bbox_fs": [
+ 105,
+ 82,
+ 506,
+ 106
+ ],
+ "lines_deleted": true
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 110,
+ 504,
+ 187
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 110,
+ 506,
+ 123
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 110,
+ 506,
+ 123
+ ],
+ "score": 1.0,
+ "content": "Graph NNs with Large Weights. Our theory suggests that the maximum singular values of weights",
+ "type": "text"
+ }
+ ],
+ "index": 2
+ },
+ {
+ "bbox": [
+ 105,
+ 120,
+ 505,
+ 133
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 120,
+ 307,
+ 133
+ ],
+ "score": 1.0,
+ "content": "in a GCN should not be smaller than a threshold",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 307,
+ 121,
+ 325,
+ 132
+ ],
+ "score": 0.9,
+ "content": "\\bar { \\lambda } ^ { - 1 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 325,
+ 120,
+ 505,
+ 133
+ ],
+ "score": 1.0,
+ "content": "because it suffers from information loss for",
+ "type": "text"
+ }
+ ],
+ "index": 3
+ },
+ {
+ "bbox": [
+ 105,
+ 131,
+ 505,
+ 146
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 131,
+ 505,
+ 146
+ ],
+ "score": 1.0,
+ "content": "node classification. On the other hand, if the scale of weights are very large, the model complexity",
+ "type": "text"
+ }
+ ],
+ "index": 4
+ },
+ {
+ "bbox": [
+ 105,
+ 143,
+ 506,
+ 156
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 143,
+ 506,
+ 156
+ ],
+ "score": 1.0,
+ "content": "of the function class represented by graph NNs increases, which may cause large generalization",
+ "type": "text"
+ }
+ ],
+ "index": 5
+ },
+ {
+ "bbox": [
+ 105,
+ 154,
+ 506,
+ 167
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 154,
+ 506,
+ 167
+ ],
+ "score": 1.0,
+ "content": "errors. Therefore, from a statistical learning theory perspective, we conjecture that the graph NNs",
+ "type": "text"
+ }
+ ],
+ "index": 6
+ },
+ {
+ "bbox": [
+ 105,
+ 165,
+ 506,
+ 179
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 165,
+ 506,
+ 179
+ ],
+ "score": 1.0,
+ "content": "with too-large weights perform poorly, too. A trade-off should exist between the expressive power",
+ "type": "text"
+ }
+ ],
+ "index": 7
+ },
+ {
+ "bbox": [
+ 106,
+ 177,
+ 505,
+ 189
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 177,
+ 505,
+ 189
+ ],
+ "score": 1.0,
+ "content": "and model complexity and there should be a “sweet spot” on the weight scale that balances the two.",
+ "type": "text"
+ }
+ ],
+ "index": 8
+ }
+ ],
+ "index": 5,
+ "bbox_fs": [
+ 105,
+ 110,
+ 506,
+ 189
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 193,
+ 505,
+ 325
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 192,
+ 505,
+ 206
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 192,
+ 505,
+ 206
+ ],
+ "score": 1.0,
+ "content": "Relation to Double Descent Phenomena. Belkin et al. (2019) pointed out that modern deep mod-",
+ "type": "text"
+ }
+ ],
+ "index": 9
+ },
+ {
+ "bbox": [
+ 105,
+ 204,
+ 505,
+ 216
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 204,
+ 505,
+ 216
+ ],
+ "score": 1.0,
+ "content": "els often have double descent risk curves: when a model is under-parameterized, a classical bias-",
+ "type": "text"
+ }
+ ],
+ "index": 10
+ },
+ {
+ "bbox": [
+ 106,
+ 214,
+ 505,
+ 228
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 214,
+ 505,
+ 228
+ ],
+ "score": 1.0,
+ "content": "variance trade-off occurs. However, once the model has a large capacity and perfectly fits the training",
+ "type": "text"
+ }
+ ],
+ "index": 11
+ },
+ {
+ "bbox": [
+ 105,
+ 226,
+ 505,
+ 239
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 226,
+ 505,
+ 239
+ ],
+ "score": 1.0,
+ "content": "data, the test error decreases as we increase the number of parameters. To the best of our knowl-",
+ "type": "text"
+ }
+ ],
+ "index": 12
+ },
+ {
+ "bbox": [
+ 106,
+ 238,
+ 505,
+ 249
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 238,
+ 505,
+ 249
+ ],
+ "score": 1.0,
+ "content": "edge, no literature reported the double descent phenomena for graph NNs (it is consistent with the",
+ "type": "text"
+ }
+ ],
+ "index": 13
+ },
+ {
+ "bbox": [
+ 105,
+ 248,
+ 505,
+ 260
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 248,
+ 505,
+ 260
+ ],
+ "score": 1.0,
+ "content": "picture of the classical U-shaped risk curve in the previous paragraph). It is known that double",
+ "type": "text"
+ }
+ ],
+ "index": 14
+ },
+ {
+ "bbox": [
+ 105,
+ 259,
+ 505,
+ 272
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 259,
+ 505,
+ 272
+ ],
+ "score": 1.0,
+ "content": "descent phenomena do not occur in some situations, especially depending on regularization types.",
+ "type": "text"
+ }
+ ],
+ "index": 15
+ },
+ {
+ "bbox": [
+ 105,
+ 270,
+ 505,
+ 282
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 270,
+ 505,
+ 282
+ ],
+ "score": 1.0,
+ "content": "For example, while Belkin et al. (2019) employed the interpolating hypothesis with the minimum",
+ "type": "text"
+ }
+ ],
+ "index": 16
+ },
+ {
+ "bbox": [
+ 105,
+ 281,
+ 505,
+ 293
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 281,
+ 505,
+ 293
+ ],
+ "score": 1.0,
+ "content": "norm, Mei & Montanari (2019) found that the double descent was alleviated or disappeared when",
+ "type": "text"
+ }
+ ],
+ "index": 17
+ },
+ {
+ "bbox": [
+ 105,
+ 291,
+ 505,
+ 305
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 291,
+ 505,
+ 305
+ ],
+ "score": 1.0,
+ "content": "they used Ridge-type regularization techniques. Therefore, one can hypothesize the over-smoothing",
+ "type": "text"
+ }
+ ],
+ "index": 18
+ },
+ {
+ "bbox": [
+ 105,
+ 303,
+ 505,
+ 316
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 303,
+ 505,
+ 316
+ ],
+ "score": 1.0,
+ "content": "is a cause or consequence of regularization that is more like a Ridge-type rather than minimum-norm",
+ "type": "text"
+ }
+ ],
+ "index": 19
+ },
+ {
+ "bbox": [
+ 106,
+ 313,
+ 167,
+ 325
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 313,
+ 167,
+ 325
+ ],
+ "score": 1.0,
+ "content": "inductive bias.",
+ "type": "text"
+ }
+ ],
+ "index": 20
+ }
+ ],
+ "index": 14.5,
+ "bbox_fs": [
+ 105,
+ 192,
+ 505,
+ 325
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 330,
+ 505,
+ 419
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 330,
+ 505,
+ 343
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 330,
+ 505,
+ 343
+ ],
+ "score": 1.0,
+ "content": "Limitations in Graph NN Architectures. Our analysis is limited to graph NNs with the ReLU",
+ "type": "text"
+ }
+ ],
+ "index": 21
+ },
+ {
+ "bbox": [
+ 106,
+ 342,
+ 505,
+ 354
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 342,
+ 505,
+ 354
+ ],
+ "score": 1.0,
+ "content": "activation function because we implicitly use the property that ReLU is a projection onto the cone",
+ "type": "text"
+ }
+ ],
+ "index": 22
+ },
+ {
+ "bbox": [
+ 107,
+ 352,
+ 505,
+ 365
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 352,
+ 145,
+ 365
+ ],
+ "score": 0.92,
+ "content": "\\{ X \\geq 0 \\}",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 146,
+ 352,
+ 505,
+ 365
+ ],
+ "score": 1.0,
+ "content": "(Appendix A, Lemma 3). This fact enables the ReLU function to get along with the non-",
+ "type": "text"
+ }
+ ],
+ "index": 23
+ },
+ {
+ "bbox": [
+ 105,
+ 363,
+ 505,
+ 376
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 363,
+ 505,
+ 376
+ ],
+ "score": 1.0,
+ "content": "negativity of eigenvectors associated with the largest eigenvalues. Therefore, it is far from trivial to",
+ "type": "text"
+ }
+ ],
+ "index": 24
+ },
+ {
+ "bbox": [
+ 105,
+ 375,
+ 505,
+ 387
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 375,
+ 505,
+ 387
+ ],
+ "score": 1.0,
+ "content": "extend our results to other activation functions such as the sigmoid function or Leaky ReLU (Maas",
+ "type": "text"
+ }
+ ],
+ "index": 25
+ },
+ {
+ "bbox": [
+ 105,
+ 385,
+ 505,
+ 398
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 385,
+ 505,
+ 398
+ ],
+ "score": 1.0,
+ "content": "et al., 2013). Another point is that our formulation considers the update operation (Gilmer et al.,",
+ "type": "text"
+ }
+ ],
+ "index": 26
+ },
+ {
+ "bbox": [
+ 105,
+ 396,
+ 506,
+ 409
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 396,
+ 506,
+ 409
+ ],
+ "score": 1.0,
+ "content": "2017) of graph NNs only and does not take readout operations into account. In particular, we cannot",
+ "type": "text"
+ }
+ ],
+ "index": 27
+ },
+ {
+ "bbox": [
+ 105,
+ 406,
+ 452,
+ 421
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 406,
+ 452,
+ 421
+ ],
+ "score": 1.0,
+ "content": "directly apply our theory to graph classification tasks in which each sample is a graph.",
+ "type": "text"
+ }
+ ],
+ "index": 28
+ }
+ ],
+ "index": 24.5,
+ "bbox_fs": [
+ 105,
+ 330,
+ 506,
+ 421
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 424,
+ 505,
+ 523
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 424,
+ 505,
+ 436
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 424,
+ 505,
+ 436
+ ],
+ "score": 1.0,
+ "content": "Over-smoothing of Residual GNNs. Considering the correspondence of graph NNs and Markov",
+ "type": "text"
+ }
+ ],
+ "index": 29
+ },
+ {
+ "bbox": [
+ 105,
+ 435,
+ 505,
+ 448
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 435,
+ 505,
+ 448
+ ],
+ "score": 1.0,
+ "content": "processes (see Appendix F), one can imagine that residual links do not contribute to alleviating",
+ "type": "text"
+ }
+ ],
+ "index": 30
+ },
+ {
+ "bbox": [
+ 105,
+ 446,
+ 505,
+ 459
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 446,
+ 505,
+ 459
+ ],
+ "score": 1.0,
+ "content": "the over-smoothing phenomena because adding residual connections to a graph NN corresponds",
+ "type": "text"
+ }
+ ],
+ "index": 31
+ },
+ {
+ "bbox": [
+ 105,
+ 457,
+ 506,
+ 470
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 457,
+ 506,
+ 470
+ ],
+ "score": 1.0,
+ "content": "to converting a Markov process to its lazy version. When a Markov process converges to a stable",
+ "type": "text"
+ }
+ ],
+ "index": 32
+ },
+ {
+ "bbox": [
+ 105,
+ 467,
+ 506,
+ 482
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 467,
+ 506,
+ 482
+ ],
+ "score": 1.0,
+ "content": "distribution, the corresponding lazy process also converges eventually under certain conditions. It",
+ "type": "text"
+ }
+ ],
+ "index": 33
+ },
+ {
+ "bbox": [
+ 106,
+ 479,
+ 505,
+ 491
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 479,
+ 505,
+ 491
+ ],
+ "score": 1.0,
+ "content": "implies that residual links might not be helpful. However, Li et al. (2019) reported that graph NNs",
+ "type": "text"
+ }
+ ],
+ "index": 34
+ },
+ {
+ "bbox": [
+ 106,
+ 491,
+ 504,
+ 502
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 491,
+ 504,
+ 502
+ ],
+ "score": 1.0,
+ "content": "with as many as 56 layers performed well if they added residual connections. Considering that,",
+ "type": "text"
+ }
+ ],
+ "index": 35
+ },
+ {
+ "bbox": [
+ 105,
+ 500,
+ 505,
+ 513
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 500,
+ 505,
+ 513
+ ],
+ "score": 1.0,
+ "content": "the situation could be more complicated than our intuitions. The analysis of the role of residual",
+ "type": "text"
+ }
+ ],
+ "index": 36
+ },
+ {
+ "bbox": [
+ 106,
+ 513,
+ 387,
+ 524
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 513,
+ 387,
+ 524
+ ],
+ "score": 1.0,
+ "content": "connections in graph NNs is a promising direction for future research.",
+ "type": "text"
+ }
+ ],
+ "index": 37
+ }
+ ],
+ "index": 33,
+ "bbox_fs": [
+ 105,
+ 424,
+ 506,
+ 524
+ ]
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 107,
+ 554,
+ 195,
+ 567
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 553,
+ 198,
+ 570
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 553,
+ 198,
+ 570
+ ],
+ "score": 1.0,
+ "content": "8 CONCLUSION",
+ "type": "text"
+ }
+ ],
+ "index": 38
+ }
+ ],
+ "index": 38
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 589,
+ 505,
+ 732
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 588,
+ 506,
+ 601
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 588,
+ 506,
+ 601
+ ],
+ "score": 1.0,
+ "content": "In this paper, to understand the empirically observed phenomena that deep non-linear graph NNs",
+ "type": "text"
+ }
+ ],
+ "index": 39
+ },
+ {
+ "bbox": [
+ 106,
+ 601,
+ 505,
+ 612
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 601,
+ 505,
+ 612
+ ],
+ "score": 1.0,
+ "content": "do not perform well, we analyzed their asymptotic behaviors by interpreting them as a dynamical",
+ "type": "text"
+ }
+ ],
+ "index": 40
+ },
+ {
+ "bbox": [
+ 105,
+ 610,
+ 506,
+ 624
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 610,
+ 506,
+ 624
+ ],
+ "score": 1.0,
+ "content": "system that includes GCN and Markov process as special cases. We gave theoretical conditions",
+ "type": "text"
+ }
+ ],
+ "index": 41
+ },
+ {
+ "bbox": [
+ 105,
+ 621,
+ 505,
+ 635
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 621,
+ 505,
+ 635
+ ],
+ "score": 1.0,
+ "content": "under which GCNs suffer from the information loss in the limit of infinite layers. Our theory directly",
+ "type": "text"
+ }
+ ],
+ "index": 42
+ },
+ {
+ "bbox": [
+ 105,
+ 632,
+ 506,
+ 647
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 632,
+ 506,
+ 647
+ ],
+ "score": 1.0,
+ "content": "related the expressive power of graph NNs and topological information of the underlying graphs via",
+ "type": "text"
+ }
+ ],
+ "index": 43
+ },
+ {
+ "bbox": [
+ 105,
+ 644,
+ 505,
+ 656
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 644,
+ 505,
+ 656
+ ],
+ "score": 1.0,
+ "content": "spectra of the Laplacian. It enabled us to leverage spectral and random graph theory to analyze the",
+ "type": "text"
+ }
+ ],
+ "index": 44
+ },
+ {
+ "bbox": [
+ 105,
+ 654,
+ 506,
+ 668
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 654,
+ 506,
+ 668
+ ],
+ "score": 1.0,
+ "content": "expressive power of graph NNs. To demonstrate this, we considered GCN on the Erdos – R ˝ enyi ´",
+ "type": "text"
+ }
+ ],
+ "index": 45
+ },
+ {
+ "bbox": [
+ 105,
+ 664,
+ 506,
+ 680
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 664,
+ 506,
+ 680
+ ],
+ "score": 1.0,
+ "content": "graph as an example and showed that when the underlying graph is sufficiently dense and large,",
+ "type": "text"
+ }
+ ],
+ "index": 46
+ },
+ {
+ "bbox": [
+ 104,
+ 676,
+ 506,
+ 691
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 676,
+ 506,
+ 691
+ ],
+ "score": 1.0,
+ "content": "a wide range of GCNs on the graph suffer from information loss. Based on the theory, we gave",
+ "type": "text"
+ }
+ ],
+ "index": 47
+ },
+ {
+ "bbox": [
+ 105,
+ 688,
+ 505,
+ 700
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 688,
+ 505,
+ 700
+ ],
+ "score": 1.0,
+ "content": "a principled guideline for how to determine the scale of weights of graph NNs and empirically",
+ "type": "text"
+ }
+ ],
+ "index": 48
+ },
+ {
+ "bbox": [
+ 106,
+ 699,
+ 505,
+ 711
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 699,
+ 505,
+ 711
+ ],
+ "score": 1.0,
+ "content": "showed that the weight normalization implied by our theory performed well in real datasets. One",
+ "type": "text"
+ }
+ ],
+ "index": 49
+ },
+ {
+ "bbox": [
+ 105,
+ 709,
+ 505,
+ 723
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 709,
+ 505,
+ 723
+ ],
+ "score": 1.0,
+ "content": "promising direction of research is to analyze the optimization and statistical properties such as the",
+ "type": "text"
+ }
+ ],
+ "index": 50
+ },
+ {
+ "bbox": [
+ 105,
+ 720,
+ 503,
+ 734
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 720,
+ 503,
+ 734
+ ],
+ "score": 1.0,
+ "content": "generalization power (Verma & Zhang, 2019) of graph NNs via spectral and random graph theories.",
+ "type": "text"
+ }
+ ],
+ "index": 51
+ }
+ ],
+ "index": 45,
+ "bbox_fs": [
+ 104,
+ 588,
+ 506,
+ 734
+ ]
+ }
+ ]
+ },
+ {
+ "preproc_blocks": [
+ {
+ "type": "title",
+ "bbox": [
+ 108,
+ 83,
+ 200,
+ 93
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 107,
+ 83,
+ 200,
+ 94
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 83,
+ 200,
+ 94
+ ],
+ "score": 1.0,
+ "content": "ACKNOWLEDGMENTS",
+ "type": "text"
+ }
+ ],
+ "index": 0
+ }
+ ],
+ "index": 0
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 101,
+ 505,
+ 156
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 100,
+ 505,
+ 115
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 100,
+ 505,
+ 115
+ ],
+ "score": 1.0,
+ "content": "We thank Katsuhiko Ishiguro for providing a part of code for the experiments, Kohei Hayashi and",
+ "type": "text"
+ }
+ ],
+ "index": 1
+ },
+ {
+ "bbox": [
+ 105,
+ 112,
+ 505,
+ 126
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 112,
+ 505,
+ 126
+ ],
+ "score": 1.0,
+ "content": "Haru Negami Oono for giving us feedback and comments on the draft, Keyulu Xu and anonymous",
+ "type": "text"
+ }
+ ],
+ "index": 2
+ },
+ {
+ "bbox": [
+ 105,
+ 123,
+ 505,
+ 137
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 123,
+ 505,
+ 137
+ ],
+ "score": 1.0,
+ "content": "reviewers for fruitful discussions via OpenReview, and Ryuta Osawa for pointing out errors and",
+ "type": "text"
+ }
+ ],
+ "index": 3
+ },
+ {
+ "bbox": [
+ 105,
+ 134,
+ 505,
+ 147
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 134,
+ 505,
+ 147
+ ],
+ "score": 1.0,
+ "content": "suggesting improvements of the paper. TS was partially supported by JSPS KAKENHI (15H05707,",
+ "type": "text"
+ }
+ ],
+ "index": 4
+ },
+ {
+ "bbox": [
+ 106,
+ 145,
+ 380,
+ 158
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 145,
+ 380,
+ 158
+ ],
+ "score": 1.0,
+ "content": "18K19793, and 18H03201), Japan Digital Design, and JST CREST.",
+ "type": "text"
+ }
+ ],
+ "index": 5
+ }
+ ],
+ "index": 3
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 107,
+ 173,
+ 175,
+ 185
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 174,
+ 176,
+ 186
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 174,
+ 176,
+ 186
+ ],
+ "score": 1.0,
+ "content": "REFERENCES",
+ "type": "text"
+ }
+ ],
+ "index": 6
+ }
+ ],
+ "index": 6
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 192,
+ 505,
+ 246
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 191,
+ 505,
+ 205
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 191,
+ 505,
+ 205
+ ],
+ "score": 1.0,
+ "content": "Takuya Akiba, Shotaro Sano, Toshihiko Yanase, Takeru Ohta, and Masanori Koyama. Optuna:",
+ "type": "text"
+ }
+ ],
+ "index": 7
+ },
+ {
+ "bbox": [
+ 115,
+ 203,
+ 505,
+ 216
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 203,
+ 505,
+ 216
+ ],
+ "score": 1.0,
+ "content": "A next-generation hyperparameter optimization framework. In Proceedings of the 25th ACM",
+ "type": "text"
+ }
+ ],
+ "index": 8
+ },
+ {
+ "bbox": [
+ 114,
+ 212,
+ 506,
+ 228
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 114,
+ 212,
+ 506,
+ 228
+ ],
+ "score": 1.0,
+ "content": "SIGKDD International Conference on Knowledge Discovery & Data Mining, KDD ’19, pp.",
+ "type": "text"
+ }
+ ],
+ "index": 9
+ },
+ {
+ "bbox": [
+ 116,
+ 225,
+ 504,
+ 237
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 116,
+ 225,
+ 504,
+ 237
+ ],
+ "score": 1.0,
+ "content": "2623–2631, New York, NY, USA, 2019. ACM. ISBN 978-1-4503-6201-6. doi: 10.1145/3292500.",
+ "type": "text"
+ }
+ ],
+ "index": 10
+ },
+ {
+ "bbox": [
+ 115,
+ 235,
+ 156,
+ 249
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 235,
+ 156,
+ 249
+ ],
+ "score": 1.0,
+ "content": "3330701.",
+ "type": "text"
+ }
+ ],
+ "index": 11
+ }
+ ],
+ "index": 9
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 254,
+ 503,
+ 277
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 254,
+ 504,
+ 267
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 254,
+ 504,
+ 267
+ ],
+ "score": 1.0,
+ "content": "Dana Angluin and Leslie G Valiant. Fast probabilistic algorithms for hamiltonian circuits and match-",
+ "type": "text"
+ }
+ ],
+ "index": 12
+ },
+ {
+ "bbox": [
+ 115,
+ 266,
+ 399,
+ 277
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 266,
+ 399,
+ 277
+ ],
+ "score": 1.0,
+ "content": "ings. Journal of Computer and system Sciences, 18(2):155–193, 1979.",
+ "type": "text"
+ }
+ ],
+ "index": 13
+ }
+ ],
+ "index": 12.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 284,
+ 502,
+ 307
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 283,
+ 504,
+ 298
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 283,
+ 504,
+ 298
+ ],
+ "score": 1.0,
+ "content": "Andrew R Barron. Universal approximation bounds for superpositions of a sigmoidal function.",
+ "type": "text"
+ }
+ ],
+ "index": 14
+ },
+ {
+ "bbox": [
+ 116,
+ 295,
+ 377,
+ 307
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 116,
+ 295,
+ 377,
+ 307
+ ],
+ "score": 1.0,
+ "content": "IEEE Transactions on Information theory, 39(3):930–945, 1993.",
+ "type": "text"
+ }
+ ],
+ "index": 15
+ }
+ ],
+ "index": 14.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 313,
+ 505,
+ 358
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 314,
+ 505,
+ 326
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 314,
+ 505,
+ 326
+ ],
+ "score": 1.0,
+ "content": "Peter W Battaglia, Jessica B Hamrick, Victor Bapst, Alvaro Sanchez-Gonzalez, Vinicius Zambaldi,",
+ "type": "text"
+ }
+ ],
+ "index": 16
+ },
+ {
+ "bbox": [
+ 116,
+ 325,
+ 505,
+ 337
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 116,
+ 325,
+ 505,
+ 337
+ ],
+ "score": 1.0,
+ "content": "Mateusz Malinowski, Andrea Tacchetti, David Raposo, Adam Santoro, Ryan Faulkner, et al.",
+ "type": "text"
+ }
+ ],
+ "index": 17
+ },
+ {
+ "bbox": [
+ 114,
+ 335,
+ 506,
+ 349
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 114,
+ 335,
+ 506,
+ 349
+ ],
+ "score": 1.0,
+ "content": "Relational inductive biases, deep learning, and graph networks. arXiv preprint arXiv:1806.01261,",
+ "type": "text"
+ }
+ ],
+ "index": 18
+ },
+ {
+ "bbox": [
+ 115,
+ 346,
+ 142,
+ 358
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 346,
+ 142,
+ 358
+ ],
+ "score": 1.0,
+ "content": "2018.",
+ "type": "text"
+ }
+ ],
+ "index": 19
+ }
+ ],
+ "index": 17.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 365,
+ 505,
+ 399
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 364,
+ 504,
+ 378
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 364,
+ 504,
+ 378
+ ],
+ "score": 1.0,
+ "content": "Mikhail Belkin, Daniel Hsu, Siyuan Ma, and Soumik Mandal. Reconciling modern machine-",
+ "type": "text"
+ }
+ ],
+ "index": 20
+ },
+ {
+ "bbox": [
+ 115,
+ 375,
+ 505,
+ 390
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 375,
+ 505,
+ 390
+ ],
+ "score": 1.0,
+ "content": "learning practice and the classical bias–variance trade-off. Proceedings of the National Academy",
+ "type": "text"
+ }
+ ],
+ "index": 21
+ },
+ {
+ "bbox": [
+ 115,
+ 388,
+ 285,
+ 399
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 388,
+ 285,
+ 399
+ ],
+ "score": 1.0,
+ "content": "of Sciences, 116(32):15849–15854, 2019.",
+ "type": "text"
+ }
+ ],
+ "index": 22
+ }
+ ],
+ "index": 21
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 406,
+ 506,
+ 439
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 104,
+ 405,
+ 505,
+ 420
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 405,
+ 505,
+ 420
+ ],
+ "score": 1.0,
+ "content": "James S. Bergstra, Remi Bardenet, Yoshua Bengio, and Bal ´ azs K ´ egl. Algorithms for hyper- ´",
+ "type": "text"
+ }
+ ],
+ "index": 23
+ },
+ {
+ "bbox": [
+ 115,
+ 417,
+ 505,
+ 430
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 417,
+ 505,
+ 430
+ ],
+ "score": 1.0,
+ "content": "parameter optimization. In Advances in Neural Information Processing Systems 24, pp. 2546–",
+ "type": "text"
+ }
+ ],
+ "index": 24
+ },
+ {
+ "bbox": [
+ 116,
+ 428,
+ 264,
+ 439
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 116,
+ 428,
+ 264,
+ 439
+ ],
+ "score": 1.0,
+ "content": "2554. Curran Associates, Inc., 2011.",
+ "type": "text"
+ }
+ ],
+ "index": 25
+ }
+ ],
+ "index": 24
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 446,
+ 504,
+ 480
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 446,
+ 506,
+ 460
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 446,
+ 506,
+ 460
+ ],
+ "score": 1.0,
+ "content": "Jie Chen, Tengfei Ma, and Cao Xiao. FastGCN: Fast learning with graph convolutional networks",
+ "type": "text"
+ }
+ ],
+ "index": 26
+ },
+ {
+ "bbox": [
+ 116,
+ 458,
+ 505,
+ 470
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 116,
+ 458,
+ 505,
+ 470
+ ],
+ "score": 1.0,
+ "content": "via importance sampling. In International Conference on Learning Representations, 2018a. URL",
+ "type": "text"
+ }
+ ],
+ "index": 27
+ },
+ {
+ "bbox": [
+ 116,
+ 469,
+ 366,
+ 482
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 116,
+ 469,
+ 302,
+ 482
+ ],
+ "score": 1.0,
+ "content": "https://openreview.net/forum?id",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 302,
+ 470,
+ 309,
+ 478
+ ],
+ "score": 0.54,
+ "content": "\\underline { { \\underline { { \\mathbf { \\Pi } } } } } =",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 309,
+ 469,
+ 366,
+ 482
+ ],
+ "score": 1.0,
+ "content": "rytstxWAW.",
+ "type": "text"
+ }
+ ],
+ "index": 28
+ }
+ ],
+ "index": 27
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 487,
+ 505,
+ 531
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 487,
+ 506,
+ 500
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 487,
+ 506,
+ 500
+ ],
+ "score": 1.0,
+ "content": "Minmin Chen, Jeffrey Pennington, and Samuel Schoenholz. Dynamical isometry and a mean field",
+ "type": "text"
+ }
+ ],
+ "index": 29
+ },
+ {
+ "bbox": [
+ 115,
+ 498,
+ 505,
+ 511
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 498,
+ 505,
+ 511
+ ],
+ "score": 1.0,
+ "content": "theory of RNNs: Gating enables signal propagation in recurrent neural networks. In Proceedings",
+ "type": "text"
+ }
+ ],
+ "index": 30
+ },
+ {
+ "bbox": [
+ 117,
+ 510,
+ 505,
+ 522
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 117,
+ 510,
+ 505,
+ 522
+ ],
+ "score": 1.0,
+ "content": "of the 35th International Conference on Machine Learning, volume 80 of Proceedings of Machine",
+ "type": "text"
+ }
+ ],
+ "index": 31
+ },
+ {
+ "bbox": [
+ 115,
+ 520,
+ 356,
+ 533
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 520,
+ 356,
+ 533
+ ],
+ "score": 1.0,
+ "content": "Learning Research, pp. 873–882. PMLR, 10–15 Jul 2018b.",
+ "type": "text"
+ }
+ ],
+ "index": 32
+ }
+ ],
+ "index": 30.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 539,
+ 505,
+ 584
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 540,
+ 505,
+ 551
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 540,
+ 505,
+ 551
+ ],
+ "score": 1.0,
+ "content": "Anna Choromanska, Yann LeCun, and Gerard Ben Arous. Open problem: The landscape of the ´",
+ "type": "text"
+ }
+ ],
+ "index": 33
+ },
+ {
+ "bbox": [
+ 115,
+ 550,
+ 506,
+ 563
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 550,
+ 506,
+ 563
+ ],
+ "score": 1.0,
+ "content": "loss surfaces of multilayer networks. In Peter Grunwald, Elad Hazan, and Satyen Kale (eds.), ¨",
+ "type": "text"
+ }
+ ],
+ "index": 34
+ },
+ {
+ "bbox": [
+ 116,
+ 561,
+ 506,
+ 574
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 116,
+ 561,
+ 506,
+ 574
+ ],
+ "score": 1.0,
+ "content": "Proceedings of The 28th Conference on Learning Theory, volume 40 of Proceedings of Machine",
+ "type": "text"
+ }
+ ],
+ "index": 35
+ },
+ {
+ "bbox": [
+ 115,
+ 572,
+ 319,
+ 584
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 572,
+ 319,
+ 584
+ ],
+ "score": 1.0,
+ "content": "Learning Research, pp. 1756–1760. PMLR, 2015.",
+ "type": "text"
+ }
+ ],
+ "index": 36
+ }
+ ],
+ "index": 34.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 590,
+ 503,
+ 613
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 589,
+ 505,
+ 605
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 589,
+ 505,
+ 605
+ ],
+ "score": 1.0,
+ "content": "Fan Chung and Linyuan Lu. Connected components in random graphs with given expected degree",
+ "type": "text"
+ }
+ ],
+ "index": 37
+ },
+ {
+ "bbox": [
+ 115,
+ 602,
+ 349,
+ 614
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 602,
+ 349,
+ 614
+ ],
+ "score": 1.0,
+ "content": "sequences. Annals of combinatorics, 6(2):125–145, 2002.",
+ "type": "text"
+ }
+ ],
+ "index": 38
+ }
+ ],
+ "index": 37.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 620,
+ 504,
+ 643
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 619,
+ 506,
+ 633
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 619,
+ 506,
+ 633
+ ],
+ "score": 1.0,
+ "content": "Fan Chung and Mary Radcliffe. On the spectra of general random graphs. the electronic journal of",
+ "type": "text"
+ }
+ ],
+ "index": 39
+ },
+ {
+ "bbox": [
+ 115,
+ 631,
+ 249,
+ 643
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 631,
+ 249,
+ 643
+ ],
+ "score": 1.0,
+ "content": "combinatorics, 18(1):215, 2011.",
+ "type": "text"
+ }
+ ],
+ "index": 40
+ }
+ ],
+ "index": 39.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 650,
+ 503,
+ 673
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 648,
+ 505,
+ 664
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 648,
+ 505,
+ 664
+ ],
+ "score": 1.0,
+ "content": "Fan Chung, Linyuan Lu, and Van Vu. The spectra of random graphs with given expected degrees.",
+ "type": "text"
+ }
+ ],
+ "index": 41
+ },
+ {
+ "bbox": [
+ 115,
+ 660,
+ 291,
+ 673
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 660,
+ 291,
+ 673
+ ],
+ "score": 1.0,
+ "content": "Internet Mathematics, 1(3):257–275, 2004.",
+ "type": "text"
+ }
+ ],
+ "index": 42
+ }
+ ],
+ "index": 41.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 680,
+ 503,
+ 703
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 678,
+ 505,
+ 693
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 678,
+ 505,
+ 693
+ ],
+ "score": 1.0,
+ "content": "Fan RK Chung and Fan Chung Graham. Spectral graph theory. Number 92 in CBMS Regional",
+ "type": "text"
+ }
+ ],
+ "index": 43
+ },
+ {
+ "bbox": [
+ 116,
+ 691,
+ 405,
+ 702
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 116,
+ 691,
+ 405,
+ 702
+ ],
+ "score": 1.0,
+ "content": "Conference Series in Mathematics. American Mathematical Soc., 1997.",
+ "type": "text"
+ }
+ ],
+ "index": 44
+ }
+ ],
+ "index": 43.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 709,
+ 503,
+ 731
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 709,
+ 505,
+ 723
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 709,
+ 322,
+ 723
+ ],
+ "score": 1.0,
+ "content": "Amin Coja-Oghlan. On the laplacian eigenvalues of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 322,
+ 711,
+ 342,
+ 722
+ ],
+ "score": 0.9,
+ "content": "G _ { n , p }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 343,
+ 709,
+ 505,
+ 723
+ ],
+ "score": 1.0,
+ "content": ". Combinatorics, Probability and Com-",
+ "type": "text"
+ }
+ ],
+ "index": 45
+ },
+ {
+ "bbox": [
+ 114,
+ 721,
+ 236,
+ 731
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 114,
+ 721,
+ 236,
+ 731
+ ],
+ "score": 1.0,
+ "content": "puting, 16(6):923–946, 2007.",
+ "type": "text"
+ }
+ ],
+ "index": 46
+ }
+ ],
+ "index": 45.5
+ }
+ ],
+ "page_idx": 9,
+ "page_size": [
+ 612,
+ 792
+ ],
+ "discarded_blocks": [
+ {
+ "type": "discarded",
+ "bbox": [
+ 108,
+ 27,
+ 293,
+ 37
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 26,
+ 294,
+ 38
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 26,
+ 294,
+ 38
+ ],
+ "score": 1.0,
+ "content": "Published as a conference paper at ICLR 2020",
+ "type": "text"
+ }
+ ]
+ }
+ ]
+ },
+ {
+ "type": "discarded",
+ "bbox": [
+ 301,
+ 751,
+ 311,
+ 760
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 299,
+ 750,
+ 313,
+ 765
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 299,
+ 750,
+ 313,
+ 765
+ ],
+ "score": 1.0,
+ "content": "",
+ "type": "text",
+ "height": 15,
+ "width": 14
+ }
+ ]
+ }
+ ]
+ }
+ ],
+ "para_blocks": [
+ {
+ "type": "title",
+ "bbox": [
+ 108,
+ 83,
+ 200,
+ 93
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 107,
+ 83,
+ 200,
+ 94
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 83,
+ 200,
+ 94
+ ],
+ "score": 1.0,
+ "content": "ACKNOWLEDGMENTS",
+ "type": "text"
+ }
+ ],
+ "index": 0
+ }
+ ],
+ "index": 0
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 101,
+ 505,
+ 156
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 100,
+ 505,
+ 115
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 100,
+ 505,
+ 115
+ ],
+ "score": 1.0,
+ "content": "We thank Katsuhiko Ishiguro for providing a part of code for the experiments, Kohei Hayashi and",
+ "type": "text"
+ }
+ ],
+ "index": 1
+ },
+ {
+ "bbox": [
+ 105,
+ 112,
+ 505,
+ 126
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 112,
+ 505,
+ 126
+ ],
+ "score": 1.0,
+ "content": "Haru Negami Oono for giving us feedback and comments on the draft, Keyulu Xu and anonymous",
+ "type": "text"
+ }
+ ],
+ "index": 2
+ },
+ {
+ "bbox": [
+ 105,
+ 123,
+ 505,
+ 137
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 123,
+ 505,
+ 137
+ ],
+ "score": 1.0,
+ "content": "reviewers for fruitful discussions via OpenReview, and Ryuta Osawa for pointing out errors and",
+ "type": "text"
+ }
+ ],
+ "index": 3
+ },
+ {
+ "bbox": [
+ 105,
+ 134,
+ 505,
+ 147
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 134,
+ 505,
+ 147
+ ],
+ "score": 1.0,
+ "content": "suggesting improvements of the paper. TS was partially supported by JSPS KAKENHI (15H05707,",
+ "type": "text"
+ }
+ ],
+ "index": 4
+ },
+ {
+ "bbox": [
+ 106,
+ 145,
+ 380,
+ 158
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 145,
+ 380,
+ 158
+ ],
+ "score": 1.0,
+ "content": "18K19793, and 18H03201), Japan Digital Design, and JST CREST.",
+ "type": "text"
+ }
+ ],
+ "index": 5
+ }
+ ],
+ "index": 3,
+ "bbox_fs": [
+ 105,
+ 100,
+ 505,
+ 158
+ ]
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 107,
+ 173,
+ 175,
+ 185
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 174,
+ 176,
+ 186
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 174,
+ 176,
+ 186
+ ],
+ "score": 1.0,
+ "content": "REFERENCES",
+ "type": "text"
+ }
+ ],
+ "index": 6
+ }
+ ],
+ "index": 6
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 192,
+ 505,
+ 246
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 191,
+ 505,
+ 205
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 191,
+ 505,
+ 205
+ ],
+ "score": 1.0,
+ "content": "Takuya Akiba, Shotaro Sano, Toshihiko Yanase, Takeru Ohta, and Masanori Koyama. Optuna:",
+ "type": "text"
+ }
+ ],
+ "index": 7
+ },
+ {
+ "bbox": [
+ 115,
+ 203,
+ 505,
+ 216
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 203,
+ 505,
+ 216
+ ],
+ "score": 1.0,
+ "content": "A next-generation hyperparameter optimization framework. In Proceedings of the 25th ACM",
+ "type": "text"
+ }
+ ],
+ "index": 8
+ },
+ {
+ "bbox": [
+ 114,
+ 212,
+ 506,
+ 228
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 114,
+ 212,
+ 506,
+ 228
+ ],
+ "score": 1.0,
+ "content": "SIGKDD International Conference on Knowledge Discovery & Data Mining, KDD ’19, pp.",
+ "type": "text"
+ }
+ ],
+ "index": 9
+ },
+ {
+ "bbox": [
+ 116,
+ 225,
+ 504,
+ 237
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 116,
+ 225,
+ 504,
+ 237
+ ],
+ "score": 1.0,
+ "content": "2623–2631, New York, NY, USA, 2019. ACM. ISBN 978-1-4503-6201-6. doi: 10.1145/3292500.",
+ "type": "text"
+ }
+ ],
+ "index": 10
+ },
+ {
+ "bbox": [
+ 115,
+ 235,
+ 156,
+ 249
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 235,
+ 156,
+ 249
+ ],
+ "score": 1.0,
+ "content": "3330701.",
+ "type": "text"
+ }
+ ],
+ "index": 11
+ }
+ ],
+ "index": 9,
+ "bbox_fs": [
+ 106,
+ 191,
+ 506,
+ 249
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 254,
+ 503,
+ 277
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 254,
+ 504,
+ 267
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 254,
+ 504,
+ 267
+ ],
+ "score": 1.0,
+ "content": "Dana Angluin and Leslie G Valiant. Fast probabilistic algorithms for hamiltonian circuits and match-",
+ "type": "text"
+ }
+ ],
+ "index": 12
+ },
+ {
+ "bbox": [
+ 115,
+ 266,
+ 399,
+ 277
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 266,
+ 399,
+ 277
+ ],
+ "score": 1.0,
+ "content": "ings. Journal of Computer and system Sciences, 18(2):155–193, 1979.",
+ "type": "text"
+ }
+ ],
+ "index": 13
+ }
+ ],
+ "index": 12.5,
+ "bbox_fs": [
+ 106,
+ 254,
+ 504,
+ 277
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 284,
+ 502,
+ 307
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 283,
+ 504,
+ 298
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 283,
+ 504,
+ 298
+ ],
+ "score": 1.0,
+ "content": "Andrew R Barron. Universal approximation bounds for superpositions of a sigmoidal function.",
+ "type": "text"
+ }
+ ],
+ "index": 14
+ },
+ {
+ "bbox": [
+ 116,
+ 295,
+ 377,
+ 307
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 116,
+ 295,
+ 377,
+ 307
+ ],
+ "score": 1.0,
+ "content": "IEEE Transactions on Information theory, 39(3):930–945, 1993.",
+ "type": "text"
+ }
+ ],
+ "index": 15
+ }
+ ],
+ "index": 14.5,
+ "bbox_fs": [
+ 105,
+ 283,
+ 504,
+ 307
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 313,
+ 505,
+ 358
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 314,
+ 505,
+ 326
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 314,
+ 505,
+ 326
+ ],
+ "score": 1.0,
+ "content": "Peter W Battaglia, Jessica B Hamrick, Victor Bapst, Alvaro Sanchez-Gonzalez, Vinicius Zambaldi,",
+ "type": "text"
+ }
+ ],
+ "index": 16
+ },
+ {
+ "bbox": [
+ 116,
+ 325,
+ 505,
+ 337
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 116,
+ 325,
+ 505,
+ 337
+ ],
+ "score": 1.0,
+ "content": "Mateusz Malinowski, Andrea Tacchetti, David Raposo, Adam Santoro, Ryan Faulkner, et al.",
+ "type": "text"
+ }
+ ],
+ "index": 17
+ },
+ {
+ "bbox": [
+ 114,
+ 335,
+ 506,
+ 349
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 114,
+ 335,
+ 506,
+ 349
+ ],
+ "score": 1.0,
+ "content": "Relational inductive biases, deep learning, and graph networks. arXiv preprint arXiv:1806.01261,",
+ "type": "text"
+ }
+ ],
+ "index": 18
+ },
+ {
+ "bbox": [
+ 115,
+ 346,
+ 142,
+ 358
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 346,
+ 142,
+ 358
+ ],
+ "score": 1.0,
+ "content": "2018.",
+ "type": "text"
+ }
+ ],
+ "index": 19
+ }
+ ],
+ "index": 17.5,
+ "bbox_fs": [
+ 106,
+ 314,
+ 506,
+ 358
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 365,
+ 505,
+ 399
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 364,
+ 504,
+ 378
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 364,
+ 504,
+ 378
+ ],
+ "score": 1.0,
+ "content": "Mikhail Belkin, Daniel Hsu, Siyuan Ma, and Soumik Mandal. Reconciling modern machine-",
+ "type": "text"
+ }
+ ],
+ "index": 20
+ },
+ {
+ "bbox": [
+ 115,
+ 375,
+ 505,
+ 390
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 375,
+ 505,
+ 390
+ ],
+ "score": 1.0,
+ "content": "learning practice and the classical bias–variance trade-off. Proceedings of the National Academy",
+ "type": "text"
+ }
+ ],
+ "index": 21
+ },
+ {
+ "bbox": [
+ 115,
+ 388,
+ 285,
+ 399
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 388,
+ 285,
+ 399
+ ],
+ "score": 1.0,
+ "content": "of Sciences, 116(32):15849–15854, 2019.",
+ "type": "text"
+ }
+ ],
+ "index": 22
+ }
+ ],
+ "index": 21,
+ "bbox_fs": [
+ 105,
+ 364,
+ 505,
+ 399
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 406,
+ 506,
+ 439
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 104,
+ 405,
+ 505,
+ 420
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 405,
+ 505,
+ 420
+ ],
+ "score": 1.0,
+ "content": "James S. Bergstra, Remi Bardenet, Yoshua Bengio, and Bal ´ azs K ´ egl. Algorithms for hyper- ´",
+ "type": "text"
+ }
+ ],
+ "index": 23
+ },
+ {
+ "bbox": [
+ 115,
+ 417,
+ 505,
+ 430
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 417,
+ 505,
+ 430
+ ],
+ "score": 1.0,
+ "content": "parameter optimization. In Advances in Neural Information Processing Systems 24, pp. 2546–",
+ "type": "text"
+ }
+ ],
+ "index": 24
+ },
+ {
+ "bbox": [
+ 116,
+ 428,
+ 264,
+ 439
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 116,
+ 428,
+ 264,
+ 439
+ ],
+ "score": 1.0,
+ "content": "2554. Curran Associates, Inc., 2011.",
+ "type": "text"
+ }
+ ],
+ "index": 25
+ }
+ ],
+ "index": 24,
+ "bbox_fs": [
+ 104,
+ 405,
+ 505,
+ 439
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 446,
+ 504,
+ 480
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 446,
+ 506,
+ 460
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 446,
+ 506,
+ 460
+ ],
+ "score": 1.0,
+ "content": "Jie Chen, Tengfei Ma, and Cao Xiao. FastGCN: Fast learning with graph convolutional networks",
+ "type": "text"
+ }
+ ],
+ "index": 26
+ },
+ {
+ "bbox": [
+ 116,
+ 458,
+ 505,
+ 470
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 116,
+ 458,
+ 505,
+ 470
+ ],
+ "score": 1.0,
+ "content": "via importance sampling. In International Conference on Learning Representations, 2018a. URL",
+ "type": "text"
+ }
+ ],
+ "index": 27
+ },
+ {
+ "bbox": [
+ 116,
+ 469,
+ 366,
+ 482
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 116,
+ 469,
+ 302,
+ 482
+ ],
+ "score": 1.0,
+ "content": "https://openreview.net/forum?id",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 302,
+ 470,
+ 309,
+ 478
+ ],
+ "score": 0.54,
+ "content": "\\underline { { \\underline { { \\mathbf { \\Pi } } } } } =",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 309,
+ 469,
+ 366,
+ 482
+ ],
+ "score": 1.0,
+ "content": "rytstxWAW.",
+ "type": "text"
+ }
+ ],
+ "index": 28
+ }
+ ],
+ "index": 27,
+ "bbox_fs": [
+ 105,
+ 446,
+ 506,
+ 482
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 487,
+ 505,
+ 531
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 487,
+ 506,
+ 500
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 487,
+ 506,
+ 500
+ ],
+ "score": 1.0,
+ "content": "Minmin Chen, Jeffrey Pennington, and Samuel Schoenholz. Dynamical isometry and a mean field",
+ "type": "text"
+ }
+ ],
+ "index": 29
+ },
+ {
+ "bbox": [
+ 115,
+ 498,
+ 505,
+ 511
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 498,
+ 505,
+ 511
+ ],
+ "score": 1.0,
+ "content": "theory of RNNs: Gating enables signal propagation in recurrent neural networks. In Proceedings",
+ "type": "text"
+ }
+ ],
+ "index": 30
+ },
+ {
+ "bbox": [
+ 117,
+ 510,
+ 505,
+ 522
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 117,
+ 510,
+ 505,
+ 522
+ ],
+ "score": 1.0,
+ "content": "of the 35th International Conference on Machine Learning, volume 80 of Proceedings of Machine",
+ "type": "text"
+ }
+ ],
+ "index": 31
+ },
+ {
+ "bbox": [
+ 115,
+ 520,
+ 356,
+ 533
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 520,
+ 356,
+ 533
+ ],
+ "score": 1.0,
+ "content": "Learning Research, pp. 873–882. PMLR, 10–15 Jul 2018b.",
+ "type": "text"
+ }
+ ],
+ "index": 32
+ }
+ ],
+ "index": 30.5,
+ "bbox_fs": [
+ 105,
+ 487,
+ 506,
+ 533
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 539,
+ 505,
+ 584
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 540,
+ 505,
+ 551
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 540,
+ 505,
+ 551
+ ],
+ "score": 1.0,
+ "content": "Anna Choromanska, Yann LeCun, and Gerard Ben Arous. Open problem: The landscape of the ´",
+ "type": "text"
+ }
+ ],
+ "index": 33
+ },
+ {
+ "bbox": [
+ 115,
+ 550,
+ 506,
+ 563
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 550,
+ 506,
+ 563
+ ],
+ "score": 1.0,
+ "content": "loss surfaces of multilayer networks. In Peter Grunwald, Elad Hazan, and Satyen Kale (eds.), ¨",
+ "type": "text"
+ }
+ ],
+ "index": 34
+ },
+ {
+ "bbox": [
+ 116,
+ 561,
+ 506,
+ 574
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 116,
+ 561,
+ 506,
+ 574
+ ],
+ "score": 1.0,
+ "content": "Proceedings of The 28th Conference on Learning Theory, volume 40 of Proceedings of Machine",
+ "type": "text"
+ }
+ ],
+ "index": 35
+ },
+ {
+ "bbox": [
+ 115,
+ 572,
+ 319,
+ 584
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 572,
+ 319,
+ 584
+ ],
+ "score": 1.0,
+ "content": "Learning Research, pp. 1756–1760. PMLR, 2015.",
+ "type": "text"
+ }
+ ],
+ "index": 36
+ }
+ ],
+ "index": 34.5,
+ "bbox_fs": [
+ 106,
+ 540,
+ 506,
+ 584
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 590,
+ 503,
+ 613
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 589,
+ 505,
+ 605
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 589,
+ 505,
+ 605
+ ],
+ "score": 1.0,
+ "content": "Fan Chung and Linyuan Lu. Connected components in random graphs with given expected degree",
+ "type": "text"
+ }
+ ],
+ "index": 37
+ },
+ {
+ "bbox": [
+ 115,
+ 602,
+ 349,
+ 614
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 602,
+ 349,
+ 614
+ ],
+ "score": 1.0,
+ "content": "sequences. Annals of combinatorics, 6(2):125–145, 2002.",
+ "type": "text"
+ }
+ ],
+ "index": 38
+ }
+ ],
+ "index": 37.5,
+ "bbox_fs": [
+ 105,
+ 589,
+ 505,
+ 614
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 620,
+ 504,
+ 643
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 619,
+ 506,
+ 633
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 619,
+ 506,
+ 633
+ ],
+ "score": 1.0,
+ "content": "Fan Chung and Mary Radcliffe. On the spectra of general random graphs. the electronic journal of",
+ "type": "text"
+ }
+ ],
+ "index": 39
+ },
+ {
+ "bbox": [
+ 115,
+ 631,
+ 249,
+ 643
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 631,
+ 249,
+ 643
+ ],
+ "score": 1.0,
+ "content": "combinatorics, 18(1):215, 2011.",
+ "type": "text"
+ }
+ ],
+ "index": 40
+ }
+ ],
+ "index": 39.5,
+ "bbox_fs": [
+ 106,
+ 619,
+ 506,
+ 643
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 650,
+ 503,
+ 673
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 648,
+ 505,
+ 664
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 648,
+ 505,
+ 664
+ ],
+ "score": 1.0,
+ "content": "Fan Chung, Linyuan Lu, and Van Vu. The spectra of random graphs with given expected degrees.",
+ "type": "text"
+ }
+ ],
+ "index": 41
+ },
+ {
+ "bbox": [
+ 115,
+ 660,
+ 291,
+ 673
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 660,
+ 291,
+ 673
+ ],
+ "score": 1.0,
+ "content": "Internet Mathematics, 1(3):257–275, 2004.",
+ "type": "text"
+ }
+ ],
+ "index": 42
+ }
+ ],
+ "index": 41.5,
+ "bbox_fs": [
+ 105,
+ 648,
+ 505,
+ 673
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 680,
+ 503,
+ 703
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 678,
+ 505,
+ 693
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 678,
+ 505,
+ 693
+ ],
+ "score": 1.0,
+ "content": "Fan RK Chung and Fan Chung Graham. Spectral graph theory. Number 92 in CBMS Regional",
+ "type": "text"
+ }
+ ],
+ "index": 43
+ },
+ {
+ "bbox": [
+ 116,
+ 691,
+ 405,
+ 702
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 116,
+ 691,
+ 405,
+ 702
+ ],
+ "score": 1.0,
+ "content": "Conference Series in Mathematics. American Mathematical Soc., 1997.",
+ "type": "text"
+ }
+ ],
+ "index": 44
+ }
+ ],
+ "index": 43.5,
+ "bbox_fs": [
+ 105,
+ 678,
+ 505,
+ 702
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 709,
+ 503,
+ 731
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 709,
+ 505,
+ 723
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 709,
+ 322,
+ 723
+ ],
+ "score": 1.0,
+ "content": "Amin Coja-Oghlan. On the laplacian eigenvalues of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 322,
+ 711,
+ 342,
+ 722
+ ],
+ "score": 0.9,
+ "content": "G _ { n , p }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 343,
+ 709,
+ 505,
+ 723
+ ],
+ "score": 1.0,
+ "content": ". Combinatorics, Probability and Com-",
+ "type": "text"
+ }
+ ],
+ "index": 45
+ },
+ {
+ "bbox": [
+ 114,
+ 721,
+ 236,
+ 731
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 114,
+ 721,
+ 236,
+ 731
+ ],
+ "score": 1.0,
+ "content": "puting, 16(6):923–946, 2007.",
+ "type": "text"
+ }
+ ],
+ "index": 46
+ }
+ ],
+ "index": 45.5,
+ "bbox_fs": [
+ 106,
+ 709,
+ 505,
+ 731
+ ]
+ }
+ ]
+ },
+ {
+ "preproc_blocks": [
+ {
+ "type": "text",
+ "bbox": [
+ 105,
+ 82,
+ 503,
+ 105
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 82,
+ 505,
+ 96
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 82,
+ 505,
+ 96
+ ],
+ "score": 1.0,
+ "content": "George Cybenko. Approximation by superpositions of a sigmoidal function. Mathematics of control,",
+ "type": "text"
+ }
+ ],
+ "index": 0
+ },
+ {
+ "bbox": [
+ 115,
+ 93,
+ 284,
+ 105
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 93,
+ 284,
+ 105
+ ],
+ "score": 1.0,
+ "content": "signals and systems, 2(4):303–314, 1989.",
+ "type": "text"
+ }
+ ],
+ "index": 1
+ }
+ ],
+ "index": 0.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 113,
+ 503,
+ 146
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 113,
+ 504,
+ 125
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 113,
+ 504,
+ 125
+ ],
+ "score": 1.0,
+ "content": "Michael Defferrard, Xavier Bresson, and Pierre Vandergheynst. Convolutional neural networks¨",
+ "type": "text"
+ }
+ ],
+ "index": 2
+ },
+ {
+ "bbox": [
+ 114,
+ 123,
+ 505,
+ 137
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 114,
+ 123,
+ 505,
+ 137
+ ],
+ "score": 1.0,
+ "content": "on graphs with fast localized spectral filtering. In Advances in Neural Information Processing",
+ "type": "text"
+ }
+ ],
+ "index": 3
+ },
+ {
+ "bbox": [
+ 116,
+ 135,
+ 353,
+ 147
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 116,
+ 135,
+ 353,
+ 147
+ ],
+ "score": 1.0,
+ "content": "Systems 29, pp. 3844–3852. Curran Associates, Inc., 2016.",
+ "type": "text"
+ }
+ ],
+ "index": 4
+ }
+ ],
+ "index": 3
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 154,
+ 505,
+ 199
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 154,
+ 505,
+ 167
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 154,
+ 505,
+ 167
+ ],
+ "score": 1.0,
+ "content": "David K Duvenaud, Dougal Maclaurin, Jorge Iparraguirre, Rafael Bombarell, Timothy Hirzel, Alan",
+ "type": "text"
+ }
+ ],
+ "index": 5
+ },
+ {
+ "bbox": [
+ 116,
+ 166,
+ 505,
+ 178
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 116,
+ 166,
+ 505,
+ 178
+ ],
+ "score": 1.0,
+ "content": "Aspuru-Guzik, and Ryan P Adams. Convolutional networks on graphs for learning molecular",
+ "type": "text"
+ }
+ ],
+ "index": 6
+ },
+ {
+ "bbox": [
+ 116,
+ 177,
+ 504,
+ 189
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 116,
+ 177,
+ 504,
+ 189
+ ],
+ "score": 1.0,
+ "content": "fingerprints. In Advances in Neural Information Processing Systems 28, pp. 2224–2232. Curran",
+ "type": "text"
+ }
+ ],
+ "index": 7
+ },
+ {
+ "bbox": [
+ 116,
+ 187,
+ 209,
+ 199
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 116,
+ 187,
+ 209,
+ 199
+ ],
+ "score": 1.0,
+ "content": "Associates, Inc., 2015.",
+ "type": "text"
+ }
+ ],
+ "index": 8
+ }
+ ],
+ "index": 6.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 207,
+ 502,
+ 230
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 206,
+ 505,
+ 219
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 206,
+ 505,
+ 219
+ ],
+ "score": 1.0,
+ "content": "Paul Erdos and Alfr ¨ ed R ´ enyi. On random graphs I. ´ Publicationes Mathematicae (Debrecen), 6:",
+ "type": "text"
+ }
+ ],
+ "index": 9
+ },
+ {
+ "bbox": [
+ 116,
+ 218,
+ 181,
+ 230
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 116,
+ 218,
+ 181,
+ 230
+ ],
+ "score": 1.0,
+ "content": "290–297, 1959.",
+ "type": "text"
+ }
+ ],
+ "index": 10
+ }
+ ],
+ "index": 9.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 237,
+ 501,
+ 249
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 236,
+ 501,
+ 251
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 236,
+ 501,
+ 251
+ ],
+ "score": 1.0,
+ "content": "Edgar N Gilbert. Random graphs. The Annals of Mathematical Statistics, 30(4):1141–1144, 1959.",
+ "type": "text"
+ }
+ ],
+ "index": 11
+ }
+ ],
+ "index": 11
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 257,
+ 503,
+ 280
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 256,
+ 505,
+ 271
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 256,
+ 505,
+ 271
+ ],
+ "score": 1.0,
+ "content": "C Lee Giles, Kurt D Bollacker, and Steve Lawrence. Citeseer: an automatic citation indexing system.",
+ "type": "text"
+ }
+ ],
+ "index": 12
+ },
+ {
+ "bbox": [
+ 115,
+ 268,
+ 474,
+ 280
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 268,
+ 474,
+ 280
+ ],
+ "score": 1.0,
+ "content": "In Proceedings of the third ACM conference on Digital libraries, pp. 89–98. ACM, 1998.",
+ "type": "text"
+ }
+ ],
+ "index": 13
+ }
+ ],
+ "index": 12.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 287,
+ 505,
+ 332
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 288,
+ 505,
+ 300
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 288,
+ 505,
+ 300
+ ],
+ "score": 1.0,
+ "content": "Justin Gilmer, Samuel S. Schoenholz, Patrick F. Riley, Oriol Vinyals, and George E. Dahl. Neural",
+ "type": "text"
+ }
+ ],
+ "index": 14
+ },
+ {
+ "bbox": [
+ 115,
+ 299,
+ 506,
+ 312
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 299,
+ 506,
+ 312
+ ],
+ "score": 1.0,
+ "content": "message passing for quantum chemistry. In Proceedings of the 34th International Conference on",
+ "type": "text"
+ }
+ ],
+ "index": 15
+ },
+ {
+ "bbox": [
+ 115,
+ 309,
+ 505,
+ 322
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 309,
+ 505,
+ 322
+ ],
+ "score": 1.0,
+ "content": "Machine Learning, volume 70 of Proceedings of Machine Learning Research, pp. 1263–1272.",
+ "type": "text"
+ }
+ ],
+ "index": 16
+ },
+ {
+ "bbox": [
+ 115,
+ 320,
+ 174,
+ 333
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 320,
+ 174,
+ 333
+ ],
+ "score": 1.0,
+ "content": "PMLR, 2017.",
+ "type": "text"
+ }
+ ],
+ "index": 17
+ }
+ ],
+ "index": 15.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 105,
+ 339,
+ 504,
+ 363
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 104,
+ 337,
+ 506,
+ 355
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 337,
+ 506,
+ 355
+ ],
+ "score": 1.0,
+ "content": "Torben Hagerup and Christine Rub. A guided tour of Chernoff bounds. ¨ Information processing",
+ "type": "text"
+ }
+ ],
+ "index": 18
+ },
+ {
+ "bbox": [
+ 116,
+ 351,
+ 235,
+ 363
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 116,
+ 351,
+ 235,
+ 363
+ ],
+ "score": 1.0,
+ "content": "letters, 33(6):305–308, 1990.",
+ "type": "text"
+ }
+ ],
+ "index": 19
+ }
+ ],
+ "index": 18.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 370,
+ 505,
+ 404
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 369,
+ 505,
+ 384
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 369,
+ 505,
+ 384
+ ],
+ "score": 1.0,
+ "content": "Will Hamilton, Zhitao Ying, and Jure Leskovec. Inductive representation learning on large graphs.",
+ "type": "text"
+ }
+ ],
+ "index": 20
+ },
+ {
+ "bbox": [
+ 115,
+ 381,
+ 506,
+ 395
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 381,
+ 506,
+ 395
+ ],
+ "score": 1.0,
+ "content": "In Advances in Neural Information Processing Systems 30, pp. 1024–1034. Curran Associates,",
+ "type": "text"
+ }
+ ],
+ "index": 21
+ },
+ {
+ "bbox": [
+ 115,
+ 392,
+ 163,
+ 405
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 392,
+ 163,
+ 405
+ ],
+ "score": 1.0,
+ "content": "Inc., 2017.",
+ "type": "text"
+ }
+ ],
+ "index": 22
+ }
+ ],
+ "index": 21
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 105,
+ 412,
+ 503,
+ 435
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 412,
+ 505,
+ 425
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 412,
+ 505,
+ 425
+ ],
+ "score": 1.0,
+ "content": "Mikael Henaff, Joan Bruna, and Yann LeCun. Deep convolutional networks on graph-structured",
+ "type": "text"
+ }
+ ],
+ "index": 23
+ },
+ {
+ "bbox": [
+ 116,
+ 423,
+ 302,
+ 436
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 116,
+ 423,
+ 302,
+ 436
+ ],
+ "score": 1.0,
+ "content": "data. arXiv preprint arXiv:1506.05163, 2015.",
+ "type": "text"
+ }
+ ],
+ "index": 24
+ }
+ ],
+ "index": 23.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 442,
+ 504,
+ 466
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 442,
+ 505,
+ 455
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 442,
+ 505,
+ 455
+ ],
+ "score": 1.0,
+ "content": "Kurt Hornik. Approximation capabilities of multilayer feedforward networks. Neural networks, 4",
+ "type": "text"
+ }
+ ],
+ "index": 25
+ },
+ {
+ "bbox": [
+ 115,
+ 453,
+ 196,
+ 466
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 453,
+ 196,
+ 466
+ ],
+ "score": 1.0,
+ "content": "(2):251–257, 1991.",
+ "type": "text"
+ }
+ ],
+ "index": 26
+ }
+ ],
+ "index": 25.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 473,
+ 504,
+ 496
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 473,
+ 505,
+ 486
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 473,
+ 505,
+ 486
+ ],
+ "score": 1.0,
+ "content": "Kurt Hornik, Maxwell Stinchcombe, and Halbert White. Multilayer feedforward networks are uni-",
+ "type": "text"
+ }
+ ],
+ "index": 27
+ },
+ {
+ "bbox": [
+ 117,
+ 485,
+ 361,
+ 496
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 117,
+ 485,
+ 361,
+ 496
+ ],
+ "score": 1.0,
+ "content": "versal approximators. Neural networks, 2(5):359–366, 1989.",
+ "type": "text"
+ }
+ ],
+ "index": 28
+ }
+ ],
+ "index": 27.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 504,
+ 504,
+ 527
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 503,
+ 504,
+ 517
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 503,
+ 504,
+ 517
+ ],
+ "score": 1.0,
+ "content": "CD Johnson. Stabilization of linear dynamical systems with respect to arbitrary linear subspaces.",
+ "type": "text"
+ }
+ ],
+ "index": 29
+ },
+ {
+ "bbox": [
+ 116,
+ 515,
+ 416,
+ 527
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 116,
+ 515,
+ 416,
+ 527
+ ],
+ "score": 1.0,
+ "content": "Journal of Mathematical Analysis and Applications, 44(1):175–186, 1973.",
+ "type": "text"
+ }
+ ],
+ "index": 30
+ }
+ ],
+ "index": 29.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 534,
+ 504,
+ 558
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 534,
+ 505,
+ 547
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 534,
+ 505,
+ 547
+ ],
+ "score": 1.0,
+ "content": "Diederik P Kingma and Jimmy Ba. Adam: A method for stochastic optimization. In International",
+ "type": "text"
+ }
+ ],
+ "index": 31
+ },
+ {
+ "bbox": [
+ 116,
+ 545,
+ 310,
+ 558
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 116,
+ 545,
+ 310,
+ 558
+ ],
+ "score": 1.0,
+ "content": "Conference on Learning Representations, 2015.",
+ "type": "text"
+ }
+ ],
+ "index": 32
+ }
+ ],
+ "index": 31.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 564,
+ 504,
+ 588
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 564,
+ 505,
+ 578
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 564,
+ 505,
+ 578
+ ],
+ "score": 1.0,
+ "content": "Thomas N. Kipf and Max Welling. Semi-supervised classification with graph convolutional net-",
+ "type": "text"
+ }
+ ],
+ "index": 33
+ },
+ {
+ "bbox": [
+ 117,
+ 577,
+ 406,
+ 588
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 117,
+ 577,
+ 406,
+ 588
+ ],
+ "score": 1.0,
+ "content": "works. In International Conference on Learning Representations, 2017.",
+ "type": "text"
+ }
+ ],
+ "index": 34
+ }
+ ],
+ "index": 33.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 595,
+ 503,
+ 630
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 595,
+ 505,
+ 609
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 595,
+ 505,
+ 609
+ ],
+ "score": 1.0,
+ "content": "Alex Krizhevsky, Ilya Sutskever, and Geoffrey E Hinton. ImageNet classification with deep convo-",
+ "type": "text"
+ }
+ ],
+ "index": 35
+ },
+ {
+ "bbox": [
+ 116,
+ 607,
+ 505,
+ 619
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 116,
+ 607,
+ 505,
+ 619
+ ],
+ "score": 1.0,
+ "content": "lutional neural networks. In Advances in Neural Information Processing Systems 25, pp. 1097–",
+ "type": "text"
+ }
+ ],
+ "index": 36
+ },
+ {
+ "bbox": [
+ 117,
+ 618,
+ 264,
+ 629
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 117,
+ 618,
+ 264,
+ 629
+ ],
+ "score": 1.0,
+ "content": "1105. Curran Associates, Inc., 2012.",
+ "type": "text"
+ }
+ ],
+ "index": 37
+ }
+ ],
+ "index": 36
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 105,
+ 637,
+ 504,
+ 660
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 637,
+ 505,
+ 650
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 637,
+ 505,
+ 650
+ ],
+ "score": 1.0,
+ "content": "Yann A LeCun, Leon Bottou, Genevieve B Orr, and Klaus-Robert M ´ uller. Efficient backprop. In ¨",
+ "type": "text"
+ }
+ ],
+ "index": 38
+ },
+ {
+ "bbox": [
+ 115,
+ 648,
+ 370,
+ 661
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 648,
+ 370,
+ 661
+ ],
+ "score": 1.0,
+ "content": "Neural networks: Tricks of the trade, pp. 9–48. Springer, 2012.",
+ "type": "text"
+ }
+ ],
+ "index": 39
+ }
+ ],
+ "index": 38.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 667,
+ 505,
+ 701
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 666,
+ 506,
+ 682
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 666,
+ 506,
+ 682
+ ],
+ "score": 1.0,
+ "content": "Guohao Li, Matthias Muller, Ali Thabet, and Bernard Ghanem. Deepgcns: Can gcns go as deep as",
+ "type": "text"
+ }
+ ],
+ "index": 40
+ },
+ {
+ "bbox": [
+ 115,
+ 678,
+ 505,
+ 691
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 678,
+ 505,
+ 691
+ ],
+ "score": 1.0,
+ "content": "cnns? In Proceedings of the IEEE International Conference on Computer Vision, pp. 9267–9276,",
+ "type": "text"
+ }
+ ],
+ "index": 41
+ },
+ {
+ "bbox": [
+ 115,
+ 689,
+ 143,
+ 702
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 689,
+ 143,
+ 702
+ ],
+ "score": 1.0,
+ "content": "2019.",
+ "type": "text"
+ }
+ ],
+ "index": 42
+ }
+ ],
+ "index": 41
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 709,
+ 503,
+ 732
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 107,
+ 709,
+ 505,
+ 722
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 709,
+ 505,
+ 722
+ ],
+ "score": 1.0,
+ "content": "Qimai Li, Zhichao Han, and Xiao-Ming Wu. Deeper insights into graph convolutional networks for",
+ "type": "text"
+ }
+ ],
+ "index": 43
+ },
+ {
+ "bbox": [
+ 116,
+ 721,
+ 491,
+ 732
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 116,
+ 721,
+ 491,
+ 732
+ ],
+ "score": 1.0,
+ "content": "semi-supervised learning. In Thirty-Second AAAI Conference on Artificial Intelligence, 2018.",
+ "type": "text"
+ }
+ ],
+ "index": 44
+ }
+ ],
+ "index": 43.5
+ }
+ ],
+ "page_idx": 10,
+ "page_size": [
+ 612,
+ 792
+ ],
+ "discarded_blocks": [
+ {
+ "type": "discarded",
+ "bbox": [
+ 107,
+ 27,
+ 294,
+ 37
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 26,
+ 294,
+ 38
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 26,
+ 294,
+ 38
+ ],
+ "score": 1.0,
+ "content": "Published as a conference paper at ICLR 2020",
+ "type": "text"
+ }
+ ]
+ }
+ ]
+ },
+ {
+ "type": "discarded",
+ "bbox": [
+ 300,
+ 751,
+ 310,
+ 760
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 299,
+ 750,
+ 312,
+ 765
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 299,
+ 750,
+ 312,
+ 765
+ ],
+ "score": 1.0,
+ "content": "",
+ "type": "text",
+ "height": 15,
+ "width": 13
+ }
+ ]
+ }
+ ]
+ }
+ ],
+ "para_blocks": [
+ {
+ "type": "text",
+ "bbox": [
+ 105,
+ 82,
+ 503,
+ 105
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 82,
+ 505,
+ 96
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 82,
+ 505,
+ 96
+ ],
+ "score": 1.0,
+ "content": "George Cybenko. Approximation by superpositions of a sigmoidal function. Mathematics of control,",
+ "type": "text"
+ }
+ ],
+ "index": 0
+ },
+ {
+ "bbox": [
+ 115,
+ 93,
+ 284,
+ 105
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 93,
+ 284,
+ 105
+ ],
+ "score": 1.0,
+ "content": "signals and systems, 2(4):303–314, 1989.",
+ "type": "text"
+ }
+ ],
+ "index": 1
+ }
+ ],
+ "index": 0.5,
+ "bbox_fs": [
+ 105,
+ 82,
+ 505,
+ 105
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 113,
+ 503,
+ 146
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 113,
+ 504,
+ 125
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 113,
+ 504,
+ 125
+ ],
+ "score": 1.0,
+ "content": "Michael Defferrard, Xavier Bresson, and Pierre Vandergheynst. Convolutional neural networks¨",
+ "type": "text"
+ }
+ ],
+ "index": 2
+ },
+ {
+ "bbox": [
+ 114,
+ 123,
+ 505,
+ 137
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 114,
+ 123,
+ 505,
+ 137
+ ],
+ "score": 1.0,
+ "content": "on graphs with fast localized spectral filtering. In Advances in Neural Information Processing",
+ "type": "text"
+ }
+ ],
+ "index": 3
+ },
+ {
+ "bbox": [
+ 116,
+ 135,
+ 353,
+ 147
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 116,
+ 135,
+ 353,
+ 147
+ ],
+ "score": 1.0,
+ "content": "Systems 29, pp. 3844–3852. Curran Associates, Inc., 2016.",
+ "type": "text"
+ }
+ ],
+ "index": 4
+ }
+ ],
+ "index": 3,
+ "bbox_fs": [
+ 106,
+ 113,
+ 505,
+ 147
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 154,
+ 505,
+ 199
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 154,
+ 505,
+ 167
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 154,
+ 505,
+ 167
+ ],
+ "score": 1.0,
+ "content": "David K Duvenaud, Dougal Maclaurin, Jorge Iparraguirre, Rafael Bombarell, Timothy Hirzel, Alan",
+ "type": "text"
+ }
+ ],
+ "index": 5
+ },
+ {
+ "bbox": [
+ 116,
+ 166,
+ 505,
+ 178
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 116,
+ 166,
+ 505,
+ 178
+ ],
+ "score": 1.0,
+ "content": "Aspuru-Guzik, and Ryan P Adams. Convolutional networks on graphs for learning molecular",
+ "type": "text"
+ }
+ ],
+ "index": 6
+ },
+ {
+ "bbox": [
+ 116,
+ 177,
+ 504,
+ 189
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 116,
+ 177,
+ 504,
+ 189
+ ],
+ "score": 1.0,
+ "content": "fingerprints. In Advances in Neural Information Processing Systems 28, pp. 2224–2232. Curran",
+ "type": "text"
+ }
+ ],
+ "index": 7
+ },
+ {
+ "bbox": [
+ 116,
+ 187,
+ 209,
+ 199
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 116,
+ 187,
+ 209,
+ 199
+ ],
+ "score": 1.0,
+ "content": "Associates, Inc., 2015.",
+ "type": "text"
+ }
+ ],
+ "index": 8
+ }
+ ],
+ "index": 6.5,
+ "bbox_fs": [
+ 106,
+ 154,
+ 505,
+ 199
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 207,
+ 502,
+ 230
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 206,
+ 505,
+ 219
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 206,
+ 505,
+ 219
+ ],
+ "score": 1.0,
+ "content": "Paul Erdos and Alfr ¨ ed R ´ enyi. On random graphs I. ´ Publicationes Mathematicae (Debrecen), 6:",
+ "type": "text"
+ }
+ ],
+ "index": 9
+ },
+ {
+ "bbox": [
+ 116,
+ 218,
+ 181,
+ 230
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 116,
+ 218,
+ 181,
+ 230
+ ],
+ "score": 1.0,
+ "content": "290–297, 1959.",
+ "type": "text"
+ }
+ ],
+ "index": 10
+ }
+ ],
+ "index": 9.5,
+ "bbox_fs": [
+ 105,
+ 206,
+ 505,
+ 230
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 237,
+ 501,
+ 249
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 236,
+ 501,
+ 251
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 236,
+ 501,
+ 251
+ ],
+ "score": 1.0,
+ "content": "Edgar N Gilbert. Random graphs. The Annals of Mathematical Statistics, 30(4):1141–1144, 1959.",
+ "type": "text"
+ }
+ ],
+ "index": 11
+ }
+ ],
+ "index": 11,
+ "bbox_fs": [
+ 106,
+ 236,
+ 501,
+ 251
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 257,
+ 503,
+ 280
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 256,
+ 505,
+ 271
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 256,
+ 505,
+ 271
+ ],
+ "score": 1.0,
+ "content": "C Lee Giles, Kurt D Bollacker, and Steve Lawrence. Citeseer: an automatic citation indexing system.",
+ "type": "text"
+ }
+ ],
+ "index": 12
+ },
+ {
+ "bbox": [
+ 115,
+ 268,
+ 474,
+ 280
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 268,
+ 474,
+ 280
+ ],
+ "score": 1.0,
+ "content": "In Proceedings of the third ACM conference on Digital libraries, pp. 89–98. ACM, 1998.",
+ "type": "text"
+ }
+ ],
+ "index": 13
+ }
+ ],
+ "index": 12.5,
+ "bbox_fs": [
+ 106,
+ 256,
+ 505,
+ 280
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 287,
+ 505,
+ 332
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 288,
+ 505,
+ 300
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 288,
+ 505,
+ 300
+ ],
+ "score": 1.0,
+ "content": "Justin Gilmer, Samuel S. Schoenholz, Patrick F. Riley, Oriol Vinyals, and George E. Dahl. Neural",
+ "type": "text"
+ }
+ ],
+ "index": 14
+ },
+ {
+ "bbox": [
+ 115,
+ 299,
+ 506,
+ 312
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 299,
+ 506,
+ 312
+ ],
+ "score": 1.0,
+ "content": "message passing for quantum chemistry. In Proceedings of the 34th International Conference on",
+ "type": "text"
+ }
+ ],
+ "index": 15
+ },
+ {
+ "bbox": [
+ 115,
+ 309,
+ 505,
+ 322
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 309,
+ 505,
+ 322
+ ],
+ "score": 1.0,
+ "content": "Machine Learning, volume 70 of Proceedings of Machine Learning Research, pp. 1263–1272.",
+ "type": "text"
+ }
+ ],
+ "index": 16
+ },
+ {
+ "bbox": [
+ 115,
+ 320,
+ 174,
+ 333
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 320,
+ 174,
+ 333
+ ],
+ "score": 1.0,
+ "content": "PMLR, 2017.",
+ "type": "text"
+ }
+ ],
+ "index": 17
+ }
+ ],
+ "index": 15.5,
+ "bbox_fs": [
+ 105,
+ 288,
+ 506,
+ 333
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 105,
+ 339,
+ 504,
+ 363
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 104,
+ 337,
+ 506,
+ 355
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 337,
+ 506,
+ 355
+ ],
+ "score": 1.0,
+ "content": "Torben Hagerup and Christine Rub. A guided tour of Chernoff bounds. ¨ Information processing",
+ "type": "text"
+ }
+ ],
+ "index": 18
+ },
+ {
+ "bbox": [
+ 116,
+ 351,
+ 235,
+ 363
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 116,
+ 351,
+ 235,
+ 363
+ ],
+ "score": 1.0,
+ "content": "letters, 33(6):305–308, 1990.",
+ "type": "text"
+ }
+ ],
+ "index": 19
+ }
+ ],
+ "index": 18.5,
+ "bbox_fs": [
+ 104,
+ 337,
+ 506,
+ 363
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 370,
+ 505,
+ 404
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 369,
+ 505,
+ 384
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 369,
+ 505,
+ 384
+ ],
+ "score": 1.0,
+ "content": "Will Hamilton, Zhitao Ying, and Jure Leskovec. Inductive representation learning on large graphs.",
+ "type": "text"
+ }
+ ],
+ "index": 20
+ },
+ {
+ "bbox": [
+ 115,
+ 381,
+ 506,
+ 395
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 381,
+ 506,
+ 395
+ ],
+ "score": 1.0,
+ "content": "In Advances in Neural Information Processing Systems 30, pp. 1024–1034. Curran Associates,",
+ "type": "text"
+ }
+ ],
+ "index": 21
+ },
+ {
+ "bbox": [
+ 115,
+ 392,
+ 163,
+ 405
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 392,
+ 163,
+ 405
+ ],
+ "score": 1.0,
+ "content": "Inc., 2017.",
+ "type": "text"
+ }
+ ],
+ "index": 22
+ }
+ ],
+ "index": 21,
+ "bbox_fs": [
+ 105,
+ 369,
+ 506,
+ 405
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 105,
+ 412,
+ 503,
+ 435
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 412,
+ 505,
+ 425
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 412,
+ 505,
+ 425
+ ],
+ "score": 1.0,
+ "content": "Mikael Henaff, Joan Bruna, and Yann LeCun. Deep convolutional networks on graph-structured",
+ "type": "text"
+ }
+ ],
+ "index": 23
+ },
+ {
+ "bbox": [
+ 116,
+ 423,
+ 302,
+ 436
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 116,
+ 423,
+ 302,
+ 436
+ ],
+ "score": 1.0,
+ "content": "data. arXiv preprint arXiv:1506.05163, 2015.",
+ "type": "text"
+ }
+ ],
+ "index": 24
+ }
+ ],
+ "index": 23.5,
+ "bbox_fs": [
+ 106,
+ 412,
+ 505,
+ 436
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 442,
+ 504,
+ 466
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 442,
+ 505,
+ 455
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 442,
+ 505,
+ 455
+ ],
+ "score": 1.0,
+ "content": "Kurt Hornik. Approximation capabilities of multilayer feedforward networks. Neural networks, 4",
+ "type": "text"
+ }
+ ],
+ "index": 25
+ },
+ {
+ "bbox": [
+ 115,
+ 453,
+ 196,
+ 466
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 453,
+ 196,
+ 466
+ ],
+ "score": 1.0,
+ "content": "(2):251–257, 1991.",
+ "type": "text"
+ }
+ ],
+ "index": 26
+ }
+ ],
+ "index": 25.5,
+ "bbox_fs": [
+ 106,
+ 442,
+ 505,
+ 466
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 473,
+ 504,
+ 496
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 473,
+ 505,
+ 486
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 473,
+ 505,
+ 486
+ ],
+ "score": 1.0,
+ "content": "Kurt Hornik, Maxwell Stinchcombe, and Halbert White. Multilayer feedforward networks are uni-",
+ "type": "text"
+ }
+ ],
+ "index": 27
+ },
+ {
+ "bbox": [
+ 117,
+ 485,
+ 361,
+ 496
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 117,
+ 485,
+ 361,
+ 496
+ ],
+ "score": 1.0,
+ "content": "versal approximators. Neural networks, 2(5):359–366, 1989.",
+ "type": "text"
+ }
+ ],
+ "index": 28
+ }
+ ],
+ "index": 27.5,
+ "bbox_fs": [
+ 106,
+ 473,
+ 505,
+ 496
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 504,
+ 504,
+ 527
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 503,
+ 504,
+ 517
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 503,
+ 504,
+ 517
+ ],
+ "score": 1.0,
+ "content": "CD Johnson. Stabilization of linear dynamical systems with respect to arbitrary linear subspaces.",
+ "type": "text"
+ }
+ ],
+ "index": 29
+ },
+ {
+ "bbox": [
+ 116,
+ 515,
+ 416,
+ 527
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 116,
+ 515,
+ 416,
+ 527
+ ],
+ "score": 1.0,
+ "content": "Journal of Mathematical Analysis and Applications, 44(1):175–186, 1973.",
+ "type": "text"
+ }
+ ],
+ "index": 30
+ }
+ ],
+ "index": 29.5,
+ "bbox_fs": [
+ 105,
+ 503,
+ 504,
+ 527
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 534,
+ 504,
+ 558
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 534,
+ 505,
+ 547
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 534,
+ 505,
+ 547
+ ],
+ "score": 1.0,
+ "content": "Diederik P Kingma and Jimmy Ba. Adam: A method for stochastic optimization. In International",
+ "type": "text"
+ }
+ ],
+ "index": 31
+ },
+ {
+ "bbox": [
+ 116,
+ 545,
+ 310,
+ 558
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 116,
+ 545,
+ 310,
+ 558
+ ],
+ "score": 1.0,
+ "content": "Conference on Learning Representations, 2015.",
+ "type": "text"
+ }
+ ],
+ "index": 32
+ }
+ ],
+ "index": 31.5,
+ "bbox_fs": [
+ 106,
+ 534,
+ 505,
+ 558
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 564,
+ 504,
+ 588
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 564,
+ 505,
+ 578
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 564,
+ 505,
+ 578
+ ],
+ "score": 1.0,
+ "content": "Thomas N. Kipf and Max Welling. Semi-supervised classification with graph convolutional net-",
+ "type": "text"
+ }
+ ],
+ "index": 33
+ },
+ {
+ "bbox": [
+ 117,
+ 577,
+ 406,
+ 588
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 117,
+ 577,
+ 406,
+ 588
+ ],
+ "score": 1.0,
+ "content": "works. In International Conference on Learning Representations, 2017.",
+ "type": "text"
+ }
+ ],
+ "index": 34
+ }
+ ],
+ "index": 33.5,
+ "bbox_fs": [
+ 106,
+ 564,
+ 505,
+ 588
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 595,
+ 503,
+ 630
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 595,
+ 505,
+ 609
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 595,
+ 505,
+ 609
+ ],
+ "score": 1.0,
+ "content": "Alex Krizhevsky, Ilya Sutskever, and Geoffrey E Hinton. ImageNet classification with deep convo-",
+ "type": "text"
+ }
+ ],
+ "index": 35
+ },
+ {
+ "bbox": [
+ 116,
+ 607,
+ 505,
+ 619
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 116,
+ 607,
+ 505,
+ 619
+ ],
+ "score": 1.0,
+ "content": "lutional neural networks. In Advances in Neural Information Processing Systems 25, pp. 1097–",
+ "type": "text"
+ }
+ ],
+ "index": 36
+ },
+ {
+ "bbox": [
+ 117,
+ 618,
+ 264,
+ 629
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 117,
+ 618,
+ 264,
+ 629
+ ],
+ "score": 1.0,
+ "content": "1105. Curran Associates, Inc., 2012.",
+ "type": "text"
+ }
+ ],
+ "index": 37
+ }
+ ],
+ "index": 36,
+ "bbox_fs": [
+ 105,
+ 595,
+ 505,
+ 629
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 105,
+ 637,
+ 504,
+ 660
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 637,
+ 505,
+ 650
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 637,
+ 505,
+ 650
+ ],
+ "score": 1.0,
+ "content": "Yann A LeCun, Leon Bottou, Genevieve B Orr, and Klaus-Robert M ´ uller. Efficient backprop. In ¨",
+ "type": "text"
+ }
+ ],
+ "index": 38
+ },
+ {
+ "bbox": [
+ 115,
+ 648,
+ 370,
+ 661
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 648,
+ 370,
+ 661
+ ],
+ "score": 1.0,
+ "content": "Neural networks: Tricks of the trade, pp. 9–48. Springer, 2012.",
+ "type": "text"
+ }
+ ],
+ "index": 39
+ }
+ ],
+ "index": 38.5,
+ "bbox_fs": [
+ 106,
+ 637,
+ 505,
+ 661
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 667,
+ 505,
+ 701
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 666,
+ 506,
+ 682
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 666,
+ 506,
+ 682
+ ],
+ "score": 1.0,
+ "content": "Guohao Li, Matthias Muller, Ali Thabet, and Bernard Ghanem. Deepgcns: Can gcns go as deep as",
+ "type": "text"
+ }
+ ],
+ "index": 40
+ },
+ {
+ "bbox": [
+ 115,
+ 678,
+ 505,
+ 691
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 678,
+ 505,
+ 691
+ ],
+ "score": 1.0,
+ "content": "cnns? In Proceedings of the IEEE International Conference on Computer Vision, pp. 9267–9276,",
+ "type": "text"
+ }
+ ],
+ "index": 41
+ },
+ {
+ "bbox": [
+ 115,
+ 689,
+ 143,
+ 702
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 689,
+ 143,
+ 702
+ ],
+ "score": 1.0,
+ "content": "2019.",
+ "type": "text"
+ }
+ ],
+ "index": 42
+ }
+ ],
+ "index": 41,
+ "bbox_fs": [
+ 105,
+ 666,
+ 506,
+ 702
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 709,
+ 503,
+ 732
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 107,
+ 709,
+ 505,
+ 722
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 709,
+ 505,
+ 722
+ ],
+ "score": 1.0,
+ "content": "Qimai Li, Zhichao Han, and Xiao-Ming Wu. Deeper insights into graph convolutional networks for",
+ "type": "text"
+ }
+ ],
+ "index": 43
+ },
+ {
+ "bbox": [
+ 116,
+ 721,
+ 491,
+ 732
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 116,
+ 721,
+ 491,
+ 732
+ ],
+ "score": 1.0,
+ "content": "semi-supervised learning. In Thirty-Second AAAI Conference on Artificial Intelligence, 2018.",
+ "type": "text"
+ }
+ ],
+ "index": 44
+ }
+ ],
+ "index": 43.5,
+ "bbox_fs": [
+ 107,
+ 709,
+ 505,
+ 732
+ ]
+ }
+ ]
+ },
+ {
+ "preproc_blocks": [
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 82,
+ 504,
+ 105
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 107,
+ 82,
+ 506,
+ 95
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 82,
+ 506,
+ 95
+ ],
+ "score": 1.0,
+ "content": "Yujia Li, Richard Zemel, and Marc Brockschmidt. Gated graph sequence neural networks. In",
+ "type": "text"
+ }
+ ],
+ "index": 0
+ },
+ {
+ "bbox": [
+ 116,
+ 93,
+ 365,
+ 106
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 116,
+ 93,
+ 365,
+ 106
+ ],
+ "score": 1.0,
+ "content": "International Conference on Learning Representations, 2016.",
+ "type": "text"
+ }
+ ],
+ "index": 1
+ }
+ ],
+ "index": 0.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 108,
+ 111,
+ 504,
+ 145
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 110,
+ 504,
+ 124
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 110,
+ 504,
+ 124
+ ],
+ "score": 1.0,
+ "content": "Sitao Luan, Mingde Zhao, Xiao-Wen Chang, and Doina Precup. Break the ceiling: Stronger multi-",
+ "type": "text"
+ }
+ ],
+ "index": 2
+ },
+ {
+ "bbox": [
+ 116,
+ 123,
+ 505,
+ 135
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 116,
+ 123,
+ 505,
+ 135
+ ],
+ "score": 1.0,
+ "content": "scale deep graph convolutional networks. In Advances in Neural Information Processing Systems",
+ "type": "text"
+ }
+ ],
+ "index": 3
+ },
+ {
+ "bbox": [
+ 115,
+ 133,
+ 329,
+ 145
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 133,
+ 329,
+ 145
+ ],
+ "score": 1.0,
+ "content": "32, pp. 10943–10953. Curran Associates, Inc., 2019.",
+ "type": "text"
+ }
+ ],
+ "index": 4
+ }
+ ],
+ "index": 3
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 150,
+ 503,
+ 185
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 151,
+ 505,
+ 163
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 151,
+ 505,
+ 163
+ ],
+ "score": 1.0,
+ "content": "Andrew L. Maas, Awni Y. Hannun, and Andrew Y. Ng. Rectifier nonlinearities improve neural net-",
+ "type": "text"
+ }
+ ],
+ "index": 5
+ },
+ {
+ "bbox": [
+ 115,
+ 161,
+ 505,
+ 176
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 161,
+ 505,
+ 176
+ ],
+ "score": 1.0,
+ "content": "work acoustic models. In in ICML Workshop on Deep Learning for Audio, Speech and Language",
+ "type": "text"
+ }
+ ],
+ "index": 6
+ },
+ {
+ "bbox": [
+ 116,
+ 173,
+ 191,
+ 186
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 116,
+ 173,
+ 191,
+ 186
+ ],
+ "score": 1.0,
+ "content": "Processing, 2013.",
+ "type": "text"
+ }
+ ],
+ "index": 7
+ }
+ ],
+ "index": 6
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 108,
+ 190,
+ 504,
+ 225
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 190,
+ 505,
+ 204
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 190,
+ 505,
+ 204
+ ],
+ "score": 1.0,
+ "content": "Andrew Kachites McCallum, Kamal Nigam, Jason Rennie, and Kristie Seymore. Automating the",
+ "type": "text"
+ }
+ ],
+ "index": 8
+ },
+ {
+ "bbox": [
+ 115,
+ 202,
+ 505,
+ 214
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 202,
+ 505,
+ 214
+ ],
+ "score": 1.0,
+ "content": "construction of internet portals with machine learning. Information Retrieval, 3(2):127–163,",
+ "type": "text"
+ }
+ ],
+ "index": 9
+ },
+ {
+ "bbox": [
+ 115,
+ 212,
+ 142,
+ 225
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 212,
+ 142,
+ 225
+ ],
+ "score": 1.0,
+ "content": "2000.",
+ "type": "text"
+ }
+ ],
+ "index": 10
+ }
+ ],
+ "index": 9
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 230,
+ 504,
+ 254
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 230,
+ 505,
+ 244
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 230,
+ 505,
+ 244
+ ],
+ "score": 1.0,
+ "content": "Song Mei and Andrea Montanari. The generalization error of random features regression: Precise",
+ "type": "text"
+ }
+ ],
+ "index": 11
+ },
+ {
+ "bbox": [
+ 115,
+ 242,
+ 436,
+ 254
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 242,
+ 436,
+ 254
+ ],
+ "score": 1.0,
+ "content": "asymptotics and double descent curve. arXiv preprint arXiv:1908.05355, 2019.",
+ "type": "text"
+ }
+ ],
+ "index": 12
+ }
+ ],
+ "index": 11.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 259,
+ 443,
+ 272
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 259,
+ 442,
+ 273
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 259,
+ 442,
+ 273
+ ],
+ "score": 1.0,
+ "content": "Carl D Meyer. Matrix analysis and applied linear algebra, volume 71. Siam, 2000.",
+ "type": "text"
+ }
+ ],
+ "index": 13
+ }
+ ],
+ "index": 13
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 278,
+ 503,
+ 301
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 278,
+ 504,
+ 291
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 278,
+ 504,
+ 291
+ ],
+ "score": 1.0,
+ "content": "Hrushikesh Narhar Mhaskar. Approximation properties of a multilayered feedforward artificial neu-",
+ "type": "text"
+ }
+ ],
+ "index": 14
+ },
+ {
+ "bbox": [
+ 116,
+ 289,
+ 411,
+ 301
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 116,
+ 289,
+ 411,
+ 301
+ ],
+ "score": 1.0,
+ "content": "ral network. Advances in Computational Mathematics, 1(1):61–80, 1993.",
+ "type": "text"
+ }
+ ],
+ "index": 15
+ }
+ ],
+ "index": 14.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 306,
+ 502,
+ 330
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 306,
+ 504,
+ 320
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 306,
+ 504,
+ 320
+ ],
+ "score": 1.0,
+ "content": "Hai Nguyen, Shinichi Maeda, and Kenta Oono. Semi-supervised learning of hierarchical represen-",
+ "type": "text"
+ }
+ ],
+ "index": 16
+ },
+ {
+ "bbox": [
+ 115,
+ 318,
+ 486,
+ 331
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 318,
+ 486,
+ 331
+ ],
+ "score": 1.0,
+ "content": "tations of molecules using neural message passing. arXiv preprint arXiv:1711.10168, 2017.",
+ "type": "text"
+ }
+ ],
+ "index": 17
+ }
+ ],
+ "index": 16.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 335,
+ 504,
+ 359
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 335,
+ 505,
+ 348
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 335,
+ 505,
+ 348
+ ],
+ "score": 1.0,
+ "content": "James R Norris. Markov chains. Number 2 in Cambridge Series in Statistical and Probabilistic",
+ "type": "text"
+ }
+ ],
+ "index": 18
+ },
+ {
+ "bbox": [
+ 116,
+ 346,
+ 312,
+ 359
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 116,
+ 346,
+ 312,
+ 359
+ ],
+ "score": 1.0,
+ "content": "Mathematics. Cambridge university press, 1998.",
+ "type": "text"
+ }
+ ],
+ "index": 19
+ }
+ ],
+ "index": 18.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 364,
+ 504,
+ 388
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 365,
+ 504,
+ 378
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 365,
+ 504,
+ 378
+ ],
+ "score": 1.0,
+ "content": "Hoang NT and Takanori Maehara. Revisiting graph neural networks: All we have is low-pass filters.",
+ "type": "text"
+ }
+ ],
+ "index": 20
+ },
+ {
+ "bbox": [
+ 116,
+ 376,
+ 279,
+ 388
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 116,
+ 376,
+ 279,
+ 388
+ ],
+ "score": 1.0,
+ "content": "arXiv preprint arXiv:1905.09550, 2019.",
+ "type": "text"
+ }
+ ],
+ "index": 21
+ }
+ ],
+ "index": 20.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 393,
+ 504,
+ 417
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 394,
+ 504,
+ 407
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 394,
+ 504,
+ 407
+ ],
+ "score": 1.0,
+ "content": "Kenta Oono and Taiji Suzuki. Approximation and non-parametric estimation of ResNet-type con-",
+ "type": "text"
+ }
+ ],
+ "index": 22
+ },
+ {
+ "bbox": [
+ 116,
+ 405,
+ 391,
+ 417
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 116,
+ 405,
+ 391,
+ 417
+ ],
+ "score": 1.0,
+ "content": "volutional neural networks. arXiv preprint arXiv:1903.10047, 2019.",
+ "type": "text"
+ }
+ ],
+ "index": 23
+ }
+ ],
+ "index": 22.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 423,
+ 503,
+ 446
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 421,
+ 505,
+ 436
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 421,
+ 505,
+ 436
+ ],
+ "score": 1.0,
+ "content": "Sinno Jialin Pan and Qiang Yang. A survey on transfer learning. IEEE Transactions on knowledge",
+ "type": "text"
+ }
+ ],
+ "index": 24
+ },
+ {
+ "bbox": [
+ 116,
+ 434,
+ 311,
+ 446
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 116,
+ 434,
+ 311,
+ 446
+ ],
+ "score": 1.0,
+ "content": "and data engineering, 22(10):1345–1359, 2010.",
+ "type": "text"
+ }
+ ],
+ "index": 25
+ }
+ ],
+ "index": 24.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 452,
+ 502,
+ 475
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 451,
+ 504,
+ 464
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 451,
+ 504,
+ 464
+ ],
+ "score": 1.0,
+ "content": "Philipp Petersen and Felix Voigtlaender. Equivalence of approximation by convolutional neural",
+ "type": "text"
+ }
+ ],
+ "index": 26
+ },
+ {
+ "bbox": [
+ 115,
+ 462,
+ 442,
+ 475
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 462,
+ 442,
+ 475
+ ],
+ "score": 1.0,
+ "content": "networks and fully-connected networks. arXiv preprint arXiv:1809.00973, 2018.",
+ "type": "text"
+ }
+ ],
+ "index": 27
+ }
+ ],
+ "index": 26.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 480,
+ 502,
+ 504
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 480,
+ 503,
+ 493
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 480,
+ 503,
+ 493
+ ],
+ "score": 1.0,
+ "content": "Franco Scarselli, Marco Gori, Ah Chung Tsoi, Markus Hagenbuchner, and Gabriele Monfardini.",
+ "type": "text"
+ }
+ ],
+ "index": 28
+ },
+ {
+ "bbox": [
+ 116,
+ 491,
+ 496,
+ 504
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 116,
+ 491,
+ 496,
+ 504
+ ],
+ "score": 1.0,
+ "content": "The graph neural network model. IEEE Transactions on Neural Networks, 20(1):61–80, 2009.",
+ "type": "text"
+ }
+ ],
+ "index": 29
+ }
+ ],
+ "index": 28.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 108,
+ 509,
+ 504,
+ 544
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 509,
+ 505,
+ 522
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 509,
+ 505,
+ 522
+ ],
+ "score": 1.0,
+ "content": "Michael Schlichtkrull, Thomas N Kipf, Peter Bloem, Rianne Van Den Berg, Ivan Titov, and Max",
+ "type": "text"
+ }
+ ],
+ "index": 30
+ },
+ {
+ "bbox": [
+ 116,
+ 520,
+ 505,
+ 533
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 116,
+ 520,
+ 505,
+ 533
+ ],
+ "score": 1.0,
+ "content": "Welling. Modeling relational data with graph convolutional networks. In European Semantic Web",
+ "type": "text"
+ }
+ ],
+ "index": 31
+ },
+ {
+ "bbox": [
+ 117,
+ 532,
+ 286,
+ 544
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 117,
+ 532,
+ 286,
+ 544
+ ],
+ "score": 1.0,
+ "content": "Conference, pp. 593–607. Springer, 2018.",
+ "type": "text"
+ }
+ ],
+ "index": 32
+ }
+ ],
+ "index": 31
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 105,
+ 549,
+ 503,
+ 573
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 549,
+ 505,
+ 562
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 549,
+ 505,
+ 562
+ ],
+ "score": 1.0,
+ "content": "Prithviraj Sen, Galileo Namata, Mustafa Bilgic, Lise Getoor, Brian Galligher, and Tina Eliassi-Rad.",
+ "type": "text"
+ }
+ ],
+ "index": 33
+ },
+ {
+ "bbox": [
+ 117,
+ 561,
+ 416,
+ 573
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 117,
+ 561,
+ 416,
+ 573
+ ],
+ "score": 1.0,
+ "content": "Collective classification in network data. AI magazine, 29(3):93–93, 2008.",
+ "type": "text"
+ }
+ ],
+ "index": 34
+ }
+ ],
+ "index": 33.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 578,
+ 504,
+ 601
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 107,
+ 579,
+ 504,
+ 591
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 579,
+ 504,
+ 591
+ ],
+ "score": 1.0,
+ "content": "Sho Sonoda and Noboru Murata. Neural network with unbounded activation functions is universal",
+ "type": "text"
+ }
+ ],
+ "index": 35
+ },
+ {
+ "bbox": [
+ 115,
+ 590,
+ 460,
+ 602
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 590,
+ 460,
+ 602
+ ],
+ "score": 1.0,
+ "content": "approximator. Applied and Computational Harmonic Analysis, 43(2):233–268, 2017.",
+ "type": "text"
+ }
+ ],
+ "index": 36
+ }
+ ],
+ "index": 35.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 108,
+ 607,
+ 504,
+ 641
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 607,
+ 506,
+ 621
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 607,
+ 506,
+ 621
+ ],
+ "score": 1.0,
+ "content": "Nitish Srivastava, Geoffrey Hinton, Alex Krizhevsky, Ilya Sutskever, and Ruslan Salakhutdinov.",
+ "type": "text"
+ }
+ ],
+ "index": 37
+ },
+ {
+ "bbox": [
+ 115,
+ 618,
+ 506,
+ 632
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 618,
+ 506,
+ 632
+ ],
+ "score": 1.0,
+ "content": "Dropout: A simple way to prevent neural networks from overfitting. Journal of Machine Learning",
+ "type": "text"
+ }
+ ],
+ "index": 38
+ },
+ {
+ "bbox": [
+ 116,
+ 630,
+ 245,
+ 641
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 116,
+ 630,
+ 245,
+ 641
+ ],
+ "score": 1.0,
+ "content": "Research, 15:1929–1958, 2014.",
+ "type": "text"
+ }
+ ],
+ "index": 39
+ }
+ ],
+ "index": 38
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 647,
+ 504,
+ 681
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 104,
+ 645,
+ 506,
+ 662
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 645,
+ 506,
+ 662
+ ],
+ "score": 1.0,
+ "content": "Matus Telgarsky. Benefits of depth in neural networks. In 29th Annual Conference on Learning",
+ "type": "text"
+ }
+ ],
+ "index": 40
+ },
+ {
+ "bbox": [
+ 116,
+ 659,
+ 505,
+ 671
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 116,
+ 659,
+ 505,
+ 671
+ ],
+ "score": 1.0,
+ "content": "Theory, volume 49 of Proceedings of Machine Learning Research, pp. 1517–1539. PMLR, 23–",
+ "type": "text"
+ }
+ ],
+ "index": 41
+ },
+ {
+ "bbox": [
+ 115,
+ 669,
+ 170,
+ 681
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 669,
+ 170,
+ 681
+ ],
+ "score": 1.0,
+ "content": "26 Jun 2016.",
+ "type": "text"
+ }
+ ],
+ "index": 42
+ }
+ ],
+ "index": 41
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 687,
+ 505,
+ 732
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 687,
+ 505,
+ 701
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 687,
+ 505,
+ 701
+ ],
+ "score": 1.0,
+ "content": "Seiya Tokui, Kenta Oono, Shohei Hido, and Justin Clayton. Chainer: a next-generation open",
+ "type": "text"
+ }
+ ],
+ "index": 43
+ },
+ {
+ "bbox": [
+ 115,
+ 699,
+ 505,
+ 711
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 699,
+ 505,
+ 711
+ ],
+ "score": 1.0,
+ "content": "source framework for deep learning. In Proceedings of Workshop on Machine Learning Sys-",
+ "type": "text"
+ }
+ ],
+ "index": 44
+ },
+ {
+ "bbox": [
+ 115,
+ 709,
+ 505,
+ 723
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 709,
+ 505,
+ 723
+ ],
+ "score": 1.0,
+ "content": "tems (LearningSys) in The Twenty-ninth Annual Conference on Neural Information Processing",
+ "type": "text"
+ }
+ ],
+ "index": 45
+ },
+ {
+ "bbox": [
+ 114,
+ 720,
+ 208,
+ 732
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 114,
+ 720,
+ 208,
+ 732
+ ],
+ "score": 1.0,
+ "content": "Systems (NIPS), 2015.",
+ "type": "text"
+ }
+ ],
+ "index": 46
+ }
+ ],
+ "index": 44.5
+ }
+ ],
+ "page_idx": 11,
+ "page_size": [
+ 612,
+ 792
+ ],
+ "discarded_blocks": [
+ {
+ "type": "discarded",
+ "bbox": [
+ 108,
+ 27,
+ 294,
+ 37
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 26,
+ 293,
+ 38
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 26,
+ 293,
+ 38
+ ],
+ "score": 1.0,
+ "content": "Published as a conference paper at ICLR 2020",
+ "type": "text"
+ }
+ ]
+ }
+ ]
+ },
+ {
+ "type": "discarded",
+ "bbox": [
+ 300,
+ 751,
+ 311,
+ 760
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 299,
+ 750,
+ 313,
+ 764
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 299,
+ 750,
+ 313,
+ 764
+ ],
+ "score": 1.0,
+ "content": "12",
+ "type": "text"
+ }
+ ]
+ }
+ ]
+ }
+ ],
+ "para_blocks": [
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 82,
+ 504,
+ 105
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 107,
+ 82,
+ 506,
+ 95
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 82,
+ 506,
+ 95
+ ],
+ "score": 1.0,
+ "content": "Yujia Li, Richard Zemel, and Marc Brockschmidt. Gated graph sequence neural networks. In",
+ "type": "text"
+ }
+ ],
+ "index": 0
+ },
+ {
+ "bbox": [
+ 116,
+ 93,
+ 365,
+ 106
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 116,
+ 93,
+ 365,
+ 106
+ ],
+ "score": 1.0,
+ "content": "International Conference on Learning Representations, 2016.",
+ "type": "text"
+ }
+ ],
+ "index": 1
+ }
+ ],
+ "index": 0.5,
+ "bbox_fs": [
+ 107,
+ 82,
+ 506,
+ 106
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 108,
+ 111,
+ 504,
+ 145
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 110,
+ 504,
+ 124
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 110,
+ 504,
+ 124
+ ],
+ "score": 1.0,
+ "content": "Sitao Luan, Mingde Zhao, Xiao-Wen Chang, and Doina Precup. Break the ceiling: Stronger multi-",
+ "type": "text"
+ }
+ ],
+ "index": 2
+ },
+ {
+ "bbox": [
+ 116,
+ 123,
+ 505,
+ 135
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 116,
+ 123,
+ 505,
+ 135
+ ],
+ "score": 1.0,
+ "content": "scale deep graph convolutional networks. In Advances in Neural Information Processing Systems",
+ "type": "text"
+ }
+ ],
+ "index": 3
+ },
+ {
+ "bbox": [
+ 115,
+ 133,
+ 329,
+ 145
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 133,
+ 329,
+ 145
+ ],
+ "score": 1.0,
+ "content": "32, pp. 10943–10953. Curran Associates, Inc., 2019.",
+ "type": "text"
+ }
+ ],
+ "index": 4
+ }
+ ],
+ "index": 3,
+ "bbox_fs": [
+ 106,
+ 110,
+ 505,
+ 145
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 150,
+ 503,
+ 185
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 151,
+ 505,
+ 163
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 151,
+ 505,
+ 163
+ ],
+ "score": 1.0,
+ "content": "Andrew L. Maas, Awni Y. Hannun, and Andrew Y. Ng. Rectifier nonlinearities improve neural net-",
+ "type": "text"
+ }
+ ],
+ "index": 5
+ },
+ {
+ "bbox": [
+ 115,
+ 161,
+ 505,
+ 176
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 161,
+ 505,
+ 176
+ ],
+ "score": 1.0,
+ "content": "work acoustic models. In in ICML Workshop on Deep Learning for Audio, Speech and Language",
+ "type": "text"
+ }
+ ],
+ "index": 6
+ },
+ {
+ "bbox": [
+ 116,
+ 173,
+ 191,
+ 186
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 116,
+ 173,
+ 191,
+ 186
+ ],
+ "score": 1.0,
+ "content": "Processing, 2013.",
+ "type": "text"
+ }
+ ],
+ "index": 7
+ }
+ ],
+ "index": 6,
+ "bbox_fs": [
+ 106,
+ 151,
+ 505,
+ 186
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 108,
+ 190,
+ 504,
+ 225
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 190,
+ 505,
+ 204
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 190,
+ 505,
+ 204
+ ],
+ "score": 1.0,
+ "content": "Andrew Kachites McCallum, Kamal Nigam, Jason Rennie, and Kristie Seymore. Automating the",
+ "type": "text"
+ }
+ ],
+ "index": 8
+ },
+ {
+ "bbox": [
+ 115,
+ 202,
+ 505,
+ 214
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 202,
+ 505,
+ 214
+ ],
+ "score": 1.0,
+ "content": "construction of internet portals with machine learning. Information Retrieval, 3(2):127–163,",
+ "type": "text"
+ }
+ ],
+ "index": 9
+ },
+ {
+ "bbox": [
+ 115,
+ 212,
+ 142,
+ 225
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 212,
+ 142,
+ 225
+ ],
+ "score": 1.0,
+ "content": "2000.",
+ "type": "text"
+ }
+ ],
+ "index": 10
+ }
+ ],
+ "index": 9,
+ "bbox_fs": [
+ 105,
+ 190,
+ 505,
+ 225
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 230,
+ 504,
+ 254
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 230,
+ 505,
+ 244
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 230,
+ 505,
+ 244
+ ],
+ "score": 1.0,
+ "content": "Song Mei and Andrea Montanari. The generalization error of random features regression: Precise",
+ "type": "text"
+ }
+ ],
+ "index": 11
+ },
+ {
+ "bbox": [
+ 115,
+ 242,
+ 436,
+ 254
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 242,
+ 436,
+ 254
+ ],
+ "score": 1.0,
+ "content": "asymptotics and double descent curve. arXiv preprint arXiv:1908.05355, 2019.",
+ "type": "text"
+ }
+ ],
+ "index": 12
+ }
+ ],
+ "index": 11.5,
+ "bbox_fs": [
+ 106,
+ 230,
+ 505,
+ 254
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 259,
+ 443,
+ 272
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 259,
+ 442,
+ 273
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 259,
+ 442,
+ 273
+ ],
+ "score": 1.0,
+ "content": "Carl D Meyer. Matrix analysis and applied linear algebra, volume 71. Siam, 2000.",
+ "type": "text"
+ }
+ ],
+ "index": 13
+ }
+ ],
+ "index": 13,
+ "bbox_fs": [
+ 105,
+ 259,
+ 442,
+ 273
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 278,
+ 503,
+ 301
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 278,
+ 504,
+ 291
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 278,
+ 504,
+ 291
+ ],
+ "score": 1.0,
+ "content": "Hrushikesh Narhar Mhaskar. Approximation properties of a multilayered feedforward artificial neu-",
+ "type": "text"
+ }
+ ],
+ "index": 14
+ },
+ {
+ "bbox": [
+ 116,
+ 289,
+ 411,
+ 301
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 116,
+ 289,
+ 411,
+ 301
+ ],
+ "score": 1.0,
+ "content": "ral network. Advances in Computational Mathematics, 1(1):61–80, 1993.",
+ "type": "text"
+ }
+ ],
+ "index": 15
+ }
+ ],
+ "index": 14.5,
+ "bbox_fs": [
+ 106,
+ 278,
+ 504,
+ 301
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 306,
+ 502,
+ 330
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 306,
+ 504,
+ 320
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 306,
+ 504,
+ 320
+ ],
+ "score": 1.0,
+ "content": "Hai Nguyen, Shinichi Maeda, and Kenta Oono. Semi-supervised learning of hierarchical represen-",
+ "type": "text"
+ }
+ ],
+ "index": 16
+ },
+ {
+ "bbox": [
+ 115,
+ 318,
+ 486,
+ 331
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 318,
+ 486,
+ 331
+ ],
+ "score": 1.0,
+ "content": "tations of molecules using neural message passing. arXiv preprint arXiv:1711.10168, 2017.",
+ "type": "text"
+ }
+ ],
+ "index": 17
+ }
+ ],
+ "index": 16.5,
+ "bbox_fs": [
+ 105,
+ 306,
+ 504,
+ 331
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 335,
+ 504,
+ 359
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 335,
+ 505,
+ 348
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 335,
+ 505,
+ 348
+ ],
+ "score": 1.0,
+ "content": "James R Norris. Markov chains. Number 2 in Cambridge Series in Statistical and Probabilistic",
+ "type": "text"
+ }
+ ],
+ "index": 18
+ },
+ {
+ "bbox": [
+ 116,
+ 346,
+ 312,
+ 359
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 116,
+ 346,
+ 312,
+ 359
+ ],
+ "score": 1.0,
+ "content": "Mathematics. Cambridge university press, 1998.",
+ "type": "text"
+ }
+ ],
+ "index": 19
+ }
+ ],
