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Embeddings were aligned using Procrustes rotation.", "type": "text" } ], "index": 36 } ], "index": 34.5 } ], "index": 32.75 } ] }, { "preproc_blocks": [ { "type": "text", "bbox": [ 107, 82, 505, 137 ], "lines": [ { "bbox": [ 106, 83, 505, 95 ], "spans": [ { "bbox": [ 106, 83, 252, 95 ], "score": 1.0, "content": "methods typically use node features", "type": "text" }, { "bbox": [ 252, 83, 262, 93 ], "score": 0.3, "content": "\\mathbf { X }", "type": "inline_equation" }, { "bbox": [ 263, 83, 292, 95 ], "score": 1.0, "content": "of size", "type": "text" }, { "bbox": [ 292, 83, 321, 93 ], "score": 0.91, "content": "n \\times D", "type": "inline_equation" }, { "bbox": [ 321, 83, 348, 95 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 349, 83, 382, 94 ], "score": 0.93, "content": "n = | \\mathcal { V } |", "type": "inline_equation" }, { "bbox": [ 383, 83, 505, 95 ], "score": 1.0, "content": "and employ a neural network,", "type": "text" } ], "index": 0 }, { "bbox": [ 104, 90, 507, 109 ], "spans": [ { "bbox": [ 104, 90, 418, 109 ], "score": 1.0, "content": "usually a graph convolutional network (GCN) (Kipf & Welling, 2017), for the", "type": "text" }, { "bbox": [ 418, 93, 464, 104 ], "score": 0.91, "content": "\\mathbf { R } ^ { D } \\to \\mathbf { R } ^ { d }", "type": "inline_equation" }, { "bbox": [ 465, 90, 507, 109 ], "score": 1.0, "content": "mapping.", "type": "text" } ], "index": 1 }, { "bbox": [ 105, 102, 506, 118 ], "spans": [ { "bbox": [ 105, 102, 506, 118 ], "score": 1.0, "content": "GCL methods also pull connected nodes together, sometimes explicitly through their loss function,", "type": "text" } ], "index": 2 }, { "bbox": [ 105, 114, 505, 129 ], "spans": [ { "bbox": [ 105, 114, 505, 129 ], "score": 1.0, "content": "but also implicitly through the GCN architecture (Trivedi et al., 2022; Wang et al., 2023; Guo et al.,", "type": "text" } ], "index": 3 }, { "bbox": [ 105, 124, 136, 140 ], "spans": [ { "bbox": [ 105, 124, 136, 140 ], "score": 1.0, "content": "2023).", "type": "text" } ], "index": 4 } ], "index": 2 }, { "type": "text", "bbox": [ 106, 143, 505, 221 ], "lines": [ { "bbox": [ 105, 142, 506, 156 ], "spans": [ { "bbox": [ 105, 142, 506, 156 ], "score": 1.0, "content": "Recent work (Kruiger et al., 2017; Zhu et al., 2020a; Zhong et al., 2023; Bohm et al., 2022) pointed ¨", "type": "text" } ], "index": 5 }, { "bbox": [ 105, 154, 505, 167 ], "spans": [ { "bbox": [ 105, 154, 476, 167 ], "score": 1.0, "content": "out deep connections between graph layout and neighbor embedding algorithms such as", "type": "text" }, { "bbox": [ 477, 155, 482, 164 ], "score": 0.72, "content": "t", "type": "inline_equation" }, { "bbox": [ 482, 154, 505, 167 ], "score": 1.0, "content": "-SNE", "type": "text" } ], "index": 6 }, { "bbox": [ 106, 165, 504, 178 ], "spans": [ { "bbox": [ 106, 165, 504, 178 ], "score": 1.0, "content": "(Van der Maaten & Hinton, 2008) or UMAP (McInnes et al., 2018), which are based on neigh-", "type": "text" } ], "index": 7 }, { "bbox": [ 106, 177, 504, 188 ], "spans": [ { "bbox": [ 106, 177, 504, 188 ], "score": 1.0, "content": "borhood preservation. In parallel, another line of work explored connections between neighbor", "type": "text" } ], "index": 8 }, { "bbox": [ 105, 187, 505, 200 ], "spans": [ { "bbox": [ 105, 187, 505, 200 ], "score": 1.0, "content": "embeddings and contrastive learning (Damrich et al., 2022; Bohm et al., 2023; Hu et al., 2023). ¨", "type": "text" } ], "index": 9 }, { "bbox": [ 105, 198, 505, 211 ], "spans": [ { "bbox": [ 105, 198, 505, 211 ], "score": 1.0, "content": "This raises the question to what extent neighbor embedding and contrastive neighbor embedding", "type": "text" } ], "index": 10 }, { "bbox": [ 106, 208, 403, 223 ], "spans": [ { "bbox": [ 106, 208, 403, 223 ], "score": 1.0, "content": "algorithms (see Section 3) can be useful for graph representation learning.", "type": "text" } ], "index": 11 } ], "index": 8 }, { "type": "text", "bbox": [ 107, 226, 505, 292 ], "lines": [ { "bbox": [ 105, 226, 505, 239 ], "spans": [ { "bbox": [ 105, 226, 477, 239 ], "score": 1.0, "content": "In this work, we answer this question. We introduce a novel graph layout algorithm, graph", "type": "text" }, { "bbox": [ 477, 227, 482, 236 ], "score": 0.58, "content": "t", "type": "inline_equation" }, { "bbox": [ 482, 226, 505, 239 ], "score": 1.0, "content": "-SNE", "type": "text" } ], "index": 12 }, { "bbox": [ 105, 237, 505, 249 ], "spans": [ { "bbox": [ 105, 237, 505, 249 ], "score": 1.0, "content": "(Figure 1), and show that it strongly outperforms existing methods. We also introduce a novel,", "type": "text" } ], "index": 13 }, { "bbox": [ 105, 247, 506, 261 ], "spans": [ { "bbox": [ 105, 247, 506, 261 ], "score": 1.0, "content": "augmentation-free, GCL algorithm, graph CNE (Figure 1), based on the framework for contrastive", "type": "text" } ], "index": 14 }, { "bbox": [ 106, 259, 505, 271 ], "spans": [ { "bbox": [ 106, 259, 505, 271 ], "score": 1.0, "content": "neighbor embeddings, and show that it reaches competitive GCL performance without using GCNs.", "type": "text" } ], "index": 15 }, { "bbox": [ 105, 268, 506, 284 ], "spans": [ { "bbox": [ 105, 268, 506, 284 ], "score": 1.0, "content": "Conceptually, we present a single coherent framework for node-level graph representation learning,", "type": "text" } ], "index": 16 }, { "bbox": [ 106, 281, 440, 294 ], "spans": [ { "bbox": [ 106, 281, 440, 294 ], "score": 1.0, "content": "tying together graph layouts, graph contrastive learning, and neighbor embeddings.", "type": "text" } ], "index": 17 } ], "index": 14.5 }, { "type": "title", "bbox": [ 108, 307, 211, 320 ], "lines": [ { "bbox": [ 104, 306, 213, 323 ], "spans": [ { "bbox": [ 104, 306, 213, 323 ], "score": 1.0, "content": "2 RELATED WORK", "type": "text" } ], "index": 18 } ], "index": 18 }, { "type": "text", "bbox": [ 107, 332, 505, 421 ], "lines": [ { "bbox": [ 106, 333, 505, 345 ], "spans": [ { "bbox": [ 106, 333, 505, 345 ], "score": 1.0, "content": "Graph layouts Graph layout algorithms have traditionally been based on spring models, where", "type": "text" } ], "index": 19 }, { "bbox": [ 105, 343, 505, 356 ], "spans": [ { "bbox": [ 105, 343, 332, 356 ], "score": 1.0, "content": "every connected pair of nodes feels an attractive force", "type": "text" }, { "bbox": [ 333, 344, 345, 355 ], "score": 0.9, "content": "F _ { a }", "type": "inline_equation" }, { "bbox": [ 345, 343, 505, 356 ], "score": 1.0, "content": "and all pairs of nodes feel a repulsive", "type": "text" } ], "index": 20 }, { "bbox": [ 104, 353, 504, 370 ], "spans": [ { "bbox": [ 104, 353, 129, 370 ], "score": 1.0, "content": "force", "type": "text" }, { "bbox": [ 129, 355, 141, 366 ], "score": 0.86, "content": "F _ { r }", "type": "inline_equation" }, { "bbox": [ 142, 353, 409, 370 ], "score": 1.0, "content": "(force-directed graph layouts). Many algorithms can be written as", "type": "text" }, { "bbox": [ 410, 355, 447, 368 ], "score": 0.92, "content": "F _ { a } = d _ { i j } ^ { a }", "type": "inline_equation" }, { "bbox": [ 448, 353, 466, 370 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 466, 355, 504, 368 ], "score": 0.92, "content": "F _ { r } = d _ { i j } ^ { r }", "type": "inline_equation" } ], "index": 21 }, { "bbox": [ 105, 365, 505, 379 ], "spans": [ { "bbox": [ 105, 365, 198, 379 ], "score": 1.0, "content": "(Noack, 2007), where", "type": "text" }, { "bbox": [ 198, 366, 212, 378 ], "score": 0.9, "content": "d _ { i j }", "type": "inline_equation" }, { "bbox": [ 212, 365, 505, 379 ], "score": 1.0, "content": "is the embedding distance between nodes. For example, Fruchterman–", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 377, 505, 389 ], "spans": [ { "bbox": [ 105, 377, 296, 389 ], "score": 1.0, "content": "Reingold algorithm, also known as FDP, uses", "type": "text" }, { "bbox": [ 297, 377, 363, 388 ], "score": 0.92, "content": "a = 2 , r = - 1", "type": "inline_equation" }, { "bbox": [ 363, 377, 505, 389 ], "score": 1.0, "content": "(Fruchterman & Reingold, 1991);", "type": "text" } ], "index": 23 }, { "bbox": [ 105, 387, 506, 401 ], "spans": [ { "bbox": [ 105, 387, 177, 401 ], "score": 1.0, "content": "ForceAtlas2 uses", "type": "text" }, { "bbox": [ 178, 388, 238, 399 ], "score": 0.92, "content": "a = 1 , r = - 1", "type": "inline_equation" }, { "bbox": [ 238, 387, 380, 401 ], "score": 1.0, "content": "(Jacomy et al., 2014); LinLog uses", "type": "text" }, { "bbox": [ 381, 388, 441, 399 ], "score": 0.93, "content": "a = 0 , r = - 1", "type": "inline_equation" }, { "bbox": [ 441, 387, 506, 401 ], "score": 1.0, "content": "(Noack, 2007).", "type": "text" } ], "index": 24 }, { "bbox": [ 105, 398, 506, 411 ], "spans": [ { "bbox": [ 105, 398, 506, 411 ], "score": 1.0, "content": "Efficient implementations can be based on Barnes–Hut approximation of the repulsive forces, as in", "type": "text" } ], "index": 25 }, { "bbox": [ 106, 410, 490, 422 ], "spans": [ { "bbox": [ 106, 410, 490, 422 ], "score": 1.0, "content": "SFDP (Hu, 2005). Relationship to neighbour embeddings was discussed by Bohm et al. (2022). ¨", "type": "text" } ], "index": 26 } ], "index": 22.5 }, { "type": "text", "bbox": [ 107, 432, 505, 532 ], "lines": [ { "bbox": [ 106, 433, 505, 446 ], "spans": [ { "bbox": [ 106, 433, 225, 446 ], "score": 1.0, "content": "Graph layouts inspired by", "type": "text" }, { "bbox": [ 226, 434, 231, 443 ], "score": 0.67, "content": "t", "type": "inline_equation" }, { "bbox": [ 231, 433, 505, 446 ], "score": 1.0, "content": "-SNE Several recent graph layout algorithms have been inspired", "type": "text" } ], "index": 27 }, { "bbox": [ 105, 443, 506, 457 ], "spans": [ { "bbox": [ 105, 443, 450, 457 ], "score": 1.0, "content": "by neighbor embeddings. tsNET (Kruiger et al., 2017) applied modified version of", "type": "text" }, { "bbox": [ 450, 444, 455, 454 ], "score": 0.68, "content": "t", "type": "inline_equation" }, { "bbox": [ 455, 443, 506, 457 ], "score": 1.0, "content": "-SNE to the", "type": "text" } ], "index": 28 }, { "bbox": [ 104, 454, 506, 468 ], "spans": [ { "bbox": [ 104, 454, 506, 468 ], "score": 1.0, "content": "pairwise shortest path distances between all nodes. DRGraph (Zhu et al., 2020a) made tsNET faster", "type": "text" } ], "index": 29 }, { "bbox": [ 105, 465, 504, 479 ], "spans": [ { "bbox": [ 105, 465, 309, 479 ], "score": 1.0, "content": "by using negative sampling (Mikolov et al., 2013).", "type": "text" }, { "bbox": [ 309, 466, 315, 476 ], "score": 0.7, "content": "t", "type": "inline_equation" }, { "bbox": [ 315, 465, 491, 479 ], "score": 1.0, "content": "-FDP (Zhong et al., 2023) suggested custom", "type": "text" }, { "bbox": [ 491, 466, 504, 477 ], "score": 0.88, "content": "F _ { a }", "type": "inline_equation" } ], "index": 30 }, { "bbox": [ 105, 477, 505, 490 ], "spans": [ { "bbox": [ 105, 477, 123, 490 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 123, 477, 135, 488 ], "score": 0.88, "content": "F _ { r }", "type": "inline_equation" }, { "bbox": [ 135, 477, 209, 490 ], "score": 1.0, "content": "forces inspired by", "type": "text" }, { "bbox": [ 209, 478, 214, 487 ], "score": 0.69, "content": "t", "type": "inline_equation" }, { "bbox": [ 214, 477, 505, 490 ], "score": 1.0, "content": "-SNE and adopted interpolation-based approximation of Linderman et al.", "type": "text" } ], "index": 31 }, { "bbox": [ 105, 486, 505, 502 ], "spans": [ { "bbox": [ 105, 486, 284, 502 ], "score": 1.0, "content": "(2019). Below we will show that our graph", "type": "text" }, { "bbox": [ 284, 488, 289, 497 ], "score": 0.66, "content": "t", "type": "inline_equation" }, { "bbox": [ 289, 486, 443, 502 ], "score": 1.0, "content": "-SNE outperforms both DRGraph and", "type": "text" }, { "bbox": [ 443, 488, 448, 498 ], "score": 0.72, "content": "t", "type": "inline_equation" }, { "bbox": [ 448, 486, 505, 502 ], "score": 1.0, "content": "-FDP. Finally,", "type": "text" } ], "index": 32 }, { "bbox": [ 106, 499, 505, 511 ], "spans": [ { "bbox": [ 106, 499, 346, 511 ], "score": 1.0, "content": "Leow et al. (2019) also suggested an algorithm called ‘graph", "type": "text" }, { "bbox": [ 346, 500, 351, 509 ], "score": 0.57, "content": "t", "type": "inline_equation" }, { "bbox": [ 351, 499, 505, 511 ], "score": 1.0, "content": "-SNE’, that used a graph convolutional", "type": "text" } ], "index": 33 }, { "bbox": [ 104, 508, 506, 524 ], "spans": [ { "bbox": [ 104, 508, 477, 524 ], "score": 1.0, "content": "network (Kipf & Welling, 2017) to build a parametric mapping optimizing a combination of", "type": "text" }, { "bbox": [ 477, 510, 482, 520 ], "score": 0.73, "content": "t", "type": "inline_equation" }, { "bbox": [ 482, 508, 506, 524 ], "score": 1.0, "content": "-SNE", "type": "text" } ], "index": 34 }, { "bbox": [ 105, 520, 505, 533 ], "spans": [ { "bbox": [ 105, 520, 474, 533 ], "score": 1.0, "content": "losses on node features and on shortest graph distances; it has almost no relation to our graph", "type": "text" }, { "bbox": [ 474, 522, 479, 530 ], "score": 0.65, "content": "t { \\cdot }", "type": "inline_equation" }, { "bbox": [ 480, 520, 505, 533 ], "score": 1.0, "content": "-SNE.", "type": "text" } ], "index": 35 } ], "index": 31 }, { "type": "text", "bbox": [ 107, 543, 505, 676 ], "lines": [ { "bbox": [ 105, 543, 505, 557 ], "spans": [ { "bbox": [ 105, 543, 505, 557 ], "score": 1.0, "content": "Node-level graph contrastive learning The basic principle behind contrastive learning is to learn", "type": "text" } ], "index": 36 }, { "bbox": [ 105, 555, 505, 568 ], "spans": [ { "bbox": [ 105, 555, 505, 568 ], "score": 1.0, "content": "data representation by contrasting pairs of observations that are similar to each other (positive pairs)", "type": "text" } ], "index": 37 }, { "bbox": [ 105, 565, 505, 579 ], "spans": [ { "bbox": [ 105, 565, 505, 579 ], "score": 1.0, "content": "with those that are dissimilar to each other (negative pairs). In computer vision, positive pairs are", "type": "text" } ], "index": 38 }, { "bbox": [ 104, 576, 506, 591 ], "spans": [ { "bbox": [ 104, 576, 506, 591 ], "score": 1.0, "content": "generated via data augmentation, e.g. in SimCLR (Chen et al., 2020). Graph contrastive learning", "type": "text" } ], "index": 39 }, { "bbox": [ 105, 588, 505, 600 ], "spans": [ { "bbox": [ 105, 588, 505, 600 ], "score": 1.0, "content": "(GCL) requires node features (as input to the network) and can be graph-level or node-level, de-", "type": "text" } ], "index": 40 }, { "bbox": [ 105, 599, 505, 612 ], "spans": [ { "bbox": [ 105, 599, 505, 612 ], "score": 1.0, "content": "pending on whether representations are obtained for a set of graphs or for the set of nodes of a single", "type": "text" } ], "index": 41 }, { "bbox": [ 104, 609, 506, 623 ], "spans": [ { "bbox": [ 104, 609, 506, 623 ], "score": 1.0, "content": "graph. Graph-level GCL is based on graph augmentations, such as node dropping or edge pertur-", "type": "text" } ], "index": 42 }, { "bbox": [ 105, 620, 506, 633 ], "spans": [ { "bbox": [ 105, 620, 506, 633 ], "score": 1.0, "content": "bation, e.g. in GraphCL (You et al., 2020). Prominent examples of node-level GCL algorithms that", "type": "text" } ], "index": 43 }, { "bbox": [ 105, 632, 505, 644 ], "spans": [ { "bbox": [ 105, 632, 505, 644 ], "score": 1.0, "content": "are also based on graph augmentations include GRACE (Zhu et al., 2020b), GCA (Zhu et al., 2021),", "type": "text" } ], "index": 44 }, { "bbox": [ 105, 641, 505, 655 ], "spans": [ { "bbox": [ 105, 641, 505, 655 ], "score": 1.0, "content": "MVGRL (Hassani & Khasahmadi, 2020), DGI (Velickovic et al., 2019), BGRL (Thakoor et al.,", "type": "text" } ], "index": 45 }, { "bbox": [ 105, 652, 506, 667 ], "spans": [ { "bbox": [ 105, 652, 506, 667 ], "score": 1.0, "content": "2021), CCA-SSG (Zhang et al., 2021), etc. All of them use graph convolutional networks (GCN) to", "type": "text" } ], "index": 46 }, { "bbox": [ 105, 665, 211, 677 ], "spans": [ { "bbox": [ 105, 665, 211, 677 ], "score": 1.0, "content": "create graph embeddings.", "type": "text" } ], "index": 47 } ], "index": 41.5 }, { "type": "text", "bbox": [ 108, 687, 504, 732 ], "lines": [ { "bbox": [ 106, 687, 505, 700 ], "spans": [ { "bbox": [ 106, 687, 505, 700 ], "score": 1.0, "content": "Augmentation-free node-level GCL A general problem with domain-agnostic graph augmenta-", "type": "text" } ], "index": 48 }, { "bbox": [ 105, 699, 506, 712 ], "spans": [ { "bbox": [ 105, 699, 506, 712 ], "score": 1.0, "content": "tions is that they can have unpredictable effects on graph semantics (Trivedi et al., 2022), as even", "type": "text" } ], "index": 49 }, { "bbox": [ 105, 710, 505, 722 ], "spans": [ { "bbox": [ 105, 710, 505, 722 ], "score": 1.0, "content": "minor augmentations can potentially result in a semantically different graph. This motivated devel-", "type": "text" } ], "index": 50 }, { "bbox": [ 105, 721, 506, 733 ], "spans": [ { "bbox": [ 105, 721, 506, 733 ], "score": 1.0, "content": "opment of augmentation-free GCL methods. Here positive pairs are pairs of nodes that are located", "type": "text" } ], "index": 51 } ], "index": 49.5 } ], "page_idx": 1, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 107, 27, 308, 37 ], "lines": [ { "bbox": [ 107, 26, 308, 38 ], "spans": [ { "bbox": [ 107, 26, 308, 38 ], "score": 1.0, "content": "Under review as a conference paper at ICLR 2024", "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, 137 ], "lines": [ { "bbox": [ 106, 83, 505, 95 ], "spans": [ { "bbox": [ 106, 83, 252, 95 ], "score": 1.0, "content": "methods typically use node features", "type": "text" }, { "bbox": [ 252, 83, 262, 93 ], "score": 0.3, "content": "\\mathbf { X }", "type": "inline_equation" }, { "bbox": [ 263, 83, 292, 95 ], "score": 1.0, "content": "of size", "type": "text" }, { "bbox": [ 292, 83, 321, 93 ], "score": 0.91, "content": "n \\times D", "type": "inline_equation" }, { "bbox": [ 321, 83, 348, 95 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 349, 83, 382, 94 ], "score": 0.93, "content": "n = | \\mathcal { V } |", "type": "inline_equation" }, { "bbox": [ 383, 83, 505, 95 ], "score": 1.0, "content": "and employ a neural network,", "type": "text" } ], "index": 0 }, { "bbox": [ 104, 90, 507, 109 ], "spans": [ { "bbox": [ 104, 90, 418, 109 ], "score": 1.0, "content": "usually a graph convolutional network (GCN) (Kipf & Welling, 2017), for the", "type": "text" }, { "bbox": [ 418, 93, 464, 104 ], "score": 0.91, "content": "\\mathbf { R } ^ { D } \\to \\mathbf { R } ^ { d }", "type": "inline_equation" }, { "bbox": [ 465, 90, 507, 109 ], "score": 1.0, "content": "mapping.", "type": "text" } ], "index": 1 }, { "bbox": [ 105, 102, 506, 118 ], "spans": [ { "bbox": [ 105, 102, 506, 118 ], "score": 1.0, "content": "GCL methods also pull connected nodes together, sometimes explicitly through their loss function,", "type": "text" } ], "index": 2 }, { "bbox": [ 105, 114, 505, 129 ], "spans": [ { "bbox": [ 105, 114, 505, 129 ], "score": 1.0, "content": "but also implicitly through the GCN architecture (Trivedi et al., 2022; Wang et al., 2023; Guo et al.,", "type": "text" } ], "index": 3 }, { "bbox": [ 105, 124, 136, 140 ], "spans": [ { "bbox": [ 105, 124, 136, 140 ], "score": 1.0, "content": "2023).", "type": "text" } ], "index": 4 } ], "index": 2, "bbox_fs": [ 104, 83, 507, 140 ] }, { "type": "text", "bbox": [ 106, 143, 505, 221 ], "lines": [ { "bbox": [ 105, 142, 506, 156 ], "spans": [ { "bbox": [ 105, 142, 506, 156 ], "score": 1.0, "content": "Recent work (Kruiger et al., 2017; Zhu et al., 2020a; Zhong et al., 2023; Bohm et al., 2022) pointed ¨", "type": "text" } ], "index": 5 }, { "bbox": [ 105, 154, 505, 167 ], "spans": [ { "bbox": [ 105, 154, 476, 167 ], "score": 1.0, "content": "out deep connections between graph layout and neighbor embedding algorithms such as", "type": "text" }, { "bbox": [ 477, 155, 482, 164 ], "score": 0.72, "content": "t", "type": "inline_equation" }, { "bbox": [ 482, 154, 505, 167 ], "score": 1.0, "content": "-SNE", "type": "text" } ], "index": 6 }, { "bbox": [ 106, 165, 504, 178 ], "spans": [ { "bbox": [ 106, 165, 504, 178 ], "score": 1.0, "content": "(Van der Maaten & Hinton, 2008) or UMAP (McInnes et al., 2018), which are based on neigh-", "type": "text" } ], "index": 7 }, { "bbox": [ 106, 177, 504, 188 ], "spans": [ { "bbox": [ 106, 177, 504, 188 ], "score": 1.0, "content": "borhood preservation. In parallel, another line of work explored connections between neighbor", "type": "text" } ], "index": 8 }, { "bbox": [ 105, 187, 505, 200 ], "spans": [ { "bbox": [ 105, 187, 505, 200 ], "score": 1.0, "content": "embeddings and contrastive learning (Damrich et al., 2022; Bohm et al., 2023; Hu et al., 2023). ¨", "type": "text" } ], "index": 9 }, { "bbox": [ 105, 198, 505, 211 ], "spans": [ { "bbox": [ 105, 198, 505, 211 ], "score": 1.0, "content": "This raises the question to what extent neighbor embedding and contrastive neighbor embedding", "type": "text" } ], "index": 10 }, { "bbox": [ 106, 208, 403, 223 ], "spans": [ { "bbox": [ 106, 208, 403, 223 ], "score": 1.0, "content": "algorithms (see Section 3) can be useful for graph representation learning.", "type": "text" } ], "index": 11 } ], "index": 8, "bbox_fs": [ 105, 142, 506, 223 ] }, { "type": "text", "bbox": [ 107, 226, 505, 292 ], "lines": [ { "bbox": [ 105, 226, 505, 239 ], "spans": [ { "bbox": [ 105, 226, 477, 239 ], "score": 1.0, "content": "In this work, we answer this question. We introduce a novel graph layout algorithm, graph", "type": "text" }, { "bbox": [ 477, 227, 482, 236 ], "score": 0.58, "content": "t", "type": "inline_equation" }, { "bbox": [ 482, 226, 505, 239 ], "score": 1.0, "content": "-SNE", "type": "text" } ], "index": 12 }, { "bbox": [ 105, 237, 505, 249 ], "spans": [ { "bbox": [ 105, 237, 505, 249 ], "score": 1.0, "content": "(Figure 1), and show that it strongly outperforms existing methods. We also introduce a novel,", "type": "text" } ], "index": 13 }, { "bbox": [ 105, 247, 506, 261 ], "spans": [ { "bbox": [ 105, 247, 506, 261 ], "score": 1.0, "content": "augmentation-free, GCL algorithm, graph CNE (Figure 1), based on the framework for contrastive", "type": "text" } ], "index": 14 }, { "bbox": [ 106, 259, 505, 271 ], "spans": [ { "bbox": [ 106, 259, 505, 271 ], "score": 1.0, "content": "neighbor embeddings, and show that it reaches competitive GCL performance without using GCNs.", "type": "text" } ], "index": 15 }, { "bbox": [ 105, 268, 506, 284 ], "spans": [ { "bbox": [ 105, 268, 506, 284 ], "score": 1.0, "content": "Conceptually, we present a single coherent framework for node-level graph representation learning,", "type": "text" } ], "index": 16 }, { "bbox": [ 106, 281, 440, 294 ], "spans": [ { "bbox": [ 106, 281, 440, 294 ], "score": 1.0, "content": "tying together graph layouts, graph contrastive learning, and neighbor embeddings.", "type": "text" } ], "index": 17 } ], "index": 14.5, "bbox_fs": [ 105, 226, 506, 294 ] }, { "type": "title", "bbox": [ 108, 307, 211, 320 ], "lines": [ { "bbox": [ 104, 306, 213, 323 ], "spans": [ { "bbox": [ 104, 306, 213, 323 ], "score": 1.0, "content": "2 RELATED WORK", "type": "text" } ], "index": 18 } ], "index": 18 }, { "type": "text", "bbox": [ 107, 332, 505, 421 ], "lines": [ { "bbox": [ 106, 333, 505, 345 ], "spans": [ { "bbox": [ 106, 333, 505, 345 ], "score": 1.0, "content": "Graph layouts Graph layout algorithms have traditionally been based on spring models, where", "type": "text" } ], "index": 19 }, { "bbox": [ 105, 343, 505, 356 ], "spans": [ { "bbox": [ 105, 343, 332, 356 ], "score": 1.0, "content": "every connected pair of nodes feels an attractive force", "type": "text" }, { "bbox": [ 333, 344, 345, 355 ], "score": 0.9, "content": "F _ { a }", "type": "inline_equation" }, { "bbox": [ 345, 343, 505, 356 ], "score": 1.0, "content": "and all pairs of nodes feel a repulsive", "type": "text" } ], "index": 20 }, { "bbox": [ 104, 353, 504, 370 ], "spans": [ { "bbox": [ 104, 353, 129, 370 ], "score": 1.0, "content": "force", "type": "text" }, { "bbox": [ 129, 355, 141, 366 ], "score": 0.86, "content": "F _ { r }", "type": "inline_equation" }, { "bbox": [ 142, 353, 409, 370 ], "score": 1.0, "content": "(force-directed graph layouts). Many algorithms can be written as", "type": "text" }, { "bbox": [ 410, 355, 447, 368 ], "score": 0.92, "content": "F _ { a } = d _ { i j } ^ { a }", "type": "inline_equation" }, { "bbox": [ 448, 353, 466, 370 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 466, 355, 504, 368 ], "score": 0.92, "content": "F _ { r } = d _ { i j } ^ { r }", "type": "inline_equation" } ], "index": 21 }, { "bbox": [ 105, 365, 505, 379 ], "spans": [ { "bbox": [ 105, 365, 198, 379 ], "score": 1.0, "content": "(Noack, 2007), where", "type": "text" }, { "bbox": [ 198, 366, 212, 378 ], "score": 0.9, "content": "d _ { i j }", "type": "inline_equation" }, { "bbox": [ 212, 365, 505, 379 ], "score": 1.0, "content": "is the embedding distance between nodes. For example, Fruchterman–", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 377, 505, 389 ], "spans": [ { "bbox": [ 105, 377, 296, 389 ], "score": 1.0, "content": "Reingold algorithm, also known as FDP, uses", "type": "text" }, { "bbox": [ 297, 377, 363, 388 ], "score": 0.92, "content": "a = 2 , r = - 1", "type": "inline_equation" }, { "bbox": [ 363, 377, 505, 389 ], "score": 1.0, "content": "(Fruchterman & Reingold, 1991);", "type": "text" } ], "index": 23 }, { "bbox": [ 105, 387, 506, 401 ], "spans": [ { "bbox": [ 105, 387, 177, 401 ], "score": 1.0, "content": "ForceAtlas2 uses", "type": "text" }, { "bbox": [ 178, 388, 238, 399 ], "score": 0.92, "content": "a = 1 , r = - 1", "type": "inline_equation" }, { "bbox": [ 238, 387, 380, 401 ], "score": 1.0, "content": "(Jacomy et al., 2014); LinLog uses", "type": "text" }, { "bbox": [ 381, 388, 441, 399 ], "score": 0.93, "content": "a = 0 , r = - 1", "type": "inline_equation" }, { "bbox": [ 441, 387, 506, 401 ], "score": 1.0, "content": "(Noack, 2007).", "type": "text" } ], "index": 24 }, { "bbox": [ 105, 398, 506, 411 ], "spans": [ { "bbox": [ 105, 398, 506, 411 ], "score": 1.0, "content": "Efficient implementations can be based on Barnes–Hut approximation of the repulsive forces, as in", "type": "text" } ], "index": 25 }, { "bbox": [ 106, 410, 490, 422 ], "spans": [ { "bbox": [ 106, 410, 490, 422 ], "score": 1.0, "content": "SFDP (Hu, 2005). Relationship to neighbour embeddings was discussed by Bohm et al. (2022). ¨", "type": "text" } ], "index": 26 } ], "index": 22.5, "bbox_fs": [ 104, 333, 506, 422 ] }, { "type": "text", "bbox": [ 107, 432, 505, 532 ], "lines": [ { "bbox": [ 106, 433, 505, 446 ], "spans": [ { "bbox": [ 106, 433, 225, 446 ], "score": 1.0, "content": "Graph layouts inspired by", "type": "text" }, { "bbox": [ 226, 434, 231, 443 ], "score": 0.67, "content": "t", "type": "inline_equation" }, { "bbox": [ 231, 433, 505, 446 ], "score": 1.0, "content": "-SNE Several recent graph layout algorithms have been inspired", "type": "text" } ], "index": 27 }, { "bbox": [ 105, 443, 506, 457 ], "spans": [ { "bbox": [ 105, 443, 450, 457 ], "score": 1.0, "content": "by neighbor embeddings. tsNET (Kruiger et al., 2017) applied modified version of", "type": "text" }, { "bbox": [ 450, 444, 455, 454 ], "score": 0.68, "content": "t", "type": "inline_equation" }, { "bbox": [ 455, 443, 506, 457 ], "score": 1.0, "content": "-SNE to the", "type": "text" } ], "index": 28 }, { "bbox": [ 104, 454, 506, 468 ], "spans": [ { "bbox": [ 104, 454, 506, 468 ], "score": 1.0, "content": "pairwise shortest path distances between all nodes. DRGraph (Zhu et al., 2020a) made tsNET