diff --git "a/parse/test/89l6VLPrin/89l6VLPrin_middle.json" "b/parse/test/89l6VLPrin/89l6VLPrin_middle.json" new file mode 100644--- /dev/null +++ "b/parse/test/89l6VLPrin/89l6VLPrin_middle.json" @@ -0,0 +1,36601 @@ +{ + "pdf_info": [ + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 80, + 435, + 116 + ], + "lines": [ + { + "bbox": [ + 106, + 79, + 437, + 99 + ], + "spans": [ + { + "bbox": [ + 106, + 79, + 437, + 99 + ], + "score": 1.0, + "content": "GRAPH LAYOUTS AND GRAPH CONTRASTIVE", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 102, + 413, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 102, + 413, + 118 + ], + "score": 1.0, + "content": "LEARNING VIA NEIGHBOUR EMBEDDINGS", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 112, + 136, + 244, + 157 + ], + "lines": [ + { + "bbox": [ + 113, + 136, + 201, + 147 + ], + "spans": [ + { + "bbox": [ + 113, + 136, + 201, + 147 + ], + "score": 1.0, + "content": "Anonymous authors", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 112, + 146, + 245, + 159 + ], + "spans": [ + { + "bbox": [ + 112, + 146, + 245, + 159 + ], + "score": 1.0, + "content": "Paper under double-blind review", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2.5 + }, + { + "type": "title", + "bbox": [ + 278, + 186, + 333, + 199 + ], + "lines": [ + { + "bbox": [ + 277, + 186, + 335, + 200 + ], + "spans": [ + { + "bbox": [ + 277, + 186, + 335, + 200 + ], + "score": 1.0, + "content": "ABSTRACT", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 143, + 210, + 468, + 375 + ], + "lines": [ + { + "bbox": [ + 141, + 210, + 470, + 223 + ], + "spans": [ + { + "bbox": [ + 141, + 210, + 470, + 223 + ], + "score": 1.0, + "content": "In node-level graph representation learning, there are two distinct paradigms. One", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 141, + 221, + 469, + 234 + ], + "spans": [ + { + "bbox": [ + 141, + 221, + 469, + 234 + ], + "score": 1.0, + "content": "is known as graph layouts, where nodes are embedded into 2D space for visualiza-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 141, + 233, + 469, + 244 + ], + "spans": [ + { + "bbox": [ + 141, + 233, + 469, + 244 + ], + "score": 1.0, + "content": "tion purposes. Another is graph contrastive learning, where nodes are parametri-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 141, + 243, + 470, + 255 + ], + "spans": [ + { + "bbox": [ + 141, + 243, + 470, + 255 + ], + "score": 1.0, + "content": "cally embedded into a high-dimensional vector space based on node features. In", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 141, + 254, + 469, + 266 + ], + "spans": [ + { + "bbox": [ + 141, + 254, + 469, + 266 + ], + "score": 1.0, + "content": "this work, we show that these two paradigms are intimately related, and that both", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 141, + 266, + 469, + 277 + ], + "spans": [ + { + "bbox": [ + 141, + 266, + 469, + 277 + ], + "score": 1.0, + "content": "can be successfully approached via neighbour embedding methods. First, we in-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 276, + 470, + 288 + ], + "spans": [ + { + "bbox": [ + 141, + 276, + 200, + 288 + ], + "score": 1.0, + "content": "troduce graph", + "type": "text" + }, + { + "bbox": [ + 200, + 277, + 205, + 286 + ], + "score": 0.52, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 276, + 470, + 288 + ], + "score": 1.0, + "content": "-SNE for two-dimensional graph drawing, and show that the result-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 287, + 470, + 300 + ], + "spans": [ + { + "bbox": [ + 141, + 287, + 470, + 300 + ], + "score": 1.0, + "content": "ing layouts outperform all existing algorithms in terms of local structure preser-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 298, + 470, + 311 + ], + "spans": [ + { + "bbox": [ + 141, + 298, + 236, + 311 + ], + "score": 1.0, + "content": "vation, as measured by", + "type": "text" + }, + { + "bbox": [ + 236, + 298, + 258, + 308 + ], + "score": 0.45, + "content": "k \\mathbf { N N }", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 298, + 470, + 311 + ], + "score": 1.0, + "content": "classification accuracy. Second, we introduce graph", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 310, + 469, + 320 + ], + "spans": [ + { + "bbox": [ + 141, + 310, + 469, + 320 + ], + "score": 1.0, + "content": "contrastive neighbor embedding (graph CNE), which uses a fully-connected neu-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 319, + 469, + 333 + ], + "spans": [ + { + "bbox": [ + 141, + 319, + 469, + 333 + ], + "score": 1.0, + "content": "ral network (MLP) to transform graph node features into an embedding space by", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 331, + 470, + 343 + ], + "spans": [ + { + "bbox": [ + 141, + 331, + 470, + 343 + ], + "score": 1.0, + "content": "optimizing the contrastive InfoNCE objective. We show that graph CNE, while", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 142, + 343, + 469, + 354 + ], + "spans": [ + { + "bbox": [ + 142, + 343, + 469, + 354 + ], + "score": 1.0, + "content": "being conceptually simpler than most existing graph contrastive learning meth-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 142, + 353, + 469, + 365 + ], + "spans": [ + { + "bbox": [ + 142, + 353, + 469, + 365 + ], + "score": 1.0, + "content": "ods, produces competitive node representations and outperforms state-of-the-art", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 362, + 393, + 377 + ], + "spans": [ + { + "bbox": [ + 141, + 362, + 393, + 377 + ], + "score": 1.0, + "content": "MLP-based methods in terms of linear classification accuracy.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 12 + }, + { + "type": "title", + "bbox": [ + 108, + 394, + 206, + 406 + ], + "lines": [ + { + "bbox": [ + 105, + 393, + 208, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 208, + 410 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 418, + 505, + 496 + ], + "lines": [ + { + "bbox": [ + 105, + 418, + 506, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 506, + 432 + ], + "score": 1.0, + "content": "Many real-world datasets, ranging from molecule structure to citation networks come in form of", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 429, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 380, + 443 + ], + "score": 1.0, + "content": "graphs. As graphs are abstract objects consisting of a set of nodes", + "type": "text" + }, + { + "bbox": [ + 380, + 430, + 389, + 439 + ], + "score": 0.78, + "content": "\\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 429, + 468, + 443 + ], + "score": 1.0, + "content": "and a set of edges", + "type": "text" + }, + { + "bbox": [ + 468, + 430, + 475, + 439 + ], + "score": 0.72, + "content": "\\mathcal { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 429, + 505, + 443 + ], + "score": 1.0, + "content": ", graph", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 439, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 398, + 453 + ], + "score": 1.0, + "content": "representation learning, i.e. embedding graph nodes into a vector space", + "type": "text" + }, + { + "bbox": [ + 398, + 440, + 411, + 451 + ], + "score": 0.88, + "content": "\\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 439, + 505, + 453 + ], + "score": 1.0, + "content": ", is a popular approach", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 450, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 505, + 465 + ], + "score": 1.0, + "content": "in machine learning. Traditionally, a distinction is made between graph layout (or graph drawing)", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 461, + 505, + 476 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 250, + 476 + ], + "score": 1.0, + "content": "methods, which embed nodes into", + "type": "text" + }, + { + "bbox": [ + 250, + 462, + 263, + 472 + ], + "score": 0.86, + "content": "\\mathbb { R } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 461, + 505, + 476 + ], + "score": 1.0, + "content": "for visualization purposes, and graph contrastive learning", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 473, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 505, + 486 + ], + "score": 1.0, + "content": "methods, which use higher-dimensional embeddings more suitable for downstream analysis, such", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 483, + 227, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 227, + 497 + ], + "score": 1.0, + "content": "as classification or clustering.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 107, + 501, + 504, + 524 + ], + "lines": [ + { + "bbox": [ + 105, + 501, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 158, + 514 + ], + "score": 1.0, + "content": "For a graph", + "type": "text" + }, + { + "bbox": [ + 158, + 501, + 210, + 513 + ], + "score": 0.93, + "content": "G = ( \\nu , \\mathcal { E } )", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 501, + 505, + 514 + ], + "score": 1.0, + "content": ", graph layout methods usually only take into account its structure and", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 511, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 505, + 524 + ], + "score": 1.0, + "content": "obtain the layout by pulling together connected nodes. In contrast, graph contrastive learning (GCL)", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28.5 + }, + { + "type": "image", + "bbox": [ + 119, + 543, + 498, + 667 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 119, + 543, + 498, + 667 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 119, + 543, + 498, + 667 + ], + "spans": [ + { + "bbox": [ + 119, + 543, + 498, + 667 + ], + "score": 0.932, + "type": "image", + "image_path": "774b4b73abf66e7348ef56ad4f320e1e7f9e39a429d420e356684d04036a786b.jpg" + } + ] + } + ], + "index": 31, + "virtual_lines": [ + { + "bbox": [ + 119, + 543, + 498, + 584.3333333333334 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 119, + 584.3333333333334, + 498, + 625.6666666666667 + ], + "spans": [], + "index": 31 + }, + { + "bbox": [ + 119, + 625.6666666666667, + 498, + 667.0000000000001 + ], + "spans": [], + "index": 32 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 685, + 505, + 730 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 685, + 506, + 698 + ], + "spans": [ + { + "bbox": [ + 105, + 685, + 506, + 698 + ], + "score": 1.0, + "content": "Figure 1: 2D embeddings of the Amazon Computer (ACO) and Photo (APH) datasets, obtained", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 696, + 506, + 709 + ], + "spans": [ + { + "bbox": [ + 105, + 696, + 173, + 709 + ], + "score": 1.0, + "content": "using our graph", + "type": "text" + }, + { + "bbox": [ + 173, + 698, + 178, + 707 + ], + "score": 0.63, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 696, + 299, + 709 + ], + "score": 1.0, + "content": "-SNE and graph CNE. Graph", + "type": "text" + }, + { + "bbox": [ + 299, + 698, + 304, + 707 + ], + "score": 0.58, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 696, + 506, + 709 + ], + "score": 1.0, + "content": "-SNE is a graph layout method. Graph CNE is a", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 708, + 506, + 720 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 506, + 720 + ], + "score": 1.0, + "content": "graph contrastive learning method, mapping node features to 2D (or other dimensionality, e.g. 128D,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 718, + 471, + 731 + ], + "spans": [ + { + "bbox": [ + 105, + 718, + 471, + 731 + ], + "score": 1.0, + "content": "see Section 6) using a neural network. Embeddings were aligned using Procrustes rotation.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34.5 + } + ], + "index": 32.75 + } + ], + "page_idx": 0, + "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, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "score": 1.0, + "content": "1", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 80, + 435, + 116 + ], + "lines": [ + { + "bbox": [ + 106, + 79, + 437, + 99 + ], + "spans": [ + { + "bbox": [ + 106, + 79, + 437, + 99 + ], + "score": 1.0, + "content": "GRAPH LAYOUTS AND GRAPH CONTRASTIVE", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 102, + 413, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 102, + 413, + 118 + ], + "score": 1.0, + "content": "LEARNING VIA NEIGHBOUR EMBEDDINGS", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 112, + 136, + 244, + 157 + ], + "lines": [ + { + "bbox": [ + 113, + 136, + 201, + 147 + ], + "spans": [ + { + "bbox": [ + 113, + 136, + 201, + 147 + ], + "score": 1.0, + "content": "Anonymous authors", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 112, + 146, + 245, + 159 + ], + "spans": [ + { + "bbox": [ + 112, + 146, + 245, + 159 + ], + "score": 1.0, + "content": "Paper under double-blind review", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2.5, + "bbox_fs": [ + 112, + 136, + 245, + 159 + ] + }, + { + "type": "title", + "bbox": [ + 278, + 186, + 333, + 199 + ], + "lines": [ + { + "bbox": [ + 277, + 186, + 335, + 200 + ], + "spans": [ + { + "bbox": [ + 277, + 186, + 335, + 200 + ], + "score": 1.0, + "content": "ABSTRACT", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 143, + 210, + 468, + 375 + ], + "lines": [ + { + "bbox": [ + 141, + 210, + 470, + 223 + ], + "spans": [ + { + "bbox": [ + 141, + 210, + 470, + 223 + ], + "score": 1.0, + "content": "In node-level graph representation learning, there are two distinct paradigms. One", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 141, + 221, + 469, + 234 + ], + "spans": [ + { + "bbox": [ + 141, + 221, + 469, + 234 + ], + "score": 1.0, + "content": "is known as graph layouts, where nodes are embedded into 2D space for visualiza-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 141, + 233, + 469, + 244 + ], + "spans": [ + { + "bbox": [ + 141, + 233, + 469, + 244 + ], + "score": 1.0, + "content": "tion purposes. Another is graph contrastive learning, where nodes are parametri-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 141, + 243, + 470, + 255 + ], + "spans": [ + { + "bbox": [ + 141, + 243, + 470, + 255 + ], + "score": 1.0, + "content": "cally embedded into a high-dimensional vector space based on node features. In", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 141, + 254, + 469, + 266 + ], + "spans": [ + { + "bbox": [ + 141, + 254, + 469, + 266 + ], + "score": 1.0, + "content": "this work, we show that these two paradigms are intimately related, and that both", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 141, + 266, + 469, + 277 + ], + "spans": [ + { + "bbox": [ + 141, + 266, + 469, + 277 + ], + "score": 1.0, + "content": "can be successfully approached via neighbour embedding methods. First, we in-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 276, + 470, + 288 + ], + "spans": [ + { + "bbox": [ + 141, + 276, + 200, + 288 + ], + "score": 1.0, + "content": "troduce graph", + "type": "text" + }, + { + "bbox": [ + 200, + 277, + 205, + 286 + ], + "score": 0.52, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 276, + 470, + 288 + ], + "score": 1.0, + "content": "-SNE for two-dimensional graph drawing, and show that the result-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 287, + 470, + 300 + ], + "spans": [ + { + "bbox": [ + 141, + 287, + 470, + 300 + ], + "score": 1.0, + "content": "ing layouts outperform all existing algorithms in terms of local structure preser-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 298, + 470, + 311 + ], + "spans": [ + { + "bbox": [ + 141, + 298, + 236, + 311 + ], + "score": 1.0, + "content": "vation, as measured by", + "type": "text" + }, + { + "bbox": [ + 236, + 298, + 258, + 308 + ], + "score": 0.45, + "content": "k \\mathbf { N N }", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 298, + 470, + 311 + ], + "score": 1.0, + "content": "classification accuracy. Second, we introduce graph", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 310, + 469, + 320 + ], + "spans": [ + { + "bbox": [ + 141, + 310, + 469, + 320 + ], + "score": 1.0, + "content": "contrastive neighbor embedding (graph CNE), which uses a fully-connected neu-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 319, + 469, + 333 + ], + "spans": [ + { + "bbox": [ + 141, + 319, + 469, + 333 + ], + "score": 1.0, + "content": "ral network (MLP) to transform graph node features into an embedding space by", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 331, + 470, + 343 + ], + "spans": [ + { + "bbox": [ + 141, + 331, + 470, + 343 + ], + "score": 1.0, + "content": "optimizing the contrastive InfoNCE objective. We show that graph CNE, while", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 142, + 343, + 469, + 354 + ], + "spans": [ + { + "bbox": [ + 142, + 343, + 469, + 354 + ], + "score": 1.0, + "content": "being conceptually simpler than most existing graph contrastive learning meth-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 142, + 353, + 469, + 365 + ], + "spans": [ + { + "bbox": [ + 142, + 353, + 469, + 365 + ], + "score": 1.0, + "content": "ods, produces competitive node representations and outperforms state-of-the-art", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 362, + 393, + 377 + ], + "spans": [ + { + "bbox": [ + 141, + 362, + 393, + 377 + ], + "score": 1.0, + "content": "MLP-based methods in terms of linear classification accuracy.