+ "index": 18.5,
+ "bbox_fs": [
+ 106,
+ 335,
+ 505,
+ 359
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 364,
+ 504,
+ 388
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 365,
+ 504,
+ 378
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 365,
+ 504,
+ 378
+ ],
+ "score": 1.0,
+ "content": "Hoang NT and Takanori Maehara. Revisiting graph neural networks: All we have is low-pass filters.",
+ "type": "text"
+ }
+ ],
+ "index": 20
+ },
+ {
+ "bbox": [
+ 116,
+ 376,
+ 279,
+ 388
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 116,
+ 376,
+ 279,
+ 388
+ ],
+ "score": 1.0,
+ "content": "arXiv preprint arXiv:1905.09550, 2019.",
+ "type": "text"
+ }
+ ],
+ "index": 21
+ }
+ ],
+ "index": 20.5,
+ "bbox_fs": [
+ 106,
+ 365,
+ 504,
+ 388
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 393,
+ 504,
+ 417
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 394,
+ 504,
+ 407
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 394,
+ 504,
+ 407
+ ],
+ "score": 1.0,
+ "content": "Kenta Oono and Taiji Suzuki. Approximation and non-parametric estimation of ResNet-type con-",
+ "type": "text"
+ }
+ ],
+ "index": 22
+ },
+ {
+ "bbox": [
+ 116,
+ 405,
+ 391,
+ 417
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 116,
+ 405,
+ 391,
+ 417
+ ],
+ "score": 1.0,
+ "content": "volutional neural networks. arXiv preprint arXiv:1903.10047, 2019.",
+ "type": "text"
+ }
+ ],
+ "index": 23
+ }
+ ],
+ "index": 22.5,
+ "bbox_fs": [
+ 106,
+ 394,
+ 504,
+ 417
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 423,
+ 503,
+ 446
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 421,
+ 505,
+ 436
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 421,
+ 505,
+ 436
+ ],
+ "score": 1.0,
+ "content": "Sinno Jialin Pan and Qiang Yang. A survey on transfer learning. IEEE Transactions on knowledge",
+ "type": "text"
+ }
+ ],
+ "index": 24
+ },
+ {
+ "bbox": [
+ 116,
+ 434,
+ 311,
+ 446
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 116,
+ 434,
+ 311,
+ 446
+ ],
+ "score": 1.0,
+ "content": "and data engineering, 22(10):1345–1359, 2010.",
+ "type": "text"
+ }
+ ],
+ "index": 25
+ }
+ ],
+ "index": 24.5,
+ "bbox_fs": [
+ 106,
+ 421,
+ 505,
+ 446
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 452,
+ 502,
+ 475
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 451,
+ 504,
+ 464
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 451,
+ 504,
+ 464
+ ],
+ "score": 1.0,
+ "content": "Philipp Petersen and Felix Voigtlaender. Equivalence of approximation by convolutional neural",
+ "type": "text"
+ }
+ ],
+ "index": 26
+ },
+ {
+ "bbox": [
+ 115,
+ 462,
+ 442,
+ 475
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 462,
+ 442,
+ 475
+ ],
+ "score": 1.0,
+ "content": "networks and fully-connected networks. arXiv preprint arXiv:1809.00973, 2018.",
+ "type": "text"
+ }
+ ],
+ "index": 27
+ }
+ ],
+ "index": 26.5,
+ "bbox_fs": [
+ 106,
+ 451,
+ 504,
+ 475
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 480,
+ 502,
+ 504
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 480,
+ 503,
+ 493
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 480,
+ 503,
+ 493
+ ],
+ "score": 1.0,
+ "content": "Franco Scarselli, Marco Gori, Ah Chung Tsoi, Markus Hagenbuchner, and Gabriele Monfardini.",
+ "type": "text"
+ }
+ ],
+ "index": 28
+ },
+ {
+ "bbox": [
+ 116,
+ 491,
+ 496,
+ 504
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 116,
+ 491,
+ 496,
+ 504
+ ],
+ "score": 1.0,
+ "content": "The graph neural network model. IEEE Transactions on Neural Networks, 20(1):61–80, 2009.",
+ "type": "text"
+ }
+ ],
+ "index": 29
+ }
+ ],
+ "index": 28.5,
+ "bbox_fs": [
+ 106,
+ 480,
+ 503,
+ 504
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 108,
+ 509,
+ 504,
+ 544
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 509,
+ 505,
+ 522
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 509,
+ 505,
+ 522
+ ],
+ "score": 1.0,
+ "content": "Michael Schlichtkrull, Thomas N Kipf, Peter Bloem, Rianne Van Den Berg, Ivan Titov, and Max",
+ "type": "text"
+ }
+ ],
+ "index": 30
+ },
+ {
+ "bbox": [
+ 116,
+ 520,
+ 505,
+ 533
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 116,
+ 520,
+ 505,
+ 533
+ ],
+ "score": 1.0,
+ "content": "Welling. Modeling relational data with graph convolutional networks. In European Semantic Web",
+ "type": "text"
+ }
+ ],
+ "index": 31
+ },
+ {
+ "bbox": [
+ 117,
+ 532,
+ 286,
+ 544
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 117,
+ 532,
+ 286,
+ 544
+ ],
+ "score": 1.0,
+ "content": "Conference, pp. 593–607. Springer, 2018.",
+ "type": "text"
+ }
+ ],
+ "index": 32
+ }
+ ],
+ "index": 31,
+ "bbox_fs": [
+ 106,
+ 509,
+ 505,
+ 544
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 105,
+ 549,
+ 503,
+ 573
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 549,
+ 505,
+ 562
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 549,
+ 505,
+ 562
+ ],
+ "score": 1.0,
+ "content": "Prithviraj Sen, Galileo Namata, Mustafa Bilgic, Lise Getoor, Brian Galligher, and Tina Eliassi-Rad.",
+ "type": "text"
+ }
+ ],
+ "index": 33
+ },
+ {
+ "bbox": [
+ 117,
+ 561,
+ 416,
+ 573
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 117,
+ 561,
+ 416,
+ 573
+ ],
+ "score": 1.0,
+ "content": "Collective classification in network data. AI magazine, 29(3):93–93, 2008.",
+ "type": "text"
+ }
+ ],
+ "index": 34
+ }
+ ],
+ "index": 33.5,
+ "bbox_fs": [
+ 106,
+ 549,
+ 505,
+ 573
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 578,
+ 504,
+ 601
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 107,
+ 579,
+ 504,
+ 591
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 579,
+ 504,
+ 591
+ ],
+ "score": 1.0,
+ "content": "Sho Sonoda and Noboru Murata. Neural network with unbounded activation functions is universal",
+ "type": "text"
+ }
+ ],
+ "index": 35
+ },
+ {
+ "bbox": [
+ 115,
+ 590,
+ 460,
+ 602
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 590,
+ 460,
+ 602
+ ],
+ "score": 1.0,
+ "content": "approximator. Applied and Computational Harmonic Analysis, 43(2):233–268, 2017.",
+ "type": "text"
+ }
+ ],
+ "index": 36
+ }
+ ],
+ "index": 35.5,
+ "bbox_fs": [
+ 107,
+ 579,
+ 504,
+ 602
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 108,
+ 607,
+ 504,
+ 641
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 607,
+ 506,
+ 621
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 607,
+ 506,
+ 621
+ ],
+ "score": 1.0,
+ "content": "Nitish Srivastava, Geoffrey Hinton, Alex Krizhevsky, Ilya Sutskever, and Ruslan Salakhutdinov.",
+ "type": "text"
+ }
+ ],
+ "index": 37
+ },
+ {
+ "bbox": [
+ 115,
+ 618,
+ 506,
+ 632
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 618,
+ 506,
+ 632
+ ],
+ "score": 1.0,
+ "content": "Dropout: A simple way to prevent neural networks from overfitting. Journal of Machine Learning",
+ "type": "text"
+ }
+ ],
+ "index": 38
+ },
+ {
+ "bbox": [
+ 116,
+ 630,
+ 245,
+ 641
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 116,
+ 630,
+ 245,
+ 641
+ ],
+ "score": 1.0,
+ "content": "Research, 15:1929–1958, 2014.",
+ "type": "text"
+ }
+ ],
+ "index": 39
+ }
+ ],
+ "index": 38,
+ "bbox_fs": [
+ 105,
+ 607,
+ 506,
+ 641
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 647,
+ 504,
+ 681
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 104,
+ 645,
+ 506,
+ 662
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 645,
+ 506,
+ 662
+ ],
+ "score": 1.0,
+ "content": "Matus Telgarsky. Benefits of depth in neural networks. In 29th Annual Conference on Learning",
+ "type": "text"
+ }
+ ],
+ "index": 40
+ },
+ {
+ "bbox": [
+ 116,
+ 659,
+ 505,
+ 671
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 116,
+ 659,
+ 505,
+ 671
+ ],
+ "score": 1.0,
+ "content": "Theory, volume 49 of Proceedings of Machine Learning Research, pp. 1517–1539. PMLR, 23–",
+ "type": "text"
+ }
+ ],
+ "index": 41
+ },
+ {
+ "bbox": [
+ 115,
+ 669,
+ 170,
+ 681
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 669,
+ 170,
+ 681
+ ],
+ "score": 1.0,
+ "content": "26 Jun 2016.",
+ "type": "text"
+ }
+ ],
+ "index": 42
+ }
+ ],
+ "index": 41,
+ "bbox_fs": [
+ 104,
+ 645,
+ 506,
+ 681
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 687,
+ 505,
+ 732
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 687,
+ 505,
+ 701
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 687,
+ 505,
+ 701
+ ],
+ "score": 1.0,
+ "content": "Seiya Tokui, Kenta Oono, Shohei Hido, and Justin Clayton. Chainer: a next-generation open",
+ "type": "text"
+ }
+ ],
+ "index": 43
+ },
+ {
+ "bbox": [
+ 115,
+ 699,
+ 505,
+ 711
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 699,
+ 505,
+ 711
+ ],
+ "score": 1.0,
+ "content": "source framework for deep learning. In Proceedings of Workshop on Machine Learning Sys-",
+ "type": "text"
+ }
+ ],
+ "index": 44
+ },
+ {
+ "bbox": [
+ 115,
+ 709,
+ 505,
+ 723
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 709,
+ 505,
+ 723
+ ],
+ "score": 1.0,
+ "content": "tems (LearningSys) in The Twenty-ninth Annual Conference on Neural Information Processing",
+ "type": "text"
+ }
+ ],
+ "index": 45
+ },
+ {
+ "bbox": [
+ 114,
+ 720,
+ 208,
+ 732
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 114,
+ 720,
+ 208,
+ 732
+ ],
+ "score": 1.0,
+ "content": "Systems (NIPS), 2015.",
+ "type": "text"
+ }
+ ],
+ "index": 46
+ }
+ ],
+ "index": 44.5,
+ "bbox_fs": [
+ 105,
+ 687,
+ 505,
+ 732
+ ]
+ }
+ ]
+ },
+ {
+ "preproc_blocks": [
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 82,
+ 505,
+ 127
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 82,
+ 505,
+ 95
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 82,
+ 505,
+ 95
+ ],
+ "score": 1.0,
+ "content": "Seiya Tokui, Ryosuke Okuta, Takuya Akiba, Yusuke Niitani, Toru Ogawa, Shunta Saito, Shuji",
+ "type": "text"
+ }
+ ],
+ "index": 0
+ },
+ {
+ "bbox": [
+ 115,
+ 94,
+ 505,
+ 106
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 94,
+ 505,
+ 106
+ ],
+ "score": 1.0,
+ "content": "Suzuki, Kota Uenishi, Brian Vogel, and Hiroyuki Yamazaki Vincent. Chainer: A deep learn-",
+ "type": "text"
+ }
+ ],
+ "index": 1
+ },
+ {
+ "bbox": [
+ 115,
+ 104,
+ 506,
+ 117
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 104,
+ 506,
+ 117
+ ],
+ "score": 1.0,
+ "content": "ing framework for accelerating the research cycle. In Proceedings of the 25th ACM SIGKDD",
+ "type": "text"
+ }
+ ],
+ "index": 2
+ },
+ {
+ "bbox": [
+ 115,
+ 115,
+ 506,
+ 128
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 115,
+ 506,
+ 128
+ ],
+ "score": 1.0,
+ "content": "International Conference on Knowledge Discovery & Data Mining, pp. 2002–2011. ACM, 2019.",
+ "type": "text"
+ }
+ ],
+ "index": 3
+ }
+ ],
+ "index": 1.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 133,
+ 504,
+ 167
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 133,
+ 505,
+ 147
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 133,
+ 505,
+ 147
+ ],
+ "score": 1.0,
+ "content": "Petar Velickovi ˇ c, Guillem Cucurull, Arantxa Casanova, Adriana Romero, Pietro Li ´ o, and Yoshua `",
+ "type": "text"
+ }
+ ],
+ "index": 4
+ },
+ {
+ "bbox": [
+ 115,
+ 145,
+ 505,
+ 158
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 145,
+ 505,
+ 158
+ ],
+ "score": 1.0,
+ "content": "Bengio. Graph attention networks. In International Conference on Learning Representations,",
+ "type": "text"
+ }
+ ],
+ "index": 5
+ },
+ {
+ "bbox": [
+ 115,
+ 154,
+ 143,
+ 169
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 154,
+ 143,
+ 169
+ ],
+ "score": 1.0,
+ "content": "2018.",
+ "type": "text"
+ }
+ ],
+ "index": 6
+ }
+ ],
+ "index": 5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 174,
+ 505,
+ 209
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 174,
+ 505,
+ 188
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 174,
+ 505,
+ 188
+ ],
+ "score": 1.0,
+ "content": "Saurabh Verma and Zhi-Li Zhang. Stability and generalization of graph convolutional neural net-",
+ "type": "text"
+ }
+ ],
+ "index": 7
+ },
+ {
+ "bbox": [
+ 116,
+ 185,
+ 504,
+ 198
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 116,
+ 185,
+ 504,
+ 198
+ ],
+ "score": 1.0,
+ "content": "works. In Proceedings of the 25th ACM SIGKDD International Conference on Knowledge Dis-",
+ "type": "text"
+ }
+ ],
+ "index": 8
+ },
+ {
+ "bbox": [
+ 115,
+ 197,
+ 236,
+ 209
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 197,
+ 236,
+ 209
+ ],
+ "score": 1.0,
+ "content": "covery & Data Mining, 2019.",
+ "type": "text"
+ }
+ ],
+ "index": 9
+ }
+ ],
+ "index": 8
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 215,
+ 502,
+ 239
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 214,
+ 504,
+ 229
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 214,
+ 504,
+ 229
+ ],
+ "score": 1.0,
+ "content": "Boris Weisfeiler and Lehman A.A. A reduction of a graph to a canonical form and an algebra arising",
+ "type": "text"
+ }
+ ],
+ "index": 10
+ },
+ {
+ "bbox": [
+ 116,
+ 227,
+ 434,
+ 239
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 116,
+ 227,
+ 434,
+ 239
+ ],
+ "score": 1.0,
+ "content": "during this reduction. Nauchno-Technicheskaya Informatsia, 2(9):12–16, 1968.",
+ "type": "text"
+ }
+ ],
+ "index": 11
+ }
+ ],
+ "index": 10.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 244,
+ 503,
+ 269
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 244,
+ 505,
+ 258
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 244,
+ 505,
+ 258
+ ],
+ "score": 1.0,
+ "content": "Felix Wu, Tianyi Zhang, Amauri Holanda de Souza Jr, Christopher Fifty, Tao Yu, and Kilian Q",
+ "type": "text"
+ }
+ ],
+ "index": 12
+ },
+ {
+ "bbox": [
+ 116,
+ 255,
+ 505,
+ 269
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 116,
+ 255,
+ 505,
+ 269
+ ],
+ "score": 1.0,
+ "content": "Weinberger. Simplifying graph convolutional networks. arXiv preprint arXiv:1902.07153, 2019a.",
+ "type": "text"
+ }
+ ],
+ "index": 13
+ }
+ ],
+ "index": 12.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 275,
+ 505,
+ 298
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 274,
+ 505,
+ 288
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 274,
+ 505,
+ 288
+ ],
+ "score": 1.0,
+ "content": "Zonghan Wu, Shirui Pan, Fengwen Chen, Guodong Long, Chengqi Zhang, and Philip S Yu. A",
+ "type": "text"
+ }
+ ],
+ "index": 14
+ },
+ {
+ "bbox": [
+ 115,
+ 286,
+ 482,
+ 299
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 286,
+ 482,
+ 299
+ ],
+ "score": 1.0,
+ "content": "comprehensive survey on graph neural networks. arXiv preprint arXiv:1901.00596, 2019b.",
+ "type": "text"
+ }
+ ],
+ "index": 15
+ }
+ ],
+ "index": 14.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 304,
+ 505,
+ 350
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 305,
+ 506,
+ 318
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 305,
+ 506,
+ 318
+ ],
+ "score": 1.0,
+ "content": "Keyulu Xu, Chengtao Li, Yonglong Tian, Tomohiro Sonobe, Ken-ichi Kawarabayashi, and Stefanie",
+ "type": "text"
+ }
+ ],
+ "index": 16
+ },
+ {
+ "bbox": [
+ 115,
+ 316,
+ 506,
+ 329
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 316,
+ 506,
+ 329
+ ],
+ "score": 1.0,
+ "content": "Jegelka. Representation learning on graphs with jumping knowledge networks. In Proceedings of",
+ "type": "text"
+ }
+ ],
+ "index": 17
+ },
+ {
+ "bbox": [
+ 116,
+ 328,
+ 505,
+ 340
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 116,
+ 328,
+ 505,
+ 340
+ ],
+ "score": 1.0,
+ "content": "the 35th International Conference on Machine Learning, volume 80 of Proceedings of Machine",
+ "type": "text"
+ }
+ ],
+ "index": 18
+ },
+ {
+ "bbox": [
+ 116,
+ 338,
+ 318,
+ 351
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 116,
+ 338,
+ 318,
+ 351
+ ],
+ "score": 1.0,
+ "content": "Learning Research, pp. 5453–5462. PMLR, 2018.",
+ "type": "text"
+ }
+ ],
+ "index": 19
+ }
+ ],
+ "index": 17.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 356,
+ 503,
+ 380
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 356,
+ 505,
+ 369
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 356,
+ 505,
+ 369
+ ],
+ "score": 1.0,
+ "content": "Keyulu Xu, Weihua Hu, Jure Leskovec, and Stefanie Jegelka. How powerful are graph neural",
+ "type": "text"
+ }
+ ],
+ "index": 20
+ },
+ {
+ "bbox": [
+ 115,
+ 367,
+ 421,
+ 380
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 367,
+ 421,
+ 380
+ ],
+ "score": 1.0,
+ "content": "networks? In International Conference on Learning Representations, 2019.",
+ "type": "text"
+ }
+ ],
+ "index": 21
+ }
+ ],
+ "index": 20.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 386,
+ 503,
+ 410
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 386,
+ 505,
+ 399
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 386,
+ 505,
+ 399
+ ],
+ "score": 1.0,
+ "content": "Dmitry Yarotsky. Error bounds for approximations with deep ReLU networks. Neural Networks,",
+ "type": "text"
+ }
+ ],
+ "index": 22
+ },
+ {
+ "bbox": [
+ 116,
+ 398,
+ 194,
+ 409
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 116,
+ 398,
+ 194,
+ 409
+ ],
+ "score": 1.0,
+ "content": "94:103–114, 2017.",
+ "type": "text"
+ }
+ ],
+ "index": 23
+ }
+ ],
+ "index": 22.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 416,
+ 504,
+ 439
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 416,
+ 504,
+ 429
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 416,
+ 504,
+ 429
+ ],
+ "score": 1.0,
+ "content": "Jiawei Zhang. Gresnet: Graph residuals for reviving deep graph neural nets from suspended anima-",
+ "type": "text"
+ }
+ ],
+ "index": 24
+ },
+ {
+ "bbox": [
+ 115,
+ 428,
+ 300,
+ 439
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 428,
+ 300,
+ 439
+ ],
+ "score": 1.0,
+ "content": "tion. arXiv preprint arXiv:1909.05729, 2019.",
+ "type": "text"
+ }
+ ],
+ "index": 25
+ }
+ ],
+ "index": 24.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 446,
+ 504,
+ 479
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 446,
+ 505,
+ 459
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 446,
+ 404,
+ 459
+ ],
+ "score": 1.0,
+ "content": "Lingxiao Zhao and Leman Akoglu. Pairnorm: Tackling oversmoothing in",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 404,
+ 446,
+ 434,
+ 459
+ ],
+ "score": 0.85,
+ "content": "\\{ \\mathrm { g n n } \\} \\mathrm { s }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 434,
+ 446,
+ 505,
+ 459
+ ],
+ "score": 1.0,
+ "content": ". In International",
+ "type": "text"
+ }
+ ],
+ "index": 26
+ },
+ {
+ "bbox": [
+ 117,
+ 457,
+ 504,
+ 469
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 117,
+ 457,
+ 504,
+ 469
+ ],
+ "score": 1.0,
+ "content": "Conference on Learning Representations, 2020. URL https://openreview.net/forum?",
+ "type": "text"
+ }
+ ],
+ "index": 27
+ },
+ {
+ "bbox": [
+ 116,
+ 468,
+ 199,
+ 480
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 116,
+ 468,
+ 129,
+ 480
+ ],
+ "score": 1.0,
+ "content": "id",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 129,
+ 470,
+ 136,
+ 478
+ ],
+ "score": 0.38,
+ "content": "=",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 136,
+ 468,
+ 199,
+ 480
+ ],
+ "score": 1.0,
+ "content": "rkecl1rtwB.",
+ "type": "text"
+ }
+ ],
+ "index": 28
+ }
+ ],
+ "index": 27
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 105,
+ 486,
+ 505,
+ 510
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 485,
+ 506,
+ 500
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 485,
+ 506,
+ 500
+ ],
+ "score": 1.0,
+ "content": "Ding-Xuan Zhou. Universality of deep convolutional neural networks. arXiv preprint",
+ "type": "text"
+ }
+ ],
+ "index": 29
+ },
+ {
+ "bbox": [
+ 115,
+ 497,
+ 220,
+ 510
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 497,
+ 220,
+ 510
+ ],
+ "score": 1.0,
+ "content": "arXiv:1805.10769, 2018.",
+ "type": "text"
+ }
+ ],
+ "index": 30
+ }
+ ],
+ "index": 29.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 109,
+ 516,
+ 503,
+ 551
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 515,
+ 505,
+ 531
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 515,
+ 505,
+ 531
+ ],
+ "score": 1.0,
+ "content": "Pan Zhou and Jiashi Feng. Understanding generalization and optimization performance of deep",
+ "type": "text"
+ }
+ ],
+ "index": 31
+ },
+ {
+ "bbox": [
+ 116,
+ 528,
+ 505,
+ 540
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 116,
+ 528,
+ 505,
+ 540
+ ],
+ "score": 1.0,
+ "content": "CNNs. In Proceedings of the 35th International Conference on Machine Learning, volume 80 of",
+ "type": "text"
+ }
+ ],
+ "index": 32
+ },
+ {
+ "bbox": [
+ 116,
+ 539,
+ 418,
+ 551
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 116,
+ 539,
+ 418,
+ 551
+ ],
+ "score": 1.0,
+ "content": "Proceedings of Machine Learning Research, pp. 5960–5969. PMLR, 2018.",
+ "type": "text"
+ }
+ ],
+ "index": 33
+ }
+ ],
+ "index": 32
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 108,
+ 570,
+ 244,
+ 584
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 570,
+ 245,
+ 586
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 570,
+ 245,
+ 586
+ ],
+ "score": 1.0,
+ "content": "A PROOF OF THEOREM 1",
+ "type": "text"
+ }
+ ],
+ "index": 34
+ }
+ ],
+ "index": 34
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 596,
+ 505,
+ 630
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 596,
+ 505,
+ 608
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 596,
+ 505,
+ 608
+ ],
+ "score": 1.0,
+ "content": "As we wrote in the main article, it is enough to show the following lemmas (definition of mis-",
+ "type": "text"
+ }
+ ],
+ "index": 35
+ },
+ {
+ "bbox": [
+ 105,
+ 606,
+ 506,
+ 622
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 606,
+ 351,
+ 622
+ ],
+ "score": 1.0,
+ "content": "cellaneous variables are as in Section 3.2). Remember that",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 352,
+ 608,
+ 444,
+ 621
+ ],
+ "score": 0.92,
+ "content": "\\lambda = \\operatorname* { s u p } _ { n \\in [ N - M ] } | \\lambda _ { n } |",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 445,
+ 606,
+ 464,
+ 622
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 465,
+ 609,
+ 478,
+ 619
+ ],
+ "score": 0.86,
+ "content": "s _ { l h }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 479,
+ 606,
+ 506,
+ 622
+ ],
+ "score": 1.0,
+ "content": "is the",
+ "type": "text"
+ }
+ ],
+ "index": 36
+ },
+ {
+ "bbox": [
+ 105,
+ 619,
+ 237,
+ 632
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 619,
+ 218,
+ 632
+ ],
+ "score": 1.0,
+ "content": "maximum singular value of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 219,
+ 619,
+ 237,
+ 631
+ ],
+ "score": 0.89,
+ "content": "W _ { l h }",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 37
+ }
+ ],
+ "index": 36
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 105,
+ 632,
+ 371,
+ 646
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 104,
+ 630,
+ 372,
+ 649
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 630,
+ 187,
+ 649
+ ],
+ "score": 1.0,
+ "content": "Lemma 1. For any",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 187,
+ 633,
+ 236,
+ 644
+ ],
+ "score": 0.91,
+ "content": "X \\in \\mathbb { R } ^ { N \\times C }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 237,
+ 630,
+ 276,
+ 649
+ ],
+ "score": 1.0,
+ "content": ", we have",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 276,
+ 633,
+ 367,
+ 646
+ ],
+ "score": 0.91,
+ "content": "d _ { \\mathcal { M } } ( P X ) \\leq \\lambda d _ { \\mathcal { M } } ( X )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 368,
+ 630,
+ 372,
+ 649
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 38
+ }
+ ],
+ "index": 38
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 105,
+ 648,
+ 387,
+ 661
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 104,
+ 644,
+ 387,
+ 664
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 644,
+ 187,
+ 664
+ ],
+ "score": 1.0,
+ "content": "Lemma 2. For any",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 187,
+ 647,
+ 236,
+ 659
+ ],
+ "score": 0.9,
+ "content": "X \\in \\mathbb { R } ^ { N \\times C }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 237,
+ 644,
+ 276,
+ 664
+ ],
+ "score": 1.0,
+ "content": ", we have",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 276,
+ 648,
+ 383,
+ 661
+ ],
+ "score": 0.91,
+ "content": "d _ { \\mathcal { M } } ( X W _ { l h } ) \\leq s _ { l h } d _ { \\mathcal { M } } ( X )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 384,
+ 644,
+ 387,
+ 664
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 39
+ }
+ ],
+ "index": 39
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 663,
+ 370,
+ 676
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 104,
+ 659,
+ 367,
+ 679
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 659,
+ 187,
+ 679
+ ],
+ "score": 1.0,
+ "content": "Lemma 3. For any",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 187,
+ 663,
+ 236,
+ 674
+ ],
+ "score": 0.91,
+ "content": "X \\in \\mathbb { R } ^ { N \\times C }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 237,
+ 659,
+ 276,
+ 679
+ ],
+ "score": 1.0,
+ "content": ", we have",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 276,
+ 663,
+ 367,
+ 676
+ ],
+ "score": 0.91,
+ "content": "d _ { \\mathcal { M } } ( \\sigma ( X ) ) \\leq d _ { \\mathcal { M } } ( X )",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 40
+ }
+ ],
+ "index": 40
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 687,
+ 506,
+ 734
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 687,
+ 506,
+ 700
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 687,
+ 173,
+ 700
+ ],
+ "score": 1.0,
+ "content": "Proof of Lemma",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 174,
+ 688,
+ 179,
+ 698
+ ],
+ "score": 0.29,
+ "content": "^ { l }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 180,
+ 687,
+ 210,
+ 700
+ ],
+ "score": 1.0,
+ "content": ". Since",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 211,
+ 688,
+ 220,
+ 698
+ ],
+ "score": 0.83,
+ "content": "P",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 220,
+ 687,
+ 357,
+ 700
+ ],
+ "score": 1.0,
+ "content": "is a symmetric linear operator on",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 357,
+ 687,
+ 372,
+ 698
+ ],
+ "score": 0.89,
+ "content": "U ^ { \\perp }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 373,
+ 687,
+ 506,
+ 700
+ ],
+ "score": 1.0,
+ "content": ", we can choose the orthonormal",
+ "type": "text"
+ }
+ ],
+ "index": 41
+ },
+ {
+ "bbox": [
+ 104,
+ 697,
+ 507,
+ 713
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 697,
+ 130,
+ 713
+ ],
+ "score": 1.0,
+ "content": "basis",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 130,
+ 699,
+ 201,
+ 711
+ ],
+ "score": 0.91,
+ "content": "( e _ { m } ) _ { m = M + 1 , \\dots , N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 202,
+ 697,
+ 214,
+ 713
+ ],
+ "score": 1.0,
+ "content": "of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 215,
+ 698,
+ 230,
+ 709
+ ],
+ "score": 0.88,
+ "content": "U ^ { \\perp }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 231,
+ 697,
+ 358,
+ 713
+ ],
+ "score": 1.0,
+ "content": "consisting of the eigenvalue of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 359,
+ 699,
+ 382,
+ 711
+ ],
+ "score": 0.92,
+ "content": "P | _ { U ^ { \\perp } }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 383,
+ 697,
+ 405,
+ 713
+ ],
+ "score": 1.0,
+ "content": ". Let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 405,
+ 699,
+ 420,
+ 710
+ ],
+ "score": 0.89,
+ "content": "\\lambda _ { m }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 420,
+ 697,
+ 507,
+ 713
+ ],
+ "score": 1.0,
+ "content": "be the eigenvalue of",
+ "type": "text"
+ }
+ ],
+ "index": 42
+ },
+ {
+ "bbox": [
+ 107,
+ 708,
+ 506,
+ 722
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 710,
+ 115,
+ 720
+ ],
+ "score": 0.84,
+ "content": "P",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 116,
+ 708,
+ 155,
+ 722
+ ],
+ "score": 1.0,
+ "content": "to which",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 155,
+ 712,
+ 168,
+ 721
+ ],
+ "score": 0.85,
+ "content": "e _ { m }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 169,
+ 708,
+ 225,
+ 722
+ ],
+ "score": 1.0,
+ "content": "is associated",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 225,
+ 710,
+ 315,
+ 721
+ ],
+ "score": 0.89,
+ "content": "( m = M + 1 , \\ldots , N )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 315,
+ 708,
+ 470,
+ 722
+ ],
+ "score": 1.0,
+ "content": ". Note that since the operator norm of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 470,
+ 710,
+ 493,
+ 722
+ ],
+ "score": 0.92,
+ "content": "P | _ { U ^ { \\perp } }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 494,
+ 708,
+ 506,
+ 722
+ ],
+ "score": 1.0,
+ "content": "is",
+ "type": "text"
+ }
+ ],
+ "index": 43
+ },
+ {
+ "bbox": [
+ 106,
+ 720,
+ 506,
+ 734
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 721,
+ 114,
+ 731
+ ],
+ "score": 0.78,
+ "content": "\\lambda",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 114,
+ 720,
+ 153,
+ 734
+ ],
+ "score": 1.0,
+ "content": ", we have",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 153,
+ 721,
+ 192,
+ 732
+ ],
+ "score": 0.93,
+ "content": "| \\lambda _ { m } | \\leq \\lambda",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 192,
+ 720,
+ 220,
+ 734
+ ],
+ "score": 1.0,
+ "content": "for all",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 221,
+ 721,
+ 303,
+ 732
+ ],
+ "score": 0.88,
+ "content": "m = M + 1 , \\ldots , N",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 303,
+ 720,
+ 333,
+ 734
+ ],
+ "score": 1.0,
+ "content": ". Since",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 333,
+ 720,
+ 378,
+ 733
+ ],
+ "score": 0.93,
+ "content": "( e _ { m } ) _ { m \\in [ N ] }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 378,
+ 720,
+ 506,
+ 734
+ ],
+ "score": 1.0,
+ "content": "forms the orthonormal basis of",
+ "type": "text"
+ }
+ ],
+ "index": 44
+ }
+ ],
+ "index": 42.5
+ }
+ ],
+ "page_idx": 12,
+ "page_size": [
+ 612,
+ 792
+ ],
+ "discarded_blocks": [
+ {
+ "type": "discarded",
+ "bbox": [
+ 107,
+ 27,
+ 293,
+ 37
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 26,
+ 294,
+ 38
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 26,
+ 294,
+ 38
+ ],
+ "score": 1.0,
+ "content": "Published as a conference paper at ICLR 2020",
+ "type": "text"
+ }
+ ]
+ }
+ ]
+ },
+ {
+ "type": "discarded",
+ "bbox": [
+ 300,
+ 751,
+ 311,
+ 760
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 299,
+ 750,
+ 313,
+ 764
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 299,
+ 750,
+ 313,
+ 764
+ ],
+ "score": 1.0,
+ "content": "13",
+ "type": "text"
+ }
+ ]
+ }
+ ]
+ }
+ ],
+ "para_blocks": [
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 82,
+ 505,
+ 127
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 82,
+ 505,
+ 95
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 82,
+ 505,
+ 95
+ ],
+ "score": 1.0,
+ "content": "Seiya Tokui, Ryosuke Okuta, Takuya Akiba, Yusuke Niitani, Toru Ogawa, Shunta Saito, Shuji",
+ "type": "text"
+ }
+ ],
+ "index": 0
+ },
+ {
+ "bbox": [
+ 115,
+ 94,
+ 505,
+ 106
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 94,
+ 505,
+ 106
+ ],
+ "score": 1.0,
+ "content": "Suzuki, Kota Uenishi, Brian Vogel, and Hiroyuki Yamazaki Vincent. Chainer: A deep learn-",
+ "type": "text"
+ }
+ ],
+ "index": 1
+ },
+ {
+ "bbox": [
+ 115,
+ 104,
+ 506,
+ 117
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 104,
+ 506,
+ 117
+ ],
+ "score": 1.0,
+ "content": "ing framework for accelerating the research cycle. In Proceedings of the 25th ACM SIGKDD",
+ "type": "text"
+ }
+ ],
+ "index": 2
+ },
+ {
+ "bbox": [
+ 115,
+ 115,
+ 506,
+ 128
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 115,
+ 506,
+ 128
+ ],
+ "score": 1.0,
+ "content": "International Conference on Knowledge Discovery & Data Mining, pp. 2002–2011. ACM, 2019.",
+ "type": "text"
+ }
+ ],
+ "index": 3
+ }
+ ],
+ "index": 1.5,
+ "bbox_fs": [
+ 105,
+ 82,
+ 506,
+ 128
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 133,
+ 504,
+ 167
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 133,
+ 505,
+ 147
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 133,
+ 505,
+ 147
+ ],
+ "score": 1.0,
+ "content": "Petar Velickovi ˇ c, Guillem Cucurull, Arantxa Casanova, Adriana Romero, Pietro Li ´ o, and Yoshua `",
+ "type": "text"
+ }
+ ],
+ "index": 4
+ },
+ {
+ "bbox": [
+ 115,
+ 145,
+ 505,
+ 158
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 145,
+ 505,
+ 158
+ ],
+ "score": 1.0,
+ "content": "Bengio. Graph attention networks. In International Conference on Learning Representations,",
+ "type": "text"
+ }
+ ],
+ "index": 5
+ },
+ {
+ "bbox": [
+ 115,
+ 154,
+ 143,
+ 169
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 154,
+ 143,
+ 169
+ ],
+ "score": 1.0,
+ "content": "2018.",
+ "type": "text"
+ }
+ ],
+ "index": 6
+ }
+ ],
+ "index": 5,
+ "bbox_fs": [
+ 105,
+ 133,
+ 505,
+ 169
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 174,
+ 505,
+ 209
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 174,
+ 505,
+ 188
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 174,
+ 505,
+ 188
+ ],
+ "score": 1.0,
+ "content": "Saurabh Verma and Zhi-Li Zhang. Stability and generalization of graph convolutional neural net-",
+ "type": "text"
+ }
+ ],
+ "index": 7
+ },
+ {
+ "bbox": [
+ 116,
+ 185,
+ 504,
+ 198
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 116,
+ 185,
+ 504,
+ 198
+ ],
+ "score": 1.0,
+ "content": "works. In Proceedings of the 25th ACM SIGKDD International Conference on Knowledge Dis-",
+ "type": "text"
+ }
+ ],
+ "index": 8
+ },
+ {
+ "bbox": [
+ 115,
+ 197,
+ 236,
+ 209
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 197,
+ 236,
+ 209
+ ],
+ "score": 1.0,
+ "content": "covery & Data Mining, 2019.",
+ "type": "text"
+ }
+ ],
+ "index": 9
+ }
+ ],
+ "index": 8,
+ "bbox_fs": [
+ 106,
+ 174,
+ 505,
+ 209
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 215,
+ 502,
+ 239
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 214,
+ 504,
+ 229
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 214,
+ 504,
+ 229
+ ],
+ "score": 1.0,
+ "content": "Boris Weisfeiler and Lehman A.A. A reduction of a graph to a canonical form and an algebra arising",
+ "type": "text"
+ }
+ ],
+ "index": 10
+ },
+ {
+ "bbox": [
+ 116,
+ 227,
+ 434,
+ 239
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 116,
+ 227,
+ 434,
+ 239
+ ],
+ "score": 1.0,
+ "content": "during this reduction. Nauchno-Technicheskaya Informatsia, 2(9):12–16, 1968.",
+ "type": "text"
+ }
+ ],
+ "index": 11
+ }
+ ],
+ "index": 10.5,
+ "bbox_fs": [
+ 106,
+ 214,
+ 504,
+ 239
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 244,
+ 503,
+ 269
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 244,
+ 505,
+ 258
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 244,
+ 505,
+ 258
+ ],
+ "score": 1.0,
+ "content": "Felix Wu, Tianyi Zhang, Amauri Holanda de Souza Jr, Christopher Fifty, Tao Yu, and Kilian Q",
+ "type": "text"
+ }
+ ],
+ "index": 12
+ },
+ {
+ "bbox": [
+ 116,
+ 255,
+ 505,
+ 269
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 116,
+ 255,
+ 505,
+ 269
+ ],
+ "score": 1.0,
+ "content": "Weinberger. Simplifying graph convolutional networks. arXiv preprint arXiv:1902.07153, 2019a.",
+ "type": "text"
+ }
+ ],
+ "index": 13
+ }
+ ],
+ "index": 12.5,
+ "bbox_fs": [
+ 105,
+ 244,
+ 505,
+ 269
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 275,
+ 505,
+ 298
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 274,
+ 505,
+ 288
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 274,
+ 505,
+ 288
+ ],
+ "score": 1.0,
+ "content": "Zonghan Wu, Shirui Pan, Fengwen Chen, Guodong Long, Chengqi Zhang, and Philip S Yu. A",
+ "type": "text"
+ }
+ ],
+ "index": 14
+ },
+ {
+ "bbox": [
+ 115,
+ 286,
+ 482,
+ 299
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 286,
+ 482,
+ 299
+ ],
+ "score": 1.0,
+ "content": "comprehensive survey on graph neural networks. arXiv preprint arXiv:1901.00596, 2019b.",
+ "type": "text"
+ }
+ ],
+ "index": 15
+ }
+ ],
+ "index": 14.5,
+ "bbox_fs": [
+ 105,
+ 274,
+ 505,
+ 299
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 304,
+ 505,
+ 350
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 305,
+ 506,
+ 318
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 305,
+ 506,
+ 318
+ ],
+ "score": 1.0,
+ "content": "Keyulu Xu, Chengtao Li, Yonglong Tian, Tomohiro Sonobe, Ken-ichi Kawarabayashi, and Stefanie",
+ "type": "text"
+ }
+ ],
+ "index": 16
+ },
+ {
+ "bbox": [
+ 115,
+ 316,
+ 506,
+ 329
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 316,
+ 506,
+ 329
+ ],
+ "score": 1.0,
+ "content": "Jegelka. Representation learning on graphs with jumping knowledge networks. In Proceedings of",
+ "type": "text"
+ }
+ ],
+ "index": 17
+ },
+ {
+ "bbox": [
+ 116,
+ 328,
+ 505,
+ 340
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 116,
+ 328,
+ 505,
+ 340
+ ],
+ "score": 1.0,
+ "content": "the 35th International Conference on Machine Learning, volume 80 of Proceedings of Machine",
+ "type": "text"
+ }
+ ],
+ "index": 18
+ },
+ {
+ "bbox": [
+ 116,
+ 338,
+ 318,
+ 351
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 116,
+ 338,
+ 318,
+ 351
+ ],
+ "score": 1.0,
+ "content": "Learning Research, pp. 5453–5462. PMLR, 2018.",
+ "type": "text"
+ }
+ ],
+ "index": 19
+ }
+ ],
+ "index": 17.5,
+ "bbox_fs": [
+ 106,
+ 305,
+ 506,
+ 351
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 356,
+ 503,
+ 380
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 356,
+ 505,
+ 369
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 356,
+ 505,
+ 369
+ ],
+ "score": 1.0,
+ "content": "Keyulu Xu, Weihua Hu, Jure Leskovec, and Stefanie Jegelka. How powerful are graph neural",
+ "type": "text"
+ }
+ ],
+ "index": 20
+ },
+ {
+ "bbox": [
+ 115,
+ 367,
+ 421,
+ 380
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 367,
+ 421,
+ 380
+ ],
+ "score": 1.0,
+ "content": "networks? In International Conference on Learning Representations, 2019.",
+ "type": "text"
+ }
+ ],
+ "index": 21
+ }
+ ],
+ "index": 20.5,
+ "bbox_fs": [
+ 106,
+ 356,
+ 505,
+ 380
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 386,
+ 503,
+ 410
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 386,
+ 505,
+ 399
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 386,
+ 505,
+ 399
+ ],
+ "score": 1.0,
+ "content": "Dmitry Yarotsky. Error bounds for approximations with deep ReLU networks. Neural Networks,",
+ "type": "text"
+ }
+ ],
+ "index": 22
+ },
+ {
+ "bbox": [
+ 116,
+ 398,
+ 194,
+ 409
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 116,
+ 398,
+ 194,
+ 409
+ ],
+ "score": 1.0,
+ "content": "94:103–114, 2017.",
+ "type": "text"
+ }
+ ],
+ "index": 23
+ }
+ ],
+ "index": 22.5,
+ "bbox_fs": [
+ 106,
+ 386,
+ 505,
+ 409
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 416,
+ 504,
+ 439
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 416,
+ 504,
+ 429
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 416,
+ 504,
+ 429
+ ],
+ "score": 1.0,
+ "content": "Jiawei Zhang. Gresnet: Graph residuals for reviving deep graph neural nets from suspended anima-",
+ "type": "text"
+ }
+ ],
+ "index": 24
+ },
+ {
+ "bbox": [
+ 115,
+ 428,
+ 300,
+ 439
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 428,
+ 300,
+ 439
+ ],
+ "score": 1.0,
+ "content": "tion. arXiv preprint arXiv:1909.05729, 2019.",
+ "type": "text"
+ }
+ ],
+ "index": 25
+ }
+ ],
+ "index": 24.5,
+ "bbox_fs": [
+ 105,
+ 416,
+ 504,
+ 439
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 446,
+ 504,
+ 479
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 446,
+ 505,
+ 459
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 446,
+ 404,
+ 459
+ ],
+ "score": 1.0,
+ "content": "Lingxiao Zhao and Leman Akoglu. Pairnorm: Tackling oversmoothing in",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 404,
+ 446,
+ 434,
+ 459
+ ],
+ "score": 0.85,
+ "content": "\\{ \\mathrm { g n n } \\} \\mathrm { s }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 434,
+ 446,
+ 505,
+ 459
+ ],
+ "score": 1.0,
+ "content": ". In International",
+ "type": "text"
+ }
+ ],
+ "index": 26
+ },
+ {
+ "bbox": [
+ 117,
+ 457,
+ 504,
+ 469
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 117,
+ 457,
+ 504,
+ 469
+ ],
+ "score": 1.0,
+ "content": "Conference on Learning Representations, 2020. URL https://openreview.net/forum?",
+ "type": "text"
+ }
+ ],
+ "index": 27
+ },
+ {
+ "bbox": [
+ 116,
+ 468,
+ 199,
+ 480
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 116,
+ 468,
+ 129,
+ 480
+ ],
+ "score": 1.0,
+ "content": "id",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 129,
+ 470,
+ 136,
+ 478
+ ],
+ "score": 0.38,
+ "content": "=",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 136,
+ 468,
+ 199,
+ 480
+ ],
+ "score": 1.0,
+ "content": "rkecl1rtwB.",
+ "type": "text"
+ }
+ ],
+ "index": 28
+ }
+ ],
+ "index": 27,
+ "bbox_fs": [
+ 106,
+ 446,
+ 505,
+ 480
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 105,
+ 486,
+ 505,
+ 510
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 485,
+ 506,
+ 500
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 485,
+ 506,
+ 500
+ ],
+ "score": 1.0,
+ "content": "Ding-Xuan Zhou. Universality of deep convolutional neural networks. arXiv preprint",
+ "type": "text"
+ }
+ ],
+ "index": 29
+ },
+ {
+ "bbox": [
+ 115,
+ 497,
+ 220,
+ 510
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 497,
+ 220,
+ 510
+ ],
+ "score": 1.0,
+ "content": "arXiv:1805.10769, 2018.",
+ "type": "text"
+ }
+ ],
+ "index": 30
+ }
+ ],
+ "index": 29.5,
+ "bbox_fs": [
+ 106,
+ 485,
+ 506,
+ 510
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 109,
+ 516,
+ 503,
+ 551
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 515,
+ 505,
+ 531
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 515,
+ 505,
+ 531
+ ],
+ "score": 1.0,
+ "content": "Pan Zhou and Jiashi Feng. Understanding generalization and optimization performance of deep",
+ "type": "text"
+ }
+ ],
+ "index": 31
+ },
+ {
+ "bbox": [
+ 116,
+ 528,
+ 505,
+ 540
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 116,
+ 528,
+ 505,
+ 540
+ ],
+ "score": 1.0,
+ "content": "CNNs. In Proceedings of the 35th International Conference on Machine Learning, volume 80 of",
+ "type": "text"
+ }
+ ],
+ "index": 32
+ },
+ {
+ "bbox": [
+ 116,
+ 539,
+ 418,
+ 551
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 116,
+ 539,
+ 418,
+ 551
+ ],
+ "score": 1.0,
+ "content": "Proceedings of Machine Learning Research, pp. 5960–5969. PMLR, 2018.",
+ "type": "text"
+ }
+ ],
+ "index": 33
+ }
+ ],
+ "index": 32,
+ "bbox_fs": [
+ 105,
+ 515,
+ 505,
+ 551
+ ]
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 108,
+ 570,
+ 244,
+ 584
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 570,
+ 245,
+ 586
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 570,
+ 245,
+ 586
+ ],
+ "score": 1.0,
+ "content": "A PROOF OF THEOREM 1",
+ "type": "text"
+ }
+ ],
+ "index": 34
+ }
+ ],
+ "index": 34
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 596,
+ 505,
+ 630
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 596,
+ 505,
+ 608
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 596,
+ 505,
+ 608
+ ],
+ "score": 1.0,
+ "content": "As we wrote in the main article, it is enough to show the following lemmas (definition of mis-",
+ "type": "text"
+ }
+ ],
+ "index": 35
+ },
+ {
+ "bbox": [
+ 105,
+ 606,
+ 506,
+ 622
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 606,
+ 351,
+ 622
+ ],
+ "score": 1.0,
+ "content": "cellaneous variables are as in Section 3.2). Remember that",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 352,
+ 608,
+ 444,
+ 621
+ ],
+ "score": 0.92,
+ "content": "\\lambda = \\operatorname* { s u p } _ { n \\in [ N - M ] } | \\lambda _ { n } |",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 445,
+ 606,
+ 464,
+ 622
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 465,
+ 609,
+ 478,
+ 619
+ ],
+ "score": 0.86,
+ "content": "s _ { l h }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 479,
+ 606,
+ 506,
+ 622
+ ],
+ "score": 1.0,
+ "content": "is the",
+ "type": "text"
+ }
+ ],
+ "index": 36
+ },
+ {
+ "bbox": [
+ 105,
+ 619,
+ 237,
+ 632
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 619,
+ 218,
+ 632
+ ],
+ "score": 1.0,
+ "content": "maximum singular value of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 219,
+ 619,
+ 237,
+ 631
+ ],
+ "score": 0.89,
+ "content": "W _ { l h }",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 37
+ }
+ ],
+ "index": 36,
+ "bbox_fs": [
+ 105,
+ 596,
+ 506,
+ 632
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 105,
+ 632,
+ 371,
+ 646
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 104,
+ 630,
+ 372,
+ 649
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 630,
+ 187,
+ 649
+ ],
+ "score": 1.0,
+ "content": "Lemma 1. For any",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 187,
+ 633,
+ 236,
+ 644
+ ],
+ "score": 0.91,
+ "content": "X \\in \\mathbb { R } ^ { N \\times C }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 237,
+ 630,
+ 276,
+ 649
+ ],
+ "score": 1.0,
+ "content": ", we have",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 276,
+ 633,
+ 367,
+ 646
+ ],
+ "score": 0.91,
+ "content": "d _ { \\mathcal { M } } ( P X ) \\leq \\lambda d _ { \\mathcal { M } } ( X )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 368,
+ 630,
+ 372,
+ 649
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 38
+ }
+ ],
+ "index": 38,
+ "bbox_fs": [
+ 104,
+ 630,
+ 372,
+ 649
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 105,
+ 648,
+ 387,
+ 661
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 104,
+ 644,
+ 387,
+ 664
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 644,
+ 187,
+ 664
+ ],
+ "score": 1.0,
+ "content": "Lemma 2. For any",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 187,
+ 647,
+ 236,
+ 659
+ ],
+ "score": 0.9,
+ "content": "X \\in \\mathbb { R } ^ { N \\times C }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 237,
+ 644,
+ 276,
+ 664
+ ],
+ "score": 1.0,
+ "content": ", we have",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 276,
+ 648,
+ 383,
+ 661
+ ],
+ "score": 0.91,
+ "content": "d _ { \\mathcal { M } } ( X W _ { l h } ) \\leq s _ { l h } d _ { \\mathcal { M } } ( X )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 384,
+ 644,
+ 387,
+ 664
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 39
+ }
+ ],
+ "index": 39,
+ "bbox_fs": [
+ 104,
+ 644,
+ 387,
+ 664
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 663,
+ 370,
+ 676
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 104,
+ 659,
+ 367,
+ 679
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 659,
+ 187,
+ 679
+ ],
+ "score": 1.0,
+ "content": "Lemma 3. For any",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 187,
+ 663,
+ 236,
+ 674
+ ],
+ "score": 0.91,
+ "content": "X \\in \\mathbb { R } ^ { N \\times C }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 237,
+ 659,
+ 276,
+ 679
+ ],
+ "score": 1.0,
+ "content": ", we have",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 276,
+ 663,
+ 367,
+ 676
+ ],
+ "score": 0.91,
+ "content": "d _ { \\mathcal { M } } ( \\sigma ( X ) ) \\leq d _ { \\mathcal { M } } ( X )",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 40
+ }
+ ],
+ "index": 40,
+ "bbox_fs": [
+ 104,
+ 659,
+ 367,
+ 679
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 687,
+ 506,
+ 734
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 687,
+ 506,
+ 700
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 687,
+ 173,
+ 700
+ ],
+ "score": 1.0,
+ "content": "Proof of Lemma",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 174,
+ 688,
+ 179,
+ 698
+ ],
+ "score": 0.29,
+ "content": "^ { l }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 180,
+ 687,
+ 210,
+ 700
+ ],
+ "score": 1.0,
+ "content": ". Since",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 211,
+ 688,
+ 220,
+ 698
+ ],
+ "score": 0.83,
+ "content": "P",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 220,
+ 687,
+ 357,
+ 700
+ ],
+ "score": 1.0,
+ "content": "is a symmetric linear operator on",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 357,
+ 687,
+ 372,
+ 698
+ ],
+ "score": 0.89,
+ "content": "U ^ { \\perp }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 373,
+ 687,
+ 506,
+ 700
+ ],
+ "score": 1.0,
+ "content": ", we can choose the orthonormal",
+ "type": "text"
+ }
+ ],
+ "index": 41
+ },
+ {
+ "bbox": [
+ 104,
+ 697,
+ 507,
+ 713
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 697,
+ 130,
+ 713
+ ],
+ "score": 1.0,
+ "content": "basis",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 130,
+ 699,
+ 201,
+ 711
+ ],
+ "score": 0.91,
+ "content": "( e _ { m } ) _ { m = M + 1 , \\dots , N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 202,
+ 697,
+ 214,
+ 713
+ ],
+ "score": 1.0,
+ "content": "of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 215,
+ 698,
+ 230,
+ 709
+ ],
+ "score": 0.88,
+ "content": "U ^ { \\perp }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 231,
+ 697,
+ 358,
+ 713
+ ],
+ "score": 1.0,
+ "content": "consisting of the eigenvalue of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 359,
+ 699,
+ 382,
+ 711
+ ],
+ "score": 0.92,
+ "content": "P | _ { U ^ { \\perp } }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 383,
+ 697,
+ 405,
+ 713
+ ],
+ "score": 1.0,
+ "content": ". Let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 405,
+ 699,
+ 420,
+ 710
+ ],
+ "score": 0.89,
+ "content": "\\lambda _ { m }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 420,
+ 697,
+ 507,
+ 713
+ ],
+ "score": 1.0,
+ "content": "be the eigenvalue of",
+ "type": "text"
+ }
+ ],
+ "index": 42
+ },
+ {
+ "bbox": [
+ 107,
+ 708,
+ 506,
+ 722
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 710,
+ 115,
+ 720
+ ],
+ "score": 0.84,
+ "content": "P",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 116,
+ 708,
+ 155,
+ 722
+ ],
+ "score": 1.0,
+ "content": "to which",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 155,
+ 712,
+ 168,
+ 721
+ ],
+ "score": 0.85,
+ "content": "e _ { m }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 169,
+ 708,
+ 225,
+ 722
+ ],
+ "score": 1.0,
+ "content": "is associated",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 225,
+ 710,
+ 315,
+ 721
+ ],
+ "score": 0.89,
+ "content": "( m = M + 1 , \\ldots , N )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 315,
+ 708,
+ 470,
+ 722
+ ],
+ "score": 1.0,
+ "content": ". Note that since the operator norm of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 470,
+ 710,
+ 493,
+ 722
+ ],
+ "score": 0.92,
+ "content": "P | _ { U ^ { \\perp } }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 494,
+ 708,
+ 506,
+ 722
+ ],
+ "score": 1.0,
+ "content": "is",
+ "type": "text"
+ }
+ ],
+ "index": 43
+ },
+ {
+ "bbox": [
+ 106,
+ 720,
+ 506,
+ 734
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 721,
+ 114,
+ 731
+ ],
+ "score": 0.78,
+ "content": "\\lambda",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 114,
+ 720,
+ 153,
+ 734
+ ],
+ "score": 1.0,
+ "content": ", we have",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 153,
+ 721,
+ 192,
+ 732
+ ],
+ "score": 0.93,
+ "content": "| \\lambda _ { m } | \\leq \\lambda",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 192,
+ 720,
+ 220,
+ 734
+ ],
+ "score": 1.0,
+ "content": "for all",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 221,
+ 721,
+ 303,
+ 732
+ ],
+ "score": 0.88,
+ "content": "m = M + 1 , \\ldots , N",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 303,
+ 720,
+ 333,
+ 734
+ ],
+ "score": 1.0,
+ "content": ". Since",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 333,
+ 720,
+ 378,
+ 733
+ ],
+ "score": 0.93,
+ "content": "( e _ { m } ) _ { m \\in [ N ] }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 378,
+ 720,
+ 506,
+ 734
+ ],
+ "score": 1.0,
+ "content": "forms the orthonormal basis of",
+ "type": "text"
+ }
+ ],
+ "index": 44
+ },
+ {
+ "bbox": [
+ 106,
+ 73,
+ 510,
+ 101
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 81,
+ 122,
+ 93
+ ],
+ "score": 0.87,
+ "content": "\\mathbb { R } ^ { N }",
+ "type": "inline_equation",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 122,
+ 73,
+ 218,
+ 101
+ ],
+ "score": 1.0,
+ "content": ", we can uniquely write",
+ "type": "text",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 219,
+ 81,
+ 270,
+ 93
+ ],
+ "score": 0.92,
+ "content": "X \\in \\mathbb { R } ^ { N \\times C }",
+ "type": "inline_equation",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 270,
+ 73,
+ 282,
+ 101
+ ],
+ "score": 1.0,
+ "content": "as",
+ "type": "text",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 282,
+ 80,
+ 376,
+ 95
+ ],
+ "score": 0.93,
+ "content": "\\begin{array} { r } { X = \\sum _ { m = 1 } ^ { N } e _ { m } \\otimes w _ { m } } \\end{array}",
+ "type": "inline_equation",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 377,
+ 73,
+ 416,
+ 101
+ ],
+ "score": 1.0,
+ "content": "for some",
+ "type": "text",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 416,
+ 81,
+ 459,
+ 94
+ ],
+ "score": 0.93,
+ "content": "w _ { m } \\in \\mathbb { R } ^ { C }",
+ "type": "inline_equation",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 460,
+ 73,
+ 510,
+ 101
+ ],
+ "score": 1.0,
+ "content": ". Then, we",
+ "type": "text",
+ "cross_page": true
+ }
+ ],
+ "index": 0
+ },
+ {
+ "bbox": [
+ 99,
+ 86,
+ 512,
+ 117
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 99,
+ 86,
+ 127,
+ 117
+ ],
+ "score": 1.0,
+ "content": "have",
+ "type": "text",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 127,
+ 93,
+ 247,
+ 110
+ ],
+ "score": 0.93,
+ "content": "\\begin{array} { r } { d _ { \\mathcal { M } } ^ { 2 } ( X ) = \\sum _ { m = M + 1 } ^ { N } \\| w _ { m } \\| ^ { 2 } } \\end{array}",
+ "type": "inline_equation",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 247,
+ 86,
+ 275,
+ 117
+ ],
+ "score": 1.0,
+ "content": "where",
+ "type": "text",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 275,
+ 96,
+ 291,
+ 109
+ ],
+ "score": 0.9,
+ "content": "\\| \\cdot \\|",
+ "type": "inline_equation",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 292,
+ 86,
+ 512,
+ 117
+ ],
+ "score": 1.0,
+ "content": "m=1 m m m is the 2-norm of a vector. On the other hand, we have",
+ "type": "text",
+ "cross_page": true
+ }
+ ],
+ "index": 1
+ }
+ ],
+ "index": 42.5,
+ "bbox_fs": [
+ 104,
+ 687,
+ 507,
+ 734
+ ]
+ }
+ ]
+ },
+ {
+ "preproc_blocks": [
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 79,
+ 506,
+ 110
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 73,
+ 510,
+ 101
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 81,
+ 122,
+ 93
+ ],
+ "score": 0.87,
+ "content": "\\mathbb { R } ^ { N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 122,
+ 73,
+ 218,
+ 101
+ ],
+ "score": 1.0,
+ "content": ", we can uniquely write",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 219,
+ 81,
+ 270,
+ 93
+ ],
+ "score": 0.92,
+ "content": "X \\in \\mathbb { R } ^ { N \\times C }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 270,
+ 73,
+ 282,
+ 101
+ ],
+ "score": 1.0,
+ "content": "as",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 282,
+ 80,
+ 376,
+ 95
+ ],
+ "score": 0.93,
+ "content": "\\begin{array} { r } { X = \\sum _ { m = 1 } ^ { N } e _ { m } \\otimes w _ { m } } \\end{array}",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 377,
+ 73,
+ 416,
+ 101
+ ],
+ "score": 1.0,
+ "content": "for some",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 416,
+ 81,
+ 459,
+ 94
+ ],
+ "score": 0.93,
+ "content": "w _ { m } \\in \\mathbb { R } ^ { C }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 460,
+ 73,
+ 510,
+ 101
+ ],
+ "score": 1.0,
+ "content": ". Then, we",
+ "type": "text"
+ }
+ ],
+ "index": 0
+ },
+ {
+ "bbox": [
+ 99,
+ 86,
+ 512,
+ 117
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 99,
+ 86,
+ 127,
+ 117
+ ],
+ "score": 1.0,
+ "content": "have",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 127,
+ 93,
+ 247,
+ 110
+ ],
+ "score": 0.93,
+ "content": "\\begin{array} { r } { d _ { \\mathcal { M } } ^ { 2 } ( X ) = \\sum _ { m = M + 1 } ^ { N } \\| w _ { m } \\| ^ { 2 } } \\end{array}",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 247,
+ 86,
+ 275,
+ 117
+ ],
+ "score": 1.0,
+ "content": "where",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 275,
+ 96,
+ 291,
+ 109
+ ],
+ "score": 0.9,
+ "content": "\\| \\cdot \\|",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 292,
+ 86,
+ 512,
+ 117
+ ],
+ "score": 1.0,
+ "content": "m=1 m m m is the 2-norm of a vector. On the other hand, we have",
+ "type": "text"
+ }
+ ],
+ "index": 1
+ }
+ ],
+ "index": 0.5
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 203,
+ 116,
+ 408,
+ 223
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 203,
+ 116,
+ 408,
+ 223
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 203,
+ 116,
+ 408,
+ 223
+ ],
+ "score": 0.95,
+ "content": "\\begin{array} { l } { { \\displaystyle P X = \\sum _ { m = 1 } ^ { N } P e _ { m } \\otimes w _ { m } } } \\\\ { ~ } \\\\ { { \\displaystyle ~ = \\sum _ { m = 1 } ^ { M } P e _ { m } \\otimes w _ { m } + \\sum _ { m = M + 1 } ^ { N } P e _ { m } \\otimes w _ { m } } } \\\\ { { \\displaystyle ~ = \\sum _ { m = 1 } ^ { M } P e _ { m } \\otimes w _ { m } + \\sum _ { m = M + 1 } ^ { N } e _ { m } \\otimes \\left( \\lambda _ { m } w _ { m } \\right) } } \\end{array}",
+ "type": "interline_equation",
+ "image_path": "ebdf38f8421714663fc0ce017ea1e1607adc9bfbfba6c6df26e9662d41ab49ca.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 4,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 203,
+ 116,
+ 408,
+ 137.4
+ ],
+ "spans": [],
+ "index": 2
+ },
+ {
+ "bbox": [
+ 203,
+ 137.4,
+ 408,
+ 158.8
+ ],
+ "spans": [],
+ "index": 3
+ },
+ {
+ "bbox": [
+ 203,
+ 158.8,
+ 408,
+ 180.20000000000002
+ ],
+ "spans": [],
+ "index": 4
+ },
+ {
+ "bbox": [
+ 203,
+ 180.20000000000002,
+ 408,
+ 201.60000000000002
+ ],
+ "spans": [],
+ "index": 5
+ },
+ {
+ "bbox": [
+ 203,
+ 201.60000000000002,
+ 408,
+ 223.00000000000003
+ ],
+ "spans": [],
+ "index": 6
+ }
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 229,
+ 505,
+ 266
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 132,
+ 228,
+ 510,
+ 267
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 132,
+ 230,
+ 141,
+ 240
+ ],
+ "score": 0.81,
+ "content": "U",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 160,
+ 228,
+ 218,
+ 267
+ ],
+ "score": 1.0,
+ "content": "nvariant under . Therefore, w",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 219,
+ 230,
+ 228,
+ 240
+ ],
+ "score": 0.83,
+ "content": "P",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 228,
+ 228,
+ 245,
+ 267
+ ],
+ "score": 1.0,
+ "content": ", forhave",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 266,
+ 230,
+ 309,
+ 241
+ ],
+ "score": 0.91,
+ "content": "m \\in [ M ]",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 371,
+ 230,
+ 392,
+ 241
+ ],
+ "score": 0.9,
+ "content": "P e _ { m }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 393,
+ 228,
+ 510,
+ 267
+ ],
+ "score": 1.0,
+ "content": "as a linear combination ofThen, we obtain the desired",
+ "type": "text"
+ }
+ ],
+ "index": 8
+ },
+ {
+ "bbox": [
+ 106,
+ 241,
+ 387,
+ 257
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 243,
+ 160,
+ 255
+ ],
+ "score": 0.91,
+ "content": "e _ { n } ( n \\in [ M ] )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 246,
+ 241,
+ 387,
+ 257
+ ],
+ "score": 0.92,
+ "content": "\\begin{array} { r } { d _ { \\mathcal { M } } ^ { 2 } ( P X ) = \\sum _ { m = M + 1 } ^ { N } \\| \\lambda _ { m } w _ { m } \\| ^ { 2 } } \\end{array}",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 7
+ }
+ ],
+ "index": 7.5
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 239,
+ 272,
+ 371,
+ 397
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 239,
+ 272,
+ 371,
+ 397
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 239,
+ 272,
+ 371,
+ 397
+ ],
+ "score": 0.92,
+ "content": "\\begin{array} { r l r } { { d _ { { \\mathcal M } } ^ { 2 } ( P X ) = \\sum _ { m = M + 1 } ^ { N } \\| \\lambda _ { m } w _ { m } \\| ^ { 2 } } } \\\\ & { } & \\\\ & { } & { \\leq \\lambda ^ { 2 } \\sum _ { m = M + 1 } ^ { N } \\| w _ { m } \\| ^ { 2 } } \\\\ & { } & { \\leq \\lambda ^ { 2 } \\displaystyle \\sum _ { m = M + 1 } ^ { N } \\| w _ { m } \\| ^ { 2 } } \\\\ & { } & \\\\ & { } & { = \\lambda ^ { 2 } d _ { { \\mathcal M } } ^ { 2 } ( X ) . } \\end{array}",
+ "type": "interline_equation",
+ "image_path": "2d20c7b13c75612e652ad08f45b54ce116add66b3167800688d04acf8a22592a.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 9.5,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 239,
+ 272,
+ 371,
+ 334.5
+ ],
+ "spans": [],
+ "index": 9
+ },
+ {
+ "bbox": [
+ 239,
+ 334.5,
+ 371,
+ 397.0
+ ],
+ "spans": [],
+ "index": 10
+ }
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 104,
+ 430,
+ 471,
+ 443
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 430,
+ 471,
+ 444
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 430,
+ 322,
+ 444
+ ],
+ "score": 1.0,
+ "content": "Proof of Lemma 2. Using the same decomposition of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 322,
+ 432,
+ 332,
+ 441
+ ],
+ "score": 0.84,
+ "content": "X",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 332,
+ 430,
+ 471,
+ 444
+ ],
+ "score": 1.0,
+ "content": "as the proof in Lemma 1, we have",
+ "type": "text"
+ }
+ ],
+ "index": 11
+ }
+ ],
+ "index": 11
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 186,
+ 449,
+ 425,
+ 521
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 186,
+ 449,
+ 425,
+ 521
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 186,
+ 449,
+ 425,
+ 521
+ ],
+ "score": 0.94,
+ "content": "\\begin{array} { r l r } { { X W _ { l h } = \\sum _ { m = 1 } ^ { N } e _ { m } \\otimes ( W _ { l h } ^ { \\top } w _ { m } ) } } \\\\ & { } & { = \\sum _ { m = 1 } ^ { M } e _ { m } \\otimes ( W _ { l h } ^ { \\top } w _ { m } ) + \\sum _ { m = M + 1 } ^ { N } e _ { m } \\otimes ( W _ { l h } ^ { \\top } w _ { m } ) . } \\end{array}",
+ "type": "interline_equation",
+ "image_path": "11c62737dd7afa77a1fa1f9b149e31987b5092043d5a2cbbdcbc4dd3cd85d7da.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 13,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 186,
+ 449,
+ 425,
+ 473.0
+ ],
+ "spans": [],
+ "index": 12
+ },
+ {
+ "bbox": [
+ 186,
+ 473.0,
+ 425,
+ 497.0
+ ],
+ "spans": [],
+ "index": 13
+ },
+ {
+ "bbox": [
+ 186,
+ 497.0,
+ 425,
+ 521.0
+ ],
+ "spans": [],
+ "index": 14
+ }
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 526,
+ 186,
+ 538
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 525,
+ 186,
+ 539
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 525,
+ 186,
+ 539
+ ],
+ "score": 1.0,
+ "content": "Therefore, we have",
+ "type": "text"
+ }
+ ],
+ "index": 15
+ }
+ ],
+ "index": 15
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 232,
+ 545,
+ 378,
+ 632
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 232,
+ 545,
+ 378,
+ 632
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 232,
+ 545,
+ 378,
+ 632
+ ],
+ "score": 0.93,
+ "content": "\\begin{array} { r l r } { { d _ { \\mathcal { M } } ^ { 2 } ( X W _ { l h } ) = \\sum _ { m = M + 1 } ^ { N } \\| W _ { l h } ^ { \\top } w _ { m } \\| ^ { 2 } } } \\\\ & { } & { \\leq s _ { l h } ^ { 2 } \\sum _ { m = M + 1 } ^ { N } \\| w _ { m } \\| ^ { 2 } } \\\\ & { } & { = s _ { l h } ^ { 2 } d _ { \\mathcal { M } } ^ { 2 } ( X ) . } \\end{array}",
+ "type": "interline_equation",
+ "image_path": "1e1f7af71d03e46e2b2c1f6a93dbe12369a4668beb889066020801bff330d278.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 16.5,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 232,
+ 545,
+ 378,
+ 588.5
+ ],
+ "spans": [],
+ "index": 16
+ },
+ {
+ "bbox": [
+ 232,
+ 588.5,
+ 378,
+ 632.0
+ ],
+ "spans": [],
+ "index": 17
+ }
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 666,
+ 506,
+ 735
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 666,
+ 505,
+ 681
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 666,
+ 232,
+ 681
+ ],
+ "score": 1.0,
+ "content": "Proof of Lemma 3. We choose",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 233,
+ 668,
+ 317,
+ 680
+ ],
+ "score": 0.91,
+ "content": "( e _ { m } ) _ { m = N - M + 1 , \\dots , N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 317,
+ 666,
+ 482,
+ 681
+ ],
+ "score": 1.0,
+ "content": "as in the proof of Lemma 1. We denote",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 483,
+ 668,
+ 505,
+ 678
+ ],
+ "score": 0.85,
+ "content": "X =",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 18
+ },
+ {
+ "bbox": [
+ 107,
+ 676,
+ 506,
+ 695
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 680,
+ 177,
+ 693
+ ],
+ "score": 0.93,
+ "content": "( X _ { n c } ) _ { n \\in [ N ] , c \\in [ C ] }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 177,
+ 676,
+ 196,
+ 695
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 196,
+ 680,
+ 272,
+ 693
+ ],
+ "score": 0.92,
+ "content": "e _ { n } = ( e _ { m n } ) _ { m \\in [ N ] }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 273,
+ 676,
+ 348,
+ 695
+ ],
+ "score": 1.0,
+ "content": ", respectively. Let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 348,
+ 679,
+ 385,
+ 693
+ ],
+ "score": 0.92,
+ "content": "( e _ { c } ^ { \\prime } ) _ { c \\in [ C ] }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 386,
+ 676,
+ 486,
+ 695
+ ],
+ "score": 1.0,
+ "content": "be the standard basis of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 486,
+ 679,
+ 501,
+ 690
+ ],
+ "score": 0.87,
+ "content": "\\mathbb { R } ^ { C }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 501,
+ 676,
+ 506,
+ 695
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 19
+ },
+ {
+ "bbox": [
+ 104,
+ 689,
+ 507,
+ 706
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 689,
+ 133,
+ 706
+ ],
+ "score": 1.0,
+ "content": "Then,",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 134,
+ 693,
+ 219,
+ 705
+ ],
+ "score": 0.92,
+ "content": "( e _ { n } \\otimes e _ { c } ^ { \\prime } ) _ { n \\in [ N ] , c \\in [ C ] }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 219,
+ 689,
+ 335,
+ 706
+ ],
+ "score": 1.0,
+ "content": "is the orthonormal basis of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 335,
+ 692,
+ 363,
+ 702
+ ],
+ "score": 0.91,
+ "content": "\\mathbb { R } ^ { N \\times C }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 363,
+ 689,
+ 507,
+ 706
+ ],
+ "score": 1.0,
+ "content": ", endowed with the standard inner",
+ "type": "text"
+ }
+ ],
+ "index": 20
+ },
+ {
+ "bbox": [
+ 103,
+ 703,
+ 508,
+ 725
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 103,
+ 703,
+ 334,
+ 725
+ ],
+ "score": 1.0,
+ "content": "product as a Euclid space. Therefore, we can decompose",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 334,
+ 707,
+ 344,
+ 717
+ ],
+ "score": 0.82,
+ "content": "X",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 344,
+ 703,
+ 356,
+ 725
+ ],
+ "score": 1.0,
+ "content": "as",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 356,
+ 704,
+ 477,
+ 719
+ ],
+ "score": 0.92,
+ "content": "\\begin{array} { r } { X = \\sum _ { n = 1 } ^ { N } \\sum _ { c = 1 } ^ { C } a _ { n c } e _ { n } \\otimes e _ { c } ^ { \\prime } } \\end{array}",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 477,
+ 703,
+ 508,
+ 725
+ ],
+ "score": 1.0,
+ "content": "where",
+ "type": "text"
+ }
+ ],
+ "index": 21
+ },
+ {
+ "bbox": [
+ 106,
+ 711,
+ 501,
+ 741
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 718,
+ 274,
+ 734
+ ],
+ "score": 0.92,
+ "content": "\\begin{array} { r } { a _ { n c } = \\left. X , e _ { n } \\otimes e _ { c } ^ { \\prime } \\right. = \\sum _ { m = 1 } ^ { N } X _ { m c } e _ { m n } } \\end{array}",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 274,
+ 711,
+ 344,
+ 741
+ ],
+ "score": 1.0,
+ "content": ". Then, we have",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 344,
+ 719,
+ 501,
+ 734
+ ],
+ "score": 0.9,
+ "content": "\\begin{array} { r } { d _ { \\mathcal { M } } ^ { 2 } ( X ) = \\sum _ { n = M + 1 } ^ { N } \\| \\sum _ { c = 1 } ^ { C } a _ { n c } e _ { c } ^ { \\prime } \\| ^ { 2 } } \\end{array}",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 22
+ }
+ ],
+ "index": 20
+ }
+ ],
+ "page_idx": 13,
+ "page_size": [
+ 612,
+ 792
+ ],
+ "discarded_blocks": [
+ {
+ "type": "discarded",
+ "bbox": [
+ 106,
+ 26,
+ 294,
+ 38
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 25,
+ 294,
+ 38
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 25,
+ 294,
+ 38
+ ],
+ "score": 1.0,
+ "content": "Published as a conference paper at ICLR 2020",
+ "type": "text"
+ }
+ ]
+ }
+ ]
+ },
+ {
+ "type": "discarded",
+ "bbox": [
+ 300,
+ 751,
+ 311,
+ 760
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 299,
+ 750,
+ 313,
+ 764
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 299,
+ 750,
+ 313,
+ 764
+ ],
+ "score": 1.0,
+ "content": "14",
+ "type": "text"
+ }
+ ]
+ }
+ ]
+ },
+ {
+ "type": "discarded",
+ "bbox": [
+ 494,
+ 637,
+ 505,
+ 649
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 495,
+ 639,
+ 505,
+ 649
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 495,
+ 639,
+ 505,
+ 649
+ ],
+ "score": 0.998,
+ "content": "□",
+ "type": "text"
+ }
+ ]
+ }
+ ]
+ },
+ {
+ "type": "discarded",
+ "bbox": [
+ 494,
+ 402,
+ 505,
+ 413
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 496,
+ 404,
+ 504,
+ 412
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 496,
+ 404,
+ 504,
+ 412
+ ],
+ "score": 0.997,
+ "content": "□",
+ "type": "text"
+ }
+ ]
+ }
+ ]
+ }
+ ],
+ "para_blocks": [
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 79,
+ 506,
+ 110
+ ],
+ "lines": [],
+ "index": 0.5,
+ "bbox_fs": [
+ 99,
+ 73,
+ 512,
+ 117
+ ],
+ "lines_deleted": true
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 203,
+ 116,
+ 408,
+ 223
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 203,
+ 116,
+ 408,
+ 223
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 203,
+ 116,
+ 408,
+ 223
+ ],
+ "score": 0.95,
+ "content": "\\begin{array} { l } { { \\displaystyle P X = \\sum _ { m = 1 } ^ { N } P e _ { m } \\otimes w _ { m } } } \\\\ { ~ } \\\\ { { \\displaystyle ~ = \\sum _ { m = 1 } ^ { M } P e _ { m } \\otimes w _ { m } + \\sum _ { m = M + 1 } ^ { N } P e _ { m } \\otimes w _ { m } } } \\\\ { { \\displaystyle ~ = \\sum _ { m = 1 } ^ { M } P e _ { m } \\otimes w _ { m } + \\sum _ { m = M + 1 } ^ { N } e _ { m } \\otimes \\left( \\lambda _ { m } w _ { m } \\right) } } \\end{array}",
+ "type": "interline_equation",
+ "image_path": "ebdf38f8421714663fc0ce017ea1e1607adc9bfbfba6c6df26e9662d41ab49ca.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 4,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 203,
+ 116,
+ 408,
+ 137.4
+ ],
+ "spans": [],
+ "index": 2
+ },
+ {
+ "bbox": [
+ 203,
+ 137.4,
+ 408,
+ 158.8
+ ],
+ "spans": [],
+ "index": 3
+ },
+ {
+ "bbox": [
+ 203,
+ 158.8,
+ 408,
+ 180.20000000000002
+ ],
+ "spans": [],
+ "index": 4
+ },
+ {
+ "bbox": [
+ 203,
+ 180.20000000000002,
+ 408,
+ 201.60000000000002
+ ],
+ "spans": [],
+ "index": 5
+ },
+ {
+ "bbox": [
+ 203,
+ 201.60000000000002,
+ 408,
+ 223.00000000000003
+ ],
+ "spans": [],
+ "index": 6
+ }
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 229,
+ 505,
+ 266
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 132,
+ 228,
+ 510,
+ 267
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 132,
+ 230,
+ 141,
+ 240
+ ],
+ "score": 0.81,
+ "content": "U",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 160,
+ 228,
+ 218,
+ 267
+ ],
+ "score": 1.0,
+ "content": "nvariant under . Therefore, w",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 219,
+ 230,
+ 228,
+ 240
+ ],
+ "score": 0.83,
+ "content": "P",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 228,
+ 228,
+ 245,
+ 267
+ ],
+ "score": 1.0,
+ "content": ", forhave",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 266,
+ 230,
+ 309,
+ 241
+ ],
+ "score": 0.91,
+ "content": "m \\in [ M ]",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 371,
+ 230,
+ 392,
+ 241
+ ],
+ "score": 0.9,
+ "content": "P e _ { m }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 393,
+ 228,
+ 510,
+ 267
+ ],
+ "score": 1.0,
+ "content": "as a linear combination ofThen, we obtain the desired",
+ "type": "text"
+ }
+ ],
+ "index": 8
+ },
+ {
+ "bbox": [
+ 106,
+ 241,
+ 387,
+ 257
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 243,
+ 160,
+ 255
+ ],
+ "score": 0.91,
+ "content": "e _ { n } ( n \\in [ M ] )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 246,
+ 241,
+ 387,
+ 257
+ ],
+ "score": 0.92,
+ "content": "\\begin{array} { r } { d _ { \\mathcal { M } } ^ { 2 } ( P X ) = \\sum _ { m = M + 1 } ^ { N } \\| \\lambda _ { m } w _ { m } \\| ^ { 2 } } \\end{array}",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 7
+ }
+ ],
+ "index": 7.5,
+ "bbox_fs": [
+ 106,
+ 228,
+ 510,
+ 267
+ ]
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 239,
+ 272,
+ 371,
+ 397
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 239,
+ 272,
+ 371,
+ 397
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 239,
+ 272,
+ 371,
+ 397
+ ],
+ "score": 0.92,
+ "content": "\\begin{array} { r l r } { { d _ { { \\mathcal M } } ^ { 2 } ( P X ) = \\sum _ { m = M + 1 } ^ { N } \\| \\lambda _ { m } w _ { m } \\| ^ { 2 } } } \\\\ & { } & \\\\ & { } & { \\leq \\lambda ^ { 2 } \\sum _ { m = M + 1 } ^ { N } \\| w _ { m } \\| ^ { 2 } } \\\\ & { } & { \\leq \\lambda ^ { 2 } \\displaystyle \\sum _ { m = M + 1 } ^ { N } \\| w _ { m } \\| ^ { 2 } } \\\\ & { } & \\\\ & { } & { = \\lambda ^ { 2 } d _ { { \\mathcal M } } ^ { 2 } ( X ) . } \\end{array}",
+ "type": "interline_equation",
+ "image_path": "2d20c7b13c75612e652ad08f45b54ce116add66b3167800688d04acf8a22592a.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 9.5,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 239,
+ 272,
+ 371,
+ 334.5
+ ],
+ "spans": [],
+ "index": 9
+ },
+ {
+ "bbox": [
+ 239,
+ 334.5,
+ 371,
+ 397.0
+ ],
+ "spans": [],
+ "index": 10
+ }
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 104,
+ 430,
+ 471,
+ 443
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 430,
+ 471,
+ 444
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 430,
+ 322,
+ 444
+ ],
+ "score": 1.0,
+ "content": "Proof of Lemma 2. Using the same decomposition of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 322,
+ 432,
+ 332,
+ 441
+ ],
+ "score": 0.84,
+ "content": "X",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 332,
+ 430,
+ 471,
+ 444
+ ],
+ "score": 1.0,
+ "content": "as the proof in Lemma 1, we have",
+ "type": "text"
+ }
+ ],
+ "index": 11
+ }
+ ],
+ "index": 11,
+ "bbox_fs": [
+ 106,
+ 430,
+ 471,
+ 444
+ ]
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 186,
+ 449,
+ 425,
+ 521
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 186,
+ 449,
+ 425,
+ 521
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 186,
+ 449,
+ 425,
+ 521
+ ],
+ "score": 0.94,
+ "content": "\\begin{array} { r l r } { { X W _ { l h } = \\sum _ { m = 1 } ^ { N } e _ { m } \\otimes ( W _ { l h } ^ { \\top } w _ { m } ) } } \\\\ & { } & { = \\sum _ { m = 1 } ^ { M } e _ { m } \\otimes ( W _ { l h } ^ { \\top } w _ { m } ) + \\sum _ { m = M + 1 } ^ { N } e _ { m } \\otimes ( W _ { l h } ^ { \\top } w _ { m } ) . } \\end{array}",
+ "type": "interline_equation",
+ "image_path": "11c62737dd7afa77a1fa1f9b149e31987b5092043d5a2cbbdcbc4dd3cd85d7da.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 13,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 186,
+ 449,
+ 425,
+ 473.0
+ ],
+ "spans": [],
+ "index": 12
+ },
+ {
+ "bbox": [
+ 186,
+ 473.0,
+ 425,
+ 497.0
+ ],
+ "spans": [],
+ "index": 13
+ },
+ {
+ "bbox": [
+ 186,
+ 497.0,
+ 425,
+ 521.0
+ ],
+ "spans": [],
+ "index": 14
+ }
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 526,
+ 186,
+ 538
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 525,
+ 186,
+ 539
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 525,
+ 186,
+ 539
+ ],
+ "score": 1.0,
+ "content": "Therefore, we have",
+ "type": "text"
+ }
+ ],
+ "index": 15
+ }
+ ],
+ "index": 15,
+ "bbox_fs": [
+ 106,
+ 525,
+ 186,
+ 539
+ ]
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 232,
+ 545,
+ 378,
+ 632
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 232,
+ 545,
+ 378,
+ 632
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 232,
+ 545,
+ 378,
+ 632
+ ],
+ "score": 0.93,
+ "content": "\\begin{array} { r l r } { { d _ { \\mathcal { M } } ^ { 2 } ( X W _ { l h } ) = \\sum _ { m = M + 1 } ^ { N } \\| W _ { l h } ^ { \\top } w _ { m } \\| ^ { 2 } } } \\\\ & { } & { \\leq s _ { l h } ^ { 2 } \\sum _ { m = M + 1 } ^ { N } \\| w _ { m } \\| ^ { 2 } } \\\\ & { } & { = s _ { l h } ^ { 2 } d _ { \\mathcal { M } } ^ { 2 } ( X ) . } \\end{array}",
+ "type": "interline_equation",
+ "image_path": "1e1f7af71d03e46e2b2c1f6a93dbe12369a4668beb889066020801bff330d278.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 16.5,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 232,
+ 545,
+ 378,
+ 588.5
+ ],
+ "spans": [],
+ "index": 16
+ },
+ {
+ "bbox": [
+ 232,
+ 588.5,
+ 378,
+ 632.0
+ ],
+ "spans": [],
+ "index": 17
+ }
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 666,
+ 506,
+ 735
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 666,
+ 505,
+ 681
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 666,
+ 232,
+ 681
+ ],
+ "score": 1.0,
+ "content": "Proof of Lemma 3. We choose",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 233,
+ 668,
+ 317,
+ 680
+ ],
+ "score": 0.91,
+ "content": "( e _ { m } ) _ { m = N - M + 1 , \\dots , N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 317,
+ 666,
+ 482,
+ 681
+ ],
+ "score": 1.0,
+ "content": "as in the proof of Lemma 1. We denote",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 483,
+ 668,
+ 505,
+ 678
+ ],
+ "score": 0.85,
+ "content": "X =",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 18
+ },
+ {
+ "bbox": [
+ 107,
+ 676,
+ 506,
+ 695
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 680,
+ 177,
+ 693
+ ],
+ "score": 0.93,
+ "content": "( X _ { n c } ) _ { n \\in [ N ] , c \\in [ C ] }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 177,
+ 676,
+ 196,
+ 695
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 196,
+ 680,
+ 272,
+ 693
+ ],
+ "score": 0.92,
+ "content": "e _ { n } = ( e _ { m n } ) _ { m \\in [ N ] }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 273,
+ 676,
+ 348,
+ 695
+ ],
+ "score": 1.0,
+ "content": ", respectively. Let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 348,
+ 679,
+ 385,
+ 693
+ ],
+ "score": 0.92,
+ "content": "( e _ { c } ^ { \\prime } ) _ { c \\in [ C ] }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 386,
+ 676,
+ 486,
+ 695
+ ],
+ "score": 1.0,
+ "content": "be the standard basis of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 486,
+ 679,
+ 501,
+ 690
+ ],
+ "score": 0.87,
+ "content": "\\mathbb { R } ^ { C }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 501,
+ 676,
+ 506,
+ 695
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 19
+ },
+ {
+ "bbox": [
+ 104,
+ 689,
+ 507,
+ 706
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 689,
+ 133,
+ 706
+ ],
+ "score": 1.0,
+ "content": "Then,",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 134,
+ 693,
+ 219,
+ 705
+ ],
+ "score": 0.92,
+ "content": "( e _ { n } \\otimes e _ { c } ^ { \\prime } ) _ { n \\in [ N ] , c \\in [ C ] }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 219,
+ 689,
+ 335,
+ 706
+ ],
+ "score": 1.0,
+ "content": "is the orthonormal basis of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 335,
+ 692,
+ 363,
+ 702
+ ],
+ "score": 0.91,
+ "content": "\\mathbb { R } ^ { N \\times C }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 363,
+ 689,
+ 507,
+ 706
+ ],
+ "score": 1.0,
+ "content": ", endowed with the standard inner",
+ "type": "text"
+ }
+ ],
+ "index": 20
+ },
+ {
+ "bbox": [
+ 103,
+ 703,
+ 508,
+ 725
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 103,
+ 703,
+ 334,
+ 725
+ ],
+ "score": 1.0,
+ "content": "product as a Euclid space. Therefore, we can decompose",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 334,
+ 707,
+ 344,
+ 717
+ ],
+ "score": 0.82,
+ "content": "X",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 344,
+ 703,
+ 356,
+ 725
+ ],
+ "score": 1.0,
+ "content": "as",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 356,
+ 704,
+ 477,
+ 719
+ ],
+ "score": 0.92,
+ "content": "\\begin{array} { r } { X = \\sum _ { n = 1 } ^ { N } \\sum _ { c = 1 } ^ { C } a _ { n c } e _ { n } \\otimes e _ { c } ^ { \\prime } } \\end{array}",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 477,
+ 703,
+ 508,
+ 725
+ ],
+ "score": 1.0,
+ "content": "where",
+ "type": "text"
+ }
+ ],
+ "index": 21
+ },
+ {
+ "bbox": [
+ 106,
+ 711,
+ 501,
+ 741
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 718,
+ 274,
+ 734
+ ],
+ "score": 0.92,
+ "content": "\\begin{array} { r } { a _ { n c } = \\left. X , e _ { n } \\otimes e _ { c } ^ { \\prime } \\right. = \\sum _ { m = 1 } ^ { N } X _ { m c } e _ { m n } } \\end{array}",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 274,
+ 711,
+ 344,
+ 741
+ ],
+ "score": 1.0,
+ "content": ". Then, we have",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 344,
+ 719,
+ 501,
+ 734
+ ],
+ "score": 0.9,
+ "content": "\\begin{array} { r } { d _ { \\mathcal { M } } ^ { 2 } ( X ) = \\sum _ { n = M + 1 } ^ { N } \\| \\sum _ { c = 1 } ^ { C } a _ { n c } e _ { c } ^ { \\prime } \\| ^ { 2 } } \\end{array}",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 22
+ }
+ ],
+ "index": 20,
+ "bbox_fs": [
+ 103,
+ 666,
+ 508,
+ 741
+ ]
+ }
+ ]
+ },
+ {
+ "preproc_blocks": [
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 82,
+ 234,
+ 93
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 81,
+ 234,
+ 95
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 81,
+ 234,
+ 95
+ ],
+ "score": 1.0,
+ "content": "which is further transformed as",
+ "type": "text"
+ }
+ ],
+ "index": 0
+ }
+ ],
+ "index": 0
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 213,
+ 97,
+ 395,
+ 240
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 213,
+ 97,
+ 395,
+ 240
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 213,
+ 97,
+ 395,
+ 240
+ ],
+ "score": 0.95,
+ "content": "\\begin{array} { l } { \\displaystyle d _ { M } ^ { 2 } ( X ) = \\sum _ { n = M + 1 } ^ { N } \\left\\| \\sum _ { c = 1 } ^ { C } a _ { n c } e _ { c } ^ { t } \\right\\| ^ { 2 } } \\\\ { = \\displaystyle \\sum _ { n = M + 1 } ^ { N } \\sum _ { c = 1 } ^ { C } a _ { n c } ^ { 2 } } \\\\ { = \\displaystyle \\sum _ { c = 1 } ^ { C } \\left( \\sum _ { n = 1 } ^ { N } a _ { n c } ^ { 2 } - \\sum _ { n = 1 } ^ { M } a _ { n c } ^ { 2 } \\right) } \\\\ { = \\displaystyle \\sum _ { c = 1 } ^ { C } \\left( \\| X _ { \\mathrm { - c } } \\| ^ { 2 } - \\sum _ { n = 1 } ^ { M } ( X _ { \\mathrm { - c } } , e _ { n } ) ^ { 2 } \\right) , } \\end{array}",
+ "type": "interline_equation",
+ "image_path": "3e54bc240f5d9875acd41a1878664feda1310273d70a7eb5f0d1b1f678be3ff5.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 5.5,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 213,
+ 97,
+ 395,
+ 111.3
+ ],
+ "spans": [],
+ "index": 1
+ },
+ {
+ "bbox": [
+ 213,
+ 111.3,
+ 395,
+ 125.6
+ ],
+ "spans": [],
+ "index": 2
+ },
+ {
+ "bbox": [
+ 213,
+ 125.6,
+ 395,
+ 139.9
+ ],
+ "spans": [],
+ "index": 3
+ },
+ {
+ "bbox": [
+ 213,
+ 139.9,
+ 395,
+ 154.20000000000002
+ ],
+ "spans": [],
+ "index": 4
+ },
+ {
+ "bbox": [
+ 213,
+ 154.20000000000002,
+ 395,
+ 168.50000000000003
+ ],
+ "spans": [],
+ "index": 5
+ },
+ {
+ "bbox": [
+ 213,
+ 168.50000000000003,
+ 395,
+ 182.80000000000004
+ ],
+ "spans": [],
+ "index": 6
+ },
+ {
+ "bbox": [
+ 213,
+ 182.80000000000004,
+ 395,
+ 197.10000000000005
+ ],
+ "spans": [],
+ "index": 7
+ },
+ {
+ "bbox": [
+ 213,
+ 197.10000000000005,
+ 395,
+ 211.40000000000006
+ ],
+ "spans": [],
+ "index": 8
+ },
+ {
+ "bbox": [
+ 213,
+ 211.40000000000006,
+ 395,
+ 225.70000000000007
+ ],
+ "spans": [],
+ "index": 9
+ },
+ {
+ "bbox": [
+ 213,
+ 225.70000000000007,
+ 395,
+ 240.00000000000009
+ ],
+ "spans": [],
+ "index": 10
+ }
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 241,
+ 354,
+ 253
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 240,
+ 354,
+ 254
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 240,
+ 133,
+ 254
+ ],
+ "score": 1.0,
+ "content": "where",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 133,
+ 242,
+ 149,
+ 252
+ ],
+ "score": 0.87,
+ "content": "X _ { \\cdot c }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 150,
+ 240,
+ 174,
+ 254
+ ],
+ "score": 1.0,
+ "content": "is the",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 174,
+ 243,
+ 180,
+ 251
+ ],
+ "score": 0.8,
+ "content": "c",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 180,
+ 240,
+ 263,
+ 254
+ ],
+ "score": 1.0,
+ "content": "-th column vector of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 263,
+ 242,
+ 272,
+ 251
+ ],
+ "score": 0.84,
+ "content": "X",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 273,
+ 240,
+ 354,
+ 254
+ ],
+ "score": 1.0,
+ "content": ". Similarly, we have",
+ "type": "text"
+ }
+ ],
+ "index": 11
+ }
+ ],
+ "index": 11
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 207,
+ 256,
+ 403,
+ 290
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 207,
+ 256,
+ 403,
+ 290
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 207,
+ 256,
+ 403,
+ 290
+ ],
+ "score": 0.93,
+ "content": "d _ { \\mathcal { M } } ^ { 2 } ( \\sigma ( X ) ) = \\sum _ { c = 1 } ^ { C } \\left( \\| X _ { \\cdot c } ^ { + } \\| ^ { 2 } - \\sum _ { n = 1 } ^ { M } \\langle X _ { \\cdot c } ^ { + } , e _ { n } \\rangle ^ { 2 } \\right) ,",
+ "type": "interline_equation",
+ "image_path": "48fd613479f3c412aeb06722190f64eaf2d54146933de23b64847c8812fb608a.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 12.5,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 207,
+ 256,
+ 403,
+ 273.0
+ ],
+ "spans": [],
+ "index": 12
+ },
+ {
+ "bbox": [
+ 207,
+ 273.0,
+ 403,
+ 290.0
+ ],
+ "spans": [],
+ "index": 13
+ }
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 294,
+ 504,
+ 317
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 291,
+ 506,
+ 308
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 291,
+ 176,
+ 308
+ ],
+ "score": 1.0,
+ "content": "where we denote",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 177,
+ 294,
+ 282,
+ 307
+ ],
+ "score": 0.93,
+ "content": "\\sigma ( X ) = ( X _ { n c } ^ { + } ) _ { n \\in [ N ] , c \\in [ C ] }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 283,
+ 291,
+ 506,
+ 308
+ ],
+ "score": 1.0,
+ "content": "in shorthand. Therefore, the inequality follow from the",
+ "type": "text"
+ }
+ ],
+ "index": 14
+ },
+ {
+ "bbox": [
+ 105,
+ 304,
+ 504,
+ 318
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 304,
+ 180,
+ 318
+ ],
+ "score": 1.0,
+ "content": "following lemma.",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 497,
+ 308,
+ 504,
+ 314
+ ],
+ "score": 0.688,
+ "content": "□",
+ "type": "text"
+ }
+ ],
+ "index": 15
+ }
+ ],
+ "index": 14.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 105,
+ 321,
+ 505,
+ 360
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 101,
+ 321,
+ 506,
+ 360
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 101,
+ 327,
+ 128,
+ 360
+ ],
+ "score": 1.0,
+ "content": "fying wher",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 105,
+ 321,
+ 168,
+ 336
+ ],
+ "score": 1.0,
+ "content": "Lemma 4. Let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 128,
+ 336,
+ 160,
+ 347
+ ],
+ "score": 0.89,
+ "content": "v _ { m } \\geq 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 169,
+ 322,
+ 202,
+ 333
+ ],
+ "score": 0.91,
+ "content": "\\boldsymbol { x } \\in \\mathbb { R } ^ { N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 188,
+ 336,
+ 226,
+ 348
+ ],
+ "score": 0.9,
+ "content": "m \\in [ M ]",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 203,
+ 321,
+ 221,
+ 336
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 221,
+ 322,
+ 293,
+ 335
+ ],
+ "score": 0.91,
+ "content": "v _ { 1 } , \\hdots , v _ { M } \\in \\mathbb { R } ^ { N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 245,
+ 327,
+ 290,
+ 360
+ ],
+ "score": 1.0,
+ "content": "en, we have .",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 291,
+ 334,
+ 504,
+ 349
+ ],
+ "score": 0.89,
+ "content": "\\begin{array} { r } { \\| \\ b x \\| ^ { 2 } - \\sum _ { m = 1 } ^ { M } \\langle \\ b x , \\ b v _ { m } \\rangle ^ { 2 } \\geq \\| \\ b x ^ { + } \\| ^ { 2 } - \\sum _ { m = 1 } ^ { M } \\langle \\ b x ^ { + } , \\ b v _ { m } \\rangle ^ { 2 } } \\end{array}",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 294,
+ 321,
+ 411,
+ 336
+ ],
+ "score": 1.0,
+ "content": "be orthonormal vectors (i.e.,",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 411,
+ 323,
+ 479,
+ 335
+ ],
+ "score": 0.9,
+ "content": "\\langle v _ { m } , v _ { n } \\rangle = \\delta _ { m n } )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 479,
+ 321,
+ 506,
+ 336
+ ],
+ "score": 1.0,
+ "content": "satis-",
+ "type": "text"
+ }
+ ],
+ "index": 16
+ },
+ {
+ "bbox": [
+ 133,
+ 347,
+ 245,
+ 359
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 133,
+ 347,
+ 204,
+ 359
+ ],
+ "score": 0.89,
+ "content": "x ^ { + } : = \\operatorname* { m a x } ( x , 0 )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 219,
+ 348,
+ 245,
+ 358
+ ],
+ "score": 0.9,
+ "content": "x \\in \\mathbb { R }",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 17
+ }
+ ],
+ "index": 16.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 371,
+ 506,
+ 407
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 114,
+ 363,
+ 511,
+ 396
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 114,
+ 363,
+ 131,
+ 396
+ ],
+ "score": 1.0,
+ "content": "roof.",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 146,
+ 363,
+ 178,
+ 396
+ ],
+ "score": 1.0,
+ "content": "he value",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 178,
+ 371,
+ 272,
+ 386
+ ],
+ "score": 0.93,
+ "content": "\\begin{array} { r } { \\| y \\| ^ { 2 } - \\sum _ { m = 1 } ^ { M } \\langle y , u _ { m } \\rangle ^ { 2 } } \\end{array}",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 272,
+ 363,
+ 453,
+ 396
+ ],
+ "score": 1.0,
+ "content": "is invariant under simultaneous coordinate pe",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 460,
+ 363,
+ 511,
+ 396
+ ],
+ "score": 1.0,
+ "content": "mutation of",
+ "type": "text"
+ }
+ ],
+ "index": 18
+ },
+ {
+ "bbox": [
+ 106,
+ 385,
+ 460,
+ 396
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 387,
+ 113,
+ 396
+ ],
+ "score": 0.79,
+ "content": "y",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 131,
+ 385,
+ 145,
+ 395
+ ],
+ "score": 0.85,
+ "content": "u _ { m }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 453,
+ 387,
+ 460,
+ 394
+ ],
+ "score": 0.76,
+ "content": "x",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 19
+ },
+ {
+ "bbox": [
+ 106,
+ 395,
+ 407,
+ 407
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 396,
+ 272,
+ 407
+ ],
+ "score": 0.91,
+ "content": "x _ { 1 } \\leq \\ldots \\leq x _ { L } < 0 \\leq x _ { L + 1 } \\leq \\cdot \\cdot \\cdot \\leq x _ { N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 272,
+ 395,
+ 311,
+ 407
+ ],
+ "score": 1.0,
+ "content": "for some",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 312,
+ 396,
+ 342,
+ 406
+ ],
+ "score": 0.91,
+ "content": "L \\leq N",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 342,
+ 395,
+ 407,
+ 407
+ ],
+ "score": 1.0,
+ "content": ". Then, we have",
+ "type": "text"
+ }
+ ],
+ "index": 20
+ }
+ ],
+ "index": 19
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 253,
+ 410,
+ 358,
+ 444
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 253,
+ 410,
+ 358,
+ 444
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 253,
+ 410,
+ 358,
+ 444
+ ],
+ "score": 0.94,
+ "content": "\\| x \\| ^ { 2 } - \\| x ^ { + } \\| ^ { 2 } = \\sum _ { n = 1 } ^ { L } x _ { n } ^ { 2 } .",
+ "type": "interline_equation",
+ "image_path": "68f7d8b8c5d76abce597c2e00354310b9c41ec8fdf5c4ec93f6ccaa43c8842c1.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 21.5,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 253,
+ 410,
+ 358,
+ 427.0
+ ],
+ "spans": [],
+ "index": 21
+ },
+ {
+ "bbox": [
+ 253,
+ 427.0,
+ 358,
+ 444.0
+ ],
+ "spans": [],
+ "index": 22
+ }
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 105,
+ 447,
+ 505,
+ 470
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 444,
+ 505,
+ 461
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 444,
+ 133,
+ 461
+ ],
+ "score": 1.0,
+ "content": "When",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 133,
+ 448,
+ 162,
+ 457
+ ],
+ "score": 0.89,
+ "content": "L = 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 163,
+ 444,
+ 478,
+ 461
+ ],
+ "score": 1.0,
+ "content": ", the sum in the right hand side is treated as 0. On the other hand, writing as",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 479,
+ 448,
+ 505,
+ 459
+ ],
+ "score": 0.83,
+ "content": "v _ { m } =",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 23
+ },
+ {
+ "bbox": [
+ 107,
+ 458,
+ 258,
+ 471
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 458,
+ 155,
+ 471
+ ],
+ "score": 0.92,
+ "content": "( v _ { n m } ) _ { n \\in [ N ] }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 155,
+ 458,
+ 258,
+ 470
+ ],
+ "score": 1.0,
+ "content": ", direct calculation shows",
+ "type": "text"
+ }
+ ],
+ "index": 24
+ }
+ ],
+ "index": 23.5
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 133,
+ 473,
+ 477,
+ 514
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 133,
+ 473,
+ 477,
+ 514
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 133,
+ 473,
+ 477,
+ 514
+ ],
+ "score": 0.94,
+ "content": "\\sum _ { m = 1 } ^ { M } \\langle x , v _ { m } \\rangle ^ { 2 } - \\langle x ^ { + } , v _ { m } \\rangle ^ { 2 } = \\sum _ { m = 1 } ^ { M } \\left( \\left( \\sum _ { n = 1 } ^ { L } x _ { n } v _ { n m } \\right) ^ { 2 } - 2 \\sum _ { n = 1 } ^ { L } \\sum _ { l = L + 1 } ^ { N } x _ { n } x _ { l } v _ { n m } v _ { l m } \\right) .",
+ "type": "interline_equation",
+ "image_path": "ea22ca227c9ec044c61e3eba97e9317ef90575c82fd8b35c76fa72f45373c309.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 26,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 133,
+ 473,
+ 477,
+ 486.6666666666667
+ ],
+ "spans": [],
+ "index": 25
+ },
+ {
+ "bbox": [
+ 133,
+ 486.6666666666667,
+ 477,
+ 500.33333333333337
+ ],
+ "spans": [],
+ "index": 26
+ },
+ {
+ "bbox": [
+ 133,
+ 500.33333333333337,
+ 477,
+ 514.0
+ ],
+ "spans": [],
+ "index": 27
+ }
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 104,
+ 516,
+ 505,
+ 540
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 104,
+ 514,
+ 506,
+ 531
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 514,
+ 122,
+ 531
+ ],
+ "score": 1.0,
+ "content": "Let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 122,
+ 516,
+ 237,
+ 529
+ ],
+ "score": 0.9,
+ "content": "I _ { m } : = \\{ n \\in [ N ] \\mid v _ { n m } > 0 \\}",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 238,
+ 514,
+ 307,
+ 531
+ ],
+ "score": 1.0,
+ "content": "be the support of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 308,
+ 519,
+ 321,
+ 528
+ ],
+ "score": 0.86,
+ "content": "v _ { m }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 321,
+ 514,
+ 335,
+ 531
+ ],
+ "score": 1.0,
+ "content": "for",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 336,
+ 517,
+ 374,
+ 529
+ ],
+ "score": 0.92,
+ "content": "m \\in [ M ]",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 374,
+ 514,
+ 438,
+ 531
+ ],
+ "score": 1.0,
+ "content": ". We note that if",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 438,
+ 516,
+ 501,
+ 529
+ ],
+ "score": 0.91,
+ "content": "m \\neq m ^ { \\prime } \\in [ M ]",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 501,
+ 514,
+ 506,
+ 531
+ ],
+ "score": 1.0,
+ "content": ",",
+ "type": "text"
+ }
+ ],
+ "index": 28
+ },
+ {
+ "bbox": [
+ 105,
+ 527,
+ 381,
+ 540
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 527,
+ 141,
+ 540
+ ],
+ "score": 1.0,
+ "content": "we have",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 142,
+ 529,
+ 199,
+ 539
+ ],
+ "score": 0.91,
+ "content": "{ \\cal I } _ { m } \\cap { \\cal I } _ { m ^ { \\prime } } = \\emptyset",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 199,
+ 527,
+ 284,
+ 540
+ ],
+ "score": 1.0,
+ "content": "since if there existed",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 285,
+ 529,
+ 341,
+ 539
+ ],
+ "score": 0.91,
+ "content": "n \\in I _ { m } \\cap I _ { m ^ { \\prime } }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 341,
+ 527,
+ 381,
+ 540
+ ],
+ "score": 1.0,
+ "content": ", we have",
+ "type": "text"
+ }
+ ],
+ "index": 29
+ }
+ ],
+ "index": 28.5
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 239,
+ 543,
+ 370,
+ 557
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 239,
+ 543,
+ 370,
+ 557
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 239,
+ 543,
+ 370,
+ 557
+ ],
+ "score": 0.89,
+ "content": "0 = \\left. v _ { m } , v _ { m ^ { \\prime } } \\right. \\geq v _ { n m } v _ { n m ^ { \\prime } } > 0 ,",
+ "type": "interline_equation",
+ "image_path": "cec6e1c89fcd9f94147f760c9fab60b687a9fb58a22ddf08e549e333eb77ee26.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 30,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 239,
+ 543,
+ 370,
+ 557
+ ],
+ "spans": [],
+ "index": 30
+ }
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 560,
+ 244,
+ 571
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 558,
+ 245,
+ 573
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 558,
+ 245,
+ 573
+ ],
+ "score": 1.0,
+ "content": "which is contradictory. Therefore,",
+ "type": "text"
+ }
+ ],
+ "index": 31
+ }
+ ],
+ "index": 31
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 105,
+ 574,
+ 505,
+ 732
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 574,
+ 500,
+ 732
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 574,
+ 500,
+ 732
+ ],
+ "score": 0.86,
+ "content": "\\begin{array} { r l r } { \\displaystyle \\sum _ { m = 1 } ^ { N } \\left( \\displaystyle \\sum _ { n = 1 } ^ { L } x _ { n } v _ { n m } \\right) ^ { 2 } = \\displaystyle \\sum _ { m = 1 } ^ { N } \\left( \\displaystyle \\sum _ { n \\in L _ { I ^ { n } } \\cap [ L ] } x _ { n } v _ { n m } \\right) ^ { 2 } } & { } & \\\\ { \\displaystyle } & { \\leq \\displaystyle \\sum _ { m = 1 } ^ { N } \\left( \\displaystyle \\sum _ { n \\in L _ { I ^ { n } } \\cap [ L ] } x _ { n } ^ { 2 } \\right) \\left( \\displaystyle \\sum _ { n \\in L _ { I ^ { n } } \\cap [ L ] } x _ { n m } ^ { 2 } \\right) } & { \\displaystyle ( \\cdot \\cdot \\mathrm { C a u c h y - S c h w a r z ~ i n e q u a l i t y } ) } \\\\ { \\displaystyle } & { \\displaystyle \\leq \\displaystyle \\sum _ { m = 1 } ^ { N } \\left( \\displaystyle \\sum _ { n \\in L _ { I ^ { n } } \\cap [ L ] } x _ { n } ^ { 2 } \\right) } & { \\displaystyle ( \\cdot \\cdot | v _ { m } | ^ { 2 } = 1 ) } \\\\ { \\displaystyle } & { \\displaystyle \\leq \\displaystyle \\sum _ { n = 1 } ^ { L } x _ { n } ^ { 2 } . } \\end{array}",
+ "type": "interline_equation",
+ "image_path": "e363c202ba9b7ecee3de95f31889b854f5df5a540a429e3c029657ede6d04d16.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 33,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 105,
+ 574,
+ 505,
+ 626.6666666666666
+ ],
+ "spans": [],
+ "index": 32
+ },
+ {
+ "bbox": [
+ 105,
+ 626.6666666666666,
+ 505,
+ 679.3333333333333
+ ],
+ "spans": [],
+ "index": 33
+ },
+ {
+ "bbox": [
+ 105,
+ 679.3333333333333,
+ 505,
+ 731.9999999999999
+ ],
+ "spans": [],
+ "index": 34
+ }
+ ]
+ }
+ ],
+ "page_idx": 14,
+ "page_size": [
+ 612,
+ 792
+ ],
+ "discarded_blocks": [
+ {
+ "type": "discarded",
+ "bbox": [
+ 107,
+ 26,
+ 294,
+ 38
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 25,
+ 294,
+ 38
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 25,
+ 294,
+ 38
+ ],
+ "score": 1.0,
+ "content": "Published as a conference paper at ICLR 2020",
+ "type": "text"
+ }
+ ]
+ }
+ ]
+ },
+ {
+ "type": "discarded",
+ "bbox": [
+ 300,
+ 751,
+ 311,
+ 761
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 299,
+ 750,
+ 312,
+ 764
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 299,
+ 750,
+ 312,
+ 764
+ ],
+ "score": 1.0,
+ "content": "15",
+ "type": "text"
+ }
+ ]
+ }
+ ]
+ }
+ ],
+ "para_blocks": [
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 82,
+ 234,
+ 93
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 81,
+ 234,
+ 95
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 81,
+ 234,
+ 95
+ ],
+ "score": 1.0,
+ "content": "which is further transformed as",
+ "type": "text"
+ }
+ ],
+ "index": 0
+ }
+ ],
+ "index": 0,
+ "bbox_fs": [
+ 106,
+ 81,
+ 234,
+ 95
+ ]
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 213,
+ 97,
+ 395,
+ 240
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 213,
+ 97,
+ 395,
+ 240
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 213,
+ 97,
+ 395,
+ 240
+ ],
+ "score": 0.95,