faster", "type": "text" } ], "index": 29 }, { "bbox": [ 105, 465, 504, 479 ], "spans": [ { "bbox": [ 105, 465, 309, 479 ], "score": 1.0, "content": "by using negative sampling (Mikolov et al., 2013).", "type": "text" }, { "bbox": [ 309, 466, 315, 476 ], "score": 0.7, "content": "t", "type": "inline_equation" }, { "bbox": [ 315, 465, 491, 479 ], "score": 1.0, "content": "-FDP (Zhong et al., 2023) suggested custom", "type": "text" }, { "bbox": [ 491, 466, 504, 477 ], "score": 0.88, "content": "F _ { a }", "type": "inline_equation" } ], "index": 30 }, { "bbox": [ 105, 477, 505, 490 ], "spans": [ { "bbox": [ 105, 477, 123, 490 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 123, 477, 135, 488 ], "score": 0.88, "content": "F _ { r }", "type": "inline_equation" }, { "bbox": [ 135, 477, 209, 490 ], "score": 1.0, "content": "forces inspired by", "type": "text" }, { "bbox": [ 209, 478, 214, 487 ], "score": 0.69, "content": "t", "type": "inline_equation" }, { "bbox": [ 214, 477, 505, 490 ], "score": 1.0, "content": "-SNE and adopted interpolation-based approximation of Linderman et al.", "type": "text" } ], "index": 31 }, { "bbox": [ 105, 486, 505, 502 ], "spans": [ { "bbox": [ 105, 486, 284, 502 ], "score": 1.0, "content": "(2019). Below we will show that our graph", "type": "text" }, { "bbox": [ 284, 488, 289, 497 ], "score": 0.66, "content": "t", "type": "inline_equation" }, { "bbox": [ 289, 486, 443, 502 ], "score": 1.0, "content": "-SNE outperforms both DRGraph and", "type": "text" }, { "bbox": [ 443, 488, 448, 498 ], "score": 0.72, "content": "t", "type": "inline_equation" }, { "bbox": [ 448, 486, 505, 502 ], "score": 1.0, "content": "-FDP. Finally,", "type": "text" } ], "index": 32 }, { "bbox": [ 106, 499, 505, 511 ], "spans": [ { "bbox": [ 106, 499, 346, 511 ], "score": 1.0, "content": "Leow et al. (2019) also suggested an algorithm called ‘graph", "type": "text" }, { "bbox": [ 346, 500, 351, 509 ], "score": 0.57, "content": "t", "type": "inline_equation" }, { "bbox": [ 351, 499, 505, 511 ], "score": 1.0, "content": "-SNE’, that used a graph convolutional", "type": "text" } ], "index": 33 }, { "bbox": [ 104, 508, 506, 524 ], "spans": [ { "bbox": [ 104, 508, 477, 524 ], "score": 1.0, "content": "network (Kipf & Welling, 2017) to build a parametric mapping optimizing a combination of", "type": "text" }, { "bbox": [ 477, 510, 482, 520 ], "score": 0.73, "content": "t", "type": "inline_equation" }, { "bbox": [ 482, 508, 506, 524 ], "score": 1.0, "content": "-SNE", "type": "text" } ], "index": 34 }, { "bbox": [ 105, 520, 505, 533 ], "spans": [ { "bbox": [ 105, 520, 474, 533 ], "score": 1.0, "content": "losses on node features and on shortest graph distances; it has almost no relation to our graph", "type": "text" }, { "bbox": [ 474, 522, 479, 530 ], "score": 0.65, "content": "t { \\cdot }", "type": "inline_equation" }, { "bbox": [ 480, 520, 505, 533 ], "score": 1.0, "content": "-SNE.", "type": "text" } ], "index": 35 } ], "index": 31, "bbox_fs": [ 104, 433, 506, 533 ] }, { "type": "text", "bbox": [ 107, 543, 505, 676 ], "lines": [ { "bbox": [ 105, 543, 505, 557 ], "spans": [ { "bbox": [ 105, 543, 505, 557 ], "score": 1.0, "content": "Node-level graph contrastive learning The basic principle behind contrastive learning is to learn", "type": "text" } ], "index": 36 }, { "bbox": [ 105, 555, 505, 568 ], "spans": [ { "bbox": [ 105, 555, 505, 568 ], "score": 1.0, "content": "data representation by contrasting pairs of observations that are similar to each other (positive pairs)", "type": "text" } ], "index": 37 }, { "bbox": [ 105, 565, 505, 579 ], "spans": [ { "bbox": [ 105, 565, 505, 579 ], "score": 1.0, "content": "with those that are dissimilar to each other (negative pairs). In computer vision, positive pairs are", "type": "text" } ], "index": 38 }, { "bbox": [ 104, 576, 506, 591 ], "spans": [ { "bbox": [ 104, 576, 506, 591 ], "score": 1.0, "content": "generated via data augmentation, e.g. in SimCLR (Chen et al., 2020). Graph contrastive learning", "type": "text" } ], "index": 39 }, { "bbox": [ 105, 588, 505, 600 ], "spans": [ { "bbox": [ 105, 588, 505, 600 ], "score": 1.0, "content": "(GCL) requires node features (as input to the network) and can be graph-level or node-level, de-", "type": "text" } ], "index": 40 }, { "bbox": [ 105, 599, 505, 612 ], "spans": [ { "bbox": [ 105, 599, 505, 612 ], "score": 1.0, "content": "pending on whether representations are obtained for a set of graphs or for the set of nodes of a single", "type": "text" } ], "index": 41 }, { "bbox": [ 104, 609, 506, 623 ], "spans": [ { "bbox": [ 104, 609, 506, 623 ], "score": 1.0, "content": "graph. Graph-level GCL is based on graph augmentations, such as node dropping or edge pertur-", "type": "text" } ], "index": 42 }, { "bbox": [ 105, 620, 506, 633 ], "spans": [ { "bbox": [ 105, 620, 506, 633 ], "score": 1.0, "content": "bation, e.g. in GraphCL (You et al., 2020). Prominent examples of node-level GCL algorithms that", "type": "text" } ], "index": 43 }, { "bbox": [ 105, 632, 505, 644 ], "spans": [ { "bbox": [ 105, 632, 505, 644 ], "score": 1.0, "content": "are also based on graph augmentations include GRACE (Zhu et al., 2020b), GCA (Zhu et al., 2021),", "type": "text" } ], "index": 44 }, { "bbox": [ 105, 641, 505, 655 ], "spans": [ { "bbox": [ 105, 641, 505, 655 ], "score": 1.0, "content": "MVGRL (Hassani & Khasahmadi, 2020), DGI (Velickovic et al., 2019), BGRL (Thakoor et al.,", "type": "text" } ], "index": 45 }, { "bbox": [ 105, 652, 506, 667 ], "spans": [ { "bbox": [ 105, 652, 506, 667 ], "score": 1.0, "content": "2021), CCA-SSG (Zhang et al., 2021), etc. All of them use graph convolutional networks (GCN) to", "type": "text" } ], "index": 46 }, { "bbox": [ 105, 665, 211, 677 ], "spans": [ { "bbox": [ 105, 665, 211, 677 ], "score": 1.0, "content": "create graph embeddings.", "type": "text" } ], "index": 47 } ], "index": 41.5, "bbox_fs": [ 104, 543, 506, 677 ] }, { "type": "text", "bbox": [ 108, 687, 504, 732 ], "lines": [ { "bbox": [ 106, 687, 505, 700 ], "spans": [ { "bbox": [ 106, 687, 505, 700 ], "score": 1.0, "content": "Augmentation-free node-level GCL A general problem with domain-agnostic graph augmenta-", "type": "text" } ], "index": 48 }, { "bbox": [ 105, 699, 506, 712 ], "spans": [ { "bbox": [ 105, 699, 506, 712 ], "score": 1.0, "content": "tions is that they can have unpredictable effects on graph semantics (Trivedi et al., 2022), as even", "type": "text" } ], "index": 49 }, { "bbox": [ 105, 710, 505, 722 ], "spans": [ { "bbox": [ 105, 710, 505, 722 ], "score": 1.0, "content": "minor augmentations can potentially result in a semantically different graph. This motivated devel-", "type": "text" } ], "index": 50 }, { "bbox": [ 105, 721, 506, 733 ], "spans": [ { "bbox": [ 105, 721, 506, 733 ], "score": 1.0, "content": "opment of augmentation-free GCL methods. Here positive pairs are pairs of nodes that are located", "type": "text" } ], "index": 51 }, { "bbox": [ 106, 82, 505, 94 ], "spans": [ { "bbox": [ 106, 82, 505, 94 ], "score": 1.0, "content": "close to each other in terms of graph distance. AFGRL (Lee et al., 2022) and AF-GCL (Li et al.,", "type": "text", "cross_page": true } ], "index": 0 }, { "bbox": [ 106, 93, 504, 105 ], "spans": [ { "bbox": [ 106, 93, 437, 105 ], "score": 1.0, "content": "2023) treat nodes with small shortest path distance as candidate positives, and use", "type": "text", "cross_page": true }, { "bbox": [ 438, 94, 445, 104 ], "score": 0.76, "content": "k", "type": "inline_equation", "cross_page": true }, { "bbox": [ 445, 93, 504, 105 ], "score": 1.0, "content": "nearest neigh-", "type": "text", "cross_page": true } ], "index": 1 }, { "bbox": [ 105, 104, 505, 117 ], "spans": [ { "bbox": [ 105, 104, 505, 117 ], "score": 1.0, "content": "bors in GCN-based node representations to select actual positives. Local-GCL (Zhang et al., 2022)", "type": "text", "cross_page": true } ], "index": 2 }, { "bbox": [ 105, 115, 505, 128 ], "spans": [ { "bbox": [ 105, 115, 505, 128 ], "score": 1.0, "content": "uses all first-order graph neighbors as positives, and employs random Fourier features to approxi-", "type": "text", "cross_page": true } ], "index": 3 }, { "bbox": [ 105, 126, 471, 139 ], "spans": [ { "bbox": [ 105, 126, 128, 139 ], "score": 1.0, "content": "mate", "type": "text", "cross_page": true }, { "bbox": [ 129, 126, 156, 138 ], "score": 0.92, "content": "\\mathcal { O } ( n ^ { 2 } )", "type": "inline_equation", "cross_page": true }, { "bbox": [ 156, 126, 471, 139 ], "score": 1.0, "content": "repulsive forces. All of these methods are also based on the GCN architecture.", "type": "text", "cross_page": true } ], "index": 4 } ], "index": 49.5, "bbox_fs": [ 105, 687, 506, 733 ] } ] }, { "preproc_blocks": [ { "type": "text", "bbox": [ 107, 82, 505, 138 ], "lines": [ { "bbox": [ 106, 82, 505, 94 ], "spans": [ { "bbox": [ 106, 82, 505, 94 ], "score": 1.0, "content": "close to each other in terms of graph distance. AFGRL (Lee et al., 2022) and AF-GCL (Li et al.,", "type": "text" } ], "index": 0 }, { "bbox": [ 106, 93, 504, 105 ], "spans": [ { "bbox": [ 106, 93, 437, 105 ], "score": 1.0, "content": "2023) treat nodes with small shortest path distance as candidate positives, and use", "type": "text" }, { "bbox": [ 438, 94, 445, 104 ], "score": 0.76, "content": "k", "type": "inline_equation" }, { "bbox": [ 445, 93, 504, 105 ], "score": 1.0, "content": "nearest neigh-", "type": "text" } ], "index": 1 }, { "bbox": [ 105, 104, 505, 117 ], "spans": [ { "bbox": [ 105, 104, 505, 117 ], "score": 1.0, "content": "bors in GCN-based node representations to select actual positives. Local-GCL (Zhang et al., 2022)", "type": "text" } ], "index": 2 }, { "bbox": [ 105, 115, 505, 128 ], "spans": [ { "bbox": [ 105, 115, 505, 128 ], "score": 1.0, "content": "uses all first-order graph neighbors as positives, and employs random Fourier features to approxi-", "type": "text" } ], "index": 3 }, { "bbox": [ 105, 126, 471, 139 ], "spans": [ { "bbox": [ 105, 126, 128, 139 ], "score": 1.0, "content": "mate", "type": "text" }, { "bbox": [ 129, 126, 156, 138 ], "score": 0.92, "content": "\\mathcal { O } ( n ^ { 2 } )", "type": "inline_equation" }, { "bbox": [ 156, 126, 471, 139 ], "score": 1.0, "content": "repulsive forces. All of these methods are also based on the GCN architecture.", "type": "text" } ], "index": 4 } ], "index": 2 }, { "type": "title", "bbox": [ 108, 154, 200, 167 ], "lines": [ { "bbox": [ 104, 152, 202, 170 ], "spans": [ { "bbox": [ 104, 152, 202, 170 ], "score": 1.0, "content": "3 BACKGROUND", "type": "text" } ], "index": 5 } ], "index": 5 }, { "type": "title", "bbox": [ 108, 179, 238, 190 ], "lines": [ { "bbox": [ 106, 179, 239, 191 ], "spans": [ { "bbox": [ 106, 179, 239, 191 ], "score": 1.0, "content": "3.1 NEIGHBOR EMBEDDINGS", "type": "text" } ], "index": 6 } ], "index": 6 }, { "type": "text", "bbox": [ 106, 199, 505, 255 ], "lines": [ { "bbox": [ 106, 200, 504, 212 ], "spans": [ { "bbox": [ 106, 200, 371, 212 ], "score": 1.0, "content": "Neighbor embeddings are a family of methods aiming to embed", "type": "text" }, { "bbox": [ 372, 202, 379, 210 ], "score": 0.71, "content": "n", "type": "inline_equation" }, { "bbox": [ 380, 200, 504, 212 ], "score": 1.0, "content": "observations from some high-", "type": "text" } ], "index": 7 }, { "bbox": [ 104, 209, 505, 224 ], "spans": [ { "bbox": [ 104, 209, 212, 224 ], "score": 1.0, "content": "dimensional metric space", "type": "text" }, { "bbox": [ 212, 212, 222, 221 ], "score": 0.79, "content": "\\mathcal { X }", "type": "inline_equation" }, { "bbox": [ 222, 209, 488, 224 ], "score": 1.0, "content": "into a lower-dimensional (usually two-dimensional) vector space", "type": "text" }, { "bbox": [ 488, 210, 501, 221 ], "score": 0.87, "content": "\\bar { \\mathbb { R } } ^ { d }", "type": "inline_equation" }, { "bbox": [ 501, 209, 505, 224 ], "score": 1.0, "content": ",", "type": "text" } ], "index": 8 }, { "bbox": [ 104, 220, 506, 236 ], "spans": [ { "bbox": [ 104, 220, 506, 236 ], "score": 1.0, "content": "such that neighborhood relationships between observations are preserved in the embedding space.", "type": "text" } ], "index": 9 }, { "bbox": [ 106, 232, 503, 245 ], "spans": [ { "bbox": [ 106, 232, 147, 245 ], "score": 1.0, "content": "Typically,", "type": "text" }, { "bbox": [ 148, 233, 158, 243 ], "score": 0.81, "content": "\\mathcal { X }", "type": "inline_equation" }, { "bbox": [ 158, 232, 270, 245 ], "score": 1.0, "content": "is another real-valued space", "type": "text" }, { "bbox": [ 270, 233, 282, 243 ], "score": 0.87, "content": "\\mathbb { R } ^ { p }", "type": "inline_equation" }, { "bbox": [ 283, 232, 306, 245 ], "score": 1.0, "content": ", with", "type": "text" }, { "bbox": [ 306, 233, 333, 244 ], "score": 0.91, "content": "d \\ll p", "type": "inline_equation" }, { "bbox": [ 333, 232, 469, 245 ], "score": 1.0, "content": ". We denote the original vectors as", "type": "text" }, { "bbox": [ 469, 233, 503, 244 ], "score": 0.92, "content": "\\mathbf { x } _ { i } \\in \\mathbb { R } ^ { p }", "type": "inline_equation" } ], "index": 10 }, { "bbox": [ 105, 243, 265, 256 ], "spans": [ { "bbox": [ 105, 243, 226, 256 ], "score": 1.0, "content": "and the embedding vectors as", "type": "text" }, { "bbox": [ 227, 243, 261, 255 ], "score": 0.92, "content": "\\mathbf { y } _ { i } \\in \\bar { \\mathbb { R } } ^ { d }", "type": "inline_equation" }, { "bbox": [ 261, 243, 265, 256 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 11 } ], "index": 9 }, { "type": "text", "bbox": [ 107, 260, 504, 305 ], "lines": [ { "bbox": [ 105, 259, 506, 273 ], "spans": [ { "bbox": [ 105, 259, 327, 273 ], "score": 1.0, "content": "One of the most popular neighbor embedding methods,", "type": "text" }, { "bbox": [ 327, 262, 333, 271 ], "score": 0.76, "content": "t { \\cdot }", "type": "inline_equation" }, { "bbox": [ 333, 259, 506, 273 ], "score": 1.0, "content": "-distributed stochastic neighbor embedding", "type": "text" } ], "index": 12 }, { "bbox": [ 105, 270, 506, 285 ], "spans": [ { "bbox": [ 105, 270, 110, 285 ], "score": 1.0, "content": "(", "type": "text" }, { "bbox": [ 110, 272, 114, 282 ], "score": 0.58, "content": "t", "type": "inline_equation" }, { "bbox": [ 115, 270, 210, 285 ], "score": 1.0, "content": "-SNE; Van der Maaten", "type": "text" }, { "bbox": [ 211, 272, 221, 282 ], "score": 0.5, "content": "\\&", "type": "inline_equation" }, { "bbox": [ 221, 270, 506, 285 ], "score": 1.0, "content": "Hinton, 2008) is an extension of the stochastic neighbor embedding", "type": "text" } ], "index": 13 }, { "bbox": [ 106, 283, 506, 295 ], "spans": [ { "bbox": [ 106, 283, 344, 295 ], "score": 1.0, "content": "(SNE) originally suggested by Hinton & Roweis (2002).", "type": "text" }, { "bbox": [ 344, 283, 349, 293 ], "score": 0.66, "content": "t", "type": "inline_equation" }, { "bbox": [ 349, 283, 506, 295 ], "score": 1.0, "content": "-SNE minimizes the Kullback-Leibler", "type": "text" } ], "index": 14 }, { "bbox": [ 105, 292, 446, 308 ], "spans": [ { "bbox": [ 105, 292, 397, 308 ], "score": 1.0, "content": "divergence between the high-dimensional and low-dimensional affinities", "type": "text" }, { "bbox": [ 397, 295, 411, 306 ], "score": 0.87, "content": "p _ { i j }", "type": "inline_equation" }, { "bbox": [ 411, 292, 429, 308 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 429, 295, 441, 306 ], "score": 0.88, "content": "q _ { i j }", "type": "inline_equation" }, { "bbox": [ 442, 292, 446, 308 ], "score": 1.0, "content": ":", "type": "text" } ], "index": 15 } ], "index": 13.5 }, { "type": "interline_equation", "bbox": [ 233, 311, 378, 339 ], "lines": [ { "bbox": [ 233, 311, 378, 339 ], "spans": [ { "bbox": [ 233, 311, 378, 339 ], "score": 0.95, "content": "\\mathcal { L } = \\mathrm { { K L } } ( \\mathbf { P } \\parallel \\mathbf { Q } ) = \\sum _ { i j } p _ { i j } \\log \\frac { p _ { i j } } { q _ { i j } } .", "type": "interline_equation", "image_path": "84f384e8029d2b4f62221afe7a55436728d67f8a5b0edba25f989a97d956da3b.jpg" } ] } ], "index": 16.5, "virtual_lines": [ { "bbox": [ 233, 311, 378, 325.0 ], "spans": [], "index": 16 }, { "bbox": [ 233, 325.0, 378, 339.0 ], "spans": [], "index": 17 } ] }, { "type": "text", "bbox": [ 106, 344, 505, 378 ], "lines": [ { "bbox": [ 105, 344, 505, 357 ], "spans": [ { "bbox": [ 105, 344, 505, 357 ], "score": 1.0, "content": "Both affinity matrices are defined to be symmetric, positive, and to sum to 1. In the original algo-", "type": "text" } ], "index": 18 }, { "bbox": [ 106, 356, 505, 368 ], "spans": [ { "bbox": [ 106, 356, 132, 368 ], "score": 1.0, "content": "rithm,", "type": "text" }, { "bbox": [ 133, 356, 142, 366 ], "score": 0.62, "content": "\\mathbf { P }", "type": "inline_equation" }, { "bbox": [ 142, 356, 505, 368 ], "score": 1.0, "content": "was computed using adaptive Gaussian kernels, but almost the same results can be obtained", "type": "text" } ], "index": 19 }, { "bbox": [ 105, 367, 500, 379 ], "spans": [ { "bbox": [ 105, 367, 288, 379 ], "score": 1.0, "content": "simply by normalizing and symmetrizing the", "type": "text" }, { "bbox": [ 288, 367, 310, 377 ], "score": 0.74, "content": "k \\mathbf { N N }", "type": "inline_equation" }, { "bbox": [ 310, 367, 500, 379 ], "score": 1.0, "content": "graph adjacency matrix A (Bohm et al., 2022): ¨", "type": "text" } ], "index": 20 } ], "index": 19 }, { "type": "interline_equation", "bbox": [ 262, 383, 349, 410 ], "lines": [ { "bbox": [ 262, 383, 349, 410 ], "spans": [ { "bbox": [ 262, 383, 349, 410 ], "score": 0.96, "content": "\\mathbf { P } = { \\frac { \\mathbf { A } / k + \\mathbf { A } ^ { \\top } / k } { 2 n } } .", "type": "interline_equation", "image_path": "f689fa8500db6ec61882ece7b6b97aba88c5d29597b200782c6426f1484c3416.jpg" } ] } ], "index": 21.5, "virtual_lines": [ { "bbox": [ 262, 383, 349, 396.5 ], "spans": [], "index": 21 }, { "bbox": [ 262, 396.5, 349, 410.0 ], "spans": [], "index": 22 } ] }, { "type": "text", "bbox": [ 107, 414, 504, 447 ], "lines": [ { "bbox": [ 105, 414, 504, 427 ], "spans": [ { "bbox": [ 105, 414, 193, 427 ], "score": 1.0, "content": "Here A has element", "type": "text" }, { "bbox": [ 194, 415, 229, 426 ], "score": 0.91, "content": "a _ { i j } = 1", "type": "inline_equation" }, { "bbox": [ 229, 414, 240, 427 ], "score": 1.0, "content": "if", "type": "text" }, { "bbox": [ 241, 415, 252, 426 ], "score": 0.86, "content": "\\mathbf { x } _ { j }", "type": "inline_equation" }, { "bbox": [ 253, 414, 293, 427 ], "score": 1.0, "content": "is within", "type": "text" }, { "bbox": [ 294, 415, 300, 424 ], "score": 0.83, "content": "k", "type": "inline_equation" }, { "bbox": [ 301, 414, 389, 427 ], "score": 1.0, "content": "nearest neighbors of", "type": "text" }, { "bbox": [ 389, 416, 399, 425 ], "score": 0.85, "content": "\\mathbf { x } _ { i }", "type": "inline_equation" }, { "bbox": [ 399, 414, 497, 427 ], "score": 1.0, "content": ". Reasonable values of", "type": "text" }, { "bbox": [ 497, 415, 504, 424 ], "score": 0.83, "content": "k", "type": "inline_equation" } ], "index": 23 }, { "bbox": [ 105, 425, 506, 438 ], "spans": [ { "bbox": [ 105, 425, 361, 438 ], "score": 1.0, "content": "typically lie between 10 and 100. Low-dimensional affinities", "type": "text" }, { "bbox": [ 361, 425, 372, 437 ], "score": 0.8, "content": "\\mathbf { Q }", "type": "inline_equation" }, { "bbox": [ 372, 425, 433, 438 ], "score": 1.0, "content": "are defined in", "type": "text" }, { "bbox": [ 433, 426, 438, 435 ], "score": 0.7, "content": "t", "type": "inline_equation" }, { "bbox": [ 438, 425, 495, 438 ], "score": 1.0, "content": "-SNE using a", "type": "text" }, { "bbox": [ 496, 426, 501, 435 ], "score": 0.73, "content": "t", "type": "inline_equation" }, { "bbox": [ 501, 425, 506, 438 ], "score": 1.0, "content": "-", "type": "text" } ], "index": 24 }, { "bbox": [ 106, 436, 433, 448 ], "spans": [ { "bbox": [ 106, 436, 433, 448 ], "score": 1.0, "content": "distribution kernel with one degree of freedom, also known as the Cauchy kernel:", "type": "text" } ], "index": 25 } ], "index": 24 }, { "type": "interline_equation", "bbox": [ 236, 452, 375, 483 ], "lines": [ { "bbox": [ 236, 452, 375, 483 ], "spans": [ { "bbox": [ 236, 452, 375, 483 ], "score": 0.95, "content": "q _ { i j } = \\frac { ( 1 + \\| \\mathbf { y } _ { i } - \\mathbf { y } _ { j } \\| ^ { 2 } ) ^ { - 1 } } { \\sum _ { k \\neq l } ( 1 + \\| \\mathbf { y } _ { l } - \\mathbf { y } _ { k } \\| ^ { 2 } ) ^ { - 1 } } .", "type": "interline_equation", "image_path": "c46e8a49680d153032d10a818f4f19b1812d9912d8c69d0757d68bfeb6234671.jpg" } ] } ], "index": 26.5, "virtual_lines": [ { "bbox": [ 236, 452, 375, 467.5 ], "spans": [], "index": 26 }, { "bbox": [ 236, 467.5, 375, 483.0 ], "spans": [], "index": 27 } ] }, { "type": "text", "bbox": [ 106, 487, 504, 510 ], "lines": [ { "bbox": [ 106, 487, 504, 500 ], "spans": [ { "bbox": [ 106, 487, 504, 500 ], "score": 1.0, "content": "The original SNE algorithm used Gaussian kernel instead of Cauchy, which led to worse results", "type": "text" } ], "index": 28 }, { "bbox": [ 106, 498, 353, 510 ], "spans": [ { "bbox": [ 106, 498, 353, 510 ], "score": 1.0, "content": "when embedding high-dimensional data (Kobak et al., 2019).", "type": "text" } ], "index": 29 } ], "index": 28.5 }, { "type": "text", "bbox": [ 106, 514, 505, 582 ], "lines": [ { "bbox": [ 106, 515, 505, 528 ], "spans": [ { "bbox": [ 106, 515, 297, 528 ], "score": 1.0, "content": "Even though it is usually not presented like that,", "type": "text" }, { "bbox": [ 297, 516, 302, 525 ], "score": 0.71, "content": "t", "type": "inline_equation" }, { "bbox": [ 303, 515, 505, 528 ], "score": 1.0, "content": "-SNE can be thought of as a graph layout algorithm", "type": "text" } ], "index": 30 }, { "bbox": [ 106, 526, 505, 539 ], "spans": [ { "bbox": [ 106, 526, 120, 539 ], "score": 1.0, "content": "for", "type": "text" }, { "bbox": [ 121, 527, 142, 537 ], "score": 0.55, "content": "k \\mathbf { N N }", "type": "inline_equation" }, { "bbox": [ 143, 526, 505, 539 ], "score": 1.0, "content": "graphs, in particular after the reformulation in Equation 2. 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In practice,", "type": "text" }, { "bbox": [ 273, 549, 279, 558 ], "score": 0.68, "content": "t", "type": "inline_equation" }, { "bbox": [ 279, 548, 505, 561 ], "score": 1.0, "content": "-SNE optimization can be accelerated by an approxima-", "type": "text" } ], "index": 33 }, { "bbox": [ 105, 558, 506, 573 ], "spans": [ { "bbox": [ 105, 558, 506, 573 ], "score": 1.0, "content": "tion of the repulsive force field based on the Barnes–Hut algorithm (Van Der Maaten, 2014; Yang", "type": "text" } ], "index": 34 }, { "bbox": [ 106, 570, 335, 582 ], "spans": [ { "bbox": [ 106, 570, 335, 582 ], "score": 1.0, "content": "et al., 2013) or on interpolation (Linderman et al., 2019).", "type": "text" } ], "index": 35 } ], "index": 32.5 }, { "type": "title", "bbox": [ 109, 595, 302, 606 ], "lines": [ { "bbox": [ 106, 595, 302, 608 ], "spans": [ { "bbox": [ 106, 595, 302, 608 ], "score": 1.0, "content": "3.2 CONTRASTIVE NEIGHBOR EMBEDDINGS", "type": "text" } ], "index": 36 } ], "index": 36 }, { "type": "text", "bbox": [ 107, 615, 505, 704 ], "lines": [ { "bbox": [ 105, 615, 506, 629 ], "spans": [ { "bbox": [ 105, 615, 506, 629 ], "score": 1.0, "content": "The contrastive neighbor embedding (CNE) algorithm (Damrich et al., 2022) is a flexible framework", "type": "text" } ], "index": 37 }, { "bbox": [ 105, 627, 505, 639 ], "spans": [ { "bbox": [ 105, 627, 210, 639 ], "score": 1.0, "content": "that also operates on the", "type": "text" }, { "bbox": [ 210, 627, 232, 637 ], "score": 0.34, "content": "k \\mathbf { N N }", "type": "inline_equation" }, { "bbox": [ 233, 627, 505, 639 ], "score": 1.0, "content": "graph of the data, and optimizes the embedding in order to place", "type": "text" } ], "index": 38 }, { "bbox": [ 105, 637, 506, 650 ], "spans": [ { "bbox": [ 105, 637, 506, 650 ], "score": 1.0, "content": "connected nodes closer together than unconnected pairs of nodes. Damrich et al. (2022) considered", "type": "text" } ], "index": 39 }, { "bbox": [ 105, 648, 505, 661 ], "spans": [ { "bbox": [ 105, 648, 505, 661 ], "score": 1.0, "content": "three different loss functions: NCE (noise-contrastive estimation) (Gutmann & Hyvarinen, 2010), ¨", "type": "text" } ], "index": 40 }, { "bbox": [ 105, 659, 505, 673 ], "spans": [ { "bbox": [ 105, 659, 505, 673 ], "score": 1.0, "content": "InfoNCE (Jozefowicz et al., 2016; Oord et al., 2018), and negative sampling (Mikolov et al., 2013).", "type": "text" } ], "index": 41 }, { "bbox": [ 105, 670, 505, 684 ], "spans": [ { "bbox": [ 105, 670, 505, 684 ], "score": 1.0, "content": "These loss functions are called contrastive because they are based on contrasting edges and non-", "type": "text" } ], "index": 42 }, { "bbox": [ 105, 680, 506, 695 ], "spans": [ { "bbox": [ 105, 680, 506, 695 ], "score": 1.0, "content": "edges in the same mini-batch, and do not require a global normalization like in Equation 3. 