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 12, + "bbox_fs": [ + 141, + 210, + 470, + 377 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 394, + 206, + 406 + ], + "lines": [ + { + "bbox": [ + 105, + 393, + 208, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 208, + 410 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 418, + 505, + 496 + ], + "lines": [ + { + "bbox": [ + 105, + 418, + 506, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 506, + 432 + ], + "score": 1.0, + "content": "Many real-world datasets, ranging from molecule structure to citation networks come in form of", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 429, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 380, + 443 + ], + "score": 1.0, + "content": "graphs. As graphs are abstract objects consisting of a set of nodes", + "type": "text" + }, + { + "bbox": [ + 380, + 430, + 389, + 439 + ], + "score": 0.78, + "content": "\\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 429, + 468, + 443 + ], + "score": 1.0, + "content": "and a set of edges", + "type": "text" + }, + { + "bbox": [ + 468, + 430, + 475, + 439 + ], + "score": 0.72, + "content": "\\mathcal { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 429, + 505, + 443 + ], + "score": 1.0, + "content": ", graph", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 439, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 398, + 453 + ], + "score": 1.0, + "content": "representation learning, i.e. embedding graph nodes into a vector space", + "type": "text" + }, + { + "bbox": [ + 398, + 440, + 411, + 451 + ], + "score": 0.88, + "content": "\\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 439, + 505, + 453 + ], + "score": 1.0, + "content": ", is a popular approach", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 450, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 505, + 465 + ], + "score": 1.0, + "content": "in machine learning. Traditionally, a distinction is made between graph layout (or graph drawing)", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 461, + 505, + 476 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 250, + 476 + ], + "score": 1.0, + "content": "methods, which embed nodes into", + "type": "text" + }, + { + "bbox": [ + 250, + 462, + 263, + 472 + ], + "score": 0.86, + "content": "\\mathbb { R } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 461, + 505, + 476 + ], + "score": 1.0, + "content": "for visualization purposes, and graph contrastive learning", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 473, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 505, + 486 + ], + "score": 1.0, + "content": "methods, which use higher-dimensional embeddings more suitable for downstream analysis, such", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 483, + 227, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 227, + 497 + ], + "score": 1.0, + "content": "as classification or clustering.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 418, + 506, + 497 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 501, + 504, + 524 + ], + "lines": [ + { + "bbox": [ + 105, + 501, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 158, + 514 + ], + "score": 1.0, + "content": "For a graph", + "type": "text" + }, + { + "bbox": [ + 158, + 501, + 210, + 513 + ], + "score": 0.93, + "content": "G = ( \\nu , \\mathcal { E } )", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 501, + 505, + 514 + ], + "score": 1.0, + "content": ", graph layout methods usually only take into account its structure and", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 511, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 505, + 524 + ], + "score": 1.0, + "content": "obtain the layout by pulling together connected nodes. In contrast, graph contrastive learning (GCL)", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 501, + 505, + 524 + ] + }, + { + "type": "image", + "bbox": [ + 119, + 543, + 498, + 667 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 119, + 543, + 498, + 667 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 119, + 543, + 498, + 667 + ], + "spans": [ + { + "bbox": [ + 119, + 543, + 498, + 667 + ], + "score": 0.932, + "type": "image", + "image_path": "774b4b73abf66e7348ef56ad4f320e1e7f9e39a429d420e356684d04036a786b.jpg" + } + ] + } + ], + "index": 31, + "virtual_lines": [ + { + "bbox": [ + 119, + 543, + 498, + 584.3333333333334 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 119, + 584.3333333333334, + 498, + 625.6666666666667 + ], + "spans": [], + "index": 31 + }, + { + "bbox": [ + 119, + 625.6666666666667, + 498, + 667.0000000000001 + ], + "spans": [], + "index": 32 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 685, + 505, + 730 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 685, + 506, + 698 + ], + "spans": [ + { + "bbox": [ + 105, + 685, + 506, + 698 + ], + "score": 1.0, + "content": "Figure 1: 2D embeddings of the Amazon Computer (ACO) and Photo (APH) datasets, obtained", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 696, + 506, + 709 + ], + "spans": [ + { + "bbox": [ + 105, + 696, + 173, + 709 + ], + "score": 1.0, + "content": "using our graph", + "type": "text" + }, + { + "bbox": [ + 173, + 698, + 178, + 707 + ], + "score": 0.63, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 696, + 299, + 709 + ], + "score": 1.0, + "content": "-SNE and graph CNE. Graph", + "type": "text" + }, + { + "bbox": [ + 299, + 698, + 304, + 707 + ], + "score": 0.58, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 696, + 506, + 709 + ], + "score": 1.0, + "content": "-SNE is a graph layout method. Graph CNE is a", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 708, + 506, + 720 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 506, + 720 + ], + "score": 1.0, + "content": "graph contrastive learning method, mapping node features to 2D (or other dimensionality, e.g. 128D,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 718, + 471, + 731 + ], + "spans": [ + { + "bbox": [ + 105, + 718, + 471, + 731 + ], + "score": 1.0, + "content": "see Section 6) using a neural network. 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. 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 + }, + { + "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 + }, + { + "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 + } + ], + "index": 45.5 + } + ], + "page_idx": 2, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "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, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "score": 1.0, + "content": "3", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 138 + ], + "lines": [], + "index": 2, + "bbox_fs": [ + 105, + 82, + 505, + 139 + ], + "lines_deleted": true + }, + { + "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, + "bbox_fs": [ + 104, + 200, + 506, + 256 + ] + }, + { + "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, + "bbox_fs": [ + 105, + 259, + 506, + 308 + ] + }, + { + "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, + "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. Together, this yields six combinations, called parametric/non-parametric NC-", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 257, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 106, + 259, + 111, + 268 + ], + "score": 0.72, + "content": "t", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 112, + 257, + 171, + 271 + ], + "score": 1.0, + "content": "-SNE, InfoNC-", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 171, + 259, + 176, + 268 + ], + "score": 0.61, + "content": "\\mathbf { \\nabla } \\cdot t", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 176, + 257, + 237, + 271 + ], + "score": 1.0, + "content": "-SNE, and Neg-", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 237, + 259, + 243, + 268 + ], + "score": 0.42, + "content": "\\cdot t", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 243, + 257, + 424, + 271 + ], + "score": 1.0, + "content": "-SNE. Damrich et al. (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
", + "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": [], + "index": 7.5, + "bbox_fs": [ + 105, + 226, + 506, + 292 + ], + "lines_deleted": true + }, + { + "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, + "bbox_fs": [ + 103, + 295, + 505, + 321 + ] + }, + { + "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, + "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. This metric is similar to what is commonly used in the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 514, + 506, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 506, + 528 + ], + "score": 1.0, + "content": "literature to benchmark graph layout algorithms (Kruiger et al., 2017; Zhu et al., 2020a; Zhong et al.,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 525, + 380, + 539 + ], + "spans": [ + { + "bbox": [ + 106, + 525, + 380, + 539 + ], + "score": 1.0, + "content": "2023), and so is our main metric for measuring graph layout quality.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 542, + 505, + 609 + ], + "lines": [ + { + "bbox": [ + 106, + 542, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 106, + 542, + 125, + 555 + ], + "score": 1.0, + "content": "The", + "type": "text" + }, + { + "bbox": [ + 125, + 543, + 147, + 553 + ], + "score": 0.31, + "content": "k \\mathbf { N N }", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 542, + 505, + 555 + ], + "score": 1.0, + "content": "classification accuracy quantifies local class separation in the embedding. To calculate", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 107, + 553, + 505, + 566 + ], + "spans": [ + { + "bbox": [ + 107, + 554, + 128, + 564 + ], + "score": 0.51, + "content": "k \\mathbf { N N }", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 553, + 297, + 566 + ], + "score": 1.0, + "content": "accuracy, we split all nodes into a training", + "type": "text" + }, + { + "bbox": [ + 298, + 554, + 312, + 566 + ], + "score": 0.8, + "content": "( 2 / 3", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 553, + 417, + 566 + ], + "score": 1.0, + "content": "of all nodes) and a test set", + "type": "text" + }, + { + "bbox": [ + 418, + 554, + 433, + 565 + ], + "score": 0.83, + "content": "( 1 / 3", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 553, + 505, + 566 + ], + "score": 1.0, + "content": "of all nodes), and", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 564, + 506, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 396, + 577 + ], + "score": 1.0, + "content": "used the sklearn.neighbors.KNeighborsClassifier with", + "type": "text" + }, + { + "bbox": [ + 396, + 565, + 430, + 575 + ], + "score": 0.89, + "content": "k = 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 564, + 506, + 577 + ], + "score": 1.0, + "content": "(Pedregosa et al.,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 574, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 505, + 590 + ], + "score": 1.0, + "content": "2011). 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 + }, + { + "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 + }, + { + "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 + }, + { + "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 + } + ], + "index": 32 + } + ], + "page_idx": 4, + "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, + 751, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "score": 1.0, + "content": "5", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "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, + "bbox_fs": [ + 105, + 365, + 505, + 402 + ] + }, + { + "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, + "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. This metric is similar to what is commonly used in the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 514, + 506, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 506, + 528 + ], + "score": 1.0, + "content": "literature to benchmark graph layout algorithms (Kruiger et al., 2017; Zhu et al., 2020a; Zhong et al.,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 525, + 380, + 539 + ], + "spans": [ + { + "bbox": [ + 106, + 525, + 380, + 539 + ], + "score": 1.0, + "content": "2023), and so is our main metric for measuring graph layout quality.