+ "content": "\\begin{array} { l } { \\displaystyle d _ { M } ^ { 2 } ( X ) = \\sum _ { n = M + 1 } ^ { N } \\left\\| \\sum _ { c = 1 } ^ { C } a _ { n c } e _ { c } ^ { t } \\right\\| ^ { 2 } } \\\\ { = \\displaystyle \\sum _ { n = M + 1 } ^ { N } \\sum _ { c = 1 } ^ { C } a _ { n c } ^ { 2 } } \\\\ { = \\displaystyle \\sum _ { c = 1 } ^ { C } \\left( \\sum _ { n = 1 } ^ { N } a _ { n c } ^ { 2 } - \\sum _ { n = 1 } ^ { M } a _ { n c } ^ { 2 } \\right) } \\\\ { = \\displaystyle \\sum _ { c = 1 } ^ { C } \\left( \\| X _ { \\mathrm { - c } } \\| ^ { 2 } - \\sum _ { n = 1 } ^ { M } ( X _ { \\mathrm { - c } } , e _ { n } ) ^ { 2 } \\right) , } \\end{array}",
+ "type": "interline_equation",
+ "image_path": "3e54bc240f5d9875acd41a1878664feda1310273d70a7eb5f0d1b1f678be3ff5.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 5.5,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 213,
+ 97,
+ 395,
+ 111.3
+ ],
+ "spans": [],
+ "index": 1
+ },
+ {
+ "bbox": [
+ 213,
+ 111.3,
+ 395,
+ 125.6
+ ],
+ "spans": [],
+ "index": 2
+ },
+ {
+ "bbox": [
+ 213,
+ 125.6,
+ 395,
+ 139.9
+ ],
+ "spans": [],
+ "index": 3
+ },
+ {
+ "bbox": [
+ 213,
+ 139.9,
+ 395,
+ 154.20000000000002
+ ],
+ "spans": [],
+ "index": 4
+ },
+ {
+ "bbox": [
+ 213,
+ 154.20000000000002,
+ 395,
+ 168.50000000000003
+ ],
+ "spans": [],
+ "index": 5
+ },
+ {
+ "bbox": [
+ 213,
+ 168.50000000000003,
+ 395,
+ 182.80000000000004
+ ],
+ "spans": [],
+ "index": 6
+ },
+ {
+ "bbox": [
+ 213,
+ 182.80000000000004,
+ 395,
+ 197.10000000000005
+ ],
+ "spans": [],
+ "index": 7
+ },
+ {
+ "bbox": [
+ 213,
+ 197.10000000000005,
+ 395,
+ 211.40000000000006
+ ],
+ "spans": [],
+ "index": 8
+ },
+ {
+ "bbox": [
+ 213,
+ 211.40000000000006,
+ 395,
+ 225.70000000000007
+ ],
+ "spans": [],
+ "index": 9
+ },
+ {
+ "bbox": [
+ 213,
+ 225.70000000000007,
+ 395,
+ 240.00000000000009
+ ],
+ "spans": [],
+ "index": 10
+ }
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 241,
+ 354,
+ 253
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 240,
+ 354,
+ 254
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 240,
+ 133,
+ 254
+ ],
+ "score": 1.0,
+ "content": "where",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 133,
+ 242,
+ 149,
+ 252
+ ],
+ "score": 0.87,
+ "content": "X _ { \\cdot c }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 150,
+ 240,
+ 174,
+ 254
+ ],
+ "score": 1.0,
+ "content": "is the",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 174,
+ 243,
+ 180,
+ 251
+ ],
+ "score": 0.8,
+ "content": "c",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 180,
+ 240,
+ 263,
+ 254
+ ],
+ "score": 1.0,
+ "content": "-th column vector of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 263,
+ 242,
+ 272,
+ 251
+ ],
+ "score": 0.84,
+ "content": "X",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 273,
+ 240,
+ 354,
+ 254
+ ],
+ "score": 1.0,
+ "content": ". Similarly, we have",
+ "type": "text"
+ }
+ ],
+ "index": 11
+ }
+ ],
+ "index": 11,
+ "bbox_fs": [
+ 105,
+ 240,
+ 354,
+ 254
+ ]
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 207,
+ 256,
+ 403,
+ 290
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 207,
+ 256,
+ 403,
+ 290
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 207,
+ 256,
+ 403,
+ 290
+ ],
+ "score": 0.93,
+ "content": "d _ { \\mathcal { M } } ^ { 2 } ( \\sigma ( X ) ) = \\sum _ { c = 1 } ^ { C } \\left( \\| X _ { \\cdot c } ^ { + } \\| ^ { 2 } - \\sum _ { n = 1 } ^ { M } \\langle X _ { \\cdot c } ^ { + } , e _ { n } \\rangle ^ { 2 } \\right) ,",
+ "type": "interline_equation",
+ "image_path": "48fd613479f3c412aeb06722190f64eaf2d54146933de23b64847c8812fb608a.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 12.5,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 207,
+ 256,
+ 403,
+ 273.0
+ ],
+ "spans": [],
+ "index": 12
+ },
+ {
+ "bbox": [
+ 207,
+ 273.0,
+ 403,
+ 290.0
+ ],
+ "spans": [],
+ "index": 13
+ }
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 294,
+ 504,
+ 317
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 291,
+ 506,
+ 308
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 291,
+ 176,
+ 308
+ ],
+ "score": 1.0,
+ "content": "where we denote",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 177,
+ 294,
+ 282,
+ 307
+ ],
+ "score": 0.93,
+ "content": "\\sigma ( X ) = ( X _ { n c } ^ { + } ) _ { n \\in [ N ] , c \\in [ C ] }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 283,
+ 291,
+ 506,
+ 308
+ ],
+ "score": 1.0,
+ "content": "in shorthand. Therefore, the inequality follow from the",
+ "type": "text"
+ }
+ ],
+ "index": 14
+ },
+ {
+ "bbox": [
+ 105,
+ 304,
+ 504,
+ 318
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 304,
+ 180,
+ 318
+ ],
+ "score": 1.0,
+ "content": "following lemma.",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 497,
+ 308,
+ 504,
+ 314
+ ],
+ "score": 0.688,
+ "content": "□",
+ "type": "text"
+ }
+ ],
+ "index": 15
+ }
+ ],
+ "index": 14.5,
+ "bbox_fs": [
+ 105,
+ 291,
+ 506,
+ 318
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 105,
+ 321,
+ 505,
+ 360
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 101,
+ 321,
+ 506,
+ 360
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 101,
+ 327,
+ 128,
+ 360
+ ],
+ "score": 1.0,
+ "content": "fying wher",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 105,
+ 321,
+ 168,
+ 336
+ ],
+ "score": 1.0,
+ "content": "Lemma 4. Let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 128,
+ 336,
+ 160,
+ 347
+ ],
+ "score": 0.89,
+ "content": "v _ { m } \\geq 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 169,
+ 322,
+ 202,
+ 333
+ ],
+ "score": 0.91,
+ "content": "\\boldsymbol { x } \\in \\mathbb { R } ^ { N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 188,
+ 336,
+ 226,
+ 348
+ ],
+ "score": 0.9,
+ "content": "m \\in [ M ]",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 203,
+ 321,
+ 221,
+ 336
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 221,
+ 322,
+ 293,
+ 335
+ ],
+ "score": 0.91,
+ "content": "v _ { 1 } , \\hdots , v _ { M } \\in \\mathbb { R } ^ { N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 245,
+ 327,
+ 290,
+ 360
+ ],
+ "score": 1.0,
+ "content": "en, we have .",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 291,
+ 334,
+ 504,
+ 349
+ ],
+ "score": 0.89,
+ "content": "\\begin{array} { r } { \\| \\ b x \\| ^ { 2 } - \\sum _ { m = 1 } ^ { M } \\langle \\ b x , \\ b v _ { m } \\rangle ^ { 2 } \\geq \\| \\ b x ^ { + } \\| ^ { 2 } - \\sum _ { m = 1 } ^ { M } \\langle \\ b x ^ { + } , \\ b v _ { m } \\rangle ^ { 2 } } \\end{array}",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 294,
+ 321,
+ 411,
+ 336
+ ],
+ "score": 1.0,
+ "content": "be orthonormal vectors (i.e.,",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 411,
+ 323,
+ 479,
+ 335
+ ],
+ "score": 0.9,
+ "content": "\\langle v _ { m } , v _ { n } \\rangle = \\delta _ { m n } )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 479,
+ 321,
+ 506,
+ 336
+ ],
+ "score": 1.0,
+ "content": "satis-",
+ "type": "text"
+ }
+ ],
+ "index": 16
+ },
+ {
+ "bbox": [
+ 133,
+ 347,
+ 245,
+ 359
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 133,
+ 347,
+ 204,
+ 359
+ ],
+ "score": 0.89,
+ "content": "x ^ { + } : = \\operatorname* { m a x } ( x , 0 )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 219,
+ 348,
+ 245,
+ 358
+ ],
+ "score": 0.9,
+ "content": "x \\in \\mathbb { R }",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 17
+ }
+ ],
+ "index": 16.5,
+ "bbox_fs": [
+ 101,
+ 321,
+ 506,
+ 360
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 371,
+ 506,
+ 407
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 114,
+ 363,
+ 511,
+ 396
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 114,
+ 363,
+ 131,
+ 396
+ ],
+ "score": 1.0,
+ "content": "roof.",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 146,
+ 363,
+ 178,
+ 396
+ ],
+ "score": 1.0,
+ "content": "he value",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 178,
+ 371,
+ 272,
+ 386
+ ],
+ "score": 0.93,
+ "content": "\\begin{array} { r } { \\| y \\| ^ { 2 } - \\sum _ { m = 1 } ^ { M } \\langle y , u _ { m } \\rangle ^ { 2 } } \\end{array}",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 272,
+ 363,
+ 453,
+ 396
+ ],
+ "score": 1.0,
+ "content": "is invariant under simultaneous coordinate pe",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 460,
+ 363,
+ 511,
+ 396
+ ],
+ "score": 1.0,
+ "content": "mutation of",
+ "type": "text"
+ }
+ ],
+ "index": 18
+ },
+ {
+ "bbox": [
+ 106,
+ 385,
+ 460,
+ 396
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 387,
+ 113,
+ 396
+ ],
+ "score": 0.79,
+ "content": "y",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 131,
+ 385,
+ 145,
+ 395
+ ],
+ "score": 0.85,
+ "content": "u _ { m }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 453,
+ 387,
+ 460,
+ 394
+ ],
+ "score": 0.76,
+ "content": "x",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 19
+ },
+ {
+ "bbox": [
+ 106,
+ 395,
+ 407,
+ 407
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 396,
+ 272,
+ 407
+ ],
+ "score": 0.91,
+ "content": "x _ { 1 } \\leq \\ldots \\leq x _ { L } < 0 \\leq x _ { L + 1 } \\leq \\cdot \\cdot \\cdot \\leq x _ { N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 272,
+ 395,
+ 311,
+ 407
+ ],
+ "score": 1.0,
+ "content": "for some",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 312,
+ 396,
+ 342,
+ 406
+ ],
+ "score": 0.91,
+ "content": "L \\leq N",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 342,
+ 395,
+ 407,
+ 407
+ ],
+ "score": 1.0,
+ "content": ". Then, we have",
+ "type": "text"
+ }
+ ],
+ "index": 20
+ }
+ ],
+ "index": 19,
+ "bbox_fs": [
+ 106,
+ 363,
+ 511,
+ 407
+ ]
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 253,
+ 410,
+ 358,
+ 444
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 253,
+ 410,
+ 358,
+ 444
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 253,
+ 410,
+ 358,
+ 444
+ ],
+ "score": 0.94,
+ "content": "\\| x \\| ^ { 2 } - \\| x ^ { + } \\| ^ { 2 } = \\sum _ { n = 1 } ^ { L } x _ { n } ^ { 2 } .",
+ "type": "interline_equation",
+ "image_path": "68f7d8b8c5d76abce597c2e00354310b9c41ec8fdf5c4ec93f6ccaa43c8842c1.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 21.5,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 253,
+ 410,
+ 358,
+ 427.0
+ ],
+ "spans": [],
+ "index": 21
+ },
+ {
+ "bbox": [
+ 253,
+ 427.0,
+ 358,
+ 444.0
+ ],
+ "spans": [],
+ "index": 22
+ }
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 105,
+ 447,
+ 505,
+ 470
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 444,
+ 505,
+ 461
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 444,
+ 133,
+ 461
+ ],
+ "score": 1.0,
+ "content": "When",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 133,
+ 448,
+ 162,
+ 457
+ ],
+ "score": 0.89,
+ "content": "L = 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 163,
+ 444,
+ 478,
+ 461
+ ],
+ "score": 1.0,
+ "content": ", the sum in the right hand side is treated as 0. On the other hand, writing as",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 479,
+ 448,
+ 505,
+ 459
+ ],
+ "score": 0.83,
+ "content": "v _ { m } =",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 23
+ },
+ {
+ "bbox": [
+ 107,
+ 458,
+ 258,
+ 471
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 458,
+ 155,
+ 471
+ ],
+ "score": 0.92,
+ "content": "( v _ { n m } ) _ { n \\in [ N ] }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 155,
+ 458,
+ 258,
+ 470
+ ],
+ "score": 1.0,
+ "content": ", direct calculation shows",
+ "type": "text"
+ }
+ ],
+ "index": 24
+ }
+ ],
+ "index": 23.5,
+ "bbox_fs": [
+ 105,
+ 444,
+ 505,
+ 471
+ ]
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 133,
+ 473,
+ 477,
+ 514
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 133,
+ 473,
+ 477,
+ 514
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 133,
+ 473,
+ 477,
+ 514
+ ],
+ "score": 0.94,
+ "content": "\\sum _ { m = 1 } ^ { M } \\langle x , v _ { m } \\rangle ^ { 2 } - \\langle x ^ { + } , v _ { m } \\rangle ^ { 2 } = \\sum _ { m = 1 } ^ { M } \\left( \\left( \\sum _ { n = 1 } ^ { L } x _ { n } v _ { n m } \\right) ^ { 2 } - 2 \\sum _ { n = 1 } ^ { L } \\sum _ { l = L + 1 } ^ { N } x _ { n } x _ { l } v _ { n m } v _ { l m } \\right) .",
+ "type": "interline_equation",
+ "image_path": "ea22ca227c9ec044c61e3eba97e9317ef90575c82fd8b35c76fa72f45373c309.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 26,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 133,
+ 473,
+ 477,
+ 486.6666666666667
+ ],
+ "spans": [],
+ "index": 25
+ },
+ {
+ "bbox": [
+ 133,
+ 486.6666666666667,
+ 477,
+ 500.33333333333337
+ ],
+ "spans": [],
+ "index": 26
+ },
+ {
+ "bbox": [
+ 133,
+ 500.33333333333337,
+ 477,
+ 514.0
+ ],
+ "spans": [],
+ "index": 27
+ }
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 104,
+ 516,
+ 505,
+ 540
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 104,
+ 514,
+ 506,
+ 531
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 514,
+ 122,
+ 531
+ ],
+ "score": 1.0,
+ "content": "Let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 122,
+ 516,
+ 237,
+ 529
+ ],
+ "score": 0.9,
+ "content": "I _ { m } : = \\{ n \\in [ N ] \\mid v _ { n m } > 0 \\}",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 238,
+ 514,
+ 307,
+ 531
+ ],
+ "score": 1.0,
+ "content": "be the support of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 308,
+ 519,
+ 321,
+ 528
+ ],
+ "score": 0.86,
+ "content": "v _ { m }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 321,
+ 514,
+ 335,
+ 531
+ ],
+ "score": 1.0,
+ "content": "for",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 336,
+ 517,
+ 374,
+ 529
+ ],
+ "score": 0.92,
+ "content": "m \\in [ M ]",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 374,
+ 514,
+ 438,
+ 531
+ ],
+ "score": 1.0,
+ "content": ". We note that if",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 438,
+ 516,
+ 501,
+ 529
+ ],
+ "score": 0.91,
+ "content": "m \\neq m ^ { \\prime } \\in [ M ]",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 501,
+ 514,
+ 506,
+ 531
+ ],
+ "score": 1.0,
+ "content": ",",
+ "type": "text"
+ }
+ ],
+ "index": 28
+ },
+ {
+ "bbox": [
+ 105,
+ 527,
+ 381,
+ 540
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 527,
+ 141,
+ 540
+ ],
+ "score": 1.0,
+ "content": "we have",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 142,
+ 529,
+ 199,
+ 539
+ ],
+ "score": 0.91,
+ "content": "{ \\cal I } _ { m } \\cap { \\cal I } _ { m ^ { \\prime } } = \\emptyset",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 199,
+ 527,
+ 284,
+ 540
+ ],
+ "score": 1.0,
+ "content": "since if there existed",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 285,
+ 529,
+ 341,
+ 539
+ ],
+ "score": 0.91,
+ "content": "n \\in I _ { m } \\cap I _ { m ^ { \\prime } }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 341,
+ 527,
+ 381,
+ 540
+ ],
+ "score": 1.0,
+ "content": ", we have",
+ "type": "text"
+ }
+ ],
+ "index": 29
+ }
+ ],
+ "index": 28.5,
+ "bbox_fs": [
+ 104,
+ 514,
+ 506,
+ 540
+ ]
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 239,
+ 543,
+ 370,
+ 557
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 239,
+ 543,
+ 370,
+ 557
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 239,
+ 543,
+ 370,
+ 557
+ ],
+ "score": 0.89,
+ "content": "0 = \\left. v _ { m } , v _ { m ^ { \\prime } } \\right. \\geq v _ { n m } v _ { n m ^ { \\prime } } > 0 ,",
+ "type": "interline_equation",
+ "image_path": "cec6e1c89fcd9f94147f760c9fab60b687a9fb58a22ddf08e549e333eb77ee26.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 30,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 239,
+ 543,
+ 370,
+ 557
+ ],
+ "spans": [],
+ "index": 30
+ }
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 560,
+ 244,
+ 571
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 558,
+ 245,
+ 573
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 558,
+ 245,
+ 573
+ ],
+ "score": 1.0,
+ "content": "which is contradictory. Therefore,",
+ "type": "text"
+ }
+ ],
+ "index": 31
+ }
+ ],
+ "index": 31,
+ "bbox_fs": [
+ 106,
+ 558,
+ 245,
+ 573
+ ]
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 105,
+ 574,
+ 505,
+ 732
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 574,
+ 500,
+ 732
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 574,
+ 500,
+ 732
+ ],
+ "score": 0.86,
+ "content": "\\begin{array} { r l r } { \\displaystyle \\sum _ { m = 1 } ^ { N } \\left( \\displaystyle \\sum _ { n = 1 } ^ { L } x _ { n } v _ { n m } \\right) ^ { 2 } = \\displaystyle \\sum _ { m = 1 } ^ { N } \\left( \\displaystyle \\sum _ { n \\in L _ { I ^ { n } } \\cap [ L ] } x _ { n } v _ { n m } \\right) ^ { 2 } } & { } & \\\\ { \\displaystyle } & { \\leq \\displaystyle \\sum _ { m = 1 } ^ { N } \\left( \\displaystyle \\sum _ { n \\in L _ { I ^ { n } } \\cap [ L ] } x _ { n } ^ { 2 } \\right) \\left( \\displaystyle \\sum _ { n \\in L _ { I ^ { n } } \\cap [ L ] } x _ { n m } ^ { 2 } \\right) } & { \\displaystyle ( \\cdot \\cdot \\mathrm { C a u c h y - S c h w a r z ~ i n e q u a l i t y } ) } \\\\ { \\displaystyle } & { \\displaystyle \\leq \\displaystyle \\sum _ { m = 1 } ^ { N } \\left( \\displaystyle \\sum _ { n \\in L _ { I ^ { n } } \\cap [ L ] } x _ { n } ^ { 2 } \\right) } & { \\displaystyle ( \\cdot \\cdot | v _ { m } | ^ { 2 } = 1 ) } \\\\ { \\displaystyle } & { \\displaystyle \\leq \\displaystyle \\sum _ { n = 1 } ^ { L } x _ { n } ^ { 2 } . } \\end{array}",
+ "type": "interline_equation",
+ "image_path": "e363c202ba9b7ecee3de95f31889b854f5df5a540a429e3c029657ede6d04d16.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 33,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 105,
+ 574,
+ 505,
+ 626.6666666666666
+ ],
+ "spans": [],
+ "index": 32
+ },
+ {
+ "bbox": [
+ 105,
+ 626.6666666666666,
+ 505,
+ 679.3333333333333
+ ],
+ "spans": [],
+ "index": 33
+ },
+ {
+ "bbox": [
+ 105,
+ 679.3333333333333,
+ 505,
+ 731.9999999999999
+ ],
+ "spans": [],
+ "index": 34
+ }
+ ]
+ }
+ ]
+ },
+ {
+ "preproc_blocks": [
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 82,
+ 505,
+ 117
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 81,
+ 505,
+ 96
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 81,
+ 198,
+ 96
+ ],
+ "score": 1.0,
+ "content": "We used the fact that",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 198,
+ 83,
+ 211,
+ 93
+ ],
+ "score": 0.87,
+ "content": "I _ { m }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 212,
+ 81,
+ 288,
+ 96
+ ],
+ "score": 1.0,
+ "content": "’s are disjoint and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 288,
+ 83,
+ 330,
+ 93
+ ],
+ "score": 0.9,
+ "content": "v _ { n m } = 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 330,
+ 81,
+ 341,
+ 96
+ ],
+ "score": 1.0,
+ "content": "if",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 342,
+ 83,
+ 392,
+ 94
+ ],
+ "score": 0.9,
+ "content": "n \\not \\in \\cup _ { m } I _ { m }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 392,
+ 81,
+ 505,
+ 96
+ ],
+ "score": 1.0,
+ "content": "in the first equality above.",
+ "type": "text"
+ }
+ ],
+ "index": 0
+ },
+ {
+ "bbox": [
+ 105,
+ 92,
+ 506,
+ 108
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 92,
+ 176,
+ 108
+ ],
+ "score": 1.0,
+ "content": "Further, we have",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 177,
+ 94,
+ 249,
+ 105
+ ],
+ "score": 0.91,
+ "content": "x _ { n } x _ { l } v _ { n m } v _ { l m } \\leq 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 249,
+ 92,
+ 264,
+ 108
+ ],
+ "score": 1.0,
+ "content": "for",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 265,
+ 94,
+ 312,
+ 105
+ ],
+ "score": 0.92,
+ "content": "1 \\le n \\le L",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 313,
+ 92,
+ 331,
+ 108
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 331,
+ 94,
+ 397,
+ 105
+ ],
+ "score": 0.9,
+ "content": "L + 1 \\le l \\le N",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 398,
+ 92,
+ 478,
+ 108
+ ],
+ "score": 1.0,
+ "content": "by the definition of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 479,
+ 94,
+ 487,
+ 104
+ ],
+ "score": 0.81,
+ "content": "L",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 487,
+ 92,
+ 506,
+ 108
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ }
+ ],
+ "index": 1
+ },
+ {
+ "bbox": [
+ 106,
+ 104,
+ 357,
+ 117
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 104,
+ 177,
+ 117
+ ],
+ "score": 1.0,
+ "content": "non-negativity of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 178,
+ 106,
+ 191,
+ 115
+ ],
+ "score": 0.87,
+ "content": "v _ { m }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 191,
+ 104,
+ 357,
+ 117
+ ],
+ "score": 1.0,
+ "content": ". By combining (1), (2), and (3), we have",
+ "type": "text"
+ }
+ ],
+ "index": 2
+ }
+ ],
+ "index": 1
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 192,
+ 122,
+ 417,
+ 157
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 192,
+ 122,
+ 417,
+ 157
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 192,
+ 122,
+ 417,
+ 157
+ ],
+ "score": 0.94,
+ "content": "\\sum _ { m = 1 } ^ { M } \\langle x , v _ { m } \\rangle ^ { 2 } - \\langle x ^ { + } , v _ { m } \\rangle ^ { 2 } \\leq \\sum _ { n = 1 } ^ { L } x _ { n } ^ { 2 } = \\| x \\| ^ { 2 } - \\| x ^ { + } \\| ^ { 2 } .",
+ "type": "interline_equation",
+ "image_path": "c39fabdc21d6ccc420a7956b6414e1b3de75315041ade8eba193e361d9db97c0.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 3.5,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 192,
+ 122,
+ 417,
+ 139.5
+ ],
+ "spans": [],
+ "index": 3
+ },
+ {
+ "bbox": [
+ 192,
+ 139.5,
+ 417,
+ 157.0
+ ],
+ "spans": [],
+ "index": 4
+ }
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 193,
+ 321,
+ 206
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 194,
+ 320,
+ 207
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 194,
+ 179,
+ 207
+ ],
+ "score": 1.0,
+ "content": "Proof of Theorem",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 180,
+ 195,
+ 185,
+ 204
+ ],
+ "score": 0.37,
+ "content": "^ { l }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 185,
+ 194,
+ 320,
+ 207
+ ],
+ "score": 1.0,
+ "content": ". By Lemma 1, 2, and 3, we have",
+ "type": "text"
+ }
+ ],
+ "index": 5
+ }
+ ],
+ "index": 5
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 168,
+ 213,
+ 439,
+ 380
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 168,
+ 213,
+ 439,
+ 380
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 168,
+ 213,
+ 439,
+ 380
+ ],
+ "score": 0.93,
+ "content": "\\begin{array} { r l } { d _ { M } ( f _ { l } ( X ) ) = d _ { M } ( \\underbrace { \\sigma \\langle \\cdot \\cdot \\sigma ( \\sigma ( P X ) W _ { l } ) W _ { l ^ { 2 } } \\cdot \\cdot \\cdot \\cdot W _ { l H _ { 1 } } \\rangle } _ { H \\mathrm { ~ i n e s } } ) } & { } \\\\ & { \\leq d _ { M } ( \\underbrace { \\sigma \\langle \\cdot \\cdot \\sigma ( \\sigma ( P X ) W _ { l } ) W _ { l ^ { 2 } } \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot } _ { H \\mathrm { ~ i n e s } } ) } \\\\ & { \\leq s _ { i H _ { 1 } - 1 } d _ { M } ( \\underbrace { \\sigma \\langle \\cdot \\cdot \\sigma ( \\sigma ( P X ) W _ { l } ) W _ { l ^ { 2 } } \\cdot \\cdot \\cdot \\cdot \\cdot } _ { H - 1 \\mathrm { ~ i n e s } } ) } \\\\ & { \\cdots } \\\\ & { \\leq \\displaystyle \\left( \\prod _ { i = 1 } ^ { H _ { l } } s _ { i h } \\right) d _ { M } ( P X ) } \\\\ & { \\leq s _ { i d _ { M } ( P X ) } } \\\\ & { \\leq s _ { i H _ { 1 } } d _ { M } ( X ) . } \\end{array}",
+ "type": "interline_equation",
+ "image_path": "15704c0189b83b8ac25bed3cd3d139dd55c80d16850523239a96a4637790c552.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 7,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 168,
+ 213,
+ 439,
+ 268.6666666666667
+ ],
+ "spans": [],
+ "index": 6
+ },
+ {
+ "bbox": [
+ 168,
+ 268.6666666666667,
+ 439,
+ 324.33333333333337
+ ],
+ "spans": [],
+ "index": 7
+ },
+ {
+ "bbox": [
+ 168,
+ 324.33333333333337,
+ 439,
+ 380.00000000000006
+ ],
+ "spans": [],
+ "index": 8
+ }
+ ]
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 108,
+ 415,
+ 260,
+ 428
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 415,
+ 262,
+ 430
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 415,
+ 262,
+ 430
+ ],
+ "score": 1.0,
+ "content": "B PROOF OF PROPOSITION 1",
+ "type": "text"
+ }
+ ],
+ "index": 9
+ }
+ ],
+ "index": 9
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 441,
+ 506,
+ 513
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 440,
+ 506,
+ 454
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 441,
+ 153,
+ 454
+ ],
+ "score": 1.0,
+ "content": "Proof. Let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 153,
+ 442,
+ 222,
+ 453
+ ],
+ "score": 0.92,
+ "content": "{ \\tilde { \\mu } } _ { 1 } \\leq \\dots \\leq { \\tilde { \\mu } } _ { N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 223,
+ 441,
+ 464,
+ 454
+ ],
+ "score": 1.0,
+ "content": "be the eigenvalue of the augmented normalized Laplacian",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 464,
+ 440,
+ 473,
+ 452
+ ],
+ "score": 0.84,
+ "content": "\\tilde { \\Delta }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 474,
+ 441,
+ 506,
+ 454
+ ],
+ "score": 1.0,
+ "content": ", sorted",
+ "type": "text"
+ }
+ ],
+ "index": 10
+ },
+ {
+ "bbox": [
+ 105,
+ 453,
+ 505,
+ 468
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 454,
+ 213,
+ 468
+ ],
+ "score": 1.0,
+ "content": "in ascending order. Since",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 213,
+ 453,
+ 271,
+ 466
+ ],
+ "score": 0.93,
+ "content": "P = I _ { N } - \\tilde { \\Delta }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 271,
+ 454,
+ 361,
+ 468
+ ],
+ "score": 1.0,
+ "content": ", it is enough to show",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 361,
+ 455,
+ 450,
+ 466
+ ],
+ "score": 0.89,
+ "content": "\\tilde { \\mu } _ { 1 } = \\cdot \\cdot \\cdot = \\tilde { \\mu } _ { M } = 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 451,
+ 454,
+ 455,
+ 468
+ ],
+ "score": 1.0,
+ "content": ",",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 455,
+ 455,
+ 501,
+ 466
+ ],
+ "score": 0.81,
+ "content": "{ \\tilde { \\mu } } _ { M + 1 } > 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 502,
+ 454,
+ 505,
+ 468
+ ],
+ "score": 1.0,
+ "content": ",",
+ "type": "text"
+ }
+ ],
+ "index": 11
+ },
+ {
+ "bbox": [
+ 105,
+ 466,
+ 506,
+ 479
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 466,
+ 123,
+ 479
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 124,
+ 467,
+ 158,
+ 478
+ ],
+ "score": 0.91,
+ "content": "\\tilde { \\mu } _ { N } < 2",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 158,
+ 466,
+ 379,
+ 479
+ ],
+ "score": 1.0,
+ "content": ". For the first two, the statements are equivalent to that",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 379,
+ 466,
+ 389,
+ 477
+ ],
+ "score": 0.82,
+ "content": "\\tilde { \\Delta }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 389,
+ 466,
+ 506,
+ 479
+ ],
+ "score": 1.0,
+ "content": "is positive semi-definite and",
+ "type": "text"
+ }
+ ],
+ "index": 12
+ },
+ {
+ "bbox": [
+ 105,
+ 477,
+ 505,
+ 490
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 477,
+ 505,
+ 490
+ ],
+ "score": 1.0,
+ "content": "that the multiplicity of the eigenvalue 0 is same as the number of connected components 6. This",
+ "type": "text"
+ }
+ ],
+ "index": 13
+ },
+ {
+ "bbox": [
+ 105,
+ 488,
+ 506,
+ 502
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 488,
+ 506,
+ 502
+ ],
+ "score": 1.0,
+ "content": "is well-known for Laplacian or its normalized version (see, e.g., Chung & Graham (1997)) and the",
+ "type": "text"
+ }
+ ],
+ "index": 14
+ },
+ {
+ "bbox": [
+ 105,
+ 499,
+ 321,
+ 514
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 500,
+ 145,
+ 514
+ ],
+ "score": 1.0,
+ "content": "proof for",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 145,
+ 499,
+ 154,
+ 511
+ ],
+ "score": 0.87,
+ "content": "\\tilde { \\Delta }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 155,
+ 500,
+ 321,
+ 514
+ ],
+ "score": 1.0,
+ "content": "is similar. By direct calculation, we have",
+ "type": "text"
+ }
+ ],
+ "index": 15
+ }
+ ],
+ "index": 12.5
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 208,
+ 520,
+ 401,
+ 556
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 208,
+ 520,
+ 401,
+ 556
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 208,
+ 520,
+ 401,
+ 556
+ ],
+ "score": 0.93,
+ "content": "x ^ { \\top } \\tilde { \\Delta } x = \\frac { 1 } { 2 } \\sum _ { i , j = 1 } ^ { N } a _ { i j } \\left( \\frac { x _ { i } } { \\sqrt { d _ { i } + 1 } } - \\frac { x _ { j } } { \\sqrt { d _ { j } + 1 } } \\right) ^ { 2 }",
+ "type": "interline_equation",
+ "image_path": "912b3f39f00da229f42e6b22f48e40279df357eb9f4ff841a0f14ea1354fcf33.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 16.5,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 208,
+ 520,
+ 401,
+ 538.0
+ ],
+ "spans": [],
+ "index": 16
+ },
+ {
+ "bbox": [
+ 208,
+ 538.0,
+ 401,
+ 556.0
+ ],
+ "spans": [],
+ "index": 17
+ }
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 105,
+ 564,
+ 485,
+ 579
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 103,
+ 562,
+ 483,
+ 582
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 103,
+ 562,
+ 137,
+ 582
+ ],
+ "score": 1.0,
+ "content": "for any",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 137,
+ 564,
+ 253,
+ 579
+ ],
+ "score": 0.85,
+ "content": "x = \\lbrack x _ { 1 } \\quad \\cdot \\cdot \\cdot \\quad x _ { N } \\rbrack ^ { \\top } \\in \\mathbb { R } ^ { N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 254,
+ 562,
+ 302,
+ 582
+ ],
+ "score": 1.0,
+ "content": ". Therefore,",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 302,
+ 565,
+ 312,
+ 577
+ ],
+ "score": 0.86,
+ "content": "\\tilde { \\Delta }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 312,
+ 562,
+ 452,
+ 582
+ ],
+ "score": 1.0,
+ "content": "is positive semi-definite and hence",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 453,
+ 567,
+ 483,
+ 578
+ ],
+ "score": 0.91,
+ "content": "\\tilde { \\mu } _ { 1 } \\geq 0",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 18
+ }
+ ],
+ "index": 18
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 583,
+ 506,
+ 700
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 583,
+ 506,
+ 598
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 583,
+ 209,
+ 598
+ ],
+ "score": 1.0,
+ "content": "Suppose temporally that",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 209,
+ 586,
+ 218,
+ 595
+ ],
+ "score": 0.83,
+ "content": "G",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 218,
+ 583,
+ 289,
+ 598
+ ],
+ "score": 1.0,
+ "content": "is connected. If",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 290,
+ 584,
+ 327,
+ 595
+ ],
+ "score": 0.91,
+ "content": "\\boldsymbol { x } \\in \\mathbb { R } ^ { N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 327,
+ 583,
+ 506,
+ 598
+ ],
+ "score": 1.0,
+ "content": "is an eigenvector associated to 0, then, by",
+ "type": "text"
+ }
+ ],
+ "index": 19
+ },
+ {
+ "bbox": [
+ 106,
+ 595,
+ 505,
+ 614
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 595,
+ 232,
+ 614
+ ],
+ "score": 1.0,
+ "content": "the aformentioned calculation,",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 232,
+ 596,
+ 259,
+ 612
+ ],
+ "score": 0.91,
+ "content": "\\textstyle { \\frac { x _ { i } } { \\sqrt { d _ { i } + 1 } } }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 259,
+ 595,
+ 279,
+ 614
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 279,
+ 596,
+ 308,
+ 614
+ ],
+ "score": 0.93,
+ "content": "\\textstyle { \\frac { x _ { j } } { \\sqrt { d _ { j } + 1 } } }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 309,
+ 595,
+ 417,
+ 610
+ ],
+ "score": 1.0,
+ "content": "must be same for all pairs",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 417,
+ 596,
+ 438,
+ 608
+ ],
+ "score": 0.92,
+ "content": "( i , j )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 439,
+ 595,
+ 479,
+ 610
+ ],
+ "score": 1.0,
+ "content": "such that",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 479,
+ 597,
+ 505,
+ 609
+ ],
+ "score": 0.89,
+ "content": "a _ { i j } >",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 20
+ },
+ {
+ "bbox": [
+ 105,
+ 612,
+ 505,
+ 629
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 612,
+ 186,
+ 629
+ ],
+ "score": 1.0,
+ "content": "0. However, since",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 186,
+ 613,
+ 195,
+ 623
+ ],
+ "score": 0.8,
+ "content": "G",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 195,
+ 612,
+ 254,
+ 629
+ ],
+ "score": 1.0,
+ "content": "is connected,",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 254,
+ 614,
+ 281,
+ 628
+ ],
+ "score": 0.91,
+ "content": "\\textstyle { \\frac { x _ { i } } { \\sqrt { d _ { i } + 1 } } }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 282,
+ 612,
+ 397,
+ 629
+ ],
+ "score": 1.0,
+ "content": "must be same value for all",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 397,
+ 613,
+ 432,
+ 625
+ ],
+ "score": 0.93,
+ "content": "i \\in [ N ]",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 432,
+ 612,
+ 505,
+ 629
+ ],
+ "score": 1.0,
+ "content": ". That means the",
+ "type": "text"
+ }
+ ],
+ "index": 21
+ },
+ {
+ "bbox": [
+ 105,
+ 625,
+ 506,
+ 639
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 625,
+ 506,
+ 639
+ ],
+ "score": 1.0,
+ "content": "multiplicity of the eigenvalue 0 is 1 and any eigenvector associated to 0 must be proportional to",
+ "type": "text"
+ }
+ ],
+ "index": 22
+ },
+ {
+ "bbox": [
+ 106,
+ 636,
+ 506,
+ 652
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 637,
+ 128,
+ 649
+ ],
+ "score": 0.87,
+ "content": "{ \\tilde { D } } ^ { \\frac { 1 } { 2 } } \\mathbf { 1 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 128,
+ 636,
+ 197,
+ 652
+ ],
+ "score": 1.0,
+ "content": ". Now, suppose",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 197,
+ 639,
+ 207,
+ 649
+ ],
+ "score": 0.81,
+ "content": "G",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 207,
+ 636,
+ 226,
+ 652
+ ],
+ "score": 1.0,
+ "content": "has",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 226,
+ 639,
+ 238,
+ 649
+ ],
+ "score": 0.82,
+ "content": "M",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 239,
+ 636,
+ 336,
+ 652
+ ],
+ "score": 1.0,
+ "content": "connected components",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 336,
+ 638,
+ 385,
+ 650
+ ],
+ "score": 0.92,
+ "content": "V _ { 1 } , \\dots , V _ { M }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 385,
+ 636,
+ 410,
+ 652
+ ],
+ "score": 1.0,
+ "content": ". Let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 410,
+ 637,
+ 427,
+ 650
+ ],
+ "score": 0.91,
+ "content": "\\tilde { \\Delta } _ { m }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 427,
+ 636,
+ 506,
+ 652
+ ],
+ "score": 1.0,
+ "content": "be the augmented",
+ "type": "text"
+ }
+ ],
+ "index": 23
+ },
+ {
+ "bbox": [
+ 105,
+ 649,
+ 506,
+ 663
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 649,
+ 389,
+ 663
+ ],
+ "score": 1.0,
+ "content": "normalized Laplacians corresponding to each connected component",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 389,
+ 650,
+ 404,
+ 661
+ ],
+ "score": 0.89,
+ "content": "V _ { m }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 405,
+ 649,
+ 422,
+ 663
+ ],
+ "score": 1.0,
+ "content": "for",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 423,
+ 650,
+ 466,
+ 662
+ ],
+ "score": 0.91,
+ "content": "m \\in \\ [ M ]",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 466,
+ 649,
+ 506,
+ 663
+ ],
+ "score": 1.0,
+ "content": ". By the",
+ "type": "text"
+ }
+ ],
+ "index": 24
+ },
+ {
+ "bbox": [
+ 105,
+ 661,
+ 506,
+ 675
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 662,
+ 214,
+ 675
+ ],
+ "score": 1.0,
+ "content": "aformentioned discussion,",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 214,
+ 661,
+ 231,
+ 673
+ ],
+ "score": 0.91,
+ "content": "\\tilde { \\Delta } _ { m }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 232,
+ 662,
+ 424,
+ 675
+ ],
+ "score": 1.0,
+ "content": "has the eigenvalue 0 with multiplicity 1. Since",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 425,
+ 661,
+ 434,
+ 672
+ ],
+ "score": 0.84,
+ "content": "\\tilde { \\Delta }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 435,
+ 662,
+ 506,
+ 675
+ ],
+ "score": 1.0,
+ "content": "is the direct sum",
+ "type": "text"
+ }
+ ],
+ "index": 25
+ },
+ {
+ "bbox": [
+ 106,
+ 673,
+ 505,
+ 687
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 673,
+ 117,
+ 687
+ ],
+ "score": 1.0,
+ "content": "of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 118,
+ 673,
+ 139,
+ 687
+ ],
+ "score": 0.92,
+ "content": "\\tilde { \\Delta } _ { m } ^ { \\prime } s",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 139,
+ 673,
+ 213,
+ 687
+ ],
+ "score": 1.0,
+ "content": ", the eigenvalue of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 213,
+ 674,
+ 222,
+ 685
+ ],
+ "score": 0.86,
+ "content": "\\tilde { \\Delta }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 222,
+ 673,
+ 320,
+ 687
+ ],
+ "score": 1.0,
+ "content": "is the union of those for",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 320,
+ 673,
+ 336,
+ 686
+ ],
+ "score": 0.91,
+ "content": "\\tilde { \\Delta } _ { m }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 337,
+ 673,
+ 392,
+ 687
+ ],
+ "score": 1.0,
+ "content": "’s. Therefore,",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 392,
+ 673,
+ 401,
+ 685
+ ],
+ "score": 0.85,
+ "content": "\\tilde { \\Delta }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 402,
+ 673,
+ 505,
+ 687
+ ],
+ "score": 1.0,
+ "content": "has the eigenvalue 0 with",
+ "type": "text"
+ }
+ ],
+ "index": 26
+ },
+ {
+ "bbox": [
+ 105,
+ 686,
+ 420,
+ 700
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 686,
+ 155,
+ 700
+ ],
+ "score": 1.0,
+ "content": "multiplicity",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 156,
+ 688,
+ 168,
+ 698
+ ],
+ "score": 0.79,
+ "content": "M",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 168,
+ 686,
+ 186,
+ 700
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 186,
+ 686,
+ 240,
+ 699
+ ],
+ "score": 0.93,
+ "content": "e _ { m } = \\tilde { D } ^ { \\frac { 1 } { 2 } } \\mathbf { 1 } _ { m }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 240,
+ 686,
+ 420,
+ 700
+ ],
+ "score": 1.0,
+ "content": "’s are the orthogonal basis of the eigenspace.",
+ "type": "text"
+ }
+ ],
+ "index": 27
+ }
+ ],
+ "index": 23
+ }
+ ],
+ "page_idx": 15,
+ "page_size": [
+ 612,
+ 792
+ ],
+ "discarded_blocks": [
+ {
+ "type": "discarded",
+ "bbox": [
+ 106,
+ 711,
+ 506,
+ 732
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 117,
+ 708,
+ 506,
+ 725
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 117,
+ 708,
+ 506,
+ 725
+ ],
+ "score": 1.0,
+ "content": "6The former statement is identical to Lemma 1 and latter one is the extension of Lemma 2 of Wu et al.",
+ "type": "text"
+ }
+ ]
+ },
+ {
+ "bbox": [
+ 105,
+ 719,
+ 141,
+ 734
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 719,
+ 141,
+ 734
+ ],
+ "score": 1.0,
+ "content": "(2019a).",
+ "type": "text"
+ }
+ ]
+ }
+ ]
+ },
+ {
+ "type": "discarded",
+ "bbox": [
+ 107,
+ 26,
+ 294,
+ 38
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 25,
+ 294,
+ 38
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 25,
+ 294,
+ 38
+ ],
+ "score": 1.0,
+ "content": "Published as a conference paper at ICLR 2020",
+ "type": "text"
+ }
+ ]
+ }
+ ]
+ },
+ {
+ "type": "discarded",
+ "bbox": [
+ 300,
+ 751,
+ 311,
+ 760
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 299,
+ 750,
+ 313,
+ 764
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 299,
+ 750,
+ 313,
+ 764
+ ],
+ "score": 1.0,
+ "content": "16",
+ "type": "text"
+ }
+ ]
+ }
+ ]
+ },
+ {
+ "type": "discarded",
+ "bbox": [
+ 494,
+ 386,
+ 505,
+ 397
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 496,
+ 388,
+ 505,
+ 398
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 496,
+ 388,
+ 505,
+ 398
+ ],
+ "score": 1.0,
+ "content": "□",
+ "type": "text"
+ }
+ ]
+ }
+ ]
+ },
+ {
+ "type": "discarded",
+ "bbox": [
+ 494,
+ 164,
+ 505,
+ 175
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 496,
+ 166,
+ 504,
+ 174
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 496,
+ 166,
+ 504,
+ 174
+ ],
+ "score": 0.997,
+ "content": "□",
+ "type": "text"
+ }
+ ]
+ }
+ ]
+ }
+ ],
+ "para_blocks": [
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 82,
+ 505,
+ 117
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 81,
+ 505,
+ 96
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 81,
+ 198,
+ 96
+ ],
+ "score": 1.0,
+ "content": "We used the fact that",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 198,
+ 83,
+ 211,
+ 93
+ ],
+ "score": 0.87,
+ "content": "I _ { m }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 212,
+ 81,
+ 288,
+ 96
+ ],
+ "score": 1.0,
+ "content": "’s are disjoint and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 288,
+ 83,
+ 330,
+ 93
+ ],
+ "score": 0.9,
+ "content": "v _ { n m } = 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 330,
+ 81,
+ 341,
+ 96
+ ],
+ "score": 1.0,
+ "content": "if",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 342,
+ 83,
+ 392,
+ 94
+ ],
+ "score": 0.9,
+ "content": "n \\not \\in \\cup _ { m } I _ { m }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 392,
+ 81,
+ 505,
+ 96
+ ],
+ "score": 1.0,
+ "content": "in the first equality above.",
+ "type": "text"
+ }
+ ],
+ "index": 0
+ },
+ {
+ "bbox": [
+ 105,
+ 92,
+ 506,
+ 108
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 92,
+ 176,
+ 108
+ ],
+ "score": 1.0,
+ "content": "Further, we have",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 177,
+ 94,
+ 249,
+ 105
+ ],
+ "score": 0.91,
+ "content": "x _ { n } x _ { l } v _ { n m } v _ { l m } \\leq 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 249,
+ 92,
+ 264,
+ 108
+ ],
+ "score": 1.0,
+ "content": "for",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 265,
+ 94,
+ 312,
+ 105
+ ],
+ "score": 0.92,
+ "content": "1 \\le n \\le L",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 313,
+ 92,
+ 331,
+ 108
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 331,
+ 94,
+ 397,
+ 105
+ ],
+ "score": 0.9,
+ "content": "L + 1 \\le l \\le N",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 398,
+ 92,
+ 478,
+ 108
+ ],
+ "score": 1.0,
+ "content": "by the definition of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 479,
+ 94,
+ 487,
+ 104
+ ],
+ "score": 0.81,
+ "content": "L",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 487,
+ 92,
+ 506,
+ 108
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ }
+ ],
+ "index": 1
+ },
+ {
+ "bbox": [
+ 106,
+ 104,
+ 357,
+ 117
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 104,
+ 177,
+ 117
+ ],
+ "score": 1.0,
+ "content": "non-negativity of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 178,
+ 106,
+ 191,
+ 115
+ ],
+ "score": 0.87,
+ "content": "v _ { m }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 191,
+ 104,
+ 357,
+ 117
+ ],
+ "score": 1.0,
+ "content": ". By combining (1), (2), and (3), we have",
+ "type": "text"
+ }
+ ],
+ "index": 2
+ }
+ ],
+ "index": 1,
+ "bbox_fs": [
+ 105,
+ 81,
+ 506,
+ 117
+ ]
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 192,
+ 122,
+ 417,
+ 157
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 192,
+ 122,
+ 417,
+ 157
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 192,
+ 122,
+ 417,
+ 157
+ ],
+ "score": 0.94,
+ "content": "\\sum _ { m = 1 } ^ { M } \\langle x , v _ { m } \\rangle ^ { 2 } - \\langle x ^ { + } , v _ { m } \\rangle ^ { 2 } \\leq \\sum _ { n = 1 } ^ { L } x _ { n } ^ { 2 } = \\| x \\| ^ { 2 } - \\| x ^ { + } \\| ^ { 2 } .",
+ "type": "interline_equation",
+ "image_path": "c39fabdc21d6ccc420a7956b6414e1b3de75315041ade8eba193e361d9db97c0.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 3.5,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 192,
+ 122,
+ 417,
+ 139.5
+ ],
+ "spans": [],
+ "index": 3
+ },
+ {
+ "bbox": [
+ 192,
+ 139.5,
+ 417,
+ 157.0
+ ],
+ "spans": [],
+ "index": 4
+ }
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 193,
+ 321,
+ 206
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 194,
+ 320,
+ 207
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 194,
+ 179,
+ 207
+ ],
+ "score": 1.0,
+ "content": "Proof of Theorem",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 180,
+ 195,
+ 185,
+ 204
+ ],
+ "score": 0.37,
+ "content": "^ { l }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 185,
+ 194,
+ 320,
+ 207
+ ],
+ "score": 1.0,
+ "content": ". By Lemma 1, 2, and 3, we have",
+ "type": "text"
+ }
+ ],
+ "index": 5
+ }
+ ],
+ "index": 5,
+ "bbox_fs": [
+ 106,
+ 194,
+ 320,
+ 207
+ ]
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 168,
+ 213,
+ 439,
+ 380
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 168,
+ 213,
+ 439,
+ 380
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 168,
+ 213,
+ 439,
+ 380
+ ],
+ "score": 0.93,
+ "content": "\\begin{array} { r l } { d _ { M } ( f _ { l } ( X ) ) = d _ { M } ( \\underbrace { \\sigma \\langle \\cdot \\cdot \\sigma ( \\sigma ( P X ) W _ { l } ) W _ { l ^ { 2 } } \\cdot \\cdot \\cdot \\cdot W _ { l H _ { 1 } } \\rangle } _ { H \\mathrm { ~ i n e s } } ) } & { } \\\\ & { \\leq d _ { M } ( \\underbrace { \\sigma \\langle \\cdot \\cdot \\sigma ( \\sigma ( P X ) W _ { l } ) W _ { l ^ { 2 } } \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot } _ { H \\mathrm { ~ i n e s } } ) } \\\\ & { \\leq s _ { i H _ { 1 } - 1 } d _ { M } ( \\underbrace { \\sigma \\langle \\cdot \\cdot \\sigma ( \\sigma ( P X ) W _ { l } ) W _ { l ^ { 2 } } \\cdot \\cdot \\cdot \\cdot \\cdot } _ { H - 1 \\mathrm { ~ i n e s } } ) } \\\\ & { \\cdots } \\\\ & { \\leq \\displaystyle \\left( \\prod _ { i = 1 } ^ { H _ { l } } s _ { i h } \\right) d _ { M } ( P X ) } \\\\ & { \\leq s _ { i d _ { M } ( P X ) } } \\\\ & { \\leq s _ { i H _ { 1 } } d _ { M } ( X ) . } \\end{array}",
+ "type": "interline_equation",
+ "image_path": "15704c0189b83b8ac25bed3cd3d139dd55c80d16850523239a96a4637790c552.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 7,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 168,
+ 213,
+ 439,
+ 268.6666666666667
+ ],
+ "spans": [],
+ "index": 6
+ },
+ {
+ "bbox": [
+ 168,
+ 268.6666666666667,
+ 439,
+ 324.33333333333337
+ ],
+ "spans": [],
+ "index": 7
+ },
+ {
+ "bbox": [
+ 168,
+ 324.33333333333337,
+ 439,
+ 380.00000000000006
+ ],
+ "spans": [],
+ "index": 8
+ }
+ ]
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 108,
+ 415,
+ 260,
+ 428
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 415,
+ 262,
+ 430
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 415,
+ 262,
+ 430
+ ],
+ "score": 1.0,
+ "content": "B PROOF OF PROPOSITION 1",
+ "type": "text"
+ }
+ ],
+ "index": 9
+ }
+ ],
+ "index": 9
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 441,
+ 506,
+ 513
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 440,
+ 506,
+ 454
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 441,
+ 153,
+ 454
+ ],
+ "score": 1.0,
+ "content": "Proof. Let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 153,
+ 442,
+ 222,
+ 453
+ ],
+ "score": 0.92,
+ "content": "{ \\tilde { \\mu } } _ { 1 } \\leq \\dots \\leq { \\tilde { \\mu } } _ { N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 223,
+ 441,
+ 464,
+ 454
+ ],
+ "score": 1.0,
+ "content": "be the eigenvalue of the augmented normalized Laplacian",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 464,
+ 440,
+ 473,
+ 452
+ ],
+ "score": 0.84,
+ "content": "\\tilde { \\Delta }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 474,
+ 441,
+ 506,
+ 454
+ ],
+ "score": 1.0,
+ "content": ", sorted",
+ "type": "text"
+ }
+ ],
+ "index": 10
+ },
+ {
+ "bbox": [
+ 105,
+ 453,
+ 505,
+ 468
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 454,
+ 213,
+ 468
+ ],
+ "score": 1.0,
+ "content": "in ascending order. Since",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 213,
+ 453,
+ 271,
+ 466
+ ],
+ "score": 0.93,
+ "content": "P = I _ { N } - \\tilde { \\Delta }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 271,
+ 454,
+ 361,
+ 468
+ ],
+ "score": 1.0,
+ "content": ", it is enough to show",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 361,
+ 455,
+ 450,
+ 466
+ ],
+ "score": 0.89,
+ "content": "\\tilde { \\mu } _ { 1 } = \\cdot \\cdot \\cdot = \\tilde { \\mu } _ { M } = 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 451,
+ 454,
+ 455,
+ 468
+ ],
+ "score": 1.0,
+ "content": ",",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 455,
+ 455,
+ 501,
+ 466
+ ],
+ "score": 0.81,
+ "content": "{ \\tilde { \\mu } } _ { M + 1 } > 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 502,
+ 454,
+ 505,
+ 468
+ ],
+ "score": 1.0,
+ "content": ",",
+ "type": "text"
+ }
+ ],
+ "index": 11
+ },
+ {
+ "bbox": [
+ 105,
+ 466,
+ 506,
+ 479
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 466,
+ 123,
+ 479
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 124,
+ 467,
+ 158,
+ 478
+ ],
+ "score": 0.91,
+ "content": "\\tilde { \\mu } _ { N } < 2",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 158,
+ 466,
+ 379,
+ 479
+ ],
+ "score": 1.0,
+ "content": ". For the first two, the statements are equivalent to that",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 379,
+ 466,
+ 389,
+ 477
+ ],
+ "score": 0.82,
+ "content": "\\tilde { \\Delta }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 389,
+ 466,
+ 506,
+ 479
+ ],
+ "score": 1.0,
+ "content": "is positive semi-definite and",
+ "type": "text"
+ }
+ ],
+ "index": 12
+ },
+ {
+ "bbox": [
+ 105,
+ 477,
+ 505,
+ 490
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 477,
+ 505,
+ 490
+ ],
+ "score": 1.0,
+ "content": "that the multiplicity of the eigenvalue 0 is same as the number of connected components 6. This",
+ "type": "text"
+ }
+ ],
+ "index": 13
+ },
+ {
+ "bbox": [
+ 105,
+ 488,
+ 506,
+ 502
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 488,
+ 506,
+ 502
+ ],
+ "score": 1.0,
+ "content": "is well-known for Laplacian or its normalized version (see, e.g., Chung & Graham (1997)) and the",
+ "type": "text"
+ }
+ ],
+ "index": 14
+ },
+ {
+ "bbox": [
+ 105,
+ 499,
+ 321,
+ 514
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 500,
+ 145,
+ 514
+ ],
+ "score": 1.0,
+ "content": "proof for",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 145,
+ 499,
+ 154,
+ 511
+ ],
+ "score": 0.87,
+ "content": "\\tilde { \\Delta }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 155,
+ 500,
+ 321,
+ 514
+ ],
+ "score": 1.0,
+ "content": "is similar. By direct calculation, we have",
+ "type": "text"
+ }
+ ],
+ "index": 15
+ }
+ ],
+ "index": 12.5,
+ "bbox_fs": [
+ 105,
+ 440,
+ 506,
+ 514
+ ]
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 208,
+ 520,
+ 401,
+ 556
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 208,
+ 520,
+ 401,
+ 556
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 208,
+ 520,
+ 401,
+ 556
+ ],
+ "score": 0.93,
+ "content": "x ^ { \\top } \\tilde { \\Delta } x = \\frac { 1 } { 2 } \\sum _ { i , j = 1 } ^ { N } a _ { i j } \\left( \\frac { x _ { i } } { \\sqrt { d _ { i } + 1 } } - \\frac { x _ { j } } { \\sqrt { d _ { j } + 1 } } \\right) ^ { 2 }",
+ "type": "interline_equation",
+ "image_path": "912b3f39f00da229f42e6b22f48e40279df357eb9f4ff841a0f14ea1354fcf33.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 16.5,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 208,
+ 520,
+ 401,
+ 538.0
+ ],
+ "spans": [],
+ "index": 16
+ },
+ {
+ "bbox": [
+ 208,
+ 538.0,
+ 401,
+ 556.0
+ ],
+ "spans": [],
+ "index": 17
+ }
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 105,
+ 564,
+ 485,
+ 579
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 103,
+ 562,
+ 483,
+ 582
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 103,
+ 562,
+ 137,
+ 582
+ ],
+ "score": 1.0,
+ "content": "for any",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 137,
+ 564,
+ 253,
+ 579
+ ],
+ "score": 0.85,
+ "content": "x = \\lbrack x _ { 1 } \\quad \\cdot \\cdot \\cdot \\quad x _ { N } \\rbrack ^ { \\top } \\in \\mathbb { R } ^ { N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 254,
+ 562,
+ 302,
+ 582
+ ],
+ "score": 1.0,
+ "content": ". Therefore,",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 302,
+ 565,
+ 312,
+ 577
+ ],
+ "score": 0.86,
+ "content": "\\tilde { \\Delta }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 312,
+ 562,
+ 452,
+ 582
+ ],
+ "score": 1.0,
+ "content": "is positive semi-definite and hence",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 453,
+ 567,
+ 483,
+ 578
+ ],
+ "score": 0.91,
+ "content": "\\tilde { \\mu } _ { 1 } \\geq 0",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 18
+ }
+ ],
+ "index": 18,
+ "bbox_fs": [
+ 103,
+ 562,
+ 483,
+ 582
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 583,
+ 506,
+ 700
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 583,
+ 506,
+ 598
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 583,
+ 209,
+ 598
+ ],
+ "score": 1.0,
+ "content": "Suppose temporally that",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 209,
+ 586,
+ 218,
+ 595
+ ],
+ "score": 0.83,
+ "content": "G",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 218,
+ 583,
+ 289,
+ 598
+ ],
+ "score": 1.0,
+ "content": "is connected. If",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 290,
+ 584,
+ 327,
+ 595
+ ],
+ "score": 0.91,
+ "content": "\\boldsymbol { x } \\in \\mathbb { R } ^ { N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 327,
+ 583,
+ 506,
+ 598
+ ],
+ "score": 1.0,
+ "content": "is an eigenvector associated to 0, then, by",
+ "type": "text"
+ }
+ ],
+ "index": 19
+ },
+ {
+ "bbox": [
+ 106,
+ 595,
+ 505,
+ 614
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 595,
+ 232,
+ 614
+ ],
+ "score": 1.0,
+ "content": "the aformentioned calculation,",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 232,
+ 596,
+ 259,
+ 612
+ ],
+ "score": 0.91,
+ "content": "\\textstyle { \\frac { x _ { i } } { \\sqrt { d _ { i } + 1 } } }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 259,
+ 595,
+ 279,
+ 614
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 279,
+ 596,
+ 308,
+ 614
+ ],
+ "score": 0.93,
+ "content": "\\textstyle { \\frac { x _ { j } } { \\sqrt { d _ { j } + 1 } } }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 309,
+ 595,
+ 417,
+ 610
+ ],
+ "score": 1.0,
+ "content": "must be same for all pairs",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 417,
+ 596,
+ 438,
+ 608
+ ],
+ "score": 0.92,
+ "content": "( i , j )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 439,
+ 595,
+ 479,
+ 610
+ ],
+ "score": 1.0,
+ "content": "such that",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 479,
+ 597,
+ 505,
+ 609
+ ],
+ "score": 0.89,
+ "content": "a _ { i j } >",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 20
+ },
+ {
+ "bbox": [
+ 105,
+ 612,
+ 505,
+ 629
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 612,
+ 186,
+ 629
+ ],
+ "score": 1.0,
+ "content": "0. However, since",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 186,
+ 613,
+ 195,
+ 623
+ ],
+ "score": 0.8,
+ "content": "G",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 195,
+ 612,
+ 254,
+ 629
+ ],
+ "score": 1.0,
+ "content": "is connected,",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 254,
+ 614,
+ 281,
+ 628
+ ],
+ "score": 0.91,
+ "content": "\\textstyle { \\frac { x _ { i } } { \\sqrt { d _ { i } + 1 } } }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 282,
+ 612,
+ 397,
+ 629
+ ],
+ "score": 1.0,
+ "content": "must be same value for all",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 397,
+ 613,
+ 432,
+ 625
+ ],
+ "score": 0.93,
+ "content": "i \\in [ N ]",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 432,
+ 612,
+ 505,
+ 629
+ ],
+ "score": 1.0,
+ "content": ". That means the",
+ "type": "text"
+ }
+ ],
+ "index": 21
+ },
+ {
+ "bbox": [
+ 105,
+ 625,
+ 506,
+ 639
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 625,
+ 506,
+ 639
+ ],
+ "score": 1.0,
+ "content": "multiplicity of the eigenvalue 0 is 1 and any eigenvector associated to 0 must be proportional to",
+ "type": "text"
+ }
+ ],
+ "index": 22
+ },
+ {
+ "bbox": [
+ 106,
+ 636,
+ 506,
+ 652
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 637,
+ 128,
+ 649
+ ],
+ "score": 0.87,
+ "content": "{ \\tilde { D } } ^ { \\frac { 1 } { 2 } } \\mathbf { 1 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 128,
+ 636,
+ 197,
+ 652
+ ],
+ "score": 1.0,
+ "content": ". Now, suppose",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 197,
+ 639,
+ 207,
+ 649
+ ],
+ "score": 0.81,
+ "content": "G",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 207,
+ 636,
+ 226,
+ 652
+ ],
+ "score": 1.0,
+ "content": "has",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 226,
+ 639,
+ 238,
+ 649
+ ],
+ "score": 0.82,
+ "content": "M",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 239,
+ 636,
+ 336,
+ 652
+ ],
+ "score": 1.0,
+ "content": "connected components",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 336,
+ 638,
+ 385,
+ 650
+ ],
+ "score": 0.92,
+ "content": "V _ { 1 } , \\dots , V _ { M }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 385,
+ 636,
+ 410,
+ 652
+ ],
+ "score": 1.0,
+ "content": ". Let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 410,
+ 637,
+ 427,
+ 650
+ ],
+ "score": 0.91,
+ "content": "\\tilde { \\Delta } _ { m }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 427,
+ 636,
+ 506,
+ 652
+ ],
+ "score": 1.0,
+ "content": "be the augmented",
+ "type": "text"
+ }
+ ],
+ "index": 23
+ },
+ {
+ "bbox": [
+ 105,
+ 649,
+ 506,
+ 663
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 649,
+ 389,
+ 663
+ ],
+ "score": 1.0,
+ "content": "normalized Laplacians corresponding to each connected component",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 389,
+ 650,
+ 404,
+ 661
+ ],
+ "score": 0.89,
+ "content": "V _ { m }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 405,
+ 649,
+ 422,
+ 663
+ ],
+ "score": 1.0,
+ "content": "for",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 423,
+ 650,
+ 466,
+ 662
+ ],
+ "score": 0.91,
+ "content": "m \\in \\ [ M ]",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 466,
+ 649,
+ 506,
+ 663
+ ],
+ "score": 1.0,
+ "content": ". By the",
+ "type": "text"
+ }
+ ],
+ "index": 24
+ },
+ {
+ "bbox": [
+ 105,
+ 661,
+ 506,
+ 675
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 662,
+ 214,
+ 675
+ ],
+ "score": 1.0,
+ "content": "aformentioned discussion,",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 214,
+ 661,
+ 231,
+ 673
+ ],
+ "score": 0.91,
+ "content": "\\tilde { \\Delta } _ { m }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 232,
+ 662,
+ 424,
+ 675
+ ],
+ "score": 1.0,
+ "content": "has the eigenvalue 0 with multiplicity 1. Since",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 425,
+ 661,
+ 434,
+ 672
+ ],
+ "score": 0.84,
+ "content": "\\tilde { \\Delta }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 435,
+ 662,
+ 506,
+ 675
+ ],
+ "score": 1.0,
+ "content": "is the direct sum",
+ "type": "text"
+ }
+ ],
+ "index": 25
+ },
+ {
+ "bbox": [
+ 106,
+ 673,
+ 505,
+ 687
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 673,
+ 117,
+ 687
+ ],
+ "score": 1.0,
+ "content": "of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 118,
+ 673,
+ 139,
+ 687
+ ],
+ "score": 0.92,
+ "content": "\\tilde { \\Delta } _ { m } ^ { \\prime } s",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 139,
+ 673,
+ 213,
+ 687
+ ],
+ "score": 1.0,
+ "content": ", the eigenvalue of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 213,
+ 674,
+ 222,
+ 685
+ ],
+ "score": 0.86,
+ "content": "\\tilde { \\Delta }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 222,
+ 673,
+ 320,
+ 687
+ ],
+ "score": 1.0,
+ "content": "is the union of those for",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 320,
+ 673,
+ 336,
+ 686
+ ],
+ "score": 0.91,
+ "content": "\\tilde { \\Delta } _ { m }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 337,
+ 673,
+ 392,
+ 687
+ ],
+ "score": 1.0,
+ "content": "’s. Therefore,",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 392,
+ 673,
+ 401,
+ 685
+ ],
+ "score": 0.85,
+ "content": "\\tilde { \\Delta }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 402,
+ 673,
+ 505,
+ 687
+ ],
+ "score": 1.0,
+ "content": "has the eigenvalue 0 with",
+ "type": "text"
+ }
+ ],
+ "index": 26
+ },
+ {
+ "bbox": [
+ 105,
+ 686,
+ 420,
+ 700
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 686,
+ 155,
+ 700
+ ],
+ "score": 1.0,
+ "content": "multiplicity",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 156,
+ 688,
+ 168,
+ 698
+ ],
+ "score": 0.79,
+ "content": "M",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 168,
+ 686,
+ 186,
+ 700
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 186,
+ 686,
+ 240,
+ 699
+ ],
+ "score": 0.93,
+ "content": "e _ { m } = \\tilde { D } ^ { \\frac { 1 } { 2 } } \\mathbf { 1 } _ { m }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 240,
+ 686,
+ 420,
+ 700
+ ],
+ "score": 1.0,
+ "content": "’s are the orthogonal basis of the eigenspace.",
+ "type": "text"
+ }
+ ],
+ "index": 27
+ }
+ ],
+ "index": 23,
+ "bbox_fs": [
+ 105,
+ 583,
+ 506,
+ 700
+ ]
+ }
+ ]
+ },
+ {
+ "preproc_blocks": [
+ {
+ "type": "text",
+ "bbox": [
+ 104,
+ 82,
+ 504,
+ 107
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 80,
+ 505,
+ 95
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 80,
+ 181,
+ 95
+ ],
+ "score": 1.0,
+ "content": "Finally, we prove",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 181,
+ 83,
+ 218,
+ 94
+ ],
+ "score": 0.9,
+ "content": "\\tilde { \\mu } _ { N \\mathrm { ~ } } < \\mathrm { ~ 2 ~ }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 218,
+ 80,
+ 241,
+ 95
+ ],
+ "score": 1.0,
+ "content": ". Let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 242,
+ 84,
+ 257,
+ 94
+ ],
+ "score": 0.84,
+ "content": "\\mu _ { N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 257,
+ 80,
+ 482,
+ 95
+ ],
+ "score": 1.0,
+ "content": "be the largest eigenvalue of the normalized Laplacian",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 482,
+ 82,
+ 505,
+ 94
+ ],
+ "score": 0.83,
+ "content": "\\Delta =",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 0
+ },
+ {
+ "bbox": [
+ 107,
+ 90,
+ 414,
+ 110
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 93,
+ 185,
+ 107
+ ],
+ "score": 0.93,
+ "content": "D ^ { - \\frac { 1 } { 2 } } \\bar { ( } D - \\bar { A } ) D ^ { - \\frac { 1 } { 2 } }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 185,
+ 90,
+ 216,
+ 110
+ ],
+ "score": 1.0,
+ "content": ", where",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 216,
+ 93,
+ 279,
+ 105
+ ],
+ "score": 0.92,
+ "content": "D ^ { - \\frac { 1 } { 2 } } \\in \\mathbb { R } ^ { N \\times N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 279,
+ 90,
+ 414,
+ 110
+ ],
+ "score": 1.0,
+ "content": "is the diagonal matrix defined by",
+ "type": "text"
+ }
+ ],
+ "index": 1
+ }
+ ],
+ "index": 0.5
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 225,
+ 112,
+ 384,
+ 146
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 225,
+ 112,
+ 384,
+ 146
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 225,
+ 112,
+ 384,
+ 146
+ ],
+ "score": 0.94,
+ "content": "D _ { i i } ^ { - \\frac { 1 } { 2 } } = \\left\\{ \\begin{array} { l l } { \\deg ( i ) ^ { - \\frac { 1 } { 2 } } } & { ( \\mathrm { i f } \\deg ( i ) \\neq 0 ) } \\\\ { 0 } & { ( \\mathrm { i f } \\deg ( i ) = 0 ) } \\end{array} \\right. .",
+ "type": "interline_equation",
+ "image_path": "3d269a0ba588fd8303d9cdfd91801f8052013907dddbbf2c70e75cc2e48e6492.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 2.5,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 225,
+ 112,
+ 384,
+ 129.0
+ ],
+ "spans": [],
+ "index": 2
+ },
+ {
+ "bbox": [
+ 225,
+ 129.0,
+ 384,
+ 146.0
+ ],
+ "spans": [],
+ "index": 3
+ }
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 105,
+ 152,
+ 504,
+ 176
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 104,
+ 150,
+ 506,
+ 167
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 150,
+ 147,
+ 167
+ ],
+ "score": 1.0,
+ "content": "Note that",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 147,
+ 151,
+ 183,
+ 163
+ ],
+ "score": 0.91,
+ "content": "D ^ { - \\frac { 1 } { 2 } } D ^ { \\frac { 1 } { 2 } }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 184,
+ 150,
+ 203,
+ 167
+ ],
+ "score": 1.0,
+ "content": "nor",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 203,
+ 151,
+ 239,
+ 163
+ ],
+ "score": 0.92,
+ "content": "D ^ { \\frac { 1 } { 2 } } D ^ { - { \\frac { 1 } { 2 } } }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 239,
+ 150,
+ 386,
+ 167
+ ],
+ "score": 1.0,
+ "content": "are not equal to the identity matrix",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 387,
+ 154,
+ 400,
+ 164
+ ],
+ "score": 0.89,
+ "content": "I _ { N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 400,
+ 150,
+ 506,
+ 167
+ ],
+ "score": 1.0,
+ "content": "in general. However, we",
+ "type": "text"
+ }
+ ],
+ "index": 4
+ },
+ {
+ "bbox": [
+ 104,
+ 162,
+ 129,
+ 178
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 162,
+ 129,
+ 178
+ ],
+ "score": 1.0,
+ "content": "have",
+ "type": "text"
+ }
+ ],
+ "index": 5
+ }
+ ],
+ "index": 4.5
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 254,
+ 178,
+ 356,
+ 194
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 254,
+ 178,
+ 356,
+ 194
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 254,
+ 178,
+ 356,
+ 194
+ ],
+ "score": 0.9,
+ "content": "L = D ^ { \\frac { 1 } { 2 } } D ^ { - \\frac { 1 } { 2 } } L D ^ { - \\frac { 1 } { 2 } } D ^ { \\frac { 1 } { 2 } }",
+ "type": "interline_equation",
+ "image_path": "f06328db9cdb9c357ee18d32337ee54247ad13e04e1cc4534382a8bbdb729367.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 6,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 254,
+ 178,
+ 356,
+ 194
+ ],
+ "spans": [],
+ "index": 6
+ }
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 199,
+ 396,
+ 212
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 199,
+ 396,
+ 213
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 199,
+ 133,
+ 213
+ ],
+ "score": 1.0,
+ "content": "where",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 133,
+ 200,
+ 183,
+ 210
+ ],
+ "score": 0.91,
+ "content": "L = D - A",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 183,
+ 199,
+ 396,
+ 213
+ ],
+ "score": 1.0,
+ "content": "is the (unnormalized) Laplacian. Therefore, we have",
+ "type": "text"
+ }
+ ],
+ "index": 7
+ }
+ ],
+ "index": 7
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 162,
+ 214,
+ 452,
+ 529
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 162,
+ 214,
+ 452,
+ 529
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 162,
+ 214,
+ 452,
+ 529
+ ],
+ "score": 0.95,
+ "content": "\\begin{array} { r l } { \\alpha _ { 1 } } & { \\gamma _ { 2 } \\geq \\frac { \\alpha _ { 2 } ( \\lambda _ { 1 } ) } { \\alpha _ { 1 } } } \\\\ { = } & { \\gamma _ { 1 } \\geq \\alpha _ { 2 } \\leq \\alpha _ { 1 } } \\\\ & { - \\gamma _ { 2 } \\geq \\alpha _ { 2 } \\leq \\alpha _ { 2 } \\leq \\alpha _ { 2 } \\leq \\alpha _ { 2 } } \\\\ & { - \\gamma _ { 3 } \\geq \\alpha _ { 3 } } \\\\ { = } & { \\gamma _ { 2 } \\geq \\alpha _ { 3 } \\leq \\alpha _ { 3 } \\leq \\alpha _ { 4 } } \\\\ & { - \\gamma _ { 2 } \\geq \\alpha _ { 4 } \\leq \\alpha _ { 4 } } \\\\ & { - \\gamma _ { 3 } \\geq \\alpha _ { 4 } \\leq \\alpha _ { 5 } \\leq \\alpha _ { 4 } } \\\\ & { - \\gamma _ { 5 } \\geq \\alpha _ { 5 } \\leq \\alpha _ { 5 } \\leq \\alpha _ { 4 } } \\\\ & { - \\gamma _ { 6 } \\geq \\alpha _ { 6 } \\leq \\alpha _ { 5 } \\leq \\alpha _ { 5 } } \\\\ & { - \\alpha _ { 5 } \\leq \\alpha _ { 5 } \\leq \\alpha _ { 5 } \\leq \\alpha _ { 6 } } \\\\ & { \\gamma _ { 6 } \\geq \\alpha _ { 5 } \\leq \\alpha _ { 7 } \\leq \\alpha _ { 6 } \\leq \\alpha _ { 7 } \\leq \\alpha _ { 6 } \\leq \\alpha _ { 7 } \\leq \\alpha _ { 7 } \\leq \\alpha _ { 8 } } \\\\ & { \\gamma _ { 7 } \\geq \\alpha _ { 6 } \\leq \\alpha _ { 7 } \\leq \\alpha _ { 7 } \\leq \\alpha _ { 8 } } \\\\ & { \\alpha _ { 1 } \\leq \\alpha _ { 1 } \\leq \\alpha _ { 1 } \\leq \\alpha _ { 1 } } \\\\ & { \\gamma _ { 2 } \\leq \\alpha _ { 2 } \\leq \\alpha _ { 2 } \\leq \\alpha _ { 3 } \\leq \\alpha _ { 1 } } \\\\ & { \\gamma _ { 1 } \\geq \\alpha _ { 2 } \\leq \\alpha _ { 2 } \\leq \\alpha _ { 3 } \\leq \\alpha _ { 2 } \\leq \\alpha _ { 3 } } \\\\ & { \\gamma _ { 2 } \\leq \\alpha _ { 3 } \\leq \\alpha _ { 4 } } \\\\ & { \\alpha _ { 2 } \\leq \\alpha _ { 3 } \\leq \\alpha _ { 5 } \\leq \\alpha _ { 4 } } \\\\ & { \\gamma _ { 2 } \\leq \\alpha _ { 3 } \\leq \\alpha _ { 4 } } \\\\ & { \\gamma _ { 3 } \\geq \\alpha _ { 5 } \\leq \\alpha _ { 4 } } \\\\ & { \\gamma _ { 2 } \\geq \\alpha _ { 2 } \\leq \\alpha _ { 3 } \\leq \\alpha _ { 4 } } \\\\ & \\gamma _ { 3 } \\geq \\ \\end{array}",
+ "type": "interline_equation",
+ "image_path": "0130e9d5369d0baf35ac0feeea7a01ddaaafd449509a606e0c5a756b6400d2fd.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 18.5,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 162,
+ 214,
+ 452,
+ 228.3181818181818
+ ],
+ "spans": [],
+ "index": 8
+ },
+ {
+ "bbox": [
+ 162,
+ 228.3181818181818,
+ 452,
+ 242.63636363636363
+ ],
+ "spans": [],
+ "index": 9
+ },
+ {
+ "bbox": [
+ 162,
+ 242.63636363636363,
+ 452,
+ 256.95454545454544
+ ],
+ "spans": [],
+ "index": 10
+ },
+ {
+ "bbox": [
+ 162,
+ 256.95454545454544,
+ 452,
+ 271.27272727272725
+ ],
+ "spans": [],
+ "index": 11
+ },
+ {
+ "bbox": [
+ 162,
+ 271.27272727272725,
+ 452,
+ 285.59090909090907
+ ],
+ "spans": [],
+ "index": 12
+ },
+ {
+ "bbox": [
+ 162,
+ 285.59090909090907,
+ 452,
+ 299.9090909090909
+ ],
+ "spans": [],
+ "index": 13
+ },
+ {
+ "bbox": [
+ 162,
+ 299.9090909090909,
+ 452,
+ 314.2272727272727
+ ],
+ "spans": [],
+ "index": 14
+ },
+ {
+ "bbox": [
+ 162,
+ 314.2272727272727,
+ 452,
+ 328.5454545454545
+ ],
+ "spans": [],
+ "index": 15
+ },
+ {
+ "bbox": [
+ 162,
+ 328.5454545454545,
+ 452,
+ 342.8636363636363
+ ],
+ "spans": [],
+ "index": 16
+ },
+ {
+ "bbox": [
+ 162,
+ 342.8636363636363,
+ 452,
+ 357.18181818181813
+ ],
+ "spans": [],
+ "index": 17
+ },
+ {
+ "bbox": [
+ 162,
+ 357.18181818181813,
+ 452,
+ 371.49999999999994
+ ],
+ "spans": [],
+ "index": 18
+ },
+ {
+ "bbox": [
+ 162,
+ 371.49999999999994,
+ 452,
+ 385.81818181818176
+ ],
+ "spans": [],
+ "index": 19
+ },
+ {
+ "bbox": [
+ 162,
+ 385.81818181818176,
+ 452,
+ 400.13636363636357
+ ],
+ "spans": [],
+ "index": 20
+ },
+ {
+ "bbox": [
+ 162,
+ 400.13636363636357,
+ 452,
+ 414.4545454545454
+ ],
+ "spans": [],
+ "index": 21
+ },
+ {
+ "bbox": [
+ 162,
+ 414.4545454545454,
+ 452,
+ 428.7727272727272
+ ],
+ "spans": [],
+ "index": 22
+ },
+ {
+ "bbox": [
+ 162,
+ 428.7727272727272,
+ 452,
+ 443.090909090909
+ ],
+ "spans": [],
+ "index": 23
+ },
+ {
+ "bbox": [
+ 162,
+ 443.090909090909,
+ 452,
+ 457.4090909090908
+ ],
+ "spans": [],
+ "index": 24
+ },
+ {
+ "bbox": [
+ 162,
+ 457.4090909090908,
+ 452,
+ 471.72727272727263
+ ],
+ "spans": [],
+ "index": 25
+ },
+ {
+ "bbox": [
+ 162,
+ 471.72727272727263,
+ 452,
+ 486.04545454545445
+ ],
+ "spans": [],
+ "index": 26
+ },
+ {
+ "bbox": [
+ 162,
+ 486.04545454545445,
+ 452,
+ 500.36363636363626
+ ],
+ "spans": [],
+ "index": 27
+ },
+ {
+ "bbox": [
+ 162,
+ 500.36363636363626,
+ 452,
+ 514.6818181818181
+ ],
+ "spans": [],
+ "index": 28
+ },
+ {
+ "bbox": [
+ 162,
+ 514.6818181818181,
+ 452,
+ 529.0
+ ],
+ "spans": [],
+ "index": 29
+ }
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 537,
+ 505,
+ 603
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 536,
+ 505,
+ 559
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 540,
+ 311,
+ 558
+ ],
+ "score": 1.0,
+ "content": "Therefore, we have µ˜N ≤ µN 7. Since maxi∈[N]",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 266,
+ 536,
+ 381,
+ 559
+ ],
+ "score": 0.94,
+ "content": "\\begin{array} { r } { \\operatorname* { m a x } _ { i \\in [ N ] } \\left( \\frac { \\deg ( i ) } { \\deg ( i ) + 1 } \\right) ^ { \\frac { 1 } { 2 } } < 1 } \\end{array}",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 381,
+ 542,
+ 435,
+ 556
+ ],
+ "score": 1.0,
+ "content": ", the equality",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 435,
+ 544,
+ 479,
+ 555
+ ],
+ "score": 0.92,
+ "content": "\\tilde { \\mu } _ { N } = \\mu _ { N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 479,
+ 542,
+ 505,
+ 556
+ ],
+ "score": 1.0,
+ "content": "holds",
+ "type": "text"
+ }
+ ],
+ "index": 30
+ },
+ {
+ "bbox": [
+ 105,
+ 557,
+ 505,
+ 570
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 557,
+ 164,
+ 570
+ ],
+ "score": 1.0,
+ "content": "if and only if",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 164,
+ 558,
+ 200,
+ 569
+ ],
+ "score": 0.91,
+ "content": "\\mu _ { N } = 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 201,
+ 557,
+ 236,
+ 570
+ ],
+ "score": 1.0,
+ "content": ", that is,",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 236,
+ 558,
+ 245,
+ 568
+ ],
+ "score": 0.81,
+ "content": "G",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 246,
+ 557,
+ 264,
+ 570
+ ],
+ "score": 1.0,
+ "content": "has",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 264,
+ 558,
+ 275,
+ 568
+ ],
+ "score": 0.83,
+ "content": "N",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 275,
+ 557,
+ 505,
+ 570
+ ],
+ "score": 1.0,
+ "content": "connected components. On the other hand, it is known",
+ "type": "text"
+ }
+ ],
+ "index": 31
+ },
+ {
+ "bbox": [
+ 106,
+ 568,
+ 506,
+ 581
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 568,
+ 124,
+ 581
+ ],
+ "score": 1.0,
+ "content": "that",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 124,
+ 569,
+ 159,
+ 580
+ ],
+ "score": 0.91,
+ "content": "\\mu _ { N } \\leq 2",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 160,
+ 568,
+ 309,
+ 581
+ ],
+ "score": 1.0,
+ "content": "and the equality holds if and only if",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 309,
+ 569,
+ 319,
+ 579
+ ],
+ "score": 0.8,
+ "content": "G",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 319,
+ 568,
+ 506,
+ 581
+ ],
+ "score": 1.0,
+ "content": "has non-trivial bipartite graph as a connected",
+ "type": "text"
+ }
+ ],
+ "index": 32
+ },
+ {
+ "bbox": [
+ 105,
+ 579,
+ 506,
+ 592
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 579,
+ 224,
+ 592
+ ],
+ "score": 1.0,
+ "content": "component (see, e.g., Chung",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 224,
+ 580,
+ 233,
+ 590
+ ],
+ "score": 0.51,
+ "content": "\\&",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 234,
+ 579,
+ 351,
+ 592
+ ],
+ "score": 1.0,
+ "content": "Graham (1997)). Therefore,",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 351,
+ 580,
+ 393,
+ 591
+ ],
+ "score": 0.91,
+ "content": "\\tilde { \\mu } _ { N } = \\mu _ { N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 394,
+ 579,
+ 412,
+ 592
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 412,
+ 580,
+ 447,
+ 591
+ ],
+ "score": 0.9,
+ "content": "\\mu _ { N } = 2",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 447,
+ 579,
+ 506,
+ 592
+ ],
+ "score": 1.0,
+ "content": "does not hold",
+ "type": "text"
+ }
+ ],
+ "index": 33
+ },
+ {
+ "bbox": [
+ 106,
+ 591,
+ 506,
+ 602
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 591,
+ 227,
+ 602
+ ],
+ "score": 1.0,
+ "content": "simultaneously and we obtain",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 228,
+ 591,
+ 261,
+ 602
+ ],
+ "score": 0.9,
+ "content": "\\mu _ { N } < 2",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 261,
+ 591,
+ 264,
+ 602
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 493,
+ 591,
+ 506,
+ 602
+ ],
+ "score": 0.996,
+ "content": "□",
+ "type": "text"
+ }
+ ],
+ "index": 34
+ }
+ ],
+ "index": 32
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 107,
+ 618,
+ 480,
+ 645
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 617,
+ 482,
+ 632
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 617,
+ 482,
+ 632
+ ],
+ "score": 1.0,
+ "content": "C COUNTEREXAMPLE OF PREVIOUS STUDY ON OVER-SMOOTHING FOR",
+ "type": "text"
+ }
+ ],
+ "index": 35
+ },
+ {
+ "bbox": [
+ 126,
+ 632,
+ 234,
+ 645
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 126,
+ 632,
+ 234,
+ 645
+ ],
+ "score": 1.0,
+ "content": "NON-LINEAR GNNS",
+ "type": "text"
+ }
+ ],
+ "index": 36
+ }
+ ],
+ "index": 35.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 656,
+ 505,
+ 692
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 656,
+ 506,
+ 670
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 656,
+ 352,
+ 670
+ ],
+ "score": 1.0,
+ "content": "We restate Theorem 1 of the preprint (version2) of Luan et al.",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 352,
+ 657,
+ 383,
+ 668
+ ],
+ "score": 0.79,
+ "content": "( 2 0 1 9 ) ^ { 8 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 383,
+ 656,
+ 403,
+ 670
+ ],
+ "score": 1.0,
+ "content": ". Let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 403,
+ 657,
+ 412,
+ 667
+ ],
+ "score": 0.82,
+ "content": "G",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 413,
+ 656,
+ 506,
+ 670
+ ],
+ "score": 1.0,
+ "content": "be a simple undirected",
+ "type": "text"
+ }
+ ],
+ "index": 37
+ },
+ {
+ "bbox": [
+ 105,
+ 668,
+ 505,
+ 681
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 668,
+ 152,
+ 681
+ ],
+ "score": 1.0,
+ "content": "graph with",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 152,
+ 669,
+ 163,
+ 678
+ ],
+ "score": 0.8,
+ "content": "N",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 163,
+ 668,
+ 207,
+ 681
+ ],
+ "score": 1.0,
+ "content": "nodes and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 207,
+ 669,
+ 214,
+ 678
+ ],
+ "score": 0.8,
+ "content": "k",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 214,
+ 668,
+ 505,
+ 681
+ ],
+ "score": 1.0,
+ "content": "connected components such that it does not have a bipartite component.",
+ "type": "text"
+ }
+ ],
+ "index": 38
+ },
+ {
+ "bbox": [
+ 104,
+ 677,
+ 506,
+ 694
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 677,
+ 122,
+ 694
+ ],
+ "score": 1.0,
+ "content": "Let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 122,
+ 679,
+ 247,
+ 690
+ ],
+ "score": 0.91,
+ "content": "L = \\tilde { D } ^ { - 1 / 2 } \\tilde { A } \\tilde { D } ^ { - 1 / 2 } \\in \\mathbb { R } ^ { N \\times N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 247,
+ 677,
+ 421,
+ 694
+ ],
+ "score": 1.0,
+ "content": "be the augmented normalized Laplacian of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 422,
+ 681,
+ 431,
+ 690
+ ],
+ "score": 0.79,
+ "content": "G",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 431,
+ 677,
+ 451,
+ 694
+ ],
+ "score": 1.0,
+ "content": ". Let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 451,
+ 680,
+ 487,
+ 692
+ ],
+ "score": 0.92,
+ "content": "F \\in \\mathbb { N } _ { + }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 487,
+ 677,
+ 506,
+ 694
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ }
+ ],
+ "index": 39
+ }
+ ],
+ "index": 38
+ }
+ ],
+ "page_idx": 16,
+ "page_size": [
+ 612,
+ 792
+ ],
+ "discarded_blocks": [
+ {
+ "type": "discarded",
+ "bbox": [
+ 106,
+ 700,
+ 504,
+ 731
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 118,
+ 698,
+ 506,
+ 714
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 118,
+ 698,
+ 405,
+ 714
+ ],
+ "score": 1.0,
+ "content": "7Theorem 1 of Wu et al. (2019a) showed that this inequality strictly holds when",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 406,
+ 701,
+ 414,
+ 710
+ ],
+ "score": 0.82,
+ "content": "G",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 414,
+ 698,
+ 506,
+ 714
+ ],
+ "score": 1.0,
+ "content": "is simple and connected.",
+ "type": "text"
+ }
+ ]
+ },
+ {
+ "bbox": [
+ 106,
+ 710,
+ 234,
+ 722
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 710,
+ 234,
+ 722
+ ],
+ "score": 1.0,
+ "content": "We do not require this assumption.",
+ "type": "text"
+ }
+ ]
+ },
+ {
+ "bbox": [
+ 117,
+ 719,
+ 308,
+ 733
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 117,
+ 719,
+ 308,
+ 733
+ ],
+ "score": 1.0,
+ "content": "8https://arxiv.org/abs/1906.02174v2",
+ "type": "text"
+ }
+ ]
+ }
+ ]
+ },
+ {
+ "type": "discarded",
+ "bbox": [
+ 107,
+ 26,
+ 294,
+ 38
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 25,
+ 294,
+ 39
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 25,
+ 294,
+ 39
+ ],
+ "score": 1.0,
+ "content": "Published as a conference paper at ICLR 2020",
+ "type": "text"
+ }
+ ]
+ }
+ ]
+ },
+ {
+ "type": "discarded",
+ "bbox": [
+ 300,
+ 751,
+ 311,
+ 760
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 299,
+ 750,
+ 312,
+ 764
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 299,
+ 750,
+ 312,
+ 764
+ ],
+ "score": 1.0,
+ "content": "17",
+ "type": "text"
+ }
+ ]
+ }
+ ]
+ }
+ ],
+ "para_blocks": [
+ {
+ "type": "text",
+ "bbox": [
+ 104,
+ 82,
+ 504,
+ 107
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 80,
+ 505,
+ 95
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 80,
+ 181,
+ 95
+ ],
+ "score": 1.0,
+ "content": "Finally, we prove",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 181,
+ 83,
+ 218,
+ 94
+ ],
+ "score": 0.9,
+ "content": "\\tilde { \\mu } _ { N \\mathrm { ~ } } < \\mathrm { ~ 2 ~ }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 218,
+ 80,
+ 241,
+ 95
+ ],
+ "score": 1.0,
+ "content": ". Let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 242,
+ 84,
+ 257,
+ 94
+ ],
+ "score": 0.84,
+ "content": "\\mu _ { N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 257,
+ 80,
+ 482,
+ 95
+ ],
+ "score": 1.0,
+ "content": "be the largest eigenvalue of the normalized Laplacian",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 482,
+ 82,
+ 505,
+ 94
+ ],
+ "score": 0.83,
+ "content": "\\Delta =",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 0
+ },
+ {
+ "bbox": [
+ 107,
+ 90,
+ 414,
+ 110
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 93,
+ 185,
+ 107
+ ],
+ "score": 0.93,
+ "content": "D ^ { - \\frac { 1 } { 2 } } \\bar { ( } D - \\bar { A } ) D ^ { - \\frac { 1 } { 2 } }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 185,
+ 90,
+ 216,
+ 110
+ ],
+ "score": 1.0,
+ "content": ", where",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 216,
+ 93,
+ 279,
+ 105
+ ],
+ "score": 0.92,
+ "content": "D ^ { - \\frac { 1 } { 2 } } \\in \\mathbb { R } ^ { N \\times N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 279,
+ 90,
+ 414,
+ 110
+ ],
+ "score": 1.0,
+ "content": "is the diagonal matrix defined by",
+ "type": "text"
+ }
+ ],
+ "index": 1
+ }
+ ],
+ "index": 0.5,
+ "bbox_fs": [
+ 105,
+ 80,
+ 505,
+ 110
+ ]
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 225,
+ 112,
+ 384,
+ 146
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 225,
+ 112,
+ 384,
+ 146
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 225,
+ 112,
+ 384,
+ 146
+ ],
+ "score": 0.94,
+ "content": "D _ { i i } ^ { - \\frac { 1 } { 2 } } = \\left\\{ \\begin{array} { l l } { \\deg ( i ) ^ { - \\frac { 1 } { 2 } } } & { ( \\mathrm { i f } \\deg ( i ) \\neq 0 ) } \\\\ { 0 } & { ( \\mathrm { i f } \\deg ( i ) = 0 ) } \\end{array} \\right. .",
+ "type": "interline_equation",
+ "image_path": "3d269a0ba588fd8303d9cdfd91801f8052013907dddbbf2c70e75cc2e48e6492.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 2.5,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 225,
+ 112,
+ 384,
+ 129.0
+ ],
+ "spans": [],
+ "index": 2
+ },
+ {
+ "bbox": [
+ 225,
+ 129.0,
+ 384,
+ 146.0
+ ],
+ "spans": [],
+ "index": 3
+ }
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 105,
+ 152,
+ 504,
+ 176
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 104,
+ 150,
+ 506,
+ 167
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 150,
+ 147,
+ 167
+ ],
+ "score": 1.0,
+ "content": "Note that",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 147,
+ 151,
+ 183,
+ 163
+ ],
+ "score": 0.91,
+ "content": "D ^ { - \\frac { 1 } { 2 } } D ^ { \\frac { 1 } { 2 } }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 184,
+ 150,
+ 203,
+ 167
+ ],
+ "score": 1.0,
+ "content": "nor",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 203,
+ 151,
+ 239,
+ 163
+ ],
+ "score": 0.92,
+ "content": "D ^ { \\frac { 1 } { 2 } } D ^ { - { \\frac { 1 } { 2 } } }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 239,
+ 150,
+ 386,
+ 167
+ ],
+ "score": 1.0,
+ "content": "are not equal to the identity matrix",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 387,
+ 154,
+ 400,
+ 164
+ ],
+ "score": 0.89,
+ "content": "I _ { N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 400,
+ 150,
+ 506,
+ 167
+ ],
+ "score": 1.0,
+ "content": "in general. However, we",
+ "type": "text"
+ }
+ ],
+ "index": 4
+ },
+ {
+ "bbox": [
+ 104,
+ 162,
+ 129,
+ 178
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 162,
+ 129,
+ 178
+ ],
+ "score": 1.0,
+ "content": "have",
+ "type": "text"
+ }
+ ],
+ "index": 5
+ }
+ ],
+ "index": 4.5,
+ "bbox_fs": [
+ 104,
+ 150,
+ 506,
+ 178
+ ]
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 254,
+ 178,
+ 356,
+ 194
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 254,
+ 178,
+ 356,
+ 194
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 254,
+ 178,
+ 356,
+ 194
+ ],
+ "score": 0.9,
+ "content": "L = D ^ { \\frac { 1 } { 2 } } D ^ { - \\frac { 1 } { 2 } } L D ^ { - \\frac { 1 } { 2 } } D ^ { \\frac { 1 } { 2 } }",
+ "type": "interline_equation",
+ "image_path": "f06328db9cdb9c357ee18d32337ee54247ad13e04e1cc4534382a8bbdb729367.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 6,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 254,
+ 178,
+ 356,
+ 194
+ ],
+ "spans": [],
+ "index": 6
+ }
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 199,
+ 396,
+ 212
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 199,
+ 396,
+ 213
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 199,
+ 133,
+ 213
+ ],
+ "score": 1.0,
+ "content": "where",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 133,
+ 200,
+ 183,
+ 210
+ ],
+ "score": 0.91,
+ "content": "L = D - A",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 183,
+ 199,
+ 396,
+ 213
+ ],
+ "score": 1.0,
+ "content": "is the (unnormalized) Laplacian. Therefore, we have",
+ "type": "text"
+ }
+ ],
+ "index": 7
+ }
+ ],
+ "index": 7,
+ "bbox_fs": [
+ 106,
+ 199,
+ 396,
+ 213
+ ]
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 162,
+ 214,
+ 452,
+ 529
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 162,
+ 214,
+ 452,
+ 529
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 162,
+ 214,
+ 452,
+ 529
+ ],
+ "score": 0.95,
+ "content": "\\begin{array} { r l } { \\alpha _ { 1 } } & { \\gamma _ { 2 } \\geq \\frac { \\alpha _ { 2 } ( \\lambda _ { 1 } ) } { \\alpha _ { 1 } } } \\\\ { = } & { \\gamma _ { 1 } \\geq \\alpha _ { 2 } \\leq \\alpha _ { 1 } } \\\\ & { - \\gamma _ { 2 } \\geq \\alpha _ { 2 } \\leq \\alpha _ { 2 } \\leq \\alpha _ { 2 } \\leq \\alpha _ { 2 } } \\\\ & { - \\gamma _ { 3 } \\geq \\alpha _ { 3 } } \\\\ { = } & { \\gamma _ { 2 } \\geq \\alpha _ { 3 } \\leq \\alpha _ { 3 } \\leq \\alpha _ { 4 } } \\\\ & { - \\gamma _ { 2 } \\geq \\alpha _ { 4 } \\leq \\alpha _ { 4 } } \\\\ & { - \\gamma _ { 3 } \\geq \\alpha _ { 4 } \\leq \\alpha _ { 5 } \\leq \\alpha _ { 4 } } \\\\ & { - \\gamma _ { 5 } \\geq \\alpha _ { 5 } \\leq \\alpha _ { 5 } \\leq \\alpha _ { 4 } } \\\\ & { - \\gamma _ { 6 } \\geq \\alpha _ { 6 } \\leq \\alpha _ { 5 } \\leq \\alpha _ { 5 } } \\\\ & { - \\alpha _ { 5 } \\leq \\alpha _ { 5 } \\leq \\alpha _ { 5 } \\leq \\alpha _ { 6 } } \\\\ & { \\gamma _ { 6 } \\geq \\alpha _ { 5 } \\leq \\alpha _ { 7 } \\leq \\alpha _ { 6 } \\leq \\alpha _ { 7 } \\leq \\alpha _ { 6 } \\leq \\alpha _ { 7 } \\leq \\alpha _ { 7 } \\leq \\alpha _ { 8 } } \\\\ & { \\gamma _ { 7 } \\geq \\alpha _ { 6 } \\leq \\alpha _ { 7 } \\leq \\alpha _ { 7 } \\leq \\alpha _ { 8 } } \\\\ & { \\alpha _ { 1 } \\leq \\alpha _ { 1 } \\leq \\alpha _ { 1 } \\leq \\alpha _ { 1 } } \\\\ & { \\gamma _ { 2 } \\leq \\alpha _ { 2 } \\leq \\alpha _ { 2 } \\leq \\alpha _ { 3 } \\leq \\alpha _ { 1 } } \\\\ & { \\gamma _ { 1 } \\geq \\alpha _ { 2 } \\leq \\alpha _ { 2 } \\leq \\alpha _ { 3 } \\leq \\alpha _ { 2 } \\leq \\alpha _ { 3 } } \\\\ & { \\gamma _ { 2 } \\leq \\alpha _ { 3 } \\leq \\alpha _ { 4 } } \\\\ & { \\alpha _ { 2 } \\leq \\alpha _ { 3 } \\leq \\alpha _ { 5 } \\leq \\alpha _ { 4 } } \\\\ & { \\gamma _ { 2 } \\leq \\alpha _ { 3 } \\leq \\alpha _ { 4 } } \\\\ & { \\gamma _ { 3 } \\geq \\alpha _ { 5 } \\leq \\alpha _ { 4 } } \\\\ & { \\gamma _ { 2 } \\geq \\alpha _ { 2 } \\leq \\alpha _ { 3 } \\leq \\alpha _ { 4 } } \\\\ & \\gamma _ { 3 } \\geq \\ \\end{array}",
+ "type": "interline_equation",
+ "image_path": "0130e9d5369d0baf35ac0feeea7a01ddaaafd449509a606e0c5a756b6400d2fd.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 18.5,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 162,
+ 214,
+ 452,
+ 228.3181818181818
+ ],
+ "spans": [],
+ "index": 8
+ },
+ {
+ "bbox": [
+ 162,
+ 228.3181818181818,
+ 452,
+ 242.63636363636363
+ ],
+ "spans": [],
+ "index": 9
+ },
+ {
+ "bbox": [
+ 162,
+ 242.63636363636363,
+ 452,
+ 256.95454545454544
+ ],
+ "spans": [],
+ "index": 10
+ },
+ {
+ "bbox": [
+ 162,
+ 256.95454545454544,
+ 452,
+ 271.27272727272725
+ ],
+ "spans": [],
+ "index": 11
+ },
+ {
+ "bbox": [
+ 162,
+ 271.27272727272725,
+ 452,
+ 285.59090909090907
+ ],
+ "spans": [],
+ "index": 12
+ },
+ {
+ "bbox": [
+ 162,
+ 285.59090909090907,
+ 452,
+ 299.9090909090909
+ ],
+ "spans": [],
+ "index": 13
+ },
+ {
+ "bbox": [
+ 162,
+ 299.9090909090909,
+ 452,
+ 314.2272727272727
+ ],
+ "spans": [],
+ "index": 14
+ },
+ {
+ "bbox": [
+ 162,
+ 314.2272727272727,
+ 452,
+ 328.5454545454545
+ ],
+ "spans": [],
+ "index": 15
+ },
+ {
+ "bbox": [
+ 162,
+ 328.5454545454545,
+ 452,
+ 342.8636363636363
+ ],
+ "spans": [],
+ "index": 16
+ },
+ {
+ "bbox": [
+ 162,
+ 342.8636363636363,
+ 452,
+ 357.18181818181813
+ ],
+ "spans": [],
+ "index": 17
+ },
+ {
+ "bbox": [
+ 162,
+ 357.18181818181813,
+ 452,
+ 371.49999999999994
+ ],
+ "spans": [],
+ "index": 18
+ },
+ {
+ "bbox": [
+ 162,
+ 371.49999999999994,
+ 452,
+ 385.81818181818176
+ ],
+ "spans": [],
+ "index": 19
+ },
+ {
+ "bbox": [
+ 162,
+ 385.81818181818176,
+ 452,
+ 400.13636363636357
+ ],
+ "spans": [],
+ "index": 20
+ },
+ {
+ "bbox": [
+ 162,
+ 400.13636363636357,
+ 452,
+ 414.4545454545454
+ ],
+ "spans": [],
+ "index": 21
+ },
+ {
+ "bbox": [
+ 162,
+ 414.4545454545454,
+ 452,
+ 428.7727272727272
+ ],
+ "spans": [],
+ "index": 22
+ },
+ {
+ "bbox": [
+ 162,
+ 428.7727272727272,
+ 452,
+ 443.090909090909
+ ],
+ "spans": [],
+ "index": 23
+ },
+ {
+ "bbox": [
+ 162,
+ 443.090909090909,
+ 452,
+ 457.4090909090908
+ ],
+ "spans": [],
+ "index": 24
+ },
+ {
+ "bbox": [
+ 162,
+ 457.4090909090908,
+ 452,
+ 471.72727272727263
+ ],
+ "spans": [],
+ "index": 25
+ },
+ {
+ "bbox": [
+ 162,
+ 471.72727272727263,
+ 452,
+ 486.04545454545445
+ ],
+ "spans": [],
+ "index": 26
+ },
+ {
+ "bbox": [
+ 162,
+ 486.04545454545445,
+ 452,
+ 500.36363636363626
+ ],
+ "spans": [],
+ "index": 27
+ },
+ {
+ "bbox": [
+ 162,
+ 500.36363636363626,
+ 452,
+ 514.6818181818181
+ ],
+ "spans": [],
+ "index": 28
+ },
+ {
+ "bbox": [
+ 162,
+ 514.6818181818181,
+ 452,
+ 529.0
+ ],
+ "spans": [],
+ "index": 29
+ }
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 537,
+ 505,
+ 603
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 536,
+ 505,
+ 559
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 540,
+ 311,
+ 558
+ ],
+ "score": 1.0,
+ "content": "Therefore, we have µ˜N ≤ µN 7. Since maxi∈[N]",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 266,
+ 536,
+ 381,
+ 559
+ ],
+ "score": 0.94,
+ "content": "\\begin{array} { r } { \\operatorname* { m a x } _ { i \\in [ N ] } \\left( \\frac { \\deg ( i ) } { \\deg ( i ) + 1 } \\right) ^ { \\frac { 1 } { 2 } } < 1 } \\end{array}",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 381,
+ 542,
+ 435,
+ 556
+ ],
+ "score": 1.0,
+ "content": ", the equality",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 435,
+ 544,
+ 479,
+ 555
+ ],
+ "score": 0.92,
+ "content": "\\tilde { \\mu } _ { N } = \\mu _ { N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 479,
+ 542,
+ 505,
+ 556
+ ],
+ "score": 1.0,
+ "content": "holds",
+ "type": "text"
+ }
+ ],
+ "index": 30
+ },
+ {
+ "bbox": [
+ 105,
+ 557,
+ 505,
+ 570
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 557,
+ 164,
+ 570
+ ],
+ "score": 1.0,
+ "content": "if and only if",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 164,
+ 558,
+ 200,
+ 569
+ ],
+ "score": 0.91,
+ "content": "\\mu _ { N } = 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 201,
+ 557,
+ 236,
+ 570
+ ],
+ "score": 1.0,
+ "content": ", that is,",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 236,
+ 558,
+ 245,
+ 568
+ ],
+ "score": 0.81,
+ "content": "G",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 246,
+ 557,
+ 264,
+ 570
+ ],
+ "score": 1.0,
+ "content": "has",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 264,
+ 558,
+ 275,
+ 568
+ ],
+ "score": 0.83,
+ "content": "N",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 275,
+ 557,
+ 505,
+ 570
+ ],
+ "score": 1.0,
+ "content": "connected components. On the other hand, it is known",
+ "type": "text"
+ }
+ ],
+ "index": 31
+ },
+ {
+ "bbox": [
+ 106,
+ 568,
+ 506,
+ 581
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 568,
+ 124,
+ 581
+ ],
+ "score": 1.0,
+ "content": "that",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 124,
+ 569,
+ 159,
+ 580
+ ],
+ "score": 0.91,
+ "content": "\\mu _ { N } \\leq 2",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 160,
+ 568,
+ 309,
+ 581
+ ],
+ "score": 1.0,
+ "content": "and the equality holds if and only if",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 309,
+ 569,
+ 319,
+ 579
+ ],
+ "score": 0.8,
+ "content": "G",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 319,
+ 568,
+ 506,
+ 581
+ ],
+ "score": 1.0,
+ "content": "has non-trivial bipartite graph as a connected",
+ "type": "text"
+ }
+ ],
+ "index": 32
+ },
+ {
+ "bbox": [
+ 105,
+ 579,
+ 506,
+ 592
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 579,
+ 224,
+ 592
+ ],
+ "score": 1.0,
+ "content": "component (see, e.g., Chung",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 224,
+ 580,
+ 233,
+ 590
+ ],
+ "score": 0.51,
+ "content": "\\&",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 234,
+ 579,
+ 351,
+ 592
+ ],
+ "score": 1.0,
+ "content": "Graham (1997)). Therefore,",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 351,
+ 580,
+ 393,
+ 591
+ ],
+ "score": 0.91,
+ "content": "\\tilde { \\mu } _ { N } = \\mu _ { N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 394,
+ 579,
+ 412,
+ 592
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 412,
+ 580,
+ 447,
+ 591
+ ],
+ "score": 0.9,
+ "content": "\\mu _ { N } = 2",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 447,
+ 579,
+ 506,
+ 592
+ ],
+ "score": 1.0,
+ "content": "does not hold",
+ "type": "text"
+ }
+ ],
+ "index": 33
+ },
+ {
+ "bbox": [
+ 106,
+ 591,
+ 506,
+ 602
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 591,
+ 227,
+ 602
+ ],
+ "score": 1.0,
+ "content": "simultaneously and we obtain",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 228,
+ 591,
+ 261,
+ 602
+ ],
+ "score": 0.9,
+ "content": "\\mu _ { N } < 2",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 261,
+ 591,
+ 264,
+ 602
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 493,
+ 591,
+ 506,
+ 602
+ ],
+ "score": 0.996,
+ "content": "□",
+ "type": "text"
+ }
+ ],
+ "index": 34
+ }
+ ],
+ "index": 32,
+ "bbox_fs": [
+ 105,
+ 536,
+ 506,
+ 602
+ ]
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 107,
+ 618,
+ 480,
+ 645
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 617,
+ 482,
+ 632
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 617,
+ 482,
+ 632
+ ],
+ "score": 1.0,
+ "content": "C COUNTEREXAMPLE OF PREVIOUS STUDY ON OVER-SMOOTHING FOR",
+ "type": "text"
+ }
+ ],
+ "index": 35
+ },
+ {
+ "bbox": [
+ 126,
+ 632,
+ 234,
+ 645
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 126,
+ 632,
+ 234,
+ 645
+ ],
+ "score": 1.0,
+ "content": "NON-LINEAR GNNS",
+ "type": "text"
+ }
+ ],
+ "index": 36
+ }
+ ],
+ "index": 35.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 656,
+ 505,
+ 692
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 656,
+ 506,
+ 670
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 656,
+ 352,
+ 670
+ ],
+ "score": 1.0,
+ "content": "We restate Theorem 1 of the preprint (version2) of Luan et al.",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 352,
+ 657,
+ 383,
+ 668
+ ],
+ "score": 0.79,
+ "content": "( 2 0 1 9 ) ^ { 8 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 383,
+ 656,
+ 403,
+ 670
+ ],
+ "score": 1.0,
+ "content": ". Let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 403,
+ 657,
+ 412,
+ 667
+ ],
+ "score": 0.82,
+ "content": "G",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 413,
+ 656,
+ 506,
+ 670
+ ],
+ "score": 1.0,
+ "content": "be a simple undirected",
+ "type": "text"
+ }
+ ],
+ "index": 37
+ },
+ {
+ "bbox": [
+ 105,
+ 668,
+ 505,
+ 681
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 668,
+ 152,
+ 681
+ ],
+ "score": 1.0,
+ "content": "graph with",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 152,
+ 669,
+ 163,
+ 678
+ ],
+ "score": 0.8,
+ "content": "N",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 163,
+ 668,
+ 207,
+ 681
+ ],
+ "score": 1.0,
+ "content": "nodes and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 207,
+ 669,
+ 214,
+ 678
+ ],
+ "score": 0.8,
+ "content": "k",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 214,
+ 668,
+ 505,
+ 681
+ ],
+ "score": 1.0,
+ "content": "connected components such that it does not have a bipartite component.",
+ "type": "text"
+ }
+ ],
+ "index": 38
+ },
+ {
+ "bbox": [
+ 104,
+ 677,
+ 506,
+ 694
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 677,
+ 122,
+ 694
+ ],
+ "score": 1.0,
+ "content": "Let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 122,
+ 679,
+ 247,
+ 690
+ ],
+ "score": 0.91,
+ "content": "L = \\tilde { D } ^ { - 1 / 2 } \\tilde { A } \\tilde { D } ^ { - 1 / 2 } \\in \\mathbb { R } ^ { N \\times N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 247,
+ 677,
+ 421,
+ 694
+ ],
+ "score": 1.0,
+ "content": "be the augmented normalized Laplacian of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 422,
+ 681,
+ 431,
+ 690
+ ],
+ "score": 0.79,
+ "content": "G",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 431,
+ 677,
+ 451,
+ 694
+ ],
+ "score": 1.0,
+ "content": ". Let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 451,
+ 680,
+ 487,
+ 692
+ ],
+ "score": 0.92,
+ "content": "F \\in \\mathbb { N } _ { + }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 487,
+ 677,
+ 506,
+ 694
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ }
+ ],
+ "index": 39
+ }
+ ],
+ "index": 38,
+ "bbox_fs": [
+ 104,
+ 656,
+ 506,
+ 694
+ ]
+ }
+ ]
+ },
+ {
+ "preproc_blocks": [
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 81,
+ 505,
+ 138
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 107,
+ 79,
+ 506,
+ 96
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 81,
+ 163,
+ 93
+ ],
+ "score": 0.91,
+ "content": "W _ { n } \\in \\mathbb { R } ^ { F \\times F }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 164,
+ 79,
+ 250,
+ 96
+ ],
+ "score": 1.0,
+ "content": "be the weight of the",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 250,
+ 84,
+ 258,
+ 92
+ ],
+ "score": 0.78,
+ "content": "n",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 258,
+ 79,
+ 308,
+ 96
+ ],
+ "score": 1.0,
+ "content": "-th layer for",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 308,
+ 83,
+ 344,
+ 94
+ ],
+ "score": 0.91,
+ "content": "n \\in \\mathbb { N } _ { + }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 344,
+ 79,
+ 406,
+ 96
+ ],
+ "score": 1.0,
+ "content": ". For the input",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 407,
+ 81,
+ 459,
+ 93
+ ],
+ "score": 0.91,
+ "content": "X \\in \\mathbb { R } ^ { N \\times F }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 459,
+ 79,
+ 506,
+ 96
+ ],
+ "score": 1.0,
+ "content": ", we define",
+ "type": "text"
+ }
+ ],
+ "index": 0
+ },
+ {
+ "bbox": [
+ 104,
+ 89,
+ 504,
+ 108
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 89,
+ 150,
+ 108
+ ],
+ "score": 1.0,
+ "content": "the output",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 150,
+ 93,
+ 204,
+ 105
+ ],
+ "score": 0.93,
+ "content": "Y _ { n } \\in \\mathbb { R } ^ { N \\times F }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 204,
+ 89,
+ 232,
+ 108
+ ],
+ "score": 1.0,
+ "content": "of the",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 232,
+ 95,
+ 239,
+ 104
+ ],
+ "score": 0.77,
+ "content": "n",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 240,
+ 89,
+ 331,
+ 108
+ ],
+ "score": 1.0,
+ "content": "-th layer of a GCN by",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 331,
+ 93,
+ 467,
+ 106
+ ],
+ "score": 0.91,
+ "content": "Y _ { n } = \\sigma ( L \\cdot \\cdot \\cdot \\sigma ( L X W _ { 0 } ) \\cdot \\cdot \\cdot W _ { n } )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 468,