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In the original algo-", "type": "text" } ], "index": 18 }, { "bbox": [ 106, 356, 505, 368 ], "spans": [ { "bbox": [ 106, 356, 132, 368 ], "score": 1.0, "content": "rithm,", "type": "text" }, { "bbox": [ 133, 356, 142, 366 ], "score": 0.62, "content": "\\mathbf { P }", "type": "inline_equation" }, { "bbox": [ 142, 356, 505, 368 ], "score": 1.0, "content": "was computed using adaptive Gaussian kernels, but almost the same results can be obtained", "type": "text" } ], "index": 19 }, { "bbox": [ 105, 367, 500, 379 ], "spans": [ { "bbox": [ 105, 367, 288, 379 ], "score": 1.0, "content": "simply by normalizing and symmetrizing the", "type": "text" }, { "bbox": [ 288, 367, 310, 377 ], "score": 0.74, "content": "k \\mathbf { N N }", "type": "inline_equation" }, { "bbox": [ 310, 367, 500, 379 ], "score": 1.0, "content": "graph adjacency matrix A (Bohm et al., 2022): ¨", "type": "text" } ], "index": 20 } ], "index": 19, "bbox_fs": [ 105, 344, 505, 379 ] }, { "type": "interline_equation", "bbox": [ 262, 383, 349, 410 ], "lines": [ { "bbox": [ 262, 383, 349, 410 ], "spans": [ { "bbox": [ 262, 383, 349, 410 ], "score": 0.96, "content": "\\mathbf { P } = { \\frac { \\mathbf { A } / k + \\mathbf { A } ^ { \\top } / k } { 2 n } } .", "type": "interline_equation", "image_path": "f689fa8500db6ec61882ece7b6b97aba88c5d29597b200782c6426f1484c3416.jpg" } ] } ], "index": 21.5, "virtual_lines": [ { "bbox": [ 262, 383, 349, 396.5 ], "spans": [], "index": 21 }, { "bbox": [ 262, 396.5, 349, 410.0 ], "spans": [], "index": 22 } ] }, { "type": "text", "bbox": [ 107, 414, 504, 447 ], "lines": [ { "bbox": [ 105, 414, 504, 427 ], "spans": [ { "bbox": [ 105, 414, 193, 427 ], "score": 1.0, "content": "Here A has element", "type": "text" }, { "bbox": [ 194, 415, 229, 426 ], "score": 0.91, "content": "a _ { i j } = 1", "type": "inline_equation" }, { "bbox": [ 229, 414, 240, 427 ], "score": 1.0, "content": "if", "type": "text" }, { "bbox": [ 241, 415, 252, 426 ], "score": 0.86, "content": "\\mathbf { x } _ { j }", "type": "inline_equation" }, { "bbox": [ 253, 414, 293, 427 ], "score": 1.0, "content": "is within", "type": "text" }, { "bbox": [ 294, 415, 300, 424 ], "score": 0.83, "content": "k", "type": "inline_equation" }, { "bbox": [ 301, 414, 389, 427 ], "score": 1.0, "content": "nearest neighbors of", "type": "text" }, { "bbox": [ 389, 416, 399, 425 ], "score": 0.85, "content": "\\mathbf { x } _ { i }", "type": "inline_equation" }, { "bbox": [ 399, 414, 497, 427 ], "score": 1.0, "content": ". Reasonable values of", "type": "text" }, { "bbox": [ 497, 415, 504, 424 ], "score": 0.83, "content": "k", "type": "inline_equation" } ], "index": 23 }, { "bbox": [ 105, 425, 506, 438 ], "spans": [ { "bbox": [ 105, 425, 361, 438 ], "score": 1.0, "content": "typically lie between 10 and 100. Low-dimensional affinities", "type": "text" }, { "bbox": [ 361, 425, 372, 437 ], "score": 0.8, "content": "\\mathbf { Q }", "type": "inline_equation" }, { "bbox": [ 372, 425, 433, 438 ], "score": 1.0, "content": "are defined in", "type": "text" }, { "bbox": [ 433, 426, 438, 435 ], "score": 0.7, "content": "t", "type": "inline_equation" }, { "bbox": [ 438, 425, 495, 438 ], "score": 1.0, "content": "-SNE using a", "type": "text" }, { "bbox": [ 496, 426, 501, 435 ], "score": 0.73, "content": "t", "type": "inline_equation" }, { "bbox": [ 501, 425, 506, 438 ], "score": 1.0, "content": "-", "type": "text" } ], "index": 24 }, { "bbox": [ 106, 436, 433, 448 ], "spans": [ { "bbox": [ 106, 436, 433, 448 ], "score": 1.0, "content": "distribution kernel with one degree of freedom, also known as the Cauchy kernel:", "type": "text" } ], "index": 25 } ], "index": 24, "bbox_fs": [ 105, 414, 506, 448 ] }, { "type": "interline_equation", "bbox": [ 236, 452, 375, 483 ], "lines": [ { "bbox": [ 236, 452, 375, 483 ], "spans": [ { "bbox": [ 236, 452, 375, 483 ], "score": 0.95, "content": "q _ { i j } = \\frac { ( 1 + \\| \\mathbf { y } _ { i } - \\mathbf { y } _ { j } \\| ^ { 2 } ) ^ { - 1 } } { \\sum _ { k \\neq l } ( 1 + \\| \\mathbf { y } _ { l } - \\mathbf { y } _ { k } \\| ^ { 2 } ) ^ { - 1 } } .", "type": "interline_equation", "image_path": "c46e8a49680d153032d10a818f4f19b1812d9912d8c69d0757d68bfeb6234671.jpg" } ] } ], "index": 26.5, "virtual_lines": [ { "bbox": [ 236, 452, 375, 467.5 ], "spans": [], "index": 26 }, { "bbox": [ 236, 467.5, 375, 483.0 ], "spans": [], "index": 27 } ] }, { "type": "text", "bbox": [ 106, 487, 504, 510 ], "lines": [ { "bbox": [ 106, 487, 504, 500 ], "spans": [ { "bbox": [ 106, 487, 504, 500 ], "score": 1.0, "content": "The original SNE algorithm used Gaussian kernel instead of Cauchy, which led to worse results", "type": "text" } ], "index": 28 }, { "bbox": [ 106, 498, 353, 510 ], "spans": [ { "bbox": [ 106, 498, 353, 510 ], "score": 1.0, "content": "when embedding high-dimensional data (Kobak et al., 2019).", "type": "text" } ], "index": 29 } ], "index": 28.5, "bbox_fs": [ 106, 487, 504, 510 ] }, { "type": "text", "bbox": [ 106, 514, 505, 582 ], "lines": [ { "bbox": [ 106, 515, 505, 528 ], "spans": [ { "bbox": [ 106, 515, 297, 528 ], "score": 1.0, "content": "Even though it is usually not presented like that,", "type": "text" }, { "bbox": [ 297, 516, 302, 525 ], "score": 0.71, "content": "t", "type": "inline_equation" }, { "bbox": [ 303, 515, 505, 528 ], "score": 1.0, "content": "-SNE can be thought of as a graph layout algorithm", "type": "text" } ], "index": 30 }, { "bbox": [ 106, 526, 505, 539 ], "spans": [ { "bbox": [ 106, 526, 120, 539 ], "score": 1.0, "content": "for", "type": "text" }, { "bbox": [ 121, 527, 142, 537 ], "score": 0.55, "content": "k \\mathbf { N N }", "type": "inline_equation" }, { "bbox": [ 143, 526, 505, 539 ], "score": 1.0, "content": "graphs, in particular after the reformulation in Equation 2. During optimization, neighbor-", "type": "text" } ], "index": 31 }, { "bbox": [ 105, 537, 505, 550 ], "spans": [ { "bbox": [ 105, 537, 505, 550 ], "score": 1.0, "content": "ing nodes (sharing an edge) feel attraction, whereas all nodes feel repulsion, arising through the", "type": "text" } ], "index": 32 }, { "bbox": [ 105, 548, 505, 561 ], "spans": [ { "bbox": [ 105, 548, 273, 561 ], "score": 1.0, "content": "normalization in Equation 3. In practice,", "type": "text" }, { "bbox": [ 273, 549, 279, 558 ], "score": 0.68, "content": "t", "type": "inline_equation" }, { "bbox": [ 279, 548, 505, 561 ], "score": 1.0, "content": "-SNE optimization can be accelerated by an approxima-", "type": "text" } ], "index": 33 }, { "bbox": [ 105, 558, 506, 573 ], "spans": [ { "bbox": [ 105, 558, 506, 573 ], "score": 1.0, "content": "tion of the repulsive force field based on the Barnes–Hut algorithm (Van Der Maaten, 2014; Yang", "type": "text" } ], "index": 34 }, { "bbox": [ 106, 570, 335, 582 ], "spans": [ { "bbox": [ 106, 570, 335, 582 ], "score": 1.0, "content": "et al., 2013) or on interpolation (Linderman et al., 2019).", "type": "text" } ], "index": 35 } ], "index": 32.5, "bbox_fs": [ 105, 515, 506, 582 ] }, { "type": "title", "bbox": [ 109, 595, 302, 606 ], "lines": [ { "bbox": [ 106, 595, 302, 608 ], "spans": [ { "bbox": [ 106, 595, 302, 608 ], "score": 1.0, "content": "3.2 CONTRASTIVE NEIGHBOR EMBEDDINGS", "type": "text" } ], "index": 36 } ], "index": 36 }, { "type": "text", "bbox": [ 107, 615, 505, 704 ], "lines": [ { "bbox": [ 105, 615, 506, 629 ], "spans": [ { "bbox": [ 105, 615, 506, 629 ], "score": 1.0, "content": "The contrastive neighbor embedding (CNE) algorithm (Damrich et al., 2022) is a flexible framework", "type": "text" } ], "index": 37 }, { "bbox": [ 105, 627, 505, 639 ], "spans": [ { "bbox": [ 105, 627, 210, 639 ], "score": 1.0, "content": "that also operates on the", "type": "text" }, { "bbox": [ 210, 627, 232, 637 ], "score": 0.34, "content": "k \\mathbf { N N }", "type": "inline_equation" }, { "bbox": [ 233, 627, 505, 639 ], "score": 1.0, "content": "graph of the data, and optimizes the embedding in order to place", "type": "text" } ], "index": 38 }, { "bbox": [ 105, 637, 506, 650 ], "spans": [ { "bbox": [ 105, 637, 506, 650 ], "score": 1.0, "content": "connected nodes closer together than unconnected pairs of nodes. Damrich et al. (2022) considered", "type": "text" } ], "index": 39 }, { "bbox": [ 105, 648, 505, 661 ], "spans": [ { "bbox": [ 105, 648, 505, 661 ], "score": 1.0, "content": "three different loss functions: NCE (noise-contrastive estimation) (Gutmann & Hyvarinen, 2010), ¨", "type": "text" } ], "index": 40 }, { "bbox": [ 105, 659, 505, 673 ], "spans": [ { "bbox": [ 105, 659, 505, 673 ], "score": 1.0, "content": "InfoNCE (Jozefowicz et al., 2016; Oord et al., 2018), and negative sampling (Mikolov et al., 2013).", "type": "text" } ], "index": 41 }, { "bbox": [ 105, 670, 505, 684 ], "spans": [ { "bbox": [ 105, 670, 505, 684 ], "score": 1.0, "content": "These loss functions are called contrastive because they are based on contrasting edges and non-", "type": "text" } ], "index": 42 }, { "bbox": [ 105, 680, 506, 695 ], "spans": [ { "bbox": [ 105, 680, 506, 695 ], "score": 1.0, "content": "edges in the same mini-batch, and do not require a global normalization like in Equation 3. Using", "type": "text" } ], "index": 43 }, { "bbox": [ 105, 692, 304, 705 ], "spans": [ { "bbox": [ 105, 692, 273, 705 ], "score": 1.0, "content": "NCE and InfoNCE in CNE approximates", "type": "text" }, { "bbox": [ 274, 694, 278, 703 ], "score": 0.67, "content": "t", "type": "inline_equation" }, { "bbox": [ 279, 692, 304, 705 ], "score": 1.0, "content": "-SNE.", "type": "text" } ], "index": 44 } ], "index": 40.5, "bbox_fs": [ 105, 615, 506, 705 ] }, { "type": "text", "bbox": [ 107, 709, 504, 732 ], "lines": [ { "bbox": [ 106, 709, 505, 722 ], "spans": [ { "bbox": [ 106, 709, 505, 722 ], "score": 1.0, "content": "Damrich et al. (2022) also considered parametric embeddings, where a neural network (usually a", "type": "text" } ], "index": 45 }, { "bbox": [ 105, 720, 505, 734 ], "spans": [ { "bbox": [ 105, 720, 381, 734 ], "score": 1.0, "content": "fully-connected network) is trained to produce embedding vectors", "type": "text" }, { "bbox": [ 381, 720, 433, 732 ], "score": 0.93, "content": "\\mathbf { y } _ { i } ~ = ~ f ( \\mathbf { x } _ { i } )", "type": "inline_equation" }, { "bbox": [ 433, 720, 505, 734 ], "score": 1.0, "content": "using one of the", "type": "text" } ], "index": 46 }, { "bbox": [ 106, 226, 505, 237 ], "spans": [ { "bbox": [ 106, 226, 505, 237 ], "score": 1.0, "content": "loss function listed above. This allows to embed new observations that have not been part of the", "type": "text", "cross_page": true } ], "index": 5 }, { "bbox": [ 106, 236, 505, 249 ], "spans": [ { "bbox": [ 106, 236, 388, 249 ], "score": 1.0, "content": "training process. In contrast, non-parametric embeddings optimize", "type": "text", "cross_page": true }, { "bbox": [ 389, 238, 400, 248 ], "score": 0.84, "content": "\\mathbf { y } _ { i }", "type": "inline_equation", "cross_page": true }, { "bbox": [ 401, 236, 505, 249 ], "score": 1.0, "content": "vectors directly, without", "type": "text", "cross_page": true } ], "index": 6 }, { "bbox": [ 105, 247, 506, 260 ], "spans": [ { "bbox": [ 105, 247, 124, 260 ], "score": 1.0, "content": "any", "type": "text", "cross_page": true }, { "bbox": [ 124, 248, 142, 259 ], "score": 0.9, "content": "f ( \\cdot )", "type": "inline_equation", "cross_page": true }, { "bbox": [ 142, 247, 506, 260 ], "score": 1.0, "content": "function. 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(2022) showed that Neg-", "type": "text", "cross_page": true }, { "bbox": [ 425, 259, 430, 268 ], "score": 0.48, "content": "\\cdot t", "type": "inline_equation", "cross_page": true }, { "bbox": [ 430, 257, 505, 271 ], "score": 1.0, "content": "-SNE is equivalent", "type": "text", "cross_page": true } ], "index": 8 }, { "bbox": [ 105, 269, 505, 282 ], "spans": [ { "bbox": [ 105, 269, 290, 282 ], "score": 1.0, "content": "to UMAP (McInnes et al., 2018), while NC-", "type": "text", "cross_page": true }, { "bbox": [ 290, 270, 295, 279 ], "score": 0.52, "content": "\\cdot t .", "type": "inline_equation", "cross_page": true }, { "bbox": [ 296, 269, 505, 282 ], "score": 1.0, "content": "-SNE was first suggested by Artemenkov & Panov", "type": "text", "cross_page": true } ], "index": 9 }, { "bbox": [ 106, 280, 178, 292 ], "spans": [ { "bbox": [ 106, 280, 178, 292 ], "score": 1.0, "content": "(2020) as NCVis.", "type": "text", "cross_page": true } ], "index": 10 } ], "index": 45.5, "bbox_fs": [ 105, 709, 505, 734 ] } ] }, { "preproc_blocks": [ { "type": "table", "bbox": [ 106, 113, 505, 203 ], "blocks": [ { "type": "table_caption", "bbox": [ 106, 89, 502, 112 ], "group_id": 0, "lines": [ { "bbox": [ 106, 88, 504, 102 ], "spans": [ { "bbox": [ 106, 88, 504, 102 ], "score": 1.0, "content": "Table 1: Benchmark datasets. Columns: number of nodes in the largest connected component,", "type": "text" } ], "index": 0 }, { "bbox": [ 105, 100, 489, 113 ], "spans": [ { "bbox": [ 105, 100, 489, 113 ], "score": 1.0, "content": "number of undirected edges, edges/nodes ratio, number of node classes, feature dimensionality.", "type": "text" } ], "index": 1 } ], "index": 0.5 }, { "type": "table_body", "bbox": [ 106, 113, 505, 203 ], "group_id": 0, "lines": [ { "bbox": [ 106, 113, 505, 203 ], "spans": [ { "bbox": [ 106, 113, 505, 203 ], "score": 0.98, "html": "
DatasetAbbr.NodesEdgesE/NClassesDim.
CiteseerGraphDatasetCSR212036791.763703
CoraGraphDatasetCOR248550692.071433
AmazonCoBuyPhotoDatasetAPH7487119 04315.98745
AmazonCoBuyComputerDatasetACO13381245 77818.410767
PubmedGraphDatasetPUB19717443242.23500
ogbn-arxivARX169 34311577996.840128
", "type": "table", "image_path": "32045b7de92c000aa9741d404727a1fbdd02da3b666bb8192afd128d5335788f.jpg" } ] } ], "index": 3, "virtual_lines": [ { "bbox": [ 106, 113, 505, 143.0 ], "spans": [], "index": 2 }, { "bbox": [ 106, 143.0, 505, 173.0 ], "spans": [], "index": 3 }, { "bbox": [ 106, 173.0, 505, 203.0 ], "spans": [], "index": 4 } ] } ], "index": 1.75 }, { "type": "text", "bbox": [ 106, 225, 505, 291 ], "lines": [ { "bbox": [ 106, 226, 505, 237 ], "spans": [ { "bbox": [ 106, 226, 505, 237 ], "score": 1.0, "content": "loss function listed above. This allows to embed new observations that have not been part of the", "type": "text" } ], "index": 5 }, { "bbox": [ 106, 236, 505, 249 ], "spans": [ { "bbox": [ 106, 236, 388, 249 ], "score": 1.0, "content": "training process. In contrast, non-parametric embeddings optimize", "type": "text" }, { "bbox": [ 389, 238, 400, 248 ], "score": 0.84, "content": "\\mathbf { y } _ { i }", "type": "inline_equation" }, { "bbox": [ 401, 236, 505, 249 ], "score": 1.0, "content": "vectors directly, without", "type": "text" } ], "index": 6 }, { "bbox": [ 105, 247, 506, 260 ], "spans": [ { "bbox": [ 105, 247, 124, 260 ], "score": 1.0, "content": "any", "type": "text" }, { "bbox": [ 124, 248, 142, 259 ], "score": 0.9, "content": "f ( \\cdot )", "type": "inline_equation" }, { "bbox": [ 142, 247, 506, 260 ], "score": 1.0, "content": "function. Together, this yields six combinations, called parametric/non-parametric NC-", "type": "text" } ], "index": 7 }, { "bbox": [ 106, 257, 505, 271 ], "spans": [ { "bbox": [ 106, 259, 111, 268 ], "score": 0.72, "content": "t", "type": "inline_equation" }, { "bbox": [ 112, 257, 171, 271 ], "score": 1.0, "content": "-SNE, InfoNC-", "type": "text" }, { "bbox": [ 171, 259, 176, 268 ], "score": 0.61, "content": "\\mathbf { \\nabla } \\cdot t", "type": "inline_equation" }, { "bbox": [ 176, 257, 237, 271 ], "score": 1.0, "content": "-SNE, and Neg-", "type": "text" }, { "bbox": [ 237, 259, 243, 268 ], "score": 0.42, "content": "\\cdot t", "type": "inline_equation" }, { "bbox": [ 243, 257, 424, 271 ], "score": 1.0, "content": "-SNE. Damrich et al. (2022) showed that Neg-", "type": "text" }, { "bbox": [ 425, 259, 430, 268 ], "score": 0.48, "content": "\\cdot t", "type": "inline_equation" }, { "bbox": [ 430, 257, 505, 271 ], "score": 1.0, "content": "-SNE is equivalent", "type": "text" } ], "index": 8 }, { "bbox": [ 105, 269, 505, 282 ], "spans": [ { "bbox": [ 105, 269, 290, 282 ], "score": 1.0, "content": "to UMAP (McInnes et al., 2018), while NC-", "type": "text" }, { "bbox": [ 290, 270, 295, 279 ], "score": 0.52, "content": "\\cdot t .", "type": "inline_equation" }, { "bbox": [ 296, 269, 505, 282 ], "score": 1.0, "content": "-SNE was first suggested by Artemenkov & Panov", "type": "text" } ], "index": 9 }, { "bbox": [ 106, 280, 178, 292 ], "spans": [ { "bbox": [ 106, 280, 178, 292 ], "score": 1.0, "content": "(2020) as NCVis.", "type": "text" } ], "index": 10 } ], "index": 7.5 }, { "type": "text", "bbox": [ 106, 297, 505, 319 ], "lines": [ { "bbox": [ 104, 295, 505, 310 ], "spans": [ { "bbox": [ 104, 295, 456, 310 ], "score": 1.0, "content": "In this work we will only use the InfoNCE loss function, defined for one graph edge", "type": "text" }, { "bbox": [ 457, 298, 466, 309 ], "score": 0.72, "content": "i j", "type": "inline_equation" }, { "bbox": [ 466, 295, 505, 310 ], "score": 1.0, "content": "(positive", "type": "text" } ], "index": 11 }, { "bbox": [ 103, 308, 141, 321 ], "spans": [ { "bbox": [ 103, 308, 141, 321 ], "score": 1.0, "content": "pair) as", "type": "text" } ], "index": 12 } ], "index": 11.5 }, { "type": "interline_equation", "bbox": [ 239, 320, 371, 345 ], "lines": [ { "bbox": [ 239, 320, 371, 345 ], "spans": [ { "bbox": [ 239, 320, 371, 345 ], "score": 0.95, "content": "\\ell ( i , j ) = - \\log \\frac { q _ { i j } } { q _ { i j } + \\sum _ { k = 1 } ^ { m } q _ { i k } } ,", "type": "interline_equation", "image_path": "e7a10df242066fb864f77c049c0e98325858e908fda9983626b622d3c3ce927a.jpg" } ] } ], "index": 13, "virtual_lines": [ { "bbox": [ 239, 320, 371, 345 ], "spans": [], "index": 13 } ] }, { "type": "text", "bbox": [ 106, 349, 505, 405 ], "lines": [ { "bbox": [ 105, 348, 505, 361 ], "spans": [ { "bbox": [ 105, 348, 273, 361 ], "score": 1.0, "content": "where the sum in the denominator is over", "type": "text" }, { "bbox": [ 273, 352, 284, 359 ], "score": 0.55, "content": "m", "type": "inline_equation" }, { "bbox": [ 284, 348, 343, 361 ], "score": 1.0, "content": "negative pairs", "type": "text" }, { "bbox": [ 344, 350, 354, 360 ], "score": 0.85, "content": "i k", "type": "inline_equation" }, { "bbox": [ 354, 348, 381, 361 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 382, 350, 389, 359 ], "score": 0.82, "content": "k", "type": "inline_equation" }, { "bbox": [ 389, 348, 505, 361 ], "score": 1.0, "content": "can be drawn from all nodes", "type": "text" } ], "index": 14 }, { "bbox": [ 105, 359, 506, 373 ], "spans": [ { "bbox": [ 105, 359, 249, 373 ], "score": 1.0, "content": "in the same mini-batch apart from", "type": "text" }, { "bbox": [ 249, 361, 254, 370 ], "score": 0.69, "content": "i", "type": "inline_equation" }, { "bbox": [ 254, 359, 273, 373 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 273, 361, 280, 372 ], "score": 0.79, "content": "j", "type": "inline_equation" }, { "bbox": [ 280, 359, 399, 373 ], "score": 1.0, "content": ". One mini-batch consists of", "type": "text" }, { "bbox": [ 399, 361, 405, 370 ], "score": 0.71, "content": "b", "type": "inline_equation" }, { "bbox": [ 406, 359, 506, 373 ], "score": 1.0, "content": "graph edges, and hence", "type": "text" } ], "index": 15 }, { "bbox": [ 104, 369, 506, 385 ], "spans": [ { "bbox": [ 104, 369, 142, 385 ], "score": 1.0, "content": "contains", "type": "text" }, { "bbox": [ 142, 371, 153, 381 ], "score": 0.55, "content": "2 b", "type": "inline_equation" }, { "bbox": [ 154, 369, 314, 385 ], "score": 1.0, "content": "nodes. Therefore, for a given batch size", "type": "text" }, { "bbox": [ 314, 372, 320, 381 ], "score": 0.75, "content": "b", "type": "inline_equation" }, { "bbox": [ 320, 369, 409, 385 ], "score": 1.0, "content": ", the maximal value of", "type": "text" }, { "bbox": [ 410, 372, 420, 381 ], "score": 0.78, "content": "m", "type": "inline_equation" }, { "bbox": [ 420, 369, 430, 385 ], "score": 1.0, "content": "is", "type": "text" }, { "bbox": [ 430, 371, 457, 382 ], "score": 0.85, "content": "2 b - 2", "type": "inline_equation" }, { "bbox": [ 457, 369, 506, 385 ], "score": 1.0, "content": ". The larger", "type": "text" } ], "index": 16 }, { "bbox": [ 105, 381, 505, 395 ], "spans": [ { "bbox": [ 105, 381, 120, 395 ], "score": 1.0, "content": "the", "type": "text" }, { "bbox": [ 121, 384, 131, 392 ], "score": 0.73, "content": "m", "type": "inline_equation" }, { "bbox": [ 131, 381, 209, 395 ], "score": 1.0, "content": ", the closer InfoNC-", "type": "text" }, { "bbox": [ 209, 383, 214, 392 ], "score": 0.58, "content": "t", "type": "inline_equation" }, { "bbox": [ 214, 381, 256, 395 ], "score": 1.0, "content": "-SNE is to", "type": "text" }, { "bbox": [ 256, 383, 261, 392 ], "score": 0.74, "content": "t", "type": "inline_equation" }, { "bbox": [ 262, 381, 395, 395 ], "score": 1.0, "content": "-SNE (Damrich et al., 2022). The", "type": "text" }, { "bbox": [ 396, 383, 408, 394 ], "score": 0.86, "content": "q _ { i j }", "type": "inline_equation" }, { "bbox": [ 408, 381, 505, 395 ], "score": 1.0, "content": "affinities do not need to", "type": "text" } ], "index": 17 }, { "bbox": [ 105, 392, 270, 406 ], "spans": [ { "bbox": [ 105, 392, 270, 406 ], "score": 1.0, "content": "be normalized and are defined simply as", "type": "text" } ], "index": 18 } ], "index": 16 }, { "type": "interline_equation", "bbox": [ 249, 410, 362, 426 ], "lines": [ { "bbox": [ 249, 410, 362, 426 ], "spans": [ { "bbox": [ 249, 410, 362, 426 ], "score": 0.92, "content": "q _ { i j } = ( 1 + \\| \\mathbf { y } _ { i } - \\mathbf { y } _ { j } \\| ^ { 2 } ) ^ { - 1 } .", "type": "interline_equation", "image_path": "542f3326e12a976e1b1d73e77d6ac81f195d71cc97e9aaf70bc2c7a823f0a235.jpg" } ] } ], "index": 19, "virtual_lines": [ { "bbox": [ 249, 410, 362, 426 ], "spans": [], "index": 19 } ] }, { "type": "text", "bbox": [ 107, 432, 505, 455 ], "lines": [ { "bbox": [ 105, 432, 505, 445 ], "spans": [ { "bbox": [ 105, 432, 314, 444 ], "score": 1.0, "content": "It is easy to see that InfoNCE loss will aim to make", "type": "text" }, { "bbox": [ 315, 434, 327, 445 ], "score": 0.88, "content": "q _ { i j }", "type": "inline_equation" }, { "bbox": [ 327, 432, 360, 444 ], "score": 1.0, "content": "large if", "type": "text" }, { "bbox": [ 360, 433, 369, 444 ], "score": 0.88, "content": "i j", "type": "inline_equation" }, { "bbox": [ 369, 432, 505, 444 ], "score": 1.0, "content": "is a positive pair and small if it is", "type": "text" } ], "index": 20 }, { "bbox": [ 105, 443, 168, 455 ], "spans": [ { "bbox": [ 105, 443, 168, 455 ], "score": 1.0, "content": "a negative one.", "type": "text" } ], "index": 21 } ], "index": 20.5 }, { "type": "text", "bbox": [ 107, 460, 505, 493 ], "lines": [ { "bbox": [ 106, 460, 505, 472 ], "spans": [ { "bbox": [ 106, 460, 327, 472 ], "score": 1.0, "content": "When using high-dimensional embedding space, e.g.", "type": "text" }, { "bbox": [ 327, 460, 366, 470 ], "score": 0.89, "content": "d = 1 2 8", "type": "inline_equation" }, { "bbox": [ 366, 460, 411, 472 ], "score": 1.0, "content": "instead of", "type": "text" }, { "bbox": [ 411, 460, 438, 470 ], "score": 0.9, "content": "d = 2", "type": "inline_equation" }, { "bbox": [ 439, 460, 505, 472 ], "score": 1.0, "content": ", it makes sense", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 471, 505, 484 ], "spans": [ { "bbox": [ 105, 471, 144, 484 ], "score": 1.0, "content": "to define", "type": "text" }, { "bbox": [ 144, 473, 156, 483 ], "score": 0.87, "content": "q _ { i j }", "type": "inline_equation" }, { "bbox": [ 157, 471, 505, 484 ], "score": 1.0, "content": "using the Gaussian kernel transformation of the cosine distance (Damrich et al., 2022;", "type": "text" } ], "index": 23 }, { "bbox": [ 106, 482, 186, 493 ], "spans": [ { "bbox": [ 106, 482, 186, 493 ], "score": 1.0, "content": "Bohm et al., 2023):¨", "type": "text" } ], "index": 24 } ], "index": 23 }, { "type": "interline_equation", "bbox": [ 141, 500, 469, 527 ], "lines": [ { "bbox": [ 141, 500, 469, 527 ], "spans": [ { "bbox": [ 141, 500, 469, 527 ], "score": 0.92, "content": "q _ { i j } = \\exp \\bigl ( \\mathbf { y } _ { i } ^ { \\mathsf { T } } \\mathbf { y } _ { j } / ( \\lVert \\mathbf { y } _ { i } \\rVert \\cdot \\lVert \\mathbf { y } _ { j } \\rVert ) / \\tau \\bigr ) = \\mathrm { c o n s t } \\cdot \\exp \\Big ( - \\Big \\lVert \\frac { \\mathbf { y } _ { i } } { \\lVert \\mathbf { y } _ { i } \\rVert } - \\frac { \\mathbf { y } _ { j } } { \\lVert \\mathbf { y } _ { j } \\rVert } \\Big \\rVert ^ { 2 } \\Big / ( 2 \\tau ) \\Big ) ,", "type": "interline_equation", "image_path": "f1aeccf40854bc160dcf59105db1a796b87b92147e9ccbcdc96975959727fbc8.jpg" } ] } ], "index": 26, "virtual_lines": [ { "bbox": [ 141, 500, 469, 509.0 ], "spans": [], "index": 25 }, { "bbox": [ 141, 509.0, 469, 518.0 ], "spans": [], "index": 26 }, { "bbox": [ 141, 518.0, 469, 527.0 ], "spans": [], "index": 27 } ] }, { "type": "text", "bbox": [ 106, 533, 505, 578 ], "lines": [ { "bbox": [ 106, 533, 505, 546 ], "spans": [ { "bbox": [ 106, 533, 133, 546 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 134, 536, 141, 543 ], "score": 0.77, "content": "\\tau", "type": "inline_equation" }, { "bbox": [ 141, 533, 295, 546 ], "score": 1.0, "content": "is called the temperature (by default,", "type": "text" }, { "bbox": [ 295, 533, 331, 544 ], "score": 0.87, "content": "\\tau = 0 . 5", "type": "inline_equation" }, { "bbox": [ 331, 533, 505, 546 ], "score": 1.0, "content": "). Together with Equation 5, this gives the", "type": "text" } ], "index": 28 }, { "bbox": [ 105, 543, 505, 557 ], "spans": [ { "bbox": [ 105, 543, 505, 557 ], "score": 1.0, "content": "same loss function as used in SimCLR (Chen et al., 2020), a popular contrastive learning algo-", "type": "text" } ], "index": 29 }, { "bbox": [ 106, 555, 506, 568 ], "spans": [ { "bbox": [ 106, 556, 363, 568 ], "score": 1.0, "content": "rithm in computer vision. The only difference is that instead of", "type": "text" }, { "bbox": [ 363, 555, 385, 566 ], "score": 0.78, "content": "k \\mathbf { N N }", "type": "inline_equation" }, { "bbox": [ 385, 556, 506, 568 ], "score": 1.0, "content": "edges, SimCLR uses pairs of", "type": "text" } ], "index": 30 }, { "bbox": [ 105, 567, 252, 579 ], "spans": [ { "bbox": [ 105, 567, 252, 579 ], "score": 1.0, "content": "augmented images as positive pairs.", "type": "text" } ], "index": 31 } ], "index": 29.5 }, { "type": "title", "bbox": [ 108, 595, 244, 608 ], "lines": [ { "bbox": [ 105, 594, 245, 610 ], "spans": [ { "bbox": [ 105, 594, 245, 610 ], "score": 1.0, "content": "4 EXPERIMENTAL SETUP", "type": "text" } ], "index": 32 } ], "index": 32 }, { "type": "text", "bbox": [ 106, 621, 505, 732 ], "lines": [ { "bbox": [ 105, 621, 505, 634 ], "spans": [ { "bbox": [ 105, 621, 505, 634 ], "score": 1.0, "content": "Datasets We used six publicly available graph datasets (Table 1). All datasets were retrieved from", "type": "text" } ], "index": 33 }, { "bbox": [ 105, 633, 505, 646 ], "spans": [ { "bbox": [ 105, 633, 505, 646 ], "score": 1.0, "content": "the Deep Graph Library (Wang et al., 2019), except ogbn-arxiv, which was retrieved from the", "type": "text" } ], "index": 34 }, { "bbox": [ 106, 643, 506, 656 ], "spans": [ { "bbox": [ 106, 643, 506, 656 ], "score": 1.0, "content": "Open Graph Benchmark (Hu et al., 2020). Each dataset was treated as an unweighted undirected", "type": "text" } ], "index": 35 }, { "bbox": [ 105, 654, 506, 668 ], "spans": [ { "bbox": [ 105, 654, 506, 668 ], "score": 1.0, "content": "graph, where each node has a class label and a feature vector (typically a word embedding vector", "type": "text" } ], "index": 36 }, { "bbox": [ 106, 666, 505, 678 ], "spans": [ { "bbox": [ 106, 666, 505, 678 ], "score": 1.0, "content": "of some descriptive text about the node, such as a product review). We restricted ourselves to", "type": "text" } ], "index": 37 }, { "bbox": [ 105, 677, 506, 690 ], "spans": [ { "bbox": [ 105, 677, 506, 690 ], "score": 1.0, "content": "graphs with labeled nodes in order to use classification accuracy as the performance metric. We also", "type": "text" } ], "index": 38 }, { "bbox": [ 105, 687, 505, 700 ], "spans": [ { "bbox": [ 105, 687, 505, 700 ], "score": 1.0, "content": "restricted ourselves to graphs with feature vectors in order to use both non-parametric and parametric", "type": "text" } ], "index": 39 }, { "bbox": [ 105, 699, 505, 712 ], "spans": [ { "bbox": [ 105, 699, 505, 712 ], "score": 1.0, "content": "embeddings. In all datasets we used only the largest connected component, and excluded all self-", "type": "text" } ], "index": 40 }, { "bbox": [ 105, 710, 506, 722 ], "spans": [ { "bbox": [ 105, 710, 506, 722 ], "score": 1.0, "content": "loops if present, using NetworkX (Hagberg et al., 2008) functions connected components", "type": "text" } ], "index": 41 }, { "bbox": [ 106, 720, 209, 733 ], "spans": [ { "bbox": [ 106, 720, 209, 733 ], "score": 1.0, "content": "and selfloop edges.", "type": "text" } ], "index": 42 } ], "index": 37.5 } ], "page_idx": 3, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 107, 27, 308, 37 ], "lines": [ { "bbox": [ 107, 26, 308, 38 ], "spans": [ { "bbox": [ 107, 26, 308, 38 ], "score": 1.0, "content": "Under review as a conference paper at ICLR 2024", "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": "table", "bbox": [ 106, 113, 505, 203 ], "blocks": [ { "type": "table_caption", "bbox": [ 106, 89, 502, 112 ], "group_id": 0, "lines": [ { "bbox": [ 106, 88, 504, 102 ], "spans": [ { "bbox": [ 106, 88, 504, 102 ], "score": 1.0, "content": "Table 1: Benchmark datasets. Columns: number of nodes in the largest connected component,", "type": "text" } ], "index": 0 }, { "bbox": [ 105, 100, 489, 113 ], "spans": [ { "bbox": [ 105, 100, 489, 113 ], "score": 1.0, "content": "number of undirected edges, edges/nodes ratio, number of node classes, feature dimensionality.", "type": "text" } ], "index": 1 } ], "index": 0.5 }, { "type": "table_body", "bbox": [ 106, 113, 505, 203 ], "group_id": 0, "lines": [ { "bbox": [ 106, 113, 505, 203 ], "spans": [ { "bbox": [ 106, 113, 505, 203 ], "score": 0.98, "html": "
DatasetAbbr.NodesEdgesE/NClassesDim.