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16, + "bbox_fs": [ + 105, + 480, + 506, + 539 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 542, + 505, + 609 + ], + "lines": [ + { + "bbox": [ + 106, + 542, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 106, + 542, + 125, + 555 + ], + "score": 1.0, + "content": "The", + "type": "text" + }, + { + "bbox": [ + 125, + 543, + 147, + 553 + ], + "score": 0.31, + "content": "k \\mathbf { N N }", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 542, + 505, + 555 + ], + "score": 1.0, + "content": "classification accuracy quantifies local class separation in the embedding. To calculate", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 107, + 553, + 505, + 566 + ], + "spans": [ + { + "bbox": [ + 107, + 554, + 128, + 564 + ], + "score": 0.51, + "content": "k \\mathbf { N N }", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 553, + 297, + 566 + ], + "score": 1.0, + "content": "accuracy, we split all nodes into a training", + "type": "text" + }, + { + "bbox": [ + 298, + 554, + 312, + 566 + ], + "score": 0.8, + "content": "( 2 / 3", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 553, + 417, + 566 + ], + "score": 1.0, + "content": "of all nodes) and a test set", + "type": "text" + }, + { + "bbox": [ + 418, + 554, + 433, + 565 + ], + "score": 0.83, + "content": "( 1 / 3", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 553, + 505, + 566 + ], + "score": 1.0, + "content": "of all nodes), and", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 564, + 506, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 396, + 577 + ], + "score": 1.0, + "content": "used the sklearn.neighbors.KNeighborsClassifier with", + "type": "text" + }, + { + "bbox": [ + 396, + 565, + 430, + 575 + ], + "score": 0.89, + "content": "k = 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 564, + 506, + 577 + ], + "score": 1.0, + "content": "(Pedregosa et al.,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 574, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 505, + 590 + ], + "score": 1.0, + "content": "2011). 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. 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, + "bbox_fs": [ + 105, + 295, + 505, + 333 + ] + }, + { + "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, + "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. NCVis: noise contrastive approach for scalable visual-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 116, + 110, + 421, + 123 + ], + "spans": [ + { + "bbox": [ + 116, + 110, + 421, + 123 + ], + "score": 1.0, + "content": "ization. In Proceedings of The Web Conference 2020, pp. 2941–2947, 2020.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5 + }, + { + "type": "text", + "bbox": [ + 100, + 129, + 506, + 754 + ], + "lines": [ + { + "bbox": [ + 105, + 129, + 506, + 142 + ], + "spans": [ + { + "bbox": [ + 105, + 129, + 506, + 142 + ], + "score": 1.0, + "content": "Mikhail Belkin and Partha Niyogi. Laplacian eigenmaps for dimensionality reduction and data", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 114, + 140, + 365, + 154 + ], + "spans": [ + { + "bbox": [ + 114, + 140, + 365, + 154 + ], + "score": 1.0, + "content": "representation. Neural Computation, 15(6):1373–1396, 2003.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 158, + 504, + 171 + ], + "spans": [ + { + "bbox": [ + 105, + 158, + 504, + 171 + ], + "score": 1.0, + "content": "Anna C Belkina, Christopher O Ciccolella, Rina Anno, Richard Halpert, Josef Spidlen, and Jen-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 114, + 168, + 506, + 183 + ], + "spans": [ + { + "bbox": [ + 114, + 168, + 506, + 183 + ], + "score": 1.0, + "content": "nifer E Snyder-Cappione. Automated optimized parameters for T-distributed stochastic neighbor", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 116, + 181, + 505, + 194 + ], + "spans": [ + { + "bbox": [ + 116, + 181, + 505, + 194 + ], + "score": 1.0, + "content": "embedding improve visualization and analysis of large datasets. Nature Communications, 10(1):", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 115, + 191, + 168, + 204 + ], + "spans": [ + { + "bbox": [ + 115, + 191, + 168, + 204 + ], + "score": 1.0, + "content": "5415, 2019.", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 211, + 505, + 224 + ], + "spans": [ + { + "bbox": [ + 105, + 211, + 505, + 224 + ], + "score": 1.0, + "content": "Jan Niklas Bohm, Philipp Berens, and Dmitry Kobak. Attraction-repulsion spectrum in neighbor ¨", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 115, + 222, + 447, + 235 + ], + "spans": [ + { + "bbox": [ + 115, + 222, + 447, + 235 + ], + "score": 1.0, + "content": "embeddings. The Journal of Machine Learning Research, 23(1):4118–4149, 2022.", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 240, + 506, + 253 + ], + "spans": [ + { + "bbox": [ + 105, + 240, + 506, + 253 + ], + "score": 1.0, + "content": "Jan Niklas Bohm, Philipp Berens, and Dmitry Kobak. Unsupervised visualization of image datasets ¨", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 116, + 252, + 473, + 264 + ], + "spans": [ + { + "bbox": [ + 116, + 252, + 473, + 264 + ], + "score": 1.0, + "content": "using contrastive learning. International Conference on Learning Representations, 2023.", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 268, + 506, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 506, + 282 + ], + "score": 1.0, + "content": "Ting Chen, Simon Kornblith, Mohammad Norouzi, and Geoffrey Hinton. A simple framework for", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 114, + 279, + 506, + 295 + ], + "spans": [ + { + "bbox": [ + 114, + 279, + 506, + 295 + ], + "score": 1.0, + "content": "contrastive learning of visual representations. In International Conference on Machine Learning,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 114, + 291, + 239, + 304 + ], + "spans": [ + { + "bbox": [ + 114, + 291, + 239, + 304 + ], + "score": 1.0, + "content": "pp. 1597–1607. PMLR, 2020.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 309, + 505, + 322 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 434, + 322 + ], + "score": 1.0, + "content": "Sebastian Damrich, Niklas Bohm, Fred A Hamprecht, and Dmitry Kobak. From ¨", + "type": "text" + }, + { + "bbox": [ + 434, + 311, + 439, + 319 + ], + "score": 0.6, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 309, + 505, + 322 + ], + "score": 1.0, + "content": "-SNE to UMAP", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 115, + 321, + 506, + 334 + ], + "spans": [ + { + "bbox": [ + 115, + 321, + 506, + 334 + ], + "score": 1.0, + "content": "with contrastive learning. In The Eleventh International Conference on Learning Representations,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 115, + 331, + 142, + 345 + ], + "spans": [ + { + "bbox": [ + 115, + 331, + 142, + 345 + ], + "score": 1.0, + "content": "2022.", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 348, + 505, + 364 + ], + "spans": [ + { + "bbox": [ + 104, + 348, + 505, + 364 + ], + "score": 1.0, + "content": "Thomas MJ Fruchterman and Edward M Reingold. Graph drawing by force-directed placement.", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 116, + 361, + 366, + 374 + ], + "spans": [ + { + "bbox": [ + 116, + 361, + 366, + 374 + ], + "score": 1.0, + "content": "Software: Practice and Experience, 21(11):1129–1164, 1991.", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 379, + 506, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 506, + 392 + ], + "score": 1.0, + "content": "Xiaojun Guo, Yifei Wang, Zeming Wei, and Yisen Wang. Architecture matters: Uncovering implicit", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 115, + 390, + 505, + 404 + ], + "spans": [ + { + "bbox": [ + 115, + 390, + 505, + 404 + ], + "score": 1.0, + "content": "mechanisms in graph contrastive learning. In Thirty-seventh Conference on Neural Information", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 115, + 402, + 225, + 415 + ], + "spans": [ + { + "bbox": [ + 115, + 402, + 225, + 415 + ], + "score": 1.0, + "content": "Processing Systems, 2023.", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 420, + 505, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 505, + 433 + ], + "score": 1.0, + "content": "Michael Gutmann and Aapo Hyvarinen. Noise-contrastive estimation: A new estimation principle ¨", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 114, + 430, + 506, + 444 + ], + "spans": [ + { + "bbox": [ + 114, + 430, + 506, + 444 + ], + "score": 1.0, + "content": "for unnormalized statistical models. In Proceedings of the Thirteenth International Conference on", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 113, + 439, + 506, + 457 + ], + "spans": [ + { + "bbox": [ + 113, + 439, + 506, + 457 + ], + "score": 1.0, + "content": "Artificial Intelligence and Statistics, pp. 297–304. JMLR Workshop and Conference Proceedings,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 115, + 451, + 142, + 466 + ], + "spans": [ + { + "bbox": [ + 115, + 451, + 142, + 466 + ], + "score": 1.0, + "content": "2010.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 469, + 505, + 485 + ], + "spans": [ + { + "bbox": [ + 104, + 469, + 505, + 485 + ], + "score": 1.0, + "content": "Aric Hagberg, Pieter Swart, and Daniel S Chult. Exploring network structure, dynamics, and func-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 115, + 482, + 409, + 495 + ], + "spans": [ + { + "bbox": [ + 115, + 482, + 409, + 495 + ], + "score": 1.0, + "content": "tion using NetworkX. Technical report, Los Alamos National Lab, 2008.", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 499, + 506, + 515 + ], + "spans": [ + { + "bbox": [ + 104, + 499, + 506, + 515 + ], + "score": 1.0, + "content": "Kaveh Hassani and Amir Hosein Khasahmadi. Contrastive multi-view representation learning on", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 114, + 511, + 477, + 525 + ], + "spans": [ + { + "bbox": [ + 114, + 511, + 477, + 525 + ], + "score": 1.0, + "content": "graphs. In International Conference on Machine Learning, pp. 4116–4126. PMLR, 2020.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 528, + 505, + 544 + ], + "spans": [ + { + "bbox": [ + 104, + 528, + 505, + 544 + ], + "score": 1.0, + "content": "Geoffrey E Hinton and Sam Roweis. Stochastic neighbor embedding. Advances in Neural Informa-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 115, + 541, + 257, + 554 + ], + "spans": [ + { + "bbox": [ + 115, + 541, + 257, + 554 + ], + "score": 1.0, + "content": "tion Processing Systems, 15, 2002.", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 557, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 104, + 557, + 506, + 574 + ], + "score": 1.0, + "content": "Tianyang Hu, Zhili Liu, Fengwei Zhou, Wenjia Wang, and Weiran Huang. Your contrastive learning", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 114, + 569, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 114, + 569, + 505, + 585 + ], + "score": 1.0, + "content": "is secretly doing stochastic neighbor embedding. International Conference on Learning Repre-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 114, + 582, + 187, + 593 + ], + "spans": [ + { + "bbox": [ + 114, + 582, + 187, + 593 + ], + "score": 1.0, + "content": "sentations, 2023.", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 598, + 506, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 598, + 506, + 614 + ], + "score": 1.0, + "content": "Weihua Hu, Matthias Fey, Marinka Zitnik, Yuxiao Dong, Hongyu Ren, Bowen Liu, Michele Catasta,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 114, + 609, + 506, + 624 + ], + "spans": [ + { + "bbox": [ + 114, + 609, + 506, + 624 + ], + "score": 1.0, + "content": "and Jure Leskovec. Open graph benchmark: Datasets for machine learning on graphs. Advances", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 116, + 623, + 386, + 634 + ], + "spans": [ + { + "bbox": [ + 116, + 623, + 386, + 634 + ], + "score": 1.0, + "content": "in Neural Information Processing Systems, 33:22118–22133, 2020.", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 638, + 506, + 654 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 506, + 654 + ], + "score": 1.0, + "content": "Yifan Hu. Efficient, high-quality force-directed graph drawing. Mathematica Journal, 10(1):37–71,", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 114, + 650, + 143, + 665 + ], + "spans": [ + { + "bbox": [ + 114, + 650, + 143, + 665 + ], + "score": 1.0, + "content": "2005.", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 103, + 668, + 507, + 684 + ], + "spans": [ + { + "bbox": [ + 103, + 668, + 507, + 684 + ], + "score": 1.0, + "content": "Mathieu Jacomy, Tommaso Venturini, Sebastien Heymann, and Mathieu Bastian. ForceAtlas2, a", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 115, + 681, + 505, + 694 + ], + "spans": [ + { + "bbox": [ + 115, + 681, + 505, + 694 + ], + "score": 1.0, + "content": "continuous graph layout algorithm for handy network visualization designed for the gephi soft-", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 115, + 692, + 264, + 704 + ], + "spans": [ + { + "bbox": [ + 115, + 692, + 264, + 704 + ], + "score": 1.0, + "content": "ware. PloS One, 9(6):e98679, 2014.", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 104, + 708, + 506, + 724 + ], + "spans": [ + { + "bbox": [ + 104, + 708, + 506, + 724 + ], + "score": 1.0, + "content": "Rafal Jozefowicz, Oriol Vinyals, Mike Schuster, Noam Shazeer, and Yonghui Wu. Exploring the", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 115, + 720, + 398, + 735 + ], + "spans": [ + { + "bbox": [ + 115, + 720, + 398, + 735 + ], + "score": 1.0, + "content": "limits of language modeling. arXiv preprint arXiv:1602.02410, 2016.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 24.5 + } + ], + "page_idx": 9, + "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": [ + 301, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 313, + 765 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 313, + 765 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 15, + "width": 14 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 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. NCVis: noise contrastive approach for scalable visual-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 116, + 110, + 421, + 123 + ], + "spans": [ + { + "bbox": [ + 116, + 110, + 421, + 123 + ], + "score": 1.0, + "content": "ization. In Proceedings of The Web Conference 2020, pp. 2941–2947, 2020.