+ 89,
+ 496,
+ 108
+ ],
+ "score": 1.0,
+ "content": "where",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 496,
+ 96,
+ 504,
+ 103
+ ],
+ "score": 0.74,
+ "content": "\\sigma",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 1
+ },
+ {
+ "bbox": [
+ 103,
+ 102,
+ 506,
+ 118
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 103,
+ 102,
+ 285,
+ 118
+ ],
+ "score": 1.0,
+ "content": "is the ReLU function. We assume the input",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 286,
+ 105,
+ 296,
+ 114
+ ],
+ "score": 0.85,
+ "content": "X",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 296,
+ 102,
+ 473,
+ 118
+ ],
+ "score": 1.0,
+ "content": "is drawn from a continuous distribution on",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 473,
+ 104,
+ 501,
+ 114
+ ],
+ "score": 0.9,
+ "content": "\\mathbb { R } ^ { N \\times F }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 501,
+ 102,
+ 506,
+ 118
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 2
+ },
+ {
+ "bbox": [
+ 105,
+ 113,
+ 506,
+ 129
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 113,
+ 268,
+ 129
+ ],
+ "score": 1.0,
+ "content": "Then, the theorem claims that we have",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 268,
+ 115,
+ 367,
+ 127
+ ],
+ "score": 0.73,
+ "content": "\\begin{array} { r } { \\operatorname* { l i m } _ { n \\to \\infty } \\operatorname { r a n k } ( Y _ { n } ) = k } \\end{array}",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 367,
+ 113,
+ 506,
+ 129
+ ],
+ "score": 1.0,
+ "content": "almost surely with respect to the",
+ "type": "text"
+ }
+ ],
+ "index": 3
+ },
+ {
+ "bbox": [
+ 106,
+ 126,
+ 180,
+ 138
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 126,
+ 165,
+ 138
+ ],
+ "score": 1.0,
+ "content": "distribution of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 166,
+ 127,
+ 176,
+ 136
+ ],
+ "score": 0.82,
+ "content": "X",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 176,
+ 126,
+ 180,
+ 138
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 4
+ }
+ ],
+ "index": 2
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 142,
+ 504,
+ 166
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 141,
+ 505,
+ 157
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 141,
+ 308,
+ 157
+ ],
+ "score": 1.0,
+ "content": "We construct a conterexample. Consider a graph",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 309,
+ 144,
+ 318,
+ 154
+ ],
+ "score": 0.82,
+ "content": "G",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 318,
+ 141,
+ 375,
+ 157
+ ],
+ "score": 1.0,
+ "content": "consisting of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 375,
+ 144,
+ 406,
+ 154
+ ],
+ "score": 0.91,
+ "content": "N = 4",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 406,
+ 141,
+ 505,
+ 157
+ ],
+ "score": 1.0,
+ "content": "nodes whose adjacency",
+ "type": "text"
+ }
+ ],
+ "index": 5
+ },
+ {
+ "bbox": [
+ 105,
+ 154,
+ 145,
+ 166
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 154,
+ 145,
+ 166
+ ],
+ "score": 1.0,
+ "content": "matrix is",
+ "type": "text"
+ }
+ ],
+ "index": 6
+ }
+ ],
+ "index": 5.5
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 259,
+ 169,
+ 351,
+ 217
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 259,
+ 169,
+ 351,
+ 217
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 259,
+ 169,
+ 351,
+ 217
+ ],
+ "score": 0.94,
+ "content": "A = { \\left[ \\begin{array} { l l l l } { 1 } & { 1 } & { 1 } & { 1 } \\\\ { 1 } & { 1 } & { 1 } & { 0 } \\\\ { 1 } & { 1 } & { 1 } & { 0 } \\\\ { 1 } & { 0 } & { 0 } & { 1 } \\end{array} \\right] } ~ .",
+ "type": "interline_equation",
+ "image_path": "7a5e18ea3dc07a33bee281a590ef95fa607fa5f03975c7fdc507c908ced32b73.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 7.5,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 259,
+ 169,
+ 351,
+ 193.0
+ ],
+ "spans": [],
+ "index": 7
+ },
+ {
+ "bbox": [
+ 259,
+ 193.0,
+ 351,
+ 217.0
+ ],
+ "spans": [],
+ "index": 8
+ }
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 221,
+ 505,
+ 278
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 222,
+ 505,
+ 234
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 222,
+ 146,
+ 234
+ ],
+ "score": 1.0,
+ "content": "Note that",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 147,
+ 223,
+ 156,
+ 232
+ ],
+ "score": 0.83,
+ "content": "G",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 156,
+ 222,
+ 231,
+ 234
+ ],
+ "score": 1.0,
+ "content": "is connected (i.e.,",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 231,
+ 222,
+ 258,
+ 232
+ ],
+ "score": 0.89,
+ "content": "k = 1",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 258,
+ 222,
+ 437,
+ 234
+ ],
+ "score": 1.0,
+ "content": ") and is not bipartite. We make a GCN with",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 437,
+ 223,
+ 466,
+ 232
+ ],
+ "score": 0.91,
+ "content": "F = 3",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 466,
+ 222,
+ 505,
+ 234
+ ],
+ "score": 1.0,
+ "content": "channels",
+ "type": "text"
+ }
+ ],
+ "index": 9
+ },
+ {
+ "bbox": [
+ 106,
+ 233,
+ 505,
+ 246
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 233,
+ 237,
+ 246
+ ],
+ "score": 1.0,
+ "content": "and whose weight matrices are",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 237,
+ 234,
+ 279,
+ 245
+ ],
+ "score": 0.92,
+ "content": "W _ { n } \\ = \\ I _ { 3 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 279,
+ 233,
+ 435,
+ 246
+ ],
+ "score": 1.0,
+ "content": "(the identity matrix of size 3) for all",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 435,
+ 234,
+ 466,
+ 244
+ ],
+ "score": 0.89,
+ "content": "n \\in \\mathbb N",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 466,
+ 233,
+ 505,
+ 246
+ ],
+ "score": 1.0,
+ "content": ". For the",
+ "type": "text"
+ }
+ ],
+ "index": 10
+ },
+ {
+ "bbox": [
+ 106,
+ 244,
+ 505,
+ 256
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 244,
+ 207,
+ 256
+ ],
+ "score": 1.0,
+ "content": "distribution of the input",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 207,
+ 245,
+ 217,
+ 254
+ ],
+ "score": 0.8,
+ "content": "X",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 217,
+ 244,
+ 505,
+ 256
+ ],
+ "score": 1.0,
+ "content": ", we consider an absolutely continuous distribution with respect to the",
+ "type": "text"
+ }
+ ],
+ "index": 11
+ },
+ {
+ "bbox": [
+ 104,
+ 252,
+ 506,
+ 268
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 252,
+ 199,
+ 268
+ ],
+ "score": 1.0,
+ "content": "Lebesgue measure on",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 199,
+ 254,
+ 222,
+ 265
+ ],
+ "score": 0.9,
+ "content": "\\mathbb { R } ^ { 4 \\times 3 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 223,
+ 252,
+ 266,
+ 268
+ ],
+ "score": 1.0,
+ "content": "such that",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 266,
+ 255,
+ 339,
+ 267
+ ],
+ "score": 0.92,
+ "content": "P ( X \\geq 0 ) ~ { \\stackrel { . } { > } } ~ 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 339,
+ 252,
+ 368,
+ 268
+ ],
+ "score": 1.0,
+ "content": "(here,",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 368,
+ 255,
+ 402,
+ 266
+ ],
+ "score": 0.9,
+ "content": "X ~ \\geq ~ 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 402,
+ 252,
+ 506,
+ 268
+ ],
+ "score": 1.0,
+ "content": "means the element-wise",
+ "type": "text"
+ }
+ ],
+ "index": 12
+ },
+ {
+ "bbox": [
+ 105,
+ 266,
+ 446,
+ 279
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 266,
+ 446,
+ 279
+ ],
+ "score": 1.0,
+ "content": "comparison). For example, the standard Gaussian distribution satisfies the condition.",
+ "type": "text"
+ }
+ ],
+ "index": 13
+ }
+ ],
+ "index": 11
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 281,
+ 505,
+ 327
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 282,
+ 505,
+ 295
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 282,
+ 131,
+ 295
+ ],
+ "score": 1.0,
+ "content": "Since",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 132,
+ 283,
+ 159,
+ 294
+ ],
+ "score": 0.9,
+ "content": "L \\geq 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 160,
+ 282,
+ 200,
+ 295
+ ],
+ "score": 1.0,
+ "content": ", we have",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 200,
+ 283,
+ 248,
+ 294
+ ],
+ "score": 0.91,
+ "content": "Y _ { n } = L ^ { n } X",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 249,
+ 282,
+ 259,
+ 295
+ ],
+ "score": 1.0,
+ "content": "if",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 259,
+ 283,
+ 289,
+ 294
+ ],
+ "score": 0.9,
+ "content": "X \\geq 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 290,
+ 282,
+ 311,
+ 295
+ ],
+ "score": 1.0,
+ "content": ". Let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 312,
+ 282,
+ 364,
+ 293
+ ],
+ "score": 0.91,
+ "content": "{ \\cal L } = { \\cal P } ^ { \\top } \\Delta { \\cal P }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 364,
+ 282,
+ 468,
+ 295
+ ],
+ "score": 1.0,
+ "content": "be the diagonalization of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 469,
+ 284,
+ 477,
+ 293
+ ],
+ "score": 0.82,
+ "content": "L",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 477,
+ 282,
+ 505,
+ 295
+ ],
+ "score": 1.0,
+ "content": "where",
+ "type": "text"
+ }
+ ],
+ "index": 14
+ },
+ {
+ "bbox": [
+ 107,
+ 293,
+ 504,
+ 307
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 294,
+ 149,
+ 306
+ ],
+ "score": 0.92,
+ "content": "P \\in O ( 4 )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 149,
+ 293,
+ 311,
+ 307
+ ],
+ "score": 1.0,
+ "content": "is an orthogonal matrix of size 4. Since",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 312,
+ 294,
+ 365,
+ 306
+ ],
+ "score": 0.81,
+ "content": "\\mathrm { r a n k } ( L ) = 3",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 365,
+ 293,
+ 405,
+ 307
+ ],
+ "score": 1.0,
+ "content": ", we have",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 405,
+ 294,
+ 464,
+ 306
+ ],
+ "score": 0.91,
+ "content": "{ \\bar { \\operatorname { r a n k } } } ( \\Lambda ^ { n } ) = 3",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 464,
+ 293,
+ 496,
+ 307
+ ],
+ "score": 1.0,
+ "content": "for any",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 497,
+ 296,
+ 504,
+ 304
+ ],
+ "score": 0.72,
+ "content": "n",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 15
+ },
+ {
+ "bbox": [
+ 105,
+ 305,
+ 505,
+ 318
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 305,
+ 189,
+ 318
+ ],
+ "score": 1.0,
+ "content": "(we can assume that",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 189,
+ 305,
+ 224,
+ 316
+ ],
+ "score": 0.93,
+ "content": "\\Lambda _ { 4 4 } = 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 224,
+ 305,
+ 459,
+ 318
+ ],
+ "score": 1.0,
+ "content": "without loss of generality). Therefore, under the condition",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 459,
+ 306,
+ 487,
+ 316
+ ],
+ "score": 0.91,
+ "content": "X \\geq 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 487,
+ 305,
+ 505,
+ 318
+ ],
+ "score": 1.0,
+ "content": ", we",
+ "type": "text"
+ }
+ ],
+ "index": 16
+ },
+ {
+ "bbox": [
+ 105,
+ 315,
+ 129,
+ 328
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 315,
+ 129,
+ 328
+ ],
+ "score": 1.0,
+ "content": "have",
+ "type": "text"
+ }
+ ],
+ "index": 17
+ }
+ ],
+ "index": 15.5
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 149,
+ 331,
+ 461,
+ 366
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 149,
+ 331,
+ 461,
+ 366
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 149,
+ 331,
+ 461,
+ 366
+ ],
+ "score": 0.87,
+ "content": "\\begin{array} { r l } & { \\mathrm { r a n k } ( Y _ { n } ) = 3 \\iff \\mathrm { r a n k } ( P ^ { \\top } \\Lambda ^ { n } P X ) = 3 } \\\\ & { \\qquad \\iff X \\in \\{ P ^ { - 1 } \\left[ B \\quad v \\right] ^ { \\top } | B \\in \\mathbb { R } ^ { 3 \\times 3 } \\mathrm { i s ~ i n v e r t i b l e } , v \\in \\mathbb { R } ^ { 3 } \\} . } \\end{array}",
+ "type": "interline_equation",
+ "image_path": "a6c4bcffddae18f22db6c3010b3d23d09c517d0a82c87531f0c12ddcfa294be5.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 19,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 149,
+ 331,
+ 461,
+ 342.6666666666667
+ ],
+ "spans": [],
+ "index": 18
+ },
+ {
+ "bbox": [
+ 149,
+ 342.6666666666667,
+ 461,
+ 354.33333333333337
+ ],
+ "spans": [],
+ "index": 19
+ },
+ {
+ "bbox": [
+ 149,
+ 354.33333333333337,
+ 461,
+ 366.00000000000006
+ ],
+ "spans": [],
+ "index": 20
+ }
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 370,
+ 505,
+ 415
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 370,
+ 505,
+ 383
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 370,
+ 290,
+ 383
+ ],
+ "score": 1.0,
+ "content": "Note that the last condition is independent of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 291,
+ 373,
+ 298,
+ 380
+ ],
+ "score": 0.78,
+ "content": "n",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 298,
+ 370,
+ 505,
+ 383
+ ],
+ "score": 1.0,
+ "content": ". Since the set of invertible matrices is dense in the",
+ "type": "text"
+ }
+ ],
+ "index": 21
+ },
+ {
+ "bbox": [
+ 105,
+ 381,
+ 505,
+ 393
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 381,
+ 505,
+ 393
+ ],
+ "score": 1.0,
+ "content": "set of all matrices of the same size (with respect to the standard topology of the Euclidean space),",
+ "type": "text"
+ }
+ ],
+ "index": 22
+ },
+ {
+ "bbox": [
+ 105,
+ 392,
+ 507,
+ 406
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 392,
+ 141,
+ 406
+ ],
+ "score": 1.0,
+ "content": "we have",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 142,
+ 392,
+ 216,
+ 405
+ ],
+ "score": 0.92,
+ "content": "P ( \\{ \\mathrm { r a n k } ( Y _ { n } ) = 3 ",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 216,
+ 392,
+ 244,
+ 406
+ ],
+ "score": 1.0,
+ "content": "for all",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 245,
+ 392,
+ 298,
+ 405
+ ],
+ "score": 0.79,
+ "content": "n \\in \\mathbb { N } \\} ) > 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 298,
+ 392,
+ 381,
+ 406
+ ],
+ "score": 1.0,
+ "content": ". Therefore, we have",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 382,
+ 392,
+ 477,
+ 405
+ ],
+ "score": 0.77,
+ "content": "\\begin{array} { r } { \\operatorname* { l i m } _ { n \\infty } \\operatorname { r a n k } ( Y _ { n } ) = 3 } \\end{array}",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 477,
+ 392,
+ 507,
+ 406
+ ],
+ "score": 1.0,
+ "content": "with a",
+ "type": "text"
+ }
+ ],
+ "index": 23
+ },
+ {
+ "bbox": [
+ 105,
+ 403,
+ 505,
+ 417
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 403,
+ 193,
+ 417
+ ],
+ "score": 1.0,
+ "content": "non-zero probability.",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 494,
+ 403,
+ 505,
+ 414
+ ],
+ "score": 1.0,
+ "content": "□",
+ "type": "text"
+ }
+ ],
+ "index": 24
+ }
+ ],
+ "index": 22.5
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 108,
+ 431,
+ 245,
+ 444
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 430,
+ 245,
+ 446
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 430,
+ 245,
+ 446
+ ],
+ "score": 1.0,
+ "content": "D PROOF OF THEOREM 3",
+ "type": "text"
+ }
+ ],
+ "index": 25
+ }
+ ],
+ "index": 25
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 456,
+ 505,
+ 501
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 455,
+ 505,
+ 469
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 455,
+ 505,
+ 469
+ ],
+ "score": 1.0,
+ "content": "We follow the proof of Theorem 2 of Chung & Radcliffe (2011). The idea is to relate the spectral",
+ "type": "text"
+ }
+ ],
+ "index": 26
+ },
+ {
+ "bbox": [
+ 106,
+ 468,
+ 505,
+ 480
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 468,
+ 505,
+ 480
+ ],
+ "score": 1.0,
+ "content": "distribution of the normalized Laplacian with that of its expected version. Since we can compute",
+ "type": "text"
+ }
+ ],
+ "index": 27
+ },
+ {
+ "bbox": [
+ 105,
+ 479,
+ 505,
+ 492
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 479,
+ 505,
+ 492
+ ],
+ "score": 1.0,
+ "content": "the latter one explicitly for the Erdos-R ˝ enyi graph, we can derive the convergence of spectra. We ´",
+ "type": "text"
+ }
+ ],
+ "index": 28
+ },
+ {
+ "bbox": [
+ 105,
+ 490,
+ 483,
+ 503
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 490,
+ 483,
+ 503
+ ],
+ "score": 1.0,
+ "content": "employ this technique and derive similar conclusion for the augmented normalized Laplacian.",
+ "type": "text"
+ }
+ ],
+ "index": 29
+ }
+ ],
+ "index": 27.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 506,
+ 505,
+ 626
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 506,
+ 505,
+ 519
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 506,
+ 444,
+ 519
+ ],
+ "score": 1.0,
+ "content": "First, we consider genral random graphs not restricted to Erdos-R ˝ enyi graphs. Let ´",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 445,
+ 507,
+ 483,
+ 518
+ ],
+ "score": 0.9,
+ "content": "N \\in { \\mathbb { N } } _ { + }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 484,
+ 506,
+ 505,
+ 519
+ ],
+ "score": 1.0,
+ "content": ", and",
+ "type": "text"
+ }
+ ],
+ "index": 30
+ },
+ {
+ "bbox": [
+ 107,
+ 515,
+ 506,
+ 533
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 518,
+ 177,
+ 531
+ ],
+ "score": 0.92,
+ "content": "P = ( p _ { i j } ) _ { i , j \\in [ N ] }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 177,
+ 515,
+ 387,
+ 533
+ ],
+ "score": 1.0,
+ "content": "be a non-negative symmetric matrix (meaning that",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 387,
+ 518,
+ 421,
+ 530
+ ],
+ "score": 0.92,
+ "content": "p _ { i j } \\geq 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 421,
+ 515,
+ 455,
+ 533
+ ],
+ "score": 1.0,
+ "content": "for any",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 455,
+ 518,
+ 500,
+ 529
+ ],
+ "score": 0.9,
+ "content": "i , j \\in [ N ] )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 501,
+ 515,
+ 506,
+ 533
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 31
+ },
+ {
+ "bbox": [
+ 105,
+ 528,
+ 506,
+ 542
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 528,
+ 123,
+ 542
+ ],
+ "score": 1.0,
+ "content": "Let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 123,
+ 530,
+ 133,
+ 539
+ ],
+ "score": 0.76,
+ "content": "G",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 133,
+ 528,
+ 287,
+ 542
+ ],
+ "score": 1.0,
+ "content": "be an undirected random graph with",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 288,
+ 529,
+ 298,
+ 539
+ ],
+ "score": 0.8,
+ "content": "N",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 298,
+ 528,
+ 438,
+ 542
+ ],
+ "score": 1.0,
+ "content": "nodes such that an edge between",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 439,
+ 530,
+ 444,
+ 539
+ ],
+ "score": 0.73,
+ "content": "i",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 444,
+ 528,
+ 463,
+ 542
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 464,
+ 530,
+ 470,
+ 541
+ ],
+ "score": 0.82,
+ "content": "j",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 470,
+ 528,
+ 506,
+ 542
+ ],
+ "score": 1.0,
+ "content": "is inde-",
+ "type": "text"
+ }
+ ],
+ "index": 32
+ },
+ {
+ "bbox": [
+ 105,
+ 540,
+ 506,
+ 553
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 540,
+ 248,
+ 553
+ ],
+ "score": 1.0,
+ "content": "pendently present with probability",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 248,
+ 541,
+ 262,
+ 551
+ ],
+ "score": 0.82,
+ "content": "p _ { i j }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 262,
+ 540,
+ 284,
+ 553
+ ],
+ "score": 1.0,
+ "content": ". Let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 285,
+ 540,
+ 294,
+ 550
+ ],
+ "score": 0.8,
+ "content": "A",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 294,
+ 540,
+ 312,
+ 553
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 313,
+ 540,
+ 322,
+ 550
+ ],
+ "score": 0.82,
+ "content": "D",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 323,
+ 540,
+ 506,
+ 553
+ ],
+ "score": 1.0,
+ "content": "be the adjacency and the degree matrices of",
+ "type": "text"
+ }
+ ],
+ "index": 33
+ },
+ {
+ "bbox": [
+ 107,
+ 550,
+ 506,
+ 565
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 551,
+ 115,
+ 560
+ ],
+ "score": 0.77,
+ "content": "G",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 116,
+ 550,
+ 207,
+ 565
+ ],
+ "score": 1.0,
+ "content": ", respectively (that is,",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 208,
+ 551,
+ 275,
+ 563
+ ],
+ "score": 0.89,
+ "content": "A _ { i j } \\sim \\mathrm { B e r } ( p _ { i j } )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 276,
+ 550,
+ 485,
+ 565
+ ],
+ "score": 1.0,
+ "content": ", i.i.d.). Define the expected node degree of node",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 485,
+ 551,
+ 491,
+ 560
+ ],
+ "score": 0.7,
+ "content": "i",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 491,
+ 550,
+ 506,
+ 565
+ ],
+ "score": 1.0,
+ "content": "by",
+ "type": "text"
+ }
+ ],
+ "index": 34
+ },
+ {
+ "bbox": [
+ 106,
+ 562,
+ 507,
+ 579
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 562,
+ 174,
+ 578
+ ],
+ "score": 0.92,
+ "content": "\\begin{array} { r } { t _ { i } : = \\sum _ { j = 1 } ^ { N } p _ { i j } } \\end{array}",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 174,
+ 563,
+ 200,
+ 579
+ ],
+ "score": 1.0,
+ "content": ". Let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 200,
+ 563,
+ 263,
+ 576
+ ],
+ "score": 0.85,
+ "content": "{ \\tilde { A } } : = A + I _ { N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 263,
+ 563,
+ 269,
+ 579
+ ],
+ "score": 1.0,
+ "content": ",",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 269,
+ 563,
+ 335,
+ 576
+ ],
+ "score": 0.9,
+ "content": "{ \\tilde { D } } : = D + I _ { N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 335,
+ 563,
+ 384,
+ 579
+ ],
+ "score": 1.0,
+ "content": "and define",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 384,
+ 562,
+ 486,
+ 576
+ ],
+ "score": 0.93,
+ "content": "{ \\bar { A } } : = \\mathbb { E } [ { \\tilde { A } } ] = P + I _ { N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 486,
+ 563,
+ 507,
+ 579
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ }
+ ],
+ "index": 35
+ },
+ {
+ "bbox": [
+ 106,
+ 576,
+ 506,
+ 594
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 578,
+ 256,
+ 591
+ ],
+ "score": 0.88,
+ "content": "\\bar { D } : = \\mathbb { E } [ \\tilde { D } ] = \\mathrm { d i a g } ( t _ { 1 } , \\dots , t _ { N } ) + I _ { N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 257,
+ 576,
+ 506,
+ 594
+ ],
+ "score": 1.0,
+ "content": "correspondingly. We define the augmented normalized Lapla-",
+ "type": "text"
+ }
+ ],
+ "index": 36
+ },
+ {
+ "bbox": [
+ 104,
+ 589,
+ 503,
+ 604
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 589,
+ 126,
+ 604
+ ],
+ "score": 1.0,
+ "content": "cian",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 127,
+ 591,
+ 137,
+ 602
+ ],
+ "score": 0.83,
+ "content": "\\tilde { \\Delta }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 137,
+ 589,
+ 150,
+ 604
+ ],
+ "score": 1.0,
+ "content": "of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 151,
+ 592,
+ 160,
+ 602
+ ],
+ "score": 0.8,
+ "content": "G",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 160,
+ 589,
+ 176,
+ 604
+ ],
+ "score": 1.0,
+ "content": "by",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 176,
+ 590,
+ 280,
+ 603
+ ],
+ "score": 0.85,
+ "content": "\\tilde { \\Delta } : = I _ { N } - \\tilde { D } ^ { - \\frac { 1 } { 2 } } \\tilde { A } \\tilde { D } ^ { - \\frac { 1 } { 2 } }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 280,
+ 589,
+ 399,
+ 604
+ ],
+ "score": 1.0,
+ "content": "and its expected version by",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 399,
+ 590,
+ 503,
+ 603
+ ],
+ "score": 0.92,
+ "content": "\\bar { \\Delta } : = I _ { N } - \\bar { D } ^ { - \\frac { 1 } { 2 } } \\bar { A } \\bar { D } ^ { - \\frac { 1 } { 2 } }",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 37
+ },
+ {
+ "bbox": [
+ 104,
+ 600,
+ 505,
+ 617
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 600,
+ 223,
+ 617
+ ],
+ "score": 1.0,
+ "content": "9. For a symmetric matrix",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 223,
+ 602,
+ 266,
+ 613
+ ],
+ "score": 0.88,
+ "content": "X ~ \\in ~ \\mathbb { R } ^ { N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 266,
+ 600,
+ 505,
+ 617
+ ],
+ "score": 1.0,
+ "content": ", we define its eigenvalues, sorted in ascending order by",
+ "type": "text"
+ }
+ ],
+ "index": 38
+ },
+ {
+ "bbox": [
+ 106,
+ 613,
+ 419,
+ 627
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 613,
+ 203,
+ 626
+ ],
+ "score": 0.93,
+ "content": "\\lambda _ { 1 } ( X ) \\leq \\cdots \\leq \\lambda _ { N } ( X )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 203,
+ 613,
+ 305,
+ 627
+ ],
+ "score": 1.0,
+ "content": "and its operator norm by",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 305,
+ 613,
+ 414,
+ 627
+ ],
+ "score": 0.93,
+ "content": "\\| X \\| = \\operatorname* { i n a x } _ { n \\in [ N ] } | \\lambda _ { n } ( X ) |",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 415,
+ 613,
+ 419,
+ 627
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 39
+ }
+ ],
+ "index": 34.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 628,
+ 505,
+ 663
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 627,
+ 505,
+ 642
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 627,
+ 360,
+ 642
+ ],
+ "score": 1.0,
+ "content": "Lemma 5 (Ref. Chung & Radcliffe (2011) Theorem 2). Let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 360,
+ 629,
+ 435,
+ 641
+ ],
+ "score": 0.89,
+ "content": "\\delta : = \\mathrm { m i n } _ { n \\in [ N ] } t _ { n }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 435,
+ 627,
+ 505,
+ 642
+ ],
+ "score": 1.0,
+ "content": "be the minimum",
+ "type": "text"
+ }
+ ],
+ "index": 40
+ },
+ {
+ "bbox": [
+ 105,
+ 641,
+ 505,
+ 654
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 641,
+ 183,
+ 654
+ ],
+ "score": 1.0,
+ "content": "expected degree of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 183,
+ 641,
+ 192,
+ 651
+ ],
+ "score": 0.71,
+ "content": "G",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 192,
+ 641,
+ 211,
+ 654
+ ],
+ "score": 1.0,
+ "content": ". Set",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 211,
+ 641,
+ 310,
+ 653
+ ],
+ "score": 0.92,
+ "content": "k ( \\varepsilon ) : = 3 ( 1 + \\log ( 4 / \\varepsilon ) )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 311,
+ 641,
+ 371,
+ 654
+ ],
+ "score": 1.0,
+ "content": ". Then, for any",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 372,
+ 642,
+ 396,
+ 652
+ ],
+ "score": 0.83,
+ "content": "\\varepsilon > 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 396,
+ 641,
+ 399,
+ 654
+ ],
+ "score": 1.0,
+ "content": ",",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 400,
+ 641,
+ 487,
+ 653
+ ],
+ "score": 0.9,
+ "content": "i f { \\dot { \\delta } } + { \\dot { 1 } } > k ( \\varepsilon ) \\log N",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 487,
+ 641,
+ 505,
+ 654
+ ],
+ "score": 1.0,
+ "content": ", we",
+ "type": "text"
+ }
+ ],
+ "index": 41
+ },
+ {
+ "bbox": [
+ 105,
+ 651,
+ 129,
+ 665
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 651,
+ 129,
+ 665
+ ],
+ "score": 1.0,
+ "content": "have",
+ "type": "text"
+ }
+ ],
+ "index": 42
+ }
+ ],
+ "index": 41
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 215,
+ 667,
+ 396,
+ 696
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 215,
+ 667,
+ 396,
+ 696
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 215,
+ 667,
+ 396,
+ 696
+ ],
+ "score": 0.92,
+ "content": "\\operatorname* { m a x } _ { n \\in [ N ] } \\left| \\lambda _ { n } ( \\tilde { \\Delta } ) - \\lambda _ { n } ( \\bar { \\Delta } ) \\right| \\leq 4 \\sqrt { \\frac { 3 \\log ( 4 N / \\varepsilon ) } { \\delta + 1 } }",
+ "type": "interline_equation",
+ "image_path": "faa530371dd9306d6740988ca847f650db8a9d0bcc8578a2693011979e058fb8.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 43.5,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 215,
+ 667,
+ 396,
+ 681.5
+ ],
+ "spans": [],
+ "index": 43
+ },
+ {
+ "bbox": [
+ 215,
+ 681.5,
+ 396,
+ 696.0
+ ],
+ "spans": [],
+ "index": 44
+ }
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 702,
+ 229,
+ 714
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 701,
+ 227,
+ 715
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 701,
+ 204,
+ 715
+ ],
+ "score": 1.0,
+ "content": "with probability at least",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 204,
+ 703,
+ 227,
+ 713
+ ],
+ "score": 0.85,
+ "content": "1 - \\varepsilon",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 45
+ }
+ ],
+ "index": 45
+ }
+ ],
+ "page_idx": 17,
+ "page_size": [
+ 612,
+ 792
+ ],
+ "discarded_blocks": [
+ {
+ "type": "discarded",
+ "bbox": [
+ 107,
+ 27,
+ 293,
+ 37
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 26,
+ 294,
+ 38
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 26,
+ 294,
+ 38
+ ],
+ "score": 1.0,
+ "content": "Published as a conference paper at ICLR 2020",
+ "type": "text"
+ }
+ ]
+ }
+ ]
+ },
+ {
+ "type": "discarded",
+ "bbox": [
+ 117,
+ 720,
+ 388,
+ 732
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 118,
+ 718,
+ 388,
+ 734
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 118,
+ 718,
+ 158,
+ 734
+ ],
+ "score": 1.0,
+ "content": "9Note that",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 158,
+ 720,
+ 198,
+ 732
+ ],
+ "score": 0.92,
+ "content": "\\mathbb { E } [ \\tilde { \\Delta } ] \\neq \\bar { \\Delta }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 199,
+ 718,
+ 351,
+ 734
+ ],
+ "score": 1.0,
+ "content": "in general due to the dependence between",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 352,
+ 721,
+ 359,
+ 730
+ ],
+ "score": 0.84,
+ "content": "\\tilde { A }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 360,
+ 718,
+ 375,
+ 734
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 376,
+ 720,
+ 384,
+ 730
+ ],
+ "score": 0.85,
+ "content": "\\tilde { D }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 385,
+ 718,
+ 388,
+ 734
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ]
+ }
+ ]
+ },
+ {
+ "type": "discarded",
+ "bbox": [
+ 300,
+ 751,
+ 311,
+ 760
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 299,
+ 750,
+ 313,
+ 763
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 299,
+ 750,
+ 313,
+ 763
+ ],
+ "score": 1.0,
+ "content": "18",
+ "type": "text"
+ }
+ ]
+ }
+ ]
+ }
+ ],
+ "para_blocks": [
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 81,
+ 505,
+ 138
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 107,
+ 79,
+ 506,
+ 96
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 81,
+ 163,
+ 93
+ ],
+ "score": 0.91,
+ "content": "W _ { n } \\in \\mathbb { R } ^ { F \\times F }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 164,
+ 79,
+ 250,
+ 96
+ ],
+ "score": 1.0,
+ "content": "be the weight of the",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 250,
+ 84,
+ 258,
+ 92
+ ],
+ "score": 0.78,
+ "content": "n",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 258,
+ 79,
+ 308,
+ 96
+ ],
+ "score": 1.0,
+ "content": "-th layer for",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 308,
+ 83,
+ 344,
+ 94
+ ],
+ "score": 0.91,
+ "content": "n \\in \\mathbb { N } _ { + }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 344,
+ 79,
+ 406,
+ 96
+ ],
+ "score": 1.0,
+ "content": ". For the input",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 407,
+ 81,
+ 459,
+ 93
+ ],
+ "score": 0.91,
+ "content": "X \\in \\mathbb { R } ^ { N \\times F }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 459,
+ 79,
+ 506,
+ 96
+ ],
+ "score": 1.0,
+ "content": ", we define",
+ "type": "text"
+ }
+ ],
+ "index": 0
+ },
+ {
+ "bbox": [
+ 104,
+ 89,
+ 504,
+ 108
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 89,
+ 150,
+ 108
+ ],
+ "score": 1.0,
+ "content": "the output",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 150,
+ 93,
+ 204,
+ 105
+ ],
+ "score": 0.93,
+ "content": "Y _ { n } \\in \\mathbb { R } ^ { N \\times F }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 204,
+ 89,
+ 232,
+ 108
+ ],
+ "score": 1.0,
+ "content": "of the",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 232,
+ 95,
+ 239,
+ 104
+ ],
+ "score": 0.77,
+ "content": "n",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 240,
+ 89,
+ 331,
+ 108
+ ],
+ "score": 1.0,
+ "content": "-th layer of a GCN by",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 331,
+ 93,
+ 467,
+ 106
+ ],
+ "score": 0.91,
+ "content": "Y _ { n } = \\sigma ( L \\cdot \\cdot \\cdot \\sigma ( L X W _ { 0 } ) \\cdot \\cdot \\cdot W _ { n } )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 468,
+ 89,
+ 496,
+ 108
+ ],
+ "score": 1.0,
+ "content": "where",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 496,
+ 96,
+ 504,
+ 103
+ ],
+ "score": 0.74,
+ "content": "\\sigma",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 1
+ },
+ {
+ "bbox": [
+ 103,
+ 102,
+ 506,
+ 118
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 103,
+ 102,
+ 285,
+ 118
+ ],
+ "score": 1.0,
+ "content": "is the ReLU function. We assume the input",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 286,
+ 105,
+ 296,
+ 114
+ ],
+ "score": 0.85,
+ "content": "X",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 296,
+ 102,
+ 473,
+ 118
+ ],
+ "score": 1.0,
+ "content": "is drawn from a continuous distribution on",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 473,
+ 104,
+ 501,
+ 114
+ ],
+ "score": 0.9,
+ "content": "\\mathbb { R } ^ { N \\times F }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 501,
+ 102,
+ 506,
+ 118
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 2
+ },
+ {
+ "bbox": [
+ 105,
+ 113,
+ 506,
+ 129
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 113,
+ 268,
+ 129
+ ],
+ "score": 1.0,
+ "content": "Then, the theorem claims that we have",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 268,
+ 115,
+ 367,
+ 127
+ ],
+ "score": 0.73,
+ "content": "\\begin{array} { r } { \\operatorname* { l i m } _ { n \\to \\infty } \\operatorname { r a n k } ( Y _ { n } ) = k } \\end{array}",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 367,
+ 113,
+ 506,
+ 129
+ ],
+ "score": 1.0,
+ "content": "almost surely with respect to the",
+ "type": "text"
+ }
+ ],
+ "index": 3
+ },
+ {
+ "bbox": [
+ 106,
+ 126,
+ 180,
+ 138
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 126,
+ 165,
+ 138
+ ],
+ "score": 1.0,
+ "content": "distribution of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 166,
+ 127,
+ 176,
+ 136
+ ],
+ "score": 0.82,
+ "content": "X",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 176,
+ 126,
+ 180,
+ 138
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 4
+ }
+ ],
+ "index": 2,
+ "bbox_fs": [
+ 103,
+ 79,
+ 506,
+ 138
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 142,
+ 504,
+ 166
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 141,
+ 505,
+ 157
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 141,
+ 308,
+ 157
+ ],
+ "score": 1.0,
+ "content": "We construct a conterexample. Consider a graph",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 309,
+ 144,
+ 318,
+ 154
+ ],
+ "score": 0.82,
+ "content": "G",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 318,
+ 141,
+ 375,
+ 157
+ ],
+ "score": 1.0,
+ "content": "consisting of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 375,
+ 144,
+ 406,
+ 154
+ ],
+ "score": 0.91,
+ "content": "N = 4",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 406,
+ 141,
+ 505,
+ 157
+ ],
+ "score": 1.0,
+ "content": "nodes whose adjacency",
+ "type": "text"
+ }
+ ],
+ "index": 5
+ },
+ {
+ "bbox": [
+ 105,
+ 154,
+ 145,
+ 166
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 154,
+ 145,
+ 166
+ ],
+ "score": 1.0,
+ "content": "matrix is",
+ "type": "text"
+ }
+ ],
+ "index": 6
+ }
+ ],
+ "index": 5.5,
+ "bbox_fs": [
+ 105,
+ 141,
+ 505,
+ 166
+ ]
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 259,
+ 169,
+ 351,
+ 217
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 259,
+ 169,
+ 351,
+ 217
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 259,
+ 169,
+ 351,
+ 217
+ ],
+ "score": 0.94,
+ "content": "A = { \\left[ \\begin{array} { l l l l } { 1 } & { 1 } & { 1 } & { 1 } \\\\ { 1 } & { 1 } & { 1 } & { 0 } \\\\ { 1 } & { 1 } & { 1 } & { 0 } \\\\ { 1 } & { 0 } & { 0 } & { 1 } \\end{array} \\right] } ~ .",
+ "type": "interline_equation",
+ "image_path": "7a5e18ea3dc07a33bee281a590ef95fa607fa5f03975c7fdc507c908ced32b73.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 7.5,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 259,
+ 169,
+ 351,
+ 193.0
+ ],
+ "spans": [],
+ "index": 7
+ },
+ {
+ "bbox": [
+ 259,
+ 193.0,
+ 351,
+ 217.0
+ ],
+ "spans": [],
+ "index": 8
+ }
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 221,
+ 505,
+ 278
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 222,
+ 505,
+ 234
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 222,
+ 146,
+ 234
+ ],
+ "score": 1.0,
+ "content": "Note that",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 147,
+ 223,
+ 156,
+ 232
+ ],
+ "score": 0.83,
+ "content": "G",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 156,
+ 222,
+ 231,
+ 234
+ ],
+ "score": 1.0,
+ "content": "is connected (i.e.,",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 231,
+ 222,
+ 258,
+ 232
+ ],
+ "score": 0.89,
+ "content": "k = 1",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 258,
+ 222,
+ 437,
+ 234
+ ],
+ "score": 1.0,
+ "content": ") and is not bipartite. We make a GCN with",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 437,
+ 223,
+ 466,
+ 232
+ ],
+ "score": 0.91,
+ "content": "F = 3",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 466,
+ 222,
+ 505,
+ 234
+ ],
+ "score": 1.0,
+ "content": "channels",
+ "type": "text"
+ }
+ ],
+ "index": 9
+ },
+ {
+ "bbox": [
+ 106,
+ 233,
+ 505,
+ 246
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 233,
+ 237,
+ 246
+ ],
+ "score": 1.0,
+ "content": "and whose weight matrices are",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 237,
+ 234,
+ 279,
+ 245
+ ],
+ "score": 0.92,
+ "content": "W _ { n } \\ = \\ I _ { 3 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 279,
+ 233,
+ 435,
+ 246
+ ],
+ "score": 1.0,
+ "content": "(the identity matrix of size 3) for all",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 435,
+ 234,
+ 466,
+ 244
+ ],
+ "score": 0.89,
+ "content": "n \\in \\mathbb N",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 466,
+ 233,
+ 505,
+ 246
+ ],
+ "score": 1.0,
+ "content": ". For the",
+ "type": "text"
+ }
+ ],
+ "index": 10
+ },
+ {
+ "bbox": [
+ 106,
+ 244,
+ 505,
+ 256
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 244,
+ 207,
+ 256
+ ],
+ "score": 1.0,
+ "content": "distribution of the input",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 207,
+ 245,
+ 217,
+ 254
+ ],
+ "score": 0.8,
+ "content": "X",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 217,
+ 244,
+ 505,
+ 256
+ ],
+ "score": 1.0,
+ "content": ", we consider an absolutely continuous distribution with respect to the",
+ "type": "text"
+ }
+ ],
+ "index": 11
+ },
+ {
+ "bbox": [
+ 104,
+ 252,
+ 506,
+ 268
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 252,
+ 199,
+ 268
+ ],
+ "score": 1.0,
+ "content": "Lebesgue measure on",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 199,
+ 254,
+ 222,
+ 265
+ ],
+ "score": 0.9,
+ "content": "\\mathbb { R } ^ { 4 \\times 3 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 223,
+ 252,
+ 266,
+ 268
+ ],
+ "score": 1.0,
+ "content": "such that",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 266,
+ 255,
+ 339,
+ 267
+ ],
+ "score": 0.92,
+ "content": "P ( X \\geq 0 ) ~ { \\stackrel { . } { > } } ~ 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 339,
+ 252,
+ 368,
+ 268
+ ],
+ "score": 1.0,
+ "content": "(here,",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 368,
+ 255,
+ 402,
+ 266
+ ],
+ "score": 0.9,
+ "content": "X ~ \\geq ~ 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 402,
+ 252,
+ 506,
+ 268
+ ],
+ "score": 1.0,
+ "content": "means the element-wise",
+ "type": "text"
+ }
+ ],
+ "index": 12
+ },
+ {
+ "bbox": [
+ 105,
+ 266,
+ 446,
+ 279
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 266,
+ 446,
+ 279
+ ],
+ "score": 1.0,
+ "content": "comparison). For example, the standard Gaussian distribution satisfies the condition.",
+ "type": "text"
+ }
+ ],
+ "index": 13
+ }
+ ],
+ "index": 11,
+ "bbox_fs": [
+ 104,
+ 222,
+ 506,
+ 279
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 281,
+ 505,
+ 327
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 282,
+ 505,
+ 295
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 282,
+ 131,
+ 295
+ ],
+ "score": 1.0,
+ "content": "Since",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 132,
+ 283,
+ 159,
+ 294
+ ],
+ "score": 0.9,
+ "content": "L \\geq 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 160,
+ 282,
+ 200,
+ 295
+ ],
+ "score": 1.0,
+ "content": ", we have",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 200,
+ 283,
+ 248,
+ 294
+ ],
+ "score": 0.91,
+ "content": "Y _ { n } = L ^ { n } X",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 249,
+ 282,
+ 259,
+ 295
+ ],
+ "score": 1.0,
+ "content": "if",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 259,
+ 283,
+ 289,
+ 294
+ ],
+ "score": 0.9,
+ "content": "X \\geq 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 290,
+ 282,
+ 311,
+ 295
+ ],
+ "score": 1.0,
+ "content": ". Let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 312,
+ 282,
+ 364,
+ 293
+ ],
+ "score": 0.91,
+ "content": "{ \\cal L } = { \\cal P } ^ { \\top } \\Delta { \\cal P }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 364,
+ 282,
+ 468,
+ 295
+ ],
+ "score": 1.0,
+ "content": "be the diagonalization of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 469,
+ 284,
+ 477,
+ 293
+ ],
+ "score": 0.82,
+ "content": "L",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 477,
+ 282,
+ 505,
+ 295
+ ],
+ "score": 1.0,
+ "content": "where",
+ "type": "text"
+ }
+ ],
+ "index": 14
+ },
+ {
+ "bbox": [
+ 107,
+ 293,
+ 504,
+ 307
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 294,
+ 149,
+ 306
+ ],
+ "score": 0.92,
+ "content": "P \\in O ( 4 )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 149,
+ 293,
+ 311,
+ 307
+ ],
+ "score": 1.0,
+ "content": "is an orthogonal matrix of size 4. Since",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 312,
+ 294,
+ 365,
+ 306
+ ],
+ "score": 0.81,
+ "content": "\\mathrm { r a n k } ( L ) = 3",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 365,
+ 293,
+ 405,
+ 307
+ ],
+ "score": 1.0,
+ "content": ", we have",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 405,
+ 294,
+ 464,
+ 306
+ ],
+ "score": 0.91,
+ "content": "{ \\bar { \\operatorname { r a n k } } } ( \\Lambda ^ { n } ) = 3",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 464,
+ 293,
+ 496,
+ 307
+ ],
+ "score": 1.0,
+ "content": "for any",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 497,
+ 296,
+ 504,
+ 304
+ ],
+ "score": 0.72,
+ "content": "n",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 15
+ },
+ {
+ "bbox": [
+ 105,
+ 305,
+ 505,
+ 318
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 305,
+ 189,
+ 318
+ ],
+ "score": 1.0,
+ "content": "(we can assume that",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 189,
+ 305,
+ 224,
+ 316
+ ],
+ "score": 0.93,
+ "content": "\\Lambda _ { 4 4 } = 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 224,
+ 305,
+ 459,
+ 318
+ ],
+ "score": 1.0,
+ "content": "without loss of generality). Therefore, under the condition",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 459,
+ 306,
+ 487,
+ 316
+ ],
+ "score": 0.91,
+ "content": "X \\geq 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 487,
+ 305,
+ 505,
+ 318
+ ],
+ "score": 1.0,
+ "content": ", we",
+ "type": "text"
+ }
+ ],
+ "index": 16
+ },
+ {
+ "bbox": [
+ 105,
+ 315,
+ 129,
+ 328
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 315,
+ 129,
+ 328
+ ],
+ "score": 1.0,
+ "content": "have",
+ "type": "text"
+ }
+ ],
+ "index": 17
+ }
+ ],
+ "index": 15.5,
+ "bbox_fs": [
+ 105,
+ 282,
+ 505,
+ 328
+ ]
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 149,
+ 331,
+ 461,
+ 366
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 149,
+ 331,
+ 461,
+ 366
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 149,
+ 331,
+ 461,
+ 366
+ ],
+ "score": 0.87,
+ "content": "\\begin{array} { r l } & { \\mathrm { r a n k } ( Y _ { n } ) = 3 \\iff \\mathrm { r a n k } ( P ^ { \\top } \\Lambda ^ { n } P X ) = 3 } \\\\ & { \\qquad \\iff X \\in \\{ P ^ { - 1 } \\left[ B \\quad v \\right] ^ { \\top } | B \\in \\mathbb { R } ^ { 3 \\times 3 } \\mathrm { i s ~ i n v e r t i b l e } , v \\in \\mathbb { R } ^ { 3 } \\} . } \\end{array}",
+ "type": "interline_equation",
+ "image_path": "a6c4bcffddae18f22db6c3010b3d23d09c517d0a82c87531f0c12ddcfa294be5.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 19,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 149,
+ 331,
+ 461,
+ 342.6666666666667
+ ],
+ "spans": [],
+ "index": 18
+ },
+ {
+ "bbox": [
+ 149,
+ 342.6666666666667,
+ 461,
+ 354.33333333333337
+ ],
+ "spans": [],
+ "index": 19
+ },
+ {
+ "bbox": [
+ 149,
+ 354.33333333333337,
+ 461,
+ 366.00000000000006
+ ],
+ "spans": [],
+ "index": 20
+ }
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 370,
+ 505,
+ 415
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 370,
+ 505,
+ 383
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 370,
+ 290,
+ 383
+ ],
+ "score": 1.0,
+ "content": "Note that the last condition is independent of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 291,
+ 373,
+ 298,
+ 380
+ ],
+ "score": 0.78,
+ "content": "n",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 298,
+ 370,
+ 505,
+ 383
+ ],
+ "score": 1.0,
+ "content": ". Since the set of invertible matrices is dense in the",
+ "type": "text"
+ }
+ ],
+ "index": 21
+ },
+ {
+ "bbox": [
+ 105,
+ 381,
+ 505,
+ 393
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 381,
+ 505,
+ 393
+ ],
+ "score": 1.0,
+ "content": "set of all matrices of the same size (with respect to the standard topology of the Euclidean space),",
+ "type": "text"
+ }
+ ],
+ "index": 22
+ },
+ {
+ "bbox": [
+ 105,
+ 392,
+ 507,
+ 406
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 392,
+ 141,
+ 406
+ ],
+ "score": 1.0,
+ "content": "we have",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 142,
+ 392,
+ 216,
+ 405
+ ],
+ "score": 0.92,
+ "content": "P ( \\{ \\mathrm { r a n k } ( Y _ { n } ) = 3 ",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 216,
+ 392,
+ 244,
+ 406
+ ],
+ "score": 1.0,
+ "content": "for all",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 245,
+ 392,
+ 298,
+ 405
+ ],
+ "score": 0.79,
+ "content": "n \\in \\mathbb { N } \\} ) > 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 298,
+ 392,
+ 381,
+ 406
+ ],
+ "score": 1.0,
+ "content": ". Therefore, we have",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 382,
+ 392,
+ 477,
+ 405
+ ],
+ "score": 0.77,
+ "content": "\\begin{array} { r } { \\operatorname* { l i m } _ { n \\infty } \\operatorname { r a n k } ( Y _ { n } ) = 3 } \\end{array}",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 477,
+ 392,
+ 507,
+ 406
+ ],
+ "score": 1.0,
+ "content": "with a",
+ "type": "text"
+ }
+ ],
+ "index": 23
+ },
+ {
+ "bbox": [
+ 105,
+ 403,
+ 505,
+ 417
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 403,
+ 193,
+ 417
+ ],
+ "score": 1.0,
+ "content": "non-zero probability.",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 494,
+ 403,
+ 505,
+ 414
+ ],
+ "score": 1.0,
+ "content": "□",
+ "type": "text"
+ }
+ ],
+ "index": 24
+ }
+ ],
+ "index": 22.5,
+ "bbox_fs": [
+ 105,
+ 370,
+ 507,
+ 417
+ ]
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 108,
+ 431,
+ 245,
+ 444
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 430,
+ 245,
+ 446
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 430,
+ 245,
+ 446
+ ],
+ "score": 1.0,
+ "content": "D PROOF OF THEOREM 3",
+ "type": "text"
+ }
+ ],
+ "index": 25
+ }
+ ],
+ "index": 25
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 456,
+ 505,
+ 501
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 455,
+ 505,
+ 469
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 455,
+ 505,
+ 469
+ ],
+ "score": 1.0,
+ "content": "We follow the proof of Theorem 2 of Chung & Radcliffe (2011). The idea is to relate the spectral",
+ "type": "text"
+ }
+ ],
+ "index": 26
+ },
+ {
+ "bbox": [
+ 106,
+ 468,
+ 505,
+ 480
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 468,
+ 505,
+ 480
+ ],
+ "score": 1.0,
+ "content": "distribution of the normalized Laplacian with that of its expected version. Since we can compute",
+ "type": "text"
+ }
+ ],
+ "index": 27
+ },
+ {
+ "bbox": [
+ 105,
+ 479,
+ 505,
+ 492
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 479,
+ 505,
+ 492
+ ],
+ "score": 1.0,
+ "content": "the latter one explicitly for the Erdos-R ˝ enyi graph, we can derive the convergence of spectra. We ´",
+ "type": "text"
+ }
+ ],
+ "index": 28
+ },
+ {
+ "bbox": [
+ 105,
+ 490,
+ 483,
+ 503
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 490,
+ 483,
+ 503
+ ],
+ "score": 1.0,
+ "content": "employ this technique and derive similar conclusion for the augmented normalized Laplacian.",
+ "type": "text"
+ }
+ ],
+ "index": 29
+ }
+ ],
+ "index": 27.5,
+ "bbox_fs": [
+ 105,
+ 455,
+ 505,
+ 503
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 506,
+ 505,
+ 626
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 506,
+ 505,
+ 519
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 506,
+ 444,
+ 519
+ ],
+ "score": 1.0,
+ "content": "First, we consider genral random graphs not restricted to Erdos-R ˝ enyi graphs. Let ´",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 445,
+ 507,
+ 483,
+ 518
+ ],
+ "score": 0.9,
+ "content": "N \\in { \\mathbb { N } } _ { + }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 484,
+ 506,
+ 505,
+ 519
+ ],
+ "score": 1.0,
+ "content": ", and",
+ "type": "text"
+ }
+ ],
+ "index": 30
+ },
+ {
+ "bbox": [
+ 107,
+ 515,
+ 506,
+ 533
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 518,
+ 177,
+ 531
+ ],
+ "score": 0.92,
+ "content": "P = ( p _ { i j } ) _ { i , j \\in [ N ] }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 177,
+ 515,
+ 387,
+ 533
+ ],
+ "score": 1.0,
+ "content": "be a non-negative symmetric matrix (meaning that",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 387,
+ 518,
+ 421,
+ 530
+ ],
+ "score": 0.92,
+ "content": "p _ { i j } \\geq 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 421,
+ 515,
+ 455,
+ 533
+ ],
+ "score": 1.0,
+ "content": "for any",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 455,
+ 518,
+ 500,
+ 529
+ ],
+ "score": 0.9,
+ "content": "i , j \\in [ N ] )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 501,
+ 515,
+ 506,
+ 533
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 31
+ },
+ {
+ "bbox": [
+ 105,
+ 528,
+ 506,
+ 542
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 528,
+ 123,
+ 542
+ ],
+ "score": 1.0,
+ "content": "Let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 123,
+ 530,
+ 133,
+ 539
+ ],
+ "score": 0.76,
+ "content": "G",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 133,
+ 528,
+ 287,
+ 542
+ ],
+ "score": 1.0,
+ "content": "be an undirected random graph with",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 288,
+ 529,
+ 298,
+ 539
+ ],
+ "score": 0.8,
+ "content": "N",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 298,
+ 528,
+ 438,
+ 542
+ ],
+ "score": 1.0,
+ "content": "nodes such that an edge between",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 439,
+ 530,
+ 444,
+ 539
+ ],
+ "score": 0.73,
+ "content": "i",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 444,
+ 528,
+ 463,
+ 542
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 464,
+ 530,
+ 470,
+ 541
+ ],
+ "score": 0.82,
+ "content": "j",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 470,
+ 528,
+ 506,
+ 542
+ ],
+ "score": 1.0,
+ "content": "is inde-",
+ "type": "text"
+ }
+ ],
+ "index": 32
+ },
+ {
+ "bbox": [
+ 105,
+ 540,
+ 506,
+ 553
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 540,
+ 248,
+ 553
+ ],
+ "score": 1.0,
+ "content": "pendently present with probability",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 248,
+ 541,
+ 262,
+ 551
+ ],
+ "score": 0.82,
+ "content": "p _ { i j }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 262,
+ 540,
+ 284,
+ 553
+ ],
+ "score": 1.0,
+ "content": ". Let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 285,
+ 540,
+ 294,
+ 550
+ ],
+ "score": 0.8,
+ "content": "A",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 294,
+ 540,
+ 312,
+ 553
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 313,
+ 540,
+ 322,
+ 550
+ ],
+ "score": 0.82,
+ "content": "D",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 323,
+ 540,
+ 506,
+ 553
+ ],
+ "score": 1.0,
+ "content": "be the adjacency and the degree matrices of",
+ "type": "text"
+ }
+ ],
+ "index": 33
+ },
+ {
+ "bbox": [
+ 107,
+ 550,
+ 506,
+ 565
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 551,
+ 115,
+ 560
+ ],
+ "score": 0.77,
+ "content": "G",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 116,
+ 550,
+ 207,
+ 565
+ ],
+ "score": 1.0,
+ "content": ", respectively (that is,",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 208,
+ 551,
+ 275,
+ 563
+ ],
+ "score": 0.89,
+ "content": "A _ { i j } \\sim \\mathrm { B e r } ( p _ { i j } )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 276,
+ 550,
+ 485,
+ 565
+ ],
+ "score": 1.0,
+ "content": ", i.i.d.). Define the expected node degree of node",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 485,
+ 551,
+ 491,
+ 560
+ ],
+ "score": 0.7,
+ "content": "i",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 491,
+ 550,
+ 506,
+ 565
+ ],
+ "score": 1.0,
+ "content": "by",
+ "type": "text"
+ }
+ ],
+ "index": 34
+ },
+ {
+ "bbox": [
+ 106,
+ 562,
+ 507,
+ 579
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 562,
+ 174,
+ 578
+ ],
+ "score": 0.92,
+ "content": "\\begin{array} { r } { t _ { i } : = \\sum _ { j = 1 } ^ { N } p _ { i j } } \\end{array}",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 174,
+ 563,
+ 200,
+ 579
+ ],
+ "score": 1.0,
+ "content": ". Let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 200,
+ 563,
+ 263,
+ 576
+ ],
+ "score": 0.85,
+ "content": "{ \\tilde { A } } : = A + I _ { N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 263,
+ 563,
+ 269,
+ 579
+ ],
+ "score": 1.0,
+ "content": ",",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 269,
+ 563,
+ 335,
+ 576
+ ],
+ "score": 0.9,
+ "content": "{ \\tilde { D } } : = D + I _ { N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 335,
+ 563,
+ 384,
+ 579
+ ],
+ "score": 1.0,
+ "content": "and define",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 384,
+ 562,
+ 486,
+ 576
+ ],
+ "score": 0.93,
+ "content": "{ \\bar { A } } : = \\mathbb { E } [ { \\tilde { A } } ] = P + I _ { N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 486,
+ 563,
+ 507,
+ 579
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ }
+ ],
+ "index": 35
+ },
+ {
+ "bbox": [
+ 106,
+ 576,
+ 506,
+ 594
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 578,
+ 256,
+ 591
+ ],
+ "score": 0.88,
+ "content": "\\bar { D } : = \\mathbb { E } [ \\tilde { D } ] = \\mathrm { d i a g } ( t _ { 1 } , \\dots , t _ { N } ) + I _ { N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 257,
+ 576,
+ 506,
+ 594
+ ],
+ "score": 1.0,
+ "content": "correspondingly. We define the augmented normalized Lapla-",
+ "type": "text"
+ }
+ ],
+ "index": 36
+ },
+ {
+ "bbox": [
+ 104,
+ 589,
+ 503,
+ 604
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 589,
+ 126,
+ 604
+ ],
+ "score": 1.0,
+ "content": "cian",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 127,
+ 591,
+ 137,
+ 602
+ ],
+ "score": 0.83,
+ "content": "\\tilde { \\Delta }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 137,
+ 589,
+ 150,
+ 604
+ ],
+ "score": 1.0,
+ "content": "of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 151,
+ 592,
+ 160,
+ 602
+ ],
+ "score": 0.8,
+ "content": "G",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 160,
+ 589,
+ 176,
+ 604
+ ],
+ "score": 1.0,
+ "content": "by",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 176,
+ 590,
+ 280,
+ 603
+ ],
+ "score": 0.85,
+ "content": "\\tilde { \\Delta } : = I _ { N } - \\tilde { D } ^ { - \\frac { 1 } { 2 } } \\tilde { A } \\tilde { D } ^ { - \\frac { 1 } { 2 } }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 280,
+ 589,
+ 399,
+ 604
+ ],
+ "score": 1.0,
+ "content": "and its expected version by",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 399,
+ 590,
+ 503,
+ 603
+ ],
+ "score": 0.92,
+ "content": "\\bar { \\Delta } : = I _ { N } - \\bar { D } ^ { - \\frac { 1 } { 2 } } \\bar { A } \\bar { D } ^ { - \\frac { 1 } { 2 } }",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 37
+ },
+ {
+ "bbox": [
+ 104,
+ 600,
+ 505,
+ 617
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 600,
+ 223,
+ 617
+ ],
+ "score": 1.0,
+ "content": "9. For a symmetric matrix",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 223,
+ 602,
+ 266,
+ 613
+ ],
+ "score": 0.88,
+ "content": "X ~ \\in ~ \\mathbb { R } ^ { N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 266,
+ 600,
+ 505,
+ 617
+ ],
+ "score": 1.0,
+ "content": ", we define its eigenvalues, sorted in ascending order by",
+ "type": "text"
+ }
+ ],
+ "index": 38
+ },
+ {
+ "bbox": [
+ 106,
+ 613,
+ 419,
+ 627
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 613,
+ 203,
+ 626
+ ],
+ "score": 0.93,
+ "content": "\\lambda _ { 1 } ( X ) \\leq \\cdots \\leq \\lambda _ { N } ( X )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 203,
+ 613,
+ 305,
+ 627
+ ],
+ "score": 1.0,
+ "content": "and its operator norm by",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 305,
+ 613,
+ 414,
+ 627
+ ],
+ "score": 0.93,
+ "content": "\\| X \\| = \\operatorname* { i n a x } _ { n \\in [ N ] } | \\lambda _ { n } ( X ) |",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 415,
+ 613,
+ 419,
+ 627
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 39
+ }
+ ],
+ "index": 34.5,
+ "bbox_fs": [
+ 104,
+ 506,
+ 507,
+ 627
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 628,
+ 505,
+ 663
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 627,
+ 505,
+ 642
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 627,
+ 360,
+ 642
+ ],
+ "score": 1.0,
+ "content": "Lemma 5 (Ref. Chung & Radcliffe (2011) Theorem 2). Let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 360,
+ 629,
+ 435,
+ 641
+ ],
+ "score": 0.89,
+ "content": "\\delta : = \\mathrm { m i n } _ { n \\in [ N ] } t _ { n }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 435,
+ 627,
+ 505,
+ 642
+ ],
+ "score": 1.0,
+ "content": "be the minimum",
+ "type": "text"
+ }
+ ],
+ "index": 40
+ },
+ {
+ "bbox": [
+ 105,
+ 641,
+ 505,
+ 654
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 641,
+ 183,
+ 654
+ ],
+ "score": 1.0,
+ "content": "expected degree of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 183,
+ 641,
+ 192,
+ 651
+ ],
+ "score": 0.71,
+ "content": "G",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 192,
+ 641,
+ 211,
+ 654
+ ],
+ "score": 1.0,
+ "content": ". Set",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 211,
+ 641,
+ 310,
+ 653
+ ],
+ "score": 0.92,
+ "content": "k ( \\varepsilon ) : = 3 ( 1 + \\log ( 4 / \\varepsilon ) )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 311,
+ 641,
+ 371,
+ 654
+ ],
+ "score": 1.0,
+ "content": ". Then, for any",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 372,
+ 642,
+ 396,
+ 652
+ ],
+ "score": 0.83,
+ "content": "\\varepsilon > 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 396,
+ 641,
+ 399,
+ 654
+ ],
+ "score": 1.0,
+ "content": ",",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 400,
+ 641,
+ 487,
+ 653
+ ],
+ "score": 0.9,
+ "content": "i f { \\dot { \\delta } } + { \\dot { 1 } } > k ( \\varepsilon ) \\log N",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 487,
+ 641,
+ 505,
+ 654
+ ],
+ "score": 1.0,
+ "content": ", we",
+ "type": "text"
+ }
+ ],
+ "index": 41
+ },
+ {
+ "bbox": [
+ 105,
+ 651,
+ 129,
+ 665
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 651,
+ 129,
+ 665
+ ],
+ "score": 1.0,
+ "content": "have",
+ "type": "text"
+ }
+ ],
+ "index": 42
+ }
+ ],
+ "index": 41,
+ "bbox_fs": [
+ 105,
+ 627,
+ 505,
+ 665
+ ]
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 215,
+ 667,
+ 396,
+ 696
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 215,
+ 667,
+ 396,
+ 696
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 215,
+ 667,
+ 396,
+ 696
+ ],
+ "score": 0.92,
+ "content": "\\operatorname* { m a x } _ { n \\in [ N ] } \\left| \\lambda _ { n } ( \\tilde { \\Delta } ) - \\lambda _ { n } ( \\bar { \\Delta } ) \\right| \\leq 4 \\sqrt { \\frac { 3 \\log ( 4 N / \\varepsilon ) } { \\delta + 1 } }",
+ "type": "interline_equation",
+ "image_path": "faa530371dd9306d6740988ca847f650db8a9d0bcc8578a2693011979e058fb8.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 43.5,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 215,
+ 667,
+ 396,
+ 681.5
+ ],
+ "spans": [],
+ "index": 43
+ },
+ {
+ "bbox": [
+ 215,
+ 681.5,
+ 396,
+ 696.0
+ ],
+ "spans": [],
+ "index": 44
+ }
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 702,
+ 229,
+ 714
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 701,
+ 227,
+ 715
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 701,
+ 204,
+ 715
+ ],
+ "score": 1.0,
+ "content": "with probability at least",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 204,
+ 703,
+ 227,
+ 713
+ ],
+ "score": 0.85,
+ "content": "1 - \\varepsilon",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 45
+ }
+ ],
+ "index": 45,
+ "bbox_fs": [
+ 106,
+ 701,
+ 227,
+ 715
+ ]
+ }
+ ]
+ },
+ {
+ "preproc_blocks": [
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 80,
+ 506,
+ 127
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 104,
+ 80,
+ 507,
+ 100
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 80,
+ 258,
+ 99
+ ],
+ "score": 1.0,
+ "content": "Proof. By Weyl’s theorem, we have",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 259,
+ 80,
+ 433,
+ 100
+ ],
+ "score": 0.93,
+ "content": "\\begin{array} { r } { \\operatorname* { m a x } _ { n \\in [ N ] } \\left| \\lambda _ { n } ( \\tilde { \\Delta } ) - \\lambda _ { n } ( \\bar { \\Delta } ) \\right| \\leq \\| \\tilde { \\Delta } - \\bar { \\Delta } \\| } \\end{array}",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 434,
+ 80,
+ 507,
+ 99
+ ],
+ "score": 1.0,
+ "content": ". Therefore, it is",
+ "type": "text"
+ }
+ ],
+ "index": 0
+ },
+ {
+ "bbox": [
+ 105,
+ 97,
+ 506,
+ 114
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 97,
+ 180,
+ 114
+ ],
+ "score": 1.0,
+ "content": "enough to bound",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 181,
+ 99,
+ 222,
+ 113
+ ],
+ "score": 0.91,
+ "content": "\\| \\tilde { \\Delta } - \\bar { \\Delta } \\|",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 222,
+ 97,
+ 247,
+ 114
+ ],
+ "score": 1.0,
+ "content": ". Let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 248,
+ 100,
+ 340,
+ 112
+ ],
+ "score": 0.9,
+ "content": "C : = I _ { N } - \\bar { D } ^ { - \\frac { 1 } { 2 } } \\tilde { A } \\bar { D }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 340,
+ 97,
+ 506,
+ 114
+ ],
+ "score": 1.0,
+ "content": ". By the triangular inequality, we have",
+ "type": "text"
+ }
+ ],
+ "index": 1
+ },
+ {
+ "bbox": [
+ 107,
+ 111,
+ 414,
+ 128
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 112,
+ 248,
+ 126
+ ],
+ "score": 0.91,
+ "content": "\\| \\tilde { \\Delta } - \\bar { \\Delta } \\| \\leq \\| \\tilde { \\Delta } - C \\| + \\| C - \\bar { \\Delta } \\|",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 249,
+ 111,
+ 414,
+ 128
+ ],
+ "score": 1.0,
+ "content": ". We will bound these terms respectively.",
+ "type": "text"
+ }
+ ],
+ "index": 2
+ }
+ ],
+ "index": 1
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 105,
+ 130,
+ 504,
+ 156
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 104,
+ 130,
+ 504,
+ 145
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 130,
+ 170,
+ 144
+ ],
+ "score": 1.0,
+ "content": "First, we bound",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 170,
+ 131,
+ 205,
+ 145
+ ],
+ "score": 0.93,
+ "content": "\\| C - { \\bar { \\Delta } } \\|",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 205,
+ 130,
+ 308,
+ 144
+ ],
+ "score": 1.0,
+ "content": ". Direct calculation shows",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 308,
+ 130,
+ 428,
+ 145
+ ],
+ "score": 0.93,
+ "content": "C - \\bar { \\Delta } = - \\bar { D } ^ { - \\frac { 1 } { 2 } } ( A { - } P ) \\bar { D } ^ { - \\frac { 1 } { 2 } }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 428,
+ 130,
+ 447,
+ 144
+ ],
+ "score": 1.0,
+ "content": ". Let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 447,
+ 131,
+ 504,
+ 143
+ ],
+ "score": 0.86,
+ "content": "E ^ { i j } \\in \\mathbb { R } ^ { N \\times N }",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 3
+ },
+ {
+ "bbox": [
+ 105,
+ 142,
+ 199,
+ 157
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 142,
+ 199,
+ 157
+ ],
+ "score": 1.0,
+ "content": "be a matrix defined by",
+ "type": "text"
+ }
+ ],
+ "index": 4
+ }
+ ],
+ "index": 3.5
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 190,
+ 164,
+ 419,
+ 192
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 190,
+ 164,
+ 419,
+ 192
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 190,
+ 164,
+ 419,
+ 192
+ ],
+ "score": 0.73,
+ "content": "( E ^ { i j } ) _ { k l } = { \\left\\{ \\begin{array} { l l } { 1 } & { { \\mathrm { i f ~ } } ( i = k { \\mathrm { ~ a n d ~ } } i = l ) { \\mathrm { ~ o r ~ } } ( i = l { \\mathrm { ~ a n d ~ } } j = k ) , } \\\\ { 0 } & { { \\mathrm { o t h e r w i s e . } } } \\end{array} \\right. }",
+ "type": "interline_equation",
+ "image_path": "fe9cdfdb83f038780d2070d9493f39f5009327a9c0eb6768b53b00ddfffcfe9a.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 5,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 190,
+ 164,
+ 419,
+ 192
+ ],
+ "spans": [],
+ "index": 5
+ }
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 199,
+ 258,
+ 212
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 197,
+ 258,
+ 214
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 197,
+ 230,
+ 214
+ ],
+ "score": 1.0,
+ "content": "We define the random variable",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 231,
+ 200,
+ 245,
+ 212
+ ],
+ "score": 0.9,
+ "content": "Y _ { i j }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 245,
+ 197,
+ 258,
+ 214
+ ],
+ "score": 1.0,
+ "content": "by",
+ "type": "text"
+ }
+ ],
+ "index": 6
+ }
+ ],
+ "index": 6
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 246,
+ 219,
+ 365,
+ 248
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 246,
+ 219,
+ 365,
+ 248
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 246,
+ 219,
+ 365,
+ 248
+ ],
+ "score": 0.94,
+ "content": "Y _ { i j } : = \\frac { A _ { i j } - p _ { i j } } { \\sqrt { t _ { i } + 1 } \\sqrt { t _ { j } + 1 } } E ^ { i j } .",
+ "type": "interline_equation",
+ "image_path": "d4cfac4151ae12a2d38b39962bf8e377443e15fc5af480baf66fe0bf602e815e.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 7,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 246,
+ 219,
+ 365,
+ 248
+ ],
+ "spans": [],
+ "index": 7
+ }
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 256,
+ 505,
+ 287
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 100,
+ 250,
+ 511,
+ 294
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 100,
+ 261,
+ 185,
+ 294
+ ],
+ "score": 1.0,
+ "content": "ij Radcliffe (2011) to",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 101,
+ 250,
+ 132,
+ 279
+ ],
+ "score": 1.0,
+ "content": "Then,",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 146,
+ 250,
+ 274,
+ 279
+ ],
+ "score": 1.0,
+ "content": "’s are independent and we have",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 185,
+ 273,
+ 199,
+ 286
+ ],
+ "score": 0.89,
+ "content": "Y _ { i j }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 199,
+ 261,
+ 251,
+ 294
+ ],
+ "score": 1.0,
+ "content": "’s, we bound",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 252,
+ 273,
+ 312,
+ 286
+ ],
+ "score": 0.93,
+ "content": "\\| Y _ { i j } - \\mathbb { E } [ Y _ { i j } ] \\|",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 275,
+ 256,
+ 363,
+ 272
+ ],
+ "score": 0.94,
+ "content": "\\begin{array} { r } { C - \\bar { \\Delta } = \\sum _ { i , j = 1 } ^ { N } Y _ { i j } } \\end{array}",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 312,
+ 261,
+ 331,
+ 294
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 331,
+ 271,
+ 399,
+ 287
+ ],
+ "score": 0.92,
+ "content": "\\| \\Sigma _ { i , j = 1 } ^ { N } \\mathbb { E } [ Y _ { i j } ^ { 2 } ] \\|",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 363,
+ 250,
+ 511,
+ 279
+ ],
+ "score": 1.0,
+ "content": ". To apply Theorem 5 of Chung &",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 400,
+ 261,
+ 469,
+ 294
+ ],
+ "score": 1.0,
+ "content": ". First, we have",
+ "type": "text"
+ }
+ ],
+ "index": 8
+ }
+ ],
+ "index": 8
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 186,
+ 294,
+ 424,
+ 325
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 186,
+ 294,
+ 424,
+ 325
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 186,
+ 294,
+ 424,
+ 325
+ ],
+ "score": 0.92,
+ "content": "\\| Y _ { i j } - \\mathbb { E } [ Y _ { i j } ] \\| = \\| Y _ { i j } \\| \\leq \\frac { \\| E ^ { i j } \\| } { \\sqrt { t _ { i } + 1 } \\sqrt { t _ { j } + 1 } } \\leq ( \\delta + 1 ) ^ { - 1 } .",
+ "type": "interline_equation",
+ "image_path": "cae7c6b924677a28a81735f52c21a2db10cea67ace48042da7e719cbd90f9ece.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 9,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 186,
+ 294,
+ 424,
+ 325
+ ],
+ "spans": [],
+ "index": 9
+ }
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 331,
+ 131,
+ 343
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 330,
+ 132,
+ 344
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 330,
+ 132,
+ 344
+ ],
+ "score": 1.0,
+ "content": "Since",
+ "type": "text"
+ }
+ ],
+ "index": 10
+ }
+ ],
+ "index": 10
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 200,
+ 348,
+ 409,
+ 378
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 200,
+ 348,
+ 409,
+ 378
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 200,
+ 348,
+ 409,
+ 378
+ ],
+ "score": 0.92,
+ "content": "\\mathbb { E } [ Y _ { i j } ^ { 2 } ] = \\frac { p _ { i j } - p _ { i j } ^ { 2 } } { ( t _ { i } + 1 ) ( t _ { j } + 1 ) } \\left\\{ { E ^ { i i } + E ^ { j j } } \\begin{array} { l l } { ( \\mathrm { i f ~ } i \\ne j ) , } \\\\ { E ^ { i i } } & { ( \\mathrm { i f ~ } i = j ) , } \\end{array} \\right.",
+ "type": "interline_equation",
+ "image_path": "a52cf9e5e80dfb4e7363c299ec6286164906ebce3dececfbc73d458f5b45119c.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 11,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 200,
+ 348,
+ 409,
+ 378
+ ],
+ "spans": [],
+ "index": 11
+ }
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 386,
+ 141,
+ 397
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 384,
+ 143,
+ 398
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 384,
+ 143,
+ 398
+ ],
+ "score": 1.0,
+ "content": "we have",
+ "type": "text"
+ }
+ ],
+ "index": 12
+ }
+ ],
+ "index": 12
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 207,
+ 404,
+ 402,
+ 538
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 207,
+ 404,
+ 402,
+ 538
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 207,
+ 404,
+ 402,
+ 538
+ ],
+ "score": 0.94,
+ "content": "\\begin{array} { r } { \\left| \\displaystyle \\sum _ { i , j = 1 } ^ { N } \\mathbb { E } [ Y _ { i j } ^ { 2 } ] \\right| = \\left| \\left| \\displaystyle \\sum _ { i , j = 1 } ^ { N } \\frac { p _ { i j } - p _ { i j } ^ { 2 } } { ( t _ { i } + 1 ) ( t _ { j } + 1 ) } E ^ { i i } \\right| \\right| } \\\\ { = \\displaystyle \\operatorname* { m a x } _ { i \\in [ N ] } \\left( \\displaystyle \\sum _ { j = 1 } ^ { N } \\frac { p _ { i j } - p _ { i j } ^ { 2 } } { ( t _ { i } + 1 ) ( t _ { j } + 1 ) } \\right) } \\\\ { \\leq \\displaystyle \\operatorname* { m a x } _ { i \\in [ N ] } \\left( \\displaystyle \\sum _ { j = 1 } ^ { N } \\frac { p _ { i j } } { ( t _ { i } + 1 ) ( t _ { j } + 1 ) } \\right) } \\\\ { \\leq ( \\delta + 1 ) ^ { - 1 } . } \\end{array}",
+ "type": "interline_equation",
+ "image_path": "671d8d0807c4af3435e2fc7a59b03aa66054295c22241b527c28d8f8d4f43a38.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 16.5,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 207,
+ 404,
+ 402,
+ 420.75
+ ],
+ "spans": [],
+ "index": 13
+ },
+ {
+ "bbox": [
+ 207,
+ 420.75,
+ 402,
+ 437.5
+ ],
+ "spans": [],
+ "index": 14
+ },
+ {
+ "bbox": [
+ 207,
+ 437.5,
+ 402,
+ 454.25
+ ],
+ "spans": [],
+ "index": 15
+ },
+ {
+ "bbox": [
+ 207,
+ 454.25,
+ 402,
+ 471.0
+ ],
+ "spans": [],
+ "index": 16
+ },
+ {
+ "bbox": [
+ 207,
+ 471.0,
+ 402,
+ 487.75
+ ],
+ "spans": [],
+ "index": 17
+ },
+ {
+ "bbox": [
+ 207,
+ 487.75,
+ 402,
+ 504.5
+ ],
+ "spans": [],
+ "index": 18
+ },
+ {
+ "bbox": [
+ 207,
+ 504.5,
+ 402,
+ 521.25
+ ],
+ "spans": [],
+ "index": 19
+ },
+ {
+ "bbox": [
+ 207,
+ 521.25,
+ 402,
+ 538.0
+ ],
+ "spans": [],
+ "index": 20
+ }
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 546,
+ 505,
+ 577
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 546,
+ 505,
+ 566
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 551,
+ 148,
+ 564
+ ],
+ "score": 1.0,
+ "content": "By letting",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 149,
+ 546,
+ 222,
+ 566
+ ],
+ "score": 0.89,
+ "content": "\\begin{array} { r } { a \\gets \\sqrt { \\frac { 3 \\log ( 4 N / \\varepsilon ) } { \\delta + 1 } } } \\end{array}",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 222,
+ 546,
+ 227,
+ 566
+ ],
+ "score": 1.0,
+ "content": ",",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 227,
+ 549,
+ 293,
+ 564
+ ],
+ "score": 0.52,
+ "content": "M \\gets ( \\delta + 1 ) ^ { - 1 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 294,
+ 546,
+ 298,
+ 566
+ ],
+ "score": 1.0,
+ "content": ",",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 298,
+ 550,
+ 364,
+ 563
+ ],
+ "score": 0.87,
+ "content": "v ^ { 2 } ( \\delta + 1 ) ^ { - 1 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 364,
+ 546,
+ 505,
+ 566
+ ],
+ "score": 1.0,
+ "content": "and applying Theorem 5 of Chung",
+ "type": "text"
+ }
+ ],
+ "index": 21
+ },
+ {
+ "bbox": [
+ 105,
+ 563,
+ 223,
+ 577
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 563,
+ 223,
+ 577
+ ],
+ "score": 1.0,
+ "content": "& Radcliffe (2011), we have",
+ "type": "text"
+ }
+ ],
+ "index": 22
+ }
+ ],
+ "index": 21.5
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 171,
+ 582,
+ 439,
+ 641
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 171,
+ 582,
+ 439,
+ 641
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 171,
+ 582,
+ 439,
+ 641
+ ],
+ "score": 0.93,
+ "content": "\\begin{array} { r l } & { \\operatorname* { P r } ( \\| C - \\bar { \\Delta } \\| > a ) \\leq 2 N \\exp \\left( - \\displaystyle \\frac { a ^ { 2 } } { 2 ( \\delta + 1 ) ^ { - 1 } + 2 ( \\delta + 1 ) ^ { - 1 } a / 3 } \\right) } \\\\ & { \\phantom { \\operatorname* { P r } } \\leq 2 N \\exp \\left( - \\displaystyle \\frac { 3 \\log \\left( 4 N / \\varepsilon \\right) } { 2 ( 1 + a / 3 ) } \\right) . } \\end{array}",
+ "type": "interline_equation",
+ "image_path": "a1bcd9ba5a9bd225226118673c07e91de3eaf40dfa178bcc7be802d6733cac42.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 24,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 171,
+ 582,
+ 439,
+ 601.6666666666666
+ ],
+ "spans": [],
+ "index": 23
+ },
+ {
+ "bbox": [
+ 171,
+ 601.6666666666666,
+ 439,
+ 621.3333333333333
+ ],
+ "spans": [],
+ "index": 24
+ },
+ {
+ "bbox": [
+ 171,
+ 621.3333333333333,
+ 439,
+ 640.9999999999999
+ ],
+ "spans": [],
+ "index": 25
+ }
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 645,
+ 443,
+ 660
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 645,
+ 442,
+ 660
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 645,
+ 187,
+ 660
+ ],
+ "score": 1.0,
+ "content": "By the definition of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 187,
+ 647,
+ 205,
+ 659
+ ],
+ "score": 0.92,
+ "content": "k ( \\varepsilon )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 206,
+ 645,
+ 244,
+ 660
+ ],
+ "score": 1.0,
+ "content": ", we have",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 245,
+ 648,
+ 270,
+ 657
+ ],
+ "score": 0.89,
+ "content": "a < 1",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 270,
+ 645,
+ 279,
+ 660
+ ],
+ "score": 1.0,
+ "content": "if",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 280,
+ 647,
+ 356,
+ 659
+ ],
+ "score": 0.93,
+ "content": "\\delta + 1 > k ( \\varepsilon ) \\log n",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 356,
+ 645,
+ 397,
+ 660
+ ],
+ "score": 1.0,
+ "content": ". For such",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 397,
+ 648,
+ 403,
+ 657
+ ],
+ "score": 0.81,
+ "content": "\\delta",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 403,
+ 645,
+ 442,
+ 660
+ ],
+ "score": 1.0,
+ "content": ", we have",
+ "type": "text"
+ }
+ ],
+ "index": 26
+ }
+ ],
+ "index": 26
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 186,
+ 666,
+ 425,
+ 732
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 186,
+ 666,
+ 425,
+ 732
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 186,
+ 666,
+ 425,
+ 732
+ ],
+ "score": 0.94,
+ "content": "\\begin{array} { l } { \\displaystyle \\operatorname* { P r } ( | | C - \\bar { \\Delta } | | > a ) \\leq 2 N \\exp \\left( - \\frac { 3 \\log ( 4 N / \\varepsilon ) } { 2 ( 1 + a / 3 ) } \\right) } \\\\ { \\leq 2 N \\exp \\left( - \\log ( 4 N / \\varepsilon ) \\right) \\quad \\displaystyle ( \\because a < 1 ) } \\\\ { = \\displaystyle \\frac { \\varepsilon } { 2 } . } \\end{array}",
+ "type": "interline_equation",
+ "image_path": "fc09ff0f9c9bdeeb3f37e0f6c7e58b8b98a7adbcd430ca7cb03b465487d743d3.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 28.5,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 186,
+ 666,
+ 425,
+ 682.5
+ ],
+ "spans": [],
+ "index": 27
+ },
+ {
+ "bbox": [
+ 186,
+ 682.5,
+ 425,
+ 699.0
+ ],
+ "spans": [],
+ "index": 28
+ },
+ {
+ "bbox": [
+ 186,
+ 699.0,
+ 425,
+ 715.5
+ ],
+ "spans": [],
+ "index": 29
+ },
+ {
+ "bbox": [
+ 186,
+ 715.5,
+ 425,
+ 732.0
+ ],
+ "spans": [],
+ "index": 30
+ }
+ ]
+ }
+ ],
+ "page_idx": 18,
+ "page_size": [
+ 612,
+ 792
+ ],
+ "discarded_blocks": [
+ {
+ "type": "discarded",
+ "bbox": [
+ 106,
+ 26,
+ 294,
+ 38
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 25,
+ 294,
+ 38
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 25,
+ 294,
+ 38
+ ],
+ "score": 1.0,
+ "content": "Published as a conference paper at ICLR 2020",
+ "type": "text"
+ }
+ ]
+ }
+ ]
+ },
+ {
+ "type": "discarded",
+ "bbox": [
+ 300,
+ 751,
+ 311,
+ 760
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 299,
+ 750,
+ 312,
+ 765
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 299,
+ 750,
+ 312,
+ 765
+ ],
+ "score": 1.0,
+ "content": "19",
+ "type": "text"
+ }
+ ]
+ }
+ ]
+ }
+ ],
+ "para_blocks": [
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 80,
+ 506,
+ 127
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 104,
+ 80,
+ 507,
+ 100
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 80,
+ 258,
+ 99
+ ],
+ "score": 1.0,
+ "content": "Proof. By Weyl’s theorem, we have",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 259,
+ 80,
+ 433,
+ 100
+ ],
+ "score": 0.93,
+ "content": "\\begin{array} { r } { \\operatorname* { m a x } _ { n \\in [ N ] } \\left| \\lambda _ { n } ( \\tilde { \\Delta } ) - \\lambda _ { n } ( \\bar { \\Delta } ) \\right| \\leq \\| \\tilde { \\Delta } - \\bar { \\Delta } \\| } \\end{array}",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 434,
+ 80,
+ 507,
+ 99
+ ],
+ "score": 1.0,
+ "content": ". Therefore, it is",
+ "type": "text"
+ }
+ ],
+ "index": 0
+ },
+ {
+ "bbox": [
+ 105,
+ 97,
+ 506,
+ 114
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 97,
+ 180,
+ 114
+ ],
+ "score": 1.0,
+ "content": "enough to bound",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 181,
+ 99,
+ 222,
+ 113
+ ],
+ "score": 0.91,
+ "content": "\\| \\tilde { \\Delta } - \\bar { \\Delta } \\|",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 222,
+ 97,
+ 247,
+ 114
+ ],
+ "score": 1.0,
+ "content": ". Let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 248,
+ 100,
+ 340,
+ 112
+ ],
+ "score": 0.9,
+ "content": "C : = I _ { N } - \\bar { D } ^ { - \\frac { 1 } { 2 } } \\tilde { A } \\bar { D }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 340,
+ 97,
+ 506,
+ 114
+ ],
+ "score": 1.0,
+ "content": ". By the triangular inequality, we have",
+ "type": "text"
+ }
+ ],
+ "index": 1
+ },
+ {
+ "bbox": [
+ 107,
+ 111,
+ 414,
+ 128
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 112,
+ 248,
+ 126
+ ],
+ "score": 0.91,
+ "content": "\\| \\tilde { \\Delta } - \\bar { \\Delta } \\| \\leq \\| \\tilde { \\Delta } - C \\| + \\| C - \\bar { \\Delta } \\|",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 249,
+ 111,
+ 414,
+ 128
+ ],
+ "score": 1.0,
+ "content": ". We will bound these terms respectively.",
+ "type": "text"
+ }
+ ],
+ "index": 2
+ }
+ ],
+ "index": 1,
+ "bbox_fs": [
+ 104,
+ 80,
+ 507,
+ 128
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 105,
+ 130,
+ 504,
+ 156
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 104,
+ 130,
+ 504,
+ 145
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 130,
+ 170,
+ 144
+ ],
+ "score": 1.0,
+ "content": "First, we bound",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 170,
+ 131,
+ 205,
+ 145
+ ],
+ "score": 0.93,
+ "content": "\\| C - { \\bar { \\Delta } } \\|",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 205,
+ 130,
+ 308,
+ 144
+ ],
+ "score": 1.0,
+ "content": ". Direct calculation shows",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 308,
+ 130,
+ 428,
+ 145
+ ],
+ "score": 0.93,
+ "content": "C - \\bar { \\Delta } = - \\bar { D } ^ { - \\frac { 1 } { 2 } } ( A { - } P ) \\bar { D } ^ { - \\frac { 1 } { 2 } }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 428,
+ 130,
+ 447,
+ 144
+ ],
+ "score": 1.0,
+ "content": ". Let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 447,
+ 131,
+ 504,
+ 143
+ ],
+ "score": 0.86,
+ "content": "E ^ { i j } \\in \\mathbb { R } ^ { N \\times N }",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 3
+ },
+ {
+ "bbox": [
+ 105,
+ 142,
+ 199,
+ 157
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 142,
+ 199,
+ 157
+ ],
+ "score": 1.0,
+ "content": "be a matrix defined by",
+ "type": "text"
+ }
+ ],
+ "index": 4
+ }
+ ],
+ "index": 3.5,
+ "bbox_fs": [
+ 104,
+ 130,
+ 504,
+ 157
+ ]
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 190,
+ 164,
+ 419,
+ 192
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 190,
+ 164,
+ 419,
+ 192
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 190,
+ 164,
+ 419,
+ 192
+ ],
+ "score": 0.73,
+ "content": "( E ^ { i j } ) _ { k l } = { \\left\\{ \\begin{array} { l l } { 1 } & { { \\mathrm { i f ~ } } ( i = k { \\mathrm { ~ a n d ~ } } i = l ) { \\mathrm { ~ o r ~ } } ( i = l { \\mathrm { ~ a n d ~ } } j = k ) , } \\\\ { 0 } & { { \\mathrm { o t h e r w i s e . } } } \\end{array} \\right. }",
+ "type": "interline_equation",
+ "image_path": "fe9cdfdb83f038780d2070d9493f39f5009327a9c0eb6768b53b00ddfffcfe9a.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 5,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 190,
+ 164,
+ 419,
+ 192
+ ],
+ "spans": [],
+ "index": 5
+ }
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 199,
+ 258,
+ 212
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 197,
+ 258,
+ 214
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 197,
+ 230,
+ 214
+ ],
+ "score": 1.0,
+ "content": "We define the random variable",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 231,
+ 200,
+ 245,
+ 212
+ ],
+ "score": 0.9,
+ "content": "Y _ { i j }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 245,
+ 197,
+ 258,
+ 214
+ ],
+ "score": 1.0,
+ "content": "by",
+ "type": "text"
+ }
+ ],
+ "index": 6
+ }
+ ],
+ "index": 6,
+ "bbox_fs": [
+ 106,
+ 197,
+ 258,
+ 214
+ ]
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 246,
+ 219,
+ 365,
+ 248
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 246,
+ 219,
+ 365,
+ 248
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 246,
+ 219,
+ 365,
+ 248
+ ],
+ "score": 0.94,
+ "content": "Y _ { i j } : = \\frac { A _ { i j } - p _ { i j } } { \\sqrt { t _ { i } + 1 } \\sqrt { t _ { j } + 1 } } E ^ { i j } .",
+ "type": "interline_equation",
+ "image_path": "d4cfac4151ae12a2d38b39962bf8e377443e15fc5af480baf66fe0bf602e815e.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 7,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 246,
+ 219,
+ 365,
+ 248
+ ],
+ "spans": [],
+ "index": 7
+ }
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 256,
+ 505,
+ 287
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 100,
+ 250,
+ 511,
+ 294
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 100,
+ 261,
+ 185,
+ 294
+ ],
+ "score": 1.0,
+ "content": "ij Radcliffe (2011) to",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 101,
+ 250,
+ 132,
+ 279
+ ],
+ "score": 1.0,
+ "content": "Then,",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 146,
+ 250,
+ 274,
+ 279
+ ],
+ "score": 1.0,
+ "content": "’s are independent and we have",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 185,
+ 273,
+ 199,
+ 286
+ ],
+ "score": 0.89,
+ "content": "Y _ { i j }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 199,
+ 261,
+ 251,
+ 294
+ ],
+ "score": 1.0,
+ "content": "’s, we bound",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 252,
+ 273,
+ 312,
+ 286
+ ],
+ "score": 0.93,
+ "content": "\\| Y _ { i j } - \\mathbb { E } [ Y _ { i j } ] \\|",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 275,
+ 256,
+ 363,
+ 272
+ ],
+ "score": 0.94,
+ "content": "\\begin{array} { r } { C - \\bar { \\Delta } = \\sum _ { i , j = 1 } ^ { N } Y _ { i j } } \\end{array}",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 312,
+ 261,
+ 331,
+ 294
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 331,
+ 271,
+ 399,
+ 287
+ ],
+ "score": 0.92,
+ "content": "\\| \\Sigma _ { i , j = 1 } ^ { N } \\mathbb { E } [ Y _ { i j } ^ { 2 } ] \\|",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 363,
+ 250,
+ 511,
+ 279
+ ],
+ "score": 1.0,
+ "content": ". To apply Theorem 5 of Chung &",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 400,
+ 261,
+ 469,
+ 294
+ ],
+ "score": 1.0,
+ "content": ". First, we have",
+ "type": "text"
+ }
+ ],
+ "index": 8
+ }
+ ],
+ "index": 8,
+ "bbox_fs": [
+ 100,
+ 250,
+ 511,
+ 294
+ ]
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 186,
+ 294,
+ 424,
+ 325
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 186,
+ 294,
+ 424,
+ 325
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 186,
+ 294,
+ 424,
+ 325
+ ],
+ "score": 0.92,
+ "content": "\\| Y _ { i j } - \\mathbb { E } [ Y _ { i j } ] \\| = \\| Y _ { i j } \\| \\leq \\frac { \\| E ^ { i j } \\| } { \\sqrt { t _ { i } + 1 } \\sqrt { t _ { j } + 1 } } \\leq ( \\delta + 1 ) ^ { - 1 } .",
+ "type": "interline_equation",
+ "image_path": "cae7c6b924677a28a81735f52c21a2db10cea67ace48042da7e719cbd90f9ece.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 9,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 186,
+ 294,
+ 424,
+ 325
+ ],
+ "spans": [],
+ "index": 9
+ }
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 331,
+ 131,
+ 343
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 330,
+ 132,
+ 344
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 330,
+ 132,
+ 344
+ ],
+ "score": 1.0,
+ "content": "Since",
+ "type": "text"
+ }
+ ],
+ "index": 10
+ }
+ ],
+ "index": 10,
+ "bbox_fs": [
+ 106,
+ 330,
+ 132,
+ 344
+ ]
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 200,
+ 348,
+ 409,
+ 378
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 200,
+ 348,
+ 409,
+ 378
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 200,
+ 348,
+ 409,
+ 378
+ ],
+ "score": 0.92,
+ "content": "\\mathbb { E } [ Y _ { i j } ^ { 2 } ] = \\frac { p _ { i j } - p _ { i j } ^ { 2 } } { ( t _ { i } + 1 ) ( t _ { j } + 1 ) } \\left\\{ { E ^ { i i } + E ^ { j j } } \\begin{array} { l l } { ( \\mathrm { i f ~ } i \\ne j ) , } \\\\ { E ^ { i i } } & { ( \\mathrm { i f ~ } i = j ) , } \\end{array} \\right.",
+ "type": "interline_equation",
+ "image_path": "a52cf9e5e80dfb4e7363c299ec6286164906ebce3dececfbc73d458f5b45119c.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 11,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 200,
+ 348,
+ 409,
+ 378
+ ],
+ "spans": [],
+ "index": 11
+ }
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 386,
+ 141,
+ 397
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 384,
+ 143,
+ 398
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 384,
+ 143,
+ 398
+ ],
+ "score": 1.0,
+ "content": "we have",
+ "type": "text"
+ }
+ ],
+ "index": 12
+ }
+ ],
+ "index": 12,
+ "bbox_fs": [
+ 105,
+ 384,
+ 143,
+ 398
+ ]
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 207,
+ 404,
+ 402,
+ 538
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 207,
+ 404,
+ 402,
+ 538
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 207,
+ 404,
+ 402,
+ 538
+ ],
+ "score": 0.94,
+ "content": "\\begin{array} { r } { \\left| \\displaystyle \\sum _ { i , j = 1 } ^ { N } \\mathbb { E } [ Y _ { i j } ^ { 2 } ] \\right| = \\left| \\left| \\displaystyle \\sum _ { i , j = 1 } ^ { N } \\frac { p _ { i j } - p _ { i j } ^ { 2 } } { ( t _ { i } + 1 ) ( t _ { j } + 1 ) } E ^ { i i } \\right| \\right| } \\\\ { = \\displaystyle \\operatorname* { m a x } _ { i \\in [ N ] } \\left( \\displaystyle \\sum _ { j = 1 } ^ { N } \\frac { p _ { i j } - p _ { i j } ^ { 2 } } { ( t _ { i } + 1 ) ( t _ { j } + 1 ) } \\right) } \\\\ { \\leq \\displaystyle \\operatorname* { m a x } _ { i \\in [ N ] } \\left( \\displaystyle \\sum _ { j = 1 } ^ { N } \\frac { p _ { i j } } { ( t _ { i } + 1 ) ( t _ { j } + 1 ) } \\right) } \\\\ { \\leq ( \\delta + 1 ) ^ { - 1 } . } \\end{array}",
+ "type": "interline_equation",
+ "image_path": "671d8d0807c4af3435e2fc7a59b03aa66054295c22241b527c28d8f8d4f43a38.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 16.5,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 207,
+ 404,
+ 402,
+ 420.75
+ ],
+ "spans": [],
+ "index": 13
+ },
+ {
+ "bbox": [
+ 207,
+ 420.75,
+ 402,
+ 437.5
+ ],
+ "spans": [],
+ "index": 14
+ },
+ {
+ "bbox": [
+ 207,
+ 437.5,
+ 402,
+ 454.25
+ ],
+ "spans": [],
+ "index": 15
+ },
+ {
+ "bbox": [
+ 207,
+ 454.25,
+ 402,
+ 471.0
+ ],
+ "spans": [],
+ "index": 16
+ },
+ {
+ "bbox": [
+ 207,
+ 471.0,
+ 402,
+ 487.75
+ ],
+ "spans": [],
+ "index": 17
+ },
+ {
+ "bbox": [
+ 207,
+ 487.75,
+ 402,
+ 504.5
+ ],
+ "spans": [],
+ "index": 18
+ },
+ {
+ "bbox": [
+ 207,
+ 504.5,
+ 402,
+ 521.25
+ ],
+ "spans": [],
+ "index": 19
+ },
+ {
+ "bbox": [
+ 207,
+ 521.25,
+ 402,
+ 538.0
+ ],
+ "spans": [],
+ "index": 20
+ }
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 546,
+ 505,
+ 577
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 546,
+ 505,
+ 566
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 551,
+ 148,
+ 564
+ ],
+ "score": 1.0,
+ "content": "By letting",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 149,
+ 546,
+ 222,
+ 566
+ ],
+ "score": 0.89,
+ "content": "\\begin{array} { r } { a \\gets \\sqrt { \\frac { 3 \\log ( 4 N / \\varepsilon ) } { \\delta + 1 } } } \\end{array}",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 222,
+ 546,
+ 227,
+ 566
+ ],
+ "score": 1.0,
+ "content": ",",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 227,
+ 549,
+ 293,
+ 564
+ ],
+ "score": 0.52,
+ "content": "M \\gets ( \\delta + 1 ) ^ { - 1 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 294,
+ 546,
+ 298,
+ 566
+ ],
+ "score": 1.0,
+ "content": ",",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 298,
+ 550,
+ 364,
+ 563
+ ],
+ "score": 0.87,
+ "content": "v ^ { 2 } ( \\delta + 1 ) ^ { - 1 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 364,
+ 546,
+ 505,
+ 566
+ ],
+ "score": 1.0,
+ "content": "and applying Theorem 5 of Chung",
+ "type": "text"
+ }
+ ],
+ "index": 21
+ },
+ {
+ "bbox": [
+ 105,
+ 563,
+ 223,
+ 577
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 563,
+ 223,
+ 577
+ ],
+ "score": 1.0,
+ "content": "& Radcliffe (2011), we have",
+ "type": "text"
+ }
+ ],
+ "index": 22
+ }
+ ],
+ "index": 21.5,
+ "bbox_fs": [
+ 105,
+ 546,
+ 505,
+ 577
+ ]
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 171,
+ 582,
+ 439,
+ 641
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 171,
+ 582,
+ 439,
+ 641
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 171,
+ 582,
+ 439,
+ 641
+ ],
+ "score": 0.93,
+ "content": "\\begin{array} { r l } & { \\operatorname* { P r } ( \\| C - \\bar { \\Delta } \\| > a ) \\leq 2 N \\exp \\left( - \\displaystyle \\frac { a ^ { 2 } } { 2 ( \\delta + 1 ) ^ { - 1 } + 2 ( \\delta + 1 ) ^ { - 1 } a / 3 } \\right) } \\\\ & { \\phantom { \\operatorname* { P r } } \\leq 2 N \\exp \\left( - \\displaystyle \\frac { 3 \\log \\left( 4 N / \\varepsilon \\right) } { 2 ( 1 + a / 3 ) } \\right) . } \\end{array}",
+ "type": "interline_equation",
+ "image_path": "a1bcd9ba5a9bd225226118673c07e91de3eaf40dfa178bcc7be802d6733cac42.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 24,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 171,
+ 582,
+ 439,
+ 601.6666666666666
+ ],
+ "spans": [],
+ "index": 23
+ },
+ {
+ "bbox": [
+ 171,
+ 601.6666666666666,
+ 439,
+ 621.3333333333333
+ ],
+ "spans": [],
+ "index": 24
+ },
+ {
+ "bbox": [
+ 171,
+ 621.3333333333333,
+ 439,
+ 640.9999999999999
+ ],
+ "spans": [],
+ "index": 25
+ }
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 645,
+ 443,
+ 660
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 645,
+ 442,
+ 660
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 645,
+ 187,
+ 660
+ ],
+ "score": 1.0,
+ "content": "By the definition of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 187,
+ 647,
+ 205,
+ 659
+ ],
+ "score": 0.92,
+ "content": "k ( \\varepsilon )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 206,
+ 645,
+ 244,
+ 660
+ ],
+ "score": 1.0,
+ "content": ", we have",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 245,
+ 648,
+ 270,
+ 657
+ ],
+ "score": 0.89,
+ "content": "a < 1",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 270,
+ 645,
+ 279,
+ 660
+ ],
+ "score": 1.0,
+ "content": "if",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 280,
+ 647,
+ 356,
+ 659
+ ],
+ "score": 0.93,
+ "content": "\\delta + 1 > k ( \\varepsilon ) \\log n",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 356,
+ 645,
+ 397,
+ 660
+ ],
+ "score": 1.0,
+ "content": ". For such",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 397,
+ 648,
+ 403,
+ 657
+ ],
+ "score": 0.81,
+ "content": "\\delta",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 403,
+ 645,
+ 442,
+ 660
+ ],
+ "score": 1.0,
+ "content": ", we have",
+ "type": "text"
+ }
+ ],
+ "index": 26
+ }
+ ],
+ "index": 26,
+ "bbox_fs": [
+ 105,
+ 645,
+ 442,
+ 660
+ ]
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 186,
+ 666,
+ 425,
+ 732
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 186,
+ 666,
+ 425,
+ 732
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 186,
+ 666,
+ 425,
+ 732
+ ],
+ "score": 0.94,
+ "content": "\\begin{array} { l } { \\displaystyle \\operatorname* { P r } ( | | C - \\bar { \\Delta } | | > a ) \\leq 2 N \\exp \\left( - \\frac { 3 \\log ( 4 N / \\varepsilon ) } { 2 ( 1 + a / 3 ) } \\right) } \\\\ { \\leq 2 N \\exp \\left( - \\log ( 4 N / \\varepsilon ) \\right) \\quad \\displaystyle ( \\because a < 1 ) } \\\\ { = \\displaystyle \\frac { \\varepsilon } { 2 } . } \\end{array}",
+ "type": "interline_equation",
+ "image_path": "fc09ff0f9c9bdeeb3f37e0f6c7e58b8b98a7adbcd430ca7cb03b465487d743d3.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 28.5,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 186,
+ 666,
+ 425,
+ 682.5
+ ],
+ "spans": [],
+ "index": 27
+ },
+ {
+ "bbox": [
+ 186,
+ 682.5,
+ 425,
+ 699.0
+ ],
+ "spans": [],
+ "index": 28
+ },
+ {
+ "bbox": [
+ 186,
+ 699.0,
+ 425,
+ 715.5
+ ],
+ "spans": [],
+ "index": 29
+ },
+ {
+ "bbox": [
+ 186,
+ 715.5,
+ 425,
+ 732.0
+ ],
+ "spans": [],
+ "index": 30
+ }
+ ]
+ }
+ ]
+ },
+ {
+ "preproc_blocks": [
+ {
+ "type": "text",
+ "bbox": [
+ 105,
+ 81,
+ 505,
+ 105
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 81,
+ 505,
+ 96
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 81,
+ 174,
+ 96
+ ],
+ "score": 1.0,
+ "content": "Next, we bound",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 175,
+ 81,
+ 215,
+ 95
+ ],
+ "score": 0.93,
+ "content": "\\| \\tilde { \\Delta } - C \\|",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 215,
+ 81,
+ 270,
+ 96
+ ],
+ "score": 1.0,
+ "content": ". First, since",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 270,
+ 83,
+ 297,
+ 93
+ ],
+ "score": 0.92,
+ "content": "a < 1",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 298,
+ 81,
+ 505,
+ 96
+ ],
+ "score": 1.0,
+ "content": ", by Chernoff bound (see, e.g. Angluin & Valiant",
+ "type": "text"
+ }
+ ],
+ "index": 0
+ },
+ {
+ "bbox": [
+ 106,
+ 93,
+ 278,
+ 106
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 93,
+ 278,
+ 106
+ ],
+ "score": 1.0,
+ "content": "(1979); Hagerup & Rub (1990))), we have ¨",
+ "type": "text"
+ }
+ ],
+ "index": 1
+ }
+ ],
+ "index": 0.5
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 200,
+ 108,
+ 409,
+ 188
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 200,
+ 108,
+ 409,
+ 188
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 200,
+ 108,
+ 409,
+ 188
+ ],
+ "score": 0.94,
+ "content": "\\begin{array} { r l r } { { \\operatorname* { P r } ( | d _ { i } - t _ { i } | > a ( t _ { i } + 1 ) ) \\le 2 \\exp ( - \\frac { a ^ { 2 } ( t _ { i } + 1 ) } { 3 } ) } } \\\\ & { } & \\\\ & { } & { \\le 2 \\exp ( - \\frac { a ^ { 2 } ( \\delta + 1 ) } { 3 } ) } \\\\ & { } & { = \\frac { \\varepsilon } { 2 N } . } \\end{array}",
+ "type": "interline_equation",
+ "image_path": "74d357cc81d67fef8ac3d521a2d5baabb76cb5a42dfb3a5cb3a28999d9b09ecd.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 4,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 200,
+ 108,
+ 409,
+ 124.0
+ ],
+ "spans": [],
+ "index": 2
+ },
+ {
+ "bbox": [
+ 200,
+ 124.0,
+ 409,
+ 140.0
+ ],
+ "spans": [],
+ "index": 3
+ },
+ {
+ "bbox": [
+ 200,
+ 140.0,
+ 409,
+ 156.0
+ ],
+ "spans": [],
+ "index": 4
+ },
+ {
+ "bbox": [
+ 200,
+ 156.0,
+ 409,
+ 172.0
+ ],
+ "spans": [],
+ "index": 5
+ },
+ {
+ "bbox": [
+ 200,
+ 172.0,
+ 409,
+ 188.0
+ ],
+ "spans": [],
+ "index": 6
+ }
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 189,
+ 303,
+ 201
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 189,
+ 302,
+ 202
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 189,
+ 159,
+ 202
+ ],
+ "score": 1.0,
+ "content": "Therefore, if",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 160,
+ 189,
+ 244,
+ 201
+ ],
+ "score": 0.92,
+ "content": "| d _ { i } - t _ { i } | \\leq a ( t _ { i } + 1 )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 244,
+ 189,
+ 302,
+ 202
+ ],
+ "score": 1.0,
+ "content": ", then we have",
+ "type": "text"
+ }
+ ],
+ "index": 7
+ }
+ ],
+ "index": 7
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 165,
+ 204,
+ 445,
+ 280
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 165,
+ 204,
+ 445,
+ 280
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 165,
+ 204,
+ 445,
+ 280
+ ],
+ "score": 0.93,
+ "content": "\\begin{array} { r l r } { \\left. \\sqrt { \\displaystyle \\frac { d _ { i } + 1 } { t _ { i } + 1 } } - 1 \\right. \\le \\left. \\displaystyle \\frac { d _ { i } + 1 } { t _ { i } + 1 } - 1 \\right. } & { \\left( \\cdot \\cdot | \\sqrt { x } - 1 | \\le | x - 1 | \\mathrm { \\ f o r \\ } x \\ge 0 \\right) } & \\\\ { = \\displaystyle \\left. \\displaystyle \\frac { d _ { i } - t _ { i } } { t _ { i } + 1 } \\right. } & \\\\ { \\le a . } & \\end{array}",
+ "type": "interline_equation",
+ "image_path": "c1060e4c83d8629475673c96d2ea0f7c95c94f2a4a6eb00fc748a911f99f3501.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 9,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 165,
+ 204,
+ 445,
+ 229.33333333333334
+ ],
+ "spans": [],
+ "index": 8
+ },
+ {
+ "bbox": [
+ 165,
+ 229.33333333333334,
+ 445,
+ 254.66666666666669
+ ],
+ "spans": [],
+ "index": 9
+ },
+ {
+ "bbox": [
+ 165,
+ 254.66666666666669,
+ 445,