CiteseerGraphDatasetCSR212036791.763703
CoraGraphDatasetCOR248550692.071433
AmazonCoBuyPhotoDatasetAPH7487119 04315.98745
AmazonCoBuyComputerDatasetACO13381245 77818.410767
PubmedGraphDatasetPUB19717443242.23500
ogbn-arxivARX169 34311577996.840128
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One mini-batch consists of", "type": "text" }, { "bbox": [ 399, 361, 405, 370 ], "score": 0.71, "content": "b", "type": "inline_equation" }, { "bbox": [ 406, 359, 506, 373 ], "score": 1.0, "content": "graph edges, and hence", "type": "text" } ], "index": 15 }, { "bbox": [ 104, 369, 506, 385 ], "spans": [ { "bbox": [ 104, 369, 142, 385 ], "score": 1.0, "content": "contains", "type": "text" }, { "bbox": [ 142, 371, 153, 381 ], "score": 0.55, "content": "2 b", "type": "inline_equation" }, { "bbox": [ 154, 369, 314, 385 ], "score": 1.0, "content": "nodes. 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The larger", "type": "text" } ], "index": 16 }, { "bbox": [ 105, 381, 505, 395 ], "spans": [ { "bbox": [ 105, 381, 120, 395 ], "score": 1.0, "content": "the", "type": "text" }, { "bbox": [ 121, 384, 131, 392 ], "score": 0.73, "content": "m", "type": "inline_equation" }, { "bbox": [ 131, 381, 209, 395 ], "score": 1.0, "content": ", the closer InfoNC-", "type": "text" }, { "bbox": [ 209, 383, 214, 392 ], "score": 0.58, "content": "t", "type": "inline_equation" }, { "bbox": [ 214, 381, 256, 395 ], "score": 1.0, "content": "-SNE is to", "type": "text" }, { "bbox": [ 256, 383, 261, 392 ], "score": 0.74, "content": "t", "type": "inline_equation" }, { "bbox": [ 262, 381, 395, 395 ], "score": 1.0, "content": "-SNE (Damrich et al., 2022). The", "type": "text" }, { "bbox": [ 396, 383, 408, 394 ], "score": 0.86, "content": "q _ { i j }", "type": "inline_equation" }, { "bbox": [ 408, 381, 505, 395 ], "score": 1.0, "content": "affinities do not need to", "type": "text" } ], "index": 17 }, { "bbox": [ 105, 392, 270, 406 ], "spans": [ { "bbox": [ 105, 392, 270, 406 ], "score": 1.0, "content": "be normalized and are defined simply as", "type": "text" } ], "index": 18 } ], "index": 16, "bbox_fs": [ 104, 348, 506, 406 ] }, { "type": "interline_equation", "bbox": [ 249, 410, 362, 426 ], "lines": [ { "bbox": [ 249, 410, 362, 426 ], "spans": [ { "bbox": [ 249, 410, 362, 426 ], "score": 0.92, "content": "q _ { i j } = ( 1 + \\| \\mathbf { y } _ { i } - \\mathbf { y } _ { j } \\| ^ { 2 } ) ^ { - 1 } .", "type": "interline_equation", "image_path": "542f3326e12a976e1b1d73e77d6ac81f195d71cc97e9aaf70bc2c7a823f0a235.jpg" } ] } ], "index": 19, "virtual_lines": [ { "bbox": [ 249, 410, 362, 426 ], "spans": [], "index": 19 } ] }, { "type": "text", "bbox": [ 107, 432, 505, 455 ], "lines": [ { "bbox": [ 105, 432, 505, 445 ], "spans": [ { "bbox": [ 105, 432, 314, 444 ], "score": 1.0, "content": "It is easy to see that InfoNCE loss will aim to make", "type": "text" }, { "bbox": [ 315, 434, 327, 445 ], "score": 0.88, "content": "q _ { i j }", "type": "inline_equation" }, { "bbox": [ 327, 432, 360, 444 ], "score": 1.0, "content": "large if", "type": "text" }, { "bbox": [ 360, 433, 369, 444 ], "score": 0.88, "content": "i j", "type": "inline_equation" }, { "bbox": [ 369, 432, 505, 444 ], "score": 1.0, "content": "is a positive pair and small if it is", "type": "text" } ], "index": 20 }, { "bbox": [ 105, 443, 168, 455 ], "spans": [ { "bbox": [ 105, 443, 168, 455 ], "score": 1.0, "content": "a negative one.", "type": "text" } ], "index": 21 } ], "index": 20.5, "bbox_fs": [ 105, 432, 505, 455 ] }, { "type": "text", "bbox": [ 107, 460, 505, 493 ], "lines": [ { "bbox": [ 106, 460, 505, 472 ], "spans": [ { "bbox": [ 106, 460, 327, 472 ], "score": 1.0, "content": "When using high-dimensional embedding space, e.g.", "type": "text" }, { "bbox": [ 327, 460, 366, 470 ], "score": 0.89, "content": "d = 1 2 8", "type": "inline_equation" }, { "bbox": [ 366, 460, 411, 472 ], "score": 1.0, "content": "instead of", "type": "text" }, { "bbox": [ 411, 460, 438, 470 ], "score": 0.9, "content": "d = 2", "type": "inline_equation" }, { "bbox": [ 439, 460, 505, 472 ], "score": 1.0, "content": ", it makes sense", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 471, 505, 484 ], "spans": [ { "bbox": [ 105, 471, 144, 484 ], "score": 1.0, "content": "to define", "type": "text" }, { "bbox": [ 144, 473, 156, 483 ], "score": 0.87, "content": "q _ { i j }", "type": "inline_equation" }, { "bbox": [ 157, 471, 505, 484 ], "score": 1.0, "content": "using the Gaussian kernel transformation of the cosine distance (Damrich et al., 2022;", "type": "text" } ], "index": 23 }, { "bbox": [ 106, 482, 186, 493 ], "spans": [ { "bbox": [ 106, 482, 186, 493 ], "score": 1.0, "content": "Bohm et al., 2023):¨", "type": "text" } ], "index": 24 } ], "index": 23, "bbox_fs": [ 105, 460, 505, 493 ] }, { "type": "interline_equation", "bbox": [ 141, 500, 469, 527 ], "lines": [ { "bbox": [ 141, 500, 469, 527 ], "spans": [ { "bbox": [ 141, 500, 469, 527 ], "score": 0.92, "content": "q _ { i j } = \\exp \\bigl ( \\mathbf { y } _ { i } ^ { \\mathsf { T } } \\mathbf { y } _ { j } / ( \\lVert \\mathbf { y } _ { i } \\rVert \\cdot \\lVert \\mathbf { y } _ { j } \\rVert ) / \\tau \\bigr ) = \\mathrm { c o n s t } \\cdot \\exp \\Big ( - \\Big \\lVert \\frac { \\mathbf { y } _ { i } } { \\lVert \\mathbf { y } _ { i } \\rVert } - \\frac { \\mathbf { y } _ { j } } { \\lVert \\mathbf { y } _ { j } \\rVert } \\Big \\rVert ^ { 2 } \\Big / ( 2 \\tau ) \\Big ) ,", "type": "interline_equation", "image_path": "f1aeccf40854bc160dcf59105db1a796b87b92147e9ccbcdc96975959727fbc8.jpg" } ] } ], "index": 26, "virtual_lines": [ { "bbox": [ 141, 500, 469, 509.0 ], "spans": [], "index": 25 }, { "bbox": [ 141, 509.0, 469, 518.0 ], "spans": [], "index": 26 }, { "bbox": [ 141, 518.0, 469, 527.0 ], "spans": [], "index": 27 } ] }, { "type": "text", "bbox": [ 106, 533, 505, 578 ], "lines": [ { "bbox": [ 106, 533, 505, 546 ], "spans": [ { "bbox": [ 106, 533, 133, 546 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 134, 536, 141, 543 ], "score": 0.77, "content": "\\tau", "type": "inline_equation" }, { "bbox": [ 141, 533, 295, 546 ], "score": 1.0, "content": "is called the temperature (by default,", "type": "text" }, { "bbox": [ 295, 533, 331, 544 ], "score": 0.87, "content": "\\tau = 0 . 5", "type": "inline_equation" }, { "bbox": [ 331, 533, 505, 546 ], "score": 1.0, "content": "). Together with Equation 5, this gives the", "type": "text" } ], "index": 28 }, { "bbox": [ 105, 543, 505, 557 ], "spans": [ { "bbox": [ 105, 543, 505, 557 ], "score": 1.0, "content": "same loss function as used in SimCLR (Chen et al., 2020), a popular contrastive learning algo-", "type": "text" } ], "index": 29 }, { "bbox": [ 106, 555, 506, 568 ], "spans": [ { "bbox": [ 106, 556, 363, 568 ], "score": 1.0, "content": "rithm in computer vision. The only difference is that instead of", "type": "text" }, { "bbox": [ 363, 555, 385, 566 ], "score": 0.78, "content": "k \\mathbf { N N }", "type": "inline_equation" }, { "bbox": [ 385, 556, 506, 568 ], "score": 1.0, "content": "edges, SimCLR uses pairs of", "type": "text" } ], "index": 30 }, { "bbox": [ 105, 567, 252, 579 ], "spans": [ { "bbox": [ 105, 567, 252, 579 ], "score": 1.0, "content": "augmented images as positive pairs.", "type": "text" } ], "index": 31 } ], "index": 29.5, "bbox_fs": [ 105, 533, 506, 579 ] }, { "type": "title", "bbox": [ 108, 595, 244, 608 ], "lines": [ { "bbox": [ 105, 594, 245, 610 ], "spans": [ { "bbox": [ 105, 594, 245, 610 ], "score": 1.0, "content": "4 EXPERIMENTAL SETUP", "type": "text" } ], "index": 32 } ], "index": 32 }, { "type": "text", "bbox": [ 106, 621, 505, 732 ], "lines": [ { "bbox": [ 105, 621, 505, 634 ], "spans": [ { "bbox": [ 105, 621, 505, 634 ], "score": 1.0, "content": "Datasets We used six publicly available graph datasets (Table 1). All datasets were retrieved from", "type": "text" } ], "index": 33 }, { "bbox": [ 105, 633, 505, 646 ], "spans": [ { "bbox": [ 105, 633, 505, 646 ], "score": 1.0, "content": "the Deep Graph Library (Wang et al., 2019), except ogbn-arxiv, which was retrieved from the", "type": "text" } ], "index": 34 }, { "bbox": [ 106, 643, 506, 656 ], "spans": [ { "bbox": [ 106, 643, 506, 656 ], "score": 1.0, "content": "Open Graph Benchmark (Hu et al., 2020). Each dataset was treated as an unweighted undirected", "type": "text" } ], "index": 35 }, { "bbox": [ 105, 654, 506, 668 ], "spans": [ { "bbox": [ 105, 654, 506, 668 ], "score": 1.0, "content": "graph, where each node has a class label and a feature vector (typically a word embedding vector", "type": "text" } ], "index": 36 }, { "bbox": [ 106, 666, 505, 678 ], "spans": [ { "bbox": [ 106, 666, 505, 678 ], "score": 1.0, "content": "of some descriptive text about the node, such as a product review). We restricted ourselves to", "type": "text" } ], "index": 37 }, { "bbox": [ 105, 677, 506, 690 ], "spans": [ { "bbox": [ 105, 677, 506, 690 ], "score": 1.0, "content": "graphs with labeled nodes in order to use classification accuracy as the performance metric. We also", "type": "text" } ], "index": 38 }, { "bbox": [ 105, 687, 505, 700 ], "spans": [ { "bbox": [ 105, 687, 505, 700 ], "score": 1.0, "content": "restricted ourselves to graphs with feature vectors in order to use both non-parametric and parametric", "type": "text" } ], "index": 39 }, { "bbox": [ 105, 699, 505, 712 ], "spans": [ { "bbox": [ 105, 699, 505, 712 ], "score": 1.0, "content": "embeddings. In all datasets we used only the largest connected component, and excluded all self-", "type": "text" } ], "index": 40 }, { "bbox": [ 105, 710, 506, 722 ], "spans": [ { "bbox": [ 105, 710, 506, 722 ], "score": 1.0, "content": "loops if present, using NetworkX (Hagberg et al., 2008) functions connected components", "type": "text" } ], "index": 41 }, { "bbox": [ 106, 720, 209, 733 ], "spans": [ { "bbox": [ 106, 720, 209, 733 ], "score": 1.0, "content": "and selfloop edges.", "type": "text" } ], "index": 42 } ], "index": 37.5, "bbox_fs": [ 105, 621, 506, 733 ] } ] }, { "preproc_blocks": [ { "type": "image", "bbox": [ 104, 75, 496, 294 ], "blocks": [ { "type": "image_body", "bbox": [ 104, 75, 496, 294 ], "group_id": 0, "lines": [ { "bbox": [ 104, 75, 496, 294 ], "spans": [ { "bbox": [ 104, 75, 496, 294 ], "score": 0.973, "type": "image", "image_path": "2e5f7f25dc75ee881bccb233b1ff2c656b23adc633b4fd11cb1bd5536ccf62fd.jpg" } ] } ], "index": 1, "virtual_lines": [ { "bbox": [ 104, 75, 496, 148.0 ], "spans": [], "index": 0 }, { "bbox": [ 104, 148.0, 496, 221.0 ], "spans": [], "index": 1 }, { "bbox": [ 104, 221.0, 496, 294.0 ], "spans": [], "index": 2 } ] }, { "type": "image_caption", "bbox": [ 106, 314, 505, 348 ], "group_id": 0, "lines": [ { "bbox": [ 106, 314, 504, 326 ], "spans": [ { "bbox": [ 106, 314, 504, 326 ], "score": 1.0, "content": "Figure 2: Embeddings of the ACO and APH datasets obtained using FDP (Fruchterman & Rein-", "type": "text" } ], "index": 3 }, { "bbox": [ 105, 325, 505, 338 ], "spans": [ { "bbox": [ 105, 325, 299, 338 ], "score": 1.0, "content": "gold, 1991), DRGraph (Zhu et al., 2020a), and", "type": "text" }, { "bbox": [ 299, 326, 304, 336 ], "score": 0.72, "content": "t", "type": "inline_equation" }, { "bbox": [ 305, 325, 474, 338 ], "score": 1.0, "content": "-FDP (Zhong et al., 2023), and our graph", "type": "text" }, { "bbox": [ 474, 326, 479, 335 ], "score": 0.58, "content": "t { \\cdot }", "type": "inline_equation" }, { "bbox": [ 480, 325, 505, 338 ], "score": 1.0, "content": "-SNE.", "type": "text" } ], "index": 4 }, { "bbox": [ 106, 336, 505, 349 ], "spans": [ { "bbox": [ 106, 336, 505, 349 ], "score": 1.0, "content": "Embeddings in each row were aligned using Procrustes rotation. See Figure A.3 for all six datasets.", "type": "text" } ], "index": 5 } ], "index": 4 } ], "index": 2.5 }, { "type": "text", "bbox": [ 106, 366, 505, 400 ], "lines": [ { "bbox": [ 105, 365, 505, 380 ], "spans": [ { "bbox": [ 105, 365, 505, 380 ], "score": 1.0, "content": "Performance metrics We evaluated the performance of our methods using three performance", "type": "text" } ], "index": 6 }, { "bbox": [ 105, 378, 505, 391 ], "spans": [ { "bbox": [ 105, 378, 142, 391 ], "score": 1.0, "content": "metrics:", "type": "text" }, { "bbox": [ 142, 379, 149, 388 ], "score": 0.79, "content": "k", "type": "inline_equation" }, { "bbox": [ 149, 378, 226, 391 ], "score": 1.0, "content": "-nearest-neighbors", "type": "text" }, { "bbox": [ 226, 378, 252, 389 ], "score": 0.32, "content": "( k \\mathsf { N N } )", "type": "inline_equation" }, { "bbox": [ 253, 378, 281, 391 ], "score": 1.0, "content": "recall,", "type": "text" }, { "bbox": [ 281, 378, 303, 388 ], "score": 0.54, "content": "k \\mathbf { N N }", "type": "inline_equation" }, { "bbox": [ 303, 378, 505, 391 ], "score": 1.0, "content": "classification accuracy, and, for high-dimensional", "type": "text" } ], "index": 7 }, { "bbox": [ 106, 388, 279, 402 ], "spans": [ { "bbox": [ 106, 388, 279, 402 ], "score": 1.0, "content": "embeddings, linear classification accuracy.", "type": "text" } ], "index": 8 } ], "index": 7 }, { "type": "text", "bbox": [ 107, 405, 505, 439 ], "lines": [ { "bbox": [ 106, 406, 504, 418 ], "spans": [ { "bbox": [ 106, 406, 124, 418 ], "score": 1.0, "content": "The", "type": "text" }, { "bbox": [ 124, 406, 146, 416 ], "score": 0.69, "content": "k \\mathbf { N N }", "type": "inline_equation" }, { "bbox": [ 147, 406, 504, 418 ], "score": 1.0, "content": "recall quantifies how well local node neighborhoods are preserved in the embedding. We", "type": "text" } ], "index": 9 }, { "bbox": [ 105, 416, 505, 429 ], "spans": [ { "bbox": [ 105, 416, 505, 429 ], "score": 1.0, "content": "defined it as the average fraction of each node’s graph neighbors that are among the node’s nearest", "type": "text" } ], "index": 10 }, { "bbox": [ 106, 428, 222, 441 ], "spans": [ { "bbox": [ 106, 428, 222, 441 ], "score": 1.0, "content": "neighbors in the embedding:", "type": "text" } ], "index": 11 } ], "index": 10 }, { "type": "interline_equation", "bbox": [ 227, 443, 382, 478 ], "lines": [ { "bbox": [ 227, 443, 382, 478 ], "spans": [ { "bbox": [ 227, 443, 382, 478 ], "score": 0.95, "content": "\\mathrm { R e c a l l } = \\frac { 1 } { | \\mathcal { V } | } \\sum _ { i = 1 } ^ { | \\mathcal { V } | } \\frac { \\left| N _ { G } [ i ] \\cap N _ { E , k _ { i } } [ i ] \\right| } { k _ { i } } ,", "type": "interline_equation", "image_path": "013acbeb5f8e7077d62b07cab43f72279519c94d2eac7331ea5a660e788ee069.jpg" } ] } ], "index": 12.5, "virtual_lines": [ { "bbox": [ 227, 443, 382, 460.5 ], "spans": [], "index": 12 }, { "bbox": [ 227, 460.5, 382, 478.0 ], "spans": [], "index": 13 } ] }, { "type": "text", "bbox": [ 107, 481, 505, 538 ], "lines": [ { "bbox": [ 105, 480, 504, 495 ], "spans": [ { "bbox": [ 105, 480, 133, 495 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 134, 482, 147, 494 ], "score": 0.9, "content": "| \\nu |", "type": "inline_equation" }, { "bbox": [ 147, 480, 294, 495 ], "score": 1.0, "content": "is the number of nodes in the graph,", "type": "text" }, { "bbox": [ 294, 482, 319, 494 ], "score": 0.92, "content": "N _ { G } [ i ]", "type": "inline_equation" }, { "bbox": [ 319, 480, 390, 495 ], "score": 1.0, "content": "is the set of node", "type": "text" }, { "bbox": [ 390, 483, 395, 492 ], "score": 0.64, "content": "i", "type": "inline_equation" }, { "bbox": [ 395, 480, 472, 495 ], "score": 1.0, "content": "’s graph neighbors,", "type": "text" }, { "bbox": [ 472, 482, 504, 493 ], "score": 0.89, "content": "N _ { E , k } [ i ]", "type": "inline_equation" } ], "index": 14 }, { "bbox": [ 105, 490, 504, 507 ], "spans": [ { "bbox": [ 105, 490, 200, 507 ], "score": 1.0, "content": "denotes the set of node", "type": "text" }, { "bbox": [ 200, 494, 205, 503 ], "score": 0.71, "content": "i", "type": "inline_equation" }, { "bbox": [ 205, 490, 212, 507 ], "score": 1.0, "content": "’s", "type": "text" }, { "bbox": [ 213, 493, 220, 503 ], "score": 0.8, "content": "k", "type": "inline_equation" }, { "bbox": [ 220, 490, 451, 507 ], "score": 1.0, "content": "Euclidean nearest neighbors in the embedding space, and", "type": "text" }, { "bbox": [ 452, 493, 504, 505 ], "score": 0.9, "content": "k _ { i } = | N _ { G } [ i ] |", "type": "inline_equation" } ], "index": 15 }, { "bbox": [ 105, 504, 505, 516 ], "spans": [ { "bbox": [ 105, 504, 197, 516 ], "score": 1.0, "content": "is the number of node", "type": "text" }, { "bbox": [ 198, 505, 203, 514 ], "score": 0.71, "content": "i", "type": "inline_equation" }, { "bbox": [ 203, 504, 505, 516 ], "score": 1.0, "content": "’s graph neighbors. 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We", "type": "text" } ], "index": 9 }, { "bbox": [ 105, 416, 505, 429 ], "spans": [ { "bbox": [ 105, 416, 505, 429 ], "score": 1.0, "content": "defined it as the average fraction of each node’s graph neighbors that are among the node’s nearest", "type": "text" } ], "index": 10 }, { "bbox": [ 106, 428, 222, 441 ], "spans": [ { "bbox": [ 106, 428, 222, 441 ], "score": 1.0, "content": "neighbors in the embedding:", "type": "text" } ], "index": 11 } ], "index": 10, "bbox_fs": [ 105, 406, 505, 441 ] }, { "type": "interline_equation", "bbox": [ 227, 443, 382, 478 ], "lines": [ { "bbox": [ 227, 443, 382, 478 ], "spans": [ { "bbox": [ 227, 443, 382, 478 ], "score": 0.95, "content": "\\mathrm { R e c a l l } = \\frac { 1 } { | \\mathcal { V } | } \\sum _ { i = 1 } ^ { | \\mathcal { V } | } \\frac { \\left| N _ { G } [ i ] \\cap N _ { E , k _ { i } } [ i ] \\right| } { k _ { i } } ,", "type": "interline_equation", "image_path": "013acbeb5f8e7077d62b07cab43f72279519c94d2eac7331ea5a660e788ee069.jpg" } ] } ], "index": 12.5, "virtual_lines": [ { "bbox": [ 227, 443, 382, 460.5 ], "spans": [], "index": 12 }, { "bbox": [ 227, 460.5, 382, 478.0 ], "spans": [], "index": 13 } ] }, { "type": "text", "bbox": [ 107, 481, 505, 538 ], "lines": [ { "bbox": [ 105, 480, 504, 495 ], "spans": [ { "bbox": [ 105, 480, 133, 495 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 134, 482, 147, 494 ], "score": 0.9, "content": "| \\nu |", "type": "inline_equation" }, { "bbox": [ 147, 480, 294, 495 ], "score": 1.0, "content": "is the number of nodes in the graph,", "type": "text" }, { "bbox": [ 294, 482, 319, 494 ], "score": 0.92, "content": "N _ { G } [ i ]", "type": "inline_equation" }, { "bbox": [ 319, 480, 390, 495 ], "score": 1.0, "content": "is the set of node", "type": "text" }, { "bbox": [ 390, 483, 395, 492 ], "score": 0.64, "content": "i", "type": "inline_equation" }, { "bbox": [ 395, 480, 472, 495 ], "score": 1.0, "content": "’s graph neighbors,", "type": "text" }, { "bbox": [ 472, 482, 504, 493 ], "score": 0.89, "content": "N _ { E , k } [ i ]", "type": "inline_equation" } ], "index": 14 }, { "bbox": [ 105, 490, 504, 507 ], "spans": [ { "bbox": [ 105, 490, 200, 507 ], "score": 1.0, "content": "denotes the set of node", "type": "text" }, { "bbox": [ 200, 494, 205, 503 ], "score": 0.71, "content": "i", "type": "inline_equation" }, { "bbox": [ 205, 490, 212, 507 ], "score": 1.0, "content": "’s", "type": "text" }, { "bbox": [ 213, 493, 220, 503 ], "score": 0.8, "content": "k", "type": "inline_equation" }, { "bbox": [ 220, 490, 451, 507 ], "score": 1.0, "content": "Euclidean nearest neighbors in the embedding space, and", "type": "text" }, { "bbox": [ 452, 493, 504, 505 ], "score": 0.9, "content": "k _ { i } = | N _ { G } [ i ] |", "type": "inline_equation" } ], "index": 15 }, { "bbox": [ 105, 504, 505, 516 ], "spans": [ { "bbox": [ 105, 504, 197, 516 ], "score": 1.0, "content": "is the number of node", "type": "text" }, { "bbox": [ 198, 505, 203, 514 ], "score": 0.71, "content": "i", "type": "inline_equation" }, { "bbox": [ 203, 504, 505, 516 ], "score": 1.0, "content": "’s graph neighbors. 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Of note, we used the train/test split only for training the classifier but not for computing", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 586, 505, 600 ], "spans": [ { "bbox": [ 105, 586, 505, 600 ], "score": 1.0, "content": "the graph embedding itself. We used sklearn.preprocessing.StandardScaler to stan-", "type": "text" } ], "index": 23 }, { "bbox": [ 106, 597, 286, 610 ], "spans": [ { "bbox": [ 106, 597, 286, 610 ], "score": 1.0, "content": "dardize all features based on the training set.", "type": "text" } ], "index": 24 } ], "index": 21.5, "bbox_fs": [ 105, 542, 506, 610 ] }, { "type": "text", "bbox": [ 107, 614, 505, 648 ], "lines": [ { "bbox": [ 105, 614, 505, 627 ], "spans": [ { "bbox": [ 105, 614, 192, 627 ], "score": 1.0, "content": "For graph CNE with", "type": "text" }, { "bbox": [ 193, 615, 230, 625 ], "score": 0.89, "content": "d = 1 2 8", "type": "inline_equation" }, { "bbox": [ 230, 614, 505, 627 ], "score": 1.0, "content": ", trained using cosine distance, we experimented with using cosine-", "type": "text" } ], "index": 25 }, { "bbox": [ 105, 625, 505, 638 ], "spans": [ { "bbox": [ 105, 625, 168, 638 ], "score": 1.0, "content": "distance-based", "type": "text" }, { "bbox": [ 168, 626, 190, 636 ], "score": 0.49, "content": "k \\mathbf { N N }", "type": "inline_equation" }, { "bbox": [ 190, 625, 505, 638 ], "score": 1.0, "content": "recall and accuracy, but found that it gave very close results to the Euclidean-", "type": "text" } ], "index": 26 }, { "bbox": [ 106, 636, 408, 649 ], "spans": [ { "bbox": [ 106, 636, 167, 649 ], "score": 1.0, "content": "distance-based", "type": "text" }, { "bbox": [ 168, 637, 189, 647 ], "score": 0.41, "content": "k \\mathbf { N N }", "type": "inline_equation" }, { "bbox": [ 190, 636, 408, 649 ], "score": 1.0, "content": "evaluations (all differences below 1 percentage point).", "type": "text" } ], "index": 27 } ], "index": 26, "bbox_fs": [ 105, 614, 505, 649 ] }, { "type": "text", "bbox": [ 107, 653, 504, 687 ], "lines": [ { "bbox": [ 106, 654, 505, 665 ], "spans": [ { "bbox": [ 106, 654, 505, 665 ], "score": 1.0, "content": "For linear accuracy we used the sklearn.linear model.LogisticRegression class", "type": "text" } ], "index": 28 }, { "bbox": [ 106, 664, 505, 676 ], "spans": [ { "bbox": [ 106, 664, 241, 676 ], "score": 1.0, "content": "with no regularization (penalty", "type": "text" }, { "bbox": [ 242, 666, 249, 674 ], "score": 0.38, "content": "=", "type": "inline_equation" }, { "bbox": [ 249, 664, 505, 676 ], "score": 1.0, "content": "None) and otherwise default parameters, and the same train/test", "type": "text" } ], "index": 29 }, { "bbox": [ 106, 676, 353, 688 ], "spans": [ { "bbox": [ 106, 676, 353, 688 ], "score": 1.0, "content": "split. Features were standardized using StandardScaler.", "type": "text" } ], "index": 30 } ], "index": 29, "bbox_fs": [ 106, 654, 505, 688 ] }, { "type": "text", "bbox": [ 108, 699, 504, 732 ], "lines": [ { "bbox": [ 106, 698, 506, 712 ], "spans": [ { "bbox": [ 106, 698, 506, 712 ], "score": 1.0, "content": "Computing environment All computations were performed on a remote computing server with", "type": "text" } ], "index": 31 }, { "bbox": [ 106, 709, 505, 722 ], "spans": [ { "bbox": [ 106, 709, 505, 722 ], "score": 1.0, "content": "an Intel Xeon Gold CPU with 16 double-threaded 2.9 Ghz cores, 384 GB of RAM, and an NVIDIA", "type": "text" } ], "index": 32 }, { "bbox": [ 105, 720, 505, 733 ], "spans": [ { "bbox": [ 105, 720, 394, 733 ], "score": 1.0, "content": "RTX A6000 GPU. GPU training was used for CNE models but not for", "type": "text" }, { "bbox": [ 395, 721, 400, 730 ], "score": 0.65, "content": "t", "type": "inline_equation" }, { "bbox": [ 400, 720, 505, 733 ], "score": 1.0, "content": "-SNE. Computation times", "type": "text" } ], "index": 33 }, { "bbox": [ 106, 278, 505, 290 ], "spans": [ { "bbox": [ 106, 278, 357, 290 ], "score": 1.0, "content": "are shown in Figure A.1. For the largest dataset (ARX), graph", "type": "text", "cross_page": true }, { "bbox": [ 357, 279, 362, 288 ], "score": 0.66, "content": "t", "type": "inline_equation", "cross_page": true }, { "bbox": [ 363, 278, 505, 290 ], "score": 1.0, "content": "-SNE took around 100 seconds and", "type": "text", "cross_page": true } ], "index": 5 }, { "bbox": [ 105, 289, 253, 302 ], "spans": [ { "bbox": [ 105, 289, 253, 302 ], "score": 1.0, "content": "graph CNE took around 60 minutes.", "type": "text", "cross_page": true } ], "index": 6 } ], "index": 32, "bbox_fs": [ 105, 698, 506, 733 ] } ] }, { "preproc_blocks": [ { "type": "image", "bbox": [ 107, 81, 504, 225 ], "blocks": [ { "type": "image_body", "bbox": [ 107, 81, 504, 225 ], "group_id": 0, "lines": [ { "bbox": [ 107, 81, 504, 225 ], "spans": [ { "bbox": [ 107, 81, 504, 225 ], "score": 0.968, "type": "image", "image_path": "36d02dd4d58e27a78df1813d2a0ebc6d57cdb0d6a908f0caf5fd9adffc64418a.jpg" } ] } ], "index": 1, "virtual_lines": [ { "bbox": [ 107, 81, 504, 129.0 ], "spans": [], "index": 0 }, { "bbox": [ 107, 129.0, 504, 177.0 ], "spans": [], "index": 1 }, { "bbox": [ 107, 177.0, 504, 225.0 ], "spans": [], "index": 2 } ] }, { "type": "image_caption", "bbox": [ 106, 237, 504, 260 ], "group_id": 0, "lines": [ { "bbox": [ 105, 237, 505, 250 ], "spans": [ { "bbox": [ 105, 237, 316, 250 ], "score": 1.0, "content": "Figure 3: Performance metrics for graph layouts:", "type": "text" }, { "bbox": [ 317, 237, 339, 248 ], "score": 0.34, "content": "k \\mathbf { N N }", "type": "inline_equation" }, { "bbox": [ 339, 237, 384, 250 ], "score": 1.0, "content": "recall and", "type": "text" }, { "bbox": [ 385, 238, 407, 248 ], "score": 0.67, "content": "k \\mathbf { N N }", "type": "inline_equation" }, { "bbox": [ 407, 237, 505, 250 ], "score": 1.0, "content": "accuracy. Datasets are", "type": "text" } ], "index": 3 }, { "bbox": [ 106, 249, 474, 261 ], "spans": [ { "bbox": [ 106, 249, 474, 261 ], "score": 1.0, "content": "ordered by the increasing sample size. See Figures 2 and A.3 for the corresponding layouts.", "type": "text" } ], "index": 4 } ], "index": 3.5 } ], "index": 2.25 }, { "type": "text", "bbox": [ 106, 278, 504, 300 ], "lines": [ { "bbox": [ 106, 278, 505, 290 ], "spans": [ { "bbox": [ 106, 278, 357, 290 ], "score": 1.0, "content": "are shown in Figure A.1. For the largest dataset (ARX), graph", "type": "text" }, { "bbox": [ 357, 279, 362, 288 ], "score": 0.66, "content": "t", "type": "inline_equation" }, { "bbox": [ 363, 278, 505, 290 ], "score": 1.0, "content": "-SNE took around 100 seconds and", "type": "text" } ], "index": 5 }, { "bbox": [ 105, 289, 253, 302 ], "spans": [ { "bbox": [ 105, 289, 253, 302 ], "score": 1.0, "content": "graph CNE took around 60 minutes.", "type": "text" } ], "index": 6 } ], "index": 5.5 }, { "type": "title", "bbox": [ 107, 316, 311, 329 ], "lines": [ { "bbox": [ 105, 316, 311, 330 ], "spans": [ { "bbox": [ 105, 316, 275, 330 ], "score": 1.0, "content": "5 GRAPH LAYOUTS VIA GRAPH", "type": "text" }, { "bbox": [ 276, 317, 281, 327 ], "score": 0.59, "content": "t", "type": "inline_equation" }, { "bbox": [ 282, 316, 311, 330 ], "score": 1.0, "content": "-SNE", "type": "text" } ], "index": 7 } ], "index": 7 }, { "type": "text", "bbox": [ 106, 340, 505, 385 ], "lines": [ { "bbox": [ 105, 340, 505, 354 ], "spans": [ { "bbox": [ 105, 340, 124, 354 ], "score": 1.0, "content": "The", "type": "text" }, { "bbox": [ 125, 342, 130, 351 ], "score": 0.68, "content": "t", "type": "inline_equation" }, { "bbox": [ 131, 340, 505, 354 ], "score": 1.0, "content": "-SNE algorithm consists of two steps: first, it computes pairwise affinities between all pairs", "type": "text" } ], "index": 8 }, { "bbox": [ 105, 352, 506, 365 ], "spans": [ { "bbox": [ 105, 352, 202, 365 ], "score": 1.0, "content": "of points based on the", "type": "text" }, { "bbox": [ 202, 352, 224, 362 ], "score": 0.73, "content": "k \\mathbf { N N }", "type": "inline_equation" }, { "bbox": [ 224, 352, 506, 365 ], "score": 1.0, "content": "graph; second, it optimizes the embedding to match these affinities", "type": "text" } ], "index": 9 }, { "bbox": [ 106, 363, 505, 375 ], "spans": [ { "bbox": [ 106, 363, 204, 375 ], "score": 1.0, "content": "(Section 3.1). For graph", "type": "text" }, { "bbox": [ 205, 364, 210, 373 ], "score": 0.63, "content": "t", "type": "inline_equation" }, { "bbox": [ 210, 363, 505, 375 ], "score": 1.0, "content": "-SNE, we replace the first step and obtain the affinity matrix directly from", "type": "text" } ], "index": 10 }, { "bbox": [ 106, 374, 503, 386 ], "spans": [ { "bbox": [ 106, 374, 271, 386 ], "score": 1.0, "content": "the graph adjacency matrix. We then run", "type": "text" }, { "bbox": [ 271, 375, 276, 384 ], "score": 0.69, "content": "t", "type": "inline_equation" }, { "bbox": [ 276, 374, 503, 386 ], "score": 1.0, "content": "-SNE optimization to produce the embedding (Figure 1).", "type": "text" } ], "index": 11 } ], "index": 9.5 }, { "type": "text", "bbox": [ 106, 390, 505, 446 ], "lines": [ { "bbox": [ 106, 390, 506, 403 ], "spans": [ { "bbox": [ 106, 390, 222, 403 ], "score": 1.0, "content": "Given an unweighted graph", "type": "text" }, { "bbox": [ 222, 390, 273, 403 ], "score": 0.94, "content": "G = ( \\nu , \\mathcal { E } )", "type": "inline_equation" }, { "bbox": [ 273, 390, 362, 403 ], "score": 1.0, "content": ", its adjacency matrix", "type": "text" }, { "bbox": [ 363, 391, 373, 401 ], "score": 0.31, "content": "\\mathbf { A }", "type": "inline_equation" }, { "bbox": [ 373, 390, 457, 403 ], "score": 1.0, "content": "is defined such that", "type": "text" }, { "bbox": [ 457, 391, 495, 403 ], "score": 0.93, "content": "A _ { i j } = 1", "type": "inline_equation" }, { "bbox": [ 495, 390, 506, 403 ], "score": 1.0, "content": "if", "type": "text" } ], "index": 12 }, { "bbox": [ 107, 402, 506, 415 ], "spans": [ { "bbox": [ 107, 402, 150, 414 ], "score": 0.92, "content": "( i , j ) \\in \\mathcal { E }", "type": "inline_equation" }, { "bbox": [ 150, 402, 169, 415 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 170, 402, 208, 414 ], "score": 0.93, "content": "A _ { i j } = 0", "type": "inline_equation" }, { "bbox": [ 208, 402, 506, 415 ], "score": 1.0, "content": "otherwise. Since all graphs considered in this study are undirected, the", "type": "text" } ], "index": 13 }, { "bbox": [ 105, 412, 505, 426 ], "spans": [ { "bbox": [ 105, 412, 298, 426 ], "score": 1.0, "content": "adjacency matrix is a binary, symmetric square", "type": "text" }, { "bbox": [ 298, 414, 324, 423 ], "score": 0.9, "content": "n \\times n", "type": "inline_equation" }, { "bbox": [ 324, 412, 505, 426 ], "score": 1.0, "content": "matrix. In order to convert it into an affinity", "type": "text" } ], "index": 14 }, { "bbox": [ 105, 422, 505, 436 ], "spans": [ { "bbox": [ 105, 422, 182, 436 ], "score": 1.0, "content": "matrix suitable for", "type": "text" }, { "bbox": [ 183, 424, 188, 433 ], "score": 0.67, "content": "t", "type": "inline_equation" }, { "bbox": [ 188, 422, 307, 436 ], "score": 1.0, "content": "-SNE, we follow the standard", "type": "text" }, { "bbox": [ 307, 424, 312, 433 ], "score": 0.69, "content": "t", "type": "inline_equation" }, { "bbox": [ 313, 422, 505, 436 ], "score": 1.0, "content": "-SNE’s approach (Section 3.1): divide each row", "type": "text" } ], "index": 15 }, { "bbox": [ 106, 434, 497, 448 ], "spans": [ { "bbox": [ 106, 434, 497, 448 ], "score": 1.0, "content": "by the sum of its elements, then symmetrize the resulting matrix, and then normalize to sum to 1:", "type": "text" } ], "index": 16 } ], "index": 14 }, { "type": "interline_equation", "bbox": [ 213, 448, 396, 482 ], "lines": [ { "bbox": [ 213, 448, 396, 482 ], "spans": [ { "bbox": [ 213, 448, 396, 482 ], "score": 0.9, "content": "\\mathbf { P } = { \\frac { { \\tilde { \\mathbf { A } } } + { \\tilde { \\mathbf { A } } } ^ { \\top } } { 2 n } } , { \\mathrm { ~ w h e r e ~ } } { \\tilde { A } } _ { i j } = A _ { i j } { \\Big / } \\sum _ { k = 1 } ^ { n } A _ { i k } .", "type": "interline_equation", "image_path": "059da263f07d66cdd192e32bb1681c7b7d46a29444d542c16284af5f58184c25.jpg" } ] } ], "index": 17.5, "virtual_lines": [ { "bbox": [ 213, 448, 396, 465.0 ], "spans": [], "index": 17 }, { "bbox": [ 213, 465.0, 396, 482.0 ], "spans": [], "index": 18 } ] }, { "type": "text", "bbox": [ 107, 488, 505, 545 ], "lines": [ { "bbox": [ 105, 488, 506, 502 ], "spans": [ { "bbox": [ 105, 488, 506, 502 ], "score": 1.0, "content": "For optimization, we used the openTSNE library (Policar et al., 2019) with default parameters. It ˇ", "type": "text" } ], "index": 19 }, { "bbox": [ 106, 501, 505, 513 ], "spans": [ { "bbox": [ 106, 501, 505, 513 ], "score": 1.0, "content": "uses Laplacian Eigenmaps (Belkin & Niyogi, 2003) for initialization (Kobak & Linderman, 2021),", "type": "text" } ], "index": 20 }, { "bbox": [ 105, 510, 505, 524 ], "spans": [ { "bbox": [ 105, 510, 230, 524 ], "score": 1.0, "content": "sets the learning rate equal to", "type": "text" }, { "bbox": [ 230, 514, 237, 521 ], "score": 0.74, "content": "n", "type": "inline_equation" }, { "bbox": [ 238, 510, 505, 524 ], "score": 1.0, "content": "to achieve good convergence (Linderman & Steinerberger, 2019;", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 521, 505, 536 ], "spans": [ { "bbox": [ 105, 521, 398, 536 ], "score": 1.0, "content": "Belkina et al., 2019), and employs fast FIt-SNE algorithm that has linear", "type": "text" }, { "bbox": [ 398, 522, 421, 534 ], "score": 0.92, "content": "{ \\mathcal { O } } ( n )", "type": "inline_equation" }, { "bbox": [ 421, 521, 505, 536 ], "score": 1.0, "content": "runtime (Linderman", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 533, 159, 546 ], "spans": [ { "bbox": [ 105, 533, 159, 546 ], "score": 1.0, "content": "et al., 2019).", "type": "text" } ], "index": 23 } ], "index": 21 }, { "type": "text", "bbox": [ 106, 549, 505, 639 ], "lines": [ { "bbox": [ 106, 549, 505, 563 ], "spans": [ { "bbox": [ 106, 549, 188, 563 ], "score": 1.0, "content": "We compared graph", "type": "text" }, { "bbox": [ 189, 551, 193, 560 ], "score": 0.59, "content": "t", "type": "inline_equation" }, { "bbox": [ 194, 549, 505, 563 ], "score": 1.0, "content": "-SNE with three existing graph layout algorithms: FDP (Fruchterman & Rein-", "type": "text" } ], "index": 24 }, { "bbox": [ 105, 561, 506, 574 ], "spans": [ { "bbox": [ 105, 561, 304, 574 ], "score": 1.0, "content": "gold, 1991), DRGraph (Zhu et al., 2020a), and", "type": "text" }, { "bbox": [ 304, 562, 309, 571 ], "score": 0.75, "content": "t", "type": "inline_equation" }, { "bbox": [ 309, 561, 506, 574 ], "score": 1.0, "content": "-FDP (Zhong et al., 2023). We chose FDP be-", "type": "text" } ], "index": 25 }, { "bbox": [ 105, 572, 505, 585 ], "spans": [ { "bbox": [ 105, 572, 505, 585 ], "score": 1.0, "content": "cause it is the default layout algorithm in a popular NetworkX package (Hagberg et al., 2008).", "type": "text" } ], "index": 26 }, { "bbox": [ 106, 583, 505, 595 ], "spans": [ { "bbox": [ 106, 583, 199, 595 ], "score": 1.0, "content": "Two other algorithms,", "type": "text" }, { "bbox": [ 199, 584, 204, 593 ], "score": 0.71, "content": "t", "type": "inline_equation" }, { "bbox": [ 204, 583, 505, 595 ], "score": 1.0, "content": "-FDP and DRGraph, are recent and can be considered state-of-the-art (we", "type": "text" } ], "index": 27 }, { "bbox": [ 106, 594, 504, 606 ], "spans": [ { "bbox": [ 106, 594, 504, 606 ], "score": 1.0, "content": "did not use tsNET (Kruiger et al., 2017) for benchmarking, because it cannot embed large graphs", "type": "text" } ], "index": 28 }, { "bbox": [ 105, 605, 506, 617 ], "spans": [ { "bbox": [ 105, 605, 506, 617 ], "score": 1.0, "content": "and is outperformed by its successor DRGraph). We used the NetworkX implementation of FDP", "type": "text" } ], "index": 29 }, { "bbox": [ 106, 616, 505, 628 ], "spans": [ { "bbox": [ 106, 616, 505, 628 ], "score": 1.0, "content": "(networkx.drawing.layout.spring layout) and the original implementations of both", "type": "text" } ], "index": 30 }, { "bbox": [ 106, 626, 349, 640 ], "spans": [ { "bbox": [ 106, 628, 111, 637 ], "score": 0.76, "content": "t", "type": "inline_equation" }, { "bbox": [ 112, 626, 349, 640 ], "score": 1.0, "content": "-FDP and DRGraph, all with default parameters (Figure 2).", "type": "text" } ], "index": 31 } ], "index": 27.5 }, { "type": "text", "bbox": [ 106, 643, 505, 732 ], "lines": [ { "bbox": [ 106, 643, 505, 657 ], "spans": [ { "bbox": [ 106, 643, 191, 657 ], "score": 1.0, "content": "We found that graph", "type": "text" }, { "bbox": [ 192, 645, 196, 654 ], "score": 0.63, "content": "t", "type": "inline_equation" }, { "bbox": [ 197, 643, 457, 657 ], "score": 1.0, "content": "-SNE consistently outperformed all competitors in terms of both", "type": "text" }, { "bbox": [ 458, 644, 479, 654 ], "score": 0.36, "content": "k \\mathbf { N N }", "type": "inline_equation" }, { "bbox": [ 480, 643, 505, 657 ], "score": 1.0, "content": "recall", "type": "text" } ], "index": 32 }, { "bbox": [ 106, 655, 505, 666 ], "spans": [ { "bbox": [ 106, 655, 124, 666 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 124, 655, 146, 666 ], "score": 0.71, "content": "k \\mathbf { N N }", "type": "inline_equation" }, { "bbox": [ 147, 655, 505, 666 ], "score": 1.0, "content": "accuracy (Figure 3): it showed the highest values on all datasets, 12 out of 12 times.", "type": "text" } ], "index": 33 }, { "bbox": [ 105, 666, 506, 679 ], "spans": [ { "bbox": [ 105, 666, 506, 679 ], "score": 1.0, "content": "In agreement with the original results of Zhu et al. (2020a) and Zhong et al. (2023), we saw that", "type": "text" } ], "index": 34 }, { "bbox": [ 106, 677, 505, 689 ], "spans": [ { "bbox": [ 106, 677, 164, 689 ], "score": 1.0, "content": "DRGraph and", "type": "text" }, { "bbox": [ 164, 677, 169, 687 ], "score": 0.77, "content": "t", "type": "inline_equation" }, { "bbox": [ 169, 677, 376, 689 ], "score": 1.0, "content": "-FDP outperformed FDP in both metrics. Our graph", "type": "text" }, { "bbox": [ 376, 678, 381, 687 ], "score": 0.6, "content": "t", "type": "inline_equation" }, { "bbox": [ 381, 677, 505, 689 ], "score": 1.0, "content": "-SNE showed further improve-", "type": "text" } ], "index": 35 }, { "bbox": [ 105, 687, 505, 701 ], "spans": [ { "bbox": [ 105, 687, 269, 701 ], "score": 1.0, "content": "ment, and it was substantial: in terms of", "type": "text" }, { "bbox": [ 270, 688, 291, 698 ], "score": 0.57, "content": "k \\mathbf { N N }", "type": "inline_equation" }, { "bbox": [ 291, 687, 344, 701 ], "score": 1.0, "content": "recall, graph", "type": "text" }, { "bbox": [ 344, 689, 349, 698 ], "score": 0.69, "content": "t", "type": "inline_equation" }, { "bbox": [ 350, 687, 505, 701 ], "score": 1.0, "content": "-SNE improved on the best competitor", "type": "text" } ], "index": 36 }, { "bbox": [ 105, 698, 505, 713 ], "spans": [ { "bbox": [ 105, 698, 317, 713 ], "score": 1.0, "content": "on average by 18.2 percentage points, and in terms of", "type": "text" }, { "bbox": [ 317, 699, 339, 709 ], "score": 0.35, "content": "k \\mathbf { N N }", "type": "inline_equation" }, { "bbox": [ 339, 698, 505, 713 ], "score": 1.0, "content": "accuracy — on average by 6.7 percentage", "type": "text" } ], "index": 37 }, { "bbox": [ 105, 710, 505, 722 ], "spans": [ { "bbox": [ 105, 710, 505, 722 ], "score": 1.0, "content": "points. The improvement was particularly strong for the largest graph (ARX), where performance", "type": "text" } ], "index": 38 }, { "bbox": [ 106, 721, 264, 733 ], "spans": [ { "bbox": [ 106, 721, 264, 733 ], "score": 1.0, "content": "of other methods strongly deteriorated.", "type": "text" } ], "index": 39 } ], "index": 35.5 } ], "page_idx": 5, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 106, 27, 308, 37 ], "lines": [ { "bbox": [ 106, 26, 308, 38 ], "spans": [ { "bbox": [ 106, 26, 308, 38 ], "score": 1.0, "content": "Under review as a conference paper at ICLR 2024", "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": [ 107, 81, 504, 225 ], "blocks": [ { "type": "image_body", "bbox": [ 107, 81, 504, 225 ], "group_id": 0, "lines": [ { "bbox": [ 107, 81, 504, 225 ], "spans": [ { "bbox": [ 107, 81, 504, 225 ], "score": 0.968, "type": "image", "image_path": "36d02dd4d58e27a78df1813d2a0ebc6d57cdb0d6a908f0caf5fd9adffc64418a.jpg" } ] } ], "index": 1, "virtual_lines": [ { "bbox": [ 107, 81, 504, 129.0 ], "spans": [], "index": 0 }, { "bbox": [ 107, 129.0, 504, 177.0 ], "spans": [], "index": 1 }, { "bbox": [ 107, 177.0, 504, 225.0 ], "spans": [], "index": 2 } ] }, { "type": "image_caption", "bbox": [ 106, 237, 504, 260 ], "group_id": 0, "lines": [ { "bbox": [ 105, 237, 505, 250 ], "spans": [ { "bbox": [ 105, 237, 316, 250 ], "score": 1.0, "content": "Figure 3: Performance metrics for graph layouts:", "type": "text" }, { "bbox": [ 317, 237, 339, 248 ], "score": 0.34, "content": "k \\mathbf { N N }", "type": "inline_equation" }, { "bbox": [ 339, 237, 384, 250 ], "score": 1.0, "content": "recall and", "type": "text" }, { "bbox": [ 385, 238, 407, 248 ], "score": 0.67, "content": "k \\mathbf { N N }", "type": "inline_equation" }, { "bbox": [ 407, 237, 505, 250 ], "score": 1.0, "content": "accuracy. Datasets are", "type": "text" } ], "index": 3 }, { "bbox": [ 106, 249, 474, 261 ], "spans": [ { "bbox": [ 106, 249, 474, 261 ], "score": 1.0, "content": "ordered by the increasing sample size. See Figures 2 and A.3 for the corresponding layouts.", "type": "text" } ], "index": 4 } ], "index": 3.5 } ], "index": 2.25 }, { "type": "text", "bbox": [ 106, 278, 504, 300 ], "lines": [], "index": 5.5, "bbox_fs": [ 105, 278, 505, 302 ], "lines_deleted": true }, { "type": "title", "bbox": [ 107, 316, 311, 329 ], "lines": [ { "bbox": [ 105, 316, 311, 330 ], "spans": [ { "bbox": [ 105, 316, 275, 330 ], "score": 1.0, "content": "5 GRAPH LAYOUTS VIA GRAPH", "type": "text" }, { "bbox": [ 276, 317, 281, 327 ], "score": 0.59, "content": "t", "type": "inline_equation" }, { "bbox": [ 282, 316, 311, 330 ], "score": 1.0, "content": "-SNE", "type": "text" } ], "index": 7 } ], "index": 7 }, { "type": "text", "bbox": [ 106, 340, 505, 385 ], "lines": [ { "bbox": [ 105, 340, 505, 354 ], "spans": [ { "bbox": [ 105, 340, 124, 354 ], "score": 1.0, "content": "The", "type": "text" }, { "bbox": [ 125, 342, 130, 351 ], "score": 0.68, "content": "t", "type": "inline_equation" }, { "bbox": [ 131, 340, 505, 354 ], "score": 1.0, "content": "-SNE algorithm consists of two steps: first, it computes pairwise affinities between all pairs", "type": "text" } ], "index": 8 }, { "bbox": [ 105, 352, 506, 365 ], "spans": [ { "bbox": [ 105, 352, 202, 365 ], "score": 1.0, "content": "of points based on the", "type": "text" }, { "bbox": [ 202, 352, 224, 362 ], "score": 0.73, "content": "k \\mathbf { N N }", "type": "inline_equation" }, { "bbox": [ 224, 352, 506, 365 ], "score": 1.0, "content": "graph; second, it optimizes the embedding to match these affinities", "type": "text" } ], "index": 9 }, { "bbox": [ 106, 363, 505, 375 ], "spans": [ { "bbox": [ 106, 363, 204, 375 ], "score": 1.0, "content": "(Section 3.1). For graph", "type": "text" }, { "bbox": [ 205, 364, 210, 373 ], "score": 0.63, "content": "t", "type": "inline_equation" }, { "bbox": [ 210, 363, 505, 375 ], "score": 1.0, "content": "-SNE, we replace the first step and obtain the affinity matrix directly from", "type": "text" } ], "index": 10 }, { "bbox": [ 106, 374, 503, 386 ], "spans": [ { "bbox": [ 106, 374, 271, 386 ], "score": 1.0, "content": "the graph adjacency matrix. We then run", "type": "text" }, { "bbox": [ 271, 375, 276, 384 ], "score": 0.69, "content": "t", "type": "inline_equation" }, { "bbox": [ 276, 374, 503, 386 ], "score": 1.0, "content": "-SNE optimization to produce the embedding (Figure 1).", "type": "text" } ], "index": 11 } ], "index": 9.5, "bbox_fs": [ 105, 340, 506, 386 ] }, { "type": "text", "bbox": [ 106, 390, 505, 446 ], "lines": [ { "bbox": [ 106, 390, 506, 403 ], "spans": [ { "bbox": [ 106, 390, 222, 403 ], "score": 1.0, "content": "Given an unweighted graph", "type": "text" }, { "bbox": [ 222, 390, 273, 403 ], "score": 0.94, "content": "G = ( \\nu , \\mathcal { E } )", "type": "inline_equation" }, { "bbox": [ 273, 390, 362, 403 ], "score": 1.0, "content": ", its adjacency matrix", "type": "text" }, { "bbox": [ 363, 391, 373, 401 ], "score": 0.31, "content": "\\mathbf { A }", "type": "inline_equation" }, { "bbox": [ 373, 390, 457, 403 ], "score": 1.0, "content": "is defined such that", "type": "text" }, { "bbox": [ 457, 391, 495, 403 ], "score": 0.93, "content": "A _ { i j } = 1", "type": "inline_equation" }, { "bbox": [ 495, 390, 506, 403 ], "score": 1.0, "content": "if", "type": "text" } ], "index": 12 }, { "bbox": [ 107, 402, 506, 415 ], "spans": [ { "bbox": [ 107, 402, 150, 414 ], "score": 0.92, "content": "( i , j ) \\in \\mathcal { E }", "type": "inline_equation" }, { "bbox": [ 150, 402, 169, 415 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 170, 402, 208, 414 ], "score": 0.93, "content": "A _ { i j } = 0", "type": "inline_equation" }, { "bbox": [ 208, 402, 506, 415 ], "score": 1.0, "content": "otherwise. Since all graphs considered in this study are undirected, the", "type": "text" } ], "index": 13 }, { "bbox": [ 105, 412, 505, 426 ], "spans": [ { "bbox": [ 105, 412, 298, 426 ], "score": 1.0, "content": "adjacency matrix is a binary, symmetric square", "type": "text" }, { "bbox": [ 298, 414, 324, 423 ], "score": 0.9, "content": "n \\times n", "type": "inline_equation" }, { "bbox": [ 324, 412, 505, 426 ], "score": 1.0, "content": "matrix. In order to convert it into an affinity", "type": "text" } ], "index": 14 }, { "bbox": [ 105, 422, 505, 436 ], "spans": [ { "bbox": [ 105, 422, 182, 436 ], "score": 1.0, "content": "matrix suitable for", "type": "text" }, { "bbox": [ 183, 424, 188, 433 ], "score": 0.67, "content": "t", "type": "inline_equation" }, { "bbox": [ 188, 422, 307, 436 ], "score": 1.0, "content": "-SNE, we follow the standard", "type": "text" }, { "bbox": [ 307, 424, 312, 433 ], "score": 0.69, "content": "t", "type": "inline_equation" }, { "bbox": [ 313, 422, 505, 436 ], "score": 1.0, "content": "-SNE’s approach (Section 3.1): divide each row", "type": "text" } ], "index": 15 }, { "bbox": [ 106, 434, 497, 448 ], "spans": [ { "bbox": [ 106, 434, 497, 448 ], "score": 1.0, "content": "by the sum of its elements, then symmetrize the resulting matrix, and then normalize to sum to 1:", "type": "text" } ], "index": 16 } ], "index": 14, "bbox_fs": [ 105, 390, 506, 448 ] }, { "type": "interline_equation", "bbox": [ 213, 448, 396, 482 ], "lines": [ { "bbox": [ 213, 448, 396, 482 ], "spans": [ { "bbox": [ 213, 448, 396, 482 ], "score": 0.9, "content": "\\mathbf { P } = { \\frac { { \\tilde { \\mathbf { A } } } + { \\tilde { \\mathbf { A } } } ^ { \\top } } { 2 n } } , { \\mathrm { ~ w h e r e ~ } } { \\tilde { A } } _ { i j } = A _ { i j } { \\Big / } \\sum _ { k = 1 } ^ { n } A _ { i k } .", "type": "interline_equation", "image_path": "059da263f07d66cdd192e32bb1681c7b7d46a29444d542c16284af5f58184c25.jpg" } ] } ], "index": 17.5, "virtual_lines": [ { "bbox": [ 213, 448, 396, 465.0 ], "spans": [], "index": 17 }, { "bbox": [ 213, 465.0, 396, 482.0 ], "spans": [], "index": 18 } ] }, { "type": "text", "bbox": [ 107, 488, 505, 545 ], "lines": [ { "bbox": [ 105, 488, 506, 502 ], "spans": [ { "bbox": [ 105, 488, 506, 502 ], "score": 1.0, "content": "For optimization, we used the openTSNE library (Policar et al., 2019) with default parameters. It ˇ", "type": "text" } ], "index": 19 }, { "bbox": [ 106, 501, 505, 513 ], "spans": [ { "bbox": [ 106, 501, 505, 513 ], "score": 1.0, "content": "uses Laplacian Eigenmaps (Belkin & Niyogi, 2003) for initialization (Kobak & Linderman, 2021),", "type": "text" } ], "index": 20 }, { "bbox": [ 105, 510, 505, 524 ], "spans": [ { "bbox": [ 105, 510, 230, 524 ], "score": 1.0, "content": "sets the learning rate equal to", "type": "text" }, { "bbox": [ 230, 514, 237, 521 ], "score": 0.74, "content": "n", "type": "inline_equation" }, { "bbox": [ 238, 510, 505, 524 ], "score": 1.0, "content": "to achieve good convergence (Linderman & Steinerberger, 2019;", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 521, 505, 536 ], "spans": [ { "bbox": [ 105, 521, 398, 536 ], "score": 1.0, "content": "Belkina et al., 2019), and employs fast FIt-SNE algorithm that has linear", "type": "text" }, { "bbox": [ 398, 522, 421, 534 ], "score": 0.92, "content": "{ \\mathcal { O } } ( n )", "type": "inline_equation" }, { "bbox": [ 421, 521, 505, 536 ], "score": 1.0, "content": "runtime (Linderman", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 533, 159, 546 ], "spans": [ { "bbox": [ 105, 533, 159, 546 ], "score": 1.0, "content": "et al., 2019).", "type": "text" } ], "index": 23 } ], "index": 21, "bbox_fs": [ 105, 488, 506, 546 ] }, { "type": "text", "bbox": [ 106, 549, 505, 639 ], "lines": [ { "bbox": [ 106, 549, 505, 563 ], "spans": [ { "bbox": [ 106, 549, 188, 563 ], "score": 1.0, "content": "We compared graph", "type": "text" }, { "bbox": [ 189, 551, 193, 560 ], "score": 0.59, "content": "t", "type": "inline_equation" }, { "bbox": [ 194, 549, 505, 563 ], "score": 1.0, "content": "-SNE with three existing graph layout algorithms: FDP (Fruchterman & Rein-", "type": "text" } ], "index": 24 }, { "bbox": [ 105, 561, 506, 574 ], "spans": [ { "bbox": [ 105, 561, 304, 574 ], "score": 1.0, "content": "gold, 1991), DRGraph (Zhu et al., 2020a), and", "type": "text" }, { "bbox": [ 304, 562, 309, 571 ], "score": 0.75, "content": "t", "type": "inline_equation" }, { "bbox": [ 309, 561, 506, 574 ], "score": 1.0, "content": "-FDP (Zhong et al., 2023). We chose FDP be-", "type": "text" } ], "index": 25 }, { "bbox": [ 105, 572, 505, 585 ], "spans": [ { "bbox": [ 105, 572, 505, 585 ], "score": 1.0, "content": "cause it is the default layout algorithm in a popular NetworkX package (Hagberg et al., 2008).", "type": "text" } ], "index": 26 }, { "bbox": [ 106, 583, 505, 595 ], "spans": [ { "bbox": [ 106, 583, 199, 595 ], "score": 1.0, "content": "Two other algorithms,", "type": "text" }, { "bbox": [ 199, 584, 204, 593 ], "score": 0.71, "content": "t", "type": "inline_equation" }, { "bbox": [ 204, 583, 505, 595 ], "score": 1.0, "content": "-FDP and DRGraph, are recent and can be considered state-of-the-art (we", "type": "text" } ], "index": 27 }, { "bbox": [ 106, 594, 504, 606 ], "spans": [ { "bbox": [ 106, 594, 504, 606 ], "score": 1.0, "content": "did not use tsNET (Kruiger et al., 2017) for benchmarking, because it cannot embed large graphs", "type": "text" } ], "index": 28 }, { "bbox": [ 105, 605, 506, 617 ], "spans": [ { "bbox": [ 105, 605, 506, 617 ], "score": 1.0, "content": "and is outperformed by its successor DRGraph). We used the NetworkX implementation of FDP", "type": "text" } ], "index": 29 }, { "bbox": [ 106, 616, 505, 628 ], "spans": [ { "bbox": [ 106, 616, 505, 628 ], "score": 1.0, "content": "(networkx.drawing.layout.spring layout) and the original implementations of both", "type": "text" } ], "index": 30 }, { "bbox": [ 106, 626, 349, 640 ], "spans": [ { "bbox": [ 106, 628, 111, 637 ], "score": 0.76, "content": "t", "type": "inline_equation" }, { "bbox": [ 112, 626, 349, 640 ], "score": 1.0, "content": "-FDP and DRGraph, all with default parameters (Figure 2).", "type": "text" } ], "index": 31 } ], "index": 27.5, "bbox_fs": [ 105, 549, 506, 640 ] }, { "type": "text", "bbox": [ 106, 643, 505, 732 ], "lines": [ { "bbox": [ 106, 643, 505, 657 ], "spans": [ { "bbox": [ 106, 643, 191, 657 ], "score": 1.0, "content": "We found that graph", "type": "text" }, { "bbox": [ 192, 645, 196, 654 ], "score": 0.63, "content": "t", "type": "inline_equation" }, { "bbox": [ 197, 643, 457, 657 ], "score": 1.0, "content": "-SNE consistently outperformed all competitors in terms of both", "type": "text" }, { "bbox": [ 458, 644, 479, 654 ], "score": 0.36, "content": "k \\mathbf { N N }", "type": "inline_equation" }, { "bbox": [ 480, 643, 505, 657 ], "score": 1.0, "content": "recall", "type": "text" } ], "index": 32 }, { "bbox": [ 106, 655, 505, 666 ], "spans": [ { "bbox": [ 106, 655, 124, 666 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 124, 655, 146, 666 ], "score": 0.71, "content": "k \\mathbf { N N }", "type": "inline_equation" }, { "bbox": [ 147, 655, 505, 666 ], "score": 1.0, "content": "accuracy (Figure 3): it showed the highest values on all datasets, 12 out of 12 times.", "type": "text" } ], "index": 33 }, { "bbox": [ 105, 666, 506, 679 ], "spans": [ { "bbox": [ 105, 666, 506, 679 ], "score": 1.0, "content": "In agreement with the original results of Zhu et al. (2020a) and Zhong et al. (2023), we saw that", "type": "text" } ], "index": 34 }, { "bbox": [ 106, 677, 505, 689 ], "spans": [ { "bbox": [ 106, 677, 164, 689 ], "score": 1.0, "content": "DRGraph and", "type": "text" }, { "bbox": [ 164, 677, 169, 687 ], "score": 0.77, "content": "t", "type": "inline_equation" }, { "bbox": [ 169, 677, 376, 689 ], "score": 1.0, "content": "-FDP