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5, + "bbox_fs": [ + 106, + 99, + 505, + 123 + ] + }, + { + "type": "list", + "bbox": [ + 100, + 129, + 506, + 754 + ], + "lines": [ + { + "bbox": [ + 105, + 129, + 506, + 142 + ], + "spans": [ + { + "bbox": [ + 105, + 129, + 506, + 142 + ], + "score": 1.0, + "content": "Mikhail Belkin and Partha Niyogi. Laplacian eigenmaps for dimensionality reduction and data", + "type": "text" + } + ], + "index": 3, + "is_list_start_line": true + }, + { + "bbox": [ + 114, + 140, + 365, + 154 + ], + "spans": [ + { + "bbox": [ + 114, + 140, + 365, + 154 + ], + "score": 1.0, + "content": "representation. Neural Computation, 15(6):1373–1396, 2003.", + "type": "text" + } + ], + "index": 4, + "is_list_end_line": true + }, + { + "bbox": [ + 105, + 158, + 504, + 171 + ], + "spans": [ + { + "bbox": [ + 105, + 158, + 504, + 171 + ], + "score": 1.0, + "content": "Anna C Belkina, Christopher O Ciccolella, Rina Anno, Richard Halpert, Josef Spidlen, and Jen-", + "type": "text" + } + ], + "index": 5, + "is_list_start_line": true + }, + { + "bbox": [ + 114, + 168, + 506, + 183 + ], + "spans": [ + { + "bbox": [ + 114, + 168, + 506, + 183 + ], + "score": 1.0, + "content": "nifer E Snyder-Cappione. Automated optimized parameters for T-distributed stochastic neighbor", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 116, + 181, + 505, + 194 + ], + "spans": [ + { + "bbox": [ + 116, + 181, + 505, + 194 + ], + "score": 1.0, + "content": "embedding improve visualization and analysis of large datasets. Nature Communications, 10(1):", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 115, + 191, + 168, + 204 + ], + "spans": [ + { + "bbox": [ + 115, + 191, + 168, + 204 + ], + "score": 1.0, + "content": "5415, 2019.", + "type": "text" + } + ], + "index": 8, + "is_list_end_line": true + }, + { + "bbox": [ + 105, + 211, + 505, + 224 + ], + "spans": [ + { + "bbox": [ + 105, + 211, + 505, + 224 + ], + "score": 1.0, + "content": "Jan Niklas Bohm, Philipp Berens, and Dmitry Kobak. Attraction-repulsion spectrum in neighbor ¨", + "type": "text" + } + ], + "index": 9, + "is_list_start_line": true + }, + { + "bbox": [ + 115, + 222, + 447, + 235 + ], + "spans": [ + { + "bbox": [ + 115, + 222, + 447, + 235 + ], + "score": 1.0, + "content": "embeddings. The Journal of Machine Learning Research, 23(1):4118–4149, 2022.", + "type": "text" + } + ], + "index": 10, + "is_list_end_line": true + }, + { + "bbox": [ + 105, + 240, + 506, + 253 + ], + "spans": [ + { + "bbox": [ + 105, + 240, + 506, + 253 + ], + "score": 1.0, + "content": "Jan Niklas Bohm, Philipp Berens, and Dmitry Kobak. Unsupervised visualization of image datasets ¨", + "type": "text" + } + ], + "index": 11, + "is_list_start_line": true + }, + { + "bbox": [ + 116, + 252, + 473, + 264 + ], + "spans": [ + { + "bbox": [ + 116, + 252, + 473, + 264 + ], + "score": 1.0, + "content": "using contrastive learning. International Conference on Learning Representations, 2023.", + "type": "text" + } + ], + "index": 12, + "is_list_end_line": true + }, + { + "bbox": [ + 105, + 268, + 506, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 506, + 282 + ], + "score": 1.0, + "content": "Ting Chen, Simon Kornblith, Mohammad Norouzi, and Geoffrey Hinton. A simple framework for", + "type": "text" + } + ], + "index": 13, + "is_list_start_line": true + }, + { + "bbox": [ + 114, + 279, + 506, + 295 + ], + "spans": [ + { + "bbox": [ + 114, + 279, + 506, + 295 + ], + "score": 1.0, + "content": "contrastive learning of visual representations. In International Conference on Machine Learning,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 114, + 291, + 239, + 304 + ], + "spans": [ + { + "bbox": [ + 114, + 291, + 239, + 304 + ], + "score": 1.0, + "content": "pp. 1597–1607. PMLR, 2020.", + "type": "text" + } + ], + "index": 15, + "is_list_end_line": true + }, + { + "bbox": [ + 105, + 309, + 505, + 322 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 434, + 322 + ], + "score": 1.0, + "content": "Sebastian Damrich, Niklas Bohm, Fred A Hamprecht, and Dmitry Kobak. From ¨", + "type": "text" + }, + { + "bbox": [ + 434, + 311, + 439, + 319 + ], + "score": 0.6, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 309, + 505, + 322 + ], + "score": 1.0, + "content": "-SNE to UMAP", + "type": "text" + } + ], + "index": 16, + "is_list_start_line": true + }, + { + "bbox": [ + 115, + 321, + 506, + 334 + ], + "spans": [ + { + "bbox": [ + 115, + 321, + 506, + 334 + ], + "score": 1.0, + "content": "with contrastive learning. In The Eleventh International Conference on Learning Representations,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 115, + 331, + 142, + 345 + ], + "spans": [ + { + "bbox": [ + 115, + 331, + 142, + 345 + ], + "score": 1.0, + "content": "2022.", + "type": "text" + } + ], + "index": 18, + "is_list_end_line": true + }, + { + "bbox": [ + 104, + 348, + 505, + 364 + ], + "spans": [ + { + "bbox": [ + 104, + 348, + 505, + 364 + ], + "score": 1.0, + "content": "Thomas MJ Fruchterman and Edward M Reingold. Graph drawing by force-directed placement.", + "type": "text" + } + ], + "index": 19, + "is_list_start_line": true + }, + { + "bbox": [ + 116, + 361, + 366, + 374 + ], + "spans": [ + { + "bbox": [ + 116, + 361, + 366, + 374 + ], + "score": 1.0, + "content": "Software: Practice and Experience, 21(11):1129–1164, 1991.", + "type": "text" + } + ], + "index": 20, + "is_list_end_line": true + }, + { + "bbox": [ + 105, + 379, + 506, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 506, + 392 + ], + "score": 1.0, + "content": "Xiaojun Guo, Yifei Wang, Zeming Wei, and Yisen Wang. Architecture matters: Uncovering implicit", + "type": "text" + } + ], + "index": 21, + "is_list_start_line": true + }, + { + "bbox": [ + 115, + 390, + 505, + 404 + ], + "spans": [ + { + "bbox": [ + 115, + 390, + 505, + 404 + ], + "score": 1.0, + "content": "mechanisms in graph contrastive learning. In Thirty-seventh Conference on Neural Information", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 115, + 402, + 225, + 415 + ], + "spans": [ + { + "bbox": [ + 115, + 402, + 225, + 415 + ], + "score": 1.0, + "content": "Processing Systems, 2023.", + "type": "text" + } + ], + "index": 23, + "is_list_end_line": true + }, + { + "bbox": [ + 105, + 420, + 505, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 505, + 433 + ], + "score": 1.0, + "content": "Michael Gutmann and Aapo Hyvarinen. Noise-contrastive estimation: A new estimation principle ¨", + "type": "text" + } + ], + "index": 24, + "is_list_start_line": true + }, + { + "bbox": [ + 114, + 430, + 506, + 444 + ], + "spans": [ + { + "bbox": [ + 114, + 430, + 506, + 444 + ], + "score": 1.0, + "content": "for unnormalized statistical models. In Proceedings of the Thirteenth International Conference on", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 113, + 439, + 506, + 457 + ], + "spans": [ + { + "bbox": [ + 113, + 439, + 506, + 457 + ], + "score": 1.0, + "content": "Artificial Intelligence and Statistics, pp. 297–304. JMLR Workshop and Conference Proceedings,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 115, + 451, + 142, + 466 + ], + "spans": [ + { + "bbox": [ + 115, + 451, + 142, + 466 + ], + "score": 1.0, + "content": "2010.", + "type": "text" + } + ], + "index": 27, + "is_list_end_line": true + }, + { + "bbox": [ + 104, + 469, + 505, + 485 + ], + "spans": [ + { + "bbox": [ + 104, + 469, + 505, + 485 + ], + "score": 1.0, + "content": "Aric Hagberg, Pieter Swart, and Daniel S Chult. Exploring network structure, dynamics, and func-", + "type": "text" + } + ], + "index": 28, + "is_list_start_line": true + }, + { + "bbox": [ + 115, + 482, + 409, + 495 + ], + "spans": [ + { + "bbox": [ + 115, + 482, + 409, + 495 + ], + "score": 1.0, + "content": "tion using NetworkX. Technical report, Los Alamos National Lab, 2008.", + "type": "text" + } + ], + "index": 29, + "is_list_end_line": true + }, + { + "bbox": [ + 104, + 499, + 506, + 515 + ], + "spans": [ + { + "bbox": [ + 104, + 499, + 506, + 515 + ], + "score": 1.0, + "content": "Kaveh Hassani and Amir Hosein Khasahmadi. Contrastive multi-view representation learning on", + "type": "text" + } + ], + "index": 30, + "is_list_start_line": true + }, + { + "bbox": [ + 114, + 511, + 477, + 525 + ], + "spans": [ + { + "bbox": [ + 114, + 511, + 477, + 525 + ], + "score": 1.0, + "content": "graphs. In International Conference on Machine Learning, pp. 4116–4126. PMLR, 2020.", + "type": "text" + } + ], + "index": 31, + "is_list_end_line": true + }, + { + "bbox": [ + 104, + 528, + 505, + 544 + ], + "spans": [ + { + "bbox": [ + 104, + 528, + 505, + 544 + ], + "score": 1.0, + "content": "Geoffrey E Hinton and Sam Roweis. Stochastic neighbor embedding. Advances in Neural Informa-", + "type": "text" + } + ], + "index": 32, + "is_list_start_line": true + }, + { + "bbox": [ + 115, + 541, + 257, + 554 + ], + "spans": [ + { + "bbox": [ + 115, + 541, + 257, + 554 + ], + "score": 1.0, + "content": "tion Processing Systems, 15, 2002.", + "type": "text" + } + ], + "index": 33, + "is_list_end_line": true + }, + { + "bbox": [ + 104, + 557, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 104, + 557, + 506, + 574 + ], + "score": 1.0, + "content": "Tianyang Hu, Zhili Liu, Fengwei Zhou, Wenjia Wang, and Weiran Huang. Your contrastive learning", + "type": "text" + } + ], + "index": 34, + "is_list_start_line": true + }, + { + "bbox": [ + 114, + 569, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 114, + 569, + 505, + 585 + ], + "score": 1.0, + "content": "is secretly doing stochastic neighbor embedding. International Conference on Learning Repre-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 114, + 582, + 187, + 593 + ], + "spans": [ + { + "bbox": [ + 114, + 582, + 187, + 593 + ], + "score": 1.0, + "content": "sentations, 2023.", + "type": "text" + } + ], + "index": 36, + "is_list_end_line": true + }, + { + "bbox": [ + 105, + 598, + 506, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 598, + 506, + 614 + ], + "score": 1.0, + "content": "Weihua Hu, Matthias Fey, Marinka Zitnik, Yuxiao Dong, Hongyu Ren, Bowen Liu, Michele Catasta,", + "type": "text" + } + ], + "index": 37, + "is_list_start_line": true + }, + { + "bbox": [ + 114, + 609, + 506, + 624 + ], + "spans": [ + { + "bbox": [ + 114, + 609, + 506, + 624 + ], + "score": 1.0, + "content": "and Jure Leskovec. Open graph benchmark: Datasets for machine learning on graphs. Advances", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 116, + 623, + 386, + 634 + ], + "spans": [ + { + "bbox": [ + 116, + 623, + 386, + 634 + ], + "score": 1.0, + "content": "in Neural Information Processing Systems, 33:22118–22133, 2020.", + "type": "text" + } + ], + "index": 39, + "is_list_end_line": true + }, + { + "bbox": [ + 105, + 638, + 506, + 654 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 506, + 654 + ], + "score": 1.0, + "content": "Yifan Hu. Efficient, high-quality force-directed graph drawing. Mathematica Journal, 10(1):37–71,", + "type": "text" + } + ], + "index": 40, + "is_list_start_line": true + }, + { + "bbox": [ + 114, + 650, + 143, + 665 + ], + "spans": [ + { + "bbox": [ + 114, + 650, + 143, + 665 + ], + "score": 1.0, + "content": "2005.", + "type": "text" + } + ], + "index": 41, + "is_list_end_line": true + }, + { + "bbox": [ + 103, + 668, + 507, + 684 + ], + "spans": [ + { + "bbox": [ + 103, + 668, + 507, + 684 + ], + "score": 1.0, + "content": "Mathieu Jacomy, Tommaso Venturini, Sebastien Heymann, and Mathieu Bastian. ForceAtlas2, a", + "type": "text" + } + ], + "index": 42, + "is_list_start_line": true + }, + { + "bbox": [ + 115, + 681, + 505, + 694 + ], + "spans": [ + { + "bbox": [ + 115, + 681, + 505, + 694 + ], + "score": 1.0, + "content": "continuous graph layout algorithm for handy network visualization designed for the gephi soft-", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 115, + 692, + 264, + 704 + ], + "spans": [ + { + "bbox": [ + 115, + 692, + 264, + 704 + ], + "score": 1.0, + "content": "ware. PloS One, 9(6):e98679, 2014.", + "type": "text" + } + ], + "index": 44, + "is_list_end_line": true + }, + { + "bbox": [ + 104, + 708, + 506, + 724 + ], + "spans": [ + { + "bbox": [ + 104, + 708, + 506, + 724 + ], + "score": 1.0, + "content": "Rafal Jozefowicz, Oriol Vinyals, Mike Schuster, Noam Shazeer, and Yonghui Wu. Exploring the", + "type": "text" + } + ], + "index": 45, + "is_list_start_line": true + }, + { + "bbox": [ + 115, + 720, + 398, + 735 + ], + "spans": [ + { + "bbox": [ + 115, + 720, + 398, + 735 + ], + "score": 1.0, + "content": "limits of language modeling. arXiv preprint arXiv:1602.02410, 2016.", + "type": "text" + } + ], + "index": 46, + "is_list_end_line": true + }, + { + "bbox": [ + 105, + 81, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 506, + 96 + ], + "score": 1.0, + "content": "Diederik P Kingma and Jimmy Ba. Adam: A method for stochastic optimization. International", + "type": "text", + "cross_page": true + } + ], + "index": 0, + "is_list_start_line": true + }, + { + "bbox": [ + 114, + 92, + 310, + 108 + ], + "spans": [ + { + "bbox": [ + 114, + 92, + 310, + 108 + ], + "score": 1.0, + "content": "Conference on Learning Representations, 2015.", + "type": "text", + "cross_page": true + } + ], + "index": 1, + "is_list_end_line": true + }, + { + "bbox": [ + 105, + 112, + 506, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 112, + 506, + 127 + ], + "score": 1.0, + "content": "Thomas N Kipf and Max Welling. Semi-supervised classification with graph convolutional net-", + "type": "text", + "cross_page": true + } + ], + "index": 2, + "is_list_start_line": true + }, + { + "bbox": [ + 114, + 124, + 398, + 138 + ], + "spans": [ + { + "bbox": [ + 114, + 124, + 398, + 138 + ], + "score": 1.0, + "content": "works. International Conference for Learning Representations, 2017.", + "type": "text", + "cross_page": true + } + ], + "index": 3, + "is_list_end_line": true + }, + { + "bbox": [ + 105, + 144, + 506, + 159 + ], + "spans": [ + { + "bbox": [ + 105, + 144, + 506, + 159 + ], + "score": 1.0, + "content": "Dmitry Kobak and Philipp Berens. The art of using t-SNE for single-cell transcriptomics. Nature", + "type": "text", + "cross_page": true + } + ], + "index": 4, + "is_list_start_line": true + }, + { + "bbox": [ + 114, + 155, + 265, + 169 + ], + "spans": [ + { + "bbox": [ + 114, + 155, + 265, + 169 + ], + "score": 1.0, + "content": "Communications, 10(1):5416, 2019.", + "type": "text", + "cross_page": true + } + ], + "index": 5, + "is_list_end_line": true + }, + { + "bbox": [ + 105, + 176, + 506, + 190 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 506, + 190 + ], + "score": 1.0, + "content": "Dmitry Kobak and George C Linderman. Initialization is critical for preserving global data structure", + "type": "text", + "cross_page": true + } + ], + "index": 6, + "is_list_start_line": true + }, + { + "bbox": [ + 115, + 187, + 407, + 200 + ], + "spans": [ + { + "bbox": [ + 115, + 187, + 407, + 200 + ], + "score": 1.0, + "content": "in both t-SNE and UMAP. Nature Biotechnology, 39(2):156–157, 2021.", + "type": "text", + "cross_page": true + } + ], + "index": 7, + "is_list_end_line": true + }, + { + "bbox": [ + 105, + 205, + 506, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 205, + 506, + 221 + ], + "score": 1.0, + "content": "Dmitry Kobak, George Linderman, Stefan Steinerberger, Yuval Kluger, and Philipp Berens. Heavy-", + "type": "text", + "cross_page": true + } + ], + "index": 8, + "is_list_start_line": true + }, + { + "bbox": [ + 114, + 218, + 506, + 232 + ], + "spans": [ + { + "bbox": [ + 114, + 218, + 506, + 232 + ], + "score": 1.0, + "content": "tailed kernels reveal a finer cluster structure in t-SNE visualisations. In Joint European Conference", + "type": "text", + "cross_page": true + } + ], + "index": 9 + }, + { + "bbox": [ + 113, + 227, + 489, + 243 + ], + "spans": [ + { + "bbox": [ + 113, + 227, + 489, + 243 + ], + "score": 1.0, + "content": "on Machine Learning and Knowledge Discovery in Databases, pp. 124–139. Springer, 2019.", + "type": "text", + "cross_page": true + } + ], + "index": 10, + "is_list_end_line": true + }, + { + "bbox": [ + 105, + 247, + 506, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 506, + 263 + ], + "score": 1.0, + "content": "Johannes F Kruiger, Paulo E Rauber, Rafael Messias Martins, Andreas Kerren, Stephen Kobourov,", + "type": "text", + "cross_page": true + } + ], + "index": 11, + "is_list_start_line": true + }, + { + "bbox": [ + 114, + 258, + 506, + 276 + ], + "spans": [ + { + "bbox": [ + 114, + 258, + 506, + 276 + ], + "score": 1.0, + "content": "and Alexandru C Telea. Graph layouts by t-SNE. In Computer Graphics Forum, volume 36, pp.", + "type": "text", + "cross_page": true + } + ], + "index": 12 + }, + { + "bbox": [ + 115, + 271, + 272, + 285 + ], + "spans": [ + { + "bbox": [ + 115, + 271, + 272, + 285 + ], + "score": 1.0, + "content": "283–294. Wiley Online Library, 2017.", + "type": "text", + "cross_page": true + } + ], + "index": 13, + "is_list_end_line": true + }, + { + "bbox": [ + 105, + 289, + 506, + 305 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 506, + 305 + ], + "score": 1.0, + "content": "Namkyeong Lee, Junseok Lee, and Chanyoung Park. Augmentation-free self-supervised learning", + "type": "text", + "cross_page": true + } + ], + "index": 14, + "is_list_start_line": true + }, + { + "bbox": [ + 114, + 300, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 114, + 300, + 505, + 316 + ], + "score": 1.0, + "content": "on graphs. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 36, pp.", + "type": "text", + "cross_page": true + } + ], + "index": 15 + }, + { + "bbox": [ + 115, + 313, + 191, + 325 + ], + "spans": [ + { + "bbox": [ + 115, + 313, + 191, + 325 + ], + "score": 1.0, + "content": "7372–7380, 2022.", + "type": "text", + "cross_page": true + } + ], + "index": 16, + "is_list_end_line": true + }, + { + "bbox": [ + 105, + 331, + 506, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 506, + 347 + ], + "score": 1.0, + "content": "Yao Yang Leow, Thomas Laurent, and Xavier Bresson. GraphTSNE: a visualization technique", + "type": "text", + "cross_page": true + } + ], + "index": 17, + "is_list_start_line": true + }, + { + "bbox": [ + 114, + 344, + 506, + 358 + ], + "spans": [ + { + "bbox": [ + 114, + 344, + 506, + 358 + ], + "score": 1.0, + "content": "for graph-structured data. Representation Learning on Graphs and Manifold Workshop at the", + "type": "text", + "cross_page": true + } + ], + "index": 18 + }, + { + "bbox": [ + 114, + 355, + 366, + 369 + ], + "spans": [ + { + "bbox": [ + 114, + 355, + 366, + 369 + ], + "score": 1.0, + "content": "International Conference for Learning Representations, 2019.", + "type": "text", + "cross_page": true + } + ], + "index": 19, + "is_list_end_line": true + }, + { + "bbox": [ + 105, + 375, + 506, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 506, + 389 + ], + "score": 1.0, + "content": "Haifeng Li, Jun Cao, Jiawei Zhu, Qinyao Luo, Silu He, and Xuying Wang. Augmentation-free graph", + "type": "text", + "cross_page": true + } + ], + "index": 20, + "is_list_start_line": true + }, + { + "bbox": [ + 115, + 386, + 506, + 400 + ], + "spans": [ + { + "bbox": [ + 115, + 386, + 506, + 400 + ], + "score": 1.0, + "content": "contrastive learning of invariant-discriminative representations. IEEE Transactions on Neural", + "type": "text", + "cross_page": true + } + ], + "index": 21 + }, + { + "bbox": [ + 114, + 396, + 316, + 410 + ], + "spans": [ + { + "bbox": [ + 114, + 396, + 316, + 410 + ], + "score": 1.0, + "content": "Networks and Learning Systems, pp. 1–11, 2023.", + "type": "text", + "cross_page": true + } + ], + "index": 22, + "is_list_end_line": true + }, + { + "bbox": [ + 106, + 417, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 505, + 429 + ], + "score": 1.0, + "content": "George C Linderman and Stefan Steinerberger. Clustering with t-SNE, provably. SIAM Journal on", + "type": "text", + "cross_page": true + } + ], + "index": 23, + "is_list_start_line": true + }, + { + "bbox": [ + 115, + 429, + 322, + 441 + ], + "spans": [ + { + "bbox": [ + 115, + 429, + 322, + 441 + ], + "score": 1.0, + "content": "Mathematics of Data Science, 1(2):313–332, 2019.", + "type": "text", + "cross_page": true + } + ], + "index": 24, + "is_list_end_line": true + }, + { + "bbox": [ + 105, + 447, + 504, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 504, + 461 + ], + "score": 1.0, + "content": "George C Linderman, Manas Rachh, Jeremy G Hoskins, Stefan Steinerberger, and Yuval Kluger.", + "type": "text", + "cross_page": true + } + ], + "index": 25, + "is_list_start_line": true + }, + { + "bbox": [ + 114, + 459, + 506, + 473 + ], + "spans": [ + { + "bbox": [ + 114, + 459, + 506, + 473 + ], + "score": 1.0, + "content": "Fast interpolation-based t-SNE for improved visualization of single-cell RNA-seq data. Nature", + "type": "text", + "cross_page": true + } + ], + "index": 26 + }, + { + "bbox": [ + 115, + 471, + 245, + 483 + ], + "spans": [ + { + "bbox": [ + 115, + 471, + 245, + 483 + ], + "score": 1.0, + "content": "Methods, 16(3):243–245, 2019.", + "type": "text", + "cross_page": true + } + ], + "index": 27, + "is_list_end_line": true + }, + { + "bbox": [ + 105, + 490, + 506, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 490, + 506, + 504 + ], + "score": 1.0, + "content": "Leland McInnes, John Healy, and James Melville. UMAP: Uniform manifold approximation and", + "type": "text", + "cross_page": true + } + ], + "index": 28, + "is_list_start_line": true + }, + { + "bbox": [ + 113, + 500, + 424, + 515 + ], + "spans": [ + { + "bbox": [ + 113, + 500, + 424, + 515 + ], + "score": 1.0, + "content": "projection for dimension reduction. arXiv preprint arXiv:1802.03426, 2018.", + "type": "text", + "cross_page": true + } + ], + "index": 29, + "is_list_end_line": true + }, + { + "bbox": [ + 106, + 522, + 504, + 533 + ], + "spans": [ + { + "bbox": [ + 106, + 522, + 504, + 533 + ], + "score": 1.0, + "content": "Tomas Mikolov, Ilya Sutskever, Kai Chen, Greg S Corrado, and Jeff Dean. Distributed representa-", + "type": "text", + "cross_page": true + } + ], + "index": 30, + "is_list_start_line": true + }, + { + "bbox": [ + 114, + 532, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 114, + 532, + 506, + 546 + ], + "score": 1.0, + "content": "tions of words and phrases and their compositionality. Advances in Neural Information Processing", + "type": "text", + "cross_page": true + } + ], + "index": 31 + }, + { + "bbox": [ + 114, + 543, + 194, + 558 + ], + "spans": [ + { + "bbox": [ + 114, + 543, + 194, + 558 + ], + "score": 1.0, + "content": "Systems, 26, 2013.", + "type": "text", + "cross_page": true + } + ], + "index": 32, + "is_list_end_line": true + }, + { + "bbox": [ + 104, + 561, + 506, + 579 + ], + "spans": [ + { + "bbox": [ + 104, + 561, + 506, + 579 + ], + "score": 1.0, + "content": "Andreas Noack. Energy models for graph clustering. Journal of Graph Algorithms and Applications,", + "type": "text", + "cross_page": true + } + ], + "index": 33, + "is_list_start_line": true + }, + { + "bbox": [ + 115, + 575, + 206, + 587 + ], + "spans": [ + { + "bbox": [ + 115, + 575, + 206, + 587 + ], + "score": 1.0, + "content": "11(2):453–480, 2007.", + "type": "text", + "cross_page": true + } + ], + "index": 34, + "is_list_end_line": true + }, + { + "bbox": [ + 105, + 594, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 608 + ], + "score": 1.0, + "content": "Aaron van den Oord, Yazhe Li, and Oriol Vinyals. Representation learning with contrastive predic-", + "type": "text", + "cross_page": true + } + ], + "index": 35, + "is_list_start_line": true + }, + { + "bbox": [ + 114, + 606, + 329, + 618 + ], + "spans": [ + { + "bbox": [ + 114, + 606, + 329, + 618 + ], + "score": 1.0, + "content": "tive coding. arXiv preprint arXiv:1807.03748, 2018.", + "type": "text", + "cross_page": true + } + ], + "index": 36, + "is_list_end_line": true + }, + { + "bbox": [ + 105, + 625, + 506, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 625, + 506, + 639 + ], + "score": 1.0, + "content": "Fabian Pedregosa, Gael Varoquaux, Alexandre Gramfort, Vincent Michel, Bertrand Thirion, Olivier ¨", + "type": "text", + "cross_page": true + } + ], + "index": 37, + "is_list_start_line": true + }, + { + "bbox": [ + 114, + 635, + 506, + 650 + ], + "spans": [ + { + "bbox": [ + 114, + 635, + 506, + 650 + ], + "score": 1.0, + "content": "Grisel, Mathieu Blondel, Peter Prettenhofer, Ron Weiss, Vincent Dubourg, et al. Scikit-learn:", + "type": "text", + "cross_page": true + } + ], + "index": 38 + }, + { + "bbox": [ + 115, + 648, + 482, + 660 + ], + "spans": [ + { + "bbox": [ + 115, + 648, + 482, + 660 + ], + "score": 1.0, + "content": "Machine learning in Python. Journal of Machine Learning Research, 12:2825–2830, 2011.", + "type": "text", + "cross_page": true + } + ], + "index": 39, + "is_list_end_line": true + }, + { + "bbox": [ + 105, + 667, + 506, + 681 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 506, + 681 + ], + "score": 1.0, + "content": "Pavlin G Policar, Martin Stra ˇ zar, and Bla ˇ z Zupan. openTSNE: a modular Python library for t-SNE ˇ", + "type": "text", + "cross_page": true + } + ], + "index": 40, + "is_list_start_line": true + }, + { + "bbox": [ + 115, + 678, + 398, + 693 + ], + "spans": [ + { + "bbox": [ + 115, + 678, + 398, + 693 + ], + "score": 1.0, + "content": "dimensionality reduction and embedding. BioRxiv, pp. 731877, 2019.", + "type": "text", + "cross_page": true + } + ], + "index": 41, + "is_list_end_line": true + }, + { + "bbox": [ + 104, + 696, + 506, + 714 + ], + "spans": [ + { + "bbox": [ + 104, + 696, + 506, + 714 + ], + "score": 1.0, + "content": "Shantanu Thakoor, Corentin Tallec, Mohammad Gheshlaghi Azar, Mehdi Azabou, Eva L Dyer,", + "type": "text", + "cross_page": true + } + ], + "index": 42, + "is_list_start_line": true + }, + { + "bbox": [ + 114, + 708, + 506, + 724 + ], + "spans": [ + { + "bbox": [ + 114, + 708, + 506, + 724 + ], + "score": 1.0, + "content": "Remi Munos, Petar Velickovi ˇ c, and Michal Valko. Large-scale representation learning on graphs ´", + "type": "text", + "cross_page": true + } + ], + "index": 43 + }, + { + "bbox": [ + 116, + 720, + 354, + 735 + ], + "spans": [ + { + "bbox": [ + 116, + 720, + 354, + 735 + ], + "score": 1.0, + "content": "via bootstrapping. arXiv preprint arXiv:2102.06514, 2021.", + "type": "text", + "cross_page": true + } + ], + "index": 44, + "is_list_end_line": true + }, + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "Puja Trivedi, Ekdeep Singh Lubana, Yujun Yan, Yaoqing Yang, and Danai Koutra. Augmentations", + "type": "text", + "cross_page": true + } + ], + "index": 0, + "is_list_start_line": true + }, + { + "bbox": [ + 115, + 94, + 505, + 108 + ], + "spans": [ + { + "bbox": [ + 115, + 94, + 505, + 108 + ], + "score": 1.0, + "content": "in graph contrastive learning: Current methodological flaws & towards better practices. In Pro-", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 115, + 105, + 383, + 117 + ], + "spans": [ + { + "bbox": [ + 115, + 105, + 383, + 117 + ], + "score": 1.0, + "content": "ceedings of the ACM Web Conference 2022, pp. 1538–1549, 2022.", + "type": "text", + "cross_page": true + } + ], + "index": 2, + "is_list_end_line": true + }, + { + "bbox": [ + 105, + 122, + 505, + 137 + ], + "spans": [ + { + "bbox": [ + 105, + 122, + 505, + 137 + ], + "score": 1.0, + "content": "Laurens Van Der Maaten. Accelerating t-SNE using tree-based algorithms. The Journal of Machine", + "type": "text", + "cross_page": true + } + ], + "index": 3, + "is_list_start_line": true + }, + { + "bbox": [ + 115, + 134, + 298, + 148 + ], + "spans": [ + { + "bbox": [ + 115, + 134, + 298, + 148 + ], + "score": 1.0, + "content": "Learning Research, 15(1):3221–3245, 2014.", + "type": "text", + "cross_page": true + } + ], + "index": 4, + "is_list_end_line": true + }, + { + "bbox": [ + 104, + 153, + 505, + 168 + ], + "spans": [ + { + "bbox": [ + 104, + 153, + 505, + 168 + ], + "score": 1.0, + "content": "Laurens Van der Maaten and Geoffrey Hinton. Visualizing data using t-SNE. Journal of Machine", + "type": "text", + "cross_page": true + } + ], + "index": 5, + "is_list_start_line": true + }, + { + "bbox": [ + 115, + 165, + 248, + 177 + ], + "spans": [ + { + "bbox": [ + 115, + 165, + 248, + 177 + ], + "score": 1.0, + "content": "Learning Research, 9(11), 2008.", + "type": "text", + "cross_page": true + } + ], + "index": 6, + "is_list_end_line": true + }, + { + "bbox": [ + 105, + 183, + 505, + 197 + ], + "spans": [ + { + "bbox": [ + 105, + 183, + 505, + 197 + ], + "score": 1.0, + "content": "Petar Velickovic, William Fedus, William L Hamilton, Pietro Lio, Yoshua Bengio, and R Devon. `", + "type": "text", + "cross_page": true + } + ], + "index": 7, + "is_list_start_line": true + }, + { + "bbox": [ + 115, + 193, + 331, + 208 + ], + "spans": [ + { + "bbox": [ + 115, + 193, + 331, + 208 + ], + "score": 1.0, + "content": "676 hjelm. Deep graph infomax. ICLR, 2(3):4, 2019.", + "type": "text", + "cross_page": true + } + ], + "index": 8, + "is_list_end_line": true + }, + { + "bbox": [ + 105, + 213, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 505, + 227 + ], + "score": 1.0, + "content": "Minjie Wang, Da Zheng, Zihao Ye, Quan Gan, Mufei Li, Xiang Song, Jinjing Zhou, Chao Ma,", + "type": "text", + "cross_page": true + } + ], + "index": 9, + "is_list_start_line": true + }, + { + "bbox": [ + 115, + 224, + 506, + 238 + ], + "spans": [ + { + "bbox": [ + 115, + 224, + 506, + 238 + ], + "score": 1.0, + "content": "Lingfan Yu, Yu Gai, et al. Deep graph library: A graph-centric, highly-performant package for", + "type": "text", + "cross_page": true + } + ], + "index": 10 + }, + { + "bbox": [ + 115, + 235, + 374, + 248 + ], + "spans": [ + { + "bbox": [ + 115, + 235, + 374, + 248 + ], + "score": 1.0, + "content": "graph neural networks. arXiv preprint arXiv:1909.01315, 2019.", + "type": "text", + "cross_page": true + } + ], + "index": 11, + "is_list_end_line": true + }, + { + "bbox": [ + 105, + 253, + 505, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 505, + 268 + ], + "score": 1.0, + "content": "Yifei Wang, Qi Zhang, Tianqi Du, Jiansheng Yang, Zhouchen Lin, and Yisen Wang. A message", + "type": "text", + "cross_page": true + } + ], + "index": 12, + "is_list_start_line": true + }, + { + "bbox": [ + 115, + 265, + 506, + 279 + ], + "spans": [ + { + "bbox": [ + 115, + 265, + 506, + 279 + ], + "score": 1.0, + "content": "passing perspective on learning dynamics of contrastive learning. International Conference on", + "type": "text", + "cross_page": true + } + ], + "index": 13 + }, + { + "bbox": [ + 115, + 276, + 250, + 290 + ], + "spans": [ + { + "bbox": [ + 115, + 276, + 250, + 290 + ], + "score": 1.0, + "content": "Learning Representations, 2023.", + "type": "text", + "cross_page": true + } + ], + "index": 14, + "is_list_end_line": true + }, + { + "bbox": [ + 105, + 294, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 294, + 505, + 309 + ], + "score": 1.0, + "content": "Zhirong Yang, Jaakko Peltonen, and Samuel Kaski. Scalable optimization of neighbor embedding", + "type": "text", + "cross_page": true + } + ], + "index": 15, + "is_list_start_line": true + }, + { + "bbox": [ + 115, + 306, + 505, + 320 + ], + "spans": [ + { + "bbox": [ + 115, + 306, + 505, + 320 + ], + "score": 1.0, + "content": "for visualization. In International Conference on Machine Learning, pp. 127–135. PMLR, 2013.", + "type": "text", + "cross_page": true + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 324, + 505, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 505, + 339 + ], + "score": 1.0, + "content": "Yuning You, Tianlong Chen, Yongduo Sui, Ting Chen, Zhangyang Wang, and Yang Shen. Graph", + "type": "text", + "cross_page": true + } + ], + "index": 17, + "is_list_start_line": true + }, + { + "bbox": [ + 115, + 336, + 506, + 350 + ], + "spans": [ + { + "bbox": [ + 115, + 336, + 506, + 350 + ], + "score": 1.0, + "content": "contrastive learning with augmentations. Advances in Neural Information Processing Systems,", + "type": "text", + "cross_page": true + } + ], + "index": 18 + }, + { + "bbox": [ + 117, + 347, + 203, + 358 + ], + "spans": [ + { + "bbox": [ + 117, + 347, + 203, + 358 + ], + "score": 1.0, + "content": "33:5812–5823, 2020.", + "type": "text", + "cross_page": true + } + ], + "index": 