+ 280.0
+ ],
+ "spans": [],
+ "index": 10
+ }
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 283,
+ 253,
+ 294
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 281,
+ 253,
+ 296
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 281,
+ 253,
+ 296
+ ],
+ "score": 1.0,
+ "content": "Therefore, by union bound, we have",
+ "type": "text"
+ }
+ ],
+ "index": 11
+ }
+ ],
+ "index": 11
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 212,
+ 297,
+ 399,
+ 331
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 212,
+ 297,
+ 399,
+ 331
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 212,
+ 297,
+ 399,
+ 331
+ ],
+ "score": 0.94,
+ "content": "\\| \\bar { D } ^ { - \\frac { 1 } { 2 } } \\tilde { D } ^ { \\frac { 1 } { 2 } } - I _ { N } \\| = \\operatorname* { m a x } _ { i \\in [ N ] } \\left| \\sqrt { \\frac { d _ { i } + 1 } { t _ { i } + 1 } } - 1 \\right| \\leq a",
+ "type": "interline_equation",
+ "image_path": "277ca10271cab09840a50b713b61446378a2676bf929d31999e2bd31576d319d.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 12.5,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 212,
+ 297,
+ 399,
+ 314.0
+ ],
+ "spans": [],
+ "index": 12
+ },
+ {
+ "bbox": [
+ 212,
+ 314.0,
+ 399,
+ 331.0
+ ],
+ "spans": [],
+ "index": 13
+ }
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 108,
+ 334,
+ 504,
+ 369
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 333,
+ 505,
+ 347
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 333,
+ 203,
+ 347
+ ],
+ "score": 1.0,
+ "content": "with probability at least",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 204,
+ 334,
+ 236,
+ 346
+ ],
+ "score": 0.92,
+ "content": "1 - \\varepsilon / 2",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 236,
+ 333,
+ 505,
+ 347
+ ],
+ "score": 1.0,
+ "content": ". Further, since the eigenvalue of the augmented normalized Lapla-",
+ "type": "text"
+ }
+ ],
+ "index": 14
+ },
+ {
+ "bbox": [
+ 105,
+ 345,
+ 506,
+ 361
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 345,
+ 147,
+ 361
+ ],
+ "score": 1.0,
+ "content": "cian is in",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 148,
+ 347,
+ 168,
+ 359
+ ],
+ "score": 0.66,
+ "content": "[ 0 , 2 ]",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 168,
+ 345,
+ 332,
+ 361
+ ],
+ "score": 1.0,
+ "content": "by the proof of Proposition 1, we have",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 333,
+ 345,
+ 397,
+ 359
+ ],
+ "score": 0.94,
+ "content": "\\| I _ { N } - \\tilde { \\Delta } \\| \\leq \\bar { 1 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 398,
+ 345,
+ 506,
+ 361
+ ],
+ "score": 1.0,
+ "content": ". By combining them, we",
+ "type": "text"
+ }
+ ],
+ "index": 15
+ },
+ {
+ "bbox": [
+ 105,
+ 357,
+ 128,
+ 370
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 357,
+ 128,
+ 370
+ ],
+ "score": 1.0,
+ "content": "have",
+ "type": "text"
+ }
+ ],
+ "index": 16
+ }
+ ],
+ "index": 15
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 145,
+ 368,
+ 466,
+ 418
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 145,
+ 368,
+ 466,
+ 418
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 145,
+ 368,
+ 466,
+ 418
+ ],
+ "score": 0.93,
+ "content": "\\begin{array} { r l } & { \\| \\tilde { \\Delta } - C \\| = \\| ( \\bar { D } ^ { - \\frac { 1 } { 2 } } \\tilde { D } ^ { \\frac { 1 } { 2 } } - I _ { N } ) ( I _ { N } - \\tilde { \\Delta } ) \\tilde { D } ^ { \\frac { 1 } { 2 } } \\bar { D } ^ { - \\frac { 1 } { 2 } } + ( I _ { N } - \\tilde { \\Delta } ) ( I - \\tilde { D } ^ { \\frac { 1 } { 2 } } \\bar { D } ^ { - \\frac { 1 } { 2 } } ) \\| } \\\\ & { \\qquad \\leq \\| ( \\bar { D } ^ { - \\frac { 1 } { 2 } } \\tilde { D } ^ { \\frac { 1 } { 2 } } - I _ { N } \\| \\| \\tilde { D } ^ { \\frac { 1 } { 2 } } \\bar { D } ^ { - \\frac { 1 } { 2 } } \\| + \\| I - \\tilde { D } ^ { \\frac { 1 } { 2 } } \\bar { D } ^ { - \\frac { 1 } { 2 } } \\| } \\\\ & { \\qquad \\leq a ( a + 1 ) + a . } \\end{array}",
+ "type": "interline_equation",
+ "image_path": "b36d7b90316bd484e3acdd5b2078ffc194317e76183ac3b51ff553211a8c7236.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 18,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 145,
+ 368,
+ 466,
+ 384.6666666666667
+ ],
+ "spans": [],
+ "index": 17
+ },
+ {
+ "bbox": [
+ 145,
+ 384.6666666666667,
+ 466,
+ 401.33333333333337
+ ],
+ "spans": [],
+ "index": 18
+ },
+ {
+ "bbox": [
+ 145,
+ 401.33333333333337,
+ 466,
+ 418.00000000000006
+ ],
+ "spans": [],
+ "index": 19
+ }
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 420,
+ 213,
+ 432
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 420,
+ 213,
+ 433
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 420,
+ 213,
+ 433
+ ],
+ "score": 1.0,
+ "content": "From (5) and (6), we have",
+ "type": "text"
+ }
+ ],
+ "index": 20
+ }
+ ],
+ "index": 20
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 233,
+ 433,
+ 377,
+ 494
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 233,
+ 433,
+ 377,
+ 494
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 233,
+ 433,
+ 377,
+ 494
+ ],
+ "score": 0.94,
+ "content": "\\begin{array} { r l } & { \\| \\tilde { \\Delta } - \\bar { \\Delta } \\| \\leq \\| \\tilde { \\Delta } - C \\| + \\| C - \\bar { \\Delta } \\| } \\\\ & { \\qquad \\leq a + a ( a + 1 ) + a } \\\\ & { \\qquad \\leq a ^ { 2 } + 3 a } \\\\ & { \\qquad \\leq 4 a \\quad ( \\because a < 1 ) } \\end{array}",
+ "type": "interline_equation",
+ "image_path": "7da86c93036b713cc0974d5bb45826b01ccbb959f3adf50cc6bf75e87c2a16ef.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 21.5,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 233,
+ 433,
+ 377,
+ 463.5
+ ],
+ "spans": [],
+ "index": 21
+ },
+ {
+ "bbox": [
+ 233,
+ 463.5,
+ 377,
+ 494.0
+ ],
+ "spans": [],
+ "index": 22
+ }
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 509,
+ 294,
+ 522
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 508,
+ 295,
+ 523
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 508,
+ 203,
+ 523
+ ],
+ "score": 1.0,
+ "content": "with probability at least",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 204,
+ 510,
+ 227,
+ 520
+ ],
+ "score": 0.87,
+ "content": "1 - \\varepsilon",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 227,
+ 508,
+ 295,
+ 523
+ ],
+ "score": 1.0,
+ "content": "by union bound.",
+ "type": "text"
+ }
+ ],
+ "index": 23
+ }
+ ],
+ "index": 23
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 532,
+ 506,
+ 608
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 533,
+ 504,
+ 546
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 533,
+ 122,
+ 546
+ ],
+ "score": 1.0,
+ "content": "Let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 122,
+ 534,
+ 158,
+ 545
+ ],
+ "score": 0.91,
+ "content": "N \\in { \\mathbb { N } } _ { + }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 158,
+ 533,
+ 176,
+ 546
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 176,
+ 534,
+ 200,
+ 545
+ ],
+ "score": 0.89,
+ "content": "p > 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 201,
+ 533,
+ 351,
+ 546
+ ],
+ "score": 1.0,
+ "content": ". In the case of the Erdos-R ˝ enyi graph ´",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 352,
+ 534,
+ 374,
+ 545
+ ],
+ "score": 0.89,
+ "content": "G _ { N , p }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 374,
+ 533,
+ 433,
+ 546
+ ],
+ "score": 1.0,
+ "content": ", we should set",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 434,
+ 533,
+ 504,
+ 545
+ ],
+ "score": 0.9,
+ "content": "P = p ( J _ { N } - I _ { N } )",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 24
+ },
+ {
+ "bbox": [
+ 104,
+ 542,
+ 505,
+ 559
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 542,
+ 134,
+ 559
+ ],
+ "score": 1.0,
+ "content": "where",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 134,
+ 545,
+ 190,
+ 556
+ ],
+ "score": 0.92,
+ "content": "J _ { N } \\in \\mathbb { R } ^ { N \\times N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 190,
+ 542,
+ 348,
+ 559
+ ],
+ "score": 1.0,
+ "content": "are the all-one matrix. Then, we have",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 349,
+ 545,
+ 446,
+ 557
+ ],
+ "score": 0.89,
+ "content": "\\bar { A } = p J _ { N } + ( 1 - p ) I _ { N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 446,
+ 542,
+ 450,
+ 559
+ ],
+ "score": 1.0,
+ "content": ",",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 451,
+ 545,
+ 505,
+ 558
+ ],
+ "score": 0.89,
+ "content": "\\bar { D } = ( N p -",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 25
+ },
+ {
+ "bbox": [
+ 106,
+ 555,
+ 506,
+ 571
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 557,
+ 146,
+ 569
+ ],
+ "score": 0.91,
+ "content": "p + 1 ) I _ { N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 146,
+ 556,
+ 168,
+ 569
+ ],
+ "score": 1.0,
+ "content": ", and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 168,
+ 556,
+ 281,
+ 571
+ ],
+ "score": 0.93,
+ "content": "\\begin{array} { r } { \\bar { \\Delta } = \\frac { p } { N p - p + 1 } ( N I _ { N } - J _ { N } ) } \\end{array}",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 281,
+ 555,
+ 383,
+ 570
+ ],
+ "score": 1.0,
+ "content": ". Since the eigenvalue of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 384,
+ 558,
+ 397,
+ 568
+ ],
+ "score": 0.9,
+ "content": "J _ { N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 398,
+ 555,
+ 408,
+ 570
+ ],
+ "score": 1.0,
+ "content": "is",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 409,
+ 558,
+ 419,
+ 567
+ ],
+ "score": 0.81,
+ "content": "N",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 420,
+ 555,
+ 506,
+ 570
+ ],
+ "score": 1.0,
+ "content": "(with multiplicity 1)",
+ "type": "text"
+ }
+ ],
+ "index": 26
+ },
+ {
+ "bbox": [
+ 105,
+ 569,
+ 506,
+ 587
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 569,
+ 205,
+ 587
+ ],
+ "score": 1.0,
+ "content": "and 0 (with multiplicity",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 205,
+ 572,
+ 235,
+ 583
+ ],
+ "score": 0.87,
+ "content": "N - 1 ,",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 235,
+ 569,
+ 313,
+ 587
+ ],
+ "score": 1.0,
+ "content": "), the eigenvalue of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 313,
+ 570,
+ 323,
+ 582
+ ],
+ "score": 0.86,
+ "content": "\\bar { \\Delta }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 324,
+ 569,
+ 445,
+ 587
+ ],
+ "score": 1.0,
+ "content": "is 0 (with multiplicity 1) and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 445,
+ 570,
+ 480,
+ 586
+ ],
+ "score": 0.94,
+ "content": "\\frac { N p } { N p - p + 1 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 480,
+ 571,
+ 506,
+ 585
+ ],
+ "score": 1.0,
+ "content": "(with",
+ "type": "text"
+ }
+ ],
+ "index": 27
+ },
+ {
+ "bbox": [
+ 105,
+ 584,
+ 506,
+ 598
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 584,
+ 156,
+ 598
+ ],
+ "score": 1.0,
+ "content": "multiplicity",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 156,
+ 585,
+ 187,
+ 596
+ ],
+ "score": 0.83,
+ "content": "N - 1 )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 187,
+ 584,
+ 212,
+ 598
+ ],
+ "score": 1.0,
+ "content": ". For",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 212,
+ 585,
+ 234,
+ 597
+ ],
+ "score": 0.86,
+ "content": "G _ { N , p }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 234,
+ 584,
+ 239,
+ 598
+ ],
+ "score": 1.0,
+ "content": ",",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 239,
+ 586,
+ 245,
+ 595
+ ],
+ "score": 0.63,
+ "content": "\\delta",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 245,
+ 584,
+ 375,
+ 598
+ ],
+ "score": 1.0,
+ "content": "is the expected average degree",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 375,
+ 585,
+ 416,
+ 597
+ ],
+ "score": 0.92,
+ "content": "( N - 1 ) p",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 416,
+ 584,
+ 506,
+ 598
+ ],
+ "score": 1.0,
+ "content": ". Hence, we have the",
+ "type": "text"
+ }
+ ],
+ "index": 28
+ },
+ {
+ "bbox": [
+ 106,
+ 596,
+ 242,
+ 608
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 596,
+ 242,
+ 608
+ ],
+ "score": 1.0,
+ "content": "following lemma from Lemma 5:",
+ "type": "text"
+ }
+ ],
+ "index": 29
+ }
+ ],
+ "index": 26.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 105,
+ 610,
+ 505,
+ 638
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 609,
+ 505,
+ 626
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 609,
+ 169,
+ 626
+ ],
+ "score": 1.0,
+ "content": "Lemma 6. Let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 170,
+ 610,
+ 179,
+ 621
+ ],
+ "score": 0.81,
+ "content": "\\tilde { \\Delta }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 180,
+ 609,
+ 444,
+ 626
+ ],
+ "score": 1.0,
+ "content": "be its augmented normalized Laplacian of the Erdos-R ˝ enyi graph ´",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 444,
+ 612,
+ 466,
+ 624
+ ],
+ "score": 0.91,
+ "content": "G _ { N , p }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 467,
+ 609,
+ 505,
+ 626
+ ],
+ "score": 1.0,
+ "content": ". For any",
+ "type": "text"
+ }
+ ],
+ "index": 30
+ },
+ {
+ "bbox": [
+ 106,
+ 618,
+ 478,
+ 640
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 624,
+ 131,
+ 636
+ ],
+ "score": 0.77,
+ "content": "\\varepsilon > 0 ,",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 132,
+ 623,
+ 289,
+ 639
+ ],
+ "score": 0.87,
+ "content": "\\begin{array} { r } { i f \\frac { N p - p + 1 } { \\log N } > k ( \\varepsilon ) : = 3 ( 1 + \\log ( 4 / \\varepsilon ) ) } \\end{array}",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 290,
+ 618,
+ 413,
+ 640
+ ],
+ "score": 1.0,
+ "content": ", then, with probability at least",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 414,
+ 625,
+ 436,
+ 635
+ ],
+ "score": 0.86,
+ "content": "1 - \\varepsilon",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 437,
+ 618,
+ 478,
+ 640
+ ],
+ "score": 1.0,
+ "content": ", we have",
+ "type": "text"
+ }
+ ],
+ "index": 31
+ }
+ ],
+ "index": 30.5
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 198,
+ 642,
+ 412,
+ 675
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 198,
+ 642,
+ 412,
+ 675
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 198,
+ 642,
+ 412,
+ 675
+ ],
+ "score": 0.92,
+ "content": "\\operatorname* { m a x } _ { i = 2 , \\ldots , N } \\left| \\lambda _ { i } ( \\tilde { \\Delta } ) - \\frac { N p } { N p - p + 1 } \\right| \\leq 4 \\sqrt { \\frac { 3 \\log ( 4 N / \\varepsilon ) } { N p - p + 1 } } .",
+ "type": "interline_equation",
+ "image_path": "e0c336f319cdb5b4687e3e0d749c2480665543a37bcb8772f9437612db712e00.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 32.5,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 198,
+ 642,
+ 412,
+ 658.5
+ ],
+ "spans": [],
+ "index": 32
+ },
+ {
+ "bbox": [
+ 198,
+ 658.5,
+ 412,
+ 675.0
+ ],
+ "spans": [],
+ "index": 33
+ }
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 678,
+ 506,
+ 733
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 676,
+ 506,
+ 691
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 676,
+ 241,
+ 691
+ ],
+ "score": 1.0,
+ "content": "Corollary 4. Consider GCN on",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 241,
+ 679,
+ 263,
+ 690
+ ],
+ "score": 0.89,
+ "content": "G _ { N , p }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 264,
+ 676,
+ 287,
+ 691
+ ],
+ "score": 1.0,
+ "content": ". Let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 287,
+ 678,
+ 300,
+ 689
+ ],
+ "score": 0.82,
+ "content": "W _ { l }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 300,
+ 676,
+ 387,
+ 691
+ ],
+ "score": 1.0,
+ "content": "be the weight of the",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 388,
+ 679,
+ 392,
+ 688
+ ],
+ "score": 0.55,
+ "content": "l",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 392,
+ 676,
+ 482,
+ 691
+ ],
+ "score": 1.0,
+ "content": "-th layer of GCN and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 482,
+ 680,
+ 491,
+ 689
+ ],
+ "score": 0.79,
+ "content": "s _ { l }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 491,
+ 676,
+ 506,
+ 691
+ ],
+ "score": 1.0,
+ "content": "be",
+ "type": "text"
+ }
+ ],
+ "index": 34
+ },
+ {
+ "bbox": [
+ 105,
+ 688,
+ 505,
+ 704
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 688,
+ 235,
+ 704
+ ],
+ "score": 1.0,
+ "content": "the maximum singular value of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 235,
+ 690,
+ 248,
+ 700
+ ],
+ "score": 0.81,
+ "content": "W _ { l }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 248,
+ 688,
+ 265,
+ 704
+ ],
+ "score": 1.0,
+ "content": "for",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 265,
+ 690,
+ 297,
+ 701
+ ],
+ "score": 0.89,
+ "content": "l \\in \\mathbb { N } _ { + }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 297,
+ 688,
+ 318,
+ 704
+ ],
+ "score": 1.0,
+ "content": ". Set",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 318,
+ 691,
+ 375,
+ 703
+ ],
+ "score": 0.87,
+ "content": "s : = \\mathrm { s u p } _ { l \\in \\mathbb { N } _ { + } }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 375,
+ 688,
+ 397,
+ 704
+ ],
+ "score": 1.0,
+ "content": ". Let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 397,
+ 690,
+ 423,
+ 699
+ ],
+ "score": 0.91,
+ "content": "\\varepsilon > 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 423,
+ 688,
+ 470,
+ 704
+ ],
+ "score": 1.0,
+ "content": ". We define",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 470,
+ 689,
+ 505,
+ 701
+ ],
+ "score": 0.91,
+ "content": "k ( \\varepsilon ) : =",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 35
+ },
+ {
+ "bbox": [
+ 107,
+ 700,
+ 507,
+ 723
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 706,
+ 171,
+ 719
+ ],
+ "score": 0.9,
+ "content": "3 ( 1 + \\log ( 4 / \\varepsilon ) )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 171,
+ 700,
+ 191,
+ 723
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 191,
+ 704,
+ 345,
+ 722
+ ],
+ "score": 0.91,
+ "content": "\\begin{array} { r } { l ( N , p , \\varepsilon ) = { \\frac { 1 - p } { N p - p + 1 } } + 4 { \\sqrt { \\frac { 3 \\log ( 4 N / \\varepsilon ) } { N p - p + 1 } } } } \\end{array}",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 345,
+ 700,
+ 355,
+ 723
+ ],
+ "score": 1.0,
+ "content": ". I",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 356,
+ 705,
+ 424,
+ 721
+ ],
+ "score": 0.82,
+ "content": "\\begin{array} { r } { \\ell \\frac { N p - p + 1 } { \\log N } > k ( \\varepsilon ) } \\end{array}",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 424,
+ 700,
+ 443,
+ 723
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 443,
+ 705,
+ 501,
+ 719
+ ],
+ "score": 0.92,
+ "content": "s \\leq l ( N , \\varepsilon ) ^ { - 1 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 501,
+ 700,
+ 507,
+ 723
+ ],
+ "score": 1.0,
+ "content": ",",
+ "type": "text"
+ }
+ ],
+ "index": 36
+ },
+ {
+ "bbox": [
+ 105,
+ 718,
+ 465,
+ 734
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 718,
+ 164,
+ 734
+ ],
+ "score": 1.0,
+ "content": "then, GCN on",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 164,
+ 721,
+ 186,
+ 733
+ ],
+ "score": 0.9,
+ "content": "G _ { N , p }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 187,
+ 718,
+ 437,
+ 734
+ ],
+ "score": 1.0,
+ "content": "satisfies the assumption of Theorem 2 with probability at least",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 438,
+ 721,
+ 460,
+ 731
+ ],
+ "score": 0.86,
+ "content": "1 - \\varepsilon",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 460,
+ 718,
+ 465,
+ 734
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 37
+ }
+ ],
+ "index": 35.5
+ }
+ ],
+ "page_idx": 19,
+ "page_size": [
+ 612,
+ 792
+ ],
+ "discarded_blocks": [
+ {
+ "type": "discarded",
+ "bbox": [
+ 107,
+ 26,
+ 294,
+ 38
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 25,
+ 294,
+ 38
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 25,
+ 294,
+ 38
+ ],
+ "score": 1.0,
+ "content": "Published as a conference paper at ICLR 2020",
+ "type": "text"
+ }
+ ]
+ }
+ ]
+ },
+ {
+ "type": "discarded",
+ "bbox": [
+ 300,
+ 751,
+ 311,
+ 760
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 298,
+ 749,
+ 313,
+ 764
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 298,
+ 749,
+ 313,
+ 764
+ ],
+ "score": 1.0,
+ "content": "",
+ "type": "text",
+ "height": 15,
+ "width": 15
+ }
+ ]
+ }
+ ]
+ },
+ {
+ "type": "discarded",
+ "bbox": [
+ 494,
+ 510,
+ 505,
+ 520
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 496,
+ 512,
+ 504,
+ 520
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 496,
+ 512,
+ 504,
+ 520
+ ],
+ "score": 0.997,
+ "content": "□",
+ "type": "text"
+ }
+ ]
+ }
+ ]
+ }
+ ],
+ "para_blocks": [
+ {
+ "type": "text",
+ "bbox": [
+ 105,
+ 81,
+ 505,
+ 105
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 81,
+ 505,
+ 96
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 81,
+ 174,
+ 96
+ ],
+ "score": 1.0,
+ "content": "Next, we bound",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 175,
+ 81,
+ 215,
+ 95
+ ],
+ "score": 0.93,
+ "content": "\\| \\tilde { \\Delta } - C \\|",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 215,
+ 81,
+ 270,
+ 96
+ ],
+ "score": 1.0,
+ "content": ". First, since",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 270,
+ 83,
+ 297,
+ 93
+ ],
+ "score": 0.92,
+ "content": "a < 1",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 298,
+ 81,
+ 505,
+ 96
+ ],
+ "score": 1.0,
+ "content": ", by Chernoff bound (see, e.g. Angluin & Valiant",
+ "type": "text"
+ }
+ ],
+ "index": 0
+ },
+ {
+ "bbox": [
+ 106,
+ 93,
+ 278,
+ 106
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 93,
+ 278,
+ 106
+ ],
+ "score": 1.0,
+ "content": "(1979); Hagerup & Rub (1990))), we have ¨",
+ "type": "text"
+ }
+ ],
+ "index": 1
+ }
+ ],
+ "index": 0.5,
+ "bbox_fs": [
+ 105,
+ 81,
+ 505,
+ 106
+ ]
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 200,
+ 108,
+ 409,
+ 188
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 200,
+ 108,
+ 409,
+ 188
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 200,
+ 108,
+ 409,
+ 188
+ ],
+ "score": 0.94,
+ "content": "\\begin{array} { r l r } { { \\operatorname* { P r } ( | d _ { i } - t _ { i } | > a ( t _ { i } + 1 ) ) \\le 2 \\exp ( - \\frac { a ^ { 2 } ( t _ { i } + 1 ) } { 3 } ) } } \\\\ & { } & \\\\ & { } & { \\le 2 \\exp ( - \\frac { a ^ { 2 } ( \\delta + 1 ) } { 3 } ) } \\\\ & { } & { = \\frac { \\varepsilon } { 2 N } . } \\end{array}",
+ "type": "interline_equation",
+ "image_path": "74d357cc81d67fef8ac3d521a2d5baabb76cb5a42dfb3a5cb3a28999d9b09ecd.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 4,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 200,
+ 108,
+ 409,
+ 124.0
+ ],
+ "spans": [],
+ "index": 2
+ },
+ {
+ "bbox": [
+ 200,
+ 124.0,
+ 409,
+ 140.0
+ ],
+ "spans": [],
+ "index": 3
+ },
+ {
+ "bbox": [
+ 200,
+ 140.0,
+ 409,
+ 156.0
+ ],
+ "spans": [],
+ "index": 4
+ },
+ {
+ "bbox": [
+ 200,
+ 156.0,
+ 409,
+ 172.0
+ ],
+ "spans": [],
+ "index": 5
+ },
+ {
+ "bbox": [
+ 200,
+ 172.0,
+ 409,
+ 188.0
+ ],
+ "spans": [],
+ "index": 6
+ }
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 189,
+ 303,
+ 201
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 189,
+ 302,
+ 202
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 189,
+ 159,
+ 202
+ ],
+ "score": 1.0,
+ "content": "Therefore, if",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 160,
+ 189,
+ 244,
+ 201
+ ],
+ "score": 0.92,
+ "content": "| d _ { i } - t _ { i } | \\leq a ( t _ { i } + 1 )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 244,
+ 189,
+ 302,
+ 202
+ ],
+ "score": 1.0,
+ "content": ", then we have",
+ "type": "text"
+ }
+ ],
+ "index": 7
+ }
+ ],
+ "index": 7,
+ "bbox_fs": [
+ 106,
+ 189,
+ 302,
+ 202
+ ]
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 165,
+ 204,
+ 445,
+ 280
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 165,
+ 204,
+ 445,
+ 280
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 165,
+ 204,
+ 445,
+ 280
+ ],
+ "score": 0.93,
+ "content": "\\begin{array} { r l r } { \\left. \\sqrt { \\displaystyle \\frac { d _ { i } + 1 } { t _ { i } + 1 } } - 1 \\right. \\le \\left. \\displaystyle \\frac { d _ { i } + 1 } { t _ { i } + 1 } - 1 \\right. } & { \\left( \\cdot \\cdot | \\sqrt { x } - 1 | \\le | x - 1 | \\mathrm { \\ f o r \\ } x \\ge 0 \\right) } & \\\\ { = \\displaystyle \\left. \\displaystyle \\frac { d _ { i } - t _ { i } } { t _ { i } + 1 } \\right. } & \\\\ { \\le a . } & \\end{array}",
+ "type": "interline_equation",
+ "image_path": "c1060e4c83d8629475673c96d2ea0f7c95c94f2a4a6eb00fc748a911f99f3501.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 9,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 165,
+ 204,
+ 445,
+ 229.33333333333334
+ ],
+ "spans": [],
+ "index": 8
+ },
+ {
+ "bbox": [
+ 165,
+ 229.33333333333334,
+ 445,
+ 254.66666666666669
+ ],
+ "spans": [],
+ "index": 9
+ },
+ {
+ "bbox": [
+ 165,
+ 254.66666666666669,
+ 445,
+ 280.0
+ ],
+ "spans": [],
+ "index": 10
+ }
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 283,
+ 253,
+ 294
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 281,
+ 253,
+ 296
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 281,
+ 253,
+ 296
+ ],
+ "score": 1.0,
+ "content": "Therefore, by union bound, we have",
+ "type": "text"
+ }
+ ],
+ "index": 11
+ }
+ ],
+ "index": 11,
+ "bbox_fs": [
+ 106,
+ 281,
+ 253,
+ 296
+ ]
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 212,
+ 297,
+ 399,
+ 331
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 212,
+ 297,
+ 399,
+ 331
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 212,
+ 297,
+ 399,
+ 331
+ ],
+ "score": 0.94,
+ "content": "\\| \\bar { D } ^ { - \\frac { 1 } { 2 } } \\tilde { D } ^ { \\frac { 1 } { 2 } } - I _ { N } \\| = \\operatorname* { m a x } _ { i \\in [ N ] } \\left| \\sqrt { \\frac { d _ { i } + 1 } { t _ { i } + 1 } } - 1 \\right| \\leq a",
+ "type": "interline_equation",
+ "image_path": "277ca10271cab09840a50b713b61446378a2676bf929d31999e2bd31576d319d.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 12.5,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 212,
+ 297,
+ 399,
+ 314.0
+ ],
+ "spans": [],
+ "index": 12
+ },
+ {
+ "bbox": [
+ 212,
+ 314.0,
+ 399,
+ 331.0
+ ],
+ "spans": [],
+ "index": 13
+ }
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 108,
+ 334,
+ 504,
+ 369
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 333,
+ 505,
+ 347
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 333,
+ 203,
+ 347
+ ],
+ "score": 1.0,
+ "content": "with probability at least",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 204,
+ 334,
+ 236,
+ 346
+ ],
+ "score": 0.92,
+ "content": "1 - \\varepsilon / 2",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 236,
+ 333,
+ 505,
+ 347
+ ],
+ "score": 1.0,
+ "content": ". Further, since the eigenvalue of the augmented normalized Lapla-",
+ "type": "text"
+ }
+ ],
+ "index": 14
+ },
+ {
+ "bbox": [
+ 105,
+ 345,
+ 506,
+ 361
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 345,
+ 147,
+ 361
+ ],
+ "score": 1.0,
+ "content": "cian is in",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 148,
+ 347,
+ 168,
+ 359
+ ],
+ "score": 0.66,
+ "content": "[ 0 , 2 ]",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 168,
+ 345,
+ 332,
+ 361
+ ],
+ "score": 1.0,
+ "content": "by the proof of Proposition 1, we have",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 333,
+ 345,
+ 397,
+ 359
+ ],
+ "score": 0.94,
+ "content": "\\| I _ { N } - \\tilde { \\Delta } \\| \\leq \\bar { 1 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 398,
+ 345,
+ 506,
+ 361
+ ],
+ "score": 1.0,
+ "content": ". By combining them, we",
+ "type": "text"
+ }
+ ],
+ "index": 15
+ },
+ {
+ "bbox": [
+ 105,
+ 357,
+ 128,
+ 370
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 357,
+ 128,
+ 370
+ ],
+ "score": 1.0,
+ "content": "have",
+ "type": "text"
+ }
+ ],
+ "index": 16
+ }
+ ],
+ "index": 15,
+ "bbox_fs": [
+ 105,
+ 333,
+ 506,
+ 370
+ ]
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 145,
+ 368,
+ 466,
+ 418
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 145,
+ 368,
+ 466,
+ 418
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 145,
+ 368,
+ 466,
+ 418
+ ],
+ "score": 0.93,
+ "content": "\\begin{array} { r l } & { \\| \\tilde { \\Delta } - C \\| = \\| ( \\bar { D } ^ { - \\frac { 1 } { 2 } } \\tilde { D } ^ { \\frac { 1 } { 2 } } - I _ { N } ) ( I _ { N } - \\tilde { \\Delta } ) \\tilde { D } ^ { \\frac { 1 } { 2 } } \\bar { D } ^ { - \\frac { 1 } { 2 } } + ( I _ { N } - \\tilde { \\Delta } ) ( I - \\tilde { D } ^ { \\frac { 1 } { 2 } } \\bar { D } ^ { - \\frac { 1 } { 2 } } ) \\| } \\\\ & { \\qquad \\leq \\| ( \\bar { D } ^ { - \\frac { 1 } { 2 } } \\tilde { D } ^ { \\frac { 1 } { 2 } } - I _ { N } \\| \\| \\tilde { D } ^ { \\frac { 1 } { 2 } } \\bar { D } ^ { - \\frac { 1 } { 2 } } \\| + \\| I - \\tilde { D } ^ { \\frac { 1 } { 2 } } \\bar { D } ^ { - \\frac { 1 } { 2 } } \\| } \\\\ & { \\qquad \\leq a ( a + 1 ) + a . } \\end{array}",
+ "type": "interline_equation",
+ "image_path": "b36d7b90316bd484e3acdd5b2078ffc194317e76183ac3b51ff553211a8c7236.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 18,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 145,
+ 368,
+ 466,
+ 384.6666666666667
+ ],
+ "spans": [],
+ "index": 17
+ },
+ {
+ "bbox": [
+ 145,
+ 384.6666666666667,
+ 466,
+ 401.33333333333337
+ ],
+ "spans": [],
+ "index": 18
+ },
+ {
+ "bbox": [
+ 145,
+ 401.33333333333337,
+ 466,
+ 418.00000000000006
+ ],
+ "spans": [],
+ "index": 19
+ }
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 420,
+ 213,
+ 432
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 420,
+ 213,
+ 433
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 420,
+ 213,
+ 433
+ ],
+ "score": 1.0,
+ "content": "From (5) and (6), we have",
+ "type": "text"
+ }
+ ],
+ "index": 20
+ }
+ ],
+ "index": 20,
+ "bbox_fs": [
+ 106,
+ 420,
+ 213,
+ 433
+ ]
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 233,
+ 433,
+ 377,
+ 494
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 233,
+ 433,
+ 377,
+ 494
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 233,
+ 433,
+ 377,
+ 494
+ ],
+ "score": 0.94,
+ "content": "\\begin{array} { r l } & { \\| \\tilde { \\Delta } - \\bar { \\Delta } \\| \\leq \\| \\tilde { \\Delta } - C \\| + \\| C - \\bar { \\Delta } \\| } \\\\ & { \\qquad \\leq a + a ( a + 1 ) + a } \\\\ & { \\qquad \\leq a ^ { 2 } + 3 a } \\\\ & { \\qquad \\leq 4 a \\quad ( \\because a < 1 ) } \\end{array}",
+ "type": "interline_equation",
+ "image_path": "7da86c93036b713cc0974d5bb45826b01ccbb959f3adf50cc6bf75e87c2a16ef.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 21.5,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 233,
+ 433,
+ 377,
+ 463.5
+ ],
+ "spans": [],
+ "index": 21
+ },
+ {
+ "bbox": [
+ 233,
+ 463.5,
+ 377,
+ 494.0
+ ],
+ "spans": [],
+ "index": 22
+ }
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 509,
+ 294,
+ 522
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 508,
+ 295,
+ 523
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 508,
+ 203,
+ 523
+ ],
+ "score": 1.0,
+ "content": "with probability at least",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 204,
+ 510,
+ 227,
+ 520
+ ],
+ "score": 0.87,
+ "content": "1 - \\varepsilon",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 227,
+ 508,
+ 295,
+ 523
+ ],
+ "score": 1.0,
+ "content": "by union bound.",
+ "type": "text"
+ }
+ ],
+ "index": 23
+ }
+ ],
+ "index": 23,
+ "bbox_fs": [
+ 105,
+ 508,
+ 295,
+ 523
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 532,
+ 506,
+ 608
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 533,
+ 504,
+ 546
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 533,
+ 122,
+ 546
+ ],
+ "score": 1.0,
+ "content": "Let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 122,
+ 534,
+ 158,
+ 545
+ ],
+ "score": 0.91,
+ "content": "N \\in { \\mathbb { N } } _ { + }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 158,
+ 533,
+ 176,
+ 546
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 176,
+ 534,
+ 200,
+ 545
+ ],
+ "score": 0.89,
+ "content": "p > 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 201,
+ 533,
+ 351,
+ 546
+ ],
+ "score": 1.0,
+ "content": ". In the case of the Erdos-R ˝ enyi graph ´",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 352,
+ 534,
+ 374,
+ 545
+ ],
+ "score": 0.89,
+ "content": "G _ { N , p }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 374,
+ 533,
+ 433,
+ 546
+ ],
+ "score": 1.0,
+ "content": ", we should set",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 434,
+ 533,
+ 504,
+ 545
+ ],
+ "score": 0.9,
+ "content": "P = p ( J _ { N } - I _ { N } )",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 24
+ },
+ {
+ "bbox": [
+ 104,
+ 542,
+ 505,
+ 559
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 542,
+ 134,
+ 559
+ ],
+ "score": 1.0,
+ "content": "where",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 134,
+ 545,
+ 190,
+ 556
+ ],
+ "score": 0.92,
+ "content": "J _ { N } \\in \\mathbb { R } ^ { N \\times N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 190,
+ 542,
+ 348,
+ 559
+ ],
+ "score": 1.0,
+ "content": "are the all-one matrix. Then, we have",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 349,
+ 545,
+ 446,
+ 557
+ ],
+ "score": 0.89,
+ "content": "\\bar { A } = p J _ { N } + ( 1 - p ) I _ { N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 446,
+ 542,
+ 450,
+ 559
+ ],
+ "score": 1.0,
+ "content": ",",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 451,
+ 545,
+ 505,
+ 558
+ ],
+ "score": 0.89,
+ "content": "\\bar { D } = ( N p -",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 25
+ },
+ {
+ "bbox": [
+ 106,
+ 555,
+ 506,
+ 571
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 557,
+ 146,
+ 569
+ ],
+ "score": 0.91,
+ "content": "p + 1 ) I _ { N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 146,
+ 556,
+ 168,
+ 569
+ ],
+ "score": 1.0,
+ "content": ", and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 168,
+ 556,
+ 281,
+ 571
+ ],
+ "score": 0.93,
+ "content": "\\begin{array} { r } { \\bar { \\Delta } = \\frac { p } { N p - p + 1 } ( N I _ { N } - J _ { N } ) } \\end{array}",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 281,
+ 555,
+ 383,
+ 570
+ ],
+ "score": 1.0,
+ "content": ". Since the eigenvalue of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 384,
+ 558,
+ 397,
+ 568
+ ],
+ "score": 0.9,
+ "content": "J _ { N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 398,
+ 555,
+ 408,
+ 570
+ ],
+ "score": 1.0,
+ "content": "is",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 409,
+ 558,
+ 419,
+ 567
+ ],
+ "score": 0.81,
+ "content": "N",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 420,
+ 555,
+ 506,
+ 570
+ ],
+ "score": 1.0,
+ "content": "(with multiplicity 1)",
+ "type": "text"
+ }
+ ],
+ "index": 26
+ },
+ {
+ "bbox": [
+ 105,
+ 569,
+ 506,
+ 587
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 569,
+ 205,
+ 587
+ ],
+ "score": 1.0,
+ "content": "and 0 (with multiplicity",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 205,
+ 572,
+ 235,
+ 583
+ ],
+ "score": 0.87,
+ "content": "N - 1 ,",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 235,
+ 569,
+ 313,
+ 587
+ ],
+ "score": 1.0,
+ "content": "), the eigenvalue of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 313,
+ 570,
+ 323,
+ 582
+ ],
+ "score": 0.86,
+ "content": "\\bar { \\Delta }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 324,
+ 569,
+ 445,
+ 587
+ ],
+ "score": 1.0,
+ "content": "is 0 (with multiplicity 1) and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 445,
+ 570,
+ 480,
+ 586
+ ],
+ "score": 0.94,
+ "content": "\\frac { N p } { N p - p + 1 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 480,
+ 571,
+ 506,
+ 585
+ ],
+ "score": 1.0,
+ "content": "(with",
+ "type": "text"
+ }
+ ],
+ "index": 27
+ },
+ {
+ "bbox": [
+ 105,
+ 584,
+ 506,
+ 598
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 584,
+ 156,
+ 598
+ ],
+ "score": 1.0,
+ "content": "multiplicity",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 156,
+ 585,
+ 187,
+ 596
+ ],
+ "score": 0.83,
+ "content": "N - 1 )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 187,
+ 584,
+ 212,
+ 598
+ ],
+ "score": 1.0,
+ "content": ". For",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 212,
+ 585,
+ 234,
+ 597
+ ],
+ "score": 0.86,
+ "content": "G _ { N , p }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 234,
+ 584,
+ 239,
+ 598
+ ],
+ "score": 1.0,
+ "content": ",",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 239,
+ 586,
+ 245,
+ 595
+ ],
+ "score": 0.63,
+ "content": "\\delta",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 245,
+ 584,
+ 375,
+ 598
+ ],
+ "score": 1.0,
+ "content": "is the expected average degree",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 375,
+ 585,
+ 416,
+ 597
+ ],
+ "score": 0.92,
+ "content": "( N - 1 ) p",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 416,
+ 584,
+ 506,
+ 598
+ ],
+ "score": 1.0,
+ "content": ". Hence, we have the",
+ "type": "text"
+ }
+ ],
+ "index": 28
+ },
+ {
+ "bbox": [
+ 106,
+ 596,
+ 242,
+ 608
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 596,
+ 242,
+ 608
+ ],
+ "score": 1.0,
+ "content": "following lemma from Lemma 5:",
+ "type": "text"
+ }
+ ],
+ "index": 29
+ }
+ ],
+ "index": 26.5,
+ "bbox_fs": [
+ 104,
+ 533,
+ 506,
+ 608
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 105,
+ 610,
+ 505,
+ 638
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 609,
+ 505,
+ 626
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 609,
+ 169,
+ 626
+ ],
+ "score": 1.0,
+ "content": "Lemma 6. Let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 170,
+ 610,
+ 179,
+ 621
+ ],
+ "score": 0.81,
+ "content": "\\tilde { \\Delta }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 180,
+ 609,
+ 444,
+ 626
+ ],
+ "score": 1.0,
+ "content": "be its augmented normalized Laplacian of the Erdos-R ˝ enyi graph ´",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 444,
+ 612,
+ 466,
+ 624
+ ],
+ "score": 0.91,
+ "content": "G _ { N , p }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 467,
+ 609,
+ 505,
+ 626
+ ],
+ "score": 1.0,
+ "content": ". For any",
+ "type": "text"
+ }
+ ],
+ "index": 30
+ },
+ {
+ "bbox": [
+ 106,
+ 618,
+ 478,
+ 640
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 624,
+ 131,
+ 636
+ ],
+ "score": 0.77,
+ "content": "\\varepsilon > 0 ,",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 132,
+ 623,
+ 289,
+ 639
+ ],
+ "score": 0.87,
+ "content": "\\begin{array} { r } { i f \\frac { N p - p + 1 } { \\log N } > k ( \\varepsilon ) : = 3 ( 1 + \\log ( 4 / \\varepsilon ) ) } \\end{array}",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 290,
+ 618,
+ 413,
+ 640
+ ],
+ "score": 1.0,
+ "content": ", then, with probability at least",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 414,
+ 625,
+ 436,
+ 635
+ ],
+ "score": 0.86,
+ "content": "1 - \\varepsilon",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 437,
+ 618,
+ 478,
+ 640
+ ],
+ "score": 1.0,
+ "content": ", we have",
+ "type": "text"
+ }
+ ],
+ "index": 31
+ }
+ ],
+ "index": 30.5,
+ "bbox_fs": [
+ 105,
+ 609,
+ 505,
+ 640
+ ]
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 198,
+ 642,
+ 412,
+ 675
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 198,
+ 642,
+ 412,
+ 675
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 198,
+ 642,
+ 412,
+ 675
+ ],
+ "score": 0.92,
+ "content": "\\operatorname* { m a x } _ { i = 2 , \\ldots , N } \\left| \\lambda _ { i } ( \\tilde { \\Delta } ) - \\frac { N p } { N p - p + 1 } \\right| \\leq 4 \\sqrt { \\frac { 3 \\log ( 4 N / \\varepsilon ) } { N p - p + 1 } } .",
+ "type": "interline_equation",
+ "image_path": "e0c336f319cdb5b4687e3e0d749c2480665543a37bcb8772f9437612db712e00.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 32.5,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 198,
+ 642,
+ 412,
+ 658.5
+ ],
+ "spans": [],
+ "index": 32
+ },
+ {
+ "bbox": [
+ 198,
+ 658.5,
+ 412,
+ 675.0
+ ],
+ "spans": [],
+ "index": 33
+ }
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 678,
+ 506,
+ 733
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 676,
+ 506,
+ 691
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 676,
+ 241,
+ 691
+ ],
+ "score": 1.0,
+ "content": "Corollary 4. Consider GCN on",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 241,
+ 679,
+ 263,
+ 690
+ ],
+ "score": 0.89,
+ "content": "G _ { N , p }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 264,
+ 676,
+ 287,
+ 691
+ ],
+ "score": 1.0,
+ "content": ". Let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 287,
+ 678,
+ 300,
+ 689
+ ],
+ "score": 0.82,
+ "content": "W _ { l }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 300,
+ 676,
+ 387,
+ 691
+ ],
+ "score": 1.0,
+ "content": "be the weight of the",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 388,
+ 679,
+ 392,
+ 688
+ ],
+ "score": 0.55,
+ "content": "l",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 392,
+ 676,
+ 482,
+ 691
+ ],
+ "score": 1.0,
+ "content": "-th layer of GCN and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 482,
+ 680,
+ 491,
+ 689
+ ],
+ "score": 0.79,
+ "content": "s _ { l }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 491,
+ 676,
+ 506,
+ 691
+ ],
+ "score": 1.0,
+ "content": "be",
+ "type": "text"
+ }
+ ],
+ "index": 34
+ },
+ {
+ "bbox": [
+ 105,
+ 688,
+ 505,
+ 704
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 688,
+ 235,
+ 704
+ ],
+ "score": 1.0,
+ "content": "the maximum singular value of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 235,
+ 690,
+ 248,
+ 700
+ ],
+ "score": 0.81,
+ "content": "W _ { l }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 248,
+ 688,
+ 265,
+ 704
+ ],
+ "score": 1.0,
+ "content": "for",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 265,
+ 690,
+ 297,
+ 701
+ ],
+ "score": 0.89,
+ "content": "l \\in \\mathbb { N } _ { + }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 297,
+ 688,
+ 318,
+ 704
+ ],
+ "score": 1.0,
+ "content": ". Set",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 318,
+ 691,
+ 375,
+ 703
+ ],
+ "score": 0.87,
+ "content": "s : = \\mathrm { s u p } _ { l \\in \\mathbb { N } _ { + } }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 375,
+ 688,
+ 397,
+ 704
+ ],
+ "score": 1.0,
+ "content": ". Let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 397,
+ 690,
+ 423,
+ 699
+ ],
+ "score": 0.91,
+ "content": "\\varepsilon > 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 423,
+ 688,
+ 470,
+ 704
+ ],
+ "score": 1.0,
+ "content": ". We define",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 470,
+ 689,
+ 505,
+ 701
+ ],
+ "score": 0.91,
+ "content": "k ( \\varepsilon ) : =",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 35
+ },
+ {
+ "bbox": [
+ 107,
+ 700,
+ 507,
+ 723
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 706,
+ 171,
+ 719
+ ],
+ "score": 0.9,
+ "content": "3 ( 1 + \\log ( 4 / \\varepsilon ) )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 171,
+ 700,
+ 191,
+ 723
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 191,
+ 704,
+ 345,
+ 722
+ ],
+ "score": 0.91,
+ "content": "\\begin{array} { r } { l ( N , p , \\varepsilon ) = { \\frac { 1 - p } { N p - p + 1 } } + 4 { \\sqrt { \\frac { 3 \\log ( 4 N / \\varepsilon ) } { N p - p + 1 } } } } \\end{array}",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 345,
+ 700,
+ 355,
+ 723
+ ],
+ "score": 1.0,
+ "content": ". I",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 356,
+ 705,
+ 424,
+ 721
+ ],
+ "score": 0.82,
+ "content": "\\begin{array} { r } { \\ell \\frac { N p - p + 1 } { \\log N } > k ( \\varepsilon ) } \\end{array}",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 424,
+ 700,
+ 443,
+ 723
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 443,
+ 705,
+ 501,
+ 719
+ ],
+ "score": 0.92,
+ "content": "s \\leq l ( N , \\varepsilon ) ^ { - 1 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 501,
+ 700,
+ 507,
+ 723
+ ],
+ "score": 1.0,
+ "content": ",",
+ "type": "text"
+ }
+ ],
+ "index": 36
+ },
+ {
+ "bbox": [
+ 105,
+ 718,
+ 465,
+ 734
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 718,
+ 164,
+ 734
+ ],
+ "score": 1.0,
+ "content": "then, GCN on",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 164,
+ 721,
+ 186,
+ 733
+ ],
+ "score": 0.9,
+ "content": "G _ { N , p }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 187,
+ 718,
+ 437,
+ 734
+ ],
+ "score": 1.0,
+ "content": "satisfies the assumption of Theorem 2 with probability at least",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 438,
+ 721,
+ 460,
+ 731
+ ],
+ "score": 0.86,
+ "content": "1 - \\varepsilon",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 460,
+ 718,
+ 465,
+ 734
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 37
+ }
+ ],
+ "index": 35.5,
+ "bbox_fs": [
+ 105,
+ 676,
+ 507,
+ 734
+ ]
+ }
+ ]
+ },
+ {
+ "preproc_blocks": [
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 80,
+ 354,
+ 96
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 78,
+ 355,
+ 97
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 81,
+ 217,
+ 94
+ ],
+ "score": 1.0,
+ "content": "Proof of Theorem 3. Since",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 213,
+ 78,
+ 355,
+ 97
+ ],
+ "score": 1.0,
+ "content": "log NNp = o(1), for fixed ε, we have",
+ "type": "text"
+ }
+ ],
+ "index": 0
+ }
+ ],
+ "index": 0
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 243,
+ 99,
+ 367,
+ 125
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 243,
+ 99,
+ 367,
+ 125
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 243,
+ 99,
+ 367,
+ 125
+ ],
+ "score": 0.94,
+ "content": "\\frac { N p - p + 1 } { \\log N } > \\frac { N p } { \\log N } > k ( \\varepsilon )",
+ "type": "interline_equation",
+ "image_path": "463306171e0e7ca695b2cbb0e6dc4d97c6f902d744d5e10f5b1b37f770b5c8c8.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 1,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 243,
+ 99,
+ 367,
+ 125
+ ],
+ "spans": [],
+ "index": 1
+ }
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 128,
+ 488,
+ 143
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 126,
+ 490,
+ 144
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 129,
+ 190,
+ 142
+ ],
+ "score": 1.0,
+ "content": "for sufficiently large",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 190,
+ 130,
+ 200,
+ 140
+ ],
+ "score": 0.81,
+ "content": "N",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 200,
+ 129,
+ 238,
+ 142
+ ],
+ "score": 1.0,
+ "content": ". Further,",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 239,
+ 130,
+ 279,
+ 141
+ ],
+ "score": 0.91,
+ "content": "N p \\to \\infty",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 280,
+ 129,
+ 291,
+ 142
+ ],
+ "score": 1.0,
+ "content": "as",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 291,
+ 130,
+ 327,
+ 140
+ ],
+ "score": 0.9,
+ "content": "N \\to \\infty",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 327,
+ 129,
+ 353,
+ 142
+ ],
+ "score": 1.0,
+ "content": "whe",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 348,
+ 126,
+ 490,
+ 144
+ ],
+ "score": 1.0,
+ "content": "n log NNp = o(1). Therefore, we have",
+ "type": "text"
+ }
+ ],
+ "index": 2
+ }
+ ],
+ "index": 2
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 210,
+ 146,
+ 401,
+ 174
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 210,
+ 146,
+ 401,
+ 174
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 210,
+ 146,
+ 401,
+ 174
+ ],
+ "score": 0.93,
+ "content": "\\frac { ( 1 - p ) ^ { 2 } } { N p - p + 1 } \\leq \\frac { 1 } { N p } \\leq ( 7 - 4 \\sqrt { 3 } ) ^ { 2 } \\log \\left( \\frac { 4 N } { \\varepsilon } \\right)",
+ "type": "interline_equation",
+ "image_path": "13f440081ab3f99459b4e3aa7a438bf7b3d2fbfe37e2f3dc6de25b22a2cb82fa.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 3,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 210,
+ 146,
+ 401,
+ 174
+ ],
+ "spans": [],
+ "index": 3
+ }
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 176,
+ 234,
+ 188
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 174,
+ 235,
+ 190
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 174,
+ 190,
+ 190
+ ],
+ "score": 1.0,
+ "content": "for sufficiently large",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 190,
+ 177,
+ 200,
+ 186
+ ],
+ "score": 0.8,
+ "content": "N",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 200,
+ 174,
+ 235,
+ 190
+ ],
+ "score": 1.0,
+ "content": ". Hence.",
+ "type": "text"
+ }
+ ],
+ "index": 4
+ }
+ ],
+ "index": 4
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 219,
+ 191,
+ 392,
+ 224
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 219,
+ 191,
+ 392,
+ 224
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 219,
+ 191,
+ 392,
+ 224
+ ],
+ "score": 0.95,
+ "content": "\\frac { 1 - p } { N p - p + 1 } \\leq ( 7 - 4 \\sqrt { 3 } ) \\sqrt { \\frac { \\log ( 4 N / \\varepsilon ) } { N p - p + 1 } } .",
+ "type": "interline_equation",
+ "image_path": "805df5f48792754d6a0d9f84f4cf09222c7674b243da522e00a34824e6f6df78.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 5.5,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 219,
+ 191,
+ 392,
+ 207.5
+ ],
+ "spans": [],
+ "index": 5
+ },
+ {
+ "bbox": [
+ 219,
+ 207.5,
+ 392,
+ 224.0
+ ],
+ "spans": [],
+ "index": 6
+ }
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 228,
+ 506,
+ 260
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 227,
+ 506,
+ 248
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 228,
+ 190,
+ 246
+ ],
+ "score": 1.0,
+ "content": "Therefore, we have",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 191,
+ 227,
+ 300,
+ 248
+ ],
+ "score": 0.94,
+ "content": "\\begin{array} { r } { l ( N , p , \\varepsilon ) \\le 7 \\sqrt { \\frac { \\log ( 4 N / \\varepsilon ) } { N p - p + 1 } } } \\end{array}",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 300,
+ 228,
+ 302,
+ 248
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 304,
+ 228,
+ 364,
+ 248
+ ],
+ "score": 1.0,
+ "content": "Therefore, if",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 365,
+ 228,
+ 442,
+ 248
+ ],
+ "score": 0.94,
+ "content": "s ~ \\le ~ \\frac { 1 } { 7 } \\sqrt { \\frac { N p - p + 1 } { \\log ( 4 N / \\varepsilon ) } }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 442,
+ 228,
+ 506,
+ 248
+ ],
+ "score": 1.0,
+ "content": ", then we have",
+ "type": "text"
+ }
+ ],
+ "index": 7
+ },
+ {
+ "bbox": [
+ 107,
+ 246,
+ 178,
+ 260
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 246,
+ 174,
+ 260
+ ],
+ "score": 0.91,
+ "content": "s \\leq l ( N , p , \\varepsilon ) ^ { - 1 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 174,
+ 246,
+ 178,
+ 260
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 8
+ }
+ ],
+ "index": 7.5
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 107,
+ 274,
+ 296,
+ 288
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 273,
+ 297,
+ 290
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 273,
+ 297,
+ 290
+ ],
+ "score": 1.0,
+ "content": "E MISCELLANEOUS PROPOSITIONS",
+ "type": "text"
+ }
+ ],
+ "index": 9
+ }
+ ],
+ "index": 9
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 298,
+ 359,
+ 311
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 298,
+ 359,
+ 312
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 298,
+ 359,
+ 312
+ ],
+ "score": 1.0,
+ "content": "E.1 INVARIANCE OF ORTHOGONAL COMPLEMENT SPACE",
+ "type": "text"
+ }
+ ],
+ "index": 10
+ }
+ ],
+ "index": 10
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 318,
+ 505,
+ 353
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 104,
+ 316,
+ 505,
+ 333
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 316,
+ 185,
+ 333
+ ],
+ "score": 1.0,
+ "content": "Proposition 2. Let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 185,
+ 318,
+ 234,
+ 330
+ ],
+ "score": 0.91,
+ "content": "P \\in \\mathbb { R } ^ { N \\times N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 234,
+ 316,
+ 439,
+ 333
+ ],
+ "score": 1.0,
+ "content": "be a symmetric matrix, treated as a linear operator",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 439,
+ 319,
+ 501,
+ 330
+ ],
+ "score": 0.9,
+ "content": "P : \\mathbb { R } ^ { N } \\mathbb { R } ^ { N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 501,
+ 316,
+ 505,
+ 333
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 11
+ },
+ {
+ "bbox": [
+ 105,
+ 330,
+ 506,
+ 343
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 330,
+ 163,
+ 343
+ ],
+ "score": 1.0,
+ "content": "If a subspace",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 163,
+ 330,
+ 202,
+ 341
+ ],
+ "score": 0.91,
+ "content": "U \\subset \\mathbb { R } ^ { N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 202,
+ 330,
+ 279,
+ 343
+ ],
+ "score": 1.0,
+ "content": "is invariant under",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 279,
+ 331,
+ 288,
+ 341
+ ],
+ "score": 0.8,
+ "content": "P",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 288,
+ 330,
+ 318,
+ 343
+ ],
+ "score": 1.0,
+ "content": "(i.e., if",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 319,
+ 331,
+ 347,
+ 342
+ ],
+ "score": 0.87,
+ "content": "u \\in U",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 348,
+ 330,
+ 372,
+ 343
+ ],
+ "score": 1.0,
+ "content": ", then",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 372,
+ 331,
+ 409,
+ 342
+ ],
+ "score": 0.89,
+ "content": "P u \\in U",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 409,
+ 330,
+ 439,
+ 343
+ ],
+ "score": 1.0,
+ "content": "), then,",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 439,
+ 330,
+ 455,
+ 341
+ ],
+ "score": 0.87,
+ "content": "U ^ { \\perp }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 455,
+ 330,
+ 506,
+ 343
+ ],
+ "score": 1.0,
+ "content": "is invariant",
+ "type": "text"
+ }
+ ],
+ "index": 12
+ },
+ {
+ "bbox": [
+ 105,
+ 341,
+ 163,
+ 354
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 341,
+ 132,
+ 354
+ ],
+ "score": 1.0,
+ "content": "under",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 132,
+ 343,
+ 141,
+ 352
+ ],
+ "score": 0.79,
+ "content": "P",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 141,
+ 341,
+ 163,
+ 354
+ ],
+ "score": 1.0,
+ "content": ", too.",
+ "type": "text"
+ }
+ ],
+ "index": 13
+ }
+ ],
+ "index": 12
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 364,
+ 365,
+ 377
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 363,
+ 365,
+ 380
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 363,
+ 169,
+ 380
+ ],
+ "score": 1.0,
+ "content": "Proof. For any",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 169,
+ 365,
+ 203,
+ 376
+ ],
+ "score": 0.92,
+ "content": "u \\in U ^ { \\perp }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 203,
+ 363,
+ 221,
+ 380
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 221,
+ 366,
+ 247,
+ 376
+ ],
+ "score": 0.9,
+ "content": "v \\in U",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 248,
+ 363,
+ 317,
+ 380
+ ],
+ "score": 1.0,
+ "content": ", by symmetry of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 317,
+ 366,
+ 326,
+ 375
+ ],
+ "score": 0.85,
+ "content": "P",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 326,
+ 363,
+ 365,
+ 380
+ ],
+ "score": 1.0,
+ "content": ", we have",
+ "type": "text"
+ }
+ ],
+ "index": 14
+ }
+ ],
+ "index": 14
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 200,
+ 379,
+ 410,
+ 394
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 200,
+ 379,
+ 410,
+ 394
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 200,
+ 379,
+ 410,
+ 394
+ ],
+ "score": 0.89,
+ "content": "\\langle P u , v \\rangle = ( P u ) ^ { \\top } v = u ^ { \\top } P ^ { \\top } v = u ^ { \\top } P v = \\langle u , P v \\rangle .",
+ "type": "interline_equation",
+ "image_path": "52ab40224e5255fd51194ad825ef29b1385ab12fd7152044ad6cf0db196800a5.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 15,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 200,
+ 379,
+ 410,
+ 394
+ ],
+ "spans": [],
+ "index": 15
+ }
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 108,
+ 398,
+ 503,
+ 422
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 397,
+ 501,
+ 411
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 397,
+ 131,
+ 411
+ ],
+ "score": 1.0,
+ "content": "Since",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 131,
+ 399,
+ 140,
+ 409
+ ],
+ "score": 0.83,
+ "content": "U",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 140,
+ 397,
+ 233,
+ 411
+ ],
+ "score": 1.0,
+ "content": "is an invariant space of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 233,
+ 399,
+ 242,
+ 409
+ ],
+ "score": 0.83,
+ "content": "P",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 242,
+ 397,
+ 280,
+ 411
+ ],
+ "score": 1.0,
+ "content": ", we have",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 280,
+ 399,
+ 314,
+ 409
+ ],
+ "score": 0.9,
+ "content": "P v \\in U",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 314,
+ 397,
+ 383,
+ 411
+ ],
+ "score": 1.0,
+ "content": ". Hence, we have",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 383,
+ 399,
+ 433,
+ 411
+ ],
+ "score": 0.93,
+ "content": "\\langle u , P v \\rangle = 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 433,
+ 397,
+ 467,
+ 411
+ ],
+ "score": 1.0,
+ "content": "because",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 468,
+ 398,
+ 501,
+ 409
+ ],
+ "score": 0.91,
+ "content": "u \\in U ^ { \\perp }",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 16
+ },
+ {
+ "bbox": [
+ 106,
+ 409,
+ 506,
+ 421
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 409,
+ 149,
+ 421
+ ],
+ "score": 1.0,
+ "content": "We obtain",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 150,
+ 410,
+ 191,
+ 420
+ ],
+ "score": 0.91,
+ "content": "P u \\in U ^ { \\perp }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 191,
+ 409,
+ 271,
+ 421
+ ],
+ "score": 1.0,
+ "content": "by the definition of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 271,
+ 410,
+ 286,
+ 420
+ ],
+ "score": 0.89,
+ "content": "U ^ { \\perp }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 286,
+ 409,
+ 290,
+ 421
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 494,
+ 410,
+ 506,
+ 421
+ ],
+ "score": 1.0,
+ "content": "□",
+ "type": "text"
+ }
+ ],
+ "index": 17
+ }
+ ],
+ "index": 16.5
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 106,
+ 434,
+ 312,
+ 446
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 434,
+ 312,
+ 447
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 434,
+ 312,
+ 447
+ ],
+ "score": 1.0,
+ "content": "E.2 CONVERGENCE TO TRIVIAL FIXED POINT",
+ "type": "text"
+ }
+ ],
+ "index": 18
+ }
+ ],
+ "index": 18
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 453,
+ 506,
+ 488
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 104,
+ 452,
+ 506,
+ 468
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 452,
+ 122,
+ 468
+ ],
+ "score": 1.0,
+ "content": "Let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 123,
+ 454,
+ 172,
+ 465
+ ],
+ "score": 0.91,
+ "content": "P \\in \\mathbb { R } ^ { N \\times N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 172,
+ 452,
+ 268,
+ 468
+ ],
+ "score": 1.0,
+ "content": "be a symmetric matrix,",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 269,
+ 454,
+ 321,
+ 466
+ ],
+ "score": 0.89,
+ "content": "W _ { l } \\in \\mathbb { R } ^ { C \\times C }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 322,
+ 452,
+ 325,
+ 468
+ ],
+ "score": 1.0,
+ "content": ",",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 326,
+ 457,
+ 335,
+ 466
+ ],
+ "score": 0.72,
+ "content": "s _ { l }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 335,
+ 452,
+ 476,
+ 468
+ ],
+ "score": 1.0,
+ "content": "be the maximum singular value of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 476,
+ 455,
+ 489,
+ 466
+ ],
+ "score": 0.88,
+ "content": "W _ { l }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 490,
+ 452,
+ 506,
+ 468
+ ],
+ "score": 1.0,
+ "content": "for",
+ "type": "text"
+ }
+ ],
+ "index": 19
+ },
+ {
+ "bbox": [
+ 106,
+ 462,
+ 507,
+ 480
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 466,
+ 139,
+ 478
+ ],
+ "score": 0.91,
+ "content": "l \\in \\mathbb { N } _ { + }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 140,
+ 462,
+ 190,
+ 480
+ ],
+ "score": 1.0,
+ "content": ". We define",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 190,
+ 465,
+ 282,
+ 477
+ ],
+ "score": 0.91,
+ "content": "f _ { l } ^ { ^ { \\bullet } } : \\mathbb { R } ^ { N \\times C } \\mathbb { R } ^ { N \\times { \\dot { C } } }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 282,
+ 462,
+ 298,
+ 480
+ ],
+ "score": 1.0,
+ "content": "by",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 298,
+ 466,
+ 385,
+ 478
+ ],
+ "score": 0.93,
+ "content": "f _ { l } ( X ) : = \\sigma ( P X W _ { l } )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 385,
+ 462,
+ 415,
+ 480
+ ],
+ "score": 1.0,
+ "content": "where",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 415,
+ 468,
+ 422,
+ 476
+ ],
+ "score": 0.78,
+ "content": "\\sigma",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 423,
+ 462,
+ 507,
+ 480
+ ],
+ "score": 1.0,
+ "content": "is the element-wise",
+ "type": "text"
+ }
+ ],
+ "index": 20
+ },
+ {
+ "bbox": [
+ 105,
+ 476,
+ 172,
+ 489
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 476,
+ 172,
+ 489
+ ],
+ "score": 1.0,
+ "content": "ReLU function.",
+ "type": "text"
+ }
+ ],
+ "index": 21
+ }
+ ],
+ "index": 20
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 491,
+ 505,
+ 525
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 490,
+ 505,
+ 504
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 490,
+ 351,
+ 504
+ ],
+ "score": 1.0,
+ "content": "Proposition 3. Suppose further that the operator norm of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 351,
+ 492,
+ 360,
+ 501
+ ],
+ "score": 0.82,
+ "content": "P",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 360,
+ 490,
+ 435,
+ 504
+ ],
+ "score": 1.0,
+ "content": "is no larger than",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 436,
+ 492,
+ 443,
+ 501
+ ],
+ "score": 0.77,
+ "content": "\\lambda ,",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 443,
+ 490,
+ 505,
+ 504
+ ],
+ "score": 1.0,
+ "content": ", then we have",
+ "type": "text"
+ }
+ ],
+ "index": 22
+ },
+ {
+ "bbox": [
+ 107,
+ 500,
+ 504,
+ 517
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 502,
+ 201,
+ 514
+ ],
+ "score": 0.9,
+ "content": "\\| f _ { l } ( \\bar { \\boldsymbol { X } } ) \\| _ { \\mathrm { F } } \\leq s _ { l } \\lambda \\| \\bar { \\boldsymbol { X } } \\| _ { \\mathrm { F } }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 202,
+ 500,
+ 236,
+ 517
+ ],
+ "score": 1.0,
+ "content": "for any",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 237,
+ 502,
+ 269,
+ 514
+ ],
+ "score": 0.9,
+ "content": "l \\in \\mathbb { N } _ { + }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 270,
+ 500,
+ 346,
+ 517
+ ],
+ "score": 1.0,
+ "content": ". In particular, let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 347,
+ 503,
+ 414,
+ 516
+ ],
+ "score": 0.9,
+ "content": "s : = \\operatorname* { s u p } _ { l \\in \\mathbb { N } _ { + } } s _ { l }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 415,
+ 500,
+ 430,
+ 517
+ ],
+ "score": 1.0,
+ "content": ". If",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 431,
+ 502,
+ 463,
+ 513
+ ],
+ "score": 0.88,
+ "content": "s \\lambda < 1",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 464,
+ 500,
+ 491,
+ 517
+ ],
+ "score": 1.0,
+ "content": ", then,",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 491,
+ 502,
+ 504,
+ 513
+ ],
+ "score": 0.85,
+ "content": "X _ { l }",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 23
+ },
+ {
+ "bbox": [
+ 106,
+ 514,
+ 263,
+ 527
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 514,
+ 230,
+ 527
+ ],
+ "score": 1.0,
+ "content": "exponentially approaches 0 as",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 230,
+ 515,
+ 259,
+ 524
+ ],
+ "score": 0.88,
+ "content": "l \\infty",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 259,
+ 514,
+ 263,
+ 527
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 24
+ }
+ ],
+ "index": 23
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 537,
+ 505,
+ 583
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 537,
+ 506,
+ 550
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 537,
+ 162,
+ 550
+ ],
+ "score": 1.0,
+ "content": "Proof. Since",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 162,
+ 538,
+ 169,
+ 548
+ ],
+ "score": 0.81,
+ "content": "\\lambda",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 170,
+ 537,
+ 271,
+ 550
+ ],
+ "score": 1.0,
+ "content": "is the operator norm of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 271,
+ 538,
+ 295,
+ 550
+ ],
+ "score": 0.92,
+ "content": "P | _ { U ^ { \\perp } }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 295,
+ 537,
+ 506,
+ 550
+ ],
+ "score": 1.0,
+ "content": ", the assumption implies that the operator norm of",
+ "type": "text"
+ }
+ ],
+ "index": 25
+ },
+ {
+ "bbox": [
+ 107,
+ 547,
+ 506,
+ 561
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 549,
+ 116,
+ 559
+ ],
+ "score": 0.8,
+ "content": "P",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 116,
+ 547,
+ 211,
+ 561
+ ],
+ "score": 1.0,
+ "content": "itself is no larger than",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 211,
+ 549,
+ 218,
+ 558
+ ],
+ "score": 0.78,
+ "content": "\\lambda",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 219,
+ 547,
+ 308,
+ 561
+ ],
+ "score": 1.0,
+ "content": ". Therefore, we have",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 308,
+ 549,
+ 468,
+ 560
+ ],
+ "score": 0.9,
+ "content": "\\| P X W _ { l } \\| _ { \\mathrm { F } } \\leq \\lambda \\bar { \\| } X W _ { l } \\| _ { \\mathrm { F } } \\leq s _ { l } \\bar { \\lambda } \\| X \\| _ { \\mathrm { F } }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 468,
+ 547,
+ 506,
+ 561
+ ],
+ "score": 1.0,
+ "content": ". On the",
+ "type": "text"
+ }
+ ],
+ "index": 26
+ },
+ {
+ "bbox": [
+ 103,
+ 558,
+ 506,
+ 573
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 103,
+ 558,
+ 179,
+ 573
+ ],
+ "score": 1.0,
+ "content": "other hand, since",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 179,
+ 559,
+ 230,
+ 571
+ ],
+ "score": 0.93,
+ "content": "\\sigma ( x ) ^ { 2 } \\leq x ^ { 2 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 230,
+ 558,
+ 264,
+ 573
+ ],
+ "score": 1.0,
+ "content": "for any",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 264,
+ 560,
+ 293,
+ 570
+ ],
+ "score": 0.89,
+ "content": "x \\in \\mathbb { R }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 293,
+ 558,
+ 334,
+ 573
+ ],
+ "score": 1.0,
+ "content": ", we have",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 335,
+ 560,
+ 415,
+ 571
+ ],
+ "score": 0.88,
+ "content": "\\| \\sigma ( X ) \\| _ { \\mathrm { F } } \\leq \\| X \\| _ { F }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 415,
+ 558,
+ 449,
+ 573
+ ],
+ "score": 1.0,
+ "content": "for any",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 450,
+ 559,
+ 501,
+ 570
+ ],
+ "score": 0.9,
+ "content": "\\boldsymbol { X } ^ { \\prime } \\in \\mathbb { R } ^ { N \\times C }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 502,
+ 558,
+ 506,
+ 573
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 27
+ },
+ {
+ "bbox": [
+ 105,
+ 570,
+ 504,
+ 584
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 570,
+ 224,
+ 584
+ ],
+ "score": 1.0,
+ "content": "Combining the two, we have",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 224,
+ 570,
+ 374,
+ 583
+ ],
+ "score": 0.91,
+ "content": "\\Vert f _ { l } ( X ) \\Vert _ { \\mathrm { F } } \\leq \\Vert P X W _ { l } \\Vert _ { \\mathrm { F } } \\leq s _ { l } \\lambda \\Vert X \\Vert _ { \\mathrm { F } }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 374,
+ 570,
+ 379,
+ 584
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 496,
+ 572,
+ 504,
+ 579
+ ],
+ "score": 0.998,
+ "content": "□",
+ "type": "text"
+ }
+ ],
+ "index": 28
+ }
+ ],
+ "index": 26.5
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 107,
+ 595,
+ 253,
+ 606
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 594,
+ 254,
+ 608
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 594,
+ 254,
+ 608
+ ],
+ "score": 1.0,
+ "content": "E.3 STRICTNESS OF THEOREM 1",
+ "type": "text"
+ }
+ ],
+ "index": 29
+ }
+ ],
+ "index": 29
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 615,
+ 505,
+ 650
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 615,
+ 505,
+ 628
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 615,
+ 211,
+ 628
+ ],
+ "score": 1.0,
+ "content": "Theorem 1 implies that if",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 211,
+ 616,
+ 242,
+ 627
+ ],
+ "score": 0.91,
+ "content": "s \\lambda \\leq 1",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 242,
+ 615,
+ 345,
+ 628
+ ],
+ "score": 1.0,
+ "content": ", then, one-step transition",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 345,
+ 616,
+ 354,
+ 627
+ ],
+ "score": 0.87,
+ "content": "f _ { l }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 354,
+ 615,
+ 488,
+ 628
+ ],
+ "score": 1.0,
+ "content": "does not increase the distance to",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 488,
+ 616,
+ 501,
+ 626
+ ],
+ "score": 0.82,
+ "content": "\\mathcal { M }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 501,
+ 615,
+ 505,
+ 628
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 30
+ },
+ {
+ "bbox": [
+ 106,
+ 627,
+ 505,
+ 638
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 627,
+ 505,
+ 638
+ ],
+ "score": 1.0,
+ "content": "In this section, we first prove that this theorem is strict in the sense that, there exists a situation in",
+ "type": "text"
+ }
+ ],
+ "index": 31
+ },
+ {
+ "bbox": [
+ 106,
+ 638,
+ 483,
+ 650
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 638,
+ 133,
+ 650
+ ],
+ "score": 1.0,
+ "content": "which",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 133,
+ 638,
+ 166,
+ 649
+ ],
+ "score": 0.91,
+ "content": "s _ { l } \\lambda > 1",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 167,
+ 638,
+ 258,
+ 650
+ ],
+ "score": 1.0,
+ "content": "holds and the distance",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 258,
+ 638,
+ 275,
+ 649
+ ],
+ "score": 0.9,
+ "content": "d _ { \\mathcal { M } }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 275,
+ 638,
+ 404,
+ 650
+ ],
+ "score": 1.0,
+ "content": "increases by one-step transition",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 404,
+ 638,
+ 413,
+ 649
+ ],
+ "score": 0.86,
+ "content": "f _ { l }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 413,
+ 638,
+ 470,
+ 650
+ ],
+ "score": 1.0,
+ "content": "at some point",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 471,
+ 638,
+ 480,
+ 648
+ ],
+ "score": 0.85,
+ "content": "X",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 481,
+ 638,
+ 483,
+ 650
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 32
+ }
+ ],
+ "index": 31
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 654,
+ 384,
+ 666
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 653,
+ 385,
+ 668
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 653,
+ 121,
+ 668
+ ],
+ "score": 1.0,
+ "content": "Set",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 122,
+ 655,
+ 153,
+ 665
+ ],
+ "score": 0.82,
+ "content": "N \\gets 2",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 153,
+ 653,
+ 156,
+ 668
+ ],
+ "score": 1.0,
+ "content": ",",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 156,
+ 655,
+ 186,
+ 665
+ ],
+ "score": 0.82,
+ "content": "C \\gets 1",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 186,
+ 653,
+ 206,
+ 668
+ ],
+ "score": 1.0,
+ "content": ", and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 207,
+ 655,
+ 239,
+ 664
+ ],
+ "score": 0.91,
+ "content": "M \\gets 1",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 239,
+ 653,
+ 317,
+ 668
+ ],
+ "score": 1.0,
+ "content": "in Section 3.2. For",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 317,
+ 655,
+ 353,
+ 666
+ ],
+ "score": 0.91,
+ "content": "\\mu , \\lambda > 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 353,
+ 653,
+ 385,
+ 668
+ ],
+ "score": 1.0,
+ "content": ", we set",
+ "type": "text"
+ }
+ ],
+ "index": 33
+ }
+ ],
+ "index": 33
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 206,
+ 668,
+ 402,
+ 696
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 206,
+ 668,
+ 402,
+ 696
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 206,
+ 668,
+ 402,
+ 696
+ ],
+ "score": 0.93,
+ "content": "P [ { \\mu \\atop 0 } \\lambda ] , e [ 1 \\atop 0 ] , U \\{ [ { x \\atop y } ] \\mid y = 0 \\} .",
+ "type": "interline_equation",
+ "image_path": "31c8f337f767b389f9e29dc55525ffeb7295cd5c21158f1437be017ccbef7b58.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 34,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 206,
+ 668,
+ 402,
+ 696
+ ],
+ "spans": [],
+ "index": 34
+ }
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 698,
+ 505,
+ 732
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 698,
+ 505,
+ 711
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 698,
+ 311,
+ 711
+ ],
+ "score": 1.0,
+ "content": "Then, by definition, we can check that the 3-tuple",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 311,
+ 699,
+ 347,
+ 710
+ ],
+ "score": 0.89,
+ "content": "( P , e , U )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 347,
+ 698,
+ 505,
+ 711
+ ],
+ "score": 1.0,
+ "content": "satisfies the Assumptions 1 and 2. Set",
+ "type": "text"
+ }
+ ],
+ "index": 35
+ },
+ {
+ "bbox": [
+ 107,
+ 704,
+ 508,
+ 725
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 710,
+ 186,
+ 720
+ ],
+ "score": 0.9,
+ "content": "\\mathcal { M } : = U \\otimes \\mathbb { R } = U",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 187,
+ 704,
+ 235,
+ 725
+ ],
+ "score": 1.0,
+ "content": "and choose",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 235,
+ 710,
+ 268,
+ 720
+ ],
+ "score": 0.91,
+ "content": "W \\in \\mathbb { R }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 268,
+ 704,
+ 298,
+ 725
+ ],
+ "score": 1.0,
+ "content": "so that",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 299,
+ 710,
+ 340,
+ 720
+ ],
+ "score": 0.88,
+ "content": "W ^ { \\setminus } > \\lambda ^ { - 1 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 341,
+ 704,
+ 404,
+ 725
+ ],
+ "score": 1.0,
+ "content": ". Finally define",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 405,
+ 709,
+ 491,
+ 721
+ ],
+ "score": 0.91,
+ "content": "f : \\mathbb { R } ^ { \\mathbf { \\hat { N } } \\times C } \\mathbb { R } ^ { N \\times C }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 491,
+ 704,
+ 508,
+ 725
+ ],
+ "score": 1.0,
+ "content": "by",
+ "type": "text"
+ }
+ ],
+ "index": 36
+ },
+ {
+ "bbox": [
+ 107,
+ 720,
+ 369,
+ 733
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 721,
+ 187,
+ 732
+ ],
+ "score": 0.91,
+ "content": "f ( X ) : = \\sigma ( P X W )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 188,
+ 720,
+ 216,
+ 733
+ ],
+ "score": 1.0,
+ "content": "where",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 216,
+ 722,
+ 223,
+ 730
+ ],
+ "score": 0.79,
+ "content": "\\sigma",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 224,
+ 720,
+ 369,
+ 733
+ ],
+ "score": 1.0,
+ "content": "is the element-wise ReLU function.",
+ "type": "text"
+ }
+ ],
+ "index": 37
+ }
+ ],
+ "index": 36
+ }
+ ],
+ "page_idx": 20,
+ "page_size": [
+ 612,
+ 792
+ ],
+ "discarded_blocks": [
+ {
+ "type": "discarded",
+ "bbox": [
+ 106,
+ 26,
+ 294,
+ 38
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 25,
+ 294,
+ 38
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 25,
+ 294,
+ 38
+ ],
+ "score": 1.0,
+ "content": "Published as a conference paper at ICLR 2020",
+ "type": "text"
+ }
+ ]
+ }
+ ]
+ },
+ {
+ "type": "discarded",
+ "bbox": [
+ 300,
+ 751,
+ 310,
+ 760
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 298,
+ 750,
+ 312,
+ 765
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 298,
+ 750,
+ 312,
+ 765
+ ],
+ "score": 1.0,
+ "content": "",
+ "type": "text",
+ "height": 15,
+ "width": 14
+ }
+ ]
+ }
+ ]
+ }
+ ],
+ "para_blocks": [
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 80,
+ 354,
+ 96
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 78,
+ 355,
+ 97
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 81,
+ 217,
+ 94
+ ],
+ "score": 1.0,
+ "content": "Proof of Theorem 3. Since",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 213,
+ 78,
+ 355,
+ 97
+ ],
+ "score": 1.0,
+ "content": "log NNp = o(1), for fixed ε, we have",
+ "type": "text"
+ }
+ ],
+ "index": 0
+ }
+ ],
+ "index": 0,
+ "bbox_fs": [
+ 105,
+ 78,
+ 355,
+ 97
+ ]
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 243,
+ 99,
+ 367,
+ 125
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 243,
+ 99,
+ 367,
+ 125
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 243,
+ 99,
+ 367,
+ 125
+ ],
+ "score": 0.94,
+ "content": "\\frac { N p - p + 1 } { \\log N } > \\frac { N p } { \\log N } > k ( \\varepsilon )",
+ "type": "interline_equation",
+ "image_path": "463306171e0e7ca695b2cbb0e6dc4d97c6f902d744d5e10f5b1b37f770b5c8c8.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 1,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 243,
+ 99,
+ 367,
+ 125
+ ],
+ "spans": [],
+ "index": 1
+ }
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 128,
+ 488,
+ 143
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 126,
+ 490,
+ 144
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 129,
+ 190,
+ 142
+ ],
+ "score": 1.0,
+ "content": "for sufficiently large",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 190,
+ 130,
+ 200,
+ 140
+ ],
+ "score": 0.81,
+ "content": "N",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 200,
+ 129,
+ 238,
+ 142
+ ],
+ "score": 1.0,
+ "content": ". Further,",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 239,
+ 130,
+ 279,
+ 141
+ ],
+ "score": 0.91,
+ "content": "N p \\to \\infty",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 280,
+ 129,
+ 291,
+ 142
+ ],
+ "score": 1.0,
+ "content": "as",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 291,
+ 130,
+ 327,
+ 140
+ ],
+ "score": 0.9,
+ "content": "N \\to \\infty",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 327,
+ 129,
+ 353,
+ 142
+ ],
+ "score": 1.0,
+ "content": "whe",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 348,
+ 126,
+ 490,
+ 144
+ ],
+ "score": 1.0,
+ "content": "n log NNp = o(1). Therefore, we have",
+ "type": "text"
+ }
+ ],
+ "index": 2
+ }
+ ],
+ "index": 2,
+ "bbox_fs": [
+ 105,
+ 126,
+ 490,
+ 144
+ ]
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 210,
+ 146,
+ 401,
+ 174
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 210,
+ 146,
+ 401,
+ 174
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 210,
+ 146,
+ 401,
+ 174
+ ],
+ "score": 0.93,
+ "content": "\\frac { ( 1 - p ) ^ { 2 } } { N p - p + 1 } \\leq \\frac { 1 } { N p } \\leq ( 7 - 4 \\sqrt { 3 } ) ^ { 2 } \\log \\left( \\frac { 4 N } { \\varepsilon } \\right)",
+ "type": "interline_equation",
+ "image_path": "13f440081ab3f99459b4e3aa7a438bf7b3d2fbfe37e2f3dc6de25b22a2cb82fa.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 3,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 210,
+ 146,
+ 401,
+ 174
+ ],
+ "spans": [],
+ "index": 3
+ }
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 176,
+ 234,
+ 188
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 174,
+ 235,
+ 190
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 174,
+ 190,
+ 190
+ ],
+ "score": 1.0,
+ "content": "for sufficiently large",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 190,
+ 177,
+ 200,
+ 186
+ ],
+ "score": 0.8,
+ "content": "N",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 200,
+ 174,
+ 235,
+ 190
+ ],
+ "score": 1.0,
+ "content": ". Hence.",
+ "type": "text"
+ }
+ ],
+ "index": 4
+ }
+ ],
+ "index": 4,
+ "bbox_fs": [
+ 105,
+ 174,
+ 235,
+ 190
+ ]
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 219,
+ 191,
+ 392,
+ 224
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 219,
+ 191,
+ 392,
+ 224
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 219,
+ 191,
+ 392,
+ 224
+ ],
+ "score": 0.95,
+ "content": "\\frac { 1 - p } { N p - p + 1 } \\leq ( 7 - 4 \\sqrt { 3 } ) \\sqrt { \\frac { \\log ( 4 N / \\varepsilon ) } { N p - p + 1 } } .",
+ "type": "interline_equation",
+ "image_path": "805df5f48792754d6a0d9f84f4cf09222c7674b243da522e00a34824e6f6df78.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 5.5,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 219,
+ 191,
+ 392,
+ 207.5
+ ],
+ "spans": [],
+ "index": 5
+ },
+ {
+ "bbox": [
+ 219,
+ 207.5,
+ 392,
+ 224.0
+ ],
+ "spans": [],
+ "index": 6
+ }
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 228,
+ 506,
+ 260
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 227,
+ 506,
+ 248
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 228,
+ 190,
+ 246
+ ],
+ "score": 1.0,
+ "content": "Therefore, we have",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 191,
+ 227,
+ 300,
+ 248
+ ],
+ "score": 0.94,
+ "content": "\\begin{array} { r } { l ( N , p , \\varepsilon ) \\le 7 \\sqrt { \\frac { \\log ( 4 N / \\varepsilon ) } { N p - p + 1 } } } \\end{array}",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 300,
+ 228,
+ 302,
+ 248
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 304,
+ 228,
+ 364,
+ 248
+ ],
+ "score": 1.0,
+ "content": "Therefore, if",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 365,
+ 228,
+ 442,
+ 248
+ ],
+ "score": 0.94,
+ "content": "s ~ \\le ~ \\frac { 1 } { 7 } \\sqrt { \\frac { N p - p + 1 } { \\log ( 4 N / \\varepsilon ) } }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 442,
+ 228,
+ 506,
+ 248
+ ],
+ "score": 1.0,
+ "content": ", then we have",
+ "type": "text"
+ }
+ ],
+ "index": 7
+ },
+ {
+ "bbox": [
+ 107,
+ 246,
+ 178,
+ 260
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 246,
+ 174,
+ 260
+ ],
+ "score": 0.91,
+ "content": "s \\leq l ( N , p , \\varepsilon ) ^ { - 1 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 174,
+ 246,
+ 178,
+ 260
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 8
+ }
+ ],
+ "index": 7.5,
+ "bbox_fs": [
+ 105,
+ 227,
+ 506,
+ 260
+ ]
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 107,
+ 274,
+ 296,
+ 288
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 273,
+ 297,
+ 290
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 273,
+ 297,
+ 290
+ ],
+ "score": 1.0,
+ "content": "E MISCELLANEOUS PROPOSITIONS",
+ "type": "text"
+ }
+ ],
+ "index": 9
+ }
+ ],
+ "index": 9
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 298,
+ 359,
+ 311
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 298,
+ 359,
+ 312
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 298,
+ 359,
+ 312
+ ],
+ "score": 1.0,
+ "content": "E.1 INVARIANCE OF ORTHOGONAL COMPLEMENT SPACE",
+ "type": "text"
+ }
+ ],
+ "index": 10
+ }
+ ],
+ "index": 10,
+ "bbox_fs": [
+ 105,
+ 298,
+ 359,
+ 312
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 318,
+ 505,
+ 353
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 104,
+ 316,
+ 505,
+ 333
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 316,
+ 185,
+ 333
+ ],
+ "score": 1.0,
+ "content": "Proposition 2. Let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 185,
+ 318,
+ 234,
+ 330
+ ],
+ "score": 0.91,
+ "content": "P \\in \\mathbb { R } ^ { N \\times N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 234,
+ 316,
+ 439,
+ 333
+ ],
+ "score": 1.0,
+ "content": "be a symmetric matrix, treated as a linear operator",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 439,
+ 319,
+ 501,
+ 330
+ ],
+ "score": 0.9,
+ "content": "P : \\mathbb { R } ^ { N } \\mathbb { R } ^ { N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 501,
+ 316,
+ 505,
+ 333
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 11
+ },
+ {
+ "bbox": [
+ 105,
+ 330,
+ 506,
+ 343
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 330,
+ 163,
+ 343
+ ],
+ "score": 1.0,
+ "content": "If a subspace",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 163,
+ 330,
+ 202,
+ 341
+ ],
+ "score": 0.91,
+ "content": "U \\subset \\mathbb { R } ^ { N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 202,
+ 330,
+ 279,
+ 343
+ ],
+ "score": 1.0,
+ "content": "is invariant under",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 279,
+ 331,
+ 288,
+ 341
+ ],
+ "score": 0.8,
+ "content": "P",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 288,
+ 330,
+ 318,
+ 343
+ ],
+ "score": 1.0,
+ "content": "(i.e., if",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 319,
+ 331,
+ 347,
+ 342
+ ],
+ "score": 0.87,
+ "content": "u \\in U",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 348,
+ 330,
+ 372,
+ 343
+ ],
+ "score": 1.0,
+ "content": ", then",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 372,
+ 331,
+ 409,
+ 342
+ ],
+ "score": 0.89,
+ "content": "P u \\in U",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 409,
+ 330,
+ 439,
+ 343
+ ],
+ "score": 1.0,
+ "content": "), then,",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 439,
+ 330,
+ 455,
+ 341
+ ],
+ "score": 0.87,
+ "content": "U ^ { \\perp }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 455,
+ 330,
+ 506,
+ 343
+ ],
+ "score": 1.0,
+ "content": "is invariant",
+ "type": "text"
+ }
+ ],
+ "index": 12
+ },
+ {
+ "bbox": [
+ 105,
+ 341,
+ 163,
+ 354
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 341,
+ 132,
+ 354
+ ],
+ "score": 1.0,
+ "content": "under",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 132,
+ 343,
+ 141,
+ 352
+ ],
+ "score": 0.79,
+ "content": "P",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 141,
+ 341,
+ 163,
+ 354
+ ],
+ "score": 1.0,
+ "content": ", too.",
+ "type": "text"
+ }
+ ],
+ "index": 13
+ }
+ ],
+ "index": 12,
+ "bbox_fs": [
+ 104,
+ 316,
+ 506,
+ 354
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 364,
+ 365,
+ 377
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 363,
+ 365,
+ 380
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 363,
+ 169,
+ 380
+ ],
+ "score": 1.0,
+ "content": "Proof. For any",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 169,
+ 365,
+ 203,
+ 376
+ ],
+ "score": 0.92,
+ "content": "u \\in U ^ { \\perp }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 203,
+ 363,
+ 221,
+ 380
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 221,
+ 366,
+ 247,
+ 376
+ ],
+ "score": 0.9,
+ "content": "v \\in U",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 248,
+ 363,
+ 317,
+ 380
+ ],
+ "score": 1.0,
+ "content": ", by symmetry of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 317,
+ 366,
+ 326,
+ 375
+ ],
+ "score": 0.85,
+ "content": "P",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 326,
+ 363,
+ 365,
+ 380
+ ],
+ "score": 1.0,
+ "content": ", we have",
+ "type": "text"
+ }
+ ],
+ "index": 14
+ }
+ ],
+ "index": 14,
+ "bbox_fs": [
+ 105,
+ 363,
+ 365,
+ 380
+ ]
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 200,
+ 379,
+ 410,
+ 394
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 200,
+ 379,
+ 410,
+ 394
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 200,
+ 379,
+ 410,
+ 394
+ ],
+ "score": 0.89,
+ "content": "\\langle P u , v \\rangle = ( P u ) ^ { \\top } v = u ^ { \\top } P ^ { \\top } v = u ^ { \\top } P v = \\langle u , P v \\rangle .",
+ "type": "interline_equation",
+ "image_path": "52ab40224e5255fd51194ad825ef29b1385ab12fd7152044ad6cf0db196800a5.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 15,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 200,
+ 379,
+ 410,
+ 394
+ ],
+ "spans": [],
+ "index": 15
+ }
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 108,
+ 398,
+ 503,
+ 422
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 397,
+ 501,
+ 411
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 397,
+ 131,
+ 411
+ ],
+ "score": 1.0,
+ "content": "Since",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 131,
+ 399,
+ 140,
+ 409
+ ],
+ "score": 0.83,
+ "content": "U",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 140,
+ 397,
+ 233,
+ 411
+ ],
+ "score": 1.0,
+ "content": "is an invariant space of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 233,
+ 399,
+ 242,
+ 409
+ ],
+ "score": 0.83,
+ "content": "P",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 242,
+ 397,
+ 280,
+ 411
+ ],
+ "score": 1.0,
+ "content": ", we have",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 280,
+ 399,
+ 314,
+ 409
+ ],
+ "score": 0.9,
+ "content": "P v \\in U",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 314,
+ 397,
+ 383,
+ 411
+ ],
+ "score": 1.0,
+ "content": ". Hence, we have",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 383,
+ 399,
+ 433,
+ 411
+ ],
+ "score": 0.93,
+ "content": "\\langle u , P v \\rangle = 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 433,
+ 397,
+ 467,
+ 411
+ ],
+ "score": 1.0,
+ "content": "because",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 468,
+ 398,
+ 501,
+ 409
+ ],
+ "score": 0.91,
+ "content": "u \\in U ^ { \\perp }",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 16
+ },
+ {
+ "bbox": [
+ 106,
+ 409,
+ 506,
+ 421
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 409,
+ 149,
+ 421
+ ],
+ "score": 1.0,
+ "content": "We obtain",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 150,
+ 410,
+ 191,
+ 420
+ ],
+ "score": 0.91,
+ "content": "P u \\in U ^ { \\perp }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 191,
+ 409,
+ 271,
+ 421
+ ],
+ "score": 1.0,
+ "content": "by the definition of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 271,
+ 410,
+ 286,
+ 420
+ ],
+ "score": 0.89,
+ "content": "U ^ { \\perp }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 286,
+ 409,
+ 290,
+ 421
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 494,
+ 410,
+ 506,
+ 421
+ ],
+ "score": 1.0,
+ "content": "□",
+ "type": "text"
+ }
+ ],
+ "index": 17
+ }
+ ],
+ "index": 16.5,
+ "bbox_fs": [
+ 105,
+ 397,
+ 506,
+ 421
+ ]
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 106,
+ 434,
+ 312,
+ 446
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 434,
+ 312,
+ 447
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 434,
+ 312,
+ 447
+ ],
+ "score": 1.0,
+ "content": "E.2 CONVERGENCE TO TRIVIAL FIXED POINT",
+ "type": "text"
+ }
+ ],
+ "index": 18
+ }
+ ],
+ "index": 18
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 453,
+ 506,
+ 488
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 104,
+ 452,
+ 506,
+ 468
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 452,
+ 122,
+ 468
+ ],
+ "score": 1.0,
+ "content": "Let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 123,
+ 454,
+ 172,
+ 465
+ ],
+ "score": 0.91,
+ "content": "P \\in \\mathbb { R } ^ { N \\times N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 172,
+ 452,
+ 268,
+ 468
+ ],
+ "score": 1.0,
+ "content": "be a symmetric matrix,",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 269,
+ 454,
+ 321,
+ 466
+ ],
+ "score": 0.89,
+ "content": "W _ { l } \\in \\mathbb { R } ^ { C \\times C }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 322,
+ 452,
+ 325,
+ 468
+ ],
+ "score": 1.0,
+ "content": ",",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 326,
+ 457,
+ 335,
+ 466
+ ],
+ "score": 0.72,
+ "content": "s _ { l }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 335,
+ 452,
+ 476,
+ 468
+ ],
+ "score": 1.0,
+ "content": "be the maximum singular value of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 476,
+ 455,
+ 489,
+ 466
+ ],
+ "score": 0.88,
+ "content": "W _ { l }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 490,
+ 452,
+ 506,
+ 468
+ ],
+ "score": 1.0,
+ "content": "for",
+ "type": "text"
+ }
+ ],
+ "index": 19
+ },
+ {
+ "bbox": [
+ 106,
+ 462,
+ 507,
+ 480
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 466,
+ 139,
+ 478
+ ],
+ "score": 0.91,
+ "content": "l \\in \\mathbb { N } _ { + }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 140,
+ 462,
+ 190,
+ 480
+ ],
+ "score": 1.0,
+ "content": ". We define",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 190,
+ 465,
+ 282,
+ 477
+ ],
+ "score": 0.91,
+ "content": "f _ { l } ^ { ^ { \\bullet } } : \\mathbb { R } ^ { N \\times C } \\mathbb { R } ^ { N \\times { \\dot { C } } }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 282,
+ 462,
+ 298,
+ 480
+ ],
+ "score": 1.0,
+ "content": "by",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 298,
+ 466,
+ 385,
+ 478
+ ],
+ "score": 0.93,
+ "content": "f _ { l } ( X ) : = \\sigma ( P X W _ { l } )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 385,
+ 462,
+ 415,
+ 480
+ ],
+ "score": 1.0,
+ "content": "where",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 415,
+ 468,
+ 422,
+ 476
+ ],
+ "score": 0.78,
+ "content": "\\sigma",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 423,
+ 462,
+ 507,
+ 480
+ ],
+ "score": 1.0,
+ "content": "is the element-wise",
+ "type": "text"
+ }
+ ],
+ "index": 20
+ },
+ {
+ "bbox": [
+ 105,
+ 476,
+ 172,
+ 489
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 476,
+ 172,
+ 489
+ ],
+ "score": 1.0,
+ "content": "ReLU function.",
+ "type": "text"
+ }
+ ],
+ "index": 21
+ }
+ ],
+ "index": 20,
+ "bbox_fs": [
+ 104,
+ 452,
+ 507,
+ 489
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 491,
+ 505,
+ 525
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 490,
+ 505,
+ 504
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 490,
+ 351,
+ 504
+ ],
+ "score": 1.0,
+ "content": "Proposition 3. Suppose further that the operator norm of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 351,
+ 492,
+ 360,
+ 501
+ ],
+ "score": 0.82,
+ "content": "P",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 360,
+ 490,
+ 435,
+ 504
+ ],
+ "score": 1.0,
+ "content": "is no larger than",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 436,
+ 492,
+ 443,
+ 501
+ ],
+ "score": 0.77,
+ "content": "\\lambda ,",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 443,
+ 490,
+ 505,
+ 504
+ ],
+ "score": 1.0,
+ "content": ", then we have",
+ "type": "text"
+ }
+ ],
+ "index": 22
+ },
+ {
+ "bbox": [
+ 107,
+ 500,
+ 504,
+ 517
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 502,
+ 201,
+ 514
+ ],
+ "score": 0.9,
+ "content": "\\| f _ { l } ( \\bar { \\boldsymbol { X } } ) \\| _ { \\mathrm { F } } \\leq s _ { l } \\lambda \\| \\bar { \\boldsymbol { X } } \\| _ { \\mathrm { F } }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 202,
+ 500,
+ 236,
+ 517
+ ],
+ "score": 1.0,
+ "content": "for any",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 237,
+ 502,
+ 269,
+ 514
+ ],
+ "score": 0.9,
+ "content": "l \\in \\mathbb { N } _ { + }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 270,
+ 500,
+ 346,
+ 517
+ ],
+ "score": 1.0,
+ "content": ". In particular, let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 347,
+ 503,
+ 414,
+ 516
+ ],
+ "score": 0.9,
+ "content": "s : = \\operatorname* { s u p } _ { l \\in \\mathbb { N } _ { + } } s _ { l }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 415,
+ 500,
+ 430,
+ 517
+ ],
+ "score": 1.0,
+ "content": ". If",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 431,
+ 502,
+ 463,
+ 513
+ ],
+ "score": 0.88,
+ "content": "s \\lambda < 1",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 464,
+ 500,
+ 491,
+ 517
+ ],
+ "score": 1.0,
+ "content": ", then,",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 491,
+ 502,
+ 504,
+ 513
+ ],
+ "score": 0.85,
+ "content": "X _ { l }",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 23
+ },
+ {
+ "bbox": [
+ 106,
+ 514,
+ 263,
+ 527
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 514,
+ 230,
+ 527
+ ],
+ "score": 1.0,
+ "content": "exponentially approaches 0 as",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 230,
+ 515,
+ 259,
+ 524
+ ],
+ "score": 0.88,
+ "content": "l \\infty",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 259,
+ 514,
+ 263,
+ 527
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 24
+ }
+ ],
+ "index": 23,
+ "bbox_fs": [
+ 106,
+ 490,
+ 505,
+ 527
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 537,
+ 505,
+ 583
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 537,
+ 506,
+ 550
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 537,
+ 162,
+ 550
+ ],
+ "score": 1.0,
+ "content": "Proof. Since",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 162,
+ 538,
+ 169,
+ 548
+ ],
+ "score": 0.81,
+ "content": "\\lambda",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 170,
+ 537,
+ 271,
+ 550
+ ],
+ "score": 1.0,
+ "content": "is the operator norm of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 271,
+ 538,
+ 295,
+ 550
+ ],
+ "score": 0.92,
+ "content": "P | _ { U ^ { \\perp } }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 295,
+ 537,
+ 506,
+ 550
+ ],
+ "score": 1.0,
+ "content": ", the assumption implies that the operator norm of",
+ "type": "text"
+ }
+ ],
+ "index": 25
+ },
+ {
+ "bbox": [
+ 107,
+ 547,
+ 506,
+ 561
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 549,
+ 116,
+ 559
+ ],
+ "score": 0.8,
+ "content": "P",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 116,
+ 547,
+ 211,
+ 561
+ ],
+ "score": 1.0,
+ "content": "itself is no larger than",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 211,
+ 549,
+ 218,
+ 558
+ ],
+ "score": 0.78,
+ "content": "\\lambda",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 219,
+ 547,
+ 308,
+ 561
+ ],
+ "score": 1.0,
+ "content": ". Therefore, we have",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 308,
+ 549,
+ 468,
+ 560
+ ],
+ "score": 0.9,
+ "content": "\\| P X W _ { l } \\| _ { \\mathrm { F } } \\leq \\lambda \\bar { \\| } X W _ { l } \\| _ { \\mathrm { F } } \\leq s _ { l } \\bar { \\lambda } \\| X \\| _ { \\mathrm { F } }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 468,
+ 547,
+ 506,
+ 561
+ ],
+ "score": 1.0,
+ "content": ". On the",
+ "type": "text"
+ }
+ ],
+ "index": 26
+ },
+ {
+ "bbox": [
+ 103,
+ 558,
+ 506,
+ 573
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 103,
+ 558,
+ 179,
+ 573
+ ],
+ "score": 1.0,
+ "content": "other hand, since",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 179,
+ 559,
+ 230,
+ 571
+ ],
+ "score": 0.93,
+ "content": "\\sigma ( x ) ^ { 2 } \\leq x ^ { 2 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 230,
+ 558,
+ 264,
+ 573
+ ],
+ "score": 1.0,
+ "content": "for any",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 264,
+ 560,
+ 293,
+ 570
+ ],
+ "score": 0.89,
+ "content": "x \\in \\mathbb { R }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 293,
+ 558,
+ 334,
+ 573
+ ],
+ "score": 1.0,
+ "content": ", we have",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 335,
+ 560,
+ 415,
+ 571
+ ],
+ "score": 0.88,
+ "content": "\\| \\sigma ( X ) \\| _ { \\mathrm { F } } \\leq \\| X \\| _ { F }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 415,
+ 558,
+ 449,
+ 573
+ ],
+ "score": 1.0,
+ "content": "for any",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 450,
+ 559,
+ 501,
+ 570
+ ],
+ "score": 0.9,
+ "content": "\\boldsymbol { X } ^ { \\prime } \\in \\mathbb { R } ^ { N \\times C }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 502,
+ 558,
+ 506,
+ 573
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 27
+ },
+ {
+ "bbox": [
+ 105,
+ 570,
+ 504,
+ 584
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 570,
+ 224,
+ 584
+ ],
+ "score": 1.0,
+ "content": "Combining the two, we have",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 224,
+ 570,
+ 374,
+ 583
+ ],
+ "score": 0.91,
+ "content": "\\Vert f _ { l } ( X ) \\Vert _ { \\mathrm { F } } \\leq \\Vert P X W _ { l } \\Vert _ { \\mathrm { F } } \\leq s _ { l } \\lambda \\Vert X \\Vert _ { \\mathrm { F } }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 374,
+ 570,
+ 379,
+ 584
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 496,
+ 572,
+ 504,
+ 579
+ ],
+ "score": 0.998,
+ "content": "□",
+ "type": "text"
+ }
+ ],
+ "index": 28
+ }
+ ],
+ "index": 26.5,
+ "bbox_fs": [
+ 103,
+ 537,
+ 506,
+ 584
+ ]
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 107,
+ 595,
+ 253,
+ 606
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 594,
+ 254,
+ 608
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 594,
+ 254,
+ 608
+ ],
+ "score": 1.0,
+ "content": "E.3 STRICTNESS OF THEOREM 1",
+ "type": "text"
+ }
+ ],
+ "index": 29
+ }
+ ],
+ "index": 29
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 615,
+ 505,
+ 650
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 615,
+ 505,
+ 628
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 615,
+ 211,
+ 628
+ ],
+ "score": 1.0,
+ "content": "Theorem 1 implies that if",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 211,
+ 616,
+ 242,
+ 627
+ ],
+ "score": 0.91,
+ "content": "s \\lambda \\leq 1",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 242,
+ 615,
+ 345,
+ 628
+ ],
+ "score": 1.0,
+ "content": ", then, one-step transition",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 345,