outperformed FDP in both metrics. Our graph", "type": "text" }, { "bbox": [ 376, 678, 381, 687 ], "score": 0.6, "content": "t", "type": "inline_equation" }, { "bbox": [ 381, 677, 505, 689 ], "score": 1.0, "content": "-SNE showed further improve-", "type": "text" } ], "index": 35 }, { "bbox": [ 105, 687, 505, 701 ], "spans": [ { "bbox": [ 105, 687, 269, 701 ], "score": 1.0, "content": "ment, and it was substantial: in terms of", "type": "text" }, { "bbox": [ 270, 688, 291, 698 ], "score": 0.57, "content": "k \\mathbf { N N }", "type": "inline_equation" }, { "bbox": [ 291, 687, 344, 701 ], "score": 1.0, "content": "recall, graph", "type": "text" }, { "bbox": [ 344, 689, 349, 698 ], "score": 0.69, "content": "t", "type": "inline_equation" }, { "bbox": [ 350, 687, 505, 701 ], "score": 1.0, "content": "-SNE improved on the best competitor", "type": "text" } ], "index": 36 }, { "bbox": [ 105, 698, 505, 713 ], "spans": [ { "bbox": [ 105, 698, 317, 713 ], "score": 1.0, "content": "on average by 18.2 percentage points, and in terms of", "type": "text" }, { "bbox": [ 317, 699, 339, 709 ], "score": 0.35, "content": "k \\mathbf { N N }", "type": "inline_equation" }, { "bbox": [ 339, 698, 505, 713 ], "score": 1.0, "content": "accuracy — on average by 6.7 percentage", "type": "text" } ], "index": 37 }, { "bbox": [ 105, 710, 505, 722 ], "spans": [ { "bbox": [ 105, 710, 505, 722 ], "score": 1.0, "content": "points. The improvement was particularly strong for the largest graph (ARX), where performance", "type": "text" } ], "index": 38 }, { "bbox": [ 106, 721, 264, 733 ], "spans": [ { "bbox": [ 106, 721, 264, 733 ], "score": 1.0, "content": "of other methods strongly deteriorated.", "type": "text" } ], "index": 39 } ], "index": 35.5, "bbox_fs": [ 105, 643, 506, 733 ] } ] }, { "preproc_blocks": [ { "type": "image", "bbox": [ 108, 81, 504, 225 ], "blocks": [ { "type": "image_body", "bbox": [ 108, 81, 504, 225 ], "group_id": 0, "lines": [ { "bbox": [ 108, 81, 504, 225 ], "spans": [ { "bbox": [ 108, 81, 504, 225 ], "score": 0.968, "type": "image", "image_path": "472c51903edfd06a69067c0b12783fac6303fa85e8eb34d174add67ce3674382.jpg" } ] } ], "index": 1, "virtual_lines": [ { "bbox": [ 108, 81, 504, 129.0 ], "spans": [], "index": 0 }, { "bbox": [ 108, 129.0, 504, 177.0 ], "spans": [], "index": 1 }, { "bbox": [ 108, 177.0, 504, 225.0 ], "spans": [], "index": 2 } ] }, { "type": "image_caption", "bbox": [ 106, 237, 505, 271 ], "group_id": 0, "lines": [ { "bbox": [ 105, 237, 505, 250 ], "spans": [ { "bbox": [ 105, 237, 378, 250 ], "score": 1.0, "content": "Figure 4: Performance metrics for graph CNE compared to graph", "type": "text" }, { "bbox": [ 378, 239, 383, 248 ], "score": 0.69, "content": "t", "type": "inline_equation" }, { "bbox": [ 383, 237, 409, 250 ], "score": 1.0, "content": "-SNE:", "type": "text" }, { "bbox": [ 410, 237, 432, 248 ], "score": 0.48, "content": "k \\mathbf { N N }", "type": "inline_equation" }, { "bbox": [ 432, 237, 460, 250 ], "score": 1.0, "content": "recall,", "type": "text" }, { "bbox": [ 460, 237, 482, 248 ], "score": 0.63, "content": "k \\mathbf { N N }", "type": "inline_equation" }, { "bbox": [ 483, 237, 505, 250 ], "score": 1.0, "content": "clas-", "type": "text" } ], "index": 3 }, { "bbox": [ 106, 249, 505, 261 ], "spans": [ { "bbox": [ 106, 249, 505, 261 ], "score": 1.0, "content": "sification accuracy, and linear accuracy. Shading shows standard deviation over five CNE runs.", "type": "text" } ], "index": 4 }, { "bbox": [ 106, 259, 312, 272 ], "spans": [ { "bbox": [ 106, 259, 312, 272 ], "score": 1.0, "content": "Datasets are ordered by the increasing sample size.", "type": "text" } ], "index": 5 } ], "index": 4 } ], "index": 2.5 }, { "type": "text", "bbox": [ 107, 297, 505, 331 ], "lines": [ { "bbox": [ 105, 295, 505, 311 ], "spans": [ { "bbox": [ 105, 295, 289, 311 ], "score": 1.0, "content": "Visually, the embeddings produced by graph", "type": "text" }, { "bbox": [ 289, 299, 294, 308 ], "score": 0.64, "content": "t", "type": "inline_equation" }, { "bbox": [ 295, 295, 505, 311 ], "score": 1.0, "content": "-SNE looked similar to DRGraph embeddings (Fig-", "type": "text" } ], "index": 6 }, { "bbox": [ 105, 308, 505, 321 ], "spans": [ { "bbox": [ 105, 308, 457, 321 ], "score": 1.0, "content": "ures 2 and A.3), but showed richer within-class structure, in agreement with the higher", "type": "text" }, { "bbox": [ 458, 309, 479, 319 ], "score": 0.59, "content": "k \\mathbf { N N }", "type": "inline_equation" }, { "bbox": [ 480, 308, 505, 321 ], "score": 1.0, "content": "recall", "type": "text" } ], "index": 7 }, { "bbox": [ 105, 318, 138, 333 ], "spans": [ { "bbox": [ 105, 318, 138, 333 ], "score": 1.0, "content": "values.", "type": "text" } ], "index": 8 } ], "index": 7 }, { "type": "text", "bbox": [ 107, 336, 505, 405 ], "lines": [ { "bbox": [ 106, 336, 504, 349 ], "spans": [ { "bbox": [ 106, 336, 504, 349 ], "score": 1.0, "content": "We have also experimented with an alternative way to convert the adjacency matrix into the affinity", "type": "text" } ], "index": 9 }, { "bbox": [ 104, 345, 506, 362 ], "spans": [ { "bbox": [ 104, 345, 327, 362 ], "score": 1.0, "content": "matrix: namely, to divide A by the sum of its elements:", "type": "text" }, { "bbox": [ 327, 347, 399, 361 ], "score": 0.92, "content": "\\begin{array} { r } { \\mathbf { P } = \\mathbf { A } / \\sum _ { i j } \\hat { A _ { i j } } } \\end{array}", "type": "inline_equation" }, { "bbox": [ 399, 345, 506, 362 ], "score": 1.0, "content": ". This approach resulted in", "type": "text" } ], "index": 10 }, { "bbox": [ 105, 359, 505, 372 ], "spans": [ { "bbox": [ 105, 359, 137, 372 ], "score": 1.0, "content": "similar", "type": "text" }, { "bbox": [ 137, 360, 159, 370 ], "score": 0.68, "content": "k \\mathbf { N N }", "type": "inline_equation" }, { "bbox": [ 159, 359, 202, 372 ], "score": 1.0, "content": "recall and", "type": "text" }, { "bbox": [ 202, 360, 224, 370 ], "score": 0.73, "content": "k \\mathbf { N N }", "type": "inline_equation" }, { "bbox": [ 224, 359, 505, 372 ], "score": 1.0, "content": "accuracy values, but gave visually unpleasing embeddings, with low-", "type": "text" } ], "index": 11 }, { "bbox": [ 105, 371, 505, 384 ], "spans": [ { "bbox": [ 105, 371, 505, 384 ], "score": 1.0, "content": "degree nodes pushed out to the periphery (Figure A.2). Furthermore, we experimented with various", "type": "text" } ], "index": 12 }, { "bbox": [ 105, 381, 506, 395 ], "spans": [ { "bbox": [ 105, 381, 506, 395 ], "score": 1.0, "content": "initialization schemes, but found that on our graphs, random initialization performed very similar to", "type": "text" } ], "index": 13 }, { "bbox": [ 105, 393, 295, 405 ], "spans": [ { "bbox": [ 105, 393, 295, 405 ], "score": 1.0, "content": "the default Laplacian Eigenmaps initialization.", "type": "text" } ], "index": 14 } ], "index": 11.5 }, { "type": "title", "bbox": [ 108, 428, 453, 440 ], "lines": [ { "bbox": [ 105, 428, 455, 441 ], "spans": [ { "bbox": [ 105, 428, 455, 441 ], "score": 1.0, "content": "6 NODE-LEVEL GRAPH CONTRASTIVE LEARNING VIA GRAPH CNE", "type": "text" } ], "index": 15 } ], "index": 15 }, { "type": "text", "bbox": [ 107, 456, 505, 501 ], "lines": [ { "bbox": [ 105, 456, 505, 469 ], "spans": [ { "bbox": [ 105, 456, 149, 469 ], "score": 1.0, "content": "Similar to", "type": "text" }, { "bbox": [ 150, 457, 155, 466 ], "score": 0.67, "content": "t", "type": "inline_equation" }, { "bbox": [ 155, 456, 429, 469 ], "score": 1.0, "content": "-SNE, the CNE algorithm consists of two steps. First, it builds the", "type": "text" }, { "bbox": [ 430, 457, 452, 467 ], "score": 0.59, "content": "k \\mathbf { N N }", "type": "inline_equation" }, { "bbox": [ 452, 456, 505, 469 ], "score": 1.0, "content": "graph of the", "type": "text" } ], "index": 16 }, { "bbox": [ 105, 467, 505, 480 ], "spans": [ { "bbox": [ 105, 467, 505, 480 ], "score": 1.0, "content": "data. Second, it optimizes the embedding (in our case, parametric embedding) using a contrastive", "type": "text" } ], "index": 17 }, { "bbox": [ 105, 478, 505, 491 ], "spans": [ { "bbox": [ 105, 478, 505, 491 ], "score": 1.0, "content": "loss function such as InfoNCE to make neighbors be close in the embedding (Section 3.2). For graph", "type": "text" } ], "index": 18 }, { "bbox": [ 105, 488, 374, 502 ], "spans": [ { "bbox": [ 105, 488, 374, 502 ], "score": 1.0, "content": "CNE, we omit the first step and provide the graph to CNE directly.", "type": "text" } ], "index": 19 } ], "index": 17.5 }, { "type": "text", "bbox": [ 106, 506, 505, 627 ], "lines": [ { "bbox": [ 106, 506, 506, 519 ], "spans": [ { "bbox": [ 106, 506, 398, 519 ], "score": 1.0, "content": "We used parametric CNE models, setting the output dimensionality to", "type": "text" }, { "bbox": [ 399, 506, 428, 516 ], "score": 0.89, "content": "d = 2", "type": "inline_equation" }, { "bbox": [ 428, 506, 448, 519 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 448, 506, 487, 517 ], "score": 0.89, "content": "d = 1 2 8", "type": "inline_equation" }, { "bbox": [ 487, 506, 506, 519 ], "score": 1.0, "content": ". In", "type": "text" } ], "index": 20 }, { "bbox": [ 106, 518, 505, 529 ], "spans": [ { "bbox": [ 106, 518, 505, 529 ], "score": 1.0, "content": "both cases we used a fully-connected network (MLP), as is default in CNE, with the number of", "type": "text" } ], "index": 21 }, { "bbox": [ 104, 528, 505, 540 ], "spans": [ { "bbox": [ 104, 528, 199, 540 ], "score": 1.0, "content": "neurons in each layer", "type": "text" }, { "bbox": [ 199, 528, 340, 539 ], "score": 0.92, "content": "D \\ \\to \\ 1 0 0 \\ \\to \\ 1 0 0 \\ \\to \\ 1 0 0 \\ \\to \\ d", "type": "inline_equation" }, { "bbox": [ 341, 528, 374, 540 ], "score": 1.0, "content": ", where", "type": "text" }, { "bbox": [ 374, 529, 384, 538 ], "score": 0.8, "content": "D", "type": "inline_equation" }, { "bbox": [ 384, 528, 505, 540 ], "score": 1.0, "content": "is the number of input node", "type": "text" } ], "index": 22 }, { "bbox": [ 106, 540, 504, 551 ], "spans": [ { "bbox": [ 106, 540, 504, 551 ], "score": 1.0, "content": "features (Table 1). For both dimensionalities we used the InfoNCE loss. Following Damrich et al.", "type": "text" } ], "index": 23 }, { "bbox": [ 105, 549, 505, 563 ], "spans": [ { "bbox": [ 105, 549, 405, 563 ], "score": 1.0, "content": "(2022), we used the cosine distance and the Gaussian similarity kernel for", "type": "text" }, { "bbox": [ 405, 550, 441, 560 ], "score": 0.9, "content": "d = 1 2 8", "type": "inline_equation" }, { "bbox": [ 441, 549, 505, 563 ], "score": 1.0, "content": ", mimicking the", "type": "text" } ], "index": 24 }, { "bbox": [ 105, 561, 505, 574 ], "spans": [ { "bbox": [ 105, 561, 505, 574 ], "score": 1.0, "content": "standard SimCLR setup (Chen et al., 2020), and the Euclidean distance and the Cauchy similarity", "type": "text" } ], "index": 25 }, { "bbox": [ 105, 572, 506, 585 ], "spans": [ { "bbox": [ 105, 572, 149, 585 ], "score": 1.0, "content": "kernel for", "type": "text" }, { "bbox": [ 149, 572, 175, 582 ], "score": 0.89, "content": "d = 2", "type": "inline_equation" }, { "bbox": [ 176, 572, 277, 585 ], "score": 1.0, "content": ", mimicking the standard", "type": "text" }, { "bbox": [ 277, 573, 282, 582 ], "score": 0.7, "content": "t", "type": "inline_equation" }, { "bbox": [ 282, 572, 506, 585 ], "score": 1.0, "content": "-SNE setup. We set the number of negative samples to", "type": "text" } ], "index": 26 }, { "bbox": [ 106, 582, 505, 596 ], "spans": [ { "bbox": [ 106, 582, 410, 596 ], "score": 1.0, "content": "100 (increasing it from the default 5 improved the results), and batch size to", "type": "text" }, { "bbox": [ 411, 583, 490, 595 ], "score": 0.84, "content": "\\operatorname* { m i n } \\{ \\bar { 1 } 0 2 4 , | \\mathcal { V } | / \\bar { 1 } 0 \\}", "type": "inline_equation" }, { "bbox": [ 490, 582, 505, 596 ], "score": 1.0, "content": "(in", "type": "text" } ], "index": 27 }, { "bbox": [ 105, 594, 505, 607 ], "spans": [ { "bbox": [ 105, 594, 505, 607 ], "score": 1.0, "content": "pilot experiments we noticed that small graphs required smaller batch sizes for good convergence).", "type": "text" } ], "index": 28 }, { "bbox": [ 105, 604, 506, 618 ], "spans": [ { "bbox": [ 105, 604, 506, 618 ], "score": 1.0, "content": "The number of epochs was set to 100. Optimization parameters were left at default values: Adam", "type": "text" } ], "index": 29 }, { "bbox": [ 105, 616, 338, 628 ], "spans": [ { "bbox": [ 105, 616, 338, 628 ], "score": 1.0, "content": "optimizer (Kingma & Ba, 2015) with learning rate 0.001.", "type": "text" } ], "index": 30 } ], "index": 25 }, { "type": "text", "bbox": [ 106, 632, 505, 732 ], "lines": [ { "bbox": [ 106, 633, 505, 645 ], "spans": [ { "bbox": [ 106, 633, 189, 645 ], "score": 1.0, "content": "Compared to graph", "type": "text" }, { "bbox": [ 190, 634, 195, 643 ], "score": 0.66, "content": "t", "type": "inline_equation" }, { "bbox": [ 195, 633, 317, 645 ], "score": 1.0, "content": "-SNE, graph CNE, with both", "type": "text" }, { "bbox": [ 317, 633, 347, 643 ], "score": 0.9, "content": "d = 2", "type": "inline_equation" }, { "bbox": [ 348, 633, 367, 645 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 367, 633, 407, 644 ], "score": 0.9, "content": "d = 1 2 8", "type": "inline_equation" }, { "bbox": [ 407, 633, 456, 645 ], "score": 1.0, "content": ", had lower", "type": "text" }, { "bbox": [ 457, 633, 478, 643 ], "score": 0.77, "content": "k \\mathbf { N N }", "type": "inline_equation" }, { "bbox": [ 479, 633, 505, 645 ], "score": 1.0, "content": "recall", "type": "text" } ], "index": 31 }, { "bbox": [ 106, 643, 505, 656 ], "spans": [ { "bbox": [ 106, 643, 458, 656 ], "score": 1.0, "content": "(Figure 4). This is likely because graph CNE had to use node features, whereas graph", "type": "text" }, { "bbox": [ 459, 645, 463, 654 ], "score": 0.7, "content": "t", "type": "inline_equation" }, { "bbox": [ 464, 643, 505, 656 ], "score": 1.0, "content": "-SNE was", "type": "text" } ], "index": 32 }, { "bbox": [ 105, 655, 506, 668 ], "spans": [ { "bbox": [ 105, 655, 506, 668 ], "score": 1.0, "content": "unconstrained by them and optimized graph neighborhood preservation directly. At the same time,", "type": "text" } ], "index": 33 }, { "bbox": [ 106, 664, 505, 679 ], "spans": [ { "bbox": [ 106, 666, 128, 676 ], "score": 0.56, "content": "k \\mathbf { N N }", "type": "inline_equation" }, { "bbox": [ 128, 664, 505, 679 ], "score": 1.0, "content": "accuracy was very similar (Figure 4) on all datasets, apart from the ARX dataset. The compar-", "type": "text" } ], "index": 34 }, { "bbox": [ 105, 677, 505, 690 ], "spans": [ { "bbox": [ 105, 677, 505, 690 ], "score": 1.0, "content": "atively poor performance of graph CNE on the ARX dataset was likely due to ARX feature space", "type": "text" } ], "index": 35 }, { "bbox": [ 106, 688, 505, 700 ], "spans": [ { "bbox": [ 106, 688, 329, 700 ], "score": 1.0, "content": "showing weak class separation (Table 2); whereas graph", "type": "text" }, { "bbox": [ 330, 689, 334, 698 ], "score": 0.67, "content": "t", "type": "inline_equation" }, { "bbox": [ 335, 688, 505, 700 ], "score": 1.0, "content": "-SNE does not use node features and hence", "type": "text" } ], "index": 36 }, { "bbox": [ 105, 698, 505, 711 ], "spans": [ { "bbox": [ 105, 698, 505, 711 ], "score": 1.0, "content": "is not influenced by the feature quality. Visually, two-dimensional graph CNE embeddings looked", "type": "text" } ], "index": 37 }, { "bbox": [ 105, 709, 506, 723 ], "spans": [ { "bbox": [ 105, 709, 194, 723 ], "score": 1.0, "content": "very similar to graph", "type": "text" }, { "bbox": [ 195, 711, 199, 720 ], "score": 0.55, "content": "t", "type": "inline_equation" }, { "bbox": [ 200, 709, 506, 723 ], "score": 1.0, "content": "-SNE embeddings (Figure 1), even though the former were parametric and", "type": "text" } ], "index": 38 }, { "bbox": [ 106, 721, 231, 733 ], "spans": [ { "bbox": [ 106, 721, 231, 733 ], "score": 1.0, "content": "the latter were non-parametric.", "type": "text" } ], "index": 39 } ], "index": 35 } ], "page_idx": 6, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 107, 27, 308, 37 ], "lines": [ { "bbox": [ 107, 26, 308, 38 ], "spans": [ { "bbox": [ 107, 26, 308, 38 ], "score": 1.0, "content": "Under review as a conference paper at ICLR 2024", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 303, 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": [ 108, 81, 504, 225 ], "blocks": [ { "type": "image_body", "bbox": [ 108, 81, 504, 225 ], "group_id": 0, "lines": [ { "bbox": [ 108, 81, 504, 225 ], "spans": [ { "bbox": [ 108, 81, 504, 225 ], "score": 0.968, "type": "image", "image_path": "472c51903edfd06a69067c0b12783fac6303fa85e8eb34d174add67ce3674382.jpg" } ] } ], "index": 1, "virtual_lines": [ { "bbox": [ 108, 81, 504, 129.0 ], "spans": [], "index": 0 }, { "bbox": [ 108, 129.0, 504, 177.0 ], "spans": [], "index": 1 }, { "bbox": [ 108, 177.0, 504, 225.0 ], "spans": [], "index": 2 } ] }, { "type": "image_caption", "bbox": [ 106, 237, 505, 271 ], "group_id": 0, "lines": [ { "bbox": [ 105, 237, 505, 250 ], "spans": [ { "bbox": [ 105, 237, 378, 250 ], "score": 1.0, "content": "Figure 4: Performance metrics for graph CNE compared to graph", "type": "text" }, { "bbox": [ 378, 239, 383, 248 ], "score": 0.69, "content": "t", "type": "inline_equation" }, { "bbox": [ 383, 237, 409, 250 ], "score": 1.0, "content": "-SNE:", "type": "text" }, { "bbox": [ 410, 237, 432, 248 ], "score": 0.48, "content": "k \\mathbf { N N }", "type": "inline_equation" }, { "bbox": [ 432, 237, 460, 250 ], "score": 1.0, "content": "recall,", "type": "text" }, { "bbox": [ 460, 237, 482, 248 ], "score": 0.63, "content": "k \\mathbf { N N }", "type": "inline_equation" }, { "bbox": [ 483, 237, 505, 250 ], "score": 1.0, "content": "clas-", "type": "text" } ], "index": 3 }, { "bbox": [ 106, 249, 505, 261 ], "spans": [ { "bbox": [ 106, 249, 505, 261 ], "score": 1.0, "content": "sification accuracy, and linear accuracy. 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This approach resulted in", "type": "text" } ], "index": 10 }, { "bbox": [ 105, 359, 505, 372 ], "spans": [ { "bbox": [ 105, 359, 137, 372 ], "score": 1.0, "content": "similar", "type": "text" }, { "bbox": [ 137, 360, 159, 370 ], "score": 0.68, "content": "k \\mathbf { N N }", "type": "inline_equation" }, { "bbox": [ 159, 359, 202, 372 ], "score": 1.0, "content": "recall and", "type": "text" }, { "bbox": [ 202, 360, 224, 370 ], "score": 0.73, "content": "k \\mathbf { N N }", "type": "inline_equation" }, { "bbox": [ 224, 359, 505, 372 ], "score": 1.0, "content": "accuracy values, but gave visually unpleasing embeddings, with low-", "type": "text" } ], "index": 11 }, { "bbox": [ 105, 371, 505, 384 ], "spans": [ { "bbox": [ 105, 371, 505, 384 ], "score": 1.0, "content": "degree nodes pushed out to the periphery (Figure A.2). Furthermore, we experimented with various", "type": "text" } ], "index": 12 }, { "bbox": [ 105, 381, 506, 395 ], "spans": [ { "bbox": [ 105, 381, 506, 395 ], "score": 1.0, "content": "initialization schemes, but found that on our graphs, random initialization performed very similar to", "type": "text" } ], "index": 13 }, { "bbox": [ 105, 393, 295, 405 ], "spans": [ { "bbox": [ 105, 393, 295, 405 ], "score": 1.0, "content": "the default Laplacian Eigenmaps initialization.", "type": "text" } ], "index": 14 } ], "index": 11.5, "bbox_fs": [ 104, 336, 506, 405 ] }, { "type": "title", "bbox": [ 108, 428, 453, 440 ], "lines": [ { "bbox": [ 105, 428, 455, 441 ], "spans": [ { "bbox": [ 105, 428, 455, 441 ], "score": 1.0, "content": "6 NODE-LEVEL GRAPH CONTRASTIVE LEARNING VIA GRAPH CNE", "type": "text" } ], "index": 15 } ], "index": 15 }, { "type": "text", "bbox": [ 107, 456, 505, 501 ], "lines": [ { "bbox": [ 105, 456, 505, 469 ], "spans": [ { "bbox": [ 105, 456, 149, 469 ], "score": 1.0, "content": "Similar to", "type": "text" }, { "bbox": [ 150, 457, 155, 466 ], "score": 0.67, "content": "t", "type": "inline_equation" }, { "bbox": [ 155, 456, 429, 469 ], "score": 1.0, "content": "-SNE, the CNE algorithm consists of two steps. First, it builds the", "type": "text" }, { "bbox": [ 430, 457, 452, 467 ], "score": 0.59, "content": "k \\mathbf { N N }", "type": "inline_equation" }, { "bbox": [ 452, 456, 505, 469 ], "score": 1.0, "content": "graph of the", "type": "text" } ], "index": 16 }, { "bbox": [ 105, 467, 505, 480 ], "spans": [ { "bbox": [ 105, 467, 505, 480 ], "score": 1.0, "content": "data. Second, it optimizes the embedding (in our case, parametric embedding) using a contrastive", "type": "text" } ], "index": 17 }, { "bbox": [ 105, 478, 505, 491 ], "spans": [ { "bbox": [ 105, 478, 505, 491 ], "score": 1.0, "content": "loss function such as InfoNCE to make neighbors be close in the embedding (Section 3.2). For graph", "type": "text" } ], "index": 18 }, { "bbox": [ 105, 488, 374, 502 ], "spans": [ { "bbox": [ 105, 488, 374, 502 ], "score": 1.0, "content": "CNE, we omit the first step and provide the graph to CNE directly.", "type": "text" } ], "index": 19 } ], "index": 17.5, "bbox_fs": [ 105, 456, 505, 502 ] }, { "type": "text", "bbox": [ 106, 506, 505, 627 ], "lines": [ { "bbox": [ 106, 506, 506, 519 ], "spans": [ { "bbox": [ 106, 506, 398, 519 ], "score": 1.0, "content": "We used parametric CNE models, setting the output dimensionality to", "type": "text" }, { "bbox": [ 399, 506, 428, 516 ], "score": 0.89, "content": "d = 2", "type": "inline_equation" }, { "bbox": [ 428, 506, 448, 519 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 448, 506, 487, 517 ], "score": 0.89, "content": "d = 1 2 8", "type": "inline_equation" }, { "bbox": [ 487, 506, 506, 519 ], "score": 1.0, "content": ". In", "type": "text" } ], "index": 20 }, { "bbox": [ 106, 518, 505, 529 ], "spans": [ { "bbox": [ 106, 518, 505, 529 ], "score": 1.0, "content": "both cases we used a fully-connected network (MLP), as is default in CNE, with the number of", "type": "text" } ], "index": 21 }, { "bbox": [ 104, 528, 505, 540 ], "spans": [ { "bbox": [ 104, 528, 199, 540 ], "score": 1.0, "content": "neurons in each layer", "type": "text" }, { "bbox": [ 199, 528, 340, 539 ], "score": 0.92, "content": "D \\ \\to \\ 1 0 0 \\ \\to \\ 1 0 0 \\ \\to \\ 1 0 0 \\ \\to \\ d", "type": "inline_equation" }, { "bbox": [ 341, 528, 374, 540 ], "score": 1.0, "content": ", where", "type": "text" }, { "bbox": [ 374, 529, 384, 538 ], "score": 0.8, "content": "D", "type": "inline_equation" }, { "bbox": [ 384, 528, 505, 540 ], "score": 1.0, "content": "is the number of input node", "type": "text" } ], "index": 22 }, { "bbox": [ 106, 540, 504, 551 ], "spans": [ { "bbox": [ 106, 540, 504, 551 ], "score": 1.0, "content": "features (Table 1). For both dimensionalities we used the InfoNCE loss. Following Damrich et al.", "type": "text" } ], "index": 23 }, { "bbox": [ 105, 549, 505, 563 ], "spans": [ { "bbox": [ 105, 549, 405, 563 ], "score": 1.0, "content": "(2022), we used the cosine distance and the Gaussian similarity kernel for", "type": "text" }, { "bbox": [ 405, 550, 441, 560 ], "score": 0.9, "content": "d = 1 2 8", "type": "inline_equation" }, { "bbox": [ 441, 549, 505, 563 ], "score": 1.0, "content": ", mimicking the", "type": "text" } ], "index": 24 }, { "bbox": [ 105, 561, 505, 574 ], "spans": [ { "bbox": [ 105, 561, 505, 574 ], "score": 1.0, "content": "standard SimCLR setup (Chen et al., 2020), and the Euclidean distance and the Cauchy similarity", "type": "text" } ], "index": 25 }, { "bbox": [ 105, 572, 506, 585 ], "spans": [ { "bbox": [ 105, 572, 149, 585 ], "score": 1.0, "content": "kernel for", "type": "text" }, { "bbox": [ 149, 572, 175, 582 ], "score": 0.89, "content": "d = 2", "type": "inline_equation" }, { "bbox": [ 176, 572, 277, 585 ], "score": 1.0, "content": ", mimicking the standard", "type": "text" }, { "bbox": [ 277, 573, 282, 582 ], "score": 0.7, "content": "t", "type": "inline_equation" }, { "bbox": [ 282, 572, 506, 585 ], "score": 1.0, "content": "-SNE setup. We set the number of negative samples to", "type": "text" } ], "index": 26 }, { "bbox": [ 106, 582, 505, 596 ], "spans": [ { "bbox": [ 106, 582, 410, 596 ], "score": 1.0, "content": "100 (increasing it from the default 5 improved the results), and batch size to", "type": "text" }, { "bbox": [ 411, 583, 490, 595 ], "score": 0.84, "content": "\\operatorname* { m i n } \\{ \\bar { 1 } 0 2 4 , | \\mathcal { V } | / \\bar { 1 } 0 \\}", "type": "inline_equation" }, { "bbox": [ 490, 582, 505, 596 ], "score": 1.0, "content": "(in", "type": "text" } ], "index": 27 }, { "bbox": [ 105, 594, 505, 607 ], "spans": [ { "bbox": [ 105, 594, 505, 607 ], "score": 1.0, "content": "pilot experiments we noticed that small graphs required smaller batch sizes for good convergence).", "type": "text" } ], "index": 28 }, { "bbox": [ 105, 604, 506, 618 ], "spans": [ { "bbox": [ 105, 604, 506, 618 ], "score": 1.0, "content": "The number of epochs was set to 100. Optimization parameters were left at default values: Adam", "type": "text" } ], "index": 29 }, { "bbox": [ 105, 616, 338, 628 ], "spans": [ { "bbox": [ 105, 616, 338, 628 ], "score": 1.0, "content": "optimizer (Kingma & Ba, 2015) with learning rate 0.001.", "type": "text" } ], "index": 30 } ], "index": 25, "bbox_fs": [ 104, 506, 506, 628 ] }, { "type": "text", "bbox": [ 106, 632, 505, 732 ], "lines": [ { "bbox": [ 106, 633, 505, 645 ], "spans": [ { "bbox": [ 106, 633, 189, 645 ], "score": 1.0, "content": "Compared to graph", "type": "text" }, { "bbox": [ 190, 634, 195, 643 ], "score": 0.66, "content": "t", "type": "inline_equation" }, { "bbox": [ 195, 633, 317, 645 ], "score": 1.0, "content": "-SNE, graph CNE, with both", "type": "text" }, { "bbox": [ 317, 633, 347, 643 ], "score": 0.9, "content": "d = 2", "type": "inline_equation" }, { "bbox": [ 348, 633, 367, 645 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 367, 633, 407, 644 ], "score": 0.9, "content": "d = 1 2 8", "type": "inline_equation" }, { "bbox": [ 407, 633, 456, 645 ], "score": 1.0, "content": ", had lower", "type": "text" }, { "bbox": [ 457, 633, 478, 643 ], "score": 0.77, "content": "k \\mathbf { N N }", "type": "inline_equation" }, { "bbox": [ 479, 633, 505, 645 ], "score": 1.0, "content": "recall", "type": "text" } ], "index": 31 }, { "bbox": [ 106, 643, 505, 656 ], "spans": [ { "bbox": [ 106, 643, 458, 656 ], "score": 1.0, "content": "(Figure 