19, + "is_list_end_line": true + }, + { + "bbox": [ + 106, + 366, + 505, + 378 + ], + "spans": [ + { + "bbox": [ + 106, + 366, + 505, + 378 + ], + "score": 1.0, + "content": "Hengrui Zhang, Qitian Wu, Junchi Yan, David Wipf, and Philip S Yu. From canonical correlation", + "type": "text", + "cross_page": true + } + ], + "index": 20, + "is_list_start_line": true + }, + { + "bbox": [ + 114, + 375, + 506, + 392 + ], + "spans": [ + { + "bbox": [ + 114, + 375, + 506, + 392 + ], + "score": 1.0, + "content": "analysis to self-supervised graph neural networks. Advances in Neural Information Processing", + "type": "text", + "cross_page": true + } + ], + "index": 21 + }, + { + "bbox": [ + 115, + 387, + 222, + 401 + ], + "spans": [ + { + "bbox": [ + 115, + 387, + 222, + 401 + ], + "score": 1.0, + "content": "Systems, 34:76–89, 2021.", + "type": "text", + "cross_page": true + } + ], + "index": 22, + "is_list_end_line": true + }, + { + "bbox": [ + 105, + 406, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 506, + 420 + ], + "score": 1.0, + "content": "Hengrui Zhang, Qitian Wu, Yu Wang, Shaofeng Zhang, Junchi Yan, and Philip S Yu. Localized", + "type": "text", + "cross_page": true + } + ], + "index": 23, + "is_list_start_line": true + }, + { + "bbox": [ + 115, + 417, + 404, + 431 + ], + "spans": [ + { + "bbox": [ + 115, + 417, + 404, + 431 + ], + "score": 1.0, + "content": "contrastive learning on graphs. arXiv preprint arXiv:2212.04604, 2022.", + "type": "text", + "cross_page": true + } + ], + "index": 24, + "is_list_end_line": true + }, + { + "bbox": [ + 105, + 434, + 505, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 505, + 450 + ], + "score": 1.0, + "content": "Fahai Zhong, Mingliang Xue, Jian Zhang, Fan Zhang, Rui Ban, Oliver Deussen, and Yunhai Wang.", + "type": "text", + "cross_page": true + } + ], + "index": 25, + "is_list_start_line": true + }, + { + "bbox": [ + 115, + 447, + 505, + 461 + ], + "spans": [ + { + "bbox": [ + 115, + 447, + 505, + 461 + ], + "score": 1.0, + "content": "Force-directed graph layouts revisited: a new force based on the t-distribution. IEEE Transactions", + "type": "text", + "cross_page": true + } + ], + "index": 26 + }, + { + "bbox": [ + 115, + 459, + 310, + 471 + ], + "spans": [ + { + "bbox": [ + 115, + 459, + 310, + 471 + ], + "score": 1.0, + "content": "on Visualization and Computer Graphics, 2023.", + "type": "text", + "cross_page": true + } + ], + "index": 27, + "is_list_end_line": true + }, + { + "bbox": [ + 105, + 477, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 505, + 491 + ], + "score": 1.0, + "content": "Minfeng Zhu, Wei Chen, Yuanzhe Hu, Yuxuan Hou, Liangjun Liu, and Kaiyuan Zhang. DRGraph:", + "type": "text", + "cross_page": true + } + ], + "index": 28, + "is_list_start_line": true + }, + { + "bbox": [ + 115, + 487, + 507, + 503 + ], + "spans": [ + { + "bbox": [ + 115, + 487, + 507, + 503 + ], + "score": 1.0, + "content": "An efficient graph layout algorithm for large-scale graphs by dimensionality reduction. IEEE", + "type": "text", + "cross_page": true + } + ], + "index": 29 + }, + { + "bbox": [ + 115, + 497, + 442, + 513 + ], + "spans": [ + { + "bbox": [ + 115, + 497, + 442, + 513 + ], + "score": 1.0, + "content": "Transactions on Visualization and Computer Graphics, 27(2):1666–1676, 2020a.", + "type": "text", + "cross_page": true + } + ], + "index": 30, + "is_list_end_line": true + }, + { + "bbox": [ + 105, + 517, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 505, + 532 + ], + "score": 1.0, + "content": "Yanqiao Zhu, Yichen Xu, Feng Yu, Qiang Liu, Shu Wu, and Liang Wang. Deep graph contrastive", + "type": "text", + "cross_page": true + } + ], + "index": 31, + "is_list_start_line": true + }, + { + "bbox": [ + 114, + 529, + 381, + 542 + ], + "spans": [ + { + "bbox": [ + 114, + 529, + 381, + 542 + ], + "score": 1.0, + "content": "representation learning. arXiv preprint arXiv:2006.04131, 2020b.", + "type": "text", + "cross_page": true + } + ], + "index": 32, + "is_list_end_line": true + }, + { + "bbox": [ + 106, + 546, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 106, + 546, + 505, + 563 + ], + "score": 1.0, + "content": "Yanqiao Zhu, Yichen Xu, Feng Yu, Qiang Liu, Shu Wu, and Liang Wang. Graph contrastive learning", + "type": "text", + "cross_page": true + } + ], + "index": 33, + "is_list_start_line": true + }, + { + "bbox": [ + 115, + 559, + 503, + 573 + ], + "spans": [ + { + "bbox": [ + 115, + 559, + 503, + 573 + ], + "score": 1.0, + "content": "with adaptive augmentation. In Proceedings of the Web Conference 2021, pp. 2069–2080, 2021.", + "type": "text", + "cross_page": true + } + ], + "index": 34 + } + ], + "index": 24.5, + "bbox_fs": [ + 103, + 129, + 507, + 735 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 104, + 51, + 506, + 739 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 506, + 96 + ], + "score": 1.0, + "content": "Diederik P Kingma and Jimmy Ba. Adam: A method for stochastic optimization. International", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 114, + 92, + 310, + 108 + ], + "spans": [ + { + "bbox": [ + 114, + 92, + 310, + 108 + ], + "score": 1.0, + "content": "Conference on Learning Representations, 2015.", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 112, + 506, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 112, + 506, + 127 + ], + "score": 1.0, + "content": "Thomas N Kipf and Max Welling. Semi-supervised classification with graph convolutional net-", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 114, + 124, + 398, + 138 + ], + "spans": [ + { + "bbox": [ + 114, + 124, + 398, + 138 + ], + "score": 1.0, + "content": "works. International Conference for Learning Representations, 2017.", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 144, + 506, + 159 + ], + "spans": [ + { + "bbox": [ + 105, + 144, + 506, + 159 + ], + "score": 1.0, + "content": "Dmitry Kobak and Philipp Berens. The art of using t-SNE for single-cell transcriptomics. Nature", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 114, + 155, + 265, + 169 + ], + "spans": [ + { + "bbox": [ + 114, + 155, + 265, + 169 + ], + "score": 1.0, + "content": "Communications, 10(1):5416, 2019.", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 176, + 506, + 190 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 506, + 190 + ], + "score": 1.0, + "content": "Dmitry Kobak and George C Linderman. Initialization is critical for preserving global data structure", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 115, + 187, + 407, + 200 + ], + "spans": [ + { + "bbox": [ + 115, + 187, + 407, + 200 + ], + "score": 1.0, + "content": "in both t-SNE and UMAP. Nature Biotechnology, 39(2):156–157, 2021.", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 205, + 506, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 205, + 506, + 221 + ], + "score": 1.0, + "content": "Dmitry Kobak, George Linderman, Stefan Steinerberger, Yuval Kluger, and Philipp Berens. Heavy-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 114, + 218, + 506, + 232 + ], + "spans": [ + { + "bbox": [ + 114, + 218, + 506, + 232 + ], + "score": 1.0, + "content": "tailed kernels reveal a finer cluster structure in t-SNE visualisations. In Joint European Conference", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 113, + 227, + 489, + 243 + ], + "spans": [ + { + "bbox": [ + 113, + 227, + 489, + 243 + ], + "score": 1.0, + "content": "on Machine Learning and Knowledge Discovery in Databases, pp. 124–139. Springer, 2019.", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 247, + 506, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 506, + 263 + ], + "score": 1.0, + "content": "Johannes F Kruiger, Paulo E Rauber, Rafael Messias Martins, Andreas Kerren, Stephen Kobourov,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 114, + 258, + 506, + 276 + ], + "spans": [ + { + "bbox": [ + 114, + 258, + 506, + 276 + ], + "score": 1.0, + "content": "and Alexandru C Telea. Graph layouts by t-SNE. In Computer Graphics Forum, volume 36, pp.", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 115, + 271, + 272, + 285 + ], + "spans": [ + { + "bbox": [ + 115, + 271, + 272, + 285 + ], + "score": 1.0, + "content": "283–294. Wiley Online Library, 2017.", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 289, + 506, + 305 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 506, + 305 + ], + "score": 1.0, + "content": "Namkyeong Lee, Junseok Lee, and Chanyoung Park. Augmentation-free self-supervised learning", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 114, + 300, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 114, + 300, + 505, + 316 + ], + "score": 1.0, + "content": "on graphs. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 36, pp.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 115, + 313, + 191, + 325 + ], + "spans": [ + { + "bbox": [ + 115, + 313, + 191, + 325 + ], + "score": 1.0, + "content": "7372–7380, 2022.", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 331, + 506, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 506, + 347 + ], + "score": 1.0, + "content": "Yao Yang Leow, Thomas Laurent, and Xavier Bresson. GraphTSNE: a visualization technique", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 114, + 344, + 506, + 358 + ], + "spans": [ + { + "bbox": [ + 114, + 344, + 506, + 358 + ], + "score": 1.0, + "content": "for graph-structured data. Representation Learning on Graphs and Manifold Workshop at the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 114, + 355, + 366, + 369 + ], + "spans": [ + { + "bbox": [ + 114, + 355, + 366, + 369 + ], + "score": 1.0, + "content": "International Conference for Learning Representations, 2019.", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 375, + 506, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 506, + 389 + ], + "score": 1.0, + "content": "Haifeng Li, Jun Cao, Jiawei Zhu, Qinyao Luo, Silu He, and Xuying Wang. Augmentation-free graph", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 115, + 386, + 506, + 400 + ], + "spans": [ + { + "bbox": [ + 115, + 386, + 506, + 400 + ], + "score": 1.0, + "content": "contrastive learning of invariant-discriminative representations. IEEE Transactions on Neural", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 114, + 396, + 316, + 410 + ], + "spans": [ + { + "bbox": [ + 114, + 396, + 316, + 410 + ], + "score": 1.0, + "content": "Networks and Learning Systems, pp. 1–11, 2023.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 417, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 505, + 429 + ], + "score": 1.0, + "content": "George C Linderman and Stefan Steinerberger. Clustering with t-SNE, provably. SIAM Journal on", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 115, + 429, + 322, + 441 + ], + "spans": [ + { + "bbox": [ + 115, + 429, + 322, + 441 + ], + "score": 1.0, + "content": "Mathematics of Data Science, 1(2):313–332, 2019.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 447, + 504, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 504, + 461 + ], + "score": 1.0, + "content": "George C Linderman, Manas Rachh, Jeremy G Hoskins, Stefan Steinerberger, and Yuval Kluger.", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 114, + 459, + 506, + 473 + ], + "spans": [ + { + "bbox": [ + 114, + 459, + 506, + 473 + ], + "score": 1.0, + "content": "Fast interpolation-based t-SNE for improved visualization of single-cell RNA-seq data. Nature", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 115, + 471, + 245, + 483 + ], + "spans": [ + { + "bbox": [ + 115, + 471, + 245, + 483 + ], + "score": 1.0, + "content": "Methods, 16(3):243–245, 2019.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 490, + 506, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 490, + 506, + 504 + ], + "score": 1.0, + "content": "Leland McInnes, John Healy, and James Melville. UMAP: Uniform manifold approximation and", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 113, + 500, + 424, + 515 + ], + "spans": [ + { + "bbox": [ + 113, + 500, + 424, + 515 + ], + "score": 1.0, + "content": "projection for dimension reduction. arXiv preprint arXiv:1802.03426, 2018.", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 522, + 504, + 533 + ], + "spans": [ + { + "bbox": [ + 106, + 522, + 504, + 533 + ], + "score": 1.0, + "content": "Tomas Mikolov, Ilya Sutskever, Kai Chen, Greg S Corrado, and Jeff Dean. Distributed representa-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 114, + 532, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 114, + 532, + 506, + 546 + ], + "score": 1.0, + "content": "tions of words and phrases and their compositionality. Advances in Neural Information Processing", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 114, + 543, + 194, + 558 + ], + "spans": [ + { + "bbox": [ + 114, + 543, + 194, + 558 + ], + "score": 1.0, + "content": "Systems, 26, 2013.", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 561, + 506, + 579 + ], + "spans": [ + { + "bbox": [ + 104, + 561, + 506, + 579 + ], + "score": 1.0, + "content": "Andreas Noack. Energy models for graph clustering. Journal of Graph Algorithms and Applications,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 115, + 575, + 206, + 587 + ], + "spans": [ + { + "bbox": [ + 115, + 575, + 206, + 587 + ], + "score": 1.0, + "content": "11(2):453–480, 2007.", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 594, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 608 + ], + "score": 1.0, + "content": "Aaron van den Oord, Yazhe Li, and Oriol Vinyals. Representation learning with contrastive predic-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 114, + 606, + 329, + 618 + ], + "spans": [ + { + "bbox": [ + 114, + 606, + 329, + 618 + ], + "score": 1.0, + "content": "tive coding. arXiv preprint arXiv:1807.03748, 2018.", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 625, + 506, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 625, + 506, + 639 + ], + "score": 1.0, + "content": "Fabian Pedregosa, Gael Varoquaux, Alexandre Gramfort, Vincent Michel, Bertrand Thirion, Olivier ¨", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 114, + 635, + 506, + 650 + ], + "spans": [ + { + "bbox": [ + 114, + 635, + 506, + 650 + ], + "score": 1.0, + "content": "Grisel, Mathieu Blondel, Peter Prettenhofer, Ron Weiss, Vincent Dubourg, et al. Scikit-learn:", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 115, + 648, + 482, + 660 + ], + "spans": [ + { + "bbox": [ + 115, + 648, + 482, + 660 + ], + "score": 1.0, + "content": "Machine learning in Python. Journal of Machine Learning Research, 12:2825–2830, 2011.", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 667, + 506, + 681 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 506, + 681 + ], + "score": 1.0, + "content": "Pavlin G Policar, Martin Stra ˇ zar, and Bla ˇ z Zupan. openTSNE: a modular Python library for t-SNE ˇ", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 115, + 678, + 398, + 693 + ], + "spans": [ + { + "bbox": [ + 115, + 678, + 398, + 693 + ], + "score": 1.0, + "content": "dimensionality reduction and embedding. BioRxiv, pp. 731877, 2019.", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 104, + 696, + 506, + 714 + ], + "spans": [ + { + "bbox": [ + 104, + 696, + 506, + 714 + ], + "score": 1.0, + "content": "Shantanu Thakoor, Corentin Tallec, Mohammad Gheshlaghi Azar, Mehdi Azabou, Eva L Dyer,", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 114, + 708, + 506, + 724 + ], + "spans": [ + { + "bbox": [ + 114, + 708, + 506, + 724 + ], + "score": 1.0, + "content": "Remi Munos, Petar Velickovi ˇ c, and Michal Valko. Large-scale representation learning on graphs ´", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 116, + 720, + 354, + 735 + ], + "spans": [ + { + "bbox": [ + 116, + 720, + 354, + 735 + ], + "score": 1.0, + "content": "via bootstrapping. arXiv preprint arXiv:2102.06514, 2021.