+ 616,
+ 354,
+ 627
+ ],
+ "score": 0.87,
+ "content": "f _ { l }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 354,
+ 615,
+ 488,
+ 628
+ ],
+ "score": 1.0,
+ "content": "does not increase the distance to",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 488,
+ 616,
+ 501,
+ 626
+ ],
+ "score": 0.82,
+ "content": "\\mathcal { M }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 501,
+ 615,
+ 505,
+ 628
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 30
+ },
+ {
+ "bbox": [
+ 106,
+ 627,
+ 505,
+ 638
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 627,
+ 505,
+ 638
+ ],
+ "score": 1.0,
+ "content": "In this section, we first prove that this theorem is strict in the sense that, there exists a situation in",
+ "type": "text"
+ }
+ ],
+ "index": 31
+ },
+ {
+ "bbox": [
+ 106,
+ 638,
+ 483,
+ 650
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 638,
+ 133,
+ 650
+ ],
+ "score": 1.0,
+ "content": "which",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 133,
+ 638,
+ 166,
+ 649
+ ],
+ "score": 0.91,
+ "content": "s _ { l } \\lambda > 1",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 167,
+ 638,
+ 258,
+ 650
+ ],
+ "score": 1.0,
+ "content": "holds and the distance",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 258,
+ 638,
+ 275,
+ 649
+ ],
+ "score": 0.9,
+ "content": "d _ { \\mathcal { M } }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 275,
+ 638,
+ 404,
+ 650
+ ],
+ "score": 1.0,
+ "content": "increases by one-step transition",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 404,
+ 638,
+ 413,
+ 649
+ ],
+ "score": 0.86,
+ "content": "f _ { l }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 413,
+ 638,
+ 470,
+ 650
+ ],
+ "score": 1.0,
+ "content": "at some point",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 471,
+ 638,
+ 480,
+ 648
+ ],
+ "score": 0.85,
+ "content": "X",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 481,
+ 638,
+ 483,
+ 650
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 32
+ }
+ ],
+ "index": 31,
+ "bbox_fs": [
+ 106,
+ 615,
+ 505,
+ 650
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 654,
+ 384,
+ 666
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 653,
+ 385,
+ 668
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 653,
+ 121,
+ 668
+ ],
+ "score": 1.0,
+ "content": "Set",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 122,
+ 655,
+ 153,
+ 665
+ ],
+ "score": 0.82,
+ "content": "N \\gets 2",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 153,
+ 653,
+ 156,
+ 668
+ ],
+ "score": 1.0,
+ "content": ",",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 156,
+ 655,
+ 186,
+ 665
+ ],
+ "score": 0.82,
+ "content": "C \\gets 1",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 186,
+ 653,
+ 206,
+ 668
+ ],
+ "score": 1.0,
+ "content": ", and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 207,
+ 655,
+ 239,
+ 664
+ ],
+ "score": 0.91,
+ "content": "M \\gets 1",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 239,
+ 653,
+ 317,
+ 668
+ ],
+ "score": 1.0,
+ "content": "in Section 3.2. For",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 317,
+ 655,
+ 353,
+ 666
+ ],
+ "score": 0.91,
+ "content": "\\mu , \\lambda > 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 353,
+ 653,
+ 385,
+ 668
+ ],
+ "score": 1.0,
+ "content": ", we set",
+ "type": "text"
+ }
+ ],
+ "index": 33
+ }
+ ],
+ "index": 33,
+ "bbox_fs": [
+ 105,
+ 653,
+ 385,
+ 668
+ ]
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 206,
+ 668,
+ 402,
+ 696
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 206,
+ 668,
+ 402,
+ 696
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 206,
+ 668,
+ 402,
+ 696
+ ],
+ "score": 0.93,
+ "content": "P [ { \\mu \\atop 0 } \\lambda ] , e [ 1 \\atop 0 ] , U \\{ [ { x \\atop y } ] \\mid y = 0 \\} .",
+ "type": "interline_equation",
+ "image_path": "31c8f337f767b389f9e29dc55525ffeb7295cd5c21158f1437be017ccbef7b58.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 34,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 206,
+ 668,
+ 402,
+ 696
+ ],
+ "spans": [],
+ "index": 34
+ }
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 698,
+ 505,
+ 732
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 698,
+ 505,
+ 711
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 698,
+ 311,
+ 711
+ ],
+ "score": 1.0,
+ "content": "Then, by definition, we can check that the 3-tuple",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 311,
+ 699,
+ 347,
+ 710
+ ],
+ "score": 0.89,
+ "content": "( P , e , U )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 347,
+ 698,
+ 505,
+ 711
+ ],
+ "score": 1.0,
+ "content": "satisfies the Assumptions 1 and 2. Set",
+ "type": "text"
+ }
+ ],
+ "index": 35
+ },
+ {
+ "bbox": [
+ 107,
+ 704,
+ 508,
+ 725
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 710,
+ 186,
+ 720
+ ],
+ "score": 0.9,
+ "content": "\\mathcal { M } : = U \\otimes \\mathbb { R } = U",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 187,
+ 704,
+ 235,
+ 725
+ ],
+ "score": 1.0,
+ "content": "and choose",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 235,
+ 710,
+ 268,
+ 720
+ ],
+ "score": 0.91,
+ "content": "W \\in \\mathbb { R }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 268,
+ 704,
+ 298,
+ 725
+ ],
+ "score": 1.0,
+ "content": "so that",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 299,
+ 710,
+ 340,
+ 720
+ ],
+ "score": 0.88,
+ "content": "W ^ { \\setminus } > \\lambda ^ { - 1 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 341,
+ 704,
+ 404,
+ 725
+ ],
+ "score": 1.0,
+ "content": ". Finally define",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 405,
+ 709,
+ 491,
+ 721
+ ],
+ "score": 0.91,
+ "content": "f : \\mathbb { R } ^ { \\mathbf { \\hat { N } } \\times C } \\mathbb { R } ^ { N \\times C }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 491,
+ 704,
+ 508,
+ 725
+ ],
+ "score": 1.0,
+ "content": "by",
+ "type": "text"
+ }
+ ],
+ "index": 36
+ },
+ {
+ "bbox": [
+ 107,
+ 720,
+ 369,
+ 733
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 721,
+ 187,
+ 732
+ ],
+ "score": 0.91,
+ "content": "f ( X ) : = \\sigma ( P X W )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 188,
+ 720,
+ 216,
+ 733
+ ],
+ "score": 1.0,
+ "content": "where",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 216,
+ 722,
+ 223,
+ 730
+ ],
+ "score": 0.79,
+ "content": "\\sigma",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 224,
+ 720,
+ 369,
+ 733
+ ],
+ "score": 1.0,
+ "content": "is the element-wise ReLU function.",
+ "type": "text"
+ }
+ ],
+ "index": 37
+ }
+ ],
+ "index": 36,
+ "bbox_fs": [
+ 106,
+ 698,
+ 508,
+ 733
+ ]
+ }
+ ]
+ },
+ {
+ "preproc_blocks": [
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 105,
+ 505,
+ 146
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 105,
+ 506,
+ 120
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 105,
+ 232,
+ 120
+ ],
+ "score": 1.0,
+ "content": "Proof. By definition, we have",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 233,
+ 106,
+ 298,
+ 119
+ ],
+ "score": 0.93,
+ "content": "d _ { \\mathcal { M } } ( X ) = | x _ { 2 } |",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 298,
+ 105,
+ 506,
+ 120
+ ],
+ "score": 1.0,
+ "content": ". On the other hand, direct calculation shows that",
+ "type": "text"
+ }
+ ],
+ "index": 0
+ },
+ {
+ "bbox": [
+ 107,
+ 119,
+ 506,
+ 134
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 119,
+ 260,
+ 134
+ ],
+ "score": 0.86,
+ "content": "f _ { l } ( X ) = \\left[ ( W \\mu X _ { 1 } ) ^ { + } \\quad ( W \\lambda X _ { 2 } ) ^ { + } \\right] ^ { \\top }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 260,
+ 119,
+ 279,
+ 134
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 280,
+ 120,
+ 387,
+ 133
+ ],
+ "score": 0.91,
+ "content": "d _ { \\mathcal { M } } ( f _ { l } ( X ) ) = ( W \\lambda X _ { 2 } ) ^ { + }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 387,
+ 119,
+ 416,
+ 134
+ ],
+ "score": 1.0,
+ "content": "where",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 416,
+ 120,
+ 489,
+ 133
+ ],
+ "score": 0.91,
+ "content": "x ^ { + } : = \\operatorname* { m a x } ( x , 0 )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 489,
+ 119,
+ 506,
+ 134
+ ],
+ "score": 1.0,
+ "content": "for",
+ "type": "text"
+ }
+ ],
+ "index": 1
+ },
+ {
+ "bbox": [
+ 107,
+ 132,
+ 506,
+ 147
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 134,
+ 132,
+ 144
+ ],
+ "score": 0.87,
+ "content": "x \\in \\mathbb { R }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 133,
+ 132,
+ 162,
+ 147
+ ],
+ "score": 1.0,
+ "content": ". Since",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 162,
+ 133,
+ 203,
+ 144
+ ],
+ "score": 0.89,
+ "content": "W > \\lambda ^ { - 1 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 204,
+ 132,
+ 221,
+ 147
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 222,
+ 135,
+ 251,
+ 145
+ ],
+ "score": 0.89,
+ "content": "x _ { 2 } > 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 252,
+ 132,
+ 290,
+ 147
+ ],
+ "score": 1.0,
+ "content": ", we have",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 290,
+ 134,
+ 384,
+ 146
+ ],
+ "score": 0.91,
+ "content": "d _ { \\mathcal { M } } ( f _ { l } ( X ) ) > d _ { \\mathcal { M } } ( X )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 385,
+ 132,
+ 389,
+ 147
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 494,
+ 134,
+ 506,
+ 145
+ ],
+ "score": 0.999,
+ "content": "□",
+ "type": "text"
+ }
+ ],
+ "index": 2
+ }
+ ],
+ "index": 1
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 157,
+ 506,
+ 191
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 157,
+ 505,
+ 169
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 157,
+ 505,
+ 169
+ ],
+ "score": 1.0,
+ "content": "Next, we prove the non-strictness of Theorem 1 in the sense that there exists a situation in which",
+ "type": "text"
+ }
+ ],
+ "index": 3
+ },
+ {
+ "bbox": [
+ 107,
+ 168,
+ 506,
+ 181
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 169,
+ 140,
+ 180
+ ],
+ "score": 0.9,
+ "content": "s _ { l } \\lambda > 1",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 141,
+ 168,
+ 234,
+ 181
+ ],
+ "score": 1.0,
+ "content": "holds and the distance",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 234,
+ 169,
+ 250,
+ 180
+ ],
+ "score": 0.9,
+ "content": "d _ { \\mathcal { M } }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 250,
+ 168,
+ 348,
+ 181
+ ],
+ "score": 1.0,
+ "content": "uniformly decreases by",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 348,
+ 169,
+ 357,
+ 180
+ ],
+ "score": 0.84,
+ "content": "f _ { l }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 357,
+ 168,
+ 435,
+ 181
+ ],
+ "score": 1.0,
+ "content": ". Again, we set Set",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 435,
+ 169,
+ 467,
+ 179
+ ],
+ "score": 0.85,
+ "content": "N \\gets 2",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 467,
+ 168,
+ 470,
+ 181
+ ],
+ "score": 1.0,
+ "content": ",",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 471,
+ 169,
+ 501,
+ 179
+ ],
+ "score": 0.83,
+ "content": "C \\gets 1",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 501,
+ 168,
+ 506,
+ 181
+ ],
+ "score": 1.0,
+ "content": ",",
+ "type": "text"
+ }
+ ],
+ "index": 4
+ },
+ {
+ "bbox": [
+ 106,
+ 180,
+ 249,
+ 191
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 180,
+ 123,
+ 191
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 124,
+ 180,
+ 156,
+ 190
+ ],
+ "score": 0.89,
+ "content": "M \\gets 1",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 156,
+ 180,
+ 176,
+ 191
+ ],
+ "score": 1.0,
+ "content": ". Let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 176,
+ 180,
+ 217,
+ 191
+ ],
+ "score": 0.93,
+ "content": "\\lambda \\in ( 1 , 2 )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 217,
+ 180,
+ 249,
+ 191
+ ],
+ "score": 1.0,
+ "content": "and set",
+ "type": "text"
+ }
+ ],
+ "index": 5
+ }
+ ],
+ "index": 4
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 188,
+ 196,
+ 423,
+ 224
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 188,
+ 196,
+ 423,
+ 224
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 188,
+ 196,
+ 423,
+ 224
+ ],
+ "score": 0.94,
+ "content": "P \\frac { \\lambda } { 2 } [ \\begin{array} { c c } { 1 } & { - 1 } \\\\ { - 1 } & { 1 } \\end{array} ] , e \\frac { 1 } { \\sqrt { 2 } } [ \\begin{array} { c } { 1 } \\\\ { 1 } \\end{array} ] , U \\{ [ \\begin{array} { c } { x } \\\\ { y } \\end{array} ] | x = y \\}",
+ "type": "interline_equation",
+ "image_path": "087c9606e481a961a37b1e1babb22cba1730aec8a623132d768567bf6bb36c81.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 6,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 188,
+ 196,
+ 423,
+ 224
+ ],
+ "spans": [],
+ "index": 6
+ }
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 105,
+ 228,
+ 494,
+ 240
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 227,
+ 495,
+ 242
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 227,
+ 267,
+ 242
+ ],
+ "score": 1.0,
+ "content": "Then, we can directly show that 3-tuple",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 267,
+ 228,
+ 303,
+ 241
+ ],
+ "score": 0.93,
+ "content": "( P , e , U )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 304,
+ 227,
+ 458,
+ 242
+ ],
+ "score": 1.0,
+ "content": "satisfies the Assumptions 1 and 2. Set",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 459,
+ 229,
+ 491,
+ 239
+ ],
+ "score": 0.88,
+ "content": "W 1",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 491,
+ 227,
+ 495,
+ 242
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 7
+ }
+ ],
+ "index": 7
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 243,
+ 421,
+ 256
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 104,
+ 241,
+ 421,
+ 257
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 241,
+ 206,
+ 257
+ ],
+ "score": 1.0,
+ "content": "Proposition 5. We have",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 207,
+ 243,
+ 242,
+ 254
+ ],
+ "score": 0.9,
+ "content": "W \\lambda > 1",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 243,
+ 241,
+ 261,
+ 257
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 261,
+ 243,
+ 355,
+ 255
+ ],
+ "score": 0.91,
+ "content": "d _ { \\mathcal { M } } ( f _ { l } ( X ) ) < d _ { \\mathcal { M } } ( X )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 356,
+ 241,
+ 383,
+ 257
+ ],
+ "score": 1.0,
+ "content": "for all",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 383,
+ 243,
+ 417,
+ 254
+ ],
+ "score": 0.9,
+ "content": "X \\in \\mathbb { R } ^ { 2 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 417,
+ 241,
+ 421,
+ 257
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 8
+ }
+ ],
+ "index": 8
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 267,
+ 505,
+ 307
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 104,
+ 267,
+ 506,
+ 285
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 267,
+ 197,
+ 285
+ ],
+ "score": 1.0,
+ "content": "Proof. First, note that",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 198,
+ 267,
+ 277,
+ 285
+ ],
+ "score": 0.9,
+ "content": "\\begin{array} { r } { e ^ { \\prime } : = \\frac { 1 } { \\sqrt { 2 } } \\left[ { \\begin{array} { l l } { 1 } & { - 1 } \\end{array} } \\right] ^ { \\top } } \\end{array}",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 277,
+ 267,
+ 363,
+ 285
+ ],
+ "score": 1.0,
+ "content": "is the eigenvector of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 363,
+ 271,
+ 372,
+ 280
+ ],
+ "score": 0.84,
+ "content": "P",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 372,
+ 267,
+ 427,
+ 285
+ ],
+ "score": 1.0,
+ "content": "associated to",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 428,
+ 270,
+ 483,
+ 281
+ ],
+ "score": 0.9,
+ "content": "\\lambda \\colon P e ^ { \\prime } = \\lambda e ^ { \\prime }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 484,
+ 267,
+ 506,
+ 285
+ ],
+ "score": 1.0,
+ "content": ". For",
+ "type": "text"
+ }
+ ],
+ "index": 9
+ },
+ {
+ "bbox": [
+ 106,
+ 282,
+ 505,
+ 298
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 284,
+ 205,
+ 296
+ ],
+ "score": 0.86,
+ "content": "X = a e + b e ^ { \\prime } \\left( a , b { \\dot { \\omega } } 0 \\right)",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 205,
+ 282,
+ 296,
+ 298
+ ],
+ "score": 1.0,
+ "content": ", the distance between",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 297,
+ 284,
+ 307,
+ 294
+ ],
+ "score": 0.83,
+ "content": "X",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 307,
+ 282,
+ 326,
+ 298
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 326,
+ 284,
+ 339,
+ 294
+ ],
+ "score": 0.83,
+ "content": "\\mathcal { M }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 339,
+ 282,
+ 350,
+ 298
+ ],
+ "score": 1.0,
+ "content": "is",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 351,
+ 284,
+ 408,
+ 296
+ ],
+ "score": 0.93,
+ "content": "d _ { \\mathcal { M } } ( \\boldsymbol { X } ) = | b |",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 408,
+ 282,
+ 505,
+ 298
+ ],
+ "score": 1.0,
+ "content": ". On the other hand, by",
+ "type": "text"
+ }
+ ],
+ "index": 10
+ },
+ {
+ "bbox": [
+ 106,
+ 295,
+ 222,
+ 307
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 295,
+ 222,
+ 307
+ ],
+ "score": 1.0,
+ "content": "direct computation, we have",
+ "type": "text"
+ }
+ ],
+ "index": 11
+ }
+ ],
+ "index": 10
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 204,
+ 311,
+ 406,
+ 357
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 204,
+ 311,
+ 406,
+ 357
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 204,
+ 311,
+ 406,
+ 357
+ ],
+ "score": 0.94,
+ "content": "f ( X ) = \\sigma ( P X W ) = { \\left\\{ \\begin{array} { l l } { \\left[ 0 \\quad { \\frac { \\lambda b } { \\sqrt { 2 } } } \\right] ^ { \\top } } & { { \\mathrm { ( i f ~ } } b \\geq 0 { \\mathrm { ) } } , } \\\\ { \\left[ { \\frac { - \\lambda b } { \\sqrt { 2 } } } \\quad 0 \\right] ^ { \\top } } & { { \\mathrm { ( i f ~ } } b < 0 { \\mathrm { ) } } . } \\end{array} \\right. }",
+ "type": "interline_equation",
+ "image_path": "a159a4cc7dd5ac398b3cff74136fefb345275fef911e7b7089e5ae12ee2d25ab.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 13,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 204,
+ 311,
+ 406,
+ 326.3333333333333
+ ],
+ "spans": [],
+ "index": 12
+ },
+ {
+ "bbox": [
+ 204,
+ 326.3333333333333,
+ 406,
+ 341.66666666666663
+ ],
+ "spans": [],
+ "index": 13
+ },
+ {
+ "bbox": [
+ 204,
+ 341.66666666666663,
+ 406,
+ 356.99999999999994
+ ],
+ "spans": [],
+ "index": 14
+ }
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 361,
+ 504,
+ 385
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 361,
+ 505,
+ 374
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 361,
+ 241,
+ 374
+ ],
+ "score": 1.0,
+ "content": "Therefore, the distance between",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 241,
+ 362,
+ 265,
+ 373
+ ],
+ "score": 0.9,
+ "content": "f ( X )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 265,
+ 361,
+ 285,
+ 374
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 285,
+ 362,
+ 299,
+ 372
+ ],
+ "score": 0.84,
+ "content": "\\mathcal { M }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 299,
+ 361,
+ 311,
+ 374
+ ],
+ "score": 1.0,
+ "content": "is",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 312,
+ 361,
+ 401,
+ 374
+ ],
+ "score": 0.93,
+ "content": "d _ { \\mathcal { M } } ( f ( X ) ) = \\lambda | b | / 2",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 401,
+ 361,
+ 434,
+ 374
+ ],
+ "score": 1.0,
+ "content": ". Since",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 434,
+ 362,
+ 464,
+ 372
+ ],
+ "score": 0.88,
+ "content": "\\lambda \\ : < \\ : 2",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 464,
+ 361,
+ 505,
+ 374
+ ],
+ "score": 1.0,
+ "content": ", we have",
+ "type": "text"
+ }
+ ],
+ "index": 15
+ },
+ {
+ "bbox": [
+ 107,
+ 372,
+ 504,
+ 385
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 373,
+ 198,
+ 385
+ ],
+ "score": 0.92,
+ "content": "d _ { \\mathcal { M } } ( f ( X ) ) < d _ { \\mathcal { M } } ( X )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 199,
+ 372,
+ 230,
+ 385
+ ],
+ "score": 1.0,
+ "content": "for any",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 231,
+ 373,
+ 264,
+ 383
+ ],
+ "score": 0.9,
+ "content": "X \\in \\mathbb { R } ^ { 2 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 264,
+ 372,
+ 269,
+ 385
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 496,
+ 375,
+ 504,
+ 382
+ ],
+ "score": 0.999,
+ "content": "□",
+ "type": "text"
+ }
+ ],
+ "index": 16
+ }
+ ],
+ "index": 15.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 396,
+ 505,
+ 419
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 395,
+ 506,
+ 410
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 395,
+ 275,
+ 410
+ ],
+ "score": 1.0,
+ "content": "We have shown that the non-negativity of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 276,
+ 399,
+ 281,
+ 406
+ ],
+ "score": 0.73,
+ "content": "e",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 281,
+ 395,
+ 506,
+ 410
+ ],
+ "score": 1.0,
+ "content": "(Assumption 1) is not a redundant condition in Section",
+ "type": "text"
+ }
+ ],
+ "index": 17
+ },
+ {
+ "bbox": [
+ 104,
+ 406,
+ 126,
+ 421
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 406,
+ 126,
+ 421
+ ],
+ "score": 1.0,
+ "content": "6.1.",
+ "type": "text"
+ }
+ ],
+ "index": 18
+ }
+ ],
+ "index": 17.5
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 107,
+ 435,
+ 297,
+ 448
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 434,
+ 298,
+ 450
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 434,
+ 298,
+ 450
+ ],
+ "score": 1.0,
+ "content": "F RELATION TO MARKOV PROCESS",
+ "type": "text"
+ }
+ ],
+ "index": 19
+ }
+ ],
+ "index": 19
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 460,
+ 505,
+ 493
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 104,
+ 459,
+ 505,
+ 473
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 459,
+ 505,
+ 473
+ ],
+ "score": 1.0,
+ "content": "It is known that any Markov process on finite states converges to a unique distribution (equilibrium)",
+ "type": "text"
+ }
+ ],
+ "index": 20
+ },
+ {
+ "bbox": [
+ 105,
+ 470,
+ 505,
+ 484
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 470,
+ 505,
+ 484
+ ],
+ "score": 1.0,
+ "content": "if it is irreducible and aperiodic (see, e.g., Norris (1998)). As we see in this section, this theorem is",
+ "type": "text"
+ }
+ ],
+ "index": 21
+ },
+ {
+ "bbox": [
+ 106,
+ 482,
+ 232,
+ 495
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 482,
+ 232,
+ 495
+ ],
+ "score": 1.0,
+ "content": "the special case of Corollary 3.",
+ "type": "text"
+ }
+ ],
+ "index": 22
+ }
+ ],
+ "index": 21
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 498,
+ 505,
+ 576
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 104,
+ 497,
+ 506,
+ 512
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 497,
+ 122,
+ 512
+ ],
+ "score": 1.0,
+ "content": "Let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 122,
+ 498,
+ 192,
+ 511
+ ],
+ "score": 0.93,
+ "content": "S : = \\{ 1 , \\ldots , N \\}",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 192,
+ 497,
+ 440,
+ 512
+ ],
+ "score": 1.0,
+ "content": "be a finite discrete state space. Consider a Markov process on",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 441,
+ 500,
+ 448,
+ 509
+ ],
+ "score": 0.82,
+ "content": "S",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 449,
+ 497,
+ 506,
+ 512
+ ],
+ "score": 1.0,
+ "content": "characterized",
+ "type": "text"
+ }
+ ],
+ "index": 23
+ },
+ {
+ "bbox": [
+ 102,
+ 508,
+ 507,
+ 524
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 102,
+ 508,
+ 240,
+ 524
+ ],
+ "score": 1.0,
+ "content": "by a symmetric transition matrix",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 240,
+ 510,
+ 348,
+ 523
+ ],
+ "score": 0.92,
+ "content": "P = ( p _ { i j } ) _ { i , j \\in [ N ] } \\in \\mathbb { R } ^ { N \\times N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 349,
+ 508,
+ 388,
+ 524
+ ],
+ "score": 1.0,
+ "content": "such that",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 389,
+ 510,
+ 416,
+ 521
+ ],
+ "score": 0.91,
+ "content": "P \\geq 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 417,
+ 508,
+ 434,
+ 524
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 434,
+ 510,
+ 468,
+ 520
+ ],
+ "score": 0.91,
+ "content": "P \\mathbf { 1 } = \\mathbf { 1 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 469,
+ 508,
+ 507,
+ 524
+ ],
+ "score": 1.0,
+ "content": "where 1",
+ "type": "text"
+ }
+ ],
+ "index": 24
+ },
+ {
+ "bbox": [
+ 104,
+ 520,
+ 506,
+ 534
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 520,
+ 244,
+ 534
+ ],
+ "score": 1.0,
+ "content": "is the all-one vector. We interpret",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 244,
+ 523,
+ 258,
+ 533
+ ],
+ "score": 0.86,
+ "content": "p _ { i j }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 258,
+ 520,
+ 423,
+ 534
+ ],
+ "score": 1.0,
+ "content": "as the transition probability from a state",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 423,
+ 522,
+ 428,
+ 531
+ ],
+ "score": 0.71,
+ "content": "i",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 428,
+ 520,
+ 439,
+ 534
+ ],
+ "score": 1.0,
+ "content": "to",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 439,
+ 522,
+ 446,
+ 532
+ ],
+ "score": 0.79,
+ "content": "j",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 446,
+ 520,
+ 506,
+ 534
+ ],
+ "score": 1.0,
+ "content": ". We associate",
+ "type": "text"
+ }
+ ],
+ "index": 25
+ },
+ {
+ "bbox": [
+ 107,
+ 531,
+ 504,
+ 545
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 532,
+ 116,
+ 542
+ ],
+ "score": 0.8,
+ "content": "P",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 116,
+ 531,
+ 172,
+ 545
+ ],
+ "score": 1.0,
+ "content": "with a graph",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 172,
+ 532,
+ 242,
+ 544
+ ],
+ "score": 0.93,
+ "content": "G _ { P } = ( V _ { P } , E _ { P } )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 242,
+ 531,
+ 257,
+ 545
+ ],
+ "score": 1.0,
+ "content": "by",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 257,
+ 532,
+ 301,
+ 544
+ ],
+ "score": 0.92,
+ "content": "V _ { P } = [ N ]",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 301,
+ 531,
+ 321,
+ 545
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 321,
+ 532,
+ 370,
+ 544
+ ],
+ "score": 0.92,
+ "content": "( i , j ) \\in E _ { P }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 370,
+ 531,
+ 429,
+ 545
+ ],
+ "score": 1.0,
+ "content": "if and only if",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 429,
+ 533,
+ 463,
+ 544
+ ],
+ "score": 0.92,
+ "content": "p _ { i j } > 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 463,
+ 531,
+ 495,
+ 545
+ ],
+ "score": 1.0,
+ "content": ". Since",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 495,
+ 532,
+ 504,
+ 542
+ ],
+ "score": 0.81,
+ "content": "P",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 26
+ },
+ {
+ "bbox": [
+ 105,
+ 542,
+ 505,
+ 555
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 542,
+ 221,
+ 555
+ ],
+ "score": 1.0,
+ "content": "is symmetric, we can regard",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 222,
+ 543,
+ 237,
+ 554
+ ],
+ "score": 0.87,
+ "content": "G _ { P }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 237,
+ 542,
+ 383,
+ 555
+ ],
+ "score": 1.0,
+ "content": "as an undirected graph. We assume",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 383,
+ 543,
+ 392,
+ 553
+ ],
+ "score": 0.84,
+ "content": "P",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 392,
+ 542,
+ 505,
+ 555
+ ],
+ "score": 1.0,
+ "content": "is irreducible and aperiodic",
+ "type": "text"
+ }
+ ],
+ "index": 27
+ },
+ {
+ "bbox": [
+ 104,
+ 550,
+ 507,
+ 569
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 550,
+ 392,
+ 569
+ ],
+ "score": 1.0,
+ "content": "10. Perron – Frobenius’ theorem (see, e.g., Meyer (2000)) implies that",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 392,
+ 554,
+ 401,
+ 564
+ ],
+ "score": 0.84,
+ "content": "P",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 401,
+ 550,
+ 507,
+ 569
+ ],
+ "score": 1.0,
+ "content": "satisfy the assumption of",
+ "type": "text"
+ }
+ ],
+ "index": 28
+ },
+ {
+ "bbox": [
+ 106,
+ 564,
+ 210,
+ 578
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 564,
+ 174,
+ 578
+ ],
+ "score": 1.0,
+ "content": "Corollary 3 with",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 175,
+ 565,
+ 205,
+ 575
+ ],
+ "score": 0.9,
+ "content": "M = 1",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 205,
+ 564,
+ 210,
+ 578
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 29
+ }
+ ],
+ "index": 26
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 579,
+ 505,
+ 613
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 578,
+ 505,
+ 592
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 578,
+ 349,
+ 592
+ ],
+ "score": 1.0,
+ "content": "Proposition 6 (Perron – Frobenius). Let the eigenvalues of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 349,
+ 580,
+ 358,
+ 590
+ ],
+ "score": 0.75,
+ "content": "P",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 359,
+ 578,
+ 372,
+ 592
+ ],
+ "score": 1.0,
+ "content": "be",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 372,
+ 580,
+ 438,
+ 591
+ ],
+ "score": 0.88,
+ "content": "\\lambda _ { 1 } \\leq \\cdots \\leq \\lambda _ { N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 438,
+ 578,
+ 505,
+ 592
+ ],
+ "score": 1.0,
+ "content": ". Then, we have",
+ "type": "text"
+ }
+ ],
+ "index": 30
+ },
+ {
+ "bbox": [
+ 107,
+ 590,
+ 505,
+ 603
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 591,
+ 145,
+ 602
+ ],
+ "score": 0.67,
+ "content": "- 1 < \\lambda _ { 1 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 146,
+ 590,
+ 150,
+ 603
+ ],
+ "score": 1.0,
+ "content": ",",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 150,
+ 591,
+ 195,
+ 603
+ ],
+ "score": 0.66,
+ "content": "\\lambda _ { N - 1 } < 1",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 195,
+ 590,
+ 217,
+ 603
+ ],
+ "score": 1.0,
+ "content": ", and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 217,
+ 591,
+ 252,
+ 602
+ ],
+ "score": 0.91,
+ "content": "\\lambda _ { N } = 1",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 252,
+ 590,
+ 399,
+ 603
+ ],
+ "score": 1.0,
+ "content": ". Further, there exists unique vector",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 399,
+ 591,
+ 434,
+ 601
+ ],
+ "score": 0.91,
+ "content": "e \\in \\mathbb { R } ^ { N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 434,
+ 590,
+ 475,
+ 603
+ ],
+ "score": 1.0,
+ "content": "such that",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 475,
+ 591,
+ 501,
+ 602
+ ],
+ "score": 0.88,
+ "content": "e \\geq 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 501,
+ 590,
+ 505,
+ 603
+ ],
+ "score": 1.0,
+ "content": ",",
+ "type": "text"
+ }
+ ],
+ "index": 31
+ },
+ {
+ "bbox": [
+ 107,
+ 601,
+ 325,
+ 615
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 602,
+ 140,
+ 614
+ ],
+ "score": 0.91,
+ "content": "\\| e \\| = 1",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 140,
+ 601,
+ 162,
+ 615
+ ],
+ "score": 1.0,
+ "content": ", and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 162,
+ 604,
+ 168,
+ 612
+ ],
+ "score": 0.46,
+ "content": "e",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 168,
+ 601,
+ 325,
+ 615
+ ],
+ "score": 1.0,
+ "content": "is the eigenvector for the eivenvalue 1.",
+ "type": "text"
+ }
+ ],
+ "index": 32
+ }
+ ],
+ "index": 31
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 104,
+ 615,
+ 503,
+ 640
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 614,
+ 505,
+ 630
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 614,
+ 178,
+ 630
+ ],
+ "score": 1.0,
+ "content": "Corollary 5. Let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 179,
+ 616,
+ 307,
+ 628
+ ],
+ "score": 0.9,
+ "content": "\\begin{array} { r } { \\lambda : = \\operatorname* { m a x } _ { n = 1 , \\ldots , N - 1 } | \\lambda _ { n } | ( < 1 ) } \\end{array}",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 308,
+ 614,
+ 327,
+ 630
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 327,
+ 615,
+ 434,
+ 628
+ ],
+ "score": 0.9,
+ "content": "\\mathcal { M } : = \\{ e \\otimes w \\ | \\ w \\in \\mathbb { R } ^ { C } \\}",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 434,
+ 614,
+ 448,
+ 630
+ ],
+ "score": 1.0,
+ "content": ". If",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 448,
+ 617,
+ 479,
+ 628
+ ],
+ "score": 0.83,
+ "content": "s \\lambda < 1",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 479,
+ 614,
+ 505,
+ 630
+ ],
+ "score": 1.0,
+ "content": ", then,",
+ "type": "text"
+ }
+ ],
+ "index": 33
+ },
+ {
+ "bbox": [
+ 104,
+ 627,
+ 381,
+ 640
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 627,
+ 186,
+ 640
+ ],
+ "score": 1.0,
+ "content": "for any initial value",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 186,
+ 628,
+ 199,
+ 639
+ ],
+ "score": 0.8,
+ "content": "X _ { 1 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 199,
+ 627,
+ 203,
+ 640
+ ],
+ "score": 1.0,
+ "content": ",",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 203,
+ 628,
+ 216,
+ 639
+ ],
+ "score": 0.81,
+ "content": "X _ { l }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 216,
+ 627,
+ 321,
+ 640
+ ],
+ "score": 1.0,
+ "content": "exponentially approaches",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 322,
+ 628,
+ 334,
+ 637
+ ],
+ "score": 0.58,
+ "content": "\\mathcal { M }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 335,
+ 627,
+ 347,
+ 640
+ ],
+ "score": 1.0,
+ "content": "as",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 347,
+ 629,
+ 376,
+ 637
+ ],
+ "score": 0.8,
+ "content": "l \\infty",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 377,
+ 627,
+ 381,
+ 640
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 34
+ }
+ ],
+ "index": 33.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 648,
+ 505,
+ 694
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 104,
+ 646,
+ 505,
+ 661
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 646,
+ 145,
+ 661
+ ],
+ "score": 1.0,
+ "content": "If we set",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 145,
+ 648,
+ 175,
+ 658
+ ],
+ "score": 0.9,
+ "content": "C = 1",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 175,
+ 646,
+ 194,
+ 661
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 194,
+ 648,
+ 228,
+ 659
+ ],
+ "score": 0.92,
+ "content": "W _ { l } = 1",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 228,
+ 646,
+ 258,
+ 661
+ ],
+ "score": 1.0,
+ "content": "for all",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 258,
+ 648,
+ 290,
+ 660
+ ],
+ "score": 0.91,
+ "content": "l \\in \\mathbb { N } _ { + }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 290,
+ 646,
+ 438,
+ 661
+ ],
+ "score": 1.0,
+ "content": ", then, we can inductively show that",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 439,
+ 648,
+ 472,
+ 659
+ ],
+ "score": 0.91,
+ "content": "X _ { l } \\geq 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 472,
+ 646,
+ 505,
+ 661
+ ],
+ "score": 1.0,
+ "content": "for any",
+ "type": "text"
+ }
+ ],
+ "index": 35
+ },
+ {
+ "bbox": [
+ 106,
+ 659,
+ 505,
+ 671
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 659,
+ 131,
+ 670
+ ],
+ "score": 0.9,
+ "content": "l \\geq 2",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 131,
+ 659,
+ 249,
+ 671
+ ],
+ "score": 1.0,
+ "content": ". Therefore, we can interpret",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 250,
+ 660,
+ 262,
+ 670
+ ],
+ "score": 0.86,
+ "content": "X _ { l }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 262,
+ 659,
+ 331,
+ 671
+ ],
+ "score": 1.0,
+ "content": "as a measure on",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 332,
+ 660,
+ 339,
+ 669
+ ],
+ "score": 0.79,
+ "content": "S",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 340,
+ 659,
+ 505,
+ 671
+ ],
+ "score": 1.0,
+ "content": ". Suppose further that we take the initial",
+ "type": "text"
+ }
+ ],
+ "index": 36
+ },
+ {
+ "bbox": [
+ 105,
+ 669,
+ 505,
+ 683
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 669,
+ 131,
+ 683
+ ],
+ "score": 1.0,
+ "content": "value",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 131,
+ 671,
+ 145,
+ 681
+ ],
+ "score": 0.89,
+ "content": "X _ { 1 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 145,
+ 669,
+ 159,
+ 683
+ ],
+ "score": 1.0,
+ "content": "as",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 159,
+ 670,
+ 195,
+ 681
+ ],
+ "score": 0.92,
+ "content": "X _ { 1 } ~ \\geq ~ 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 195,
+ 669,
+ 214,
+ 683
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 214,
+ 669,
+ 259,
+ 682
+ ],
+ "score": 0.91,
+ "content": "X _ { 1 } ^ { \\top } \\mathbf { \\hat { 1 } } = 1",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 259,
+ 669,
+ 361,
+ 683
+ ],
+ "score": 1.0,
+ "content": "so that we can interpret",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 361,
+ 671,
+ 375,
+ 681
+ ],
+ "score": 0.88,
+ "content": "X _ { 1 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 375,
+ 669,
+ 505,
+ 683
+ ],
+ "score": 1.0,
+ "content": "as a probability distribution on",
+ "type": "text"
+ }
+ ],
+ "index": 37
+ },
+ {
+ "bbox": [
+ 106,
+ 680,
+ 505,
+ 695
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 682,
+ 114,
+ 691
+ ],
+ "score": 0.79,
+ "content": "S",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 115,
+ 680,
+ 265,
+ 695
+ ],
+ "score": 1.0,
+ "content": ". Then, we can inductively show that",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 266,
+ 681,
+ 298,
+ 693
+ ],
+ "score": 0.87,
+ "content": "X _ { l } \\geq 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 298,
+ 680,
+ 301,
+ 695
+ ],
+ "score": 1.0,
+ "content": ",",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 302,
+ 680,
+ 344,
+ 694
+ ],
+ "score": 0.89,
+ "content": "X _ { l } ^ { \\top } \\mathbf { 1 } = 1",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 344,
+ 680,
+ 366,
+ 695
+ ],
+ "score": 1.0,
+ "content": "(i.e.,",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 366,
+ 682,
+ 379,
+ 692
+ ],
+ "score": 0.89,
+ "content": "X _ { l }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 379,
+ 680,
+ 505,
+ 695
+ ],
+ "score": 1.0,
+ "content": "is a probability distribution on",
+ "type": "text"
+ }
+ ],
+ "index": 38
+ }
+ ],
+ "index": 36.5
+ }
+ ],
+ "page_idx": 21,
+ "page_size": [
+ 612,
+ 792
+ ],
+ "discarded_blocks": [
+ {
+ "type": "discarded",
+ "bbox": [
+ 106,
+ 701,
+ 505,
+ 732
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 114,
+ 697,
+ 506,
+ 716
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 114,
+ 697,
+ 197,
+ 716
+ ],
+ "score": 1.0,
+ "content": "10A symmetric matrix",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 197,
+ 702,
+ 205,
+ 711
+ ],
+ "score": 0.76,
+ "content": "A",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 206,
+ 697,
+ 328,
+ 716
+ ],
+ "score": 1.0,
+ "content": "is called irreducible if and only if",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 329,
+ 702,
+ 343,
+ 711
+ ],
+ "score": 0.89,
+ "content": "G _ { A }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 343,
+ 697,
+ 450,
+ 716
+ ],
+ "score": 1.0,
+ "content": "is connected. We say a graph",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 451,
+ 702,
+ 459,
+ 711
+ ],
+ "score": 0.81,
+ "content": "G",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 459,
+ 697,
+ 506,
+ 716
+ ],
+ "score": 1.0,
+ "content": "is aperiodic",
+ "type": "text"
+ }
+ ]
+ },
+ {
+ "bbox": [
+ 104,
+ 710,
+ 505,
+ 724
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 710,
+ 305,
+ 724
+ ],
+ "score": 1.0,
+ "content": "if the greatest common divisor of length of all loops in",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 306,
+ 712,
+ 314,
+ 721
+ ],
+ "score": 0.83,
+ "content": "G",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 314,
+ 710,
+ 408,
+ 724
+ ],
+ "score": 1.0,
+ "content": "is 1. A symmetric matrix",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 408,
+ 712,
+ 416,
+ 720
+ ],
+ "score": 0.81,
+ "content": "A",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 416,
+ 710,
+ 505,
+ 724
+ ],
+ "score": 1.0,
+ "content": "is aperiodic if the graph",
+ "type": "text"
+ }
+ ]
+ },
+ {
+ "bbox": [
+ 106,
+ 721,
+ 219,
+ 732
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 721,
+ 121,
+ 731
+ ],
+ "score": 0.87,
+ "content": "G _ { A }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 121,
+ 721,
+ 164,
+ 732
+ ],
+ "score": 1.0,
+ "content": "induced by",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 164,
+ 722,
+ 172,
+ 730
+ ],
+ "score": 0.8,
+ "content": "A",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 172,
+ 721,
+ 219,
+ 732
+ ],
+ "score": 1.0,
+ "content": "is aperiodic.",
+ "type": "text"
+ }
+ ]
+ }
+ ]
+ },
+ {
+ "type": "discarded",
+ "bbox": [
+ 107,
+ 26,
+ 294,
+ 38
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 25,
+ 294,
+ 38
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 25,
+ 294,
+ 38
+ ],
+ "score": 1.0,
+ "content": "Published as a conference paper at ICLR 2020",
+ "type": "text"
+ }
+ ]
+ }
+ ]
+ },
+ {
+ "type": "discarded",
+ "bbox": [
+ 300,
+ 751,
+ 311,
+ 760
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 298,
+ 750,
+ 313,
+ 765
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 298,
+ 750,
+ 313,
+ 765
+ ],
+ "score": 1.0,
+ "content": "",
+ "type": "text",
+ "height": 15,
+ "width": 15
+ }
+ ]
+ }
+ ]
+ }
+ ],
+ "para_blocks": [
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 105,
+ 505,
+ 146
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 105,
+ 506,
+ 120
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 105,
+ 232,
+ 120
+ ],
+ "score": 1.0,
+ "content": "Proof. By definition, we have",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 233,
+ 106,
+ 298,
+ 119
+ ],
+ "score": 0.93,
+ "content": "d _ { \\mathcal { M } } ( X ) = | x _ { 2 } |",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 298,
+ 105,
+ 506,
+ 120
+ ],
+ "score": 1.0,
+ "content": ". On the other hand, direct calculation shows that",
+ "type": "text"
+ }
+ ],
+ "index": 0
+ },
+ {
+ "bbox": [
+ 107,
+ 119,
+ 506,
+ 134
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 119,
+ 260,
+ 134
+ ],
+ "score": 0.86,
+ "content": "f _ { l } ( X ) = \\left[ ( W \\mu X _ { 1 } ) ^ { + } \\quad ( W \\lambda X _ { 2 } ) ^ { + } \\right] ^ { \\top }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 260,
+ 119,
+ 279,
+ 134
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 280,
+ 120,
+ 387,
+ 133
+ ],
+ "score": 0.91,
+ "content": "d _ { \\mathcal { M } } ( f _ { l } ( X ) ) = ( W \\lambda X _ { 2 } ) ^ { + }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 387,
+ 119,
+ 416,
+ 134
+ ],
+ "score": 1.0,
+ "content": "where",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 416,
+ 120,
+ 489,
+ 133
+ ],
+ "score": 0.91,
+ "content": "x ^ { + } : = \\operatorname* { m a x } ( x , 0 )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 489,
+ 119,
+ 506,
+ 134
+ ],
+ "score": 1.0,
+ "content": "for",
+ "type": "text"
+ }
+ ],
+ "index": 1
+ },
+ {
+ "bbox": [
+ 107,
+ 132,
+ 506,
+ 147
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 134,
+ 132,
+ 144
+ ],
+ "score": 0.87,
+ "content": "x \\in \\mathbb { R }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 133,
+ 132,
+ 162,
+ 147
+ ],
+ "score": 1.0,
+ "content": ". Since",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 162,
+ 133,
+ 203,
+ 144
+ ],
+ "score": 0.89,
+ "content": "W > \\lambda ^ { - 1 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 204,
+ 132,
+ 221,
+ 147
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 222,
+ 135,
+ 251,
+ 145
+ ],
+ "score": 0.89,
+ "content": "x _ { 2 } > 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 252,
+ 132,
+ 290,
+ 147
+ ],
+ "score": 1.0,
+ "content": ", we have",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 290,
+ 134,
+ 384,
+ 146
+ ],
+ "score": 0.91,
+ "content": "d _ { \\mathcal { M } } ( f _ { l } ( X ) ) > d _ { \\mathcal { M } } ( X )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 385,
+ 132,
+ 389,
+ 147
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 494,
+ 134,
+ 506,
+ 145
+ ],
+ "score": 0.999,
+ "content": "□",
+ "type": "text"
+ }
+ ],
+ "index": 2
+ }
+ ],
+ "index": 1,
+ "bbox_fs": [
+ 106,
+ 105,
+ 506,
+ 147
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 157,
+ 506,
+ 191
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 157,
+ 505,
+ 169
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 157,
+ 505,
+ 169
+ ],
+ "score": 1.0,
+ "content": "Next, we prove the non-strictness of Theorem 1 in the sense that there exists a situation in which",
+ "type": "text"
+ }
+ ],
+ "index": 3
+ },
+ {
+ "bbox": [
+ 107,
+ 168,
+ 506,
+ 181
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 169,
+ 140,
+ 180
+ ],
+ "score": 0.9,
+ "content": "s _ { l } \\lambda > 1",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 141,
+ 168,
+ 234,
+ 181
+ ],
+ "score": 1.0,
+ "content": "holds and the distance",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 234,
+ 169,
+ 250,
+ 180
+ ],
+ "score": 0.9,
+ "content": "d _ { \\mathcal { M } }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 250,
+ 168,
+ 348,
+ 181
+ ],
+ "score": 1.0,
+ "content": "uniformly decreases by",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 348,
+ 169,
+ 357,
+ 180
+ ],
+ "score": 0.84,
+ "content": "f _ { l }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 357,
+ 168,
+ 435,
+ 181
+ ],
+ "score": 1.0,
+ "content": ". Again, we set Set",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 435,
+ 169,
+ 467,
+ 179
+ ],
+ "score": 0.85,
+ "content": "N \\gets 2",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 467,
+ 168,
+ 470,
+ 181
+ ],
+ "score": 1.0,
+ "content": ",",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 471,
+ 169,
+ 501,
+ 179
+ ],
+ "score": 0.83,
+ "content": "C \\gets 1",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 501,
+ 168,
+ 506,
+ 181
+ ],
+ "score": 1.0,
+ "content": ",",
+ "type": "text"
+ }
+ ],
+ "index": 4
+ },
+ {
+ "bbox": [
+ 106,
+ 180,
+ 249,
+ 191
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 180,
+ 123,
+ 191
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 124,
+ 180,
+ 156,
+ 190
+ ],
+ "score": 0.89,
+ "content": "M \\gets 1",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 156,
+ 180,
+ 176,
+ 191
+ ],
+ "score": 1.0,
+ "content": ". Let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 176,
+ 180,
+ 217,
+ 191
+ ],
+ "score": 0.93,
+ "content": "\\lambda \\in ( 1 , 2 )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 217,
+ 180,
+ 249,
+ 191
+ ],
+ "score": 1.0,
+ "content": "and set",
+ "type": "text"
+ }
+ ],
+ "index": 5
+ }
+ ],
+ "index": 4,
+ "bbox_fs": [
+ 106,
+ 157,
+ 506,
+ 191
+ ]
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 188,
+ 196,
+ 423,
+ 224
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 188,
+ 196,
+ 423,
+ 224
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 188,
+ 196,
+ 423,
+ 224
+ ],
+ "score": 0.94,
+ "content": "P \\frac { \\lambda } { 2 } [ \\begin{array} { c c } { 1 } & { - 1 } \\\\ { - 1 } & { 1 } \\end{array} ] , e \\frac { 1 } { \\sqrt { 2 } } [ \\begin{array} { c } { 1 } \\\\ { 1 } \\end{array} ] , U \\{ [ \\begin{array} { c } { x } \\\\ { y } \\end{array} ] | x = y \\}",
+ "type": "interline_equation",
+ "image_path": "087c9606e481a961a37b1e1babb22cba1730aec8a623132d768567bf6bb36c81.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 6,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 188,
+ 196,
+ 423,
+ 224
+ ],
+ "spans": [],
+ "index": 6
+ }
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 105,
+ 228,
+ 494,
+ 240
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 227,
+ 495,
+ 242
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 227,
+ 267,
+ 242
+ ],
+ "score": 1.0,
+ "content": "Then, we can directly show that 3-tuple",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 267,
+ 228,
+ 303,
+ 241
+ ],
+ "score": 0.93,
+ "content": "( P , e , U )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 304,
+ 227,
+ 458,
+ 242
+ ],
+ "score": 1.0,
+ "content": "satisfies the Assumptions 1 and 2. Set",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 459,
+ 229,
+ 491,
+ 239
+ ],
+ "score": 0.88,
+ "content": "W 1",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 491,
+ 227,
+ 495,
+ 242
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 7
+ }
+ ],
+ "index": 7,
+ "bbox_fs": [
+ 106,
+ 227,
+ 495,
+ 242
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 243,
+ 421,
+ 256
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 104,
+ 241,
+ 421,
+ 257
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 241,
+ 206,
+ 257
+ ],
+ "score": 1.0,
+ "content": "Proposition 5. We have",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 207,
+ 243,
+ 242,
+ 254
+ ],
+ "score": 0.9,
+ "content": "W \\lambda > 1",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 243,
+ 241,
+ 261,
+ 257
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 261,
+ 243,
+ 355,
+ 255
+ ],
+ "score": 0.91,
+ "content": "d _ { \\mathcal { M } } ( f _ { l } ( X ) ) < d _ { \\mathcal { M } } ( X )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 356,
+ 241,
+ 383,
+ 257
+ ],
+ "score": 1.0,
+ "content": "for all",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 383,
+ 243,
+ 417,
+ 254
+ ],
+ "score": 0.9,
+ "content": "X \\in \\mathbb { R } ^ { 2 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 417,
+ 241,
+ 421,
+ 257
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 8
+ }
+ ],
+ "index": 8,
+ "bbox_fs": [
+ 104,
+ 241,
+ 421,
+ 257
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 267,
+ 505,
+ 307
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 104,
+ 267,
+ 506,
+ 285
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 267,
+ 197,
+ 285
+ ],
+ "score": 1.0,
+ "content": "Proof. First, note that",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 198,
+ 267,
+ 277,
+ 285
+ ],
+ "score": 0.9,
+ "content": "\\begin{array} { r } { e ^ { \\prime } : = \\frac { 1 } { \\sqrt { 2 } } \\left[ { \\begin{array} { l l } { 1 } & { - 1 } \\end{array} } \\right] ^ { \\top } } \\end{array}",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 277,
+ 267,
+ 363,
+ 285
+ ],
+ "score": 1.0,
+ "content": "is the eigenvector of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 363,
+ 271,
+ 372,
+ 280
+ ],
+ "score": 0.84,
+ "content": "P",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 372,
+ 267,
+ 427,
+ 285
+ ],
+ "score": 1.0,
+ "content": "associated to",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 428,
+ 270,
+ 483,
+ 281
+ ],
+ "score": 0.9,
+ "content": "\\lambda \\colon P e ^ { \\prime } = \\lambda e ^ { \\prime }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 484,
+ 267,
+ 506,
+ 285
+ ],
+ "score": 1.0,
+ "content": ". For",
+ "type": "text"
+ }
+ ],
+ "index": 9
+ },
+ {
+ "bbox": [
+ 106,
+ 282,
+ 505,
+ 298
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 284,
+ 205,
+ 296
+ ],
+ "score": 0.86,
+ "content": "X = a e + b e ^ { \\prime } \\left( a , b { \\dot { \\omega } } 0 \\right)",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 205,
+ 282,
+ 296,
+ 298
+ ],
+ "score": 1.0,
+ "content": ", the distance between",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 297,
+ 284,
+ 307,
+ 294
+ ],
+ "score": 0.83,
+ "content": "X",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 307,
+ 282,
+ 326,
+ 298
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 326,
+ 284,
+ 339,
+ 294
+ ],
+ "score": 0.83,
+ "content": "\\mathcal { M }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 339,
+ 282,
+ 350,
+ 298
+ ],
+ "score": 1.0,
+ "content": "is",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 351,
+ 284,
+ 408,
+ 296
+ ],
+ "score": 0.93,
+ "content": "d _ { \\mathcal { M } } ( \\boldsymbol { X } ) = | b |",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 408,
+ 282,
+ 505,
+ 298
+ ],
+ "score": 1.0,
+ "content": ". On the other hand, by",
+ "type": "text"
+ }
+ ],
+ "index": 10
+ },
+ {
+ "bbox": [
+ 106,
+ 295,
+ 222,
+ 307
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 295,
+ 222,
+ 307
+ ],
+ "score": 1.0,
+ "content": "direct computation, we have",
+ "type": "text"
+ }
+ ],
+ "index": 11
+ }
+ ],
+ "index": 10,
+ "bbox_fs": [
+ 104,
+ 267,
+ 506,
+ 307
+ ]
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 204,
+ 311,
+ 406,
+ 357
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 204,
+ 311,
+ 406,
+ 357
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 204,
+ 311,
+ 406,
+ 357
+ ],
+ "score": 0.94,
+ "content": "f ( X ) = \\sigma ( P X W ) = { \\left\\{ \\begin{array} { l l } { \\left[ 0 \\quad { \\frac { \\lambda b } { \\sqrt { 2 } } } \\right] ^ { \\top } } & { { \\mathrm { ( i f ~ } } b \\geq 0 { \\mathrm { ) } } , } \\\\ { \\left[ { \\frac { - \\lambda b } { \\sqrt { 2 } } } \\quad 0 \\right] ^ { \\top } } & { { \\mathrm { ( i f ~ } } b < 0 { \\mathrm { ) } } . } \\end{array} \\right. }",
+ "type": "interline_equation",
+ "image_path": "a159a4cc7dd5ac398b3cff74136fefb345275fef911e7b7089e5ae12ee2d25ab.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 13,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 204,
+ 311,
+ 406,
+ 326.3333333333333
+ ],
+ "spans": [],
+ "index": 12
+ },
+ {
+ "bbox": [
+ 204,
+ 326.3333333333333,
+ 406,
+ 341.66666666666663
+ ],
+ "spans": [],
+ "index": 13
+ },
+ {
+ "bbox": [
+ 204,
+ 341.66666666666663,
+ 406,
+ 356.99999999999994
+ ],
+ "spans": [],
+ "index": 14
+ }
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 361,
+ 504,
+ 385
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 361,
+ 505,
+ 374
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 361,
+ 241,
+ 374
+ ],
+ "score": 1.0,
+ "content": "Therefore, the distance between",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 241,
+ 362,
+ 265,
+ 373
+ ],
+ "score": 0.9,
+ "content": "f ( X )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 265,
+ 361,
+ 285,
+ 374
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 285,
+ 362,
+ 299,
+ 372
+ ],
+ "score": 0.84,
+ "content": "\\mathcal { M }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 299,
+ 361,
+ 311,
+ 374
+ ],
+ "score": 1.0,
+ "content": "is",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 312,
+ 361,
+ 401,
+ 374
+ ],
+ "score": 0.93,
+ "content": "d _ { \\mathcal { M } } ( f ( X ) ) = \\lambda | b | / 2",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 401,
+ 361,
+ 434,
+ 374
+ ],
+ "score": 1.0,
+ "content": ". Since",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 434,
+ 362,
+ 464,
+ 372
+ ],
+ "score": 0.88,
+ "content": "\\lambda \\ : < \\ : 2",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 464,
+ 361,
+ 505,
+ 374
+ ],
+ "score": 1.0,
+ "content": ", we have",
+ "type": "text"
+ }
+ ],
+ "index": 15
+ },
+ {
+ "bbox": [
+ 107,
+ 372,
+ 504,
+ 385
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 373,
+ 198,
+ 385
+ ],
+ "score": 0.92,
+ "content": "d _ { \\mathcal { M } } ( f ( X ) ) < d _ { \\mathcal { M } } ( X )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 199,
+ 372,
+ 230,
+ 385
+ ],
+ "score": 1.0,
+ "content": "for any",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 231,
+ 373,
+ 264,
+ 383
+ ],
+ "score": 0.9,
+ "content": "X \\in \\mathbb { R } ^ { 2 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 264,
+ 372,
+ 269,
+ 385
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 496,
+ 375,
+ 504,
+ 382
+ ],
+ "score": 0.999,
+ "content": "□",
+ "type": "text"
+ }
+ ],
+ "index": 16
+ }
+ ],
+ "index": 15.5,
+ "bbox_fs": [
+ 106,
+ 361,
+ 505,
+ 385
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 396,
+ 505,
+ 419
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 395,
+ 506,
+ 410
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 395,
+ 275,
+ 410
+ ],
+ "score": 1.0,
+ "content": "We have shown that the non-negativity of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 276,
+ 399,
+ 281,
+ 406
+ ],
+ "score": 0.73,
+ "content": "e",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 281,
+ 395,
+ 506,
+ 410
+ ],
+ "score": 1.0,
+ "content": "(Assumption 1) is not a redundant condition in Section",
+ "type": "text"
+ }
+ ],
+ "index": 17
+ },
+ {
+ "bbox": [
+ 104,
+ 406,
+ 126,
+ 421
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 406,
+ 126,
+ 421
+ ],
+ "score": 1.0,
+ "content": "6.1.",
+ "type": "text"
+ }
+ ],
+ "index": 18
+ }
+ ],
+ "index": 17.5,
+ "bbox_fs": [
+ 104,
+ 395,
+ 506,
+ 421
+ ]
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 107,
+ 435,
+ 297,
+ 448
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 434,
+ 298,
+ 450
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 434,
+ 298,
+ 450
+ ],
+ "score": 1.0,
+ "content": "F RELATION TO MARKOV PROCESS",
+ "type": "text"
+ }
+ ],
+ "index": 19
+ }
+ ],
+ "index": 19
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 460,
+ 505,
+ 493
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 104,
+ 459,
+ 505,
+ 473
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 459,
+ 505,
+ 473
+ ],
+ "score": 1.0,
+ "content": "It is known that any Markov process on finite states converges to a unique distribution (equilibrium)",
+ "type": "text"
+ }
+ ],
+ "index": 20
+ },
+ {
+ "bbox": [
+ 105,
+ 470,
+ 505,
+ 484
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 470,
+ 505,
+ 484
+ ],
+ "score": 1.0,
+ "content": "if it is irreducible and aperiodic (see, e.g., Norris (1998)). As we see in this section, this theorem is",
+ "type": "text"
+ }
+ ],
+ "index": 21
+ },
+ {
+ "bbox": [
+ 106,
+ 482,
+ 232,
+ 495
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 482,
+ 232,
+ 495
+ ],
+ "score": 1.0,
+ "content": "the special case of Corollary 3.",
+ "type": "text"
+ }
+ ],
+ "index": 22
+ }
+ ],
+ "index": 21,
+ "bbox_fs": [
+ 104,
+ 459,
+ 505,
+ 495
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 498,
+ 505,
+ 576
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 104,
+ 497,
+ 506,
+ 512
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 497,
+ 122,
+ 512
+ ],
+ "score": 1.0,
+ "content": "Let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 122,
+ 498,
+ 192,
+ 511
+ ],
+ "score": 0.93,
+ "content": "S : = \\{ 1 , \\ldots , N \\}",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 192,
+ 497,
+ 440,
+ 512
+ ],
+ "score": 1.0,
+ "content": "be a finite discrete state space. Consider a Markov process on",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 441,
+ 500,
+ 448,
+ 509
+ ],
+ "score": 0.82,
+ "content": "S",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 449,
+ 497,
+ 506,
+ 512
+ ],
+ "score": 1.0,
+ "content": "characterized",
+ "type": "text"
+ }
+ ],
+ "index": 23
+ },
+ {
+ "bbox": [
+ 102,
+ 508,
+ 507,
+ 524
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 102,
+ 508,
+ 240,
+ 524
+ ],
+ "score": 1.0,
+ "content": "by a symmetric transition matrix",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 240,
+ 510,
+ 348,
+ 523
+ ],
+ "score": 0.92,
+ "content": "P = ( p _ { i j } ) _ { i , j \\in [ N ] } \\in \\mathbb { R } ^ { N \\times N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 349,
+ 508,
+ 388,
+ 524
+ ],
+ "score": 1.0,
+ "content": "such that",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 389,
+ 510,
+ 416,
+ 521
+ ],
+ "score": 0.91,
+ "content": "P \\geq 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 417,
+ 508,
+ 434,
+ 524
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 434,
+ 510,
+ 468,
+ 520
+ ],
+ "score": 0.91,
+ "content": "P \\mathbf { 1 } = \\mathbf { 1 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 469,
+ 508,
+ 507,
+ 524
+ ],
+ "score": 1.0,
+ "content": "where 1",
+ "type": "text"
+ }
+ ],
+ "index": 24
+ },
+ {
+ "bbox": [
+ 104,
+ 520,
+ 506,
+ 534
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 520,
+ 244,
+ 534
+ ],
+ "score": 1.0,
+ "content": "is the all-one vector. We interpret",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 244,
+ 523,
+ 258,
+ 533
+ ],
+ "score": 0.86,
+ "content": "p _ { i j }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 258,
+ 520,
+ 423,
+ 534
+ ],
+ "score": 1.0,
+ "content": "as the transition probability from a state",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 423,
+ 522,
+ 428,
+ 531
+ ],
+ "score": 0.71,
+ "content": "i",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 428,
+ 520,
+ 439,
+ 534
+ ],
+ "score": 1.0,
+ "content": "to",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 439,
+ 522,
+ 446,
+ 532
+ ],
+ "score": 0.79,
+ "content": "j",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 446,
+ 520,
+ 506,
+ 534
+ ],
+ "score": 1.0,
+ "content": ". We associate",
+ "type": "text"
+ }
+ ],
+ "index": 25
+ },
+ {
+ "bbox": [
+ 107,
+ 531,
+ 504,
+ 545
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 532,
+ 116,
+ 542
+ ],
+ "score": 0.8,
+ "content": "P",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 116,
+ 531,
+ 172,
+ 545
+ ],
+ "score": 1.0,
+ "content": "with a graph",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 172,
+ 532,
+ 242,
+ 544
+ ],
+ "score": 0.93,
+ "content": "G _ { P } = ( V _ { P } , E _ { P } )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 242,
+ 531,
+ 257,
+ 545
+ ],
+ "score": 1.0,
+ "content": "by",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 257,
+ 532,
+ 301,
+ 544
+ ],
+ "score": 0.92,
+ "content": "V _ { P } = [ N ]",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 301,
+ 531,
+ 321,
+ 545
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 321,
+ 532,
+ 370,
+ 544
+ ],
+ "score": 0.92,
+ "content": "( i , j ) \\in E _ { P }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 370,
+ 531,
+ 429,
+ 545
+ ],
+ "score": 1.0,
+ "content": "if and only if",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 429,
+ 533,
+ 463,
+ 544
+ ],
+ "score": 0.92,
+ "content": "p _ { i j } > 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 463,
+ 531,
+ 495,
+ 545
+ ],
+ "score": 1.0,
+ "content": ". Since",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 495,
+ 532,
+ 504,
+ 542
+ ],
+ "score": 0.81,
+ "content": "P",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 26
+ },
+ {
+ "bbox": [
+ 105,
+ 542,
+ 505,
+ 555
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 542,
+ 221,
+ 555
+ ],
+ "score": 1.0,
+ "content": "is symmetric, we can regard",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 222,
+ 543,
+ 237,
+ 554
+ ],
+ "score": 0.87,
+ "content": "G _ { P }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 237,
+ 542,
+ 383,
+ 555
+ ],
+ "score": 1.0,
+ "content": "as an undirected graph. We assume",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 383,
+ 543,
+ 392,
+ 553
+ ],
+ "score": 0.84,
+ "content": "P",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 392,
+ 542,
+ 505,
+ 555
+ ],
+ "score": 1.0,
+ "content": "is irreducible and aperiodic",
+ "type": "text"
+ }
+ ],
+ "index": 27
+ },
+ {
+ "bbox": [
+ 104,
+ 550,
+ 507,
+ 569
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 550,
+ 392,
+ 569
+ ],
+ "score": 1.0,
+ "content": "10. Perron – Frobenius’ theorem (see, e.g., Meyer (2000)) implies that",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 392,
+ 554,
+ 401,
+ 564
+ ],
+ "score": 0.84,
+ "content": "P",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 401,
+ 550,
+ 507,
+ 569
+ ],
+ "score": 1.0,
+ "content": "satisfy the assumption of",
+ "type": "text"
+ }
+ ],
+ "index": 28
+ },
+ {
+ "bbox": [
+ 106,
+ 564,
+ 210,
+ 578
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 564,
+ 174,
+ 578
+ ],
+ "score": 1.0,
+ "content": "Corollary 3 with",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 175,
+ 565,
+ 205,
+ 575
+ ],
+ "score": 0.9,
+ "content": "M = 1",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 205,
+ 564,
+ 210,
+ 578
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 29
+ }
+ ],
+ "index": 26,
+ "bbox_fs": [
+ 102,
+ 497,
+ 507,
+ 578
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 579,
+ 505,
+ 613
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 578,
+ 505,
+ 592
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 578,
+ 349,
+ 592
+ ],
+ "score": 1.0,
+ "content": "Proposition 6 (Perron – Frobenius). Let the eigenvalues of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 349,
+ 580,
+ 358,
+ 590
+ ],
+ "score": 0.75,
+ "content": "P",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 359,
+ 578,
+ 372,
+ 592
+ ],
+ "score": 1.0,
+ "content": "be",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 372,
+ 580,
+ 438,
+ 591
+ ],
+ "score": 0.88,
+ "content": "\\lambda _ { 1 } \\leq \\cdots \\leq \\lambda _ { N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 438,
+ 578,
+ 505,
+ 592
+ ],
+ "score": 1.0,
+ "content": ". Then, we have",
+ "type": "text"
+ }
+ ],
+ "index": 30
+ },
+ {
+ "bbox": [
+ 107,
+ 590,
+ 505,
+ 603
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 591,
+ 145,
+ 602
+ ],
+ "score": 0.67,
+ "content": "- 1 < \\lambda _ { 1 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 146,
+ 590,
+ 150,
+ 603
+ ],
+ "score": 1.0,
+ "content": ",",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 150,
+ 591,
+ 195,
+ 603
+ ],
+ "score": 0.66,
+ "content": "\\lambda _ { N - 1 } < 1",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 195,
+ 590,
+ 217,
+ 603
+ ],
+ "score": 1.0,
+ "content": ", and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 217,
+ 591,
+ 252,
+ 602
+ ],
+ "score": 0.91,
+ "content": "\\lambda _ { N } = 1",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 252,
+ 590,
+ 399,
+ 603
+ ],
+ "score": 1.0,
+ "content": ". Further, there exists unique vector",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 399,
+ 591,
+ 434,
+ 601
+ ],
+ "score": 0.91,
+ "content": "e \\in \\mathbb { R } ^ { N }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 434,
+ 590,
+ 475,
+ 603
+ ],
+ "score": 1.0,
+ "content": "such that",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 475,
+ 591,
+ 501,
+ 602
+ ],
+ "score": 0.88,
+ "content": "e \\geq 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 501,
+ 590,
+ 505,
+ 603
+ ],
+ "score": 1.0,
+ "content": ",",
+ "type": "text"
+ }
+ ],
+ "index": 31
+ },
+ {
+ "bbox": [
+ 107,
+ 601,
+ 325,
+ 615
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 602,
+ 140,
+ 614
+ ],
+ "score": 0.91,
+ "content": "\\| e \\| = 1",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 140,
+ 601,
+ 162,
+ 615
+ ],
+ "score": 1.0,
+ "content": ", and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 162,
+ 604,
+ 168,
+ 612
+ ],
+ "score": 0.46,
+ "content": "e",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 168,
+ 601,
+ 325,
+ 615
+ ],
+ "score": 1.0,
+ "content": "is the eigenvector for the eivenvalue 1.",
+ "type": "text"
+ }
+ ],
+ "index": 32
+ }
+ ],
+ "index": 31,
+ "bbox_fs": [
+ 105,
+ 578,
+ 505,
+ 615
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 104,
+ 615,
+ 503,
+ 640
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 614,
+ 505,
+ 630
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 614,
+ 178,
+ 630
+ ],
+ "score": 1.0,
+ "content": "Corollary 5. Let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 179,
+ 616,
+ 307,
+ 628
+ ],
+ "score": 0.9,
+ "content": "\\begin{array} { r } { \\lambda : = \\operatorname* { m a x } _ { n = 1 , \\ldots , N - 1 } | \\lambda _ { n } | ( < 1 ) } \\end{array}",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 308,
+ 614,
+ 327,
+ 630
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 327,
+ 615,
+ 434,
+ 628
+ ],
+ "score": 0.9,
+ "content": "\\mathcal { M } : = \\{ e \\otimes w \\ | \\ w \\in \\mathbb { R } ^ { C } \\}",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 434,
+ 614,
+ 448,
+ 630
+ ],
+ "score": 1.0,
+ "content": ". If",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 448,
+ 617,
+ 479,
+ 628
+ ],
+ "score": 0.83,
+ "content": "s \\lambda < 1",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 479,
+ 614,
+ 505,
+ 630
+ ],
+ "score": 1.0,
+ "content": ", then,",
+ "type": "text"
+ }
+ ],
+ "index": 33
+ },
+ {
+ "bbox": [
+ 104,
+ 627,
+ 381,
+ 640
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 627,
+ 186,
+ 640
+ ],
+ "score": 1.0,
+ "content": "for any initial value",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 186,
+ 628,
+ 199,
+ 639
+ ],
+ "score": 0.8,
+ "content": "X _ { 1 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 199,
+ 627,
+ 203,
+ 640
+ ],
+ "score": 1.0,
+ "content": ",",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 203,
+ 628,
+ 216,
+ 639
+ ],
+ "score": 0.81,
+ "content": "X _ { l }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 216,
+ 627,
+ 321,
+ 640
+ ],
+ "score": 1.0,
+ "content": "exponentially approaches",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 322,
+ 628,
+ 334,
+ 637
+ ],
+ "score": 0.58,
+ "content": "\\mathcal { M }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 335,
+ 627,
+ 347,
+ 640
+ ],
+ "score": 1.0,
+ "content": "as",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 347,
+ 629,
+ 376,
+ 637
+ ],
+ "score": 0.8,
+ "content": "l \\infty",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 377,
+ 627,
+ 381,
+ 640
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 34
+ }
+ ],
+ "index": 33.5,
+ "bbox_fs": [
+ 104,
+ 614,
+ 505,
+ 640
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 648,
+ 505,
+ 694
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 104,
+ 646,
+ 505,
+ 661
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 646,
+ 145,
+ 661
+ ],
+ "score": 1.0,
+ "content": "If we set",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 145,
+ 648,
+ 175,
+ 658
+ ],
+ "score": 0.9,
+ "content": "C = 1",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 175,
+ 646,
+ 194,
+ 661
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 194,
+ 648,
+ 228,
+ 659
+ ],
+ "score": 0.92,
+ "content": "W _ { l } = 1",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 228,
+ 646,
+ 258,
+ 661
+ ],
+ "score": 1.0,
+ "content": "for all",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 258,
+ 648,
+ 290,
+ 660
+ ],
+ "score": 0.91,
+ "content": "l \\in \\mathbb { N } _ { + }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 290,
+ 646,
+ 438,
+ 661
+ ],
+ "score": 1.0,
+ "content": ", then, we can inductively show that",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 439,
+ 648,
+ 472,
+ 659
+ ],
+ "score": 0.91,
+ "content": "X _ { l } \\geq 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 472,
+ 646,
+ 505,
+ 661
+ ],
+ "score": 1.0,
+ "content": "for any",
+ "type": "text"
+ }
+ ],
+ "index": 35
+ },
+ {
+ "bbox": [
+ 106,
+ 659,
+ 505,
+ 671
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 659,
+ 131,
+ 670
+ ],
+ "score": 0.9,
+ "content": "l \\geq 2",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 131,
+ 659,
+ 249,
+ 671
+ ],
+ "score": 1.0,
+ "content": ". Therefore, we can interpret",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 250,
+ 660,
+ 262,
+ 670
+ ],
+ "score": 0.86,
+ "content": "X _ { l }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 262,
+ 659,
+ 331,
+ 671
+ ],
+ "score": 1.0,
+ "content": "as a measure on",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 332,
+ 660,
+ 339,
+ 669
+ ],
+ "score": 0.79,
+ "content": "S",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 340,
+ 659,
+ 505,
+ 671
+ ],
+ "score": 1.0,
+ "content": ". Suppose further that we take the initial",
+ "type": "text"
+ }
+ ],
+ "index": 36
+ },
+ {
+ "bbox": [
+ 105,
+ 669,
+ 505,
+ 683
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 669,
+ 131,
+ 683
+ ],
+ "score": 1.0,
+ "content": "value",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 131,
+ 671,
+ 145,
+ 681
+ ],
+ "score": 0.89,
+ "content": "X _ { 1 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 145,
+ 669,
+ 159,
+ 683
+ ],
+ "score": 1.0,
+ "content": "as",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 159,
+ 670,
+ 195,
+ 681
+ ],
+ "score": 0.92,
+ "content": "X _ { 1 } ~ \\geq ~ 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 195,
+ 669,
+ 214,
+ 683
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 214,
+ 669,
+ 259,
+ 682
+ ],
+ "score": 0.91,
+ "content": "X _ { 1 } ^ { \\top } \\mathbf { \\hat { 1 } } = 1",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 259,
+ 669,
+ 361,
+ 683
+ ],
+ "score": 1.0,
+ "content": "so that we can interpret",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 361,
+ 671,
+ 375,
+ 681
+ ],
+ "score": 0.88,
+ "content": "X _ { 1 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 375,
+ 669,
+ 505,
+ 683
+ ],
+ "score": 1.0,
+ "content": "as a probability distribution on",
+ "type": "text"
+ }
+ ],
+ "index": 37
+ },
+ {
+ "bbox": [
+ 106,
+ 680,
+ 505,
+ 695
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 682,
+ 114,
+ 691
+ ],
+ "score": 0.79,
+ "content": "S",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 115,
+ 680,
+ 265,
+ 695
+ ],
+ "score": 1.0,
+ "content": ". Then, we can inductively show that",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 266,
+ 681,
+ 298,
+ 693
+ ],
+ "score": 0.87,
+ "content": "X _ { l } \\geq 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 298,
+ 680,
+ 301,
+ 695
+ ],
+ "score": 1.0,
+ "content": ",",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 302,
+ 680,
+ 344,
+ 694
+ ],
+ "score": 0.89,
+ "content": "X _ { l } ^ { \\top } \\mathbf { 1 } = 1",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 344,
+ 680,
+ 366,
+ 695
+ ],
+ "score": 1.0,
+ "content": "(i.e.,",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 366,
+ 682,
+ 379,
+ 692
+ ],
+ "score": 0.89,
+ "content": "X _ { l }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 379,
+ 680,
+ 505,
+ 695
+ ],
+ "score": 1.0,
+ "content": "is a probability distribution on",
+ "type": "text"
+ }
+ ],
+ "index": 38
+ }
+ ],
+ "index": 36.5,
+ "bbox_fs": [
+ 104,
+ 646,
+ 505,
+ 695
+ ]
+ }
+ ]
+ },
+ {
+ "preproc_blocks": [
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 81,
+ 505,
+ 127
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 82,
+ 505,
+ 95
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 83,
+ 115,
+ 93
+ ],
+ "score": 0.63,
+ "content": "S",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 115,
+ 82,
+ 139,
+ 95
+ ],
+ "score": 1.0,
+ "content": "), and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 139,
+ 82,
+ 255,
+ 95
+ ],
+ "score": 0.92,
+ "content": "X _ { l + 1 } = \\sigma ( P X _ { l } W _ { l } ) = P X _ { l }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 256,
+ 82,
+ 284,
+ 95
+ ],
+ "score": 1.0,
+ "content": "for all",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 285,
+ 83,
+ 316,
+ 94
+ ],
+ "score": 0.93,
+ "content": "l \\in \\mathbb { N } _ { + }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 317,
+ 82,
+ 505,
+ 95
+ ],
+ "score": 1.0,
+ "content": ". In conclusion, the corollary is reduced to the",
+ "type": "text"
+ }
+ ],
+ "index": 0
+ },
+ {
+ "bbox": [
+ 105,
+ 93,
+ 505,
+ 106
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 93,
+ 505,
+ 106
+ ],
+ "score": 1.0,
+ "content": "fact that if a finite and discrete Markov process is irreducible and aperiodic, any initial probability",
+ "type": "text"
+ }
+ ],
+ "index": 1
+ },
+ {
+ "bbox": [
+ 105,
+ 104,
+ 505,
+ 118
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 104,
+ 421,
+ 118
+ ],
+ "score": 1.0,
+ "content": "distribution converges exponentially to an equibrilium. In addition, the the rate",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 422,
+ 105,
+ 429,
+ 114
+ ],
+ "score": 0.75,
+ "content": "\\lambda",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 429,
+ 104,
+ 505,
+ 118
+ ],
+ "score": 1.0,
+ "content": "corresponds to the",
+ "type": "text"
+ }
+ ],
+ "index": 2
+ },
+ {
+ "bbox": [
+ 105,
+ 116,
+ 250,
+ 128
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 116,
+ 250,
+ 128
+ ],
+ "score": 1.0,
+ "content": "mixing time of the Markov process.",
+ "type": "text"
+ }
+ ],
+ "index": 3
+ }
+ ],
+ "index": 1.5
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 107,
+ 142,
+ 362,
+ 155
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 141,
+ 363,
+ 157
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 141,
+ 363,
+ 157
+ ],
+ "score": 1.0,
+ "content": "G GCN DEFINED BY NORMALIZED LAPLACIAN",
+ "type": "text"
+ }
+ ],
+ "index": 4
+ }
+ ],
+ "index": 4
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 165,
+ 505,
+ 268
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 104,
+ 165,
+ 505,
+ 179
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 167,
+ 209,
+ 179
+ ],
+ "score": 1.0,
+ "content": "In Section 4, we defined",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 210,
+ 168,
+ 219,
+ 177
+ ],
+ "score": 0.81,
+ "content": "P",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 219,
+ 167,
+ 399,
+ 179
+ ],
+ "score": 1.0,
+ "content": "using the augmented normalized Laplacian",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 399,
+ 165,
+ 409,
+ 177
+ ],
+ "score": 0.85,
+ "content": "\\tilde { \\Delta }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 409,
+ 167,
+ 424,
+ 179
+ ],
+ "score": 1.0,
+ "content": "by",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 424,
+ 165,
+ 483,
+ 178
+ ],
+ "score": 0.93,
+ "content": "P = I _ { N } - \\tilde { \\Delta }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 483,
+ 167,
+ 505,
+ 179
+ ],
+ "score": 1.0,
+ "content": ". We",
+ "type": "text"
+ }
+ ],
+ "index": 5
+ },
+ {
+ "bbox": [
+ 105,
+ 178,
+ 504,
+ 189
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 178,
+ 322,
+ 189
+ ],
+ "score": 1.0,
+ "content": "can alternatively use the usual normalized Laplacian",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 322,
+ 178,
+ 332,
+ 188
+ ],
+ "score": 0.81,
+ "content": "\\Delta",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 333,
+ 178,
+ 495,
+ 189
+ ],
+ "score": 1.0,
+ "content": "instead of the augmented one to define",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 495,
+ 178,
+ 504,
+ 188
+ ],
+ "score": 0.8,
+ "content": "P",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 6
+ },
+ {
+ "bbox": [
+ 106,
+ 189,
+ 505,
+ 201
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 189,
+ 505,
+ 201
+ ],
+ "score": 1.0,
+ "content": "and want to apply the theory developed in Section 3.2. We write the normalized Laplacian version",
+ "type": "text"
+ }
+ ],
+ "index": 7
+ },
+ {
+ "bbox": [
+ 105,
+ 200,
+ 505,
+ 212
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 200,
+ 118,
+ 212
+ ],
+ "score": 1.0,
+ "content": "as",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 118,
+ 200,
+ 187,
+ 211
+ ],
+ "score": 0.92,
+ "content": "P _ { \\Delta } : = I _ { N } - \\Delta",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 187,
+ 200,
+ 397,
+ 212
+ ],
+ "score": 1.0,
+ "content": ". The only obstacle is that the smallest eigenvalue",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 397,
+ 200,
+ 408,
+ 211
+ ],
+ "score": 0.88,
+ "content": "\\lambda _ { 1 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 409,
+ 200,
+ 422,
+ 212
+ ],
+ "score": 1.0,
+ "content": "of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 423,
+ 200,
+ 437,
+ 211
+ ],
+ "score": 0.89,
+ "content": "P _ { \\Delta }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 438,
+ 200,
+ 505,
+ 212
+ ],
+ "score": 1.0,
+ "content": "can be equal to",
+ "type": "text"
+ }
+ ],
+ "index": 8
+ },
+ {
+ "bbox": [
+ 106,
+ 211,
+ 505,
+ 223
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 212,
+ 120,
+ 222
+ ],
+ "score": 0.78,
+ "content": "- 1",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 121,
+ 211,
+ 178,
+ 223
+ ],
+ "score": 1.0,
+ "content": ", while that of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 178,
+ 212,
+ 187,
+ 221
+ ],
+ "score": 0.82,
+ "content": "P",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 187,
+ 211,
+ 274,
+ 223
+ ],
+ "score": 1.0,
+ "content": "is strictly larger than",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 274,
+ 212,
+ 288,
+ 222
+ ],
+ "score": 0.81,
+ "content": "- 1",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 288,
+ 211,
+ 505,
+ 223
+ ],
+ "score": 1.0,
+ "content": "(see, Proposition 1). This corresponds to that fact the",
+ "type": "text"
+ }
+ ],
+ "index": 9
+ },
+ {
+ "bbox": [
+ 105,
+ 222,
+ 505,
+ 236
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 222,
+ 193,
+ 236
+ ],
+ "score": 1.0,
+ "content": "largest eigenvalue of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 194,
+ 222,
+ 203,
+ 233
+ ],
+ "score": 0.87,
+ "content": "\\tilde { \\Delta }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 204,
+ 222,
+ 368,
+ 236
+ ],
+ "score": 1.0,
+ "content": "is strictly smaller than 2, while that for",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 369,
+ 223,
+ 378,
+ 233
+ ],
+ "score": 0.8,
+ "content": "\\Delta",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 379,
+ 222,
+ 505,
+ 236
+ ],
+ "score": 1.0,
+ "content": "can be 2. It is known that the",
+ "type": "text"
+ }
+ ],
+ "index": 10
+ },
+ {
+ "bbox": [
+ 105,
+ 234,
+ 506,
+ 246
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 234,
+ 191,
+ 246
+ ],
+ "score": 1.0,
+ "content": "largest eigenvalue of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 191,
+ 235,
+ 201,
+ 244
+ ],
+ "score": 0.82,
+ "content": "\\Delta",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 201,
+ 234,
+ 506,
+ 246
+ ],
+ "score": 1.0,
+ "content": "is 2 if and only if the graph has a non-trivial bipartite connected component",
+ "type": "text"
+ }
+ ],
+ "index": 11
+ },
+ {
+ "bbox": [
+ 106,
+ 245,
+ 505,
+ 258
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 245,
+ 180,
+ 258
+ ],
+ "score": 1.0,
+ "content": "(see, e.g., Chung",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 180,
+ 245,
+ 189,
+ 255
+ ],
+ "score": 0.33,
+ "content": "\\&",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 190,
+ 245,
+ 505,
+ 258
+ ],
+ "score": 1.0,
+ "content": "Graham (1997)). Therefore, we can develop a theory using the normalized",
+ "type": "text"
+ }
+ ],
+ "index": 12
+ },
+ {
+ "bbox": [
+ 105,
+ 255,
+ 390,
+ 268
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 255,
+ 377,
+ 268
+ ],
+ "score": 1.0,
+ "content": "Laplacian instead of the augmented one in parallel for such a graph",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 377,
+ 257,
+ 386,
+ 266
+ ],
+ "score": 0.79,
+ "content": "G",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 387,
+ 255,
+ 390,
+ 268
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 13
+ }
+ ],
+ "index": 9
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 272,
+ 505,
+ 372
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 273,
+ 504,
+ 285
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 273,
+ 504,
+ 285
+ ],
+ "score": 1.0,
+ "content": "In Section 5, we characterized the asymptotic behavior of GCN defined by the augmented normal-",
+ "type": "text"
+ }
+ ],
+ "index": 14
+ },
+ {
+ "bbox": [
+ 105,
+ 284,
+ 505,
+ 296
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 284,
+ 505,
+ 296
+ ],
+ "score": 1.0,
+ "content": "ized Laplacian via its spectral distribution (Lemma 6 of Appendix D). We can derive a similar claim",
+ "type": "text"
+ }
+ ],
+ "index": 15
+ },
+ {
+ "bbox": [
+ 105,
+ 294,
+ 505,
+ 308
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 294,
+ 505,
+ 308
+ ],
+ "score": 1.0,
+ "content": "for GCN defined via the normalized Laplacian using the original theorem for the normalized Lapla-",
+ "type": "text"
+ }
+ ],
+ "index": 16
+ },
+ {
+ "bbox": [
+ 105,
+ 306,
+ 505,
+ 319
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 306,
+ 505,
+ 319
+ ],
+ "score": 1.0,
+ "content": "cian in Chung & Radcliffe (2011) (Theorem 7 therein). The normalized Laplacian version of GCN is",
+ "type": "text"
+ }
+ ],
+ "index": 17
+ },
+ {
+ "bbox": [
+ 106,
+ 317,
+ 505,
+ 329
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 317,
+ 505,
+ 329
+ ],
+ "score": 1.0,
+ "content": "advantegeous over the one made from the augmented one because we know its spectral distribution",
+ "type": "text"
+ }
+ ],
+ "index": 18
+ },
+ {
+ "bbox": [
+ 105,
+ 326,
+ 505,
+ 342
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 326,
+ 505,
+ 342
+ ],
+ "score": 1.0,
+ "content": "for broader range of random graphs. For example, Chung & Radcliffe (2011) proved the conver-",
+ "type": "text"
+ }
+ ],
+ "index": 19
+ },
+ {
+ "bbox": [
+ 104,
+ 338,
+ 506,
+ 352
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 338,
+ 506,
+ 352
+ ],
+ "score": 1.0,
+ "content": "gence of the spectral distribution of the normalized Laplacian for Chung-Lu’s model (Chung & Lu,",
+ "type": "text"
+ }
+ ],
+ "index": 20
+ },
+ {
+ "bbox": [
+ 104,
+ 348,
+ 506,
+ 363
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 348,
+ 506,
+ 363
+ ],
+ "score": 1.0,
+ "content": "2002), which includes power law graphs as a special case (see, Theorem 4 of Chung & Radcliffe",
+ "type": "text"
+ }
+ ],
+ "index": 21
+ },
+ {
+ "bbox": [
+ 105,
+ 360,
+ 142,
+ 374
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 360,
+ 142,
+ 374
+ ],
+ "score": 1.0,
+ "content": "(2011)).",
+ "type": "text"
+ }
+ ],
+ "index": 22
+ }
+ ],
+ "index": 18
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 107,
+ 387,
+ 316,
+ 401
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 387,
+ 316,
+ 402
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 387,
+ 316,
+ 402
+ ],
+ "score": 1.0,
+ "content": "H DETAILS OF EXPERIMENT SETTINGS",
+ "type": "text"
+ }
+ ],
+ "index": 23
+ }
+ ],
+ "index": 23
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 108,
+ 412,
+ 260,
+ 423
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 411,
+ 262,
+ 424
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 411,
+ 262,
+ 424
+ ],
+ "score": 1.0,
+ "content": "H.1 EXPERIMENT OF SECTION 6.1",
+ "type": "text"
+ }
+ ],
+ "index": 24
+ }
+ ],
+ "index": 24
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 432,
+ 505,
+ 466
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 433,
+ 505,
+ 445
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 433,
+ 203,
+ 445
+ ],
+ "score": 1.0,
+ "content": "We set the eigenvalue of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 204,
+ 433,
+ 212,
+ 443
+ ],
+ "score": 0.84,
+ "content": "P",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 213,
+ 433,
+ 222,
+ 445
+ ],
+ "score": 1.0,
+ "content": "to",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 223,
+ 433,
+ 260,
+ 444
+ ],
+ "score": 0.91,
+ "content": "\\lambda _ { 1 } = 0 . 5",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 261,
+ 433,
+ 277,
+ 445
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 277,
+ 433,
+ 315,
+ 444
+ ],
+ "score": 0.91,
+ "content": "\\lambda _ { 2 } = 1 . 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 315,
+ 433,
+ 412,
+ 445
+ ],
+ "score": 1.0,
+ "content": "and randomly generated",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 413,
+ 433,
+ 422,
+ 443
+ ],
+ "score": 0.82,
+ "content": "P",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 422,
+ 433,
+ 505,
+ 445
+ ],
+ "score": 1.0,
+ "content": "until the eigenvector",
+ "type": "text"
+ }
+ ],
+ "index": 25
+ },
+ {
+ "bbox": [
+ 107,
+ 443,
+ 504,
+ 456
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 446,
+ 113,
+ 454
+ ],
+ "score": 0.71,
+ "content": "e",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 113,
+ 443,
+ 167,
+ 456
+ ],
+ "score": 1.0,
+ "content": "associated to",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 167,
+ 444,
+ 178,
+ 455
+ ],
+ "score": 0.88,
+ "content": "\\lambda _ { 2 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 179,
+ 443,
+ 466,
+ 456
+ ],
+ "score": 1.0,
+ "content": "satisfies the condition of each case described in the main article. We set",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 466,
+ 444,
+ 504,
+ 454
+ ],
+ "score": 0.87,
+ "content": "W = 1 . 2",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 26
+ },
+ {
+ "bbox": [
+ 106,
+ 455,
+ 330,
+ 467
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 455,
+ 293,
+ 467
+ ],
+ "score": 1.0,
+ "content": "and used the following values for each case as",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 293,
+ 455,
+ 302,
+ 465
+ ],
+ "score": 0.83,
+ "content": "P",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 302,
+ 455,
+ 320,
+ 467
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 320,
+ 457,
+ 326,
+ 465
+ ],
+ "score": 0.72,
+ "content": "e",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 326,
+ 455,
+ 330,
+ 467
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 27
+ }
+ ],
+ "index": 26
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 107,
+ 478,
+ 174,
+ 489
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 477,
+ 175,
+ 491
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 477,
+ 175,
+ 491
+ ],
+ "score": 1.0,
+ "content": "H.1.1 CASE 1",
+ "type": "text"
+ }
+ ],
+ "index": 28
+ }
+ ],
+ "index": 28
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 188,
+ 497,
+ 423,
+ 526
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 188,
+ 497,
+ 423,
+ 526
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 188,
+ 497,
+ 423,
+ 526
+ ],
+ "score": 0.91,
+ "content": "P = { \\left[ 0 . 2 4 9 9 8 1 9 \\quad 0 . 2 4 9 9 8 1 9 \\right] } , \\quad e = { \\left[ \\begin{array} { l } { 0 . 7 0 2 8 3 9 2 } \\\\ { 0 . 2 4 9 9 8 1 9 } \\end{array} \\right] } .",
+ "type": "interline_equation",
+ "image_path": "a45a554cc59f5473c1277d0fc3734b6b8f02d7f173def87ef21b7d057d87c7cf.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 29.5,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 188,
+ 497,
+ 423,
+ 511.5
+ ],
+ "spans": [],
+ "index": 29
+ },
+ {
+ "bbox": [
+ 188,
+ 511.5,
+ 423,
+ 526.0
+ ],
+ "spans": [],
+ "index": 30
+ }
+ ]
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 107,
+ 533,
+ 175,
+ 545
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 532,
+ 176,
+ 547
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 532,
+ 176,
+ 547
+ ],
+ "score": 1.0,
+ "content": "H.1.2 CASE 2",
+ "type": "text"
+ }
+ ],
+ "index": 31
+ }
+ ],
+ "index": 31
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 180,
+ 553,
+ 429,
+ 582
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 180,
+ 553,
+ 429,
+ 582
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 180,
+ 553,
+ 429,
+ 582
+ ],
+ "score": 0.92,
+ "content": "P = \\left[ { \\begin{array} { c c } { 0 . 6 8 9 9 5 7 4 } & { - 0 . 2 4 2 6 8 2 7 } \\\\ { - 0 . 2 4 2 6 8 2 7 } & { 0 . 8 1 0 0 4 2 6 } \\end{array} } \\right] , \\quad e = \\left[ { \\begin{array} { c } { 0 . 6 1 6 3 7 2 3 4 } \\\\ { - 0 . 7 8 7 4 5 4 8 5 } \\end{array} } \\right] .",
+ "type": "interline_equation",
+ "image_path": "ea4a91aed77b14fa03a1a13d483c412a7308a19596269770f06c7f39a684c6ac.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 33,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 180,
+ 553,
+ 429,
+ 562.6666666666666
+ ],
+ "spans": [],
+ "index": 32
+ },
+ {
+ "bbox": [
+ 180,
+ 562.6666666666666,
+ 429,
+ 572.3333333333333
+ ],
+ "spans": [],
+ "index": 33
+ },
+ {
+ "bbox": [
+ 180,
+ 572.3333333333333,
+ 429,
+ 581.9999999999999
+ ],
+ "spans": [],
+ "index": 34
+ }
+ ]
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 108,
+ 592,
+ 262,
+ 604
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 592,
+ 262,
+ 605
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 592,
+ 262,
+ 605
+ ],
+ "score": 1.0,
+ "content": "H.2 EXPERIMENT OF SECTION 6.2",
+ "type": "text"
+ }
+ ],
+ "index": 35
+ }
+ ],
+ "index": 35
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 613,
+ 505,
+ 714
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 613,
+ 506,
+ 627
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 613,
+ 304,
+ 627
+ ],
+ "score": 1.0,
+ "content": "We randomly generated an Erdos – R ˝ enyi graph ´",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 305,
+ 614,
+ 327,
+ 625
+ ],
+ "score": 0.9,
+ "content": "G _ { N , p }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 328,
+ 613,
+ 350,
+ 627
+ ],
+ "score": 1.0,
+ "content": "with",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 350,
+ 614,
+ 396,
+ 624
+ ],
+ "score": 0.9,
+ "content": "N = 1 0 0 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 397,
+ 613,
+ 506,
+ 627
+ ],
+ "score": 1.0,
+ "content": "and randomly generated a",
+ "type": "text"
+ }
+ ],
+ "index": 36
+ },
+ {
+ "bbox": [
+ 105,
+ 624,
+ 506,
+ 637
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 624,
+ 136,
+ 637
+ ],
+ "score": 1.0,
+ "content": "one-of-",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 136,
+ 624,
+ 146,
+ 634
+ ],
+ "score": 0.8,
+ "content": "K",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 147,
+ 624,
+ 321,
+ 637
+ ],
+ "score": 1.0,
+ "content": "hot vector for each node and embed it to a",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 322,
+ 625,
+ 330,
+ 634
+ ],
+ "score": 0.82,
+ "content": "C",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 331,
+ 624,
+ 506,
+ 637
+ ],
+ "score": 1.0,
+ "content": "-dimensional vector using a random matrix",
+ "type": "text"
+ }
+ ],
+ "index": 37
+ },
+ {
+ "bbox": [
+ 106,
+ 636,
+ 504,
+ 648
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 636,
+ 467,
+ 648
+ ],
+ "score": 1.0,
+ "content": "whose elements were randomly sampled from the standard Gaussian distribution. Here,",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 468,
+ 636,
+ 504,
+ 646
+ ],
+ "score": 0.88,
+ "content": "K = 1 0",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 38
+ },
+ {
+ "bbox": [
+ 104,
+ 644,
+ 507,
+ 661
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 644,
+ 123,
+ 661
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 124,
+ 647,
+ 158,
+ 658
+ ],
+ "score": 0.9,
+ "content": "C = 3 2",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 158,
+ 644,
+ 365,
+ 661
+ ],
+ "score": 1.0,
+ "content": ". We treated the resulting single as the input signal",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 366,
+ 646,
+ 427,
+ 658
+ ],
+ "score": 0.93,
+ "content": "X ^ { ( 0 ) } \\in \\mathbb { R } ^ { N \\times C }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 428,
+ 644,
+ 507,
+ 661
+ ],
+ "score": 1.0,
+ "content": ". We constructed a",
+ "type": "text"
+ }
+ ],
+ "index": 39
+ },
+ {
+ "bbox": [
+ 105,
+ 658,
+ 506,
+ 671
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 658,
+ 151,
+ 671
+ ],
+ "score": 1.0,
+ "content": "GCN with",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 151,
+ 659,
+ 185,
+ 669
+ ],
+ "score": 0.89,
+ "content": "L = 1 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 186,
+ 658,
+ 231,
+ 671
+ ],
+ "score": 1.0,
+ "content": "layers and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 231,
+ 659,
+ 240,
+ 668
+ ],
+ "score": 0.79,
+ "content": "C",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 241,
+ 658,
+ 506,
+ 671
+ ],
+ "score": 1.0,
+ "content": "channels. All parameters were i.i.d. sampled from the Gaussian",
+ "type": "text"
+ }
+ ],
+ "index": 40
+ },
+ {
+ "bbox": [
+ 106,
+ 669,
+ 505,
+ 681
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 669,
+ 409,
+ 681
+ ],
+ "score": 1.0,
+ "content": "distribution whose standard deviation is same as the one used in LeCun et al.",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 409,
+ 669,
+ 444,
+ 681
+ ],
+ "score": 0.74,
+ "content": "( 2 0 1 2 ) ^ { 1 1 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 444,
+ 669,
+ 505,
+ 681
+ ],
+ "score": 1.0,
+ "content": "and multiplied",
+ "type": "text"
+ }
+ ],
+ "index": 41
+ },
+ {
+ "bbox": [
+ 104,
+ 680,
+ 505,
+ 693
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 680,
+ 477,
+ 693
+ ],
+ "score": 1.0,
+ "content": "a scalar to each weight matrix so that the largest singular value equals to a specified value",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 477,
+ 682,
+ 483,
+ 690
+ ],
+ "score": 0.66,
+ "content": "s",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 483,
+ 680,
+ 505,
+ 693
+ ],
+ "score": 1.0,
+ "content": ". We",
+ "type": "text"
+ }
+ ],
+ "index": 42
+ },
+ {
+ "bbox": [
+ 105,
+ 691,
+ 505,
+ 704
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 691,
+ 212,
+ 704
+ ],
+ "score": 1.0,
+ "content": "used three configurations",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 212,
+ 691,
+ 329,
+ 703
+ ],
+ "score": 0.29,
+ "content": "( p , s ) = ( 0 . 1 , 0 . 1 ) \\overset { \\cdot } { , } ( 0 . 5 , 1 . \\overset { \\cdot } { 0 } )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 329,
+ 691,
+ 384,
+ 704
+ ],
+ "score": 1.0,
+ "content": ", (0.5, 10.0).",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 384,
+ 692,
+ 391,
+ 701
+ ],
+ "score": 0.7,
+ "content": "\\lambda",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 392,
+ 691,
+ 505,
+ 704
+ ],
+ "score": 1.0,
+ "content": "of the generated GCNs are",
+ "type": "text"
+ }
+ ],
+ "index": 43
+ },
+ {
+ "bbox": [
+ 104,
+ 700,
+ 505,
+ 716
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 700,
+ 477,
+ 716
+ ],
+ "score": 1.0,
+ "content": "0.063, 0.197, 0.194, respectively. See Appendix 6.2 for the results of other configurations of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 478,
+ 702,
+ 500,
+ 714
+ ],
+ "score": 0.92,
+ "content": "( p , s )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 500,
+ 700,
+ 505,
+ 716
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 44
+ }
+ ],
+ "index": 40
+ }
+ ],
+ "page_idx": 22,
+ "page_size": [
+ 612,
+ 792
+ ],
+ "discarded_blocks": [
+ {
+ "type": "discarded",
+ "bbox": [
+ 114,
+ 721,
+ 464,
+ 732
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 115,
+ 719,
+ 465,
+ 733
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 719,
+ 465,
+ 733
+ ],
+ "score": 1.0,
+ "content": "11This is the default initialization method for weight matrices in Chainer and Chainer Chemistry.",
+ "type": "text"
+ }
+ ]
+ }
+ ]
+ },
+ {
+ "type": "discarded",
+ "bbox": [
+ 107,
+ 27,
+ 293,
+ 37
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 26,
+ 294,
+ 38
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 26,
+ 294,
+ 38
+ ],
+ "score": 1.0,
+ "content": "Published as a conference paper at ICLR 2020",
+ "type": "text"
+ }
+ ]
+ }
+ ]
+ },
+ {
+ "type": "discarded",
+ "bbox": [
+ 300,
+ 751,
+ 311,
+ 760
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 298,
+ 750,
+ 313,
+ 764
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 298,
+ 750,
+ 313,
+ 764
+ ],
+ "score": 1.0,
+ "content": "",
+ "type": "text",
+ "height": 14,
+ "width": 15
+ }
+ ]
+ }
+ ]
+ }
+ ],
+ "para_blocks": [
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 81,
+ 505,
+ 127
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 82,
+ 505,
+ 95
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 83,
+ 115,
+ 93
+ ],
+ "score": 0.63,
+ "content": "S",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 115,
+ 82,
+ 139,
+ 95
+ ],
+ "score": 1.0,
+ "content": "), and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 139,
+ 82,
+ 255,
+ 95
+ ],
+ "score": 0.92,
+ "content": "X _ { l + 1 } = \\sigma ( P X _ { l } W _ { l } ) = P X _ { l }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 256,
+ 82,
+ 284,
+ 95
+ ],
+ "score": 1.0,
+ "content": "for all",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 285,
+ 83,
+ 316,
+ 94
+ ],
+ "score": 0.93,
+ "content": "l \\in \\mathbb { N } _ { + }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 317,
+ 82,
+ 505,
+ 95
+ ],
+ "score": 1.0,
+ "content": ". In conclusion, the corollary is reduced to the",
+ "type": "text"
+ }
+ ],
+ "index": 0
+ },
+ {
+ "bbox": [
+ 105,
+ 93,
+ 505,
+ 106
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 93,
+ 505,
+ 106
+ ],
+ "score": 1.0,
+ "content": "fact that if a finite and discrete Markov process is irreducible and aperiodic, any initial probability",
+ "type": "text"
+ }
+ ],
+ "index": 1
+ },
+ {
+ "bbox": [
+ 105,
+ 104,
+ 505,
+ 118
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 104,
+ 421,
+ 118
+ ],
+ "score": 1.0,
+ "content": "distribution converges exponentially to an equibrilium. In addition, the the rate",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 422,
+ 105,
+ 429,
+ 114
+ ],
+ "score": 0.75,
+ "content": "\\lambda",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 429,
+ 104,
+ 505,
+ 118
+ ],
+ "score": 1.0,
+ "content": "corresponds to the",
+ "type": "text"
+ }
+ ],