4). This is likely because graph CNE had to use node features, whereas graph", "type": "text" }, { "bbox": [ 459, 645, 463, 654 ], "score": 0.7, "content": "t", "type": "inline_equation" }, { "bbox": [ 464, 643, 505, 656 ], "score": 1.0, "content": "-SNE was", "type": "text" } ], "index": 32 }, { "bbox": [ 105, 655, 506, 668 ], "spans": [ { "bbox": [ 105, 655, 506, 668 ], "score": 1.0, "content": "unconstrained by them and optimized graph neighborhood preservation directly. At the same time,", "type": "text" } ], "index": 33 }, { "bbox": [ 106, 664, 505, 679 ], "spans": [ { "bbox": [ 106, 666, 128, 676 ], "score": 0.56, "content": "k \\mathbf { N N }", "type": "inline_equation" }, { "bbox": [ 128, 664, 505, 679 ], "score": 1.0, "content": "accuracy was very similar (Figure 4) on all datasets, apart from the ARX dataset. The compar-", "type": "text" } ], "index": 34 }, { "bbox": [ 105, 677, 505, 690 ], "spans": [ { "bbox": [ 105, 677, 505, 690 ], "score": 1.0, "content": "atively poor performance of graph CNE on the ARX dataset was likely due to ARX feature space", "type": "text" } ], "index": 35 }, { "bbox": [ 106, 688, 505, 700 ], "spans": [ { "bbox": [ 106, 688, 329, 700 ], "score": 1.0, "content": "showing weak class separation (Table 2); whereas graph", "type": "text" }, { "bbox": [ 330, 689, 334, 698 ], "score": 0.67, "content": "t", "type": "inline_equation" }, { "bbox": [ 335, 688, 505, 700 ], "score": 1.0, "content": "-SNE does not use node features and hence", "type": "text" } ], "index": 36 }, { "bbox": [ 105, 698, 505, 711 ], "spans": [ { "bbox": [ 105, 698, 505, 711 ], "score": 1.0, "content": "is not influenced by the feature quality. Visually, two-dimensional graph CNE embeddings looked", "type": "text" } ], "index": 37 }, { "bbox": [ 105, 709, 506, 723 ], "spans": [ { "bbox": [ 105, 709, 194, 723 ], "score": 1.0, "content": "very similar to graph", "type": "text" }, { "bbox": [ 195, 711, 199, 720 ], "score": 0.55, "content": "t", "type": "inline_equation" }, { "bbox": [ 200, 709, 506, 723 ], "score": 1.0, "content": "-SNE embeddings (Figure 1), even though the former were parametric and", "type": "text" } ], "index": 38 }, { "bbox": [ 106, 721, 231, 733 ], "spans": [ { "bbox": [ 106, 721, 231, 733 ], "score": 1.0, "content": "the latter were non-parametric.", "type": "text" } ], "index": 39 } ], "index": 35, "bbox_fs": [ 105, 633, 506, 733 ] } ] }, { "preproc_blocks": [ { "type": "table", "bbox": [ 106, 168, 506, 371 ], "blocks": [ { "type": "table_caption", "bbox": [ 106, 89, 505, 167 ], "group_id": 0, "lines": [ { "bbox": [ 105, 88, 505, 103 ], "spans": [ { "bbox": [ 105, 88, 276, 103 ], "score": 1.0, "content": "Table 2: Linear classification accuracy (in", "type": "text" }, { "bbox": [ 276, 90, 286, 100 ], "score": 0.58, "content": "\\%", "type": "inline_equation" }, { "bbox": [ 286, 88, 505, 103 ], "score": 1.0, "content": ") of graph CNE and existing graph contrastive learning", "type": "text" } ], "index": 0 }, { "bbox": [ 106, 100, 505, 113 ], "spans": [ { "bbox": [ 106, 100, 469, 113 ], "score": 1.0, "content": "algorithms. Output dimensionality of CNE is indicated in brackets. The line marked by", "type": "text" }, { "bbox": [ 469, 102, 477, 110 ], "score": 0.74, "content": "\\star", "type": "inline_equation" }, { "bbox": [ 477, 100, 505, 113 ], "score": 1.0, "content": "shows", "type": "text" } ], "index": 1 }, { "bbox": [ 106, 110, 506, 124 ], "spans": [ { "bbox": [ 106, 111, 128, 122 ], "score": 0.36, "content": "k \\mathbf { N N }", "type": "inline_equation" }, { "bbox": [ 129, 110, 371, 124 ], "score": 1.0, "content": "accuracy instead of linear accuracy. CNE values are mean", "type": "text" }, { "bbox": [ 371, 112, 381, 122 ], "score": 0.62, "content": "\\pm", "type": "inline_equation" }, { "bbox": [ 382, 110, 506, 124 ], "score": 1.0, "content": "standard deviation across five", "type": "text" } ], "index": 2 }, { "bbox": [ 105, 122, 505, 135 ], "spans": [ { "bbox": [ 105, 122, 505, 135 ], "score": 1.0, "content": "training runs. Non-CNE values are taken from Zhang et al. (2022), MLP values are taken from", "type": "text" } ], "index": 3 }, { "bbox": [ 106, 134, 504, 145 ], "spans": [ { "bbox": [ 106, 134, 292, 145 ], "score": 1.0, "content": "https://openreview.net/forum?id", "type": "text" }, { "bbox": [ 292, 135, 299, 143 ], "score": 0.43, "content": "{ . } = { }", "type": "inline_equation" }, { "bbox": [ 300, 134, 399, 145 ], "score": 1.0, "content": "dSYkYNNZkV¬eId", "type": "text" }, { "bbox": [ 400, 135, 407, 143 ], "score": 0.41, "content": "\\underline { { \\underline { { \\mathbf { \\Pi } } } } } =", "type": "inline_equation" }, { "bbox": [ 407, 134, 504, 145 ], "score": 1.0, "content": "aLQzIXVy0w and Guo", "type": "text" } ], "index": 4 }, { "bbox": [ 105, 143, 505, 157 ], "spans": [ { "bbox": [ 105, 143, 505, 157 ], "score": 1.0, "content": "et al. (2023). OOM denotes out-of-memory error. Datasets are ordered by the increasing sample", "type": "text" } ], "index": 5 }, { "bbox": [ 105, 155, 417, 168 ], "spans": [ { "bbox": [ 105, 155, 417, 168 ], "score": 1.0, "content": "size. For comparison, the first row shows linear accuracy in the feature space.", "type": "text" } ], "index": 6 } ], "index": 3 }, { "type": "table_body", "bbox": [ 106, 168, 506, 371 ], "group_id": 0, "lines": [ { "bbox": [ 106, 168, 506, 371 ], "spans": [ { "bbox": [ 106, 168, 506, 371 ], "score": 0.985, "html": "
CSRCORAPHACOPUBARX
Feature space70.368.690.779.687.855.1
Graph CNE (2)65.4 ± 2.262.7 ± 6.273.2 ± 1.477.1 ±0.766.9 ± 2.341.7 ± 0.8
Graph CNE (2) *72.1 ± 1.578.1± 3.292.9 ± 0.389.0± 0.277.2 ±0.645.3 ± 0.2
Graph CNE (128)72.0 ± 1.380.0± 1.292.9 ± 0.586.8±0.784.6±0.652.9 ± 0.3
GRACE71.2 ± 0.581.9 ± 0.492.2± 0.286.3± 0.380.6±0.40OM
GCA72.1± 0.482.3 ± 0.492.5 ± 0.187.9 ± 0.380.7± 0.5OOM
MVGRL73.3 ± 0.583.5 ± 0.491.7± 0.187.5 ± 0.180.1±0.70OM
DGI71.8±0.782.3 ± 0.691.6 ± 0.283.9± 0.576.8± 0.671.2 ± 0.2
BGRL71.1±0.882.7 ± 0.693.1 ± 0.389.7 ± 0.479.6± 0.572.7± 0.2
CCA-SSG73.1 ± 0.384.2 ± 0.493.1 ± 0.188.7± 0.381.6± 0.472.3 ± 0.2
AF-GCL72.0±0.483.2± 0.292.5 ± 0.389.7±0.279.1 ± 0.8
AFGRL68.7± 0.381.3± 0.293.2 ± 0.389.9 ± 0.380.6± 0.40OM
Local-GCL73.6 ± 0.484.5± 0.493.3 ± 0.488.8±0.482.1 ± 0.571.3 ± 0.3
Local-GCL,MLP70.3± 0.678.3± 0.590.9±0.482.4±0.579.6± 0.5
GRACE, MLP65.5 ± 2.667.7 ± 0.987.9 ± 0.680.9 ± 1.283.3± 0.5
", "type": "table", "image_path": "e72560a8e6b78995d863b84f76cc6f8e4908c62d027c3ff94e9094eea925efd4.jpg" } ] } ], "index": 8, "virtual_lines": [ { "bbox": [ 106, 168, 506, 235.66666666666669 ], "spans": [], "index": 7 }, { "bbox": [ 106, 235.66666666666669, 506, 303.33333333333337 ], "spans": [], "index": 8 }, { "bbox": [ 106, 303.33333333333337, 506, 371.00000000000006 ], "spans": [], "index": 9 } ] } ], "index": 5.5 }, { "type": "text", "bbox": [ 107, 394, 505, 450 ], "lines": [ { "bbox": [ 106, 394, 505, 408 ], "spans": [ { "bbox": [ 106, 394, 205, 408 ], "score": 1.0, "content": "As expected, CNE with", "type": "text" }, { "bbox": [ 206, 395, 243, 406 ], "score": 0.9, "content": "d = 1 2 8", "type": "inline_equation" }, { "bbox": [ 243, 394, 505, 408 ], "score": 1.0, "content": ", yielded considerably higher linear classification accuracy com-", "type": "text" } ], "index": 10 }, { "bbox": [ 105, 406, 505, 419 ], "spans": [ { "bbox": [ 105, 406, 505, 419 ], "score": 1.0, "content": "pared to both 2-dimensional embeddings (Figure 4). In terms of linear accuracy, graph CNE per-", "type": "text" } ], "index": 11 }, { "bbox": [ 105, 416, 505, 430 ], "spans": [ { "bbox": [ 105, 416, 505, 430 ], "score": 1.0, "content": "formed comparably to the state-of-the-art graph contrastive learning (GCL) algorithms1 (Table 2).", "type": "text" } ], "index": 12 }, { "bbox": [ 105, 428, 505, 440 ], "spans": [ { "bbox": [ 105, 428, 505, 440 ], "score": 1.0, "content": "Graph CNE achieved the best results on one of the datasets (PUB), and had close to the best results", "type": "text" } ], "index": 13 }, { "bbox": [ 106, 439, 263, 451 ], "spans": [ { "bbox": [ 106, 439, 263, 451 ], "score": 1.0, "content": "on other datasets, apart from the ARX.", "type": "text" } ], "index": 14 } ], "index": 12 }, { "type": "text", "bbox": [ 107, 455, 505, 576 ], "lines": [ { "bbox": [ 105, 456, 505, 468 ], "spans": [ { "bbox": [ 105, 456, 505, 468 ], "score": 1.0, "content": "Note that graph CNE was at disadvantage compared to all other GCL methods listed in Table 2", "type": "text" } ], "index": 15 }, { "bbox": [ 105, 467, 505, 479 ], "spans": [ { "bbox": [ 105, 467, 505, 479 ], "score": 1.0, "content": "because it used an MLP network, whereas other GCL methods traditionally use graph convolutional", "type": "text" } ], "index": 16 }, { "bbox": [ 105, 477, 505, 491 ], "spans": [ { "bbox": [ 105, 477, 505, 491 ], "score": 1.0, "content": "networks (GCN). GCN takes the entire graph as input and uses message passing, which pulls to-", "type": "text" } ], "index": 17 }, { "bbox": [ 104, 488, 506, 502 ], "spans": [ { "bbox": [ 104, 488, 506, 502 ], "score": 1.0, "content": "gether embeddings of connected nodes and helps to obtain better embeddings. However, GCN is", "type": "text" } ], "index": 18 }, { "bbox": [ 105, 499, 505, 512 ], "spans": [ { "bbox": [ 105, 499, 505, 512 ], "score": 1.0, "content": "not able to transform one node at a time, and so a trained GCN cannot be applied to a new, held-out", "type": "text" } ], "index": 19 }, { "bbox": [ 105, 510, 505, 523 ], "spans": [ { "bbox": [ 105, 510, 505, 523 ], "score": 1.0, "content": "node. In contrast, our graph CNE with MLP can (after training) process one node at a time, which", "type": "text" } ], "index": 20 }, { "bbox": [ 105, 522, 505, 534 ], "spans": [ { "bbox": [ 105, 522, 505, 534 ], "score": 1.0, "content": "we consider more appropriate for node-level graph learning (see Discussion). There are very few", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 532, 505, 544 ], "spans": [ { "bbox": [ 105, 532, 505, 544 ], "score": 1.0, "content": "GCL results based on the MLP architecture reported in the literature. Two examples are Local-GCL", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 543, 504, 556 ], "spans": [ { "bbox": [ 105, 543, 504, 556 ], "score": 1.0, "content": "and GRACE trained with MLP architecture (reported in the OpenReview discussion of Zhang et al.", "type": "text" } ], "index": 23 }, { "bbox": [ 105, 554, 505, 568 ], "spans": [ { "bbox": [ 105, 554, 505, 568 ], "score": 1.0, "content": "(2022) and in Guo et al. (2023) respectively, Table 2): both had lower accuracy compared to our", "type": "text" } ], "index": 24 }, { "bbox": [ 105, 565, 217, 578 ], "spans": [ { "bbox": [ 105, 565, 217, 578 ], "score": 1.0, "content": "graph CNE on all datasets.", "type": "text" } ], "index": 25 } ], "index": 20 }, { "type": "text", "bbox": [ 108, 582, 505, 615 ], "lines": [ { "bbox": [ 105, 581, 505, 595 ], "spans": [ { "bbox": [ 105, 581, 505, 595 ], "score": 1.0, "content": "For the ARX graph, we did not find any existing MLP-based results. Lower performance of graph", "type": "text" } ], "index": 26 }, { "bbox": [ 106, 593, 505, 606 ], "spans": [ { "bbox": [ 106, 593, 505, 606 ], "score": 1.0, "content": "CNE compared to GCN-based GCL methods was, again, likely due to the feature space of this graph", "type": "text" } ], "index": 27 }, { "bbox": [ 105, 604, 331, 617 ], "spans": [ { "bbox": [ 105, 604, 331, 617 ], "score": 1.0, "content": "showing only weak class separation (Table 2, first row).", "type": "text" } ], "index": 28 } ], "index": 27 }, { "type": "title", "bbox": [ 108, 635, 190, 648 ], "lines": [ { "bbox": [ 104, 633, 192, 651 ], "spans": [ { "bbox": [ 104, 633, 192, 651 ], "score": 1.0, "content": "7 DISCUSSION", "type": "text" } ], "index": 29 } ], "index": 29 }, { "type": "text", "bbox": [ 107, 662, 443, 673 ], "lines": [ { "bbox": [ 105, 660, 446, 676 ], "spans": [ { "bbox": [ 105, 660, 446, 676 ], "score": 1.0, "content": "Summary Our paper makes three contributions, two practical and one conceptual:", "type": "text" } ], "index": 30 } ], "index": 30 }, { "type": "text", "bbox": [ 114, 685, 504, 707 ], "lines": [ { "bbox": [ 115, 685, 505, 697 ], "spans": [ { "bbox": [ 115, 685, 345, 697 ], "score": 1.0, "content": "i. We suggested a novel graph layout algorithm, graph", "type": "text" }, { "bbox": [ 345, 686, 350, 695 ], "score": 0.58, "content": "t", "type": "inline_equation" }, { "bbox": [ 350, 685, 505, 697 ], "score": 1.0, "content": "-SNE, and showed that it outperforms", "type": "text" } ], "index": 31 }, { "bbox": [ 127, 695, 352, 708 ], "spans": [ { "bbox": [ 127, 695, 352, 708 ], "score": 1.0, "content": "existing competitors in preserving local graph structure.", "type": "text" } ], "index": 32 } ], "index": 31.5 } ], "page_idx": 7, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 117, 721, 469, 732 ], "lines": [ { "bbox": [ 119, 720, 470, 733 ], "spans": [ { "bbox": [ 119, 720, 470, 733 ], "score": 1.0, "content": "1We did not measure their performance ourselves, but took the values directly from the literature.", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 107, 27, 308, 37 ], "lines": [ { "bbox": [ 107, 25, 308, 38 ], "spans": [ { "bbox": [ 107, 25, 308, 38 ], "score": 1.0, "content": "Under review as a conference paper at ICLR 2024", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 302, 751, 308, 760 ], "lines": [ { "bbox": [ 300, 750, 309, 761 ], "spans": [ { "bbox": [ 300, 750, 309, 761 ], "score": 1.0, "content": "8", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "table", "bbox": [ 106, 168, 506, 371 ], "blocks": [ { "type": "table_caption", "bbox": [ 106, 89, 505, 167 ], "group_id": 0, "lines": [ { "bbox": [ 105, 88, 505, 103 ], "spans": [ { "bbox": [ 105, 88, 276, 103 ], "score": 1.0, "content": "Table 2: Linear classification accuracy (in", "type": "text" }, { "bbox": [ 276, 90, 286, 100 ], "score": 0.58, "content": "\\%", "type": "inline_equation" }, { "bbox": [ 286, 88, 505, 103 ], "score": 1.0, "content": ") of graph CNE and existing graph contrastive learning", "type": "text" } ], "index": 0 }, { "bbox": [ 106, 100, 505, 113 ], "spans": [ { "bbox": [ 106, 100, 469, 113 ], "score": 1.0, "content": "algorithms. Output dimensionality of CNE is indicated in brackets. The line marked by", "type": "text" }, { "bbox": [ 469, 102, 477, 110 ], "score": 0.74, "content": "\\star", "type": "inline_equation" }, { "bbox": [ 477, 100, 505, 113 ], "score": 1.0, "content": "shows", "type": "text" } ], "index": 1 }, { "bbox": [ 106, 110, 506, 124 ], "spans": [ { "bbox": [ 106, 111, 128, 122 ], "score": 0.36, "content": "k \\mathbf { N N }", "type": "inline_equation" }, { "bbox": [ 129, 110, 371, 124 ], "score": 1.0, "content": "accuracy instead of linear accuracy. CNE values are mean", "type": "text" }, { "bbox": [ 371, 112, 381, 122 ], "score": 0.62, "content": "\\pm", "type": "inline_equation" }, { "bbox": [ 382, 110, 506, 124 ], "score": 1.0, "content": "standard deviation across five", "type": "text" } ], "index": 2 }, { "bbox": [ 105, 122, 505, 135 ], "spans": [ { "bbox": [ 105, 122, 505, 135 ], "score": 1.0, "content": "training runs. Non-CNE values are taken from Zhang et al. (2022), MLP values are taken from", "type": "text" } ], "index": 3 }, { "bbox": [ 106, 134, 504, 145 ], "spans": [ { "bbox": [ 106, 134, 292, 145 ], "score": 1.0, "content": "https://openreview.net/forum?id", "type": "text" }, { "bbox": [ 292, 135, 299, 143 ], "score": 0.43, "content": "{ . } = { }", "type": "inline_equation" }, { "bbox": [ 300, 134, 399, 145 ], "score": 1.0, "content": "dSYkYNNZkV¬eId", "type": "text" }, { "bbox": [ 400, 135, 407, 143 ], "score": 0.41, "content": "\\underline { { \\underline { { \\mathbf { \\Pi } } } } } =", "type": "inline_equation" }, { "bbox": [ 407, 134, 504, 145 ], "score": 1.0, "content": "aLQzIXVy0w and Guo", "type": "text" } ], "index": 4 }, { "bbox": [ 105, 143, 505, 157 ], "spans": [ { "bbox": [ 105, 143, 505, 157 ], "score": 1.0, "content": "et al. (2023). OOM denotes out-of-memory error. Datasets are ordered by the increasing sample", "type": "text" } ], "index": 5 }, { "bbox": [ 105, 155, 417, 168 ], "spans": [ { "bbox": [ 105, 155, 417, 168 ], "score": 1.0, "content": "size. For comparison, the first row shows linear accuracy in the feature space.", "type": "text" } ], "index": 6 } ], "index": 3 }, { "type": "table_body", "bbox": [ 106, 168, 506, 371 ], "group_id": 0, "lines": [ { "bbox": [ 106, 168, 506, 371 ], "spans": [ { "bbox": [ 106, 168, 506, 371 ], "score": 0.985, "html": "
CSRCORAPHACOPUBARX
Feature space70.368.690.779.687.855.1
Graph CNE (2)65.4 ± 2.262.7 ± 6.273.2 ± 1.477.1 ±0.766.9 ± 2.341.7 ± 0.8
Graph CNE (2) *72.1 ± 1.578.1± 3.292.9 ± 0.389.0± 0.277.2 ±0.645.3 ± 0.2
Graph CNE (128)72.0 ± 1.380.0± 1.292.9 ± 0.586.8±0.784.6±0.652.9 ± 0.3
GRACE71.2 ± 0.581.9 ± 0.492.2± 0.286.3± 0.380.6±0.40OM
GCA72.1± 0.482.3 ± 0.492.5 ± 0.187.9 ± 0.380.7± 0.5OOM
MVGRL73.3 ± 0.583.5 ± 0.491.7± 0.187.5 ± 0.180.1±0.70OM
DGI71.8±0.782.3 ± 0.691.6 ± 0.283.9± 0.576.8± 0.671.2 ± 0.2
BGRL71.1±0.882.7 ± 0.693.1 ± 0.389.7 ± 0.479.6± 0.572.7± 0.2
CCA-SSG73.1 ± 0.384.2 ± 0.493.1 ± 0.188.7± 0.381.6± 0.472.3 ± 0.2
AF-GCL72.0±0.483.2± 0.292.5 ± 0.389.7±0.279.1 ± 0.8
AFGRL68.7± 0.381.3± 0.293.2 ± 0.389.9 ± 0.380.6± 0.40OM
Local-GCL73.6 ± 0.484.5± 0.493.3 ± 0.488.8±0.482.1 ± 0.571.3 ± 0.3
Local-GCL,MLP70.3± 0.678.3± 0.590.9±0.482.4±0.579.6± 0.5
GRACE, MLP65.5 ± 2.667.7 ± 0.987.9 ± 0.680.9 ± 1.283.3± 0.5
", "type": "table", "image_path": "e72560a8e6b78995d863b84f76cc6f8e4908c62d027c3ff94e9094eea925efd4.jpg" } ] } ], "index": 8, "virtual_lines": [ { "bbox": [ 106, 168, 506, 235.66666666666669 ], "spans": [], "index": 7 }, { "bbox": [ 106, 235.66666666666669, 506, 303.33333333333337 ], "spans": [], "index": 8 }, { "bbox": [ 106, 303.33333333333337, 506, 371.00000000000006 ], "spans": [], "index": 9 } ] } ], "index": 5.5 }, { "type": "text", "bbox": [ 107, 394, 505, 450 ], "lines": [ { "bbox": [ 106, 394, 505, 408 ], "spans": [ { "bbox": [ 106, 394, 205, 408 ], "score": 1.0, "content": "As expected, CNE with", "type": "text" }, { "bbox": [ 206, 395, 243, 406 ], "score": 0.9, "content": "d = 1 2 8", "type": "inline_equation" }, { "bbox": [ 243, 394, 505, 408 ], "score": 1.0, "content": ", yielded considerably higher linear classification accuracy com-", "type": "text" } ], "index": 10 }, { "bbox": [ 105, 406, 505, 419 ], "spans": [ { "bbox": [ 105, 406, 505, 419 ], "score": 1.0, "content": "pared to both 2-dimensional embeddings (Figure 4). In terms of linear accuracy, graph CNE per-", "type": "text" } ], "index": 11 }, { "bbox": [ 105, 416, 505, 430 ], "spans": [ { "bbox": [ 105, 416, 505, 430 ], "score": 1.0, "content": "formed comparably to the state-of-the-art graph contrastive learning (GCL) algorithms1 (Table 2).", "type": "text" } ], "index": 12 }, { "bbox": [ 105, 428, 505, 440 ], "spans": [ { "bbox": [ 105, 428, 505, 440 ], "score": 1.0, "content": "Graph CNE achieved the best results on one of the datasets (PUB), and had close to the best results", "type": "text" } ], "index": 13 }, { "bbox": [ 106, 439, 263, 451 ], "spans": [ { "bbox": [ 106, 439, 263, 451 ], "score": 1.0, "content": "on other datasets, apart from the ARX.", "type": "text" } ], "index": 14 } ], "index": 12, "bbox_fs": [ 105, 394, 505, 451 ] }, { "type": "text", "bbox": [ 107, 455, 505, 576 ], "lines": [ { "bbox": [ 105, 456, 505, 468 ], "spans": [ { "bbox": [ 105, 456, 505, 468 ], "score": 1.0, "content": "Note that graph CNE was at disadvantage compared to all other GCL methods listed in Table 2", "type": "text" } ], "index": 15 }, { "bbox": [ 105, 467, 505, 479 ], "spans": [ { "bbox": [ 105, 467, 505, 479 ], "score": 1.0, "content": "because it used an MLP network, whereas other GCL methods traditionally use graph convolutional", "type": "text" } ], "index": 16 }, { "bbox": [ 105, 477, 505, 491 ], "spans": [ { "bbox": [ 105, 477, 505, 491 ], "score": 1.0, "content": "networks (GCN). GCN takes the entire graph as input and uses message passing, which pulls to-", "type": "text" } ], "index": 17 }, { "bbox": [ 104, 488, 506, 502 ], "spans": [ { "bbox": [ 104, 488, 506, 502 ], "score": 1.0, "content": "gether embeddings of connected nodes and helps to obtain better embeddings. However, GCN is", "type": "text" } ], "index": 18 }, { "bbox": [ 105, 499, 505, 512 ], "spans": [ { "bbox": [ 105, 499, 505, 512 ], "score": 1.0, "content": "not able to transform one node at a time, and so a trained GCN cannot be applied to a new, held-out", "type": "text" } ], "index": 19 }, { "bbox": [ 105, 510, 505, 523 ], "spans": [ { "bbox": [ 105, 510, 505, 523 ], "score": 1.0, "content": "node. In contrast, our graph CNE with MLP can (after training) process one node at a time, which", "type": "text" } ], "index": 20 }, { "bbox": [ 105, 522, 505, 534 ], "spans": [ { "bbox": [ 105, 522, 505, 534 ], "score": 1.0, "content": "we consider more appropriate for node-level graph learning (see Discussion). There are very few", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 532, 505, 544 ], "spans": [ { "bbox": [ 105, 532, 505, 544 ], "score": 1.0, "content": "GCL results based on the MLP architecture reported in the literature. Two examples are Local-GCL", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 543, 504, 556 ], "spans": [ { "bbox": [ 105, 543, 504, 556 ], "score": 1.0, "content": "and GRACE trained with MLP architecture (reported in the OpenReview discussion of Zhang et al.", "type": "text" } ], "index": 23 }, { "bbox": [ 105, 554, 505, 568 ], "spans": [ { "bbox": [ 105, 554, 505, 568 ], "score": 1.0, "content": "(2022) and in Guo et al. (2023) respectively, Table 2): both had lower accuracy compared to our", "type": "text" } ], "index": 24 }, { "bbox": [ 105, 565, 217, 578 ], "spans": [ { "bbox": [ 105, 565, 217, 578 ], "score": 1.0, "content": "graph CNE on all datasets.", "type": "text" } ], "index": 25 } ], "index": 20, "bbox_fs": [ 104, 456, 506, 578 ] }, { "type": "text", "bbox": [ 108, 582, 505, 615 ], "lines": [ { "bbox": [ 105, 581, 505, 595 ], "spans": [ { "bbox": [ 105, 581, 505, 595 ], "score": 1.0, "content": "For the ARX graph, we did not find any existing MLP-based results. Lower performance of graph", "type": "text" } ], "index": 26 }, { "bbox": [ 106, 593, 505, 606 ], "spans": [ { "bbox": [ 106, 593, 505, 606 ], "score": 1.0, "content": "CNE compared to GCN-based GCL methods was, again, likely due to the feature space of this graph", "type": "text" } ], "index": 27 }, { "bbox": [ 105, 604, 331, 617 ], "spans": [ { "bbox": [ 105, 604, 331, 617 ], "score": 1.0, "content": "showing only weak class separation (Table 2, first row).", "type": "text" } ], "index": 28 } ], "index": 27, "bbox_fs": [ 105, 581, 505, 617 ] }, { "type": "title", "bbox": [ 108, 635, 190, 648 ], "lines": [ { "bbox": [ 104, 633, 192, 651 ], "spans": [ { "bbox": [ 104, 633, 192, 651 ], "score": 1.0, "content": "7 DISCUSSION", "type": "text" } ], "index": 29 } ], "index": 29 }, { "type": "text", "bbox": [ 107, 662, 443, 673 ], "lines": [ { "bbox": [ 105, 660, 446, 676 ], "spans": [ { "bbox": [ 105, 660, 446, 676 ], "score": 1.0, "content": "Summary Our paper makes three contributions, two practical and one conceptual:", "type": "text" } ], "index": 30 } ], "index": 30, "bbox_fs": [ 105, 660, 446, 676 ] }, { "type": "text", "bbox": [ 114, 685, 504, 707 ], "lines": [ { "bbox": [ 115, 685, 505, 697 ], "spans": [ { "bbox": [ 115, 685, 345, 697 ], "score": 1.0, "content": "i. We suggested a novel graph layout algorithm, graph", "type": "text" }, { "bbox": [ 345, 686, 350, 695 ], "score": 0.58, "content": "t", "type": "inline_equation" }, { "bbox": [ 350, 685, 505, 697 ], "score": 1.0, "content": "-SNE, and showed that it outperforms", "type": "text" } ], "index": 31 }, { "bbox": [ 127, 695, 352, 708 ], "spans": [ { "bbox": [ 127, 695, 352, 708 ], "score": 1.0, "content": "existing competitors in preserving local graph structure.", "type": "text" } ], "index": 32 } ], "index": 31.5, "bbox_fs": [ 115, 685, 505, 708 ] } ] }, { "preproc_blocks": [ { "type": "text", "bbox": [ 110, 82, 505, 198 ], "lines": [ { "bbox": [ 113, 82, 505, 96 ], "spans": [ { "bbox": [ 113, 82, 505, 96 ], "score": 1.0, "content": "ii. We suggested a novel node-level augmentation-free graph contrastive learning algorithm,", "type": "text" } ], "index": 0 }, { "bbox": [ 125, 93, 505, 106 ], "spans": [ { "bbox": [ 125, 93, 505, 106 ], "score": 1.0, "content": "graph CNE, and showed that it achieves comparable performance to the state-of-the-art meth-", "type": "text" } ], "index": 1 }, { "bbox": [ 126, 104, 505, 117 ], "spans": [ { "bbox": [ 126, 104, 505, 117 ], "score": 1.0, "content": "ods despite using the MLP architecture, and outperforms existing MLP-based graph contrastive", "type": "text" } ], "index": 2 }, { "bbox": [ 126, 115, 194, 128 ], "spans": [ { "bbox": [ 126, 115, 194, 128 ], "score": 1.0, "content": "learning results.", "type": "text" } ], "index": 3 }, { "bbox": [ 110, 141, 505, 155 ], "spans": [ { "bbox": [ 110, 141, 505, 155 ], "score": 1.0, "content": "iii. We established a conceptual connection between graph layouts and graph contrastive learning:", "type": "text" } ], "index": 4 }, { "bbox": [ 127, 154, 505, 165 ], "spans": [ { "bbox": [ 127, 154, 505, 165 ], "score": 1.0, "content": "we argued that both are instances of graph embeddings (non-parametric 2D embedding and", "type": "text" } ], "index": 5 }, { "bbox": [ 126, 164, 505, 177 ], "spans": [ { "bbox": [ 126, 164, 505, 177 ], "score": 1.0, "content": "parametric 128D embedding), and both can be efficiently implemented using neighbor em-", "type": "text" } ], "index": 6 }, { "bbox": [ 126, 173, 506, 189 ], "spans": [ { "bbox": [ 126, 173, 506, 189 ], "score": 1.0, "content": "bedding frameworks. We suggested a new task, parametric 2D embeddings (Figure 1), as a", "type": "text" } ], "index": 7 }, { "bbox": [ 128, 185, 320, 199 ], "spans": [ { "bbox": [ 128, 185, 320, 199 ], "score": 1.0, "content": "‘missing link’ between these two existing tasks.", "type": "text" } ], "index": 8 } ], "index": 4 }, { "type": "text", "bbox": [ 107, 223, 505, 300 ], "lines": [ { "bbox": [ 105, 222, 505, 237 ], "spans": [ { "bbox": [ 105, 222, 208, 237 ], "score": 1.0, "content": "Simplicity Both graph", "type": "text" }, { "bbox": [ 208, 224, 213, 233 ], "score": 0.57, "content": "t", "type": "inline_equation" }, { "bbox": [ 213, 