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 22 + } + ], + "page_idx": 10, + "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": [ + 300, + 751, + 310, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 765 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 765 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 15, + "width": 13 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "list", + "bbox": [ + 104, + 51, + 506, + 739 + ], + "lines": [], + "index": 22, + "bbox_fs": [ + 104, + 81, + 506, + 735 + ], + "lines_deleted": true + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 104, + 72, + 506, + 572 + ], + "lines": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "Puja Trivedi, Ekdeep Singh Lubana, Yujun Yan, Yaoqing Yang, and Danai Koutra. Augmentations", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 115, + 94, + 505, + 108 + ], + "spans": [ + { + "bbox": [ + 115, + 94, + 505, + 108 + ], + "score": 1.0, + "content": "in graph contrastive learning: Current methodological flaws & towards better practices. In Pro-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 115, + 105, + 383, + 117 + ], + "spans": [ + { + "bbox": [ + 115, + 105, + 383, + 117 + ], + "score": 1.0, + "content": "ceedings of the ACM Web Conference 2022, pp. 1538–1549, 2022.", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 122, + 505, + 137 + ], + "spans": [ + { + "bbox": [ + 105, + 122, + 505, + 137 + ], + "score": 1.0, + "content": "Laurens Van Der Maaten. Accelerating t-SNE using tree-based algorithms. The Journal of Machine", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 115, + 134, + 298, + 148 + ], + "spans": [ + { + "bbox": [ + 115, + 134, + 298, + 148 + ], + "score": 1.0, + "content": "Learning Research, 15(1):3221–3245, 2014.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 104, + 153, + 505, + 168 + ], + "spans": [ + { + "bbox": [ + 104, + 153, + 505, + 168 + ], + "score": 1.0, + "content": "Laurens Van der Maaten and Geoffrey Hinton. Visualizing data using t-SNE. Journal of Machine", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 115, + 165, + 248, + 177 + ], + "spans": [ + { + "bbox": [ + 115, + 165, + 248, + 177 + ], + "score": 1.0, + "content": "Learning Research, 9(11), 2008.", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 183, + 505, + 197 + ], + "spans": [ + { + "bbox": [ + 105, + 183, + 505, + 197 + ], + "score": 1.0, + "content": "Petar Velickovic, William Fedus, William L Hamilton, Pietro Lio, Yoshua Bengio, and R Devon. `", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 115, + 193, + 331, + 208 + ], + "spans": [ + { + "bbox": [ + 115, + 193, + 331, + 208 + ], + "score": 1.0, + "content": "676 hjelm. Deep graph infomax. ICLR, 2(3):4, 2019.", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 213, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 505, + 227 + ], + "score": 1.0, + "content": "Minjie Wang, Da Zheng, Zihao Ye, Quan Gan, Mufei Li, Xiang Song, Jinjing Zhou, Chao Ma,", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 115, + 224, + 506, + 238 + ], + "spans": [ + { + "bbox": [ + 115, + 224, + 506, + 238 + ], + "score": 1.0, + "content": "Lingfan Yu, Yu Gai, et al. Deep graph library: A graph-centric, highly-performant package for", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 115, + 235, + 374, + 248 + ], + "spans": [ + { + "bbox": [ + 115, + 235, + 374, + 248 + ], + "score": 1.0, + "content": "graph neural networks. arXiv preprint arXiv:1909.01315, 2019.", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 253, + 505, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 505, + 268 + ], + "score": 1.0, + "content": "Yifei Wang, Qi Zhang, Tianqi Du, Jiansheng Yang, Zhouchen Lin, and Yisen Wang. A message", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 115, + 265, + 506, + 279 + ], + "spans": [ + { + "bbox": [ + 115, + 265, + 506, + 279 + ], + "score": 1.0, + "content": "passing perspective on learning dynamics of contrastive learning. International Conference on", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 115, + 276, + 250, + 290 + ], + "spans": [ + { + "bbox": [ + 115, + 276, + 250, + 290 + ], + "score": 1.0, + "content": "Learning Representations, 2023.", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 294, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 294, + 505, + 309 + ], + "score": 1.0, + "content": "Zhirong Yang, Jaakko Peltonen, and Samuel Kaski. Scalable optimization of neighbor embedding", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 115, + 306, + 505, + 320 + ], + "spans": [ + { + "bbox": [ + 115, + 306, + 505, + 320 + ], + "score": 1.0, + "content": "for visualization. In International Conference on Machine Learning, pp. 127–135. PMLR, 2013.", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 324, + 505, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 505, + 339 + ], + "score": 1.0, + "content": "Yuning You, Tianlong Chen, Yongduo Sui, Ting Chen, Zhangyang Wang, and Yang Shen. Graph", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 115, + 336, + 506, + 350 + ], + "spans": [ + { + "bbox": [ + 115, + 336, + 506, + 350 + ], + "score": 1.0, + "content": "contrastive learning with augmentations. Advances in Neural Information Processing Systems,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 117, + 347, + 203, + 358 + ], + "spans": [ + { + "bbox": [ + 117, + 347, + 203, + 358 + ], + "score": 1.0, + "content": "33:5812–5823, 2020.", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 366, + 505, + 378 + ], + "spans": [ + { + "bbox": [ + 106, + 366, + 505, + 378 + ], + "score": 1.0, + "content": "Hengrui Zhang, Qitian Wu, Junchi Yan, David Wipf, and Philip S Yu. From canonical correlation", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 114, + 375, + 506, + 392 + ], + "spans": [ + { + "bbox": [ + 114, + 375, + 506, + 392 + ], + "score": 1.0, + "content": "analysis to self-supervised graph neural networks. Advances in Neural Information Processing", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 115, + 387, + 222, + 401 + ], + "spans": [ + { + "bbox": [ + 115, + 387, + 222, + 401 + ], + "score": 1.0, + "content": "Systems, 34:76–89, 2021.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 406, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 506, + 420 + ], + "score": 1.0, + "content": "Hengrui Zhang, Qitian Wu, Yu Wang, Shaofeng Zhang, Junchi Yan, and Philip S Yu. Localized", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 115, + 417, + 404, + 431 + ], + "spans": [ + { + "bbox": [ + 115, + 417, + 404, + 431 + ], + "score": 1.0, + "content": "contrastive learning on graphs. arXiv preprint arXiv:2212.04604, 2022.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 434, + 505, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 505, + 450 + ], + "score": 1.0, + "content": "Fahai Zhong, Mingliang Xue, Jian Zhang, Fan Zhang, Rui Ban, Oliver Deussen, and Yunhai Wang.", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 115, + 447, + 505, + 461 + ], + "spans": [ + { + "bbox": [ + 115, + 447, + 505, + 461 + ], + "score": 1.0, + "content": "Force-directed graph layouts revisited: a new force based on the t-distribution. IEEE Transactions", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 115, + 459, + 310, + 471 + ], + "spans": [ + { + "bbox": [ + 115, + 459, + 310, + 471 + ], + "score": 1.0, + "content": "on Visualization and Computer Graphics, 2023.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 477, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 505, + 491 + ], + "score": 1.0, + "content": "Minfeng Zhu, Wei Chen, Yuanzhe Hu, Yuxuan Hou, Liangjun Liu, and Kaiyuan Zhang. DRGraph:", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 115, + 487, + 507, + 503 + ], + "spans": [ + { + "bbox": [ + 115, + 487, + 507, + 503 + ], + "score": 1.0, + "content": "An efficient graph layout algorithm for large-scale graphs by dimensionality reduction. IEEE", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 115, + 497, + 442, + 513 + ], + "spans": [ + { + "bbox": [ + 115, + 497, + 442, + 513 + ], + "score": 1.0, + "content": "Transactions on Visualization and Computer Graphics, 27(2):1666–1676, 2020a.", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 517, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 505, + 532 + ], + "score": 1.0, + "content": "Yanqiao Zhu, Yichen Xu, Feng Yu, Qiang Liu, Shu Wu, and Liang Wang. Deep graph contrastive", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 114, + 529, + 381, + 542 + ], + "spans": [ + { + "bbox": [ + 114, + 529, + 381, + 542 + ], + "score": 1.0, + "content": "representation learning. arXiv preprint arXiv:2006.04131, 2020b.", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 546, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 106, + 546, + 505, + 563 + ], + "score": 1.0, + "content": "Yanqiao Zhu, Yichen Xu, Feng Yu, Qiang Liu, Shu Wu, and Liang Wang. Graph contrastive learning", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 115, + 559, + 503, + 573 + ], + "spans": [ + { + "bbox": [ + 115, + 559, + 503, + 573 + ], + "score": 1.0, + "content": "with adaptive augmentation. In Proceedings of the Web Conference 2021, pp. 2069–2080, 2021.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 17 + } + ], + "page_idx": 11, + "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": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "score": 1.0, + "content": "12", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "list", + "bbox": [ + 104, + 72, + 506, + 572 + ], + "lines": [], + "index": 17, + "bbox_fs": [ + 104, + 83, + 507, + 573 + ], + "lines_deleted": true + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 81, + 266, + 94 + ], + "lines": [ + { + "bbox": [ + 106, + 80, + 266, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 80, + 266, + 96 + ], + "score": 1.0, + "content": "A SUPPLEMENTARY FIGURES", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "image", + "bbox": [ + 216, + 111, + 395, + 268 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 216, + 111, + 395, + 268 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 216, + 111, + 395, + 268 + ], + "spans": [ + { + "bbox": [ + 216, + 111, + 395, + 268 + ], + "score": 0.968, + "type": "image", + "image_path": "60b7d14ab4a629377a795bcc99485453a195556eaadd495831ef6798a0248ae6.jpg" + } + ] + } + ], + "index": 6.5, + "virtual_lines": [ + { + "bbox": [ + 216, + 111, + 395, + 124.08333333333333 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 216, + 124.08333333333333, + 395, + 137.16666666666666 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 216, + 137.16666666666666, + 395, + 150.25 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 216, + 150.25, + 395, + 163.33333333333334 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 216, + 163.33333333333334, + 395, + 176.41666666666669 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 216, + 176.41666666666669, + 395, + 189.50000000000003 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 216, + 189.50000000000003, + 395, + 202.58333333333337 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 216, + 202.58333333333337, + 395, + 215.6666666666667 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 216, + 215.6666666666667, + 395, + 228.75000000000006 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 216, + 228.75000000000006, + 395, + 241.8333333333334 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 216, + 241.8333333333334, + 395, + 254.91666666666674 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 216, + 254.91666666666674, + 395, + 268.00000000000006 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 286, + 505, + 354 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 287, + 505, + 300 + ], + "spans": [ + { + "bbox": [ + 106, + 287, + 272, + 300 + ], + "score": 1.0, + "content": "Figure A.1: Computation times for graph", + "type": "text" + }, + { + "bbox": [ + 273, + 288, + 278, + 298 + ], + "score": 0.61, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 278, + 287, + 505, + 300 + ], + "score": 1.0, + "content": "-SNE and graph CNE with 2 and 128 output dimensions.", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 298, + 505, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 298, + 242, + 311 + ], + "score": 1.0, + "content": "openTSNE was run on CPU with", + "type": "text" + }, + { + "bbox": [ + 243, + 299, + 294, + 309 + ], + "score": 0.85, + "content": "\\mathrm { n - j } \\mathrm { o b s = - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 298, + 505, + 311 + ], + "score": 1.0, + "content": ". CNE was run on GPU. Datasets are ordered by the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 309, + 505, + 322 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 505, + 322 + ], + "score": 1.0, + "content": "increasing number of nodes. The runtime of openTSNE (for a given number of gradient descent", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 320, + 506, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 506, + 333 + ], + "score": 1.0, + "content": "steps) grows linearly with the number of nodes. The runtime of CNE (for a given number of epochs", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 331, + 505, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 505, + 344 + ], + "score": 1.0, + "content": "and a given batch size) grows linearly with the number of edges. The Pubmed dataset (PUB) has", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 342, + 383, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 383, + 354 + ], + "score": 1.0, + "content": "fewer edges than the Amazon datasets (APH and ACO), see Table 1.