+ "index": 2
+ },
+ {
+ "bbox": [
+ 105,
+ 116,
+ 250,
+ 128
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 116,
+ 250,
+ 128
+ ],
+ "score": 1.0,
+ "content": "mixing time of the Markov process.",
+ "type": "text"
+ }
+ ],
+ "index": 3
+ }
+ ],
+ "index": 1.5,
+ "bbox_fs": [
+ 105,
+ 82,
+ 505,
+ 128
+ ]
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 107,
+ 142,
+ 362,
+ 155
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 141,
+ 363,
+ 157
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 141,
+ 363,
+ 157
+ ],
+ "score": 1.0,
+ "content": "G GCN DEFINED BY NORMALIZED LAPLACIAN",
+ "type": "text"
+ }
+ ],
+ "index": 4
+ }
+ ],
+ "index": 4
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 165,
+ 505,
+ 268
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 104,
+ 165,
+ 505,
+ 179
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 167,
+ 209,
+ 179
+ ],
+ "score": 1.0,
+ "content": "In Section 4, we defined",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 210,
+ 168,
+ 219,
+ 177
+ ],
+ "score": 0.81,
+ "content": "P",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 219,
+ 167,
+ 399,
+ 179
+ ],
+ "score": 1.0,
+ "content": "using the augmented normalized Laplacian",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 399,
+ 165,
+ 409,
+ 177
+ ],
+ "score": 0.85,
+ "content": "\\tilde { \\Delta }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 409,
+ 167,
+ 424,
+ 179
+ ],
+ "score": 1.0,
+ "content": "by",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 424,
+ 165,
+ 483,
+ 178
+ ],
+ "score": 0.93,
+ "content": "P = I _ { N } - \\tilde { \\Delta }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 483,
+ 167,
+ 505,
+ 179
+ ],
+ "score": 1.0,
+ "content": ". We",
+ "type": "text"
+ }
+ ],
+ "index": 5
+ },
+ {
+ "bbox": [
+ 105,
+ 178,
+ 504,
+ 189
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 178,
+ 322,
+ 189
+ ],
+ "score": 1.0,
+ "content": "can alternatively use the usual normalized Laplacian",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 322,
+ 178,
+ 332,
+ 188
+ ],
+ "score": 0.81,
+ "content": "\\Delta",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 333,
+ 178,
+ 495,
+ 189
+ ],
+ "score": 1.0,
+ "content": "instead of the augmented one to define",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 495,
+ 178,
+ 504,
+ 188
+ ],
+ "score": 0.8,
+ "content": "P",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 6
+ },
+ {
+ "bbox": [
+ 106,
+ 189,
+ 505,
+ 201
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 189,
+ 505,
+ 201
+ ],
+ "score": 1.0,
+ "content": "and want to apply the theory developed in Section 3.2. We write the normalized Laplacian version",
+ "type": "text"
+ }
+ ],
+ "index": 7
+ },
+ {
+ "bbox": [
+ 105,
+ 200,
+ 505,
+ 212
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 200,
+ 118,
+ 212
+ ],
+ "score": 1.0,
+ "content": "as",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 118,
+ 200,
+ 187,
+ 211
+ ],
+ "score": 0.92,
+ "content": "P _ { \\Delta } : = I _ { N } - \\Delta",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 187,
+ 200,
+ 397,
+ 212
+ ],
+ "score": 1.0,
+ "content": ". The only obstacle is that the smallest eigenvalue",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 397,
+ 200,
+ 408,
+ 211
+ ],
+ "score": 0.88,
+ "content": "\\lambda _ { 1 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 409,
+ 200,
+ 422,
+ 212
+ ],
+ "score": 1.0,
+ "content": "of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 423,
+ 200,
+ 437,
+ 211
+ ],
+ "score": 0.89,
+ "content": "P _ { \\Delta }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 438,
+ 200,
+ 505,
+ 212
+ ],
+ "score": 1.0,
+ "content": "can be equal to",
+ "type": "text"
+ }
+ ],
+ "index": 8
+ },
+ {
+ "bbox": [
+ 106,
+ 211,
+ 505,
+ 223
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 212,
+ 120,
+ 222
+ ],
+ "score": 0.78,
+ "content": "- 1",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 121,
+ 211,
+ 178,
+ 223
+ ],
+ "score": 1.0,
+ "content": ", while that of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 178,
+ 212,
+ 187,
+ 221
+ ],
+ "score": 0.82,
+ "content": "P",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 187,
+ 211,
+ 274,
+ 223
+ ],
+ "score": 1.0,
+ "content": "is strictly larger than",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 274,
+ 212,
+ 288,
+ 222
+ ],
+ "score": 0.81,
+ "content": "- 1",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 288,
+ 211,
+ 505,
+ 223
+ ],
+ "score": 1.0,
+ "content": "(see, Proposition 1). This corresponds to that fact the",
+ "type": "text"
+ }
+ ],
+ "index": 9
+ },
+ {
+ "bbox": [
+ 105,
+ 222,
+ 505,
+ 236
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 222,
+ 193,
+ 236
+ ],
+ "score": 1.0,
+ "content": "largest eigenvalue of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 194,
+ 222,
+ 203,
+ 233
+ ],
+ "score": 0.87,
+ "content": "\\tilde { \\Delta }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 204,
+ 222,
+ 368,
+ 236
+ ],
+ "score": 1.0,
+ "content": "is strictly smaller than 2, while that for",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 369,
+ 223,
+ 378,
+ 233
+ ],
+ "score": 0.8,
+ "content": "\\Delta",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 379,
+ 222,
+ 505,
+ 236
+ ],
+ "score": 1.0,
+ "content": "can be 2. It is known that the",
+ "type": "text"
+ }
+ ],
+ "index": 10
+ },
+ {
+ "bbox": [
+ 105,
+ 234,
+ 506,
+ 246
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 234,
+ 191,
+ 246
+ ],
+ "score": 1.0,
+ "content": "largest eigenvalue of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 191,
+ 235,
+ 201,
+ 244
+ ],
+ "score": 0.82,
+ "content": "\\Delta",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 201,
+ 234,
+ 506,
+ 246
+ ],
+ "score": 1.0,
+ "content": "is 2 if and only if the graph has a non-trivial bipartite connected component",
+ "type": "text"
+ }
+ ],
+ "index": 11
+ },
+ {
+ "bbox": [
+ 106,
+ 245,
+ 505,
+ 258
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 245,
+ 180,
+ 258
+ ],
+ "score": 1.0,
+ "content": "(see, e.g., Chung",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 180,
+ 245,
+ 189,
+ 255
+ ],
+ "score": 0.33,
+ "content": "\\&",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 190,
+ 245,
+ 505,
+ 258
+ ],
+ "score": 1.0,
+ "content": "Graham (1997)). Therefore, we can develop a theory using the normalized",
+ "type": "text"
+ }
+ ],
+ "index": 12
+ },
+ {
+ "bbox": [
+ 105,
+ 255,
+ 390,
+ 268
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 255,
+ 377,
+ 268
+ ],
+ "score": 1.0,
+ "content": "Laplacian instead of the augmented one in parallel for such a graph",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 377,
+ 257,
+ 386,
+ 266
+ ],
+ "score": 0.79,
+ "content": "G",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 387,
+ 255,
+ 390,
+ 268
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 13
+ }
+ ],
+ "index": 9,
+ "bbox_fs": [
+ 104,
+ 165,
+ 506,
+ 268
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 272,
+ 505,
+ 372
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 273,
+ 504,
+ 285
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 273,
+ 504,
+ 285
+ ],
+ "score": 1.0,
+ "content": "In Section 5, we characterized the asymptotic behavior of GCN defined by the augmented normal-",
+ "type": "text"
+ }
+ ],
+ "index": 14
+ },
+ {
+ "bbox": [
+ 105,
+ 284,
+ 505,
+ 296
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 284,
+ 505,
+ 296
+ ],
+ "score": 1.0,
+ "content": "ized Laplacian via its spectral distribution (Lemma 6 of Appendix D). We can derive a similar claim",
+ "type": "text"
+ }
+ ],
+ "index": 15
+ },
+ {
+ "bbox": [
+ 105,
+ 294,
+ 505,
+ 308
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 294,
+ 505,
+ 308
+ ],
+ "score": 1.0,
+ "content": "for GCN defined via the normalized Laplacian using the original theorem for the normalized Lapla-",
+ "type": "text"
+ }
+ ],
+ "index": 16
+ },
+ {
+ "bbox": [
+ 105,
+ 306,
+ 505,
+ 319
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 306,
+ 505,
+ 319
+ ],
+ "score": 1.0,
+ "content": "cian in Chung & Radcliffe (2011) (Theorem 7 therein). The normalized Laplacian version of GCN is",
+ "type": "text"
+ }
+ ],
+ "index": 17
+ },
+ {
+ "bbox": [
+ 106,
+ 317,
+ 505,
+ 329
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 317,
+ 505,
+ 329
+ ],
+ "score": 1.0,
+ "content": "advantegeous over the one made from the augmented one because we know its spectral distribution",
+ "type": "text"
+ }
+ ],
+ "index": 18
+ },
+ {
+ "bbox": [
+ 105,
+ 326,
+ 505,
+ 342
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 326,
+ 505,
+ 342
+ ],
+ "score": 1.0,
+ "content": "for broader range of random graphs. For example, Chung & Radcliffe (2011) proved the conver-",
+ "type": "text"
+ }
+ ],
+ "index": 19
+ },
+ {
+ "bbox": [
+ 104,
+ 338,
+ 506,
+ 352
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 338,
+ 506,
+ 352
+ ],
+ "score": 1.0,
+ "content": "gence of the spectral distribution of the normalized Laplacian for Chung-Lu’s model (Chung & Lu,",
+ "type": "text"
+ }
+ ],
+ "index": 20
+ },
+ {
+ "bbox": [
+ 104,
+ 348,
+ 506,
+ 363
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 348,
+ 506,
+ 363
+ ],
+ "score": 1.0,
+ "content": "2002), which includes power law graphs as a special case (see, Theorem 4 of Chung & Radcliffe",
+ "type": "text"
+ }
+ ],
+ "index": 21
+ },
+ {
+ "bbox": [
+ 105,
+ 360,
+ 142,
+ 374
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 360,
+ 142,
+ 374
+ ],
+ "score": 1.0,
+ "content": "(2011)).",
+ "type": "text"
+ }
+ ],
+ "index": 22
+ }
+ ],
+ "index": 18,
+ "bbox_fs": [
+ 104,
+ 273,
+ 506,
+ 374
+ ]
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 107,
+ 387,
+ 316,
+ 401
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 387,
+ 316,
+ 402
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 387,
+ 316,
+ 402
+ ],
+ "score": 1.0,
+ "content": "H DETAILS OF EXPERIMENT SETTINGS",
+ "type": "text"
+ }
+ ],
+ "index": 23
+ }
+ ],
+ "index": 23
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 108,
+ 412,
+ 260,
+ 423
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 411,
+ 262,
+ 424
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 411,
+ 262,
+ 424
+ ],
+ "score": 1.0,
+ "content": "H.1 EXPERIMENT OF SECTION 6.1",
+ "type": "text"
+ }
+ ],
+ "index": 24
+ }
+ ],
+ "index": 24
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 432,
+ 505,
+ 466
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 433,
+ 505,
+ 445
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 433,
+ 203,
+ 445
+ ],
+ "score": 1.0,
+ "content": "We set the eigenvalue of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 204,
+ 433,
+ 212,
+ 443
+ ],
+ "score": 0.84,
+ "content": "P",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 213,
+ 433,
+ 222,
+ 445
+ ],
+ "score": 1.0,
+ "content": "to",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 223,
+ 433,
+ 260,
+ 444
+ ],
+ "score": 0.91,
+ "content": "\\lambda _ { 1 } = 0 . 5",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 261,
+ 433,
+ 277,
+ 445
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 277,
+ 433,
+ 315,
+ 444
+ ],
+ "score": 0.91,
+ "content": "\\lambda _ { 2 } = 1 . 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 315,
+ 433,
+ 412,
+ 445
+ ],
+ "score": 1.0,
+ "content": "and randomly generated",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 413,
+ 433,
+ 422,
+ 443
+ ],
+ "score": 0.82,
+ "content": "P",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 422,
+ 433,
+ 505,
+ 445
+ ],
+ "score": 1.0,
+ "content": "until the eigenvector",
+ "type": "text"
+ }
+ ],
+ "index": 25
+ },
+ {
+ "bbox": [
+ 107,
+ 443,
+ 504,
+ 456
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 446,
+ 113,
+ 454
+ ],
+ "score": 0.71,
+ "content": "e",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 113,
+ 443,
+ 167,
+ 456
+ ],
+ "score": 1.0,
+ "content": "associated to",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 167,
+ 444,
+ 178,
+ 455
+ ],
+ "score": 0.88,
+ "content": "\\lambda _ { 2 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 179,
+ 443,
+ 466,
+ 456
+ ],
+ "score": 1.0,
+ "content": "satisfies the condition of each case described in the main article. We set",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 466,
+ 444,
+ 504,
+ 454
+ ],
+ "score": 0.87,
+ "content": "W = 1 . 2",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 26
+ },
+ {
+ "bbox": [
+ 106,
+ 455,
+ 330,
+ 467
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 455,
+ 293,
+ 467
+ ],
+ "score": 1.0,
+ "content": "and used the following values for each case as",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 293,
+ 455,
+ 302,
+ 465
+ ],
+ "score": 0.83,
+ "content": "P",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 302,
+ 455,
+ 320,
+ 467
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 320,
+ 457,
+ 326,
+ 465
+ ],
+ "score": 0.72,
+ "content": "e",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 326,
+ 455,
+ 330,
+ 467
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 27
+ }
+ ],
+ "index": 26,
+ "bbox_fs": [
+ 106,
+ 433,
+ 505,
+ 467
+ ]
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 107,
+ 478,
+ 174,
+ 489
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 477,
+ 175,
+ 491
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 477,
+ 175,
+ 491
+ ],
+ "score": 1.0,
+ "content": "H.1.1 CASE 1",
+ "type": "text"
+ }
+ ],
+ "index": 28
+ }
+ ],
+ "index": 28
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 188,
+ 497,
+ 423,
+ 526
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 188,
+ 497,
+ 423,
+ 526
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 188,
+ 497,
+ 423,
+ 526
+ ],
+ "score": 0.91,
+ "content": "P = { \\left[ 0 . 2 4 9 9 8 1 9 \\quad 0 . 2 4 9 9 8 1 9 \\right] } , \\quad e = { \\left[ \\begin{array} { l } { 0 . 7 0 2 8 3 9 2 } \\\\ { 0 . 2 4 9 9 8 1 9 } \\end{array} \\right] } .",
+ "type": "interline_equation",
+ "image_path": "a45a554cc59f5473c1277d0fc3734b6b8f02d7f173def87ef21b7d057d87c7cf.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 29.5,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 188,
+ 497,
+ 423,
+ 511.5
+ ],
+ "spans": [],
+ "index": 29
+ },
+ {
+ "bbox": [
+ 188,
+ 511.5,
+ 423,
+ 526.0
+ ],
+ "spans": [],
+ "index": 30
+ }
+ ]
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 107,
+ 533,
+ 175,
+ 545
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 532,
+ 176,
+ 547
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 532,
+ 176,
+ 547
+ ],
+ "score": 1.0,
+ "content": "H.1.2 CASE 2",
+ "type": "text"
+ }
+ ],
+ "index": 31
+ }
+ ],
+ "index": 31
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 180,
+ 553,
+ 429,
+ 582
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 180,
+ 553,
+ 429,
+ 582
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 180,
+ 553,
+ 429,
+ 582
+ ],
+ "score": 0.92,
+ "content": "P = \\left[ { \\begin{array} { c c } { 0 . 6 8 9 9 5 7 4 } & { - 0 . 2 4 2 6 8 2 7 } \\\\ { - 0 . 2 4 2 6 8 2 7 } & { 0 . 8 1 0 0 4 2 6 } \\end{array} } \\right] , \\quad e = \\left[ { \\begin{array} { c } { 0 . 6 1 6 3 7 2 3 4 } \\\\ { - 0 . 7 8 7 4 5 4 8 5 } \\end{array} } \\right] .",
+ "type": "interline_equation",
+ "image_path": "ea4a91aed77b14fa03a1a13d483c412a7308a19596269770f06c7f39a684c6ac.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 33,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 180,
+ 553,
+ 429,
+ 562.6666666666666
+ ],
+ "spans": [],
+ "index": 32
+ },
+ {
+ "bbox": [
+ 180,
+ 562.6666666666666,
+ 429,
+ 572.3333333333333
+ ],
+ "spans": [],
+ "index": 33
+ },
+ {
+ "bbox": [
+ 180,
+ 572.3333333333333,
+ 429,
+ 581.9999999999999
+ ],
+ "spans": [],
+ "index": 34
+ }
+ ]
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 108,
+ 592,
+ 262,
+ 604
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 592,
+ 262,
+ 605
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 592,
+ 262,
+ 605
+ ],
+ "score": 1.0,
+ "content": "H.2 EXPERIMENT OF SECTION 6.2",
+ "type": "text"
+ }
+ ],
+ "index": 35
+ }
+ ],
+ "index": 35
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 613,
+ 505,
+ 714
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 613,
+ 506,
+ 627
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 613,
+ 304,
+ 627
+ ],
+ "score": 1.0,
+ "content": "We randomly generated an Erdos – R ˝ enyi graph ´",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 305,
+ 614,
+ 327,
+ 625
+ ],
+ "score": 0.9,
+ "content": "G _ { N , p }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 328,
+ 613,
+ 350,
+ 627
+ ],
+ "score": 1.0,
+ "content": "with",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 350,
+ 614,
+ 396,
+ 624
+ ],
+ "score": 0.9,
+ "content": "N = 1 0 0 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 397,
+ 613,
+ 506,
+ 627
+ ],
+ "score": 1.0,
+ "content": "and randomly generated a",
+ "type": "text"
+ }
+ ],
+ "index": 36
+ },
+ {
+ "bbox": [
+ 105,
+ 624,
+ 506,
+ 637
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 624,
+ 136,
+ 637
+ ],
+ "score": 1.0,
+ "content": "one-of-",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 136,
+ 624,
+ 146,
+ 634
+ ],
+ "score": 0.8,
+ "content": "K",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 147,
+ 624,
+ 321,
+ 637
+ ],
+ "score": 1.0,
+ "content": "hot vector for each node and embed it to a",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 322,
+ 625,
+ 330,
+ 634
+ ],
+ "score": 0.82,
+ "content": "C",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 331,
+ 624,
+ 506,
+ 637
+ ],
+ "score": 1.0,
+ "content": "-dimensional vector using a random matrix",
+ "type": "text"
+ }
+ ],
+ "index": 37
+ },
+ {
+ "bbox": [
+ 106,
+ 636,
+ 504,
+ 648
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 636,
+ 467,
+ 648
+ ],
+ "score": 1.0,
+ "content": "whose elements were randomly sampled from the standard Gaussian distribution. Here,",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 468,
+ 636,
+ 504,
+ 646
+ ],
+ "score": 0.88,
+ "content": "K = 1 0",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 38
+ },
+ {
+ "bbox": [
+ 104,
+ 644,
+ 507,
+ 661
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 644,
+ 123,
+ 661
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 124,
+ 647,
+ 158,
+ 658
+ ],
+ "score": 0.9,
+ "content": "C = 3 2",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 158,
+ 644,
+ 365,
+ 661
+ ],
+ "score": 1.0,
+ "content": ". We treated the resulting single as the input signal",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 366,
+ 646,
+ 427,
+ 658
+ ],
+ "score": 0.93,
+ "content": "X ^ { ( 0 ) } \\in \\mathbb { R } ^ { N \\times C }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 428,
+ 644,
+ 507,
+ 661
+ ],
+ "score": 1.0,
+ "content": ". We constructed a",
+ "type": "text"
+ }
+ ],
+ "index": 39
+ },
+ {
+ "bbox": [
+ 105,
+ 658,
+ 506,
+ 671
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 658,
+ 151,
+ 671
+ ],
+ "score": 1.0,
+ "content": "GCN with",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 151,
+ 659,
+ 185,
+ 669
+ ],
+ "score": 0.89,
+ "content": "L = 1 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 186,
+ 658,
+ 231,
+ 671
+ ],
+ "score": 1.0,
+ "content": "layers and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 231,
+ 659,
+ 240,
+ 668
+ ],
+ "score": 0.79,
+ "content": "C",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 241,
+ 658,
+ 506,
+ 671
+ ],
+ "score": 1.0,
+ "content": "channels. All parameters were i.i.d. sampled from the Gaussian",
+ "type": "text"
+ }
+ ],
+ "index": 40
+ },
+ {
+ "bbox": [
+ 106,
+ 669,
+ 505,
+ 681
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 669,
+ 409,
+ 681
+ ],
+ "score": 1.0,
+ "content": "distribution whose standard deviation is same as the one used in LeCun et al.",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 409,
+ 669,
+ 444,
+ 681
+ ],
+ "score": 0.74,
+ "content": "( 2 0 1 2 ) ^ { 1 1 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 444,
+ 669,
+ 505,
+ 681
+ ],
+ "score": 1.0,
+ "content": "and multiplied",
+ "type": "text"
+ }
+ ],
+ "index": 41
+ },
+ {
+ "bbox": [
+ 104,
+ 680,
+ 505,
+ 693
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 680,
+ 477,
+ 693
+ ],
+ "score": 1.0,
+ "content": "a scalar to each weight matrix so that the largest singular value equals to a specified value",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 477,
+ 682,
+ 483,
+ 690
+ ],
+ "score": 0.66,
+ "content": "s",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 483,
+ 680,
+ 505,
+ 693
+ ],
+ "score": 1.0,
+ "content": ". We",
+ "type": "text"
+ }
+ ],
+ "index": 42
+ },
+ {
+ "bbox": [
+ 105,
+ 691,
+ 505,
+ 704
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 691,
+ 212,
+ 704
+ ],
+ "score": 1.0,
+ "content": "used three configurations",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 212,
+ 691,
+ 329,
+ 703
+ ],
+ "score": 0.29,
+ "content": "( p , s ) = ( 0 . 1 , 0 . 1 ) \\overset { \\cdot } { , } ( 0 . 5 , 1 . \\overset { \\cdot } { 0 } )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 329,
+ 691,
+ 384,
+ 704
+ ],
+ "score": 1.0,
+ "content": ", (0.5, 10.0).",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 384,
+ 692,
+ 391,
+ 701
+ ],
+ "score": 0.7,
+ "content": "\\lambda",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 392,
+ 691,
+ 505,
+ 704
+ ],
+ "score": 1.0,
+ "content": "of the generated GCNs are",
+ "type": "text"
+ }
+ ],
+ "index": 43
+ },
+ {
+ "bbox": [
+ 104,
+ 700,
+ 505,
+ 716
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 700,
+ 477,
+ 716
+ ],
+ "score": 1.0,
+ "content": "0.063, 0.197, 0.194, respectively. See Appendix 6.2 for the results of other configurations of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 478,
+ 702,
+ 500,
+ 714
+ ],
+ "score": 0.92,
+ "content": "( p , s )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 500,
+ 700,
+ 505,
+ 716
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 44
+ }
+ ],
+ "index": 40,
+ "bbox_fs": [
+ 104,
+ 613,
+ 507,
+ 716
+ ]
+ }
+ ]
+ },
+ {
+ "preproc_blocks": [
+ {
+ "type": "table",
+ "bbox": [
+ 152,
+ 113,
+ 458,
+ 171
+ ],
+ "blocks": [
+ {
+ "type": "table_caption",
+ "bbox": [
+ 106,
+ 90,
+ 503,
+ 112
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 104,
+ 88,
+ 505,
+ 104
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 88,
+ 293,
+ 104
+ ],
+ "score": 1.0,
+ "content": "Table 1: Dataset specifications. The threshold",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 293,
+ 90,
+ 311,
+ 101
+ ],
+ "score": 0.89,
+ "content": "\\lambda ^ { - 1 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 311,
+ 88,
+ 505,
+ 104
+ ],
+ "score": 1.0,
+ "content": "in the table indicates the upper bound of Corol-",
+ "type": "text"
+ }
+ ],
+ "index": 0
+ },
+ {
+ "bbox": [
+ 104,
+ 100,
+ 136,
+ 115
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 100,
+ 136,
+ 115
+ ],
+ "score": 1.0,
+ "content": "lary 2.",
+ "type": "text"
+ }
+ ],
+ "index": 1
+ }
+ ],
+ "index": 0.5
+ },
+ {
+ "type": "table_body",
+ "bbox": [
+ 152,
+ 113,
+ 458,
+ 171
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 152,
+ 113,
+ 458,
+ 171
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 152,
+ 113,
+ 458,
+ 171
+ ],
+ "score": 0.979,
+ "html": " | #Node | #Edge | #Class | Chance Rate | Threshold λ-1 |
| Cora | 2708 | 5429 | 6 | 30.2% | 1+3.62 × 10-3 |
| CiteSeer | 3312 | 4732 | 7 | 21.1% | 1 + 1.25 × 10-3 |
| PubMed | 19717 | 44338 | 3 | 39.9% | 1 + 9.57 × 10-3 |
",
+ "type": "table",
+ "image_path": "c6d48b13dcfd9e931066c5b04d7100274a1d08a14a6d14ae91c63888e263980c.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 3,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 152,
+ 113,
+ 458,
+ 132.33333333333334
+ ],
+ "spans": [],
+ "index": 2
+ },
+ {
+ "bbox": [
+ 152,
+ 132.33333333333334,
+ 458,
+ 151.66666666666669
+ ],
+ "spans": [],
+ "index": 3
+ },
+ {
+ "bbox": [
+ 152,
+ 151.66666666666669,
+ 458,
+ 171.00000000000003
+ ],
+ "spans": [],
+ "index": 4
+ }
+ ]
+ }
+ ],
+ "index": 1.75
+ },
+ {
+ "type": "table",
+ "bbox": [
+ 156,
+ 216,
+ 454,
+ 284
+ ],
+ "blocks": [
+ {
+ "type": "table_caption",
+ "bbox": [
+ 105,
+ 193,
+ 505,
+ 215
+ ],
+ "group_id": 1,
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 192,
+ 506,
+ 207
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 192,
+ 401,
+ 207
+ ],
+ "score": 1.0,
+ "content": "Table 2: Dataset specifications for noisy citation networks. The threshold",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 401,
+ 193,
+ 419,
+ 204
+ ],
+ "score": 0.89,
+ "content": "\\lambda ^ { - 1 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 419,
+ 192,
+ 506,
+ 207
+ ],
+ "score": 1.0,
+ "content": "in the table indicates",
+ "type": "text"
+ }
+ ],
+ "index": 5
+ },
+ {
+ "bbox": [
+ 106,
+ 204,
+ 235,
+ 217
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 204,
+ 235,
+ 217
+ ],
+ "score": 1.0,
+ "content": "the upper bound of Corollary 2.",
+ "type": "text"
+ }
+ ],
+ "index": 6
+ }
+ ],
+ "index": 5.5
+ },
+ {
+ "type": "table_body",
+ "bbox": [
+ 156,
+ 216,
+ 454,
+ 284
+ ],
+ "group_id": 1,
+ "lines": [
+ {
+ "bbox": [
+ 156,
+ 216,
+ 454,
+ 284
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 156,
+ 216,
+ 454,
+ 284
+ ],
+ "score": 0.979,
+ "html": " | Original Dataset | #Edge Added | Threshold λ-1 |
| Noisy Cora 2500 | Cora | 2495 | 1.11 |
| Noisy ( Cora 5000 | Cora | 4988 | 1.15 |
| Noisy CiteSeer | CiteSeer | 4991 | 1.13 |
| Noisy PubMed | PubMed | 24993 | 1.17 |
",
+ "type": "table",
+ "image_path": "d8e780ecbc0fa4e44cbb222c9e07cff879326aa7fcb26c3f08983e08f637de59.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 8,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 156,
+ 216,
+ 454,
+ 238.66666666666666
+ ],
+ "spans": [],
+ "index": 7
+ },
+ {
+ "bbox": [
+ 156,
+ 238.66666666666666,
+ 454,
+ 261.3333333333333
+ ],
+ "spans": [],
+ "index": 8
+ },
+ {
+ "bbox": [
+ 156,
+ 261.3333333333333,
+ 454,
+ 284.0
+ ],
+ "spans": [],
+ "index": 9
+ }
+ ]
+ }
+ ],
+ "index": 6.75
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 107,
+ 306,
+ 261,
+ 317
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 305,
+ 263,
+ 318
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 305,
+ 263,
+ 318
+ ],
+ "score": 1.0,
+ "content": "H.3 EXPERIMENT OF SECTION 6.3",
+ "type": "text"
+ }
+ ],
+ "index": 10
+ }
+ ],
+ "index": 10
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 107,
+ 327,
+ 182,
+ 339
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 327,
+ 183,
+ 340
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 327,
+ 183,
+ 340
+ ],
+ "score": 1.0,
+ "content": "H.3.1 DATASET",
+ "type": "text"
+ }
+ ],
+ "index": 11
+ }
+ ],
+ "index": 11
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 347,
+ 504,
+ 392
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 347,
+ 506,
+ 360
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 347,
+ 506,
+ 360
+ ],
+ "score": 1.0,
+ "content": "We used the Cora (McCallum et al., 2000; Sen et al., 2008), CiteSeer (Giles et al., 1998; Sen et al.,",
+ "type": "text"
+ }
+ ],
+ "index": 12
+ },
+ {
+ "bbox": [
+ 105,
+ 358,
+ 506,
+ 371
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 358,
+ 506,
+ 371
+ ],
+ "score": 1.0,
+ "content": "2008), and PubMed(Sen et al., 2008) datasets for experiments. We obtained the preprocessed dataset",
+ "type": "text"
+ }
+ ],
+ "index": 13
+ },
+ {
+ "bbox": [
+ 105,
+ 369,
+ 506,
+ 383
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 369,
+ 282,
+ 383
+ ],
+ "score": 1.0,
+ "content": "from the code repository of Kipf & Welling",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 283,
+ 369,
+ 317,
+ 381
+ ],
+ "score": 0.32,
+ "content": "( 2 0 1 7 ) ^ { \\hat { 1 } 2 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 317,
+ 369,
+ 506,
+ 383
+ ],
+ "score": 1.0,
+ "content": ". Table 1 summarizes specifications of datasets",
+ "type": "text"
+ }
+ ],
+ "index": 14
+ },
+ {
+ "bbox": [
+ 106,
+ 381,
+ 325,
+ 393
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 381,
+ 325,
+ 393
+ ],
+ "score": 1.0,
+ "content": "and their noisy version (explained in the next section).",
+ "type": "text"
+ }
+ ],
+ "index": 15
+ }
+ ],
+ "index": 13.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 397,
+ 505,
+ 486
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 396,
+ 506,
+ 411
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 396,
+ 506,
+ 411
+ ],
+ "score": 1.0,
+ "content": "The Cora dataset is a citation network dataset consisting of 2708 papers and 5429 links. Each paper",
+ "type": "text"
+ }
+ ],
+ "index": 16
+ },
+ {
+ "bbox": [
+ 105,
+ 409,
+ 506,
+ 421
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 409,
+ 506,
+ 421
+ ],
+ "score": 1.0,
+ "content": "is represented as the occurence of 1433 unique words and is associated to one of 7 genres (Case",
+ "type": "text"
+ }
+ ],
+ "index": 17
+ },
+ {
+ "bbox": [
+ 106,
+ 420,
+ 504,
+ 431
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 420,
+ 504,
+ 431
+ ],
+ "score": 1.0,
+ "content": "Based, Genetic Algorithms, Neural Networks, Probabilistic Methods, Reinforcement Learning, Rule",
+ "type": "text"
+ }
+ ],
+ "index": 18
+ },
+ {
+ "bbox": [
+ 106,
+ 431,
+ 505,
+ 443
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 431,
+ 505,
+ 443
+ ],
+ "score": 1.0,
+ "content": "Learning, Theory). The graph made from the citation links has 78 connected components and the",
+ "type": "text"
+ }
+ ],
+ "index": 19
+ },
+ {
+ "bbox": [
+ 105,
+ 442,
+ 505,
+ 454
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 442,
+ 455,
+ 454
+ ],
+ "score": 1.0,
+ "content": "smallest positive eigenvalue of the augmented Normalized Laplacian is approximately",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 455,
+ 442,
+ 505,
+ 452
+ ],
+ "score": 0.89,
+ "content": "\\tilde { \\mu } = 3 . 6 2 \\times",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 20
+ },
+ {
+ "bbox": [
+ 106,
+ 451,
+ 507,
+ 466
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 452,
+ 128,
+ 463
+ ],
+ "score": 0.89,
+ "content": "1 0 ^ { - 3 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 128,
+ 451,
+ 312,
+ 466
+ ],
+ "score": 1.0,
+ "content": ". Therefore, the upper bound of Theorem 2 is",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 312,
+ 452,
+ 467,
+ 464
+ ],
+ "score": 0.92,
+ "content": "\\lambda ^ { - 1 } = ( \\bar { 1 - \\mu } ) ^ { - 1 } \\approx \\bar { 1 } + 3 . 6 2 \\times \\bar { 1 0 ^ { - 3 } }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 467,
+ 451,
+ 507,
+ 466
+ ],
+ "score": 1.0,
+ "content": ". 818 out",
+ "type": "text"
+ }
+ ],
+ "index": 21
+ },
+ {
+ "bbox": [
+ 105,
+ 463,
+ 506,
+ 476
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 463,
+ 506,
+ 476
+ ],
+ "score": 1.0,
+ "content": "of 2708 samples are labelled as “Probabilistic Methods”, which is the largest proportion. Therefore,",
+ "type": "text"
+ }
+ ],
+ "index": 22
+ },
+ {
+ "bbox": [
+ 106,
+ 474,
+ 262,
+ 487
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 474,
+ 178,
+ 487
+ ],
+ "score": 1.0,
+ "content": "the chance rate is",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 178,
+ 474,
+ 258,
+ 486
+ ],
+ "score": 0.88,
+ "content": "8 1 8 / 2 7 0 8 = 3 0 . 2 \\%",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 259,
+ 474,
+ 262,
+ 487
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 23
+ }
+ ],
+ "index": 19.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 491,
+ 505,
+ 568
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 491,
+ 506,
+ 504
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 491,
+ 506,
+ 504
+ ],
+ "score": 1.0,
+ "content": "The CiteSeer dataset is a citation network dataset consisting of 3312 papers and 4732 links. Each",
+ "type": "text"
+ }
+ ],
+ "index": 24
+ },
+ {
+ "bbox": [
+ 105,
+ 502,
+ 505,
+ 515
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 502,
+ 505,
+ 515
+ ],
+ "score": 1.0,
+ "content": "paper is represented as the occurence of 3703 unique words and is associated to one of 6 genres",
+ "type": "text"
+ }
+ ],
+ "index": 25
+ },
+ {
+ "bbox": [
+ 105,
+ 513,
+ 505,
+ 526
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 513,
+ 505,
+ 526
+ ],
+ "score": 1.0,
+ "content": "(Agents, AI, DB, IR, ML, HCI). The graph made from the citation links has 438 connected compo-",
+ "type": "text"
+ }
+ ],
+ "index": 26
+ },
+ {
+ "bbox": [
+ 105,
+ 524,
+ 504,
+ 537
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 524,
+ 504,
+ 537
+ ],
+ "score": 1.0,
+ "content": "nents and the smallest positive eigenvalue of the augmented Normalized Laplacian is approximately",
+ "type": "text"
+ }
+ ],
+ "index": 27
+ },
+ {
+ "bbox": [
+ 106,
+ 533,
+ 505,
+ 548
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 535,
+ 175,
+ 547
+ ],
+ "score": 0.92,
+ "content": "\\tilde { \\mu } = 1 . 2 5 \\times 1 0 ^ { - 3 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 175,
+ 533,
+ 354,
+ 548
+ ],
+ "score": 1.0,
+ "content": ". Therefore, the upper bound of Theorem 2 is",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 354,
+ 534,
+ 501,
+ 547
+ ],
+ "score": 0.92,
+ "content": "\\lambda ^ { - 1 } = ( 1 - \\dot { \\tilde { \\mu } } ) ^ { - 1 } \\approx 1 + \\dot { 1 } . 2 5 \\times 1 0 ^ { - \\dot { 3 } }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 501,
+ 533,
+ 505,
+ 548
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 28
+ },
+ {
+ "bbox": [
+ 105,
+ 545,
+ 506,
+ 559
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 545,
+ 506,
+ 559
+ ],
+ "score": 1.0,
+ "content": "701 out of 2708 samples are labelled as “IR”, which is the largest proportion. Therefore, the chance",
+ "type": "text"
+ }
+ ],
+ "index": 29
+ },
+ {
+ "bbox": [
+ 105,
+ 557,
+ 217,
+ 569
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 557,
+ 134,
+ 569
+ ],
+ "score": 1.0,
+ "content": "rate is",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 135,
+ 557,
+ 214,
+ 569
+ ],
+ "score": 0.87,
+ "content": "7 0 1 / 3 3 1 2 = \\hat { 2 } 1 . 1 \\%",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 214,
+ 557,
+ 217,
+ 569
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 30
+ }
+ ],
+ "index": 27
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 573,
+ 505,
+ 662
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 574,
+ 505,
+ 586
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 574,
+ 505,
+ 586
+ ],
+ "score": 1.0,
+ "content": "The PubMed dataset is a citation network dataset consisting of 19717 papers and 44338 links. Each",
+ "type": "text"
+ }
+ ],
+ "index": 31
+ },
+ {
+ "bbox": [
+ 104,
+ 585,
+ 506,
+ 598
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 585,
+ 506,
+ 598
+ ],
+ "score": 1.0,
+ "content": "paper is represented as the occurence of 500 unique words and is associated to one of 3 genres",
+ "type": "text"
+ }
+ ],
+ "index": 32
+ },
+ {
+ "bbox": [
+ 106,
+ 596,
+ 505,
+ 608
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 596,
+ 468,
+ 608
+ ],
+ "score": 1.0,
+ "content": "(“Diabetes Mellitus, Experimental”, “Diabetes Mellitus Type 1”, “Diabetes Mellitus Type",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 468,
+ 596,
+ 479,
+ 606
+ ],
+ "score": 0.5,
+ "content": "2 ^ { \\circ }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 479,
+ 596,
+ 505,
+ 608
+ ],
+ "score": 1.0,
+ "content": "). The",
+ "type": "text"
+ }
+ ],
+ "index": 33
+ },
+ {
+ "bbox": [
+ 104,
+ 606,
+ 506,
+ 621
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 606,
+ 506,
+ 621
+ ],
+ "score": 1.0,
+ "content": "graph made from the citation links has 438 connected components and the smallest positive eigen-",
+ "type": "text"
+ }
+ ],
+ "index": 34
+ },
+ {
+ "bbox": [
+ 105,
+ 617,
+ 506,
+ 630
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 617,
+ 366,
+ 630
+ ],
+ "score": 1.0,
+ "content": "value of the augmented Normalized Laplacian is approximately",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 366,
+ 617,
+ 440,
+ 629
+ ],
+ "score": 0.9,
+ "content": "\\tilde { \\mu } = 9 . 4 8 \\times 1 0 ^ { - 3 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 440,
+ 617,
+ 506,
+ 630
+ ],
+ "score": 1.0,
+ "content": ". Therefore, the",
+ "type": "text"
+ }
+ ],
+ "index": 35
+ },
+ {
+ "bbox": [
+ 104,
+ 627,
+ 506,
+ 642
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 627,
+ 224,
+ 642
+ ],
+ "score": 1.0,
+ "content": "upper bound of Theorem 2 is",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 225,
+ 628,
+ 376,
+ 641
+ ],
+ "score": 0.93,
+ "content": "\\lambda ^ { - 1 } = ( \\bar { 1 ^ { - } } \\tilde { \\mu } ) ^ { - 1 } \\approx \\bar { 1 ^ { + } } 9 . 5 7 \\times \\bar { 1 0 ^ { - 3 } }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 376,
+ 627,
+ 506,
+ 642
+ ],
+ "score": 1.0,
+ "content": ". 7875 out of 19717 samples are",
+ "type": "text"
+ }
+ ],
+ "index": 36
+ },
+ {
+ "bbox": [
+ 105,
+ 639,
+ 506,
+ 653
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 639,
+ 252,
+ 653
+ ],
+ "score": 1.0,
+ "content": "labelled as “Diabetes Mellitus Type",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 253,
+ 641,
+ 263,
+ 650
+ ],
+ "score": 0.7,
+ "content": "2 ^ { \\circ }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 264,
+ 639,
+ 506,
+ 653
+ ],
+ "score": 1.0,
+ "content": ", which is the largest proportion. Therefore, the chance rate",
+ "type": "text"
+ }
+ ],
+ "index": 37
+ },
+ {
+ "bbox": [
+ 105,
+ 650,
+ 209,
+ 663
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 650,
+ 116,
+ 663
+ ],
+ "score": 1.0,
+ "content": "is",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 116,
+ 651,
+ 206,
+ 663
+ ],
+ "score": 0.85,
+ "content": "7 8 7 5 / 1 9 7 1 7 = 3 9 . 9 \\%",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 207,
+ 650,
+ 209,
+ 663
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 38
+ }
+ ],
+ "index": 34.5
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 107,
+ 676,
+ 267,
+ 688
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 675,
+ 268,
+ 690
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 675,
+ 268,
+ 690
+ ],
+ "score": 1.0,
+ "content": "H.3.2 NOISY CITATION NETWORKS",
+ "type": "text"
+ }
+ ],
+ "index": 39
+ }
+ ],
+ "index": 39
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 697,
+ 375,
+ 709
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 696,
+ 375,
+ 709
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 696,
+ 375,
+ 709
+ ],
+ "score": 1.0,
+ "content": "We summarize the properties of noisy citation networks in Table 2.",
+ "type": "text"
+ }
+ ],
+ "index": 40
+ }
+ ],
+ "index": 40
+ }
+ ],
+ "page_idx": 23,
+ "page_size": [
+ 612,
+ 792
+ ],
+ "discarded_blocks": [
+ {
+ "type": "discarded",
+ "bbox": [
+ 117,
+ 721,
+ 224,
+ 732
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 115,
+ 718,
+ 226,
+ 735
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 718,
+ 226,
+ 735
+ ],
+ "score": 1.0,
+ "content": "12https://github.com/tkipf/gcn",
+ "type": "text"
+ }
+ ]
+ }
+ ]
+ },
+ {
+ "type": "discarded",
+ "bbox": [
+ 107,
+ 27,
+ 293,
+ 37
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 26,
+ 294,
+ 38
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 26,
+ 294,
+ 38
+ ],
+ "score": 1.0,
+ "content": "Published as a conference paper at ICLR 2020",
+ "type": "text"
+ }
+ ]
+ }
+ ]
+ },
+ {
+ "type": "discarded",
+ "bbox": [
+ 300,
+ 751,
+ 311,
+ 760
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 298,
+ 750,
+ 313,
+ 765
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 298,
+ 750,
+ 313,
+ 765
+ ],
+ "score": 1.0,
+ "content": "",
+ "type": "text",
+ "height": 15,
+ "width": 15
+ }
+ ]
+ }
+ ]
+ }
+ ],
+ "para_blocks": [
+ {
+ "type": "table",
+ "bbox": [
+ 152,
+ 113,
+ 458,
+ 171
+ ],
+ "blocks": [
+ {
+ "type": "table_caption",
+ "bbox": [
+ 106,
+ 90,
+ 503,
+ 112
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 104,
+ 88,
+ 505,
+ 104
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 88,
+ 293,
+ 104
+ ],
+ "score": 1.0,
+ "content": "Table 1: Dataset specifications. The threshold",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 293,
+ 90,
+ 311,
+ 101
+ ],
+ "score": 0.89,
+ "content": "\\lambda ^ { - 1 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 311,
+ 88,
+ 505,
+ 104
+ ],
+ "score": 1.0,
+ "content": "in the table indicates the upper bound of Corol-",
+ "type": "text"
+ }
+ ],
+ "index": 0
+ },
+ {
+ "bbox": [
+ 104,
+ 100,
+ 136,
+ 115
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 100,
+ 136,
+ 115
+ ],
+ "score": 1.0,
+ "content": "lary 2.",
+ "type": "text"
+ }
+ ],
+ "index": 1
+ }
+ ],
+ "index": 0.5
+ },
+ {
+ "type": "table_body",
+ "bbox": [
+ 152,
+ 113,
+ 458,
+ 171
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 152,
+ 113,
+ 458,
+ 171
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 152,
+ 113,
+ 458,
+ 171
+ ],
+ "score": 0.979,
+ "html": " | #Node | #Edge | #Class | Chance Rate | Threshold λ-1 |
| Cora | 2708 | 5429 | 6 | 30.2% | 1+3.62 × 10-3 |
| CiteSeer | 3312 | 4732 | 7 | 21.1% | 1 + 1.25 × 10-3 |
| PubMed | 19717 | 44338 | 3 | 39.9% | 1 + 9.57 × 10-3 |
",
+ "type": "table",
+ "image_path": "c6d48b13dcfd9e931066c5b04d7100274a1d08a14a6d14ae91c63888e263980c.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 3,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 152,
+ 113,
+ 458,
+ 132.33333333333334
+ ],
+ "spans": [],
+ "index": 2
+ },
+ {
+ "bbox": [
+ 152,
+ 132.33333333333334,
+ 458,
+ 151.66666666666669
+ ],
+ "spans": [],
+ "index": 3
+ },
+ {
+ "bbox": [
+ 152,
+ 151.66666666666669,
+ 458,
+ 171.00000000000003
+ ],
+ "spans": [],
+ "index": 4
+ }
+ ]
+ }
+ ],
+ "index": 1.75
+ },
+ {
+ "type": "table",
+ "bbox": [
+ 156,
+ 216,
+ 454,
+ 284
+ ],
+ "blocks": [
+ {
+ "type": "table_caption",
+ "bbox": [
+ 105,
+ 193,
+ 505,
+ 215
+ ],
+ "group_id": 1,
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 192,
+ 506,
+ 207
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 192,
+ 401,
+ 207
+ ],
+ "score": 1.0,
+ "content": "Table 2: Dataset specifications for noisy citation networks. The threshold",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 401,
+ 193,
+ 419,
+ 204
+ ],
+ "score": 0.89,
+ "content": "\\lambda ^ { - 1 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 419,
+ 192,
+ 506,
+ 207
+ ],
+ "score": 1.0,
+ "content": "in the table indicates",
+ "type": "text"
+ }
+ ],
+ "index": 5
+ },
+ {
+ "bbox": [
+ 106,
+ 204,
+ 235,
+ 217
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 204,
+ 235,
+ 217
+ ],
+ "score": 1.0,
+ "content": "the upper bound of Corollary 2.",
+ "type": "text"
+ }
+ ],
+ "index": 6
+ }
+ ],
+ "index": 5.5
+ },
+ {
+ "type": "table_body",
+ "bbox": [
+ 156,
+ 216,
+ 454,
+ 284
+ ],
+ "group_id": 1,
+ "lines": [
+ {
+ "bbox": [
+ 156,
+ 216,
+ 454,
+ 284
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 156,
+ 216,
+ 454,
+ 284
+ ],
+ "score": 0.979,
+ "html": " | Original Dataset | #Edge Added | Threshold λ-1 |
| Noisy Cora 2500 | Cora | 2495 | 1.11 |
| Noisy ( Cora 5000 | Cora | 4988 | 1.15 |
| Noisy CiteSeer | CiteSeer | 4991 | 1.13 |
| Noisy PubMed | PubMed | 24993 | 1.17 |
",
+ "type": "table",
+ "image_path": "d8e780ecbc0fa4e44cbb222c9e07cff879326aa7fcb26c3f08983e08f637de59.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 8,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 156,
+ 216,
+ 454,
+ 238.66666666666666
+ ],
+ "spans": [],
+ "index": 7
+ },
+ {
+ "bbox": [
+ 156,
+ 238.66666666666666,
+ 454,
+ 261.3333333333333
+ ],
+ "spans": [],
+ "index": 8
+ },
+ {
+ "bbox": [
+ 156,
+ 261.3333333333333,
+ 454,
+ 284.0
+ ],
+ "spans": [],
+ "index": 9
+ }
+ ]
+ }
+ ],
+ "index": 6.75
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 107,
+ 306,
+ 261,
+ 317
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 305,
+ 263,
+ 318
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 305,
+ 263,
+ 318
+ ],
+ "score": 1.0,
+ "content": "H.3 EXPERIMENT OF SECTION 6.3",
+ "type": "text"
+ }
+ ],
+ "index": 10
+ }
+ ],
+ "index": 10
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 107,
+ 327,
+ 182,
+ 339
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 327,
+ 183,
+ 340
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 327,
+ 183,
+ 340
+ ],
+ "score": 1.0,
+ "content": "H.3.1 DATASET",
+ "type": "text"
+ }
+ ],
+ "index": 11
+ }
+ ],
+ "index": 11
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 347,
+ 504,
+ 392
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 347,
+ 506,
+ 360
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 347,
+ 506,
+ 360
+ ],
+ "score": 1.0,
+ "content": "We used the Cora (McCallum et al., 2000; Sen et al., 2008), CiteSeer (Giles et al., 1998; Sen et al.,",
+ "type": "text"
+ }
+ ],
+ "index": 12
+ },
+ {
+ "bbox": [
+ 105,
+ 358,
+ 506,
+ 371
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 358,
+ 506,
+ 371
+ ],
+ "score": 1.0,
+ "content": "2008), and PubMed(Sen et al., 2008) datasets for experiments. We obtained the preprocessed dataset",
+ "type": "text"
+ }
+ ],
+ "index": 13
+ },
+ {
+ "bbox": [
+ 105,
+ 369,
+ 506,
+ 383
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 369,
+ 282,
+ 383
+ ],
+ "score": 1.0,
+ "content": "from the code repository of Kipf & Welling",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 283,
+ 369,
+ 317,
+ 381
+ ],
+ "score": 0.32,
+ "content": "( 2 0 1 7 ) ^ { \\hat { 1 } 2 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 317,
+ 369,
+ 506,
+ 383
+ ],
+ "score": 1.0,
+ "content": ". Table 1 summarizes specifications of datasets",
+ "type": "text"
+ }
+ ],
+ "index": 14
+ },
+ {
+ "bbox": [
+ 106,
+ 381,
+ 325,
+ 393
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 381,
+ 325,
+ 393
+ ],
+ "score": 1.0,
+ "content": "and their noisy version (explained in the next section).",
+ "type": "text"
+ }
+ ],
+ "index": 15
+ }
+ ],
+ "index": 13.5,
+ "bbox_fs": [
+ 105,
+ 347,
+ 506,
+ 393
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 397,
+ 505,
+ 486
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 396,
+ 506,
+ 411
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 396,
+ 506,
+ 411
+ ],
+ "score": 1.0,
+ "content": "The Cora dataset is a citation network dataset consisting of 2708 papers and 5429 links. Each paper",
+ "type": "text"
+ }
+ ],
+ "index": 16
+ },
+ {
+ "bbox": [
+ 105,
+ 409,
+ 506,
+ 421
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 409,
+ 506,
+ 421
+ ],
+ "score": 1.0,
+ "content": "is represented as the occurence of 1433 unique words and is associated to one of 7 genres (Case",
+ "type": "text"
+ }
+ ],
+ "index": 17
+ },
+ {
+ "bbox": [
+ 106,
+ 420,
+ 504,
+ 431
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 420,
+ 504,
+ 431
+ ],
+ "score": 1.0,
+ "content": "Based, Genetic Algorithms, Neural Networks, Probabilistic Methods, Reinforcement Learning, Rule",
+ "type": "text"
+ }
+ ],
+ "index": 18
+ },
+ {
+ "bbox": [
+ 106,
+ 431,
+ 505,
+ 443
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 431,
+ 505,
+ 443
+ ],
+ "score": 1.0,
+ "content": "Learning, Theory). The graph made from the citation links has 78 connected components and the",
+ "type": "text"
+ }
+ ],
+ "index": 19
+ },
+ {
+ "bbox": [
+ 105,
+ 442,
+ 505,
+ 454
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 442,
+ 455,
+ 454
+ ],
+ "score": 1.0,
+ "content": "smallest positive eigenvalue of the augmented Normalized Laplacian is approximately",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 455,
+ 442,
+ 505,
+ 452
+ ],
+ "score": 0.89,
+ "content": "\\tilde { \\mu } = 3 . 6 2 \\times",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 20
+ },
+ {
+ "bbox": [
+ 106,
+ 451,
+ 507,
+ 466
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 452,
+ 128,
+ 463
+ ],
+ "score": 0.89,
+ "content": "1 0 ^ { - 3 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 128,
+ 451,
+ 312,
+ 466
+ ],
+ "score": 1.0,
+ "content": ". Therefore, the upper bound of Theorem 2 is",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 312,
+ 452,
+ 467,
+ 464
+ ],
+ "score": 0.92,
+ "content": "\\lambda ^ { - 1 } = ( \\bar { 1 - \\mu } ) ^ { - 1 } \\approx \\bar { 1 } + 3 . 6 2 \\times \\bar { 1 0 ^ { - 3 } }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 467,
+ 451,
+ 507,
+ 466
+ ],
+ "score": 1.0,
+ "content": ". 818 out",
+ "type": "text"
+ }
+ ],
+ "index": 21
+ },
+ {
+ "bbox": [
+ 105,
+ 463,
+ 506,
+ 476
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 463,
+ 506,
+ 476
+ ],
+ "score": 1.0,
+ "content": "of 2708 samples are labelled as “Probabilistic Methods”, which is the largest proportion. Therefore,",
+ "type": "text"
+ }
+ ],
+ "index": 22
+ },
+ {
+ "bbox": [
+ 106,
+ 474,
+ 262,
+ 487
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 474,
+ 178,
+ 487
+ ],
+ "score": 1.0,
+ "content": "the chance rate is",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 178,
+ 474,
+ 258,
+ 486
+ ],
+ "score": 0.88,
+ "content": "8 1 8 / 2 7 0 8 = 3 0 . 2 \\%",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 259,
+ 474,
+ 262,
+ 487
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 23
+ }
+ ],
+ "index": 19.5,
+ "bbox_fs": [
+ 105,
+ 396,
+ 507,
+ 487
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 491,
+ 505,
+ 568
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 491,
+ 506,
+ 504
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 491,
+ 506,
+ 504
+ ],
+ "score": 1.0,
+ "content": "The CiteSeer dataset is a citation network dataset consisting of 3312 papers and 4732 links. Each",
+ "type": "text"
+ }
+ ],
+ "index": 24
+ },
+ {
+ "bbox": [
+ 105,
+ 502,
+ 505,
+ 515
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 502,
+ 505,
+ 515
+ ],
+ "score": 1.0,
+ "content": "paper is represented as the occurence of 3703 unique words and is associated to one of 6 genres",
+ "type": "text"
+ }
+ ],
+ "index": 25
+ },
+ {
+ "bbox": [
+ 105,
+ 513,
+ 505,
+ 526
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 513,
+ 505,
+ 526
+ ],
+ "score": 1.0,
+ "content": "(Agents, AI, DB, IR, ML, HCI). The graph made from the citation links has 438 connected compo-",
+ "type": "text"
+ }
+ ],
+ "index": 26
+ },
+ {
+ "bbox": [
+ 105,
+ 524,
+ 504,
+ 537
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 524,
+ 504,
+ 537
+ ],
+ "score": 1.0,
+ "content": "nents and the smallest positive eigenvalue of the augmented Normalized Laplacian is approximately",
+ "type": "text"
+ }
+ ],
+ "index": 27
+ },
+ {
+ "bbox": [
+ 106,
+ 533,
+ 505,
+ 548
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 535,
+ 175,
+ 547
+ ],
+ "score": 0.92,
+ "content": "\\tilde { \\mu } = 1 . 2 5 \\times 1 0 ^ { - 3 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 175,
+ 533,
+ 354,
+ 548
+ ],
+ "score": 1.0,
+ "content": ". Therefore, the upper bound of Theorem 2 is",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 354,
+ 534,
+ 501,
+ 547
+ ],
+ "score": 0.92,
+ "content": "\\lambda ^ { - 1 } = ( 1 - \\dot { \\tilde { \\mu } } ) ^ { - 1 } \\approx 1 + \\dot { 1 } . 2 5 \\times 1 0 ^ { - \\dot { 3 } }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 501,
+ 533,
+ 505,
+ 548
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 28
+ },
+ {
+ "bbox": [
+ 105,
+ 545,
+ 506,
+ 559
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 545,
+ 506,
+ 559
+ ],
+ "score": 1.0,
+ "content": "701 out of 2708 samples are labelled as “IR”, which is the largest proportion. Therefore, the chance",
+ "type": "text"
+ }
+ ],
+ "index": 29
+ },
+ {
+ "bbox": [
+ 105,
+ 557,
+ 217,
+ 569
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 557,
+ 134,
+ 569
+ ],
+ "score": 1.0,
+ "content": "rate is",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 135,
+ 557,
+ 214,
+ 569
+ ],
+ "score": 0.87,
+ "content": "7 0 1 / 3 3 1 2 = \\hat { 2 } 1 . 1 \\%",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 214,
+ 557,
+ 217,
+ 569
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 30
+ }
+ ],
+ "index": 27,
+ "bbox_fs": [
+ 105,
+ 491,
+ 506,
+ 569
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 573,
+ 505,
+ 662
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 574,
+ 505,
+ 586
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 574,
+ 505,
+ 586
+ ],
+ "score": 1.0,
+ "content": "The PubMed dataset is a citation network dataset consisting of 19717 papers and 44338 links. Each",
+ "type": "text"
+ }
+ ],
+ "index": 31
+ },
+ {
+ "bbox": [
+ 104,
+ 585,
+ 506,
+ 598
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 585,
+ 506,
+ 598
+ ],
+ "score": 1.0,
+ "content": "paper is represented as the occurence of 500 unique words and is associated to one of 3 genres",
+ "type": "text"
+ }
+ ],
+ "index": 32
+ },
+ {
+ "bbox": [
+ 106,
+ 596,
+ 505,
+ 608
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 596,
+ 468,
+ 608
+ ],
+ "score": 1.0,
+ "content": "(“Diabetes Mellitus, Experimental”, “Diabetes Mellitus Type 1”, “Diabetes Mellitus Type",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 468,
+ 596,
+ 479,
+ 606
+ ],
+ "score": 0.5,
+ "content": "2 ^ { \\circ }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 479,
+ 596,
+ 505,
+ 608
+ ],
+ "score": 1.0,
+ "content": "). The",
+ "type": "text"
+ }
+ ],
+ "index": 33
+ },
+ {
+ "bbox": [
+ 104,
+ 606,
+ 506,
+ 621
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 606,
+ 506,
+ 621
+ ],
+ "score": 1.0,
+ "content": "graph made from the citation links has 438 connected components and the smallest positive eigen-",
+ "type": "text"
+ }
+ ],
+ "index": 34
+ },
+ {
+ "bbox": [
+ 105,
+ 617,
+ 506,
+ 630
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 617,
+ 366,
+ 630
+ ],
+ "score": 1.0,
+ "content": "value of the augmented Normalized Laplacian is approximately",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 366,
+ 617,
+ 440,
+ 629
+ ],
+ "score": 0.9,
+ "content": "\\tilde { \\mu } = 9 . 4 8 \\times 1 0 ^ { - 3 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 440,
+ 617,
+ 506,
+ 630
+ ],
+ "score": 1.0,
+ "content": ". Therefore, the",
+ "type": "text"
+ }
+ ],
+ "index": 35
+ },
+ {
+ "bbox": [
+ 104,
+ 627,
+ 506,
+ 642
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 627,
+ 224,
+ 642
+ ],
+ "score": 1.0,
+ "content": "upper bound of Theorem 2 is",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 225,
+ 628,
+ 376,
+ 641
+ ],
+ "score": 0.93,
+ "content": "\\lambda ^ { - 1 } = ( \\bar { 1 ^ { - } } \\tilde { \\mu } ) ^ { - 1 } \\approx \\bar { 1 ^ { + } } 9 . 5 7 \\times \\bar { 1 0 ^ { - 3 } }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 376,
+ 627,
+ 506,
+ 642
+ ],
+ "score": 1.0,
+ "content": ". 7875 out of 19717 samples are",
+ "type": "text"
+ }
+ ],
+ "index": 36
+ },
+ {
+ "bbox": [
+ 105,
+ 639,
+ 506,
+ 653
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 639,
+ 252,
+ 653
+ ],
+ "score": 1.0,
+ "content": "labelled as “Diabetes Mellitus Type",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 253,
+ 641,
+ 263,
+ 650
+ ],
+ "score": 0.7,
+ "content": "2 ^ { \\circ }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 264,
+ 639,
+ 506,
+ 653
+ ],
+ "score": 1.0,
+ "content": ", which is the largest proportion. Therefore, the chance rate",
+ "type": "text"
+ }
+ ],
+ "index": 37
+ },
+ {
+ "bbox": [
+ 105,
+ 650,
+ 209,
+ 663
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 650,
+ 116,
+ 663
+ ],
+ "score": 1.0,
+ "content": "is",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 116,
+ 651,
+ 206,
+ 663
+ ],
+ "score": 0.85,
+ "content": "7 8 7 5 / 1 9 7 1 7 = 3 9 . 9 \\%",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 207,
+ 650,
+ 209,
+ 663
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 38
+ }
+ ],
+ "index": 34.5,
+ "bbox_fs": [
+ 104,
+ 574,
+ 506,
+ 663
+ ]
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 107,
+ 676,
+ 267,
+ 688
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 675,
+ 268,
+ 690
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 675,
+ 268,
+ 690
+ ],
+ "score": 1.0,
+ "content": "H.3.2 NOISY CITATION NETWORKS",
+ "type": "text"
+ }
+ ],
+ "index": 39
+ }
+ ],
+ "index": 39
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 697,
+ 375,
+ 709
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 696,
+ 375,
+ 709
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 696,
+ 375,
+ 709
+ ],
+ "score": 1.0,
+ "content": "We summarize the properties of noisy citation networks in Table 2.",
+ "type": "text"
+ }
+ ],
+ "index": 40
+ }
+ ],
+ "index": 40,
+ "bbox_fs": [
+ 106,
+ 696,
+ 375,
+ 709
+ ]
+ }
+ ]
+ },
+ {
+ "preproc_blocks": [
+ {
+ "type": "image",
+ "bbox": [
+ 124,
+ 86,
+ 486,
+ 323
+ ],
+ "blocks": [
+ {
+ "type": "image_body",
+ "bbox": [
+ 124,
+ 86,
+ 486,
+ 323
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 124,
+ 86,
+ 486,
+ 323
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 124,
+ 86,
+ 486,
+ 323
+ ],
+ "score": 0.976,
+ "type": "image",
+ "image_path": "33c624416452a1afbbaf1d3a06a17fe54c6e9a677c7771baa460eb82489179e5.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 1,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 124,
+ 86,
+ 486,
+ 165.0
+ ],
+ "spans": [],
+ "index": 0
+ },
+ {
+ "bbox": [
+ 124,
+ 165.0,
+ 486,
+ 244.0
+ ],
+ "spans": [],
+ "index": 1
+ },
+ {
+ "bbox": [
+ 124,
+ 244.0,
+ 486,
+ 323.0
+ ],
+ "spans": [],
+ "index": 2
+ }
+ ]
+ },
+ {
+ "type": "image_caption",
+ "bbox": [
+ 106,
+ 342,
+ 505,
+ 375
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 342,
+ 505,
+ 354
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 342,
+ 505,
+ 354
+ ],
+ "score": 1.0,
+ "content": "Figure 5: Spectral distribution of Laplacian for the citation network datasets. Left: normalized",
+ "type": "text"
+ }
+ ],
+ "index": 3
+ },
+ {
+ "bbox": [
+ 105,
+ 354,
+ 505,
+ 366
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 354,
+ 505,
+ 366
+ ],
+ "score": 1.0,
+ "content": "Laplacian. Right: augmented normalized Laplacian. Top: Cora and Noisy Cora (2500, 5000).",
+ "type": "text"
+ }
+ ],
+ "index": 4
+ },
+ {
+ "bbox": [
+ 105,
+ 363,
+ 261,
+ 376
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 363,
+ 261,
+ 376
+ ],
+ "score": 1.0,
+ "content": "Bottom: CiteSeer and Noisy CiteSeer.",
+ "type": "text"
+ }
+ ],
+ "index": 5
+ }
+ ],
+ "index": 4
+ }
+ ],
+ "index": 2.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 397,
+ 505,
+ 508
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 396,
+ 505,
+ 411
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 396,
+ 505,
+ 411
+ ],
+ "score": 1.0,
+ "content": "We created two datasets from the Cora dataset: Noisy Cora 2500 and Noisy Cora 5000. Noisy",
+ "type": "text"
+ }
+ ],
+ "index": 6
+ },
+ {
+ "bbox": [
+ 105,
+ 408,
+ 505,
+ 421
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 408,
+ 505,
+ 421
+ ],
+ "score": 1.0,
+ "content": "Cora 2500 is made from the Cora dataset by uniformly randomly adding 2500 edges, respectively.",
+ "type": "text"
+ }
+ ],
+ "index": 7
+ },
+ {
+ "bbox": [
+ 106,
+ 419,
+ 506,
+ 432
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 419,
+ 506,
+ 432
+ ],
+ "score": 1.0,
+ "content": "Since some random edges are overlapped with existing edges, the number of newly-added edges",
+ "type": "text"
+ }
+ ],
+ "index": 8
+ },
+ {
+ "bbox": [
+ 105,
+ 430,
+ 505,
+ 444
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 430,
+ 505,
+ 444
+ ],
+ "score": 1.0,
+ "content": "is 2495 in total. We only changed the underlying graph from the Cora dataset and did not change",
+ "type": "text"
+ }
+ ],
+ "index": 9
+ },
+ {
+ "bbox": [
+ 105,
+ 441,
+ 506,
+ 454
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 441,
+ 506,
+ 454
+ ],
+ "score": 1.0,
+ "content": "word occurences (feature vectors) and genres (labels). The underlying graph of the Noisy Cora",
+ "type": "text"
+ }
+ ],
+ "index": 10
+ },
+ {
+ "bbox": [
+ 104,
+ 451,
+ 506,
+ 466
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 451,
+ 426,
+ 466
+ ],
+ "score": 1.0,
+ "content": "dataset has two connected components and the smallest positive eigenvalue is",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 426,
+ 452,
+ 501,
+ 464
+ ],
+ "score": 0.91,
+ "content": "\\tilde { \\mu } \\approx 9 . 6 2 \\times 1 0 ^ { - 2 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 501,
+ 451,
+ 506,
+ 466
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 11
+ },
+ {
+ "bbox": [
+ 105,
+ 462,
+ 506,
+ 476
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 462,
+ 506,
+ 476
+ ],
+ "score": 1.0,
+ "content": "Therefore, the threshold of the maximum singular values of in Theorem 2 has been increased to",
+ "type": "text"
+ }
+ ],
+ "index": 12
+ },
+ {
+ "bbox": [
+ 106,
+ 473,
+ 506,
+ 488
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 474,
+ 210,
+ 487
+ ],
+ "score": 0.92,
+ "content": "\\lambda ^ { - 1 } = ( 1 - \\tilde { \\mu } ) ^ { - 1 } \\approx 1 . 1 1",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 211,
+ 473,
+ 506,
+ 488
+ ],
+ "score": 1.0,
+ "content": ". Similarly, Noisy Cora 5000 was made by adding 5000 edges uniformaly",
+ "type": "text"
+ }
+ ],
+ "index": 13
+ },
+ {
+ "bbox": [
+ 105,
+ 484,
+ 505,
+ 498
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 484,
+ 505,
+ 498
+ ],
+ "score": 1.0,
+ "content": "randomly. The number of newly added edges is 4988 and the graph is connected (i.e., it has only 1",
+ "type": "text"
+ }
+ ],
+ "index": 14
+ },
+ {
+ "bbox": [
+ 104,
+ 496,
+ 489,
+ 510
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 496,
+ 202,
+ 510
+ ],
+ "score": 1.0,
+ "content": "connected component).",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 202,
+ 497,
+ 210,
+ 508
+ ],
+ "score": 0.82,
+ "content": "\\tilde { \\mu }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 210,
+ 496,
+ 227,
+ 510
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 228,
+ 497,
+ 235,
+ 507
+ ],
+ "score": 0.78,
+ "content": "\\lambda",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 235,
+ 496,
+ 251,
+ 510
+ ],
+ "score": 1.0,
+ "content": "are",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 251,
+ 496,
+ 322,
+ 508
+ ],
+ "score": 0.92,
+ "content": "\\tilde { \\mu } \\approx 1 . \\bar { 3 } 2 \\times 1 0 ^ { - 1 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 322,
+ 496,
+ 340,
+ 510
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 340,
+ 496,
+ 433,
+ 509
+ ],
+ "score": 0.92,
+ "content": "\\lambda = \\bar { ( 1 - \\mu ) } ^ { - 1 } \\approx 1 . 1 5",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 434,
+ 496,
+ 489,
+ 510
+ ],
+ "score": 1.0,
+ "content": ", respectively.",
+ "type": "text"
+ }
+ ],
+ "index": 15
+ }
+ ],
+ "index": 10.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 513,
+ 505,
+ 601
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 513,
+ 505,
+ 526
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 513,
+ 505,
+ 526
+ ],
+ "score": 1.0,
+ "content": "We made the noisy version of CiteSeer (Noisy CiteSeer) and PubMed (Noisy PubMed), in the sim-",
+ "type": "text"
+ }
+ ],
+ "index": 16
+ },
+ {
+ "bbox": [
+ 105,
+ 524,
+ 505,
+ 536
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 524,
+ 505,
+ 536
+ ],
+ "score": 1.0,
+ "content": "ilar way, by adding 5000 and 25000 edges uniformly randomly to the datasets. Since some random",
+ "type": "text"
+ }
+ ],
+ "index": 17
+ },
+ {
+ "bbox": [
+ 105,
+ 535,
+ 505,
+ 548
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 535,
+ 505,
+ 548
+ ],
+ "score": 1.0,
+ "content": "edges were overlapped with existing edges, 4991 and 24993 edges are newly added, respectively.",
+ "type": "text"
+ }
+ ],
+ "index": 18
+ },
+ {
+ "bbox": [
+ 106,
+ 546,
+ 505,
+ 559
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 546,
+ 449,
+ 559
+ ],
+ "score": 1.0,
+ "content": "This manipulation reduced the number of connected component of the graph to 3.",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 449,
+ 547,
+ 457,
+ 557
+ ],
+ "score": 0.73,
+ "content": "\\tilde { \\mu }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 457,
+ 546,
+ 505,
+ 559
+ ],
+ "score": 1.0,
+ "content": "is approxi-",
+ "type": "text"
+ }
+ ],
+ "index": 19
+ },
+ {
+ "bbox": [
+ 105,
+ 556,
+ 506,
+ 570
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 556,
+ 137,
+ 570
+ ],
+ "score": 1.0,
+ "content": "mately",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 137,
+ 556,
+ 189,
+ 568
+ ],
+ "score": 0.93,
+ "content": "1 . \\dot { 1 1 } \\times 1 0 ^ { - 1 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 189,
+ 556,
+ 278,
+ 570
+ ],
+ "score": 1.0,
+ "content": "(Noisy CiteSeer) and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 279,
+ 556,
+ 330,
+ 568
+ ],
+ "score": 0.91,
+ "content": "1 . 4 3 \\times 1 0 ^ { - 1 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 331,
+ 556,
+ 419,
+ 570
+ ],
+ "score": 1.0,
+ "content": "(Noisy PubMed) and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 420,
+ 557,
+ 494,
+ 569
+ ],
+ "score": 0.93,
+ "content": "\\lambda ^ { - 1 } = { \\bf \\bar { \\Phi } } ( 1 - \\bar { \\tilde { \\mu } } ) ^ { - 1 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 495,
+ 556,
+ 506,
+ 570
+ ],
+ "score": 1.0,
+ "content": "is",
+ "type": "text"
+ }
+ ],
+ "index": 20
+ },
+ {
+ "bbox": [
+ 105,
+ 568,
+ 505,
+ 582
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 568,
+ 505,
+ 582
+ ],
+ "score": 1.0,
+ "content": "approximately 1.13 (Noisy CiteSeer) and 1.17 (Noisy PubMed), respectively. Figure 5 (right) shows",
+ "type": "text"
+ }
+ ],
+ "index": 21
+ },
+ {
+ "bbox": [
+ 105,
+ 578,
+ 505,
+ 592
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 578,
+ 505,
+ 592
+ ],
+ "score": 1.0,
+ "content": "the spectral distribution of the augmented normalized Laplacian For comparison, we show in Figure",
+ "type": "text"
+ }
+ ],
+ "index": 22
+ },
+ {
+ "bbox": [
+ 104,
+ 589,
+ 429,
+ 602
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 589,
+ 429,
+ 602
+ ],
+ "score": 1.0,
+ "content": "5 (left) the spectral distribution of the normalized Laplacian for these datasets13.",
+ "type": "text"
+ }
+ ],
+ "index": 23
+ }
+ ],
+ "index": 19.5
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 108,
+ 614,
+ 245,
+ 626
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 615,
+ 247,
+ 627
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 615,
+ 247,
+ 627
+ ],
+ "score": 1.0,
+ "content": "H.3.3 MODEL ARCHITECTURE",
+ "type": "text"
+ }
+ ],
+ "index": 24
+ }
+ ],
+ "index": 24
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 635,
+ 505,
+ 701
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 634,
+ 506,
+ 647
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 634,
+ 506,
+ 647
+ ],
+ "score": 1.0,
+ "content": "We used a GCN consisting of a single node embedding layer, one to nine graph convolution layers,",
+ "type": "text"
+ }
+ ],
+ "index": 25
+ },
+ {
+ "bbox": [
+ 105,
+ 645,
+ 506,
+ 658
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 645,
+ 506,
+ 658
+ ],
+ "score": 1.0,
+ "content": "and a readout operation (Gilmer et al., 2017), which is a linear transformation common to all nodes",
+ "type": "text"
+ }
+ ],
+ "index": 26
+ },
+ {
+ "bbox": [
+ 105,
+ 657,
+ 505,
+ 668
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 657,
+ 505,
+ 668
+ ],
+ "score": 1.0,
+ "content": "in our case. We applied softmax function to the output of GCN. The output dimension of GCN",
+ "type": "text"
+ }
+ ],
+ "index": 27
+ },
+ {
+ "bbox": [
+ 105,
+ 667,
+ 505,
+ 680
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 667,
+ 505,
+ 680
+ ],
+ "score": 1.0,
+ "content": "is same as the number of classes (i.e., seven for Noisy Cora 2500/5000, six for Noisy CiteSeer,",
+ "type": "text"
+ }
+ ],
+ "index": 28
+ },
+ {
+ "bbox": [
+ 105,
+ 677,
+ 506,
+ 691
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 677,
+ 506,
+ 691
+ ],
+ "score": 1.0,
+ "content": "and three for Noisy PubMed). We treated the number of units in each graph convolution layer as",
+ "type": "text"
+ }
+ ],
+ "index": 29
+ },
+ {
+ "bbox": [
+ 105,
+ 689,
+ 505,
+ 702
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 689,
+ 409,
+ 702
+ ],
+ "score": 1.0,
+ "content": "a hyperparameter. Optionally, we specified the maximum singular values",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 410,
+ 691,
+ 416,
+ 699
+ ],
+ "score": 0.62,
+ "content": "s",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 416,
+ 689,
+ 505,
+ 702
+ ],
+ "score": 1.0,
+ "content": "of graph convolution",
+ "type": "text"
+ }
+ ],
+ "index": 30
+ }
+ ],
+ "index": 27.5
+ }
+ ],
+ "page_idx": 24,
+ "page_size": [
+ 612,
+ 792
+ ],
+ "discarded_blocks": [
+ {
+ "type": "discarded",
+ "bbox": [
+ 109,
+ 712,
+ 505,
+ 731
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 114,
+ 708,
+ 506,
+ 725
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 114,
+ 708,
+ 506,
+ 725
+ ],
+ "score": 1.0,
+ "content": "13Due to computational resource problems, we cannot compute the spectral distributions for PubMed and",
+ "type": "text"
+ }
+ ]
+ },
+ {
+ "bbox": [
+ 106,
+ 720,
+ 164,
+ 733
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 720,
+ 164,
+ 733
+ ],
+ "score": 1.0,
+ "content": "Noisy Pubmed.",
+ "type": "text"
+ }
+ ]
+ }
+ ]
+ },
+ {
+ "type": "discarded",
+ "bbox": [
+ 108,
+ 27,
+ 293,
+ 37
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 26,
+ 294,
+ 38
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 26,
+ 294,
+ 38
+ ],
+ "score": 1.0,
+ "content": "Published as a conference paper at ICLR 2020",
+ "type": "text"
+ }
+ ]
+ }
+ ]
+ },
+ {
+ "type": "discarded",
+ "bbox": [
+ 300,
+ 751,
+ 311,
+ 760
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 298,
+ 750,
+ 313,
+ 764
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 298,
+ 750,
+ 313,
+ 764
+ ],
+ "score": 1.0,
+ "content": "",
+ "type": "text",
+ "height": 14,
+ "width": 15
+ }
+ ]
+ }
+ ]
+ }
+ ],
+ "para_blocks": [
+ {
+ "type": "image",
+ "bbox": [
+ 124,
+ 86,
+ 486,
+ 323
+ ],
+ "blocks": [
+ {
+ "type": "image_body",
+ "bbox": [
+ 124,
+ 86,
+ 486,
+ 323
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 124,
+ 86,
+ 486,
+ 323
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 124,
+ 86,
+ 486,
+ 323
+ ],
+ "score": 0.976,
+ "type": "image",
+ "image_path": "33c624416452a1afbbaf1d3a06a17fe54c6e9a677c7771baa460eb82489179e5.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 1,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 124,
+ 86,
+ 486,
+ 165.0
+ ],
+ "spans": [],
+ "index": 0
+ },
+ {
+ "bbox": [
+ 124,
+ 165.0,
+ 486,
+ 244.0
+ ],
+ "spans": [],
+ "index": 1
+ },
+ {
+ "bbox": [
+ 124,
+ 244.0,
+ 486,
+ 323.0
+ ],
+ "spans": [],
+ "index": 2
+ }
+ ]
+ },
+ {
+ "type": "image_caption",
+ "bbox": [
+ 106,
+ 342,
+ 505,
+ 375
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 342,
+ 505,
+ 354
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 342,
+ 505,
+ 354
+ ],
+ "score": 1.0,
+ "content": "Figure 5: Spectral distribution of Laplacian for the citation network datasets. Left: normalized",
+ "type": "text"
+ }
+ ],
+ "index": 3
+ },
+ {
+ "bbox": [
+ 105,
+ 354,
+ 505,
+ 366
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 354,
+ 505,
+ 366
+ ],
+ "score": 1.0,
+ "content": "Laplacian. Right: augmented normalized Laplacian. Top: Cora and Noisy Cora (2500, 5000).",
+ "type": "text"
+ }
+ ],
+ "index": 4
+ },
+ {
+ "bbox": [
+ 105,
+ 363,
+ 261,
+ 376
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 363,
+ 261,
+ 376
+ ],
+ "score": 1.0,
+ "content": "Bottom: CiteSeer and Noisy CiteSeer.",
+ "type": "text"
+ }
+ ],
+ "index": 5
+ }
+ ],
+ "index": 4
+ }
+ ],
+ "index": 2.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 397,
+ 505,
+ 508
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 396,
+ 505,
+ 411
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 396,
+ 505,
+ 411
+ ],
+ "score": 1.0,
+ "content": "We created two datasets from the Cora dataset: Noisy Cora 2500 and Noisy Cora 5000. Noisy",
+ "type": "text"
+ }
+ ],
+ "index": 6
+ },
+ {
+ "bbox": [
+ 105,
+ 408,
+ 505,
+ 421
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 408,
+ 505,
+ 421
+ ],
+ "score": 1.0,
+ "content": "Cora 2500 is made from the Cora dataset by uniformly randomly adding 2500 edges, respectively.",
+ "type": "text"
+ }
+ ],
+ "index": 7
+ },
+ {
+ "bbox": [
+ 106,
+ 419,
+ 506,
+ 432
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 419,
+ 506,
+ 432
+ ],
+ "score": 1.0,
+ "content": "Since some random edges are overlapped with existing edges, the number of newly-added edges",
+ "type": "text"
+ }
+ ],
+ "index": 8
+ },
+ {
+ "bbox": [
+ 105,
+ 430,
+ 505,
+ 444
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 430,
+ 505,
+ 444
+ ],
+ "score": 1.0,
+ "content": "is 2495 in total. We only changed the underlying graph from the Cora dataset and did not change",
+ "type": "text"
+ }
+ ],
+ "index": 9
+ },
+ {
+ "bbox": [
+ 105,
+ 441,
+ 506,
+ 454
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 441,
+ 506,
+ 454
+ ],
+ "score": 1.0,
+ "content": "word occurences (feature vectors) and genres (labels). The underlying graph of the Noisy Cora",
+ "type": "text"
+ }
+ ],
+ "index": 10
+ },
+ {
+ "bbox": [
+ 104,
+ 451,
+ 506,
+ 466
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 451,
+ 426,
+ 466
+ ],
+ "score": 1.0,
+ "content": "dataset has two connected components and the smallest positive eigenvalue is",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 426,
+ 452,
+ 501,
+ 464
+ ],
+ "score": 0.91,
+ "content": "\\tilde { \\mu } \\approx 9 . 6 2 \\times 1 0 ^ { - 2 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 501,
+ 451,
+ 506,
+ 466
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 11
+ },
+ {
+ "bbox": [
+ 105,
+ 462,
+ 506,
+ 476
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 462,
+ 506,
+ 476
+ ],
+ "score": 1.0,
+ "content": "Therefore, the threshold of the maximum singular values of in Theorem 2 has been increased to",
+ "type": "text"
+ }
+ ],
+ "index": 12
+ },
+ {
+ "bbox": [
+ 106,
+ 473,
+ 506,
+ 488
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 474,
+ 210,
+ 487
+ ],
+ "score": 0.92,
+ "content": "\\lambda ^ { - 1 } = ( 1 - \\tilde { \\mu } ) ^ { - 1 } \\approx 1 . 1 1",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 211,
+ 473,
+ 506,
+ 488
+ ],
+ "score": 1.0,
+ "content": ". Similarly, Noisy Cora 5000 was made by adding 5000 edges uniformaly",
+ "type": "text"
+ }
+ ],
+ "index": 13
+ },
+ {
+ "bbox": [
+ 105,
+ 484,
+ 505,
+ 498
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 484,
+ 505,
+ 498
+ ],
+ "score": 1.0,
+ "content": "randomly. The number of newly added edges is 4988 and the graph is connected (i.e., it has only 1",
+ "type": "text"
+ }
+ ],
+ "index": 14
+ },
+ {
+ "bbox": [
+ 104,
+ 496,
+ 489,
+ 510
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 496,
+ 202,
+ 510
+ ],
+ "score": 1.0,
+ "content": "connected component).",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 202,
+ 497,
+ 210,
+ 508
+ ],
+ "score": 0.82,
+ "content": "\\tilde { \\mu }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 210,
+ 496,
+ 227,
+ 510
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 228,
+ 497,
+ 235,
+ 507
+ ],
+ "score": 0.78,
+ "content": "\\lambda",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 235,
+ 496,
+ 251,
+ 510
+ ],
+ "score": 1.0,
+ "content": "are",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 251,
+ 496,
+ 322,
+ 508
+ ],
+ "score": 0.92,
+ "content": "\\tilde { \\mu } \\approx 1 . \\bar { 3 } 2 \\times 1 0 ^ { - 1 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 322,
+ 496,
+ 340,
+ 510
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 340,
+ 496,
+ 433,
+ 509
+ ],
+ "score": 0.92,
+ "content": "\\lambda = \\bar { ( 1 - \\mu ) } ^ { - 1 } \\approx 1 . 1 5",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 434,
+ 496,
+ 489,
+ 510
+ ],
+ "score": 1.0,
+ "content": ", respectively.",
+ "type": "text"
+ }
+ ],
+ "index": 15
+ }
+ ],
+ "index": 10.5,
+ "bbox_fs": [
+ 104,
+ 396,
+ 506,
+ 510
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 513,
+ 505,
+ 601
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 513,
+ 505,
+ 526
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 513,
+ 505,
+ 526
+ ],
+ "score": 1.0,
+ "content": "We made the noisy version of CiteSeer (Noisy CiteSeer) and PubMed (Noisy PubMed), in the sim-",
+ "type": "text"
+ }
+ ],
+ "index": 16
+ },
+ {
+ "bbox": [
+ 105,
+ 524,
+ 505,
+ 536
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 524,
+ 505,
+ 536
+ ],
+ "score": 1.0,
+ "content": "ilar way, by adding 5000 and 25000 edges uniformly randomly to the datasets. Since some random",
+ "type": "text"
+ }
+ ],
+ "index": 17
+ },
+ {
+ "bbox": [
+ 105,
+ 535,
+ 505,
+ 548
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 535,
+ 505,
+ 548
+ ],
+ "score": 1.0,
+ "content": "edges were overlapped with existing edges, 4991 and 24993 edges are newly added, respectively.",
+ "type": "text"
+ }
+ ],
+ "index": 18
+ },
+ {
+ "bbox": [
+ 106,
+ 546,
+ 505,
+ 559
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 546,
+ 449,
+ 559
+ ],
+ "score": 1.0,
+ "content": "This manipulation reduced the number of connected component of the graph to 3.",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 449,
+ 547,
+ 457,
+ 557
+ ],
+ "score": 0.73,
+ "content": "\\tilde { \\mu }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 457,
+ 546,
+ 505,
+ 559
+ ],
+ "score": 1.0,
+ "content": "is approxi-",
+ "type": "text"
+ }
+ ],
+ "index": 19
+ },
+ {
+ "bbox": [
+ 105,
+ 556,
+ 506,
+ 570
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 556,
+ 137,
+ 570
+ ],
+ "score": 1.0,
+ "content": "mately",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 137,
+ 556,
+ 189,
+ 568
+ ],
+ "score": 0.93,
+ "content": "1 . \\dot { 1 1 } \\times 1 0 ^ { - 1 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 189,
+ 556,
+ 278,
+ 570
+ ],
+ "score": 1.0,
+ "content": "(Noisy CiteSeer) and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 279,
+ 556,
+ 330,
+ 568
+ ],
+ "score": 0.91,
+ "content": "1 . 4 3 \\times 1 0 ^ { - 1 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 331,
+ 556,
+ 419,
+ 570
+ ],
+ "score": 1.0,
+ "content": "(Noisy PubMed) and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 420,
+ 557,
+ 494,
+ 569
+ ],
+ "score": 0.93,
+ "content": "\\lambda ^ { - 1 } = { \\bf \\bar { \\Phi } } ( 1 - \\bar { \\tilde { \\mu } } ) ^ { - 1 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 495,
+ 556,
+ 506,
+ 570
+ ],
+ "score": 1.0,
+ "content": "is",
+ "type": "text"
+ }
+ ],
+ "index": 20
+ },
+ {
+ "bbox": [
+ 105,
+ 568,
+ 505,
+ 582
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 568,
+ 505,
+ 582
+ ],
+ "score": 1.0,
+ "content": "approximately 1.13 (Noisy CiteSeer) and 1.17 (Noisy PubMed), respectively. Figure 5 (right) shows",
+ "type": "text"
+ }
+ ],
+ "index": 21
+ },
+ {
+ "bbox": [
+ 105,
+ 578,
+ 505,
+ 592
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 578,
+ 505,
+ 592
+ ],
+ "score": 1.0,
+ "content": "the spectral distribution of the augmented normalized Laplacian For comparison, we show in Figure",
+ "type": "text"
+ }
+ ],
+ "index": 22
+ },
+ {
+ "bbox": [
+ 104,
+ 589,
+ 429,
+ 602
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 589,
+ 429,
+ 602
+ ],
+ "score": 1.0,
+ "content": "5 (left) the spectral distribution of the normalized Laplacian for these datasets13.",
+ "type": "text"
+ }
+ ],
+ "index": 23
+ }
+ ],
+ "index": 19.5,
+ "bbox_fs": [
+ 104,
+ 513,
+ 506,
+ 602
+ ]
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 108,
+ 614,
+ 245,
+ 626
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 615,
+ 247,
+ 627
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 615,
+ 247,
+ 627
+ ],
+ "score": 1.0,
+ "content": "H.3.3 MODEL ARCHITECTURE",
+ "type": "text"
+ }
+ ],
+ "index": 24
+ }
+ ],
+ "index": 24
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 635,
+ 505,
+ 701
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 634,
+ 506,
+ 647
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 634,
+ 506,
+ 647
+ ],
+ "score": 1.0,
+ "content": "We used a GCN consisting of a single node embedding layer, one to nine graph convolution layers,",
+ "type": "text"
+ }
+ ],
+ "index": 25
+ },
+ {
+ "bbox": [
+ 105,
+ 645,
+ 506,
+ 658
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 645,
+ 506,
+ 658
+ ],
+ "score": 1.0,
+ "content": "and a readout operation (Gilmer et al., 2017), which is a linear transformation common to all nodes",
+ "type": "text"
+ }
+ ],
+ "index": 26
+ },
+ {
+ "bbox": [
+ 105,
+ 657,
+ 505,
+ 668
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 657,
+ 505,
+ 668
+ ],
+ "score": 1.0,
+ "content": "in our case. We applied softmax function to the output of GCN. The output dimension of GCN",
+ "type": "text"
+ }
+ ],
+ "index": 27
+ },
+ {
+ "bbox": [
+ 105,
+ 667,
+ 505,
+ 680
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 667,
+ 505,
+ 680
+ ],
+ "score": 1.0,
+ "content": "is same as the number of classes (i.e., seven for Noisy Cora 2500/5000, six for Noisy CiteSeer,",
+ "type": "text"
+ }
+ ],
+ "index": 28
+ },
+ {
+ "bbox": [
+ 105,
+ 677,
+ 506,
+ 691
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 677,
+ 506,
+ 691
+ ],
+ "score": 1.0,
+ "content": "and three for Noisy PubMed). We treated the number of units in each graph convolution layer as",
+ "type": "text"
+ }
+ ],
+ "index": 29
+ },
+ {
+ "bbox": [
+ 105,
+ 689,
+ 505,
+ 702
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 689,
+ 409,
+ 702
+ ],
+ "score": 1.0,
+ "content": "a hyperparameter. Optionally, we specified the maximum singular values",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 410,
+ 691,
+ 416,
+ 699
+ ],
+ "score": 0.62,
+ "content": "s",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 416,
+ 689,
+ 505,
+ 702
+ ],
+ "score": 1.0,
+ "content": "of graph convolution",
+ "type": "text"
+ }
+ ],
+ "index": 30
+ },
+ {
+ "bbox": [
+ 105,
+ 212,
+ 505,
+ 225
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 212,
+ 194,
+ 225
+ ],
+ "score": 1.0,
+ "content": "layers. The choice of",
+ "type": "text",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 195,
+ 216,
+ 201,
+ 223
+ ],
+ "score": 0.75,
+ "content": "s",
+ "type": "inline_equation",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 201,
+ 212,
+ 321,
+ 225
+ ],
+ "score": 1.0,
+ "content": "is either 0.5 (smaller than 1),",
+ "type": "text",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 321,
+ 214,
+ 331,
+ 224
+ ],
+ "score": 0.83,
+ "content": "s _ { 1 }",
+ "type": "inline_equation",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 332,
+ 212,
+ 394,
+ 225
+ ],
+ "score": 1.0,
+ "content": "(in the interval",
+ "type": "text",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 395,
+ 213,
+ 460,
+ 225
+ ],
+ "score": 0.9,
+ "content": "\\{ 1 \\leq s < \\lambda ^ { - 1 } \\}",
+ "type": "inline_equation",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 461,
+ 212,
+ 505,
+ 225
+ ],
+ "score": 1.0,
+ "content": "), 3 and 10",
+ "type": "text",
+ "cross_page": true
+ }
+ ],
+ "index": 6
+ },
+ {
+ "bbox": [
+ 106,
+ 223,
+ 505,
+ 236
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 223,
+ 156,
+ 236
+ ],
+ "score": 1.0,
+ "content": "(larger than",
+ "type": "text",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 156,
+ 223,
+ 174,
+ 235
+ ],
+ "score": 0.92,
+ "content": "\\lambda ^ { - 1 }",
+ "type": "inline_equation",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 174,
+ 223,
+ 220,
+ 236
+ ],
+ "score": 1.0,
+ "content": "). We used",
+ "type": "text",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 221,
+ 225,
+ 264,
+ 235
+ ],
+ "score": 0.9,
+ "content": "s _ { 1 } = 1 . 0 5",
+ "type": "inline_equation",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 264,
+ 223,
+ 505,
+ 236
+ ],
+ "score": 1.0,
+ "content": "for Noisy Cora 2500, Noisy CiteSeer, and Noisy PubMed,",
+ "type": "text",
+ "cross_page": true
+ }
+ ],
+ "index": 7
+ },
+ {
+ "bbox": [
+ 106,
+ 235,
+ 505,
+ 248
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 235,
+ 123,
+ 248
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 123,
+ 236,
+ 160,
+ 246
+ ],
+ "score": 0.9,
+ "content": "s _ { 1 } = 1 . 1",
+ "type": "inline_equation",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 160,
+ 235,
+ 355,
+ 248
+ ],
+ "score": 1.0,
+ "content": "for Noisy Cora 5000 and Noisy CiteSeer so that",
+ "type": "text",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 355,
+ 236,
+ 366,
+ 246
+ ],
+ "score": 0.85,
+ "content": "s _ { 1 }",
+ "type": "inline_equation",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 366,
+ 235,
+ 505,
+ 248
+ ],
+ "score": 1.0,
+ "content": "is not close to the edges of the the",
+ "type": "text",
+ "cross_page": true
+ }
+ ],
+ "index": 8
+ },
+ {
+ "bbox": [
+ 106,
+ 245,
+ 207,
+ 258
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 245,
+ 139,
+ 258
+ ],
+ "score": 1.0,
+ "content": "interval",
+ "type": "text",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 140,
+ 246,
+ 203,
+ 258
+ ],
+ "score": 0.93,
+ "content": "\\{ 1 \\leq s < \\lambda ^ { - 1 } \\}",
+ "type": "inline_equation",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 203,
+ 245,
+ 207,
+ 258
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text",
+ "cross_page": true
+ }
+ ],
+ "index": 9
+ }
+ ],
+ "index": 27.5,
+ "bbox_fs": [
+ 105,
+ 634,
+ 506,
+ 702
+ ]
+ }
+ ]
+ },
+ {
+ "preproc_blocks": [
+ {
+ "type": "table",
+ "bbox": [
+ 177,
+ 124,
+ 434,
+ 192
+ ],
+ "blocks": [
+ {
+ "type": "table_caption",
+ "bbox": [
+ 106,
+ 91,
+ 504,
+ 124
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 90,
+ 505,
+ 104
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 90,
+ 354,
+ 104
+ ],
+ "score": 1.0,
+ "content": "Table 3: Hyperparameters of the experiment in Section 6.3.",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 354,
+ 90,
+ 455,
+ 103
+ ],
+ "score": 0.88,
+ "content": "X \\sim \\mathrm { L o g U n i f } [ 1 0 ^ { a } , 1 0 ^ { b } ]",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 455,
+ 90,
+ 505,
+ 104
+ ],
+ "score": 1.0,
+ "content": "denotes the",
+ "type": "text"
+ }
+ ],
+ "index": 0
+ },
+ {
+ "bbox": [
+ 105,
+ 101,
+ 506,
+ 115
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 101,
+ 174,
+ 115
+ ],
+ "score": 1.0,
+ "content": "random variable",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 174,
+ 102,
+ 208,
+ 114
+ ],
+ "score": 0.92,
+ "content": "\\log _ { 1 0 } X",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 208,
+ 101,
+ 353,
+ 115
+ ],
+ "score": 1.0,
+ "content": "obeys the uniform distribution over",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 353,
+ 102,
+ 374,
+ 114
+ ],
+ "score": 0.87,
+ "content": "[ a , b ]",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 374,
+ 101,
+ 506,
+ 115
+ ],
+ "score": 1.0,
+ "content": ". “Learning rate” corresponds to",
+ "type": "text"
+ }
+ ],
+ "index": 1
+ },
+ {
+ "bbox": [
+ 107,
+ 113,
+ 373,
+ 125
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 115,
+ 114,
+ 123
+ ],
+ "score": 0.71,
+ "content": "\\alpha",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 114,
+ 113,
+ 373,
+ 125
+ ],
+ "score": 1.0,
+ "content": "when “Optimization algorithm” is Adam (Kingma & Ba, 2015).",
+ "type": "text"
+ }
+ ],
+ "index": 2
+ }
+ ],
+ "index": 1
+ },
+ {
+ "type": "table_body",
+ "bbox": [
+ 177,
+ 124,
+ 434,
+ 192
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 177,
+ 124,
+ 434,
+ 192
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 177,
+ 124,
+ 434,
+ 192
+ ],
+ "score": 0.979,
+ "html": "| Name | Value |
| Unit size | {10,20,..., 500} |
| Epoch | {10,20,...,100} |
| Optimization algorithm | {SGD,MomentumSGD,Adam} |
| Learning rate | LogUnif[10-5,10-2] |
",
+ "type": "table",
+ "image_path": "80ab11f7788859aa5cff8fafb447d24232326adb5a8f87e277a4cff910d30034.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 4,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 177,
+ 124,
+ 434,
+ 146.66666666666666
+ ],
+ "spans": [],
+ "index": 3
+ },
+ {
+ "bbox": [
+ 177,
+ 146.66666666666666,
+ 434,
+ 169.33333333333331
+ ],
+ "spans": [],
+ "index": 4
+ },
+ {
+ "bbox": [
+ 177,
+ 169.33333333333331,
+ 434,
+ 191.99999999999997
+ ],
+ "spans": [],
+ "index": 5
+ }
+ ]
+ }
+ ],
+ "index": 2.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 213,
+ 505,
+ 258
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 212,
+ 505,
+ 225
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 212,
+ 194,
+ 225
+ ],
+ "score": 1.0,
+ "content": "layers. The choice of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 195,
+ 216,
+ 201,
+ 223
+ ],
+ "score": 0.75,
+ "content": "s",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 201,
+ 212,
+ 321,
+ 225
+ ],
+ "score": 1.0,
+ "content": "is either 0.5 (smaller than 1),",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 321,
+ 214,
+ 331,
+ 224
+ ],
+ "score": 0.83,
+ "content": "s _ { 1 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 332,
+ 212,
+ 394,
+ 225
+ ],
+ "score": 1.0,
+ "content": "(in the interval",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 395,
+ 213,
+ 460,
+ 225
+ ],
+ "score": 0.9,
+ "content": "\\{ 1 \\leq s < \\lambda ^ { - 1 } \\}",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 461,
+ 212,
+ 505,
+ 225
+ ],
+ "score": 1.0,
+ "content": "), 3 and 10",
+ "type": "text"
+ }
+ ],
+ "index": 6
+ },
+ {
+ "bbox": [
+ 106,
+ 223,
+ 505,
+ 236
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 223,
+ 156,
+ 236
+ ],
+ "score": 1.0,
+ "content": "(larger than",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 156,
+ 223,
+ 174,
+ 235
+ ],
+ "score": 0.92,
+ "content": "\\lambda ^ { - 1 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 174,
+ 223,
+ 220,
+ 236
+ ],
+ "score": 1.0,
+ "content": "). We used",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 221,
+ 225,
+ 264,
+ 235
+ ],
+ "score": 0.9,
+ "content": "s _ { 1 } = 1 . 0 5",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 264,
+ 223,
+ 505,
+ 236
+ ],
+ "score": 1.0,
+ "content": "for Noisy Cora 2500, Noisy CiteSeer, and Noisy PubMed,",
+ "type": "text"
+ }
+ ],
+ "index": 7
+ },
+ {
+ "bbox": [
+ 106,
+ 235,
+ 505,
+ 248
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 235,
+ 123,
+ 248
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 123,
+ 236,
+ 160,
+ 246
+ ],
+ "score": 0.9,
+ "content": "s _ { 1 } = 1 . 1",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 160,
+ 235,
+ 355,
+ 248
+ ],
+ "score": 1.0,
+ "content": "for Noisy Cora 5000 and Noisy CiteSeer so that",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 355,
+ 236,
+ 366,
+ 246
+ ],
+ "score": 0.85,
+ "content": "s _ { 1 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 366,
+ 235,
+ 505,
+ 248
+ ],
+ "score": 1.0,
+ "content": "is not close to the edges of the the",
+ "type": "text"
+ }
+ ],
+ "index": 8
+ },
+ {
+ "bbox": [
+ 106,
+ 245,
+ 207,
+ 258
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 245,
+ 139,
+ 258
+ ],
+ "score": 1.0,
+ "content": "interval",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 140,
+ 246,
+ 203,
+ 258
+ ],
+ "score": 0.93,
+ "content": "\\{ 1 \\leq s < \\lambda ^ { - 1 } \\}",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 203,
+ 245,
+ 207,
+ 258
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 9
+ }
+ ],
+ "index": 7.5
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 109,
+ 270,
+ 320,
+ 282
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 270,
+ 322,
+ 283
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 270,
+ 322,
+ 283
+ ],
+ "score": 1.0,
+ "content": "H.3.4 PERFORMANCE EVALUATION PROCEDURE",
+ "type": "text"
+ }
+ ],
+ "index": 10
+ }
+ ],
+ "index": 10
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 289,
+ 505,
+ 378
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 289,
+ 505,
+ 303
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 289,
+ 505,
+ 303
+ ],
+ "score": 1.0,
+ "content": "We split all nodes in a graph (either Noisy Cora 2500/5000 or Noisy CiteSeer) into training, val-",
+ "type": "text"
+ }
+ ],
+ "index": 11
+ },
+ {
+ "bbox": [
+ 105,
+ 300,
+ 505,
+ 313
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 300,
+ 505,
+ 313
+ ],
+ "score": 1.0,
+ "content": "idation, and test sets. Data split is the same as the one done by Kipf & Welling (2017). This",
+ "type": "text"
+ }
+ ],
+ "index": 12
+ },
+ {
+ "bbox": [
+ 105,
+ 312,
+ 505,
+ 325
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 312,
+ 505,
+ 325
+ ],
+ "score": 1.0,
+ "content": "is a transductive learning (Pan & Yang, 2010) setting because we can use node properties of the",
+ "type": "text"
+ }
+ ],
+ "index": 13
+ },
+ {
+ "bbox": [
+ 105,
+ 322,
+ 505,
+ 336
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 322,
+ 505,
+ 336
+ ],
+ "score": 1.0,
+ "content": "validation and test data during training. We trained the model three times for each choice of hyper-",
+ "type": "text"
+ }
+ ],
+ "index": 14
+ },
+ {
+ "bbox": [
+ 105,
+ 334,
+ 505,
+ 346
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 334,
+ 505,
+ 346
+ ],
+ "score": 1.0,
+ "content": "paremeters using the training set and defined the objective function as the average accuracy on the",
+ "type": "text"
+ }
+ ],
+ "index": 15
+ },
+ {
+ "bbox": [
+ 105,
+ 344,
+ 505,
+ 358
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 344,
+ 505,
+ 358
+ ],
+ "score": 1.0,
+ "content": "validation set. We chose the combination of hyperparameters that achieves the best value of objec-",
+ "type": "text"
+ }
+ ],
+ "index": 16
+ },
+ {
+ "bbox": [
+ 105,
+ 356,
+ 505,
+ 369
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 356,
+ 505,
+ 369
+ ],
+ "score": 1.0,
+ "content": "tive function. We evaluate the accuracy of the test dataset three times using the chosen combination",
+ "type": "text"
+ }
+ ],
+ "index": 17
+ },
+ {
+ "bbox": [
+ 106,
+ 367,
+ 408,
+ 379
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 367,
+ 408,
+ 379
+ ],
+ "score": 1.0,
+ "content": "of hyperparameters and computed their average and the standard deviation.",
+ "type": "text"
+ }
+ ],
+ "index": 18
+ }
+ ],
+ "index": 14.5
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 107,
+ 391,
+ 186,
+ 402
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 391,
+ 187,
+ 404
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 391,
+ 187,
+ 404
+ ],
+ "score": 1.0,
+ "content": "H.3.5 TRAINING",
+ "type": "text"
+ }
+ ],
+ "index": 19
+ }
+ ],
+ "index": 19
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 411,
+ 505,
+ 466
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 411,
+ 505,
+ 423
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 411,
+ 505,
+ 423
+ ],
+ "score": 1.0,
+ "content": "At initialization, we sampled parameters from the i.i.d. Gaussian distribution. If the scale of maxi-",
+ "type": "text"
+ }
+ ],
+ "index": 20
+ },
+ {
+ "bbox": [
+ 106,
+ 423,
+ 505,
+ 434
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 423,
+ 191,
+ 434
+ ],
+ "score": 1.0,
+ "content": "mum singular values",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 192,
+ 424,
+ 198,
+ 432
+ ],
+ "score": 0.58,
+ "content": "s",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 198,
+ 423,
+ 505,
+ 434
+ ],
+ "score": 1.0,
+ "content": "was specified, we subsequently scaled weight matrices of graph convolution",
+ "type": "text"
+ }
+ ],
+ "index": 21
+ },
+ {
+ "bbox": [
+ 106,
+ 433,
+ 505,
+ 445
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 433,
+ 370,
+ 445
+ ],
+ "score": 1.0,
+ "content": "layers so that their maximum singular values were normalized to",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 371,
+ 435,
+ 376,
+ 443
+ ],
+ "score": 0.66,
+ "content": "s",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 377,
+ 433,
+ 505,
+ 445
+ ],
+ "score": 1.0,
+ "content": ". The loss function was defined",
+ "type": "text"
+ }
+ ],
+ "index": 22
+ },
+ {
+ "bbox": [
+ 106,
+ 444,
+ 505,
+ 456
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 444,
+ 505,
+ 456
+ ],
+ "score": 1.0,
+ "content": "as the sum of the cross entropy loss for all training nodes. We train the model using the one of",
+ "type": "text"
+ }
+ ],
+ "index": 23
+ },
+ {
+ "bbox": [
+ 105,
+ 455,
+ 343,
+ 468
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 455,
+ 343,
+ 468
+ ],
+ "score": 1.0,
+ "content": "gradient-based optimization methods described in Table 3.",
+ "type": "text"
+ }
+ ],
+ "index": 24
+ }
+ ],
+ "index": 22
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 108,
+ 479,
+ 224,
+ 491
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 480,
+ 225,
+ 491
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 480,
+ 225,
+ 491
+ ],
+ "score": 1.0,
+ "content": "H.3.6 HYPERPRAMETERS",
+ "type": "text"
+ }
+ ],
+ "index": 25
+ }
+ ],
+ "index": 25
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 499,
+ 504,
+ 532
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 498,
+ 504,
+ 512
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 498,
+ 504,
+ 512
+ ],
+ "score": 1.0,
+ "content": "Table 3 shows the set of hyperparameters from which we chose. Since we compute the repre-",
+ "type": "text"
+ }
+ ],
+ "index": 26
+ },
+ {
+ "bbox": [
+ 105,
+ 508,
+ 505,
+ 524
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 508,
+ 505,
+ 524
+ ],
+ "score": 1.0,
+ "content": "sentations of all nodes at once at each iteration, each epoch consists of 1 iteration. We employ",
+ "type": "text"
+ }
+ ],
+ "index": 27
+ },
+ {
+ "bbox": [
+ 105,
+ 520,
+ 464,
+ 533
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 520,
+ 464,
+ 533
+ ],
+ "score": 1.0,
+ "content": "Tree-structured Parzen Estimator (Bergstra et al., 2011) for hyperparameter optimization.",
+ "type": "text"
+ }
+ ],
+ "index": 28
+ }
+ ],
+ "index": 27
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 108,
+ 546,
+ 219,
+ 557
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 545,
+ 221,
+ 558
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 545,
+ 221,
+ 558
+ ],
+ "score": 1.0,
+ "content": "H.3.7 IMPLEMENTATION",
+ "type": "text"
+ }
+ ],
+ "index": 29
+ }
+ ],
+ "index": 29
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 565,
+ 505,
+ 643
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 563,
+ 506,
+ 579
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 563,
+ 506,
+ 579
+ ],
+ "score": 1.0,
+ "content": "We used Chainer Chemistry14, which is an extension library for the deep learning framework",
+ "type": "text"
+ }
+ ],
+ "index": 30
+ },
+ {
+ "bbox": [
+ 105,
+ 575,
+ 505,
+ 590
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 575,
+ 505,
+ 590
+ ],
+ "score": 1.0,
+ "content": "Chainer (Tokui et al., 2015; 2019), to implement GCNs and Optuna (Akiba et al., 2019) for hy-",
+ "type": "text"
+ }
+ ],
+ "index": 31
+ },
+ {
+ "bbox": [
+ 105,
+ 587,
+ 505,
+ 600
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 587,
+ 505,
+ 600
+ ],
+ "score": 1.0,
+ "content": "perparameter tuning. We conducted experiments in a signel machine which has 2 Intel(R) Xeon(R)",
+ "type": "text"
+ }
+ ],
+ "index": 32
+ },
+ {
+ "bbox": [
+ 105,
+ 598,
+ 505,
+ 610
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 598,
+ 505,
+ 610
+ ],
+ "score": 1.0,
+ "content": "Gold 6136 CPU@3.00GHz (24 cores), 192 GB memory (DDR4), and 3 GPGPUs (NVIDIA Tesla",
+ "type": "text"
+ }
+ ],
+ "index": 33
+ },
+ {
+ "bbox": [
+ 105,
+ 609,
+ 505,
+ 622
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 609,
+ 258,
+ 622
+ ],
+ "score": 1.0,
+ "content": "V100). Our implementation achieved",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 258,
+ 609,
+ 285,
+ 620
+ ],
+ "score": 0.87,
+ "content": "6 8 . 1 \\%",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 286,
+ 609,
+ 505,
+ 622
+ ],
+ "score": 1.0,
+ "content": "with Dropout (Srivastava et al., 2014) (2 graph convo-",
+ "type": "text"
+ }
+ ],
+ "index": 34
+ },
+ {
+ "bbox": [
+ 104,
+ 619,
+ 506,
+ 633
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 619,
+ 178,
+ 633
+ ],
+ "score": 1.0,
+ "content": "lution layers) and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 179,
+ 620,
+ 206,
+ 631
+ ],
+ "score": 0.87,
+ "content": "6 4 . 2 \\%",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 207,
+ 619,
+ 506,
+ 633
+ ],
+ "score": 1.0,
+ "content": "without Dropout (1 graph convolution layer) on the test dataset. These are",
+ "type": "text"
+ }
+ ],
+ "index": 35
+ },
+ {
+ "bbox": [
+ 105,
+ 631,
+ 505,
+ 644
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 631,
+ 505,
+ 644
+ ],
+ "score": 1.0,
+ "content": "slightly worse than the accuracy reported in Kipf & Welling (2017), but are still comparable with it.",
+ "type": "text"
+ }
+ ],
+ "index": 36
+ }
+ ],
+ "index": 33
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 108,
+ 657,
+ 261,
+ 668
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 657,
+ 262,
+ 669
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 657,
+ 262,
+ 669
+ ],
+ "score": 1.0,
+ "content": "H.4 EXPERIMENT OF SECTION 6.4",
+ "type": "text"
+ }
+ ],
+ "index": 37
+ }
+ ],
+ "index": 37
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 109,
+ 678,