222, 505, 237 ], "score": 1.0, "content": "-SNE and graph CNE are remarkably simple, because they use existing", "type": "text" } ], "index": 9 }, { "bbox": [ 106, 234, 505, 246 ], "spans": [ { "bbox": [ 106, 235, 111, 244 ], "score": 0.7, "content": "t", "type": "inline_equation" }, { "bbox": [ 111, 234, 505, 246 ], "score": 1.0, "content": "-SNE and CNE machinery out of the box. This is in stark contrast with competing algorithms. For", "type": "text" } ], "index": 10 }, { "bbox": [ 105, 245, 505, 257 ], "spans": [ { "bbox": [ 105, 245, 324, 257 ], "score": 1.0, "content": "example, existing graph layout algorithms inspired by", "type": "text" }, { "bbox": [ 324, 246, 329, 255 ], "score": 0.68, "content": "t", "type": "inline_equation" }, { "bbox": [ 330, 245, 505, 257 ], "score": 1.0, "content": "-SNE, such as tsNET (Kruiger et al., 2017),", "type": "text" } ], "index": 11 }, { "bbox": [ 105, 255, 506, 270 ], "spans": [ { "bbox": [ 105, 255, 250, 270 ], "score": 1.0, "content": "DRGraph (Zhu et al., 2020a), and", "type": "text" }, { "bbox": [ 250, 257, 255, 266 ], "score": 0.74, "content": "t", "type": "inline_equation" }, { "bbox": [ 255, 255, 506, 270 ], "score": 1.0, "content": "-FDP (Zhong et al., 2023), all develop their own machinery,", "type": "text" } ], "index": 12 }, { "bbox": [ 105, 266, 505, 280 ], "spans": [ { "bbox": [ 105, 266, 335, 280 ], "score": 1.0, "content": "implementation, and approximations, and deviate from", "type": "text" }, { "bbox": [ 335, 268, 340, 277 ], "score": 0.65, "content": "t", "type": "inline_equation" }, { "bbox": [ 341, 266, 505, 280 ], "score": 1.0, "content": "-SNE in many different nontrivial ways", "type": "text" } ], "index": 13 }, { "bbox": [ 105, 278, 505, 291 ], "spans": [ { "bbox": [ 105, 278, 348, 291 ], "score": 1.0, "content": "(see Section 2). However, as we demonstrated, simply using", "type": "text" }, { "bbox": [ 349, 279, 353, 288 ], "score": 0.58, "content": "t", "type": "inline_equation" }, { "bbox": [ 354, 278, 420, 291 ], "score": 1.0, "content": "-SNE (via graph", "type": "text" }, { "bbox": [ 420, 279, 425, 288 ], "score": 0.58, "content": "t", "type": "inline_equation" }, { "bbox": [ 425, 278, 505, 291 ], "score": 1.0, "content": "-SNE), outperforms", "type": "text" } ], "index": 14 }, { "bbox": [ 105, 289, 257, 302 ], "spans": [ { "bbox": [ 105, 289, 257, 302 ], "score": 1.0, "content": "all of them in terms of layout quality.", "type": "text" } ], "index": 15 } ], "index": 12 }, { "type": "text", "bbox": [ 107, 305, 505, 405 ], "lines": [ { "bbox": [ 106, 304, 505, 319 ], "spans": [ { "bbox": [ 106, 304, 505, 319 ], "score": 1.0, "content": "Similarly, in node-level graph contrastive learning (GCL), the focus has been on developing graph", "type": "text" } ], "index": 16 }, { "bbox": [ 105, 317, 506, 330 ], "spans": [ { "bbox": [ 105, 317, 506, 330 ], "score": 1.0, "content": "augmentations (see Section 2), following the contrastive learning paradigm in computer vision that", "type": "text" } ], "index": 17 }, { "bbox": [ 105, 327, 506, 341 ], "spans": [ { "bbox": [ 105, 327, 506, 341 ], "score": 1.0, "content": "is based on image augmentations. Augmentation-free GCL methods such as AFGRL (Lee et al.,", "type": "text" } ], "index": 18 }, { "bbox": [ 105, 339, 505, 352 ], "spans": [ { "bbox": [ 105, 339, 505, 352 ], "score": 1.0, "content": "2022) and AF-GCL (Li et al., 2023) instead rely on complex heuristics to select positive pairs. Our", "type": "text" } ], "index": 19 }, { "bbox": [ 105, 350, 506, 362 ], "spans": [ { "bbox": [ 105, 350, 506, 362 ], "score": 1.0, "content": "approach is conceptually much simpler, as it uses the InfoNCE loss function with graph edges as", "type": "text" } ], "index": 20 }, { "bbox": [ 105, 360, 505, 373 ], "spans": [ { "bbox": [ 105, 360, 505, 373 ], "score": 1.0, "content": "positive pairs, and nothing else. The closest method in the literature is Local-GCL (Zhang et al.,", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 370, 506, 385 ], "spans": [ { "bbox": [ 105, 370, 506, 385 ], "score": 1.0, "content": "2022), which also uses graph edges as positive pairs. The difference is that Local-GCL uses an", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 381, 506, 396 ], "spans": [ { "bbox": [ 105, 381, 248, 396 ], "score": 1.0, "content": "approximation scheme to deal with", "type": "text" }, { "bbox": [ 248, 381, 275, 394 ], "score": 0.92, "content": "O ( n ^ { 2 } )", "type": "inline_equation" }, { "bbox": [ 276, 381, 506, 396 ], "score": 1.0, "content": "repulsive forces, whereas we use the standard contrastive", "type": "text" } ], "index": 23 }, { "bbox": [ 105, 394, 381, 407 ], "spans": [ { "bbox": [ 105, 394, 381, 407 ], "score": 1.0, "content": "learning approach of within-batch repulsion, which is much simpler.", "type": "text" } ], "index": 24 } ], "index": 20 }, { "type": "text", "bbox": [ 107, 410, 505, 498 ], "lines": [ { "bbox": [ 105, 410, 505, 423 ], "spans": [ { "bbox": [ 105, 410, 505, 423 ], "score": 1.0, "content": "All of the existing GCL methods, including Local-GCL, employ graph convolutional neural net-", "type": "text" } ], "index": 25 }, { "bbox": [ 105, 421, 506, 434 ], "spans": [ { "bbox": [ 105, 421, 506, 434 ], "score": 1.0, "content": "works (GCNs). Recent work argued that the reason many GCL algorithms work well has little to", "type": "text" } ], "index": 26 }, { "bbox": [ 105, 432, 506, 445 ], "spans": [ { "bbox": [ 105, 432, 506, 445 ], "score": 1.0, "content": "do with the specific augmentations or heuristics they use, but rather is due to their GCN architecture", "type": "text" } ], "index": 27 }, { "bbox": [ 105, 442, 506, 456 ], "spans": [ { "bbox": [ 105, 442, 506, 456 ], "score": 1.0, "content": "(Trivedi et al., 2022; Guo et al., 2023). GCN uses message passing between graph nodes, which", "type": "text" } ], "index": 28 }, { "bbox": [ 106, 455, 505, 467 ], "spans": [ { "bbox": [ 106, 455, 505, 467 ], "score": 1.0, "content": "implicitly makes representations of connected node pairs more similar. In other words, in GCL al-", "type": "text" } ], "index": 29 }, { "bbox": [ 105, 465, 506, 477 ], "spans": [ { "bbox": [ 105, 465, 506, 477 ], "score": 1.0, "content": "gorithms employing GCNs, it is the GCN that does the heavy lifting, and not the specifics of the", "type": "text" } ], "index": 30 }, { "bbox": [ 105, 475, 506, 489 ], "spans": [ { "bbox": [ 105, 475, 506, 489 ], "score": 1.0, "content": "GCL algorithm. In contrast, our graph CNE uses an MLP network, and nevertheless performed", "type": "text" } ], "index": 31 }, { "bbox": [ 105, 487, 505, 499 ], "spans": [ { "bbox": [ 105, 487, 505, 499 ], "score": 1.0, "content": "similarly well. See below on why we think MLP is a more suitable choice for node-level GCL tasks.", "type": "text" } ], "index": 32 } ], "index": 28.5 }, { "type": "text", "bbox": [ 107, 524, 505, 601 ], "lines": [ { "bbox": [ 105, 524, 505, 537 ], "spans": [ { "bbox": [ 105, 524, 505, 537 ], "score": 1.0, "content": "Limitations In this work, we focused on complex real-world graphs and have purposefully not", "type": "text" } ], "index": 33 }, { "bbox": [ 105, 534, 505, 548 ], "spans": [ { "bbox": [ 105, 534, 174, 548 ], "score": 1.0, "content": "tested our graph", "type": "text" }, { "bbox": [ 174, 536, 180, 545 ], "score": 0.53, "content": "t", "type": "inline_equation" }, { "bbox": [ 180, 534, 505, 548 ], "score": 1.0, "content": "-SNE on simple planar graphs or 3D mesh graphs that are often used for bench-", "type": "text" } ], "index": 34 }, { "bbox": [ 105, 546, 505, 559 ], "spans": [ { "bbox": [ 105, 546, 343, 559 ], "score": 1.0, "content": "marking graph layout algorithms. We suspect that graph", "type": "text" }, { "bbox": [ 343, 547, 349, 556 ], "score": 0.65, "content": "t", "type": "inline_equation" }, { "bbox": [ 349, 546, 505, 559 ], "score": 1.0, "content": "-SNE would perform suboptimally on", "type": "text" } ], "index": 35 }, { "bbox": [ 105, 557, 505, 569 ], "spans": [ { "bbox": [ 105, 557, 170, 569 ], "score": 1.0, "content": "such graphs, as", "type": "text" }, { "bbox": [ 170, 558, 175, 567 ], "score": 0.69, "content": "t", "type": "inline_equation" }, { "bbox": [ 176, 557, 505, 569 ], "score": 1.0, "content": "-SNE is known to have troubles with embedding simple 2D manifolds such as the", "type": "text" } ], "index": 36 }, { "bbox": [ 105, 568, 506, 581 ], "spans": [ { "bbox": [ 105, 568, 506, 581 ], "score": 1.0, "content": "Swiss roll. To some extent this can be addressed by increasing the degree of freedom parameter of", "type": "text" } ], "index": 37 }, { "bbox": [ 105, 579, 505, 591 ], "spans": [ { "bbox": [ 105, 579, 121, 591 ], "score": 1.0, "content": "the", "type": "text" }, { "bbox": [ 121, 580, 126, 589 ], "score": 0.74, "content": "t", "type": "inline_equation" }, { "bbox": [ 126, 579, 505, 591 ], "score": 1.0, "content": "-distribution or using the Gaussian kernel instead (Kobak et al., 2019), and/or by increasing the", "type": "text" } ], "index": 38 }, { "bbox": [ 105, 590, 450, 602 ], "spans": [ { "bbox": [ 105, 590, 450, 602 ], "score": 1.0, "content": "exaggeration value (Kobak & Berens, 2019; Bohm et al., 2022; Damrich et al., 2022). ¨", "type": "text" } ], "index": 39 } ], "index": 36 }, { "type": "text", "bbox": [ 107, 606, 504, 673 ], "lines": [ { "bbox": [ 105, 606, 506, 619 ], "spans": [ { "bbox": [ 105, 606, 506, 619 ], "score": 1.0, "content": "Our graph CNE relies on the MLP and we did not experiment with GCN architecture. This, however,", "type": "text" } ], "index": 40 }, { "bbox": [ 105, 618, 506, 631 ], "spans": [ { "bbox": [ 105, 618, 506, 631 ], "score": 1.0, "content": "is not a limitation but a purposeful design choice: we think that GCN, whereas very meaningful", "type": "text" } ], "index": 41 }, { "bbox": [ 105, 628, 505, 641 ], "spans": [ { "bbox": [ 105, 628, 505, 641 ], "score": 1.0, "content": "for graph-level learning, is less applicable for node-level learning, where one may want to apply", "type": "text" } ], "index": 42 }, { "bbox": [ 106, 640, 506, 651 ], "spans": [ { "bbox": [ 106, 640, 506, 651 ], "score": 1.0, "content": "the trained model to a set of new objects (based on their node features). With GCN, this is not", "type": "text" } ], "index": 43 }, { "bbox": [ 105, 650, 506, 663 ], "spans": [ { "bbox": [ 105, 650, 506, 663 ], "score": 1.0, "content": "possible, as it requires the entire graph to be passed in at the same time. We therefore consider MLP", "type": "text" } ], "index": 44 }, { "bbox": [ 105, 662, 310, 673 ], "spans": [ { "bbox": [ 105, 662, 310, 673 ], "score": 1.0, "content": "architecture more appropriate for node-level GCL.", "type": "text" } ], "index": 45 } ], "index": 42.5 }, { "type": "text", "bbox": [ 108, 699, 504, 731 ], "lines": [ { "bbox": [ 105, 698, 505, 712 ], "spans": [ { "bbox": [ 105, 698, 505, 712 ], "score": 1.0, "content": "Take-home message We showed that graph layouts and graph contrastive learning are intimately", "type": "text" } ], "index": 46 }, { "bbox": [ 106, 709, 505, 721 ], "spans": [ { "bbox": [ 106, 709, 505, 721 ], "score": 1.0, "content": "related and can be approached by existing neighbour embedding frameworks, surpassing state-of-", "type": "text" } ], "index": 47 }, { "bbox": [ 106, 720, 166, 732 ], "spans": [ { "bbox": [ 106, 720, 166, 732 ], "score": 1.0, "content": "the-art results.", "type": "text" } ], "index": 48 } ], "index": 47 } ], "page_idx": 8, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 107, 27, 308, 37 ], "lines": [ { "bbox": [ 107, 26, 308, 38 ], "spans": [ { "bbox": [ 107, 26, 308, 38 ], "score": 1.0, "content": "Under review as a conference paper at ICLR 2024", "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": "list", "bbox": [ 110, 82, 505, 198 ], "lines": [ { "bbox": [ 113, 82, 505, 96 ], "spans": [ { "bbox": [ 113, 82, 505, 96 ], "score": 1.0, "content": "ii. We suggested a novel node-level augmentation-free graph contrastive learning algorithm,", "type": "text" } ], "index": 0, "is_list_start_line": true }, { "bbox": [ 125, 93, 505, 106 ], "spans": [ { "bbox": [ 125, 93, 505, 106 ], "score": 1.0, "content": "graph CNE, and showed that it achieves comparable performance to the state-of-the-art meth-", "type": "text" } ], "index": 1 }, { "bbox": [ 126, 104, 505, 117 ], "spans": [ { "bbox": [ 126, 104, 505, 117 ], "score": 1.0, "content": "ods despite using the MLP architecture, and outperforms existing MLP-based graph contrastive", "type": "text" } ], "index": 2 }, { "bbox": [ 126, 115, 194, 128 ], "spans": [ { "bbox": [ 126, 115, 194, 128 ], "score": 1.0, "content": "learning results.", "type": "text" } ], "index": 3, "is_list_end_line": true }, { "bbox": [ 110, 141, 505, 155 ], "spans": [ { "bbox": [ 110, 141, 505, 155 ], "score": 1.0, "content": "iii. We established a conceptual connection between graph layouts and graph contrastive learning:", "type": "text" } ], "index": 4, "is_list_start_line": true }, { "bbox": [ 127, 154, 505, 165 ], "spans": [ { "bbox": [ 127, 154, 505, 165 ], "score": 1.0, "content": "we argued that both are instances of graph embeddings (non-parametric 2D embedding and", "type": "text" } ], "index": 5 }, { "bbox": [ 126, 164, 505, 177 ], "spans": [ { "bbox": [ 126, 164, 505, 177 ], "score": 1.0, "content": "parametric 128D embedding), and both can be efficiently implemented using neighbor em-", "type": "text" } ], "index": 6 }, { "bbox": [ 126, 173, 506, 189 ], "spans": [ { "bbox": [ 126, 173, 506, 189 ], "score": 1.0, "content": "bedding frameworks. We suggested a new task, parametric 2D embeddings (Figure 1), as a", "type": "text" } ], "index": 7 }, { "bbox": [ 128, 185, 320, 199 ], "spans": [ { "bbox": [ 128, 185, 320, 199 ], "score": 1.0, "content": "‘missing link’ between these two existing tasks.", "type": "text" } ], "index": 8, "is_list_end_line": true } ], "index": 4, "bbox_fs": [ 110, 82, 506, 199 ] }, { "type": "text", "bbox": [ 107, 223, 505, 300 ], "lines": [ { "bbox": [ 105, 222, 505, 237 ], "spans": [ { "bbox": [ 105, 222, 208, 237 ], "score": 1.0, "content": "Simplicity Both graph", "type": "text" }, { "bbox": [ 208, 224, 213, 233 ], "score": 0.57, "content": "t", "type": "inline_equation" }, { "bbox": [ 213, 222, 505, 237 ], "score": 1.0, "content": "-SNE and graph CNE are remarkably simple, because they use existing", "type": "text" } ], "index": 9 }, { "bbox": [ 106, 234, 505, 246 ], "spans": [ { "bbox": [ 106, 235, 111, 244 ], "score": 0.7, "content": "t", "type": "inline_equation" }, { "bbox": [ 111, 234, 505, 246 ], "score": 1.0, "content": "-SNE and CNE machinery out of the box. This is in stark contrast with competing algorithms. For", "type": "text" } ], "index": 10 }, { "bbox": [ 105, 245, 505, 257 ], "spans": [ { "bbox": [ 105, 245, 324, 257 ], "score": 1.0, "content": "example, existing graph layout algorithms inspired by", "type": "text" }, { "bbox": [ 324, 246, 329, 255 ], "score": 0.68, "content": "t", "type": "inline_equation" }, { "bbox": [ 330, 245, 505, 257 ], "score": 1.0, "content": "-SNE, such as tsNET (Kruiger et al., 2017),", "type": "text" } ], "index": 11 }, { "bbox": [ 105, 255, 506, 270 ], "spans": [ { "bbox": [ 105, 255, 250, 270 ], "score": 1.0, "content": "DRGraph (Zhu et al., 2020a), and", "type": "text" }, { "bbox": [ 250, 257, 255, 266 ], "score": 0.74, "content": "t", "type": "inline_equation" }, { "bbox": [ 255, 255, 506, 270 ], "score": 1.0, "content": "-FDP (Zhong et al., 2023), all develop their own machinery,", "type": "text" } ], "index": 12 }, { "bbox": [ 105, 266, 505, 280 ], "spans": [ { "bbox": [ 105, 266, 335, 280 ], "score": 1.0, "content": "implementation, and approximations, and deviate from", "type": "text" }, { "bbox": [ 335, 268, 340, 277 ], "score": 0.65, "content": "t", "type": "inline_equation" }, { "bbox": [ 341, 266, 505, 280 ], "score": 1.0, "content": "-SNE in many different nontrivial ways", "type": "text" } ], "index": 13 }, { "bbox": [ 105, 278, 505, 291 ], "spans": [ { "bbox": [ 105, 278, 348, 291 ], "score": 1.0, "content": "(see Section 2). However, as we demonstrated, simply using", "type": "text" }, { "bbox": [ 349, 279, 353, 288 ], "score": 0.58, "content": "t", "type": "inline_equation" }, { "bbox": [ 354, 278, 420, 291 ], "score": 1.0, "content": "-SNE (via graph", "type": "text" }, { "bbox": [ 420, 279, 425, 288 ], "score": 0.58, "content": "t", "type": "inline_equation" }, { "bbox": [ 425, 278, 505, 291 ], "score": 1.0, "content": "-SNE), outperforms", "type": "text" } ], "index": 14 }, { "bbox": [ 105, 289, 257, 302 ], "spans": [ { "bbox": [ 105, 289, 257, 302 ], "score": 1.0, "content": "all of them in terms of layout quality.", "type": "text" } ], "index": 15 } ], "index": 12, "bbox_fs": [ 105, 222, 506, 302 ] }, { "type": "text", "bbox": [ 107, 305, 505, 405 ], "lines": [ { "bbox": [ 106, 304, 505, 319 ], "spans": [ { "bbox": [ 106, 304, 505, 319 ], "score": 1.0, "content": "Similarly, in node-level graph contrastive learning (GCL), the focus has been on developing graph", "type": "text" } ], "index": 16 }, { "bbox": [ 105, 317, 506, 330 ], "spans": [ { "bbox": [ 105, 317, 506, 330 ], "score": 1.0, "content": "augmentations (see Section 2), following the contrastive learning paradigm in computer vision that", "type": "text" } ], "index": 17 }, { "bbox": [ 105, 327, 506, 341 ], "spans": [ { "bbox": [ 105, 327, 506, 341 ], "score": 1.0, "content": "is based on image augmentations. Augmentation-free GCL methods such as AFGRL (Lee et al.,", "type": "text" } ], "index": 18 }, { "bbox": [ 105, 339, 505, 352 ], "spans": [ { "bbox": [ 105, 339, 505, 352 ], "score": 1.0, "content": "2022) and AF-GCL (Li et al., 2023) instead rely on complex heuristics to select positive pairs. Our", "type": "text" } ], "index": 19 }, { "bbox": [ 105, 350, 506, 362 ], "spans": [ { "bbox": [ 105, 350, 506, 362 ], "score": 1.0, "content": "approach is conceptually much simpler, as it uses the InfoNCE loss function with graph edges as", "type": "text" } ], "index": 20 }, { "bbox": [ 105, 360, 505, 373 ], "spans": [ { "bbox": [ 105, 360, 505, 373 ], "score": 1.0, "content": "positive pairs, and nothing else. The closest method in the literature is Local-GCL (Zhang et al.,", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 370, 506, 385 ], "spans": [ { "bbox": [ 105, 370, 506, 385 ], "score": 1.0, "content": "2022), which also uses graph edges as positive pairs. The difference is that Local-GCL uses an", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 381, 506, 396 ], "spans": [ { "bbox": [ 105, 381, 248, 396 ], "score": 1.0, "content": "approximation scheme to deal with", "type": "text" }, { "bbox": [ 248, 381, 275, 394 ], "score": 0.92, "content": "O ( n ^ { 2 } )", "type": "inline_equation" }, { "bbox": [ 276, 381, 506, 396 ], "score": 1.0, "content": "repulsive forces, whereas we use the standard contrastive", "type": "text" } ], "index": 23 }, { "bbox": [ 105, 394, 381, 407 ], "spans": [ { "bbox": [ 105, 394, 381, 407 ], "score": 1.0, "content": "learning approach of within-batch repulsion, which is much simpler.", "type": "text" } ], "index": 24 } ], "index": 20, "bbox_fs": [ 105, 304, 506, 407 ] }, { "type": "text", "bbox": [ 107, 410, 505, 498 ], "lines": [ { "bbox": [ 105, 410, 505, 423 ], "spans": [ { "bbox": [ 105, 410, 505, 423 ], "score": 1.0, "content": "All of the existing GCL methods, including Local-GCL, employ graph convolutional neural net-", "type": "text" } ], "index": 25 }, { "bbox": [ 105, 421, 506, 434 ], "spans": [ { "bbox": [ 105, 421, 506, 434 ], "score": 1.0, "content": "works (GCNs). Recent work argued that the reason many GCL algorithms work well has little to", "type": "text" } ], "index": 26 }, { "bbox": [ 105, 432, 506, 445 ], "spans": [ { "bbox": [ 105, 432, 506, 445 ], "score": 1.0, "content": "do with the specific augmentations or heuristics they use, but rather is due to their GCN architecture", "type": "text" } ], "index": 27 }, { "bbox": [ 105, 442, 506, 456 ], "spans": [ { "bbox": [ 105, 442, 506, 456 ], "score": 1.0, "content": "(Trivedi et al., 2022; Guo et al., 2023). GCN uses message passing between graph nodes, which", "type": "text" } ], "index": 28 }, { "bbox": [ 106, 455, 505, 467 ], "spans": [ { "bbox": [ 106, 455, 505, 467 ], "score": 1.0, "content": "implicitly makes representations of connected node pairs more similar. In other words, in GCL al-", "type": "text" } ], "index": 29 }, { "bbox": [ 105, 465, 506, 477 ], "spans": [ { "bbox": [ 105, 465, 506, 477 ], "score": 1.0, "content": "gorithms employing GCNs, it is the GCN that does the heavy lifting, and not the specifics of the", "type": "text" } ], "index": 30 }, { "bbox": [ 105, 475, 506, 489 ], "spans": [ { "bbox": [ 105, 475, 506, 489 ], "score": 1.0, "content": "GCL algorithm. In contrast, our graph CNE uses an MLP network, and nevertheless performed", "type": "text" } ], "index": 31 }, { "bbox": [ 105, 487, 505, 499 ], "spans": [ { "bbox": [ 105, 487, 505, 499 ], "score": 1.0, "content": "similarly well. See below on why we think MLP is a more suitable choice for node-level GCL tasks.", "type": "text" } ], "index": 32 } ], "index": 28.5, "bbox_fs": [ 105, 410, 506, 499 ] }, { "type": "text", "bbox": [ 107, 524, 505, 601 ], "lines": [ { "bbox": [ 105, 524, 505, 537 ], "spans": [ { "bbox": [ 105, 524, 505, 537 ], "score": 1.0, "content": "Limitations In this work, we focused on complex real-world graphs and have purposefully not", "type": "text" } ], "index": 33 }, { "bbox": [ 105, 534, 505, 548 ], "spans": [ { "bbox": [ 105, 534, 174, 548 ], "score": 1.0, "content": "tested our graph", "type": "text" }, { "bbox": [ 174, 536, 180, 545 ], "score": 0.53, "content": "t", "type": "inline_equation" }, { "bbox": [ 180, 534, 505, 548 ], "score": 1.0, "content": "-SNE on simple planar graphs or 3D mesh graphs that are often used for bench-", "type": "text" } ], "index": 34 }, { "bbox": [ 105, 546, 505, 559 ], "spans": [ { "bbox": [ 105, 546, 343, 559 ], "score": 1.0, "content": "marking graph layout algorithms. We suspect that graph", "type": "text" }, { "bbox": [ 343, 547, 349, 556 ], "score": 0.65, "content": "t", "type": "inline_equation" }, { "bbox": [ 349, 546, 505, 559 ], "score": 1.0, "content": "-SNE would perform suboptimally on", "type": "text" } ], "index": 35 }, { "bbox": [ 105, 557, 505, 569 ], "spans": [ { "bbox": [ 105, 557, 170, 569 ], "score": 1.0, "content": "such graphs, as", "type": "text" }, { "bbox": [ 170, 558, 175, 567 ], "score": 0.69, "content": "t", "type": "inline_equation" }, { "bbox": [ 176, 557, 505, 569 ], "score": 1.0, "content": "-SNE is known to have troubles with embedding simple 2D manifolds such as the", "type": "text" } ], "index": 36 }, { "bbox": [ 105, 568, 506, 581 ], "spans": [ { "bbox": [ 105, 568, 506, 581 ], "score": 1.0, "content": "Swiss roll. To some extent this can be addressed by increasing the degree of freedom parameter of", "type": "text" } ], "index": 37 }, { "bbox": [ 105, 579, 505, 591 ], "spans": [ { "bbox": [ 105, 579, 121, 591 ], "score": 1.0, "content": "the", "type": "text" }, { "bbox": [ 121, 580, 126, 589 ], "score": 0.74, "content": "t", "type": "inline_equation" }, { "bbox": [ 126, 579, 505, 591 ], "score": 1.0, "content": "-distribution or using the Gaussian kernel instead (Kobak et al., 2019), and/or by increasing the", "type": "text" } ], "index": 38 }, { "bbox": [ 105, 590, 450, 602 ], "spans": [ { "bbox": [ 105, 590, 450, 602 ], "score": 1.0, "content": "exaggeration value (Kobak & Berens, 2019; Bohm et al., 2022; Damrich et al., 2022). ¨", "type": "text" } ], "index": 39 } ], "index": 36, "bbox_fs": [ 105, 524, 506, 602 ] }, { "type": "text", "bbox": [ 107, 606, 504, 673 ], "lines": [ { "bbox": [ 105, 606, 506, 619 ], "spans": [ { "bbox": [ 105, 606, 506, 619 ], "score": 1.0, "content": "Our graph CNE relies on the MLP and we did not experiment with GCN architecture. This, however,", "type": "text" } ], "index": 40 }, { "bbox": [ 105, 618, 506, 631 ], "spans": [ { "bbox": [ 105, 618, 506, 631 ], "score": 1.0, "content": "is not a limitation but a purposeful design choice: we think that GCN, whereas very meaningful", "type": "text" } ], "index": 41 }, { "bbox": [ 105, 628, 505, 641 ], "spans": [ { "bbox": [ 105, 628, 505, 641 ], "score": 1.0, "content": "for graph-level learning, is less applicable for node-level learning, where one may want to apply", "type": "text" } ], "index": 42 }, { "bbox": [ 106, 640, 506, 651 ], "spans": [ { "bbox": [ 106, 640, 506, 651 ], "score": 1.0, "content": "the trained model to a set of new objects (based on their node features). With GCN, this is not", "type": "text" } ], "index": 43 }, { "bbox": [ 105, 650, 506, 663 ], "spans": [ { "bbox": [ 105, 650, 506, 663 ], "score": 1.0, "content": "possible, as it requires the entire graph to be passed in at the same time. We therefore consider MLP", "type": "text" } ], "index": 44 }, { "bbox": [ 105, 662, 310, 673 ], "spans": [ { "bbox": [ 105, 662, 310, 673 ], "score": 1.0, "content": "architecture more appropriate for node-level GCL.", "type": "text" } ], "index": 45 } ], "index": 42.5, "bbox_fs": [ 105, 606, 506, 673 ] }, { "type": "text", "bbox": [ 108, 699, 504, 731 ], "lines": [ { "bbox": [ 105, 698, 505, 712 ], "spans": [ { "bbox": [ 105, 698, 505, 712 ], "score": 1.0, "content": "Take-home message We showed that graph layouts and graph contrastive learning are intimately", "type": "text" } ], "index": 46 }, { "bbox": [ 106, 709, 505, 721 ], "spans": [ { "bbox": [ 106, 709, 505, 721 ], "score": 1.0, "content": "related and can be approached by existing neighbour embedding frameworks, surpassing state-of-", "type": "text" } ], "index": 47 }, { "bbox": [ 106, 720, 166, 732 ], "spans": [ { "bbox": [ 106, 720, 166, 732 ], "score": 1.0, "content": "the-art results.", "type": "text" } ], "index": 48 } ], "index": 47, "bbox_fs": [ 105, 698, 505, 732 ] } ] }, { "preproc_blocks": [ { "type": "title", "bbox": [ 108, 81, 175, 93 ], "lines": [ { "bbox": [ 106, 81, 176, 95 ], "spans": [ { "bbox": [ 106, 81, 176, 95 ], "score": 1.0, "content": "REFERENCES", "type": "text" } ], "index": 0 } ], "index": 0 }, { "type": "text", "bbox": [ 107, 100, 503, 123 ], "lines": [ { "bbox": [ 106, 99, 505, 112 ], "spans": [ { "bbox": [ 106, 99, 505, 112 ], "score": 1.0, "content": "Aleksandr Artemenkov and Maxim Panov. 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