+ 504,
+ 711
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 107,
+ 677,
+ 506,
+ 691
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 677,
+ 506,
+ 691
+ ],
+ "score": 1.0,
+ "content": "The experiment settings are almost same as the experiment in Section 6.3. The only difference is that",
+ "type": "text"
+ }
+ ],
+ "index": 38
+ },
+ {
+ "bbox": [
+ 107,
+ 689,
+ 505,
+ 701
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 689,
+ 505,
+ 701
+ ],
+ "score": 1.0,
+ "content": "we did not train the node embedding layer, which we put before convolution layers of a GCN, while",
+ "type": "text"
+ }
+ ],
+ "index": 39
+ },
+ {
+ "bbox": [
+ 106,
+ 699,
+ 505,
+ 712
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 699,
+ 505,
+ 712
+ ],
+ "score": 1.0,
+ "content": "we did in Section 6.3. This is because we wanted to see the the effect of convolution operations",
+ "type": "text"
+ }
+ ],
+ "index": 40
+ }
+ ],
+ "index": 39
+ }
+ ],
+ "page_idx": 25,
+ "page_size": [
+ 612,
+ 792
+ ],
+ "discarded_blocks": [
+ {
+ "type": "discarded",
+ "bbox": [
+ 115,
+ 722,
+ 399,
+ 732
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 115,
+ 718,
+ 401,
+ 736
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 115,
+ 718,
+ 401,
+ 736
+ ],
+ "score": 1.0,
+ "content": "14https://github.com/pfnet-research/chainer-chemistry",
+ "type": "text"
+ }
+ ]
+ }
+ ]
+ },
+ {
+ "type": "discarded",
+ "bbox": [
+ 107,
+ 27,
+ 293,
+ 37
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 26,
+ 294,
+ 38
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 26,
+ 294,
+ 38
+ ],
+ "score": 1.0,
+ "content": "Published as a conference paper at ICLR 2020",
+ "type": "text"
+ }
+ ]
+ }
+ ]
+ },
+ {
+ "type": "discarded",
+ "bbox": [
+ 300,
+ 751,
+ 311,
+ 760
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 298,
+ 749,
+ 313,
+ 764
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 298,
+ 749,
+ 313,
+ 764
+ ],
+ "score": 1.0,
+ "content": "",
+ "type": "text",
+ "height": 15,
+ "width": 15
+ }
+ ]
+ }
+ ]
+ }
+ ],
+ "para_blocks": [
+ {
+ "type": "table",
+ "bbox": [
+ 177,
+ 124,
+ 434,
+ 192
+ ],
+ "blocks": [
+ {
+ "type": "table_caption",
+ "bbox": [
+ 106,
+ 91,
+ 504,
+ 124
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 90,
+ 505,
+ 104
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 90,
+ 354,
+ 104
+ ],
+ "score": 1.0,
+ "content": "Table 3: Hyperparameters of the experiment in Section 6.3.",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 354,
+ 90,
+ 455,
+ 103
+ ],
+ "score": 0.88,
+ "content": "X \\sim \\mathrm { L o g U n i f } [ 1 0 ^ { a } , 1 0 ^ { b } ]",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 455,
+ 90,
+ 505,
+ 104
+ ],
+ "score": 1.0,
+ "content": "denotes the",
+ "type": "text"
+ }
+ ],
+ "index": 0
+ },
+ {
+ "bbox": [
+ 105,
+ 101,
+ 506,
+ 115
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 101,
+ 174,
+ 115
+ ],
+ "score": 1.0,
+ "content": "random variable",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 174,
+ 102,
+ 208,
+ 114
+ ],
+ "score": 0.92,
+ "content": "\\log _ { 1 0 } X",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 208,
+ 101,
+ 353,
+ 115
+ ],
+ "score": 1.0,
+ "content": "obeys the uniform distribution over",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 353,
+ 102,
+ 374,
+ 114
+ ],
+ "score": 0.87,
+ "content": "[ a , b ]",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 374,
+ 101,
+ 506,
+ 115
+ ],
+ "score": 1.0,
+ "content": ". “Learning rate” corresponds to",
+ "type": "text"
+ }
+ ],
+ "index": 1
+ },
+ {
+ "bbox": [
+ 107,
+ 113,
+ 373,
+ 125
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 115,
+ 114,
+ 123
+ ],
+ "score": 0.71,
+ "content": "\\alpha",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 114,
+ 113,
+ 373,
+ 125
+ ],
+ "score": 1.0,
+ "content": "when “Optimization algorithm” is Adam (Kingma & Ba, 2015).",
+ "type": "text"
+ }
+ ],
+ "index": 2
+ }
+ ],
+ "index": 1
+ },
+ {
+ "type": "table_body",
+ "bbox": [
+ 177,
+ 124,
+ 434,
+ 192
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 177,
+ 124,
+ 434,
+ 192
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 177,
+ 124,
+ 434,
+ 192
+ ],
+ "score": 0.979,
+ "html": "| Name | Value |
| Unit size | {10,20,..., 500} |
| Epoch | {10,20,...,100} |
| Optimization algorithm | {SGD,MomentumSGD,Adam} |
| Learning rate | LogUnif[10-5,10-2] |
",
+ "type": "table",
+ "image_path": "80ab11f7788859aa5cff8fafb447d24232326adb5a8f87e277a4cff910d30034.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 4,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 177,
+ 124,
+ 434,
+ 146.66666666666666
+ ],
+ "spans": [],
+ "index": 3
+ },
+ {
+ "bbox": [
+ 177,
+ 146.66666666666666,
+ 434,
+ 169.33333333333331
+ ],
+ "spans": [],
+ "index": 4
+ },
+ {
+ "bbox": [
+ 177,
+ 169.33333333333331,
+ 434,
+ 191.99999999999997
+ ],
+ "spans": [],
+ "index": 5
+ }
+ ]
+ }
+ ],
+ "index": 2.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 213,
+ 505,
+ 258
+ ],
+ "lines": [],
+ "index": 7.5,
+ "bbox_fs": [
+ 105,
+ 212,
+ 505,
+ 258
+ ],
+ "lines_deleted": true
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 109,
+ 270,
+ 320,
+ 282
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 270,
+ 322,
+ 283
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 270,
+ 322,
+ 283
+ ],
+ "score": 1.0,
+ "content": "H.3.4 PERFORMANCE EVALUATION PROCEDURE",
+ "type": "text"
+ }
+ ],
+ "index": 10
+ }
+ ],
+ "index": 10
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 289,
+ 505,
+ 378
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 289,
+ 505,
+ 303
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 289,
+ 505,
+ 303
+ ],
+ "score": 1.0,
+ "content": "We split all nodes in a graph (either Noisy Cora 2500/5000 or Noisy CiteSeer) into training, val-",
+ "type": "text"
+ }
+ ],
+ "index": 11
+ },
+ {
+ "bbox": [
+ 105,
+ 300,
+ 505,
+ 313
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 300,
+ 505,
+ 313
+ ],
+ "score": 1.0,
+ "content": "idation, and test sets. Data split is the same as the one done by Kipf & Welling (2017). This",
+ "type": "text"
+ }
+ ],
+ "index": 12
+ },
+ {
+ "bbox": [
+ 105,
+ 312,
+ 505,
+ 325
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 312,
+ 505,
+ 325
+ ],
+ "score": 1.0,
+ "content": "is a transductive learning (Pan & Yang, 2010) setting because we can use node properties of the",
+ "type": "text"
+ }
+ ],
+ "index": 13
+ },
+ {
+ "bbox": [
+ 105,
+ 322,
+ 505,
+ 336
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 322,
+ 505,
+ 336
+ ],
+ "score": 1.0,
+ "content": "validation and test data during training. We trained the model three times for each choice of hyper-",
+ "type": "text"
+ }
+ ],
+ "index": 14
+ },
+ {
+ "bbox": [
+ 105,
+ 334,
+ 505,
+ 346
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 334,
+ 505,
+ 346
+ ],
+ "score": 1.0,
+ "content": "paremeters using the training set and defined the objective function as the average accuracy on the",
+ "type": "text"
+ }
+ ],
+ "index": 15
+ },
+ {
+ "bbox": [
+ 105,
+ 344,
+ 505,
+ 358
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 344,
+ 505,
+ 358
+ ],
+ "score": 1.0,
+ "content": "validation set. We chose the combination of hyperparameters that achieves the best value of objec-",
+ "type": "text"
+ }
+ ],
+ "index": 16
+ },
+ {
+ "bbox": [
+ 105,
+ 356,
+ 505,
+ 369
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 356,
+ 505,
+ 369
+ ],
+ "score": 1.0,
+ "content": "tive function. We evaluate the accuracy of the test dataset three times using the chosen combination",
+ "type": "text"
+ }
+ ],
+ "index": 17
+ },
+ {
+ "bbox": [
+ 106,
+ 367,
+ 408,
+ 379
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 367,
+ 408,
+ 379
+ ],
+ "score": 1.0,
+ "content": "of hyperparameters and computed their average and the standard deviation.",
+ "type": "text"
+ }
+ ],
+ "index": 18
+ }
+ ],
+ "index": 14.5,
+ "bbox_fs": [
+ 105,
+ 289,
+ 505,
+ 379
+ ]
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 107,
+ 391,
+ 186,
+ 402
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 391,
+ 187,
+ 404
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 391,
+ 187,
+ 404
+ ],
+ "score": 1.0,
+ "content": "H.3.5 TRAINING",
+ "type": "text"
+ }
+ ],
+ "index": 19
+ }
+ ],
+ "index": 19
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 411,
+ 505,
+ 466
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 411,
+ 505,
+ 423
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 411,
+ 505,
+ 423
+ ],
+ "score": 1.0,
+ "content": "At initialization, we sampled parameters from the i.i.d. Gaussian distribution. If the scale of maxi-",
+ "type": "text"
+ }
+ ],
+ "index": 20
+ },
+ {
+ "bbox": [
+ 106,
+ 423,
+ 505,
+ 434
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 423,
+ 191,
+ 434
+ ],
+ "score": 1.0,
+ "content": "mum singular values",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 192,
+ 424,
+ 198,
+ 432
+ ],
+ "score": 0.58,
+ "content": "s",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 198,
+ 423,
+ 505,
+ 434
+ ],
+ "score": 1.0,
+ "content": "was specified, we subsequently scaled weight matrices of graph convolution",
+ "type": "text"
+ }
+ ],
+ "index": 21
+ },
+ {
+ "bbox": [
+ 106,
+ 433,
+ 505,
+ 445
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 433,
+ 370,
+ 445
+ ],
+ "score": 1.0,
+ "content": "layers so that their maximum singular values were normalized to",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 371,
+ 435,
+ 376,
+ 443
+ ],
+ "score": 0.66,
+ "content": "s",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 377,
+ 433,
+ 505,
+ 445
+ ],
+ "score": 1.0,
+ "content": ". The loss function was defined",
+ "type": "text"
+ }
+ ],
+ "index": 22
+ },
+ {
+ "bbox": [
+ 106,
+ 444,
+ 505,
+ 456
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 444,
+ 505,
+ 456
+ ],
+ "score": 1.0,
+ "content": "as the sum of the cross entropy loss for all training nodes. We train the model using the one of",
+ "type": "text"
+ }
+ ],
+ "index": 23
+ },
+ {
+ "bbox": [
+ 105,
+ 455,
+ 343,
+ 468
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 455,
+ 343,
+ 468
+ ],
+ "score": 1.0,
+ "content": "gradient-based optimization methods described in Table 3.",
+ "type": "text"
+ }
+ ],
+ "index": 24
+ }
+ ],
+ "index": 22,
+ "bbox_fs": [
+ 105,
+ 411,
+ 505,
+ 468
+ ]
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 108,
+ 479,
+ 224,
+ 491
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 480,
+ 225,
+ 491
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 480,
+ 225,
+ 491
+ ],
+ "score": 1.0,
+ "content": "H.3.6 HYPERPRAMETERS",
+ "type": "text"
+ }
+ ],
+ "index": 25
+ }
+ ],
+ "index": 25
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 499,
+ 504,
+ 532
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 498,
+ 504,
+ 512
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 498,
+ 504,
+ 512
+ ],
+ "score": 1.0,
+ "content": "Table 3 shows the set of hyperparameters from which we chose. Since we compute the repre-",
+ "type": "text"
+ }
+ ],
+ "index": 26
+ },
+ {
+ "bbox": [
+ 105,
+ 508,
+ 505,
+ 524
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 508,
+ 505,
+ 524
+ ],
+ "score": 1.0,
+ "content": "sentations of all nodes at once at each iteration, each epoch consists of 1 iteration. We employ",
+ "type": "text"
+ }
+ ],
+ "index": 27
+ },
+ {
+ "bbox": [
+ 105,
+ 520,
+ 464,
+ 533
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 520,
+ 464,
+ 533
+ ],
+ "score": 1.0,
+ "content": "Tree-structured Parzen Estimator (Bergstra et al., 2011) for hyperparameter optimization.",
+ "type": "text"
+ }
+ ],
+ "index": 28
+ }
+ ],
+ "index": 27,
+ "bbox_fs": [
+ 105,
+ 498,
+ 505,
+ 533
+ ]
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 108,
+ 546,
+ 219,
+ 557
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 545,
+ 221,
+ 558
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 545,
+ 221,
+ 558
+ ],
+ "score": 1.0,
+ "content": "H.3.7 IMPLEMENTATION",
+ "type": "text"
+ }
+ ],
+ "index": 29
+ }
+ ],
+ "index": 29
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 565,
+ 505,
+ 643
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 563,
+ 506,
+ 579
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 563,
+ 506,
+ 579
+ ],
+ "score": 1.0,
+ "content": "We used Chainer Chemistry14, which is an extension library for the deep learning framework",
+ "type": "text"
+ }
+ ],
+ "index": 30
+ },
+ {
+ "bbox": [
+ 105,
+ 575,
+ 505,
+ 590
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 575,
+ 505,
+ 590
+ ],
+ "score": 1.0,
+ "content": "Chainer (Tokui et al., 2015; 2019), to implement GCNs and Optuna (Akiba et al., 2019) for hy-",
+ "type": "text"
+ }
+ ],
+ "index": 31
+ },
+ {
+ "bbox": [
+ 105,
+ 587,
+ 505,
+ 600
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 587,
+ 505,
+ 600
+ ],
+ "score": 1.0,
+ "content": "perparameter tuning. We conducted experiments in a signel machine which has 2 Intel(R) Xeon(R)",
+ "type": "text"
+ }
+ ],
+ "index": 32
+ },
+ {
+ "bbox": [
+ 105,
+ 598,
+ 505,
+ 610
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 598,
+ 505,
+ 610
+ ],
+ "score": 1.0,
+ "content": "Gold 6136 CPU@3.00GHz (24 cores), 192 GB memory (DDR4), and 3 GPGPUs (NVIDIA Tesla",
+ "type": "text"
+ }
+ ],
+ "index": 33
+ },
+ {
+ "bbox": [
+ 105,
+ 609,
+ 505,
+ 622
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 609,
+ 258,
+ 622
+ ],
+ "score": 1.0,
+ "content": "V100). Our implementation achieved",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 258,
+ 609,
+ 285,
+ 620
+ ],
+ "score": 0.87,
+ "content": "6 8 . 1 \\%",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 286,
+ 609,
+ 505,
+ 622
+ ],
+ "score": 1.0,
+ "content": "with Dropout (Srivastava et al., 2014) (2 graph convo-",
+ "type": "text"
+ }
+ ],
+ "index": 34
+ },
+ {
+ "bbox": [
+ 104,
+ 619,
+ 506,
+ 633
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 619,
+ 178,
+ 633
+ ],
+ "score": 1.0,
+ "content": "lution layers) and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 179,
+ 620,
+ 206,
+ 631
+ ],
+ "score": 0.87,
+ "content": "6 4 . 2 \\%",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 207,
+ 619,
+ 506,
+ 633
+ ],
+ "score": 1.0,
+ "content": "without Dropout (1 graph convolution layer) on the test dataset. These are",
+ "type": "text"
+ }
+ ],
+ "index": 35
+ },
+ {
+ "bbox": [
+ 105,
+ 631,
+ 505,
+ 644
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 631,
+ 505,
+ 644
+ ],
+ "score": 1.0,
+ "content": "slightly worse than the accuracy reported in Kipf & Welling (2017), but are still comparable with it.",
+ "type": "text"
+ }
+ ],
+ "index": 36
+ }
+ ],
+ "index": 33,
+ "bbox_fs": [
+ 104,
+ 563,
+ 506,
+ 644
+ ]
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 108,
+ 657,
+ 261,
+ 668
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 657,
+ 262,
+ 669
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 657,
+ 262,
+ 669
+ ],
+ "score": 1.0,
+ "content": "H.4 EXPERIMENT OF SECTION 6.4",
+ "type": "text"
+ }
+ ],
+ "index": 37
+ }
+ ],
+ "index": 37
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 109,
+ 678,
+ 504,
+ 711
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 107,
+ 677,
+ 506,
+ 691
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 677,
+ 506,
+ 691
+ ],
+ "score": 1.0,
+ "content": "The experiment settings are almost same as the experiment in Section 6.3. The only difference is that",
+ "type": "text"
+ }
+ ],
+ "index": 38
+ },
+ {
+ "bbox": [
+ 107,
+ 689,
+ 505,
+ 701
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 689,
+ 505,
+ 701
+ ],
+ "score": 1.0,
+ "content": "we did not train the node embedding layer, which we put before convolution layers of a GCN, while",
+ "type": "text"
+ }
+ ],
+ "index": 39
+ },
+ {
+ "bbox": [
+ 106,
+ 699,
+ 505,
+ 712
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 699,
+ 505,
+ 712
+ ],
+ "score": 1.0,
+ "content": "we did in Section 6.3. This is because we wanted to see the the effect of convolution operations",
+ "type": "text"
+ }
+ ],
+ "index": 40
+ }
+ ],
+ "index": 39,
+ "bbox_fs": [
+ 106,
+ 677,
+ 506,
+ 712
+ ]
+ }
+ ]
+ },
+ {
+ "preproc_blocks": [
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 82,
+ 504,
+ 105
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 82,
+ 505,
+ 95
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 82,
+ 505,
+ 95
+ ],
+ "score": 1.0,
+ "content": "on the perpendicular component of signals, while we interested in the prediction accuracy in real",
+ "type": "text"
+ }
+ ],
+ "index": 0
+ },
+ {
+ "bbox": [
+ 106,
+ 94,
+ 284,
+ 106
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 94,
+ 284,
+ 106
+ ],
+ "score": 1.0,
+ "content": "training settings in the previous experiment.",
+ "type": "text"
+ }
+ ],
+ "index": 1
+ }
+ ],
+ "index": 0.5
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 106,
+ 121,
+ 312,
+ 134
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 120,
+ 312,
+ 135
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 120,
+ 312,
+ 135
+ ],
+ "score": 1.0,
+ "content": "I ADDITIONAL EXPERIMENT RESULTS",
+ "type": "text"
+ }
+ ],
+ "index": 2
+ }
+ ],
+ "index": 2
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 108,
+ 145,
+ 257,
+ 157
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 145,
+ 258,
+ 158
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 145,
+ 258,
+ 158
+ ],
+ "score": 1.0,
+ "content": "I.1 EXPERIMENT OF SECTION 6.1",
+ "type": "text"
+ }
+ ],
+ "index": 3
+ }
+ ],
+ "index": 3
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 166,
+ 504,
+ 189
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 166,
+ 505,
+ 179
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 166,
+ 208,
+ 179
+ ],
+ "score": 1.0,
+ "content": "We show the vector field",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 209,
+ 167,
+ 218,
+ 176
+ ],
+ "score": 0.73,
+ "content": "V",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 218,
+ 166,
+ 265,
+ 179
+ ],
+ "score": 1.0,
+ "content": "for various",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 266,
+ 167,
+ 277,
+ 177
+ ],
+ "score": 0.66,
+ "content": "W",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 278,
+ 166,
+ 505,
+ 179
+ ],
+ "score": 1.0,
+ "content": "in Figure 6 (Case 1) and Figure 7 (Case 2). Parameters",
+ "type": "text"
+ }
+ ],
+ "index": 4
+ },
+ {
+ "bbox": [
+ 105,
+ 177,
+ 502,
+ 190
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 177,
+ 149,
+ 190
+ ],
+ "score": 1.0,
+ "content": "other than",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 149,
+ 178,
+ 161,
+ 187
+ ],
+ "score": 0.65,
+ "content": "W",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 162,
+ 177,
+ 502,
+ 190
+ ],
+ "score": 1.0,
+ "content": "are same as experiments in Section 6.1 (detail values are available in Appendix H.1).",
+ "type": "text"
+ }
+ ],
+ "index": 5
+ }
+ ],
+ "index": 4.5
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 108,
+ 202,
+ 257,
+ 213
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 201,
+ 259,
+ 214
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 201,
+ 259,
+ 214
+ ],
+ "score": 1.0,
+ "content": "I.2 EXPERIMENT OF SECTION 6.2",
+ "type": "text"
+ }
+ ],
+ "index": 6
+ }
+ ],
+ "index": 6
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 222,
+ 505,
+ 267
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 222,
+ 505,
+ 236
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 222,
+ 505,
+ 236
+ ],
+ "score": 1.0,
+ "content": "Figure 8 shows the relative log distance of signals and their upper bound for various edge probability",
+ "type": "text"
+ }
+ ],
+ "index": 7
+ },
+ {
+ "bbox": [
+ 106,
+ 234,
+ 506,
+ 246
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 236,
+ 113,
+ 245
+ ],
+ "score": 0.77,
+ "content": "p",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 113,
+ 234,
+ 248,
+ 246
+ ],
+ "score": 1.0,
+ "content": "and the maximum singular value",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 249,
+ 236,
+ 255,
+ 244
+ ],
+ "score": 0.67,
+ "content": "s",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 255,
+ 234,
+ 506,
+ 246
+ ],
+ "score": 1.0,
+ "content": ". Note that we generate a new graph for each configuration of",
+ "type": "text"
+ }
+ ],
+ "index": 8
+ },
+ {
+ "bbox": [
+ 107,
+ 245,
+ 505,
+ 258
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 245,
+ 129,
+ 257
+ ],
+ "score": 0.9,
+ "content": "( p , s )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 129,
+ 245,
+ 457,
+ 258
+ ],
+ "score": 1.0,
+ "content": ". Therefore, different configurations may have different graphs and hence different",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 457,
+ 245,
+ 464,
+ 255
+ ],
+ "score": 0.75,
+ "content": "\\lambda",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 464,
+ 245,
+ 505,
+ 258
+ ],
+ "score": 1.0,
+ "content": "even they",
+ "type": "text"
+ }
+ ],
+ "index": 9
+ },
+ {
+ "bbox": [
+ 105,
+ 257,
+ 282,
+ 268
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 257,
+ 225,
+ 268
+ ],
+ "score": 1.0,
+ "content": "have a same edge probability",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 225,
+ 258,
+ 231,
+ 268
+ ],
+ "score": 0.8,
+ "content": "p",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 232,
+ 257,
+ 282,
+ 268
+ ],
+ "score": 1.0,
+ "content": "in common.",
+ "type": "text"
+ }
+ ],
+ "index": 10
+ }
+ ],
+ "index": 8.5
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 108,
+ 280,
+ 257,
+ 292
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 280,
+ 259,
+ 293
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 280,
+ 259,
+ 293
+ ],
+ "score": 1.0,
+ "content": "I.3 EXPERIMENT OF SECTION 6.3",
+ "type": "text"
+ }
+ ],
+ "index": 11
+ }
+ ],
+ "index": 11
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 108,
+ 301,
+ 243,
+ 312
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 301,
+ 243,
+ 314
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 301,
+ 243,
+ 314
+ ],
+ "score": 1.0,
+ "content": "I.3.1 PREDICTIVE ACCURACY",
+ "type": "text"
+ }
+ ],
+ "index": 12
+ }
+ ],
+ "index": 12
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 320,
+ 504,
+ 354
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 320,
+ 506,
+ 333
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 320,
+ 506,
+ 333
+ ],
+ "score": 1.0,
+ "content": "Figure 9 shows the comparison of predictive performance in terms the maximum singular value",
+ "type": "text"
+ }
+ ],
+ "index": 13
+ },
+ {
+ "bbox": [
+ 105,
+ 330,
+ 505,
+ 345
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 330,
+ 505,
+ 345
+ ],
+ "score": 1.0,
+ "content": "and layer size when the dataset is Noisy Cora 5000 (left) and Noisy Citeseer (right), respectively.",
+ "type": "text"
+ }
+ ],
+ "index": 14
+ },
+ {
+ "bbox": [
+ 106,
+ 343,
+ 271,
+ 353
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 343,
+ 271,
+ 353
+ ],
+ "score": 1.0,
+ "content": "Concrete values are available in Table 4.",
+ "type": "text"
+ }
+ ],
+ "index": 15
+ }
+ ],
+ "index": 14
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 106,
+ 366,
+ 340,
+ 377
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 365,
+ 341,
+ 379
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 365,
+ 341,
+ 379
+ ],
+ "score": 1.0,
+ "content": "I.3.2 TRANSITION OF MAXIMUM SINGULAR VALUES",
+ "type": "text"
+ }
+ ],
+ "index": 16
+ }
+ ],
+ "index": 16
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 385,
+ 505,
+ 419
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 384,
+ 505,
+ 398
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 384,
+ 505,
+ 398
+ ],
+ "score": 1.0,
+ "content": "Figure 10 – 13 show the transition of weight of graph convolution layers during training when the",
+ "type": "text"
+ }
+ ],
+ "index": 17
+ },
+ {
+ "bbox": [
+ 105,
+ 396,
+ 505,
+ 409
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 396,
+ 505,
+ 409
+ ],
+ "score": 1.0,
+ "content": "dataset is Noisy Cora 2500, Noisy Cora 5000, and Noisy CiteSeer, respectively. We note that the",
+ "type": "text"
+ }
+ ],
+ "index": 18
+ },
+ {
+ "bbox": [
+ 105,
+ 407,
+ 505,
+ 420
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 407,
+ 505,
+ 420
+ ],
+ "score": 1.0,
+ "content": "result of 3-layered GCN from the Noisy Cora 2500 is identical to Figure 3 (right) of the main article.",
+ "type": "text"
+ }
+ ],
+ "index": 19
+ }
+ ],
+ "index": 18
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 108,
+ 432,
+ 257,
+ 443
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 432,
+ 259,
+ 444
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 432,
+ 259,
+ 444
+ ],
+ "score": 1.0,
+ "content": "I.4 EXPERIMENT OF SECTION 6.4",
+ "type": "text"
+ }
+ ],
+ "index": 20
+ }
+ ],
+ "index": 20
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 452,
+ 505,
+ 541
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 452,
+ 505,
+ 466
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 452,
+ 505,
+ 466
+ ],
+ "score": 1.0,
+ "content": "Figure 14 shows the logarithm of relative perpendicular component and prediction accuracy on",
+ "type": "text"
+ }
+ ],
+ "index": 21
+ },
+ {
+ "bbox": [
+ 106,
+ 464,
+ 505,
+ 476
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 464,
+ 505,
+ 476
+ ],
+ "score": 1.0,
+ "content": "Noisy Cora, Noisy CiteSeer, and Noisy PubMed datasets. We use Pearson R as a correlation co-",
+ "type": "text"
+ }
+ ],
+ "index": 22
+ },
+ {
+ "bbox": [
+ 105,
+ 474,
+ 505,
+ 488
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 474,
+ 505,
+ 488
+ ],
+ "score": 1.0,
+ "content": "efficient. If GCNs have only one layer, it has more large relative perpendicular components (cor-",
+ "type": "text"
+ }
+ ],
+ "index": 23
+ },
+ {
+ "bbox": [
+ 105,
+ 486,
+ 505,
+ 498
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 486,
+ 505,
+ 498
+ ],
+ "score": 1.0,
+ "content": "responding to right points in the figures) than GCNs which have other number of layers. The cor-",
+ "type": "text"
+ }
+ ],
+ "index": 24
+ },
+ {
+ "bbox": [
+ 104,
+ 496,
+ 506,
+ 509
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 496,
+ 506,
+ 509
+ ],
+ "score": 1.0,
+ "content": "relation between the logarithm of relative perpendicular components and prediction accuracies are",
+ "type": "text"
+ }
+ ],
+ "index": 25
+ },
+ {
+ "bbox": [
+ 106,
+ 507,
+ 506,
+ 520
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 507,
+ 217,
+ 518
+ ],
+ "score": 0.82,
+ "content": "0 . 8 2 7 ( p { = } 6 . 8 9 0 \\times \\bar { 1 0 } ^ { - 6 } )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 217,
+ 507,
+ 290,
+ 520
+ ],
+ "score": 1.0,
+ "content": "for Noisy Cora,",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 290,
+ 507,
+ 400,
+ 519
+ ],
+ "score": 0.91,
+ "content": "\\mathrm { { 0 . 5 2 4 } } ( p \\ = \\ 1 . { \\bar { 7 } } 7 1 \\times 1 0 ^ { - 2 } )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 401,
+ 507,
+ 506,
+ 520
+ ],
+ "score": 1.0,
+ "content": "for Noisy CiteSeer, and",
+ "type": "text"
+ }
+ ],
+ "index": 26
+ },
+ {
+ "bbox": [
+ 106,
+ 518,
+ 507,
+ 532
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 519,
+ 212,
+ 531
+ ],
+ "score": 0.82,
+ "content": "0 . 6 7 9 ( p = 1 . 0 0 2 \\times 1 0 ^ { - 3 } )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 212,
+ 518,
+ 507,
+ 532
+ ],
+ "score": 1.0,
+ "content": "for Noisy PubMed, if we treat the one-layer case as outliers and remove",
+ "type": "text"
+ }
+ ],
+ "index": 27
+ },
+ {
+ "bbox": [
+ 105,
+ 529,
+ 132,
+ 542
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 529,
+ 132,
+ 542
+ ],
+ "score": 1.0,
+ "content": "them.",
+ "type": "text"
+ }
+ ],
+ "index": 28
+ }
+ ],
+ "index": 24.5
+ }
+ ],
+ "page_idx": 26,
+ "page_size": [
+ 612,
+ 792
+ ],
+ "discarded_blocks": [
+ {
+ "type": "discarded",
+ "bbox": [
+ 107,
+ 27,
+ 293,
+ 37
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 26,
+ 294,
+ 38
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 26,
+ 294,
+ 38
+ ],
+ "score": 1.0,
+ "content": "Published as a conference paper at ICLR 2020",
+ "type": "text"
+ }
+ ]
+ }
+ ]
+ },
+ {
+ "type": "discarded",
+ "bbox": [
+ 300,
+ 751,
+ 311,
+ 760
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 298,
+ 750,
+ 313,
+ 765
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 298,
+ 750,
+ 313,
+ 765
+ ],
+ "score": 1.0,
+ "content": "",
+ "type": "text",
+ "height": 15,
+ "width": 15
+ }
+ ]
+ }
+ ]
+ }
+ ],
+ "para_blocks": [
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 82,
+ 504,
+ 105
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 82,
+ 505,
+ 95
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 82,
+ 505,
+ 95
+ ],
+ "score": 1.0,
+ "content": "on the perpendicular component of signals, while we interested in the prediction accuracy in real",
+ "type": "text"
+ }
+ ],
+ "index": 0
+ },
+ {
+ "bbox": [
+ 106,
+ 94,
+ 284,
+ 106
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 94,
+ 284,
+ 106
+ ],
+ "score": 1.0,
+ "content": "training settings in the previous experiment.",
+ "type": "text"
+ }
+ ],
+ "index": 1
+ }
+ ],
+ "index": 0.5,
+ "bbox_fs": [
+ 106,
+ 82,
+ 505,
+ 106
+ ]
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 106,
+ 121,
+ 312,
+ 134
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 120,
+ 312,
+ 135
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 120,
+ 312,
+ 135
+ ],
+ "score": 1.0,
+ "content": "I ADDITIONAL EXPERIMENT RESULTS",
+ "type": "text"
+ }
+ ],
+ "index": 2
+ }
+ ],
+ "index": 2
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 108,
+ 145,
+ 257,
+ 157
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 145,
+ 258,
+ 158
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 145,
+ 258,
+ 158
+ ],
+ "score": 1.0,
+ "content": "I.1 EXPERIMENT OF SECTION 6.1",
+ "type": "text"
+ }
+ ],
+ "index": 3
+ }
+ ],
+ "index": 3
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 166,
+ 504,
+ 189
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 166,
+ 505,
+ 179
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 166,
+ 208,
+ 179
+ ],
+ "score": 1.0,
+ "content": "We show the vector field",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 209,
+ 167,
+ 218,
+ 176
+ ],
+ "score": 0.73,
+ "content": "V",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 218,
+ 166,
+ 265,
+ 179
+ ],
+ "score": 1.0,
+ "content": "for various",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 266,
+ 167,
+ 277,
+ 177
+ ],
+ "score": 0.66,
+ "content": "W",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 278,
+ 166,
+ 505,
+ 179
+ ],
+ "score": 1.0,
+ "content": "in Figure 6 (Case 1) and Figure 7 (Case 2). Parameters",
+ "type": "text"
+ }
+ ],
+ "index": 4
+ },
+ {
+ "bbox": [
+ 105,
+ 177,
+ 502,
+ 190
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 177,
+ 149,
+ 190
+ ],
+ "score": 1.0,
+ "content": "other than",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 149,
+ 178,
+ 161,
+ 187
+ ],
+ "score": 0.65,
+ "content": "W",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 162,
+ 177,
+ 502,
+ 190
+ ],
+ "score": 1.0,
+ "content": "are same as experiments in Section 6.1 (detail values are available in Appendix H.1).",
+ "type": "text"
+ }
+ ],
+ "index": 5
+ }
+ ],
+ "index": 4.5,
+ "bbox_fs": [
+ 105,
+ 166,
+ 505,
+ 190
+ ]
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 108,
+ 202,
+ 257,
+ 213
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 201,
+ 259,
+ 214
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 201,
+ 259,
+ 214
+ ],
+ "score": 1.0,
+ "content": "I.2 EXPERIMENT OF SECTION 6.2",
+ "type": "text"
+ }
+ ],
+ "index": 6
+ }
+ ],
+ "index": 6
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 222,
+ 505,
+ 267
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 222,
+ 505,
+ 236
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 222,
+ 505,
+ 236
+ ],
+ "score": 1.0,
+ "content": "Figure 8 shows the relative log distance of signals and their upper bound for various edge probability",
+ "type": "text"
+ }
+ ],
+ "index": 7
+ },
+ {
+ "bbox": [
+ 106,
+ 234,
+ 506,
+ 246
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 236,
+ 113,
+ 245
+ ],
+ "score": 0.77,
+ "content": "p",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 113,
+ 234,
+ 248,
+ 246
+ ],
+ "score": 1.0,
+ "content": "and the maximum singular value",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 249,
+ 236,
+ 255,
+ 244
+ ],
+ "score": 0.67,
+ "content": "s",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 255,
+ 234,
+ 506,
+ 246
+ ],
+ "score": 1.0,
+ "content": ". Note that we generate a new graph for each configuration of",
+ "type": "text"
+ }
+ ],
+ "index": 8
+ },
+ {
+ "bbox": [
+ 107,
+ 245,
+ 505,
+ 258
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 245,
+ 129,
+ 257
+ ],
+ "score": 0.9,
+ "content": "( p , s )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 129,
+ 245,
+ 457,
+ 258
+ ],
+ "score": 1.0,
+ "content": ". Therefore, different configurations may have different graphs and hence different",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 457,
+ 245,
+ 464,
+ 255
+ ],
+ "score": 0.75,
+ "content": "\\lambda",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 464,
+ 245,
+ 505,
+ 258
+ ],
+ "score": 1.0,
+ "content": "even they",
+ "type": "text"
+ }
+ ],
+ "index": 9
+ },
+ {
+ "bbox": [
+ 105,
+ 257,
+ 282,
+ 268
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 257,
+ 225,
+ 268
+ ],
+ "score": 1.0,
+ "content": "have a same edge probability",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 225,
+ 258,
+ 231,
+ 268
+ ],
+ "score": 0.8,
+ "content": "p",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 232,
+ 257,
+ 282,
+ 268
+ ],
+ "score": 1.0,
+ "content": "in common.",
+ "type": "text"
+ }
+ ],
+ "index": 10
+ }
+ ],
+ "index": 8.5,
+ "bbox_fs": [
+ 105,
+ 222,
+ 506,
+ 268
+ ]
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 108,
+ 280,
+ 257,
+ 292
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 280,
+ 259,
+ 293
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 280,
+ 259,
+ 293
+ ],
+ "score": 1.0,
+ "content": "I.3 EXPERIMENT OF SECTION 6.3",
+ "type": "text"
+ }
+ ],
+ "index": 11
+ }
+ ],
+ "index": 11
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 108,
+ 301,
+ 243,
+ 312
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 301,
+ 243,
+ 314
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 301,
+ 243,
+ 314
+ ],
+ "score": 1.0,
+ "content": "I.3.1 PREDICTIVE ACCURACY",
+ "type": "text"
+ }
+ ],
+ "index": 12
+ }
+ ],
+ "index": 12
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 320,
+ 504,
+ 354
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 320,
+ 506,
+ 333
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 320,
+ 506,
+ 333
+ ],
+ "score": 1.0,
+ "content": "Figure 9 shows the comparison of predictive performance in terms the maximum singular value",
+ "type": "text"
+ }
+ ],
+ "index": 13
+ },
+ {
+ "bbox": [
+ 105,
+ 330,
+ 505,
+ 345
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 330,
+ 505,
+ 345
+ ],
+ "score": 1.0,
+ "content": "and layer size when the dataset is Noisy Cora 5000 (left) and Noisy Citeseer (right), respectively.",
+ "type": "text"
+ }
+ ],
+ "index": 14
+ },
+ {
+ "bbox": [
+ 106,
+ 343,
+ 271,
+ 353
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 343,
+ 271,
+ 353
+ ],
+ "score": 1.0,
+ "content": "Concrete values are available in Table 4.",
+ "type": "text"
+ }
+ ],
+ "index": 15
+ }
+ ],
+ "index": 14,
+ "bbox_fs": [
+ 105,
+ 320,
+ 506,
+ 353
+ ]
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 106,
+ 366,
+ 340,
+ 377
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 365,
+ 341,
+ 379
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 365,
+ 341,
+ 379
+ ],
+ "score": 1.0,
+ "content": "I.3.2 TRANSITION OF MAXIMUM SINGULAR VALUES",
+ "type": "text"
+ }
+ ],
+ "index": 16
+ }
+ ],
+ "index": 16
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 385,
+ 505,
+ 419
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 384,
+ 505,
+ 398
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 384,
+ 505,
+ 398
+ ],
+ "score": 1.0,
+ "content": "Figure 10 – 13 show the transition of weight of graph convolution layers during training when the",
+ "type": "text"
+ }
+ ],
+ "index": 17
+ },
+ {
+ "bbox": [
+ 105,
+ 396,
+ 505,
+ 409
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 396,
+ 505,
+ 409
+ ],
+ "score": 1.0,
+ "content": "dataset is Noisy Cora 2500, Noisy Cora 5000, and Noisy CiteSeer, respectively. We note that the",
+ "type": "text"
+ }
+ ],
+ "index": 18
+ },
+ {
+ "bbox": [
+ 105,
+ 407,
+ 505,
+ 420
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 407,
+ 505,
+ 420
+ ],
+ "score": 1.0,
+ "content": "result of 3-layered GCN from the Noisy Cora 2500 is identical to Figure 3 (right) of the main article.",
+ "type": "text"
+ }
+ ],
+ "index": 19
+ }
+ ],
+ "index": 18,
+ "bbox_fs": [
+ 105,
+ 384,
+ 505,
+ 420
+ ]
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 108,
+ 432,
+ 257,
+ 443
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 432,
+ 259,
+ 444
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 432,
+ 259,
+ 444
+ ],
+ "score": 1.0,
+ "content": "I.4 EXPERIMENT OF SECTION 6.4",
+ "type": "text"
+ }
+ ],
+ "index": 20
+ }
+ ],
+ "index": 20
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 452,
+ 505,
+ 541
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 452,
+ 505,
+ 466
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 452,
+ 505,
+ 466
+ ],
+ "score": 1.0,
+ "content": "Figure 14 shows the logarithm of relative perpendicular component and prediction accuracy on",
+ "type": "text"
+ }
+ ],
+ "index": 21
+ },
+ {
+ "bbox": [
+ 106,
+ 464,
+ 505,
+ 476
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 464,
+ 505,
+ 476
+ ],
+ "score": 1.0,
+ "content": "Noisy Cora, Noisy CiteSeer, and Noisy PubMed datasets. We use Pearson R as a correlation co-",
+ "type": "text"
+ }
+ ],
+ "index": 22
+ },
+ {
+ "bbox": [
+ 105,
+ 474,
+ 505,
+ 488
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 474,
+ 505,
+ 488
+ ],
+ "score": 1.0,
+ "content": "efficient. If GCNs have only one layer, it has more large relative perpendicular components (cor-",
+ "type": "text"
+ }
+ ],
+ "index": 23
+ },
+ {
+ "bbox": [
+ 105,
+ 486,
+ 505,
+ 498
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 486,
+ 505,
+ 498
+ ],
+ "score": 1.0,
+ "content": "responding to right points in the figures) than GCNs which have other number of layers. The cor-",
+ "type": "text"
+ }
+ ],
+ "index": 24
+ },
+ {
+ "bbox": [
+ 104,
+ 496,
+ 506,
+ 509
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 496,
+ 506,
+ 509
+ ],
+ "score": 1.0,
+ "content": "relation between the logarithm of relative perpendicular components and prediction accuracies are",
+ "type": "text"
+ }
+ ],
+ "index": 25
+ },
+ {
+ "bbox": [
+ 106,
+ 507,
+ 506,
+ 520
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 507,
+ 217,
+ 518
+ ],
+ "score": 0.82,
+ "content": "0 . 8 2 7 ( p { = } 6 . 8 9 0 \\times \\bar { 1 0 } ^ { - 6 } )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 217,
+ 507,
+ 290,
+ 520
+ ],
+ "score": 1.0,
+ "content": "for Noisy Cora,",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 290,
+ 507,
+ 400,
+ 519
+ ],
+ "score": 0.91,
+ "content": "\\mathrm { { 0 . 5 2 4 } } ( p \\ = \\ 1 . { \\bar { 7 } } 7 1 \\times 1 0 ^ { - 2 } )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 401,
+ 507,
+ 506,
+ 520
+ ],
+ "score": 1.0,
+ "content": "for Noisy CiteSeer, and",
+ "type": "text"
+ }
+ ],
+ "index": 26
+ },
+ {
+ "bbox": [
+ 106,
+ 518,
+ 507,
+ 532
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 519,
+ 212,
+ 531
+ ],
+ "score": 0.82,
+ "content": "0 . 6 7 9 ( p = 1 . 0 0 2 \\times 1 0 ^ { - 3 } )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 212,
+ 518,
+ 507,
+ 532
+ ],
+ "score": 1.0,
+ "content": "for Noisy PubMed, if we treat the one-layer case as outliers and remove",
+ "type": "text"
+ }
+ ],
+ "index": 27
+ },
+ {
+ "bbox": [
+ 105,
+ 529,
+ 132,
+ 542
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 529,
+ 132,
+ 542
+ ],
+ "score": 1.0,
+ "content": "them.",
+ "type": "text"
+ }
+ ],
+ "index": 28
+ }
+ ],
+ "index": 24.5,
+ "bbox_fs": [
+ 104,
+ 452,
+ 507,
+ 542
+ ]
+ }
+ ]
+ },
+ {
+ "preproc_blocks": [
+ {
+ "type": "image",
+ "bbox": [
+ 114,
+ 144,
+ 480,
+ 630
+ ],
+ "blocks": [
+ {
+ "type": "image_body",
+ "bbox": [
+ 114,
+ 144,
+ 480,
+ 630
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 114,
+ 144,
+ 480,
+ 630
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 114,
+ 144,
+ 480,
+ 630
+ ],
+ "score": 0.964,
+ "type": "image",
+ "image_path": "a88890af3061b7f6c52d0927100734218d69193a2aa1e144aab4bb8cd705f753.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 1,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 114,
+ 144,
+ 480,
+ 306.0
+ ],
+ "spans": [],
+ "index": 0
+ },
+ {
+ "bbox": [
+ 114,
+ 306.0,
+ 480,
+ 468.0
+ ],
+ "spans": [],
+ "index": 1
+ },
+ {
+ "bbox": [
+ 114,
+ 468.0,
+ 480,
+ 630.0
+ ],
+ "spans": [],
+ "index": 2
+ }
+ ]
+ },
+ {
+ "type": "image_caption",
+ "bbox": [
+ 106,
+ 638,
+ 505,
+ 672
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 638,
+ 504,
+ 650
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 638,
+ 199,
+ 650
+ ],
+ "score": 1.0,
+ "content": "Figure 6: Vector field",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 199,
+ 639,
+ 209,
+ 649
+ ],
+ "score": 0.76,
+ "content": "V",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 209,
+ 638,
+ 258,
+ 650
+ ],
+ "score": 1.0,
+ "content": "for various",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 259,
+ 639,
+ 271,
+ 649
+ ],
+ "score": 0.71,
+ "content": "W",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 271,
+ 638,
+ 365,
+ 650
+ ],
+ "score": 1.0,
+ "content": "for Case 1. Top left:",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 365,
+ 639,
+ 407,
+ 649
+ ],
+ "score": 0.89,
+ "content": "W = 0 . 5",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 407,
+ 638,
+ 459,
+ 650
+ ],
+ "score": 1.0,
+ "content": ". Top right:",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 460,
+ 639,
+ 501,
+ 649
+ ],
+ "score": 0.87,
+ "content": "W = 1 . 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 502,
+ 638,
+ 504,
+ 650
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 3
+ },
+ {
+ "bbox": [
+ 105,
+ 649,
+ 505,
+ 662
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 649,
+ 157,
+ 662
+ ],
+ "score": 1.0,
+ "content": "Middle left:",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 158,
+ 650,
+ 196,
+ 660
+ ],
+ "score": 0.9,
+ "content": "W = 1 . 2",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 197,
+ 649,
+ 410,
+ 662
+ ],
+ "score": 1.0,
+ "content": "(same as Figure 1 in the main article). Middle right:",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 410,
+ 650,
+ 449,
+ 660
+ ],
+ "score": 0.89,
+ "content": "W = 1 . 5",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 450,
+ 649,
+ 505,
+ 662
+ ],
+ "score": 1.0,
+ "content": ". Bottom left:",
+ "type": "text"
+ }
+ ],
+ "index": 4
+ },
+ {
+ "bbox": [
+ 106,
+ 659,
+ 325,
+ 672
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 660,
+ 145,
+ 671
+ ],
+ "score": 0.89,
+ "content": "W = 2 . 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 145,
+ 659,
+ 205,
+ 672
+ ],
+ "score": 1.0,
+ "content": ". Bottom right:",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 206,
+ 660,
+ 244,
+ 671
+ ],
+ "score": 0.9,
+ "content": "W = 4 . 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 244,
+ 659,
+ 325,
+ 672
+ ],
+ "score": 1.0,
+ "content": ". Best view in color.",
+ "type": "text"
+ }
+ ],
+ "index": 5
+ }
+ ],
+ "index": 4
+ }
+ ],
+ "index": 2.5
+ }
+ ],
+ "page_idx": 27,
+ "page_size": [
+ 612,
+ 792
+ ],
+ "discarded_blocks": [
+ {
+ "type": "discarded",
+ "bbox": [
+ 107,
+ 27,
+ 293,
+ 37
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 26,
+ 294,
+ 38
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 26,
+ 294,
+ 38
+ ],
+ "score": 1.0,
+ "content": "Published as a conference paper at ICLR 2020",
+ "type": "text"
+ }
+ ]
+ }
+ ]
+ },
+ {
+ "type": "discarded",
+ "bbox": [
+ 300,
+ 751,
+ 311,
+ 760
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 298,
+ 750,
+ 313,
+ 763
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 298,
+ 750,
+ 313,
+ 763
+ ],
+ "score": 1.0,
+ "content": "28",
+ "type": "text"
+ }
+ ]
+ }
+ ]
+ }
+ ],
+ "para_blocks": [
+ {
+ "type": "image",
+ "bbox": [
+ 114,
+ 144,
+ 480,
+ 630
+ ],
+ "blocks": [
+ {
+ "type": "image_body",
+ "bbox": [
+ 114,
+ 144,
+ 480,
+ 630
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 114,
+ 144,
+ 480,
+ 630
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 114,
+ 144,
+ 480,
+ 630
+ ],
+ "score": 0.964,
+ "type": "image",
+ "image_path": "a88890af3061b7f6c52d0927100734218d69193a2aa1e144aab4bb8cd705f753.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 1,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 114,
+ 144,
+ 480,
+ 306.0
+ ],
+ "spans": [],
+ "index": 0
+ },
+ {
+ "bbox": [
+ 114,
+ 306.0,
+ 480,
+ 468.0
+ ],
+ "spans": [],
+ "index": 1
+ },
+ {
+ "bbox": [
+ 114,
+ 468.0,
+ 480,
+ 630.0
+ ],
+ "spans": [],
+ "index": 2
+ }
+ ]
+ },
+ {
+ "type": "image_caption",
+ "bbox": [
+ 106,
+ 638,
+ 505,
+ 672
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 638,
+ 504,
+ 650
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 638,
+ 199,
+ 650
+ ],
+ "score": 1.0,
+ "content": "Figure 6: Vector field",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 199,
+ 639,
+ 209,
+ 649
+ ],
+ "score": 0.76,
+ "content": "V",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 209,
+ 638,
+ 258,
+ 650
+ ],
+ "score": 1.0,
+ "content": "for various",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 259,
+ 639,
+ 271,
+ 649
+ ],
+ "score": 0.71,
+ "content": "W",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 271,
+ 638,
+ 365,
+ 650
+ ],
+ "score": 1.0,
+ "content": "for Case 1. Top left:",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 365,
+ 639,
+ 407,
+ 649
+ ],
+ "score": 0.89,
+ "content": "W = 0 . 5",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 407,
+ 638,
+ 459,
+ 650
+ ],
+ "score": 1.0,
+ "content": ". Top right:",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 460,
+ 639,
+ 501,
+ 649
+ ],
+ "score": 0.87,
+ "content": "W = 1 . 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 502,
+ 638,
+ 504,
+ 650
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 3
+ },
+ {
+ "bbox": [
+ 105,
+ 649,
+ 505,
+ 662
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 649,
+ 157,
+ 662
+ ],
+ "score": 1.0,
+ "content": "Middle left:",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 158,
+ 650,
+ 196,
+ 660
+ ],
+ "score": 0.9,
+ "content": "W = 1 . 2",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 197,
+ 649,
+ 410,
+ 662
+ ],
+ "score": 1.0,
+ "content": "(same as Figure 1 in the main article). Middle right:",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 410,
+ 650,
+ 449,
+ 660
+ ],
+ "score": 0.89,
+ "content": "W = 1 . 5",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 450,
+ 649,
+ 505,
+ 662
+ ],
+ "score": 1.0,
+ "content": ". Bottom left:",
+ "type": "text"
+ }
+ ],
+ "index": 4
+ },
+ {
+ "bbox": [
+ 106,
+ 659,
+ 325,
+ 672
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 660,
+ 145,
+ 671
+ ],
+ "score": 0.89,
+ "content": "W = 2 . 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 145,
+ 659,
+ 205,
+ 672
+ ],
+ "score": 1.0,
+ "content": ". Bottom right:",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 206,
+ 660,
+ 244,
+ 671
+ ],
+ "score": 0.9,
+ "content": "W = 4 . 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 244,
+ 659,
+ 325,
+ 672
+ ],
+ "score": 1.0,
+ "content": ". Best view in color.",
+ "type": "text"
+ }
+ ],
+ "index": 5
+ }
+ ],
+ "index": 4
+ }
+ ],
+ "index": 2.5
+ }
+ ]
+ },
+ {
+ "preproc_blocks": [
+ {
+ "type": "image",
+ "bbox": [
+ 114,
+ 143,
+ 480,
+ 630
+ ],
+ "blocks": [
+ {
+ "type": "image_body",
+ "bbox": [
+ 114,
+ 143,
+ 480,
+ 630
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 114,
+ 143,
+ 480,
+ 630
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 114,
+ 143,
+ 480,
+ 630
+ ],
+ "score": 0.962,
+ "type": "image",
+ "image_path": "55e8e2cb9210bdfb7542bd78186fe6fc29fb504e4a82ce013518941d8c8ae0bc.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 1,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 114,
+ 143,
+ 480,
+ 305.33333333333337
+ ],
+ "spans": [],
+ "index": 0
+ },
+ {
+ "bbox": [
+ 114,
+ 305.33333333333337,
+ 480,
+ 467.66666666666674
+ ],
+ "spans": [],
+ "index": 1
+ },
+ {
+ "bbox": [
+ 114,
+ 467.66666666666674,
+ 480,
+ 630.0000000000001
+ ],
+ "spans": [],
+ "index": 2
+ }
+ ]
+ },
+ {
+ "type": "image_caption",
+ "bbox": [
+ 106,
+ 638,
+ 505,
+ 672
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 637,
+ 505,
+ 651
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 637,
+ 199,
+ 651
+ ],
+ "score": 1.0,
+ "content": "Figure 7: Vector field",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 199,
+ 639,
+ 209,
+ 649
+ ],
+ "score": 0.76,
+ "content": "V",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 209,
+ 637,
+ 258,
+ 651
+ ],
+ "score": 1.0,
+ "content": "for various",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 259,
+ 639,
+ 271,
+ 649
+ ],
+ "score": 0.71,
+ "content": "W",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 271,
+ 637,
+ 365,
+ 651
+ ],
+ "score": 1.0,
+ "content": "for Case 2. Top left:",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 365,
+ 639,
+ 407,
+ 649
+ ],
+ "score": 0.89,
+ "content": "W = 0 . 5",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 407,
+ 637,
+ 459,
+ 651
+ ],
+ "score": 1.0,
+ "content": ". Top right:",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 460,
+ 639,
+ 501,
+ 649
+ ],
+ "score": 0.87,
+ "content": "W = 1 . 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 502,
+ 637,
+ 505,
+ 651
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 3
+ },
+ {
+ "bbox": [
+ 105,
+ 649,
+ 504,
+ 661
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 649,
+ 157,
+ 661
+ ],
+ "score": 1.0,
+ "content": "Middle left:",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 158,
+ 650,
+ 196,
+ 660
+ ],
+ "score": 0.9,
+ "content": "W = 1 . 2",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 197,
+ 649,
+ 410,
+ 661
+ ],
+ "score": 1.0,
+ "content": "(same as Figure 1 in the main article). Middle right:",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 410,
+ 650,
+ 449,
+ 660
+ ],
+ "score": 0.89,
+ "content": "W = 1 . 5",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 450,
+ 649,
+ 504,
+ 661
+ ],
+ "score": 1.0,
+ "content": ". Bottom left:",
+ "type": "text"
+ }
+ ],
+ "index": 4
+ },
+ {
+ "bbox": [
+ 106,
+ 660,
+ 325,
+ 672
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 660,
+ 145,
+ 671
+ ],
+ "score": 0.89,
+ "content": "W = 2 . 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 145,
+ 660,
+ 205,
+ 672
+ ],
+ "score": 1.0,
+ "content": ". Bottom right:",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 206,
+ 660,
+ 244,
+ 671
+ ],
+ "score": 0.9,
+ "content": "W = 4 . 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 244,
+ 660,
+ 325,
+ 672
+ ],
+ "score": 1.0,
+ "content": ". Best view in color.",
+ "type": "text"
+ }
+ ],
+ "index": 5
+ }
+ ],
+ "index": 4
+ }
+ ],
+ "index": 2.5
+ }
+ ],
+ "page_idx": 28,
+ "page_size": [
+ 612,
+ 792
+ ],
+ "discarded_blocks": [
+ {
+ "type": "discarded",
+ "bbox": [
+ 107,
+ 27,
+ 293,
+ 37
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 26,
+ 294,
+ 38
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 26,
+ 294,
+ 38
+ ],
+ "score": 1.0,
+ "content": "Published as a conference paper at ICLR 2020",
+ "type": "text"
+ }
+ ]
+ }
+ ]
+ },
+ {
+ "type": "discarded",
+ "bbox": [
+ 300,
+ 751,
+ 311,
+ 760
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 298,
+ 750,
+ 313,
+ 764
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 298,
+ 750,
+ 313,
+ 764
+ ],
+ "score": 1.0,
+ "content": "",
+ "type": "text",
+ "height": 14,
+ "width": 15
+ }
+ ]
+ }
+ ]
+ }
+ ],
+ "para_blocks": [
+ {
+ "type": "image",
+ "bbox": [
+ 114,
+ 143,
+ 480,
+ 630
+ ],
+ "blocks": [
+ {
+ "type": "image_body",
+ "bbox": [
+ 114,
+ 143,
+ 480,
+ 630
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 114,
+ 143,
+ 480,
+ 630
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 114,
+ 143,
+ 480,
+ 630
+ ],
+ "score": 0.962,
+ "type": "image",
+ "image_path": "55e8e2cb9210bdfb7542bd78186fe6fc29fb504e4a82ce013518941d8c8ae0bc.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 1,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 114,
+ 143,
+ 480,
+ 305.33333333333337
+ ],
+ "spans": [],
+ "index": 0
+ },
+ {
+ "bbox": [
+ 114,
+ 305.33333333333337,
+ 480,
+ 467.66666666666674
+ ],
+ "spans": [],
+ "index": 1
+ },
+ {
+ "bbox": [
+ 114,
+ 467.66666666666674,
+ 480,
+ 630.0000000000001
+ ],
+ "spans": [],
+ "index": 2
+ }
+ ]
+ },
+ {
+ "type": "image_caption",
+ "bbox": [
+ 106,
+ 638,
+ 505,
+ 672
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 637,
+ 505,
+ 651
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 637,
+ 199,
+ 651
+ ],
+ "score": 1.0,
+ "content": "Figure 7: Vector field",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 199,
+ 639,
+ 209,
+ 649
+ ],
+ "score": 0.76,
+ "content": "V",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 209,
+ 637,
+ 258,
+ 651
+ ],
+ "score": 1.0,
+ "content": "for various",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 259,
+ 639,
+ 271,
+ 649
+ ],
+ "score": 0.71,
+ "content": "W",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 271,
+ 637,
+ 365,
+ 651
+ ],
+ "score": 1.0,
+ "content": "for Case 2. Top left:",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 365,
+ 639,
+ 407,
+ 649
+ ],
+ "score": 0.89,
+ "content": "W = 0 . 5",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 407,
+ 637,
+ 459,
+ 651
+ ],
+ "score": 1.0,
+ "content": ". Top right:",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 460,
+ 639,
+ 501,
+ 649
+ ],
+ "score": 0.87,
+ "content": "W = 1 . 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 502,
+ 637,
+ 505,
+ 651
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 3
+ },
+ {
+ "bbox": [
+ 105,
+ 649,
+ 504,
+ 661
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 649,
+ 157,
+ 661
+ ],
+ "score": 1.0,
+ "content": "Middle left:",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 158,
+ 650,
+ 196,
+ 660
+ ],
+ "score": 0.9,
+ "content": "W = 1 . 2",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 197,
+ 649,
+ 410,
+ 661
+ ],
+ "score": 1.0,
+ "content": "(same as Figure 1 in the main article). Middle right:",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 410,
+ 650,
+ 449,
+ 660
+ ],
+ "score": 0.89,
+ "content": "W = 1 . 5",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 450,
+ 649,
+ 504,
+ 661
+ ],
+ "score": 1.0,
+ "content": ". Bottom left:",
+ "type": "text"
+ }
+ ],
+ "index": 4
+ },
+ {
+ "bbox": [
+ 106,
+ 660,
+ 325,
+ 672
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 660,
+ 145,
+ 671
+ ],
+ "score": 0.89,
+ "content": "W = 2 . 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 145,
+ 660,
+ 205,
+ 672
+ ],
+ "score": 1.0,
+ "content": ". Bottom right:",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 206,
+ 660,
+ 244,
+ 671
+ ],
+ "score": 0.9,
+ "content": "W = 4 . 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 244,
+ 660,
+ 325,
+ 672
+ ],
+ "score": 1.0,
+ "content": ". Best view in color.",
+ "type": "text"
+ }
+ ],
+ "index": 5
+ }
+ ],
+ "index": 4
+ }
+ ],
+ "index": 2.5
+ }
+ ]
+ },
+ {
+ "preproc_blocks": [
+ {
+ "type": "table",
+ "bbox": [
+ 111,
+ 199,
+ 500,
+ 642
+ ],
+ "blocks": [
+ {
+ "type": "table_caption",
+ "bbox": [
+ 105,
+ 175,
+ 504,
+ 208
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 175,
+ 505,
+ 189
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 175,
+ 505,
+ 189
+ ],
+ "score": 1.0,
+ "content": "Table 4: Comparison of performance in terms of maximum singular value of weights and layer size.",
+ "type": "text"
+ }
+ ],
+ "index": 0
+ },
+ {
+ "bbox": [
+ 106,
+ 186,
+ 473,
+ 199
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 186,
+ 473,
+ 199
+ ],
+ "score": 1.0,
+ "content": "“U” in the right most column indicates the accuracy of GCN without weight normalization.",
+ "type": "text"
+ }
+ ],
+ "index": 1
+ }
+ ],
+ "index": 0.5
+ },
+ {
+ "type": "table_body",
+ "bbox": [
+ 111,
+ 199,
+ 500,
+ 642
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 111,
+ 199,
+ 500,
+ 642
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 111,
+ 199,
+ 500,
+ 642
+ ],
+ "score": 0.861,
+ "html": "| Noisy Cora 2500 |
|
| Depth | | 1.05 | 3 | 10 | U |
| 1 | 0.389 ± 0.101 | 0.429 ± 0.090 | 0.552 ± 0.014 | 0.632 ± 0.007 | 0.587 ± 0.008 |
| 3 | 0.273 ± 0.051 | 0.309 ± 0.017 | 0.580± 0.058 | 0.661 ± 0.003 | 0.494 ± 0.041 |
| 5 | 0.319 ± 0.000 | 0.267 ± 0.059 | 0.462 ± 0.065 | 0.602 ± 0.004 | 0.326 ± 0.029 |
| 7 | 0.261 ± 0.076 | 0.262 ±0.080 | 0.407 ± 0.021 | 0.501 ± 0.017 | 0.279 ± 0.129 |
| 9 | 0.261 ± 0.080 | 0.319 ± 0.000 | 0.284 ± 0.109 | 0.443 ± 0.014 | 0.319 ± 0.000 |
| Noisy Cora 5000 |
| Maximum Singular Value |
| Depth | 1 | 1.1 | 3 | 10 | U |
| 1 | 0.301 ± 0.080 | 0.333 ± 0.099 | 0.557 ± 0.004 | 0.561 ± 0.019 | 0.555 ± 0.016 |
| 3 | 0.245 ± 0.066 | 0.247 ± 0.076 | 0.370 ± 0.041 | 0.587 ± 0.009 | 0.286 ± 0.066 |
| 5 | 0.274± 0.048 | 0.237 ± 0.070 | 0.257 ± 0.076 | 0.535 ± 0.031 | 0.319 ± 0.000 |
| 7 | 0.263 ± 0.080 | 0.297 ± 0.031 | 0.260 ± 0.074 | 0.339 ± 0.060 | 0.319 ± 0.000 |
| 9 | 0.262 ± 0.081 | 0.258 ± 0.064 | 0.262 ± 0.080 | 0.261 ±0.082 | 0.318 ± 0.002 |
| Noisy CiteSeer |
| Maximum Singular Value |
| Depth | 0.5 | 1.1 3 | | 10 | U |
| 1 | 0.461 ± 0.018 | 0.467 ± 0.012 | 0.490 ± 0.016 | 0.494 ± 0.006 | 0.495 ± 0.009 |
| 3 | 0.438 ± 0.027 | 0.436 ± 0.010 | 0.450 ± 0.019 | 0.462 ± 0.007 | 0.417 ± 0.061 |
| 5 | 0.285 ± 0.008 | 0.371 ± 0.016 | 0.373 ± 0.011 | 0.425 ± 0.007 | 0.380 ± 0.024 |
| 7 | 0.213 ± 0.006 | 0.282 ± 0.011 | 0.309 ± 0.012 | 0.385 ± 0.007 | 0.308 ± 0.012 |
| 9 | 0.182 ± 0.005 | 0.242 ± 0.030 | 0.303 ± 0.021 | 0.325 ± 0.003 | 0.229 ± 0.033 |
| Noisy Pubmed |
| Maximum Singular Value |
| Depth | 0.5 | 1.1 | 3 | 10 | U |
| 1 | 0.488 ± 0.039 | 0.636 ± 0.006 | 0.641 ± 0.010 | 0.632 ± 0.002 | 0.631 ± 0.010 |
| 3 | 0.442 ± 0.027 | 0.426 ± 0.026 | 0.658 ± 0.004 | 0.661 ± 0.005 | 0.631 ± 0.013 |
| 5 | 0.431 ± 0.033 | 0.431 ± 0.034 | 0.561 ± 0.083 | 0.641 ± 0.004 | 0.424 ± 0.093 |
| 7 | 0.428 ± 0.032 | 0.443 ± 0.051 | 0.449 ± 0.035 | 0.619 ± 0.011 | 0.440 ± 0.041 |
| 9 | 0.413 ± 0.009 | 0.438 ± 0.039 | 0.539 ± 0.052 | 0.569 ± 0.042 | 0.473 ± 0.031 |
",
+ "type": "table",
+ "image_path": "f96cfa9e68c23c4032c8a3be2585c35040745266b8bd544118d3a828d0808a9b.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 3,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 111,
+ 199,
+ 500,
+ 346.66666666666663
+ ],
+ "spans": [],
+ "index": 2
+ },
+ {
+ "bbox": [
+ 111,
+ 346.66666666666663,
+ 500,
+ 494.33333333333326
+ ],
+ "spans": [],
+ "index": 3
+ },
+ {
+ "bbox": [
+ 111,
+ 494.33333333333326,
+ 500,
+ 641.9999999999999
+ ],
+ "spans": [],
+ "index": 4
+ }
+ ]
+ }
+ ],
+ "index": 1.75
+ }
+ ],
+ "page_idx": 29,
+ "page_size": [
+ 612,
+ 792
+ ],
+ "discarded_blocks": [
+ {
+ "type": "discarded",
+ "bbox": [
+ 107,
+ 27,
+ 294,
+ 37
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 26,
+ 294,
+ 38
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 26,
+ 294,
+ 38
+ ],
+ "score": 1.0,
+ "content": "Published as a conference paper at ICLR 2020",
+ "type": "text"
+ }
+ ]
+ }
+ ]
+ },
+ {
+ "type": "discarded",
+ "bbox": [
+ 300,
+ 751,
+ 311,
+ 760
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 298,
+ 750,
+ 313,
+ 763
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 298,
+ 750,
+ 313,
+ 763
+ ],
+ "score": 1.0,
+ "content": "30",
+ "type": "text"
+ }
+ ]
+ }
+ ]
+ }
+ ],
+ "para_blocks": [
+ {
+ "type": "table",
+ "bbox": [
+ 111,
+ 199,
+ 500,
+ 642
+ ],
+ "blocks": [
+ {
+ "type": "table_caption",
+ "bbox": [
+ 105,
+ 175,
+ 504,
+ 208
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 175,
+ 505,
+ 189
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 175,
+ 505,
+ 189
+ ],
+ "score": 1.0,
+ "content": "Table 4: Comparison of performance in terms of maximum singular value of weights and layer size.",
+ "type": "text"
+ }
+ ],
+ "index": 0
+ },
+ {
+ "bbox": [
+ 106,
+ 186,
+ 473,
+ 199
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 186,
+ 473,
+ 199
+ ],
+ "score": 1.0,
+ "content": "“U” in the right most column indicates the accuracy of GCN without weight normalization.",
+ "type": "text"
+ }
+ ],
+ "index": 1
+ }
+ ],
+ "index": 0.5
+ },
+ {
+ "type": "table_body",
+ "bbox": [
+ 111,
+ 199,
+ 500,
+ 642
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 111,
+ 199,
+ 500,
+ 642
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 111,
+ 199,
+ 500,
+ 642
+ ],
+ "score": 0.861,
+ "html": "| Noisy Cora 2500 |
|
| Depth | | 1.05 | 3 | 10 | U |
| 1 | 0.389 ± 0.101 | 0.429 ± 0.090 | 0.552 ± 0.014 | 0.632 ± 0.007 | 0.587 ± 0.008 |
| 3 | 0.273 ± 0.051 | 0.309 ± 0.017 | 0.580± 0.058 | 0.661 ± 0.003 | 0.494 ± 0.041 |
| 5 | 0.319 ± 0.000 | 0.267 ± 0.059 | 0.462 ± 0.065 | 0.602 ± 0.004 | 0.326 ± 0.029 |
| 7 | 0.261 ± 0.076 | 0.262 ±0.080 | 0.407 ± 0.021 | 0.501 ± 0.017 | 0.279 ± 0.129 |
| 9 | 0.261 ± 0.080 | 0.319 ± 0.000 | 0.284 ± 0.109 | 0.443 ± 0.014 | 0.319 ± 0.000 |
| Noisy Cora 5000 |
| Maximum Singular Value |
| Depth | 1 | 1.1 | 3 | 10 | U |
| 1 | 0.301 ± 0.080 | 0.333 ± 0.099 | 0.557 ± 0.004 | 0.561 ± 0.019 | 0.555 ± 0.016 |
| 3 | 0.245 ± 0.066 | 0.247 ± 0.076 | 0.370 ± 0.041 | 0.587 ± 0.009 | 0.286 ± 0.066 |
| 5 | 0.274± 0.048 | 0.237 ± 0.070 | 0.257 ± 0.076 | 0.535 ± 0.031 | 0.319 ± 0.000 |
| 7 | 0.263 ± 0.080 | 0.297 ± 0.031 | 0.260 ± 0.074 | 0.339 ± 0.060 | 0.319 ± 0.000 |
| 9 | 0.262 ± 0.081 | 0.258 ± 0.064 | 0.262 ± 0.080 | 0.261 ±0.082 | 0.318 ± 0.002 |
| Noisy CiteSeer |
| Maximum Singular Value |
| Depth | 0.5 | 1.1 3 | | 10 | U |
| 1 | 0.461 ± 0.018 | 0.467 ± 0.012 | 0.490 ± 0.016 | 0.494 ± 0.006 | 0.495 ± 0.009 |
| 3 | 0.438 ± 0.027 | 0.436 ± 0.010 | 0.450 ± 0.019 | 0.462 ± 0.007 | 0.417 ± 0.061 |
| 5 | 0.285 ± 0.008 | 0.371 ± 0.016 | 0.373 ± 0.011 | 0.425 ± 0.007 | 0.380 ± 0.024 |
| 7 | 0.213 ± 0.006 | 0.282 ± 0.011 | 0.309 ± 0.012 | 0.385 ± 0.007 | 0.308 ± 0.012 |
| 9 | 0.182 ± 0.005 | 0.242 ± 0.030 | 0.303 ± 0.021 | 0.325 ± 0.003 | 0.229 ± 0.033 |
| Noisy Pubmed |
| Maximum Singular Value |
| Depth | 0.5 | 1.1 | 3 | 10 | U |
| 1 | 0.488 ± 0.039 | 0.636 ± 0.006 | 0.641 ± 0.010 | 0.632 ± 0.002 | 0.631 ± 0.010 |
| 3 | 0.442 ± 0.027 | 0.426 ± 0.026 | 0.658 ± 0.004 | 0.661 ± 0.005 | 0.631 ± 0.013 |
| 5 | 0.431 ± 0.033 | 0.431 ± 0.034 | 0.561 ± 0.083 | 0.641 ± 0.004 | 0.424 ± 0.093 |
| 7 | 0.428 ± 0.032 | 0.443 ± 0.051 | 0.449 ± 0.035 | 0.619 ± 0.011 | 0.440 ± 0.041 |
| 9 | 0.413 ± 0.009 | 0.438 ± 0.039 | 0.539 ± 0.052 | 0.569 ± 0.042 | 0.473 ± 0.031 |
",
+ "type": "table",
+ "image_path": "f96cfa9e68c23c4032c8a3be2585c35040745266b8bd544118d3a828d0808a9b.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 3,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 111,
+ 199,
+ 500,
+ 346.66666666666663
+ ],
+ "spans": [],
+ "index": 2
+ },
+ {
+ "bbox": [
+ 111,
+ 346.66666666666663,
+ 500,
+ 494.33333333333326
+ ],
+ "spans": [],
+ "index": 3
+ },
+ {
+ "bbox": [
+ 111,
+ 494.33333333333326,
+ 500,
+ 641.9999999999999
+ ],
+ "spans": [],
+ "index": 4
+ }
+ ]
+ }
+ ],
+ "index": 1.75
+ }
+ ]
+ },
+ {
+ "preproc_blocks": [
+ {
+ "type": "image",
+ "bbox": [
+ 117,
+ 211,
+ 488,
+ 524
+ ],
+ "blocks": [
+ {
+ "type": "image_body",
+ "bbox": [
+ 117,
+ 211,
+ 488,
+ 524
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 117,
+ 211,
+ 488,
+ 524
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 117,
+ 211,
+ 488,
+ 524
+ ],
+ "score": 0.976,
+ "type": "image",
+ "image_path": "35ca2d11400a46fce3bca8218b146c3b3f571e569a0c3840aef09cad2b8441e8.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 1,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 117,
+ 211,
+ 488,
+ 315.3333333333333
+ ],
+ "spans": [],
+ "index": 0
+ },
+ {
+ "bbox": [
+ 117,
+ 315.3333333333333,
+ 488,
+ 419.66666666666663
+ ],
+ "spans": [],
+ "index": 1
+ },
+ {
+ "bbox": [
+ 117,
+ 419.66666666666663,
+ 488,
+ 524.0
+ ],
+ "spans": [],
+ "index": 2
+ }
+ ]
+ },
+ {
+ "type": "image_caption",
+ "bbox": [
+ 106,
+ 539,
+ 506,
+ 604
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 539,
+ 505,
+ 552
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 539,
+ 223,
+ 552
+ ],
+ "score": 1.0,
+ "content": "Figure 8: The actual distance",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 223,
+ 540,
+ 239,
+ 551
+ ],
+ "score": 0.9,
+ "content": "d _ { \\mathcal { M } }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 240,
+ 539,
+ 325,
+ 552
+ ],
+ "score": 1.0,
+ "content": "to the invariant space",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 325,
+ 540,
+ 338,
+ 550
+ ],
+ "score": 0.8,
+ "content": "\\mathcal { M }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 339,
+ 539,
+ 505,
+ 552
+ ],
+ "score": 1.0,
+ "content": "and the upper bound inferred by Theorem",
+ "type": "text"
+ }
+ ],
+ "index": 3
+ },
+ {
+ "bbox": [
+ 105,
+ 550,
+ 506,
+ 564
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 550,
+ 200,
+ 564
+ ],
+ "score": 1.0,
+ "content": "1. The edge probability",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 200,
+ 552,
+ 207,
+ 563
+ ],
+ "score": 0.8,
+ "content": "p",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 208,
+ 550,
+ 475,
+ 564
+ ],
+ "score": 1.0,
+ "content": "takes 0.01(top), 0.1, 0.9(bottom) and the maximum singular value",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 476,
+ 553,
+ 482,
+ 561
+ ],
+ "score": 0.78,
+ "content": "s",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 482,
+ 550,
+ 506,
+ 564
+ ],
+ "score": 1.0,
+ "content": "takes",
+ "type": "text"
+ }
+ ],
+ "index": 4
+ },
+ {
+ "bbox": [
+ 106,
+ 561,
+ 506,
+ 583
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 565,
+ 202,
+ 578
+ ],
+ "score": 0.72,
+ "content": "0 . 1 ( \\mathrm { l e f t } ) , 1 . 0 , 1 0 ( \\mathrm { r i g h t } )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 203,
+ 561,
+ 402,
+ 583
+ ],
+ "score": 1.0,
+ "content": ". Blue lines are the log relative distance defined by",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 402,
+ 562,
+ 487,
+ 581
+ ],
+ "score": 0.94,
+ "content": "\\begin{array} { r } { y ( l ) : = \\log \\frac { d _ { \\mathcal { M } } ( X ^ { ( l ) } ) } { d _ { \\mathcal { M } } ( X ^ { ( 0 ) } ) } } \\end{array}",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 487,
+ 561,
+ 506,
+ 583
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ }
+ ],
+ "index": 5
+ },
+ {
+ "bbox": [
+ 105,
+ 579,
+ 506,
+ 594
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 580,
+ 250,
+ 594
+ ],
+ "score": 1.0,
+ "content": "orange dotted lines are upper bound",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 251,
+ 580,
+ 320,
+ 593
+ ],
+ "score": 0.92,
+ "content": "y ( l ) : = l \\log ( s \\lambda )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 320,
+ 580,
+ 350,
+ 594
+ ],
+ "score": 1.0,
+ "content": ", where",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 350,
+ 579,
+ 371,
+ 591
+ ],
+ "score": 0.9,
+ "content": "X ^ { ( 0 ) }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 371,
+ 580,
+ 461,
+ 594
+ ],
+ "score": 1.0,
+ "content": "is the input signal and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 461,
+ 580,
+ 480,
+ 591
+ ],
+ "score": 0.9,
+ "content": "X ^ { ( l ) }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 480,
+ 580,
+ 506,
+ 594
+ ],
+ "score": 1.0,
+ "content": "is the",
+ "type": "text"
+ }
+ ],
+ "index": 6
+ },
+ {
+ "bbox": [
+ 106,
+ 592,
+ 279,
+ 605
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 592,
+ 160,
+ 605
+ ],
+ "score": 1.0,
+ "content": "output of the",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 160,
+ 593,
+ 164,
+ 602
+ ],
+ "score": 0.69,
+ "content": "l",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 165,
+ 592,
+ 279,
+ 605
+ ],
+ "score": 1.0,
+ "content": "-th layer. Best view in color.",
+ "type": "text"
+ }
+ ],
+ "index": 7
+ }
+ ],
+ "index": 5
+ }
+ ],
+ "index": 3.0
+ }
+ ],
+ "page_idx": 30,
+ "page_size": [
+ 612,
+ 792
+ ],
+ "discarded_blocks": [
+ {
+ "type": "discarded",
+ "bbox": [
+ 107,
+ 27,
+ 293,
+ 37
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 26,
+ 294,
+ 38
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 26,
+ 294,
+ 38
+ ],
+ "score": 1.0,
+ "content": "Published as a conference paper at ICLR 2020",
+ "type": "text"
+ }
+ ]
+ }
+ ]
+ },
+ {
+ "type": "discarded",
+ "bbox": [
+ 300,
+ 751,
+ 310,
+ 760
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 299,
+ 750,
+ 312,
+ 764
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 299,
+ 750,
+ 312,
+ 764
+ ],
+ "score": 1.0,
+ "content": "",
+ "type": "text",
+ "height": 14,
+ "width": 13
+ }
+ ]
+ }
+ ]
+ }
+ ],
+ "para_blocks": [
+ {
+ "type": "image",
+ "bbox": [
+ 117,
+ 211,
+ 488,
+ 524
+ ],
+ "blocks": [
+ {
+ "type": "image_body",
+ "bbox": [
+ 117,
+ 211,
+ 488,
+ 524
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 117,
+ 211,
+ 488,
+ 524
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 117,
+ 211,
+ 488,
+ 524
+ ],
+ "score": 0.976,
+ "type": "image",
+ "image_path": "35ca2d11400a46fce3bca8218b146c3b3f571e569a0c3840aef09cad2b8441e8.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 1,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 117,
+ 211,
+ 488,
+ 315.3333333333333
+ ],
+ "spans": [],
+ "index": 0
+ },
+ {
+ "bbox": [
+ 117,
+ 315.3333333333333,
+ 488,
+ 419.66666666666663
+ ],
+ "spans": [],
+ "index": 1
+ },
+ {
+ "bbox": [
+ 117,
+ 419.66666666666663,
+ 488,
+ 524.0
+ ],
+ "spans": [],
+ "index": 2
+ }
+ ]
+ },
+ {
+ "type": "image_caption",
+ "bbox": [
+ 106,
+ 539,
+ 506,
+ 604
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 539,
+ 505,
+ 552
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 539,
+ 223,
+ 552
+ ],
+ "score": 1.0,
+ "content": "Figure 8: The actual distance",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 223,
+ 540,
+ 239,
+ 551
+ ],
+ "score": 0.9,
+ "content": "d _ { \\mathcal { M } }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 240,
+ 539,
+ 325,
+ 552
+ ],
+ "score": 1.0,
+ "content": "to the invariant space",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 325,
+ 540,
+ 338,
+ 550
+ ],
+ "score": 0.8,
+ "content": "\\mathcal { M }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 339,
+ 539,
+ 505,
+ 552
+ ],
+ "score": 1.0,
+ "content": "and the upper bound inferred by Theorem",
+ "type": "text"
+ }
+ ],
+ "index": 3
+ },
+ {
+ "bbox": [
+ 105,
+ 550,
+ 506,
+ 564
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 550,
+ 200,
+ 564
+ ],
+ "score": 1.0,
+ "content": "1. The edge probability",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 200,
+ 552,
+ 207,
+ 563
+ ],
+ "score": 0.8,
+ "content": "p",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 208,
+ 550,
+ 475,
+ 564
+ ],
+ "score": 1.0,
+ "content": "takes 0.01(top), 0.1, 0.9(bottom) and the maximum singular value",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 476,
+ 553,
+ 482,
+ 561
+ ],
+ "score": 0.78,
+ "content": "s",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 482,
+ 550,
+ 506,
+ 564
+ ],
+ "score": 1.0,
+ "content": "takes",
+ "type": "text"
+ }
+ ],
+ "index": 4
+ },
+ {
+ "bbox": [
+ 106,
+ 561,
+ 506,
+ 583
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 565,
+ 202,
+ 578
+ ],
+ "score": 0.72,
+ "content": "0 . 1 ( \\mathrm { l e f t } ) , 1 . 0 , 1 0 ( \\mathrm { r i g h t } )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 203,
+ 561,
+ 402,
+ 583
+ ],
+ "score": 1.0,
+ "content": ". Blue lines are the log relative distance defined by",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 402,
+ 562,
+ 487,
+ 581
+ ],
+ "score": 0.94,
+ "content": "\\begin{array} { r } { y ( l ) : = \\log \\frac { d _ { \\mathcal { M } } ( X ^ { ( l ) } ) } { d _ { \\mathcal { M } } ( X ^ { ( 0 ) } ) } } \\end{array}",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 487,
+ 561,
+ 506,
+ 583
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ }
+ ],
+ "index": 5
+ },
+ {
+ "bbox": [
+ 105,
+ 579,
+ 506,
+ 594
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 580,
+ 250,
+ 594
+ ],
+ "score": 1.0,
+ "content": "orange dotted lines are upper bound",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 251,
+ 580,
+ 320,
+ 593
+ ],
+ "score": 0.92,
+ "content": "y ( l ) : = l \\log ( s \\lambda )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 320,
+ 580,
+ 350,
+ 594
+ ],
+ "score": 1.0,
+ "content": ", where",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 350,
+ 579,
+ 371,
+ 591
+ ],
+ "score": 0.9,
+ "content": "X ^ { ( 0 ) }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 371,
+ 580,
+ 461,
+ 594
+ ],
+ "score": 1.0,
+ "content": "is the input signal and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 461,
+ 580,
+ 480,
+ 591
+ ],
+ "score": 0.9,
+ "content": "X ^ { ( l ) }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 480,
+ 580,
+ 506,
+ 594
+ ],
+ "score": 1.0,
+ "content": "is the",
+ "type": "text"
+ }
+ ],
+ "index": 6
+ },
+ {
+ "bbox": [
+ 106,
+ 592,
+ 279,
+ 605
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 592,
+ 160,
+ 605
+ ],
+ "score": 1.0,
+ "content": "output of the",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 160,
+ 593,
+ 164,
+ 602
+ ],
+ "score": 0.69,
+ "content": "l",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 165,
+ 592,
+ 279,
+ 605
+ ],
+ "score": 1.0,
+ "content": "-th layer. Best view in color.",
+ "type": "text"
+ }
+ ],
+ "index": 7
+ }
+ ],
+ "index": 5
+ }
+ ],
+ "index": 3.0
+ }
+ ]
+ },
+ {
+ "preproc_blocks": [
+ {
+ "type": "image",
+ "bbox": [
+ 122,
+ 93,
+ 488,
+ 660
+ ],
+ "blocks": [
+ {
+ "type": "image_body",
+ "bbox": [
+ 122,
+ 93,
+ 488,
+ 660
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 122,
+ 93,
+ 488,
+ 660
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 122,
+ 93,
+ 488,
+ 660
+ ],
+ "score": 0.961,
+ "type": "image",
+ "image_path": "e5a0ea6bac9839dd91cc7cc13ba7f7a7b0ff470526d0f4afb7388e35011ae897.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 1,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 122,
+ 93,
+ 488,
+ 282.0
+ ],
+ "spans": [],
+ "index": 0
+ },
+ {
+ "bbox": [
+ 122,
+ 282.0,
+ 488,
+ 471.0
+ ],
+ "spans": [],
+ "index": 1
+ },
+ {
+ "bbox": [
+ 122,
+ 471.0,
+ 488,
+ 660.0
+ ],
+ "spans": [],
+ "index": 2
+ }
+ ]
+ },
+ {
+ "type": "image_caption",
+ "bbox": [
+ 106,
+ 670,
+ 506,
+ 716
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 671,
+ 506,
+ 684
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 671,
+ 506,
+ 684
+ ],
+ "score": 1.0,
+ "content": "Figure 9: Effect of the maximum singular values of weights on predictive performance. Horizontal",
+ "type": "text"
+ }
+ ],
+ "index": 3
+ },
+ {
+ "bbox": [
+ 106,
+ 682,
+ 505,
+ 694
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 682,
+ 261,
+ 694
+ ],
+ "score": 1.0,
+ "content": "dotted lines indicate the chance rates",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 261,
+ 682,
+ 289,
+ 692
+ ],
+ "score": 0.85,
+ "content": "3 0 . 2 \\%",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 289,
+ 682,
+ 378,
+ 694
+ ],
+ "score": 1.0,
+ "content": "for Noisy Cora 5000,",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 379,
+ 682,
+ 406,
+ 692
+ ],
+ "score": 0.84,
+ "content": "2 1 . 2 \\%",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 406,
+ 682,
+ 505,
+ 694
+ ],
+ "score": 1.0,
+ "content": "for Noisy CiteSeer, and",
+ "type": "text"
+ }
+ ],
+ "index": 4
+ },
+ {
+ "bbox": [
+ 106,
+ 693,
+ 505,
+ 705
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 693,
+ 134,
+ 703
+ ],
+ "score": 0.85,
+ "content": "3 9 . 9 \\%",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 134,
+ 693,
+ 505,
+ 705
+ ],
+ "score": 1.0,
+ "content": "for Noisy PubMed). The error bar is the standard deviation of 3 trials. Left: Noisy Cora",
+ "type": "text"
+ }
+ ],
+ "index": 5
+ },
+ {
+ "bbox": [
+ 105,
+ 703,
+ 403,
+ 717
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 703,
+ 403,
+ 717
+ ],
+ "score": 1.0,
+ "content": "5000. Right: Noisy CiteSeer. Bottom: Noisy Pubmed. Best view in color.",
+ "type": "text"
+ }
+ ],
+ "index": 6
+ }
+ ],
+ "index": 4.5
+ }
+ ],
+ "index": 2.75
+ }
+ ],
+ "page_idx": 31,
+ "page_size": [
+ 612,
+ 792
+ ],
+ "discarded_blocks": [
+ {
+ "type": "discarded",
+ "bbox": [
+ 107,
+ 27,
+ 293,
+ 37
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 26,
+ 294,
+ 38
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 26,
+ 294,
+ 38
+ ],
+ "score": 1.0,
+ "content": "Published as a conference paper at ICLR 2020",
+ "type": "text"
+ }
+ ]
+ }
+ ]
+ },
+ {
+ "type": "discarded",
+ "bbox": [
+ 300,
+ 751,
+ 311,
+ 761
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 298,
+ 750,
+ 313,
+ 765
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 298,
+ 750,
+ 313,
+ 765
+ ],
+ "score": 1.0,
+ "content": "",
+ "type": "text",
+ "height": 15,
+ "width": 15
+ }
+ ]
+ }
+ ]
+ }
+ ],
+ "para_blocks": [
+ {
+ "type": "image",
+ "bbox": [
+ 122,
+ 93,
+ 488,
+ 660
+ ],
+ "blocks": [
+ {
+ "type": "image_body",
+ "bbox": [
+ 122,
+ 93,
+ 488,
+ 660
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 122,
+ 93,
+ 488,
+ 660
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 122,
+ 93,
+ 488,
+ 660
+ ],
+ "score": 0.961,
+ "type": "image",
+ "image_path": "e5a0ea6bac9839dd91cc7cc13ba7f7a7b0ff470526d0f4afb7388e35011ae897.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 1,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 122,
+ 93,
+ 488,
+ 282.0
+ ],
+ "spans": [],
+ "index": 0
+ },
+ {
+ "bbox": [
+ 122,
+ 282.0,
+ 488,
+ 471.0
+ ],
+ "spans": [],
+ "index": 1
+ },
+ {
+ "bbox": [
+ 122,
+ 471.0,
+ 488,
+ 660.0
+ ],
+ "spans": [],
+ "index": 2
+ }
+ ]
+ },
+ {
+ "type": "image_caption",
+ "bbox": [
+ 106,
+ 670,
+ 506,
+ 716
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 671,
+ 506,
+ 684
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 671,
+ 506,
+ 684
+ ],
+ "score": 1.0,
+ "content": "Figure 9: Effect of the maximum singular values of weights on predictive performance. Horizontal",
+ "type": "text"
+ }
+ ],
+ "index": 3
+ },
+ {
+ "bbox": [
+ 106,
+ 682,
+ 505,
+ 694
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 682,
+ 261,
+ 694
+ ],
+ "score": 1.0,
+ "content": "dotted lines indicate the chance rates",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 261,
+ 682,
+ 289,
+ 692
+ ],
+ "score": 0.85,
+ "content": "3 0 . 2 \\%",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 289,
+ 682,
+ 378,
+ 694
+ ],
+ "score": 1.0,
+ "content": "for Noisy Cora 5000,",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 379,
+ 682,
+ 406,
+ 692
+ ],
+ "score": 0.84,
+ "content": "2 1 . 2 \\%",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 406,
+ 682,
+ 505,
+ 694
+ ],
+ "score": 1.0,
+ "content": "for Noisy CiteSeer, and",
+ "type": "text"
+ }
+ ],
+ "index": 4
+ },
+ {
+ "bbox": [
+ 106,
+ 693,
+ 505,
+ 705
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 693,
+ 134,
+ 703
+ ],
+ "score": 0.85,
+ "content": "3 9 . 9 \\%",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 134,
+ 693,
+ 505,
+ 705
+ ],
+ "score": 1.0,
+ "content": "for Noisy PubMed). The error bar is the standard deviation of 3 trials. Left: Noisy Cora",
+ "type": "text"
+ }
+ ],
+ "index": 5
+ },
+ {
+ "bbox": [
+ 105,
+ 703,
+ 403,
+ 717
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 703,
+ 403,
+ 717
+ ],
+ "score": 1.0,
+ "content": "5000. Right: Noisy CiteSeer. Bottom: Noisy Pubmed. Best view in color.",
+ "type": "text"
+ }
+ ],
+ "index": 6
+ }
+ ],
+ "index": 4.5
+ }
+ ],
+ "index": 2.75
+ }
+ ]
+ },
+ {
+ "preproc_blocks": [
+ {
+ "type": "image",
+ "bbox": [
+ 117,
+ 192,
+ 495,
+ 581
+ ],
+ "blocks": [
+ {
+ "type": "image_body",
+ "bbox": [
+ 117,
+ 192,
+ 495,
+ 581
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 117,
+ 192,
+ 495,
+ 581
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 117,
+ 192,
+ 495,
+ 581
+ ],
+ "score": 0.977,
+ "type": "image",
+ "image_path": "fbb64a2ad3ffb4819bd64e9f80ed2a25542f265d2c003b561d981ebe46bf9d83.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 1,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 117,
+ 192,
+ 495,
+ 321.66666666666663
+ ],
+ "spans": [],
+ "index": 0
+ },
+ {
+ "bbox": [
+ 117,
+ 321.66666666666663,
+ 495,
+ 451.33333333333326
+ ],
+ "spans": [],
+ "index": 1
+ },
+ {
+ "bbox": [
+ 117,
+ 451.33333333333326,
+ 495,
+ 580.9999999999999
+ ],
+ "spans": [],
+ "index": 2
+ }
+ ]
+ },
+ {
+ "type": "image_caption",
+ "bbox": [
+ 106,
+ 600,
+ 504,
+ 623
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 600,
+ 504,
+ 613
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 600,
+ 504,
+ 613
+ ],
+ "score": 1.0,
+ "content": "Figure 10: Transition of maximum singular values of GCN during training using Noisy Cora 2500.",
+ "type": "text"
+ }
+ ],
+ "index": 3
+ },
+ {
+ "bbox": [
+ 106,
+ 611,
+ 436,
+ 624
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 611,
+ 436,
+ 624
+ ],
+ "score": 1.0,
+ "content": "Top left: 1 layer. Top right: 5 layers. Bottom left: 7 layers. Bottom right: 9 layers.",
+ "type": "text"
+ }
+ ],
+ "index": 4
+ }
+ ],
+ "index": 3.5
+ }
+ ],
+ "index": 2.25
+ }
+ ],
+ "page_idx": 32,
+ "page_size": [
+ 612,
+ 792
+ ],
+ "discarded_blocks": [
+ {
+ "type": "discarded",
+ "bbox": [
+ 107,
+ 27,
+ 293,
+ 37
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 26,
+ 294,
+ 38
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 26,
+ 294,
+ 38
+ ],
+ "score": 1.0,
+ "content": "Published as a conference paper at ICLR 2020",
+ "type": "text"
+ }
+ ]
+ }
+ ]
+ },
+ {
+ "type": "discarded",
+ "bbox": [
+ 300,
+ 751,
+ 311,
+ 760
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 298,
+ 750,
+ 312,
+ 763
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 298,
+ 750,
+ 312,
+ 763
+ ],
+ "score": 1.0,
+ "content": "33",
+ "type": "text"
+ }
+ ]
+ }
+ ]
+ },
+ {
+ "type": "discarded",
+ "bbox": [
+ 270,
+ 187,
+ 341,
+ 197
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 269,
+ 185,
+ 342,
+ 199
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 269,
+ 185,
+ 342,
+ 199
+ ],
+ "score": 1.0,
+ "content": "Noisy Cora 2500",
+ "type": "text"
+ }
+ ]
+ }
+ ]
+ }
+ ],
+ "para_blocks": [
+ {
+ "type": "image",
+ "bbox": [
+ 117,
+ 192,
+ 495,
+ 581
+ ],
+ "blocks": [
+ {
+ "type": "image_body",
+ "bbox": [
+ 117,
+ 192,
+ 495,
+ 581
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 117,
+ 192,
+ 495,
+ 581
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 117,
+ 192,
+ 495,
+ 581
+ ],
+ "score": 0.977,
+ "type": "image",
+ "image_path": "fbb64a2ad3ffb4819bd64e9f80ed2a25542f265d2c003b561d981ebe46bf9d83.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 1,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 117,
+ 192,
+ 495,
+ 321.66666666666663
+ ],
+ "spans": [],
+ "index": 0
+ },
+ {
+ "bbox": [
+ 117,
+ 321.66666666666663,
+ 495,
+ 451.33333333333326
+ ],
+ "spans": [],
+ "index": 1
+ },
+ {
+ "bbox": [
+ 117,
+ 451.33333333333326,
+ 495,
+ 580.9999999999999
+ ],
+ "spans": [],
+ "index": 2
+ }
+ ]
+ },
+ {
+ "type": "image_caption",
+ "bbox": [
+ 106,
+ 600,
+ 504,
+ 623
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 600,
+ 504,
+ 613
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 600,
+ 504,
+ 613
+ ],
+ "score": 1.0,
+ "content": "Figure 10: Transition of maximum singular values of GCN during training using Noisy Cora 2500.",
+ "type": "text"
+ }
+ ],
+ "index": 3
+ },
+ {
+ "bbox": [
+ 106,
+ 611,
+ 436,
+ 624
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 611,
+ 436,
+ 624
+ ],
+ "score": 1.0,
+ "content": "Top left: 1 layer. Top right: 5 layers. Bottom left: 7 layers. Bottom right: 9 layers.",
+ "type": "text"
+ }
+ ],
+ "index": 4
+ }
+ ],
+ "index": 3.5
+ }
+ ],
+ "index": 2.25
+ }
+ ]
+ },
+ {
+ "preproc_blocks": [
+ {
+ "type": "image",
+ "bbox": [
+ 114,
+ 193,
+ 493,
+ 582
+ ],
+ "blocks": [
+ {
+ "type": "image_body",
+ "bbox": [
+ 114,
+ 193,
+ 493,
+ 582
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 114,
+ 193,
+ 493,
+ 582
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 114,
+ 193,
+ 493,
+ 582
+ ],
+ "score": 0.976,
+ "type": "image",
+ "image_path": "1f6937585cdf3a009373b92bd7684c89717d7794c270d219f6efef36b2ea759b.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 1,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 114,
+ 193,
+ 493,
+ 322.66666666666663
+ ],
+ "spans": [],
+ "index": 0
+ },
+ {
+ "bbox": [
+ 114,
+ 322.66666666666663,
+ 493,
+ 452.33333333333326
+ ],
+ "spans": [],
+ "index": 1
+ },
+ {
+ "bbox": [
+ 114,
+ 452.33333333333326,
+ 493,
+ 581.9999999999999
+ ],
+ "spans": [],
+ "index": 2
+ }
+ ]
+ },
+ {
+ "type": "image_caption",
+ "bbox": [
+ 105,
+ 600,
+ 504,
+ 624
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 600,
+ 504,
+ 613
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 600,
+ 504,
+ 613
+ ],
+ "score": 1.0,
+ "content": "Figure 11: Transition of maximum singular values of GCN during training using Noisy Cora 5000.",
+ "type": "text"
+ }
+ ],
+ "index": 3
+ },
+ {
+ "bbox": [
+ 105,
+ 611,
+ 505,
+ 624
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 611,
+ 505,
+ 624
+ ],
+ "score": 1.0,
+ "content": "Top left: 1 layer. Top right: 3 layers. Middle left: 5 layers. Middle right: 7 layers. Bottom: 9 layers.",
+ "type": "text"
+ }
+ ],
+ "index": 4
+ }
+ ],
+ "index": 3.5
+ }
+ ],
+ "index": 2.25
+ }
+ ],
+ "page_idx": 33,
+ "page_size": [
+ 612,
+ 792
+ ],
+ "discarded_blocks": [
+ {
+ "type": "discarded",
+ "bbox": [
+ 107,
+ 27,
+ 293,
+ 37
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 26,
+ 294,
+ 38
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 26,
+ 294,
+ 38
+ ],
+ "score": 1.0,
+ "content": "Published as a conference paper at ICLR 2020",
+ "type": "text"
+ }
+ ]
+ }
+ ]
+ },
+ {
+ "type": "discarded",
+ "bbox": [
+ 300,
+ 751,
+ 311,
+ 760
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 298,
+ 749,
+ 313,
+ 764
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 298,
+ 749,
+ 313,
+ 764
+ ],
+ "score": 1.0,
+ "content": "",
+ "type": "text",
+ "height": 15,
+ "width": 15
+ }
+ ]
+ }
+ ]
+ },
+ {
+ "type": "discarded",
+ "bbox": [
+ 270,
+ 187,
+ 341,
+ 197
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 269,
+ 185,
+ 342,
+ 199
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 269,
+ 185,
+ 342,
+ 199
+ ],
+ "score": 1.0,
+ "content": "Noisy Cora 5000",
+ "type": "text"
+ }
+ ]
+ }
+ ]
+ }
+ ],
+ "para_blocks": [
+ {
+ "type": "image",
+ "bbox": [
+ 114,
+ 193,
+ 493,
+ 582
+ ],
+ "blocks": [
+ {
+ "type": "image_body",
+ "bbox": [
+ 114,
+ 193,
+ 493,
+ 582
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 114,
+ 193,
+ 493,
+ 582
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 114,
+ 193,
+ 493,
+ 582
+ ],
+ "score": 0.976,
+ "type": "image",
+ "image_path": "1f6937585cdf3a009373b92bd7684c89717d7794c270d219f6efef36b2ea759b.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 1,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 114,
+ 193,
+ 493,
+ 322.66666666666663
+ ],
+ "spans": [],
+ "index": 0
+ },
+ {
+ "bbox": [
+ 114,
+ 322.66666666666663,
+ 493,
+ 452.33333333333326
+ ],
+ "spans": [],
+ "index": 1
+ },
+ {
+ "bbox": [
+ 114,
+ 452.33333333333326,
+ 493,
+ 581.9999999999999
+ ],
+ "spans": [],
+ "index": 2
+ }
+ ]
+ },
+ {
+ "type": "image_caption",
+ "bbox": [
+ 105,
+ 600,
+ 504,
+ 624
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 600,
+ 504,
+ 613
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 600,
+ 504,
+ 613
+ ],
+ "score": 1.0,
+ "content": "Figure 11: Transition of maximum singular values of GCN during training using Noisy Cora 5000.",
+ "type": "text"
+ }
+ ],
+ "index": 3
+ },
+ {
+ "bbox": [
+ 105,
+ 611,
+ 505,
+ 624
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 611,
+ 505,
+ 624
+ ],
+ "score": 1.0,
+ "content": "Top left: 1 layer. Top right: 3 layers. Middle left: 5 layers. Middle right: 7 layers. Bottom: 9 layers.",
+ "type": "text"
+ }
+ ],
+ "index": 4
+ }
+ ],
+ "index": 3.5
+ }
+ ],
+ "index": 2.25
+ }
+ ]
+ },
+ {
+ "preproc_blocks": [
+ {
+ "type": "image",
+ "bbox": [
+ 117,
+ 192,
+ 495,
+ 581
+ ],
+ "blocks": [
+ {
+ "type": "image_body",
+ "bbox": [
+ 117,
+ 192,
+ 495,
+ 581
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 117,
+ 192,
+ 495,
+ 581
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 117,
+ 192,
+ 495,
+ 581
+ ],
+ "score": 0.977,
+ "type": "image",
+ "image_path": "370a0e6bddfbd9236178e95fde9473ffa8af793fdb6a431485b1c118dd5fa30b.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 1,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 117,
+ 192,
+ 495,
+ 321.66666666666663
+ ],
+ "spans": [],
+ "index": 0
+ },
+ {
+ "bbox": [
+ 117,
+ 321.66666666666663,
+ 495,
+ 451.33333333333326
+ ],
+ "spans": [],
+ "index": 1
+ },
+ {
+ "bbox": [
+ 117,
+ 451.33333333333326,
+ 495,
+ 580.9999999999999
+ ],
+ "spans": [],
+ "index": 2
+ }
+ ]
+ },
+ {
+ "type": "image_caption",
+ "bbox": [
+ 106,
+ 600,
+ 504,
+ 623
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 600,
+ 505,
+ 613
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 600,
+ 505,
+ 613
+ ],
+ "score": 1.0,
+ "content": "Figure 12: Transition of maximum singular values of GCN during training using Noisy CiteSeer.",
+ "type": "text"
+ }
+ ],
+ "index": 3
+ },
+ {
+ "bbox": [
+ 106,
+ 611,
+ 505,
+ 624
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 611,
+ 505,
+ 624
+ ],
+ "score": 1.0,
+ "content": "Top left: 1 layer. Top right: 3 layers. Middle left: 5 layers. Middle right: 7 layers. Bottom: 9 layers.",
+ "type": "text"
+ }
+ ],
+ "index": 4
+ }
+ ],
+ "index": 3.5
+ }
+ ],
+ "index": 2.25
+ }
+ ],
+ "page_idx": 34,
+ "page_size": [
+ 612,
+ 792
+ ],
+ "discarded_blocks": [
+ {
+ "type": "discarded",
+ "bbox": [
+ 107,
+ 27,
+ 293,
+ 37
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 26,
+ 294,
+ 38
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 26,
+ 294,
+ 38
+ ],
+ "score": 1.0,
+ "content": "Published as a conference paper at ICLR 2020",
+ "type": "text"
+ }
+ ]
+ }
+ ]
+ },
+ {
+ "type": "discarded",
+ "bbox": [
+ 300,
+ 751,
+ 311,
+ 760
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 298,
+ 749,
+ 313,
+ 764
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 298,
+ 749,
+ 313,
+ 764
+ ],
+ "score": 1.0,
+ "content": "",
+ "type": "text",
+ "height": 15,
+ "width": 15
+ }
+ ]
+ }
+ ]
+ }
+ ],
+ "para_blocks": [
+ {
+ "type": "image",
+ "bbox": [
+ 117,
+ 192,
+ 495,
+ 581
+ ],
+ "blocks": [
+ {
+ "type": "image_body",
+ "bbox": [
+ 117,
+ 192,
+ 495,
+ 581
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 117,
+ 192,
+ 495,
+ 581
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 117,
+ 192,
+ 495,
+ 581
+ ],
+ "score": 0.977,
+ "type": "image",
+ "image_path": "370a0e6bddfbd9236178e95fde9473ffa8af793fdb6a431485b1c118dd5fa30b.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 1,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 117,
+ 192,
+ 495,
+ 321.66666666666663
+ ],
+ "spans": [],
+ "index": 0
+ },
+ {
+ "bbox": [
+ 117,
+ 321.66666666666663,
+ 495,
+ 451.33333333333326
+ ],
+ "spans": [],
+ "index": 1
+ },
+ {
+ "bbox": [
+ 117,
+ 451.33333333333326,
+ 495,
+ 580.9999999999999
+ ],
+ "spans": [],
+ "index": 2
+ }
+ ]
+ },
+ {
+ "type": "image_caption",
+ "bbox": [
+ 106,
+ 600,
+ 504,
+ 623
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 600,
+ 505,
+ 613
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 600,
+ 505,
+ 613
+ ],
+ "score": 1.0,
+ "content": "Figure 12: Transition of maximum singular values of GCN during training using Noisy CiteSeer.",
+ "type": "text"
+ }
+ ],
+ "index": 3
+ },
+ {
+ "bbox": [
+ 106,
+ 611,
+ 505,
+ 624
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 611,
+ 505,
+ 624
+ ],
+ "score": 1.0,
+ "content": "Top left: 1 layer. Top right: 3 layers. Middle left: 5 layers. Middle right: 7 layers. Bottom: 9 layers.",
+ "type": "text"
+ }
+ ],
+ "index": 4
+ }
+ ],
+ "index": 3.5
+ }
+ ],
+ "index": 2.25
+ }
+ ]
+ },
+ {
+ "preproc_blocks": [
+ {
+ "type": "image",
+ "bbox": [
+ 117,
+ 191,
+ 493,
+ 581
+ ],
+ "blocks": [
+ {
+ "type": "image_body",
+ "bbox": [
+ 117,
+ 191,
+ 493,
+ 581
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 117,
+ 191,
+ 493,
+ 581
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 117,
+ 191,
+ 493,
+ 581
+ ],
+ "score": 0.977,
+ "type": "image",
+ "image_path": "8af84ea100c71eb26aa2e941cecfd27c1136c5885b58d83ae02109b6e1ef5f4c.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 1,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 117,
+ 191,
+ 493,
+ 321.0
+ ],
+ "spans": [],
+ "index": 0
+ },
+ {
+ "bbox": [
+ 117,
+ 321.0,
+ 493,
+ 451.0
+ ],
+ "spans": [],
+ "index": 1
+ },
+ {
+ "bbox": [
+ 117,
+ 451.0,
+ 493,
+ 581.0
+ ],
+ "spans": [],
+ "index": 2
+ }
+ ]
+ },
+ {
+ "type": "image_caption",
+ "bbox": [
+ 106,
+ 600,
+ 504,
+ 623
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 600,
+ 505,
+ 613
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 600,
+ 505,
+ 613
+ ],
+ "score": 1.0,
+ "content": "Figure 13: Transition of maximum singular values of GCN during training using Noisy PubMed.",
+ "type": "text"
+ }
+ ],
+ "index": 3
+ },
+ {
+ "bbox": [
+ 105,
+ 610,
+ 506,
+ 625
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 610,
+ 506,
+ 625
+ ],
+ "score": 1.0,
+ "content": "Top left: 1 layer. Top right: 3 layers. Middle left: 5 layers. Middle right: 7 layers. Bottom: 9 layers.",
+ "type": "text"
+ }
+ ],
+ "index": 4
+ }
+ ],
+ "index": 3.5
+ }
+ ],
+ "index": 2.25
+ }
+ ],
+ "page_idx": 35,
+ "page_size": [
+ 612,
+ 792
+ ],
+ "discarded_blocks": [
+ {
+ "type": "discarded",
+ "bbox": [
+ 107,
+ 27,
+ 293,
+ 37
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 26,
+ 294,
+ 38
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 26,
+ 294,
+ 38
+ ],
+ "score": 1.0,
+ "content": "Published as a conference paper at ICLR 2020",
+ "type": "text"
+ }
+ ]
+ }
+ ]
+ },
+ {
+ "type": "discarded",
+ "bbox": [
+ 300,
+ 751,
+ 311,
+ 760
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 298,
+ 750,
+ 313,
+ 763
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 298,
+ 750,
+ 313,
+ 763
+ ],
+ "score": 1.0,
+ "content": "36",
+ "type": "text"
+ }
+ ]
+ }
+ ]
+ }
+ ],
+ "para_blocks": [
+ {
+ "type": "image",
+ "bbox": [
+ 117,
+ 191,
+ 493,
+ 581
+ ],
+ "blocks": [
+ {
+ "type": "image_body",
+ "bbox": [
+ 117,
+ 191,
+ 493,
+ 581
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 117,
+ 191,
+ 493,
+ 581
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 117,
+ 191,
+ 493,
+ 581
+ ],
+ "score": 0.977,
+ "type": "image",
+ "image_path": "8af84ea100c71eb26aa2e941cecfd27c1136c5885b58d83ae02109b6e1ef5f4c.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 1,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 117,
+ 191,
+ 493,
+ 321.0
+ ],
+ "spans": [],
+ "index": 0
+ },
+ {
+ "bbox": [
+ 117,
+ 321.0,
+ 493,
+ 451.0
+ ],
+ "spans": [],
+ "index": 1
+ },
+ {
+ "bbox": [
+ 117,
+ 451.0,
+ 493,
+ 581.0
+ ],
+ "spans": [],
+ "index": 2
+ }
+ ]
+ },
+ {
+ "type": "image_caption",
+ "bbox": [
+ 106,
+ 600,
+ 504,
+ 623
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 600,
+ 505,
+ 613
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 600,
+ 505,
+ 613
+ ],
+ "score": 1.0,
+ "content": "Figure 13: Transition of maximum singular values of GCN during training using Noisy PubMed.",
+ "type": "text"
+ }
+ ],
+ "index": 3
+ },
+ {
+ "bbox": [
+ 105,
+ 610,
+ 506,
+ 625
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 610,
+ 506,
+ 625
+ ],
+ "score": 1.0,
+ "content": "Top left: 1 layer. Top right: 3 layers. Middle left: 5 layers. Middle right: 7 layers. Bottom: 9 layers.",
+ "type": "text"
+ }
+ ],
+ "index": 4
+ }
+ ],
+ "index": 3.5
+ }
+ ],
+ "index": 2.25
+ }
+ ]
+ },
+ {
+ "preproc_blocks": [
+ {
+ "type": "image",
+ "bbox": [
+ 122,
+ 299,
+ 487,
+ 477
+ ],
+ "blocks": [
+ {
+ "type": "image_body",
+ "bbox": [
+ 122,
+ 299,
+ 487,
+ 477
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 122,
+ 299,
+ 487,
+ 477
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 122,
+ 299,
+ 487,
+ 477
+ ],
+ "score": 0.849,
+ "type": "image",
+ "image_path": "8d0df2fb98721b929112213b1bfbe51f28bd6ef4fb7b8349a5cec2e5bf926177.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 1,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 122,
+ 299,
+ 487,
+ 358.3333333333333
+ ],
+ "spans": [],
+ "index": 0
+ },
+ {
+ "bbox": [
+ 122,
+ 358.3333333333333,
+ 487,
+ 417.66666666666663
+ ],
+ "spans": [],
+ "index": 1
+ },
+ {
+ "bbox": [
+ 122,
+ 417.66666666666663,
+ 487,
+ 476.99999999999994
+ ],
+ "spans": [],
+ "index": 2
+ }
+ ]
+ },
+ {
+ "type": "image_caption",
+ "bbox": [
+ 106,
+ 493,
+ 506,
+ 516
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 492,
+ 505,
+ 507
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 492,
+ 505,
+ 507
+ ],
+ "score": 1.0,
+ "content": "Figure 14: Logarithm of relative perpendicular component and prediction accuracy. Left: Noisy",
+ "type": "text"
+ }
+ ],
+ "index": 3
+ },
+ {
+ "bbox": [
+ 106,
+ 504,
+ 502,
+ 516
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 504,
+ 239,
+ 516
+ ],
+ "score": 1.0,
+ "content": "CiteSeer. Right: Noisy PubMed.",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 239,
+ 506,
+ 246,
+ 516
+ ],
+ "score": 0.78,
+ "content": "p",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 246,
+ 504,
+ 347,
+ 516
+ ],
+ "score": 1.0,
+ "content": "in the title represents the",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 347,
+ 506,
+ 354,
+ 516
+ ],
+ "score": 0.8,
+ "content": "p",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 354,
+ 504,
+ 502,
+ 516
+ ],
+ "score": 1.0,
+ "content": "-value for the Pearson R coefficients.",
+ "type": "text"
+ }
+ ],
+ "index": 4
+ }
+ ],
+ "index": 3.5
+ }
+ ],
+ "index": 2.25
+ }
+ ],
+ "page_idx": 36,
+ "page_size": [
+ 612,
+ 792
+ ],
+ "discarded_blocks": [
+ {
+ "type": "discarded",
+ "bbox": [
+ 107,
+ 27,
+ 294,
+ 37
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 26,
+ 294,
+ 38
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 26,
+ 294,
+ 38
+ ],
+ "score": 1.0,
+ "content": "Published as a conference paper at ICLR 2020",
+ "type": "text"
+ }
+ ]
+ }
+ ]
+ },
+ {
+ "type": "discarded",
+ "bbox": [
+ 300,
+ 751,
+ 311,
+ 760
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 298,
+ 750,
+ 313,
+ 764
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 298,
+ 750,
+ 313,
+ 764
+ ],
+ "score": 1.0,
+ "content": "",
+ "type": "text",
+ "height": 14,
+ "width": 15
+ }
+ ]
+ }
+ ]
+ }
+ ],
+ "para_blocks": [
+ {
+ "type": "image",
+ "bbox": [
+ 122,
+ 299,
+ 487,
+ 477
+ ],
+ "blocks": [
+ {
+ "type": "image_body",
+ "bbox": [
+ 122,
+ 299,
+ 487,
+ 477
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 122,
+ 299,
+ 487,
+ 477
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 122,
+ 299,
+ 487,
+ 477
+ ],
+ "score": 0.849,
+ "type": "image",
+ "image_path": "8d0df2fb98721b929112213b1bfbe51f28bd6ef4fb7b8349a5cec2e5bf926177.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 1,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 122,
+ 299,
+ 487,
+ 358.3333333333333
+ ],
+ "spans": [],
+ "index": 0
+ },
+ {
+ "bbox": [
+ 122,
+ 358.3333333333333,
+ 487,
+ 417.66666666666663
+ ],
+ "spans": [],
+ "index": 1
+ },
+ {
+ "bbox": [
+ 122,
+ 417.66666666666663,
+ 487,
+ 476.99999999999994
+ ],
+ "spans": [],
+ "index": 2
+ }
+ ]
+ },
+ {
+ "type": "image_caption",
+ "bbox": [
+ 106,
+ 493,
+ 506,
+ 516
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 492,
+ 505,
+ 507
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 492,
+ 505,
+ 507
+ ],
+ "score": 1.0,
+ "content": "Figure 14: Logarithm of relative perpendicular component and prediction accuracy. Left: Noisy",
+ "type": "text"
+ }
+ ],
+ "index": 3
+ },
+ {
+ "bbox": [
+ 106,
+ 504,
+ 502,
+ 516
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 504,
+ 239,
+ 516
+ ],
+ "score": 1.0,
+ "content": "CiteSeer. Right: Noisy PubMed.",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 239,
+ 506,
+ 246,
+ 516
+ ],
+ "score": 0.78,
+ "content": "p",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 246,
+ 504,
+ 347,
+ 516
+ ],
+ "score": 1.0,
+ "content": "in the title represents the",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 347,
+ 506,
+ 354,
+ 516
+ ],
+ "score": 0.8,
+ "content": "p",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 354,
+ 504,
+ 502,
+ 516
+ ],
+ "score": 1.0,
+ "content": "-value for the Pearson R coefficients.",
+ "type": "text"
+ }
+ ],
+ "index": 4
+ }
+ ],
+ "index": 3.5
+ }
+ ],
+ "index": 2.25
+ }
+ ]
+ }
+ ],
+ "_backend": "pipeline",
+ "_version_name": "2.2.2"
+}
\ No newline at end of file