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 15.5 + } + ], + "index": 11.0 + } + ], + "page_idx": 12, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "score": 1.0, + "content": "13", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 106, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2024", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 81, + 266, + 94 + ], + "lines": [ + { + "bbox": [ + 106, + 80, + 266, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 80, + 266, + 96 + ], + "score": 1.0, + "content": "A SUPPLEMENTARY FIGURES", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "image", + "bbox": [ + 216, + 111, + 395, + 268 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 216, + 111, + 395, + 268 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 216, + 111, + 395, + 268 + ], + "spans": [ + { + "bbox": [ + 216, + 111, + 395, + 268 + ], + "score": 0.968, + "type": "image", + "image_path": "60b7d14ab4a629377a795bcc99485453a195556eaadd495831ef6798a0248ae6.jpg" + } + ] + } + ], + "index": 6.5, + "virtual_lines": [ + { + "bbox": [ + 216, + 111, + 395, + 124.08333333333333 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 216, + 124.08333333333333, + 395, + 137.16666666666666 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 216, + 137.16666666666666, + 395, + 150.25 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 216, + 150.25, + 395, + 163.33333333333334 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 216, + 163.33333333333334, + 395, + 176.41666666666669 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 216, + 176.41666666666669, + 395, + 189.50000000000003 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 216, + 189.50000000000003, + 395, + 202.58333333333337 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 216, + 202.58333333333337, + 395, + 215.6666666666667 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 216, + 215.6666666666667, + 395, + 228.75000000000006 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 216, + 228.75000000000006, + 395, + 241.8333333333334 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 216, + 241.8333333333334, + 395, + 254.91666666666674 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 216, + 254.91666666666674, + 395, + 268.00000000000006 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 286, + 505, + 354 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 287, + 505, + 300 + ], + "spans": [ + { + "bbox": [ + 106, + 287, + 272, + 300 + ], + "score": 1.0, + "content": "Figure A.1: Computation times for graph", + "type": "text" + }, + { + "bbox": [ + 273, + 288, + 278, + 298 + ], + "score": 0.61, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 278, + 287, + 505, + 300 + ], + "score": 1.0, + "content": "-SNE and graph CNE with 2 and 128 output dimensions.", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 298, + 505, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 298, + 242, + 311 + ], + "score": 1.0, + "content": "openTSNE was run on CPU with", + "type": "text" + }, + { + "bbox": [ + 243, + 299, + 294, + 309 + ], + "score": 0.85, + "content": "\\mathrm { n - j } \\mathrm { o b s = - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 298, + 505, + 311 + ], + "score": 1.0, + "content": ". CNE was run on GPU. Datasets are ordered by the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 309, + 505, + 322 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 505, + 322 + ], + "score": 1.0, + "content": "increasing number of nodes. The runtime of openTSNE (for a given number of gradient descent", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 320, + 506, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 506, + 333 + ], + "score": 1.0, + "content": "steps) grows linearly with the number of nodes. The runtime of CNE (for a given number of epochs", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 331, + 505, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 505, + 344 + ], + "score": 1.0, + "content": "and a given batch size) grows linearly with the number of edges. The Pubmed dataset (PUB) has", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 342, + 383, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 383, + 354 + ], + "score": 1.0, + "content": "fewer edges than the Amazon datasets (APH and ACO), see Table 1.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 15.5 + } + ], + "index": 11.0 + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 174, + 237, + 446, + 519 + ], + "blocks": [ + { + "type": "image_caption", + "bbox": [ + 174, + 230, + 269, + 239 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 174, + 230, + 270, + 239 + ], + "spans": [ + { + "bbox": [ + 174, + 230, + 270, + 239 + ], + "score": 1.0, + "content": "ACO, per-node normalization", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "image_caption", + "bbox": [ + 326, + 231, + 434, + 239 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 326, + 230, + 434, + 240 + ], + "spans": [ + { + "bbox": [ + 326, + 230, + 434, + 240 + ], + "score": 1.0, + "content": "ACO, whole-matrix normalization", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 1 + }, + { + "type": "image_body", + "bbox": [ + 174, + 237, + 446, + 519 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 174, + 237, + 446, + 519 + ], + "spans": [ + { + "bbox": [ + 174, + 237, + 446, + 519 + ], + "score": 0.796, + "type": "image", + "image_path": "3e059afd2ab2d936a50c61c839b21eda3094285bf8417f9e3a5dc2974b23345b.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 174, + 237, + 446, + 249.26086956521738 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 174, + 249.26086956521738, + 446, + 261.52173913043475 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 174, + 261.52173913043475, + 446, + 273.78260869565213 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 174, + 273.78260869565213, + 446, + 286.0434782608695 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 174, + 286.0434782608695, + 446, + 298.3043478260869 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 174, + 298.3043478260869, + 446, + 310.56521739130426 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 174, + 310.56521739130426, + 446, + 322.82608695652164 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 174, + 322.82608695652164, + 446, + 335.086956521739 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 174, + 335.086956521739, + 446, + 347.3478260869564 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 174, + 347.3478260869564, + 446, + 359.60869565217376 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 174, + 359.60869565217376, + 446, + 371.86956521739114 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 174, + 371.86956521739114, + 446, + 384.1304347826085 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 174, + 384.1304347826085, + 446, + 396.3913043478259 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 174, + 396.3913043478259, + 446, + 408.65217391304327 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 174, + 408.65217391304327, + 446, + 420.91304347826065 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 174, + 420.91304347826065, + 446, + 433.173913043478 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 174, + 433.173913043478, + 446, + 445.4347826086954 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 174, + 445.4347826086954, + 446, + 457.6956521739128 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 174, + 457.6956521739128, + 446, + 469.95652173913015 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 174, + 469.95652173913015, + 446, + 482.21739130434753 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 174, + 482.21739130434753, + 446, + 494.4782608695649 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 174, + 494.4782608695649, + 446, + 506.7391304347823 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 174, + 506.7391304347823, + 446, + 518.9999999999997 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 546, + 505, + 579 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 545, + 505, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 185, + 559 + ], + "score": 1.0, + "content": "Figure A.2: Graph", + "type": "text" + }, + { + "bbox": [ + 186, + 547, + 191, + 556 + ], + "score": 0.57, + "content": "t { \\cdot }", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 545, + 505, + 559 + ], + "score": 1.0, + "content": "-SNE visualizations of ACO and APH datasets using per-node normalization", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 556, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 505, + 570 + ], + "score": 1.0, + "content": "of the adjacency matrix (default) and whole-matrix normalization. Embeddings in each row were", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 568, + 242, + 580 + ], + "spans": [ + { + "bbox": [ + 106, + 568, + 242, + 580 + ], + "score": 1.0, + "content": "aligned using Procrustes rotation.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26 + } + ], + "index": 7.0 + } + ], + "page_idx": 13, + "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": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 313, + 763 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 313, + 763 + ], + "score": 1.0, + "content": "14", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 174, + 237, + 446, + 519 + ], + "blocks": [ + { + "type": "image_caption", + "bbox": [ + 174, + 230, + 269, + 239 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 174, + 230, + 270, + 239 + ], + "spans": [ + { + "bbox": [ + 174, + 230, + 270, + 239 + ], + "score": 1.0, + "content": "ACO, per-node normalization", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "image_caption", + "bbox": [ + 326, + 231, + 434, + 239 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 326, + 230, + 434, + 240 + ], + "spans": [ + { + "bbox": [ + 326, + 230, + 434, + 240 + ], + "score": 1.0, + "content": "ACO, whole-matrix normalization", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 1 + }, + { + "type": "image_body", + "bbox": [ + 174, + 237, + 446, + 519 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 174, + 237, + 446, + 519 + ], + "spans": [ + { + "bbox": [ + 174, + 237, + 446, + 519 + ], + "score": 0.796, + "type": "image", + "image_path": "3e059afd2ab2d936a50c61c839b21eda3094285bf8417f9e3a5dc2974b23345b.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 174, + 237, + 446, + 249.26086956521738 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 174, + 249.26086956521738, + 446, + 261.52173913043475 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 174, + 261.52173913043475, + 446, + 273.78260869565213 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 174, + 273.78260869565213, + 446, + 286.0434782608695 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 174, + 286.0434782608695, + 446, + 298.3043478260869 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 174, + 298.3043478260869, + 446, + 310.56521739130426 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 174, + 310.56521739130426, + 446, + 322.82608695652164 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 174, + 322.82608695652164, + 446, + 335.086956521739 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 174, + 335.086956521739, + 446, + 347.3478260869564 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 174, + 347.3478260869564, + 446, + 359.60869565217376 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 174, + 359.60869565217376, + 446, + 371.86956521739114 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 174, + 371.86956521739114, + 446, + 384.1304347826085 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 174, + 384.1304347826085, + 446, + 396.3913043478259 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 174, + 396.3913043478259, + 446, + 408.65217391304327 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 174, + 408.65217391304327, + 446, + 420.91304347826065 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 174, + 420.91304347826065, + 446, + 433.173913043478 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 174, + 433.173913043478, + 446, + 445.4347826086954 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 174, + 445.4347826086954, + 446, + 457.6956521739128 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 174, + 457.6956521739128, + 446, + 469.95652173913015 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 174, + 469.95652173913015, + 446, + 482.21739130434753 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 174, + 482.21739130434753, + 446, + 494.4782608695649 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 174, + 494.4782608695649, + 446, + 506.7391304347823 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 174, + 506.7391304347823, + 446, + 518.9999999999997 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 546, + 505, + 579 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 545, + 505, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 185, + 559 + ], + "score": 1.0, + "content": "Figure A.2: Graph", + "type": "text" + }, + { + "bbox": [ + 186, + 547, + 191, + 556 + ], + "score": 0.57, + "content": "t { \\cdot }", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 545, + 505, + 559 + ], + "score": 1.0, + "content": "-SNE visualizations of ACO and APH datasets using per-node normalization", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 556, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 505, + 570 + ], + "score": 1.0, + "content": "of the adjacency matrix (default) and whole-matrix normalization. Embeddings in each row were", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 568, + 242, + 580 + ], + "spans": [ + { + "bbox": [ + 106, + 568, + 242, + 580 + ], + "score": 1.0, + "content": "aligned using Procrustes rotation.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26 + } + ], + "index": 7.0 + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 106, + 90, + 489, + 655 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 106, + 90, + 489, + 655 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 90, + 489, + 655 + ], + "spans": [ + { + "bbox": [ + 106, + 90, + 489, + 655 + ], + "score": 0.968, + "type": "image", + "image_path": "2e3a4e97efb0b5a71ebb8d946eb97878711824e2f09632ab76746aa0e846c712.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 106, + 90, + 489, + 278.33333333333337 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 106, + 278.33333333333337, + 489, + 466.66666666666674 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 106, + 466.66666666666674, + 489, + 655.0000000000001 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 669, + 505, + 703 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 669, + 504, + 682 + ], + "spans": [ + { + "bbox": [ + 106, + 669, + 504, + 682 + ], + "score": 1.0, + "content": "Figure A.3: Embeddings of all considered datasets obtained using FDP (Fruchterman & Reingold,", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 679, + 505, + 693 + ], + "spans": [ + { + "bbox": [ + 106, + 679, + 271, + 693 + ], + "score": 1.0, + "content": "1991), DRGraph (Zhu et al., 2020a), and", + "type": "text" + }, + { + "bbox": [ + 271, + 681, + 276, + 691 + ], + "score": 0.76, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 679, + 440, + 693 + ], + "score": 1.0, + "content": "-FDP (Zhong et al., 2023), and our graph", + "type": "text" + }, + { + "bbox": [ + 440, + 681, + 445, + 690 + ], + "score": 0.52, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 679, + 505, + 693 + ], + "score": 1.0, + "content": "-SNE. Embed-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 691, + 337, + 704 + ], + "spans": [ + { + "bbox": [ + 106, + 691, + 337, + 704 + ], + "score": 1.0, + "content": "dings in each row were aligned using Procrustes rotation.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + } + ], + "page_idx": 14, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "score": 1.0, + "content": "15", + "type": "text" + } + ] + } + ] + }, + { + "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" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 106, + 90, + 489, + 655 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 106, + 90, + 489, + 655 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 90, + 489, + 655 + ], + "spans": [ + { + "bbox": [ + 106, + 90, + 489, + 655 + ], + "score": 0.968, + "type": "image", + "image_path": "2e3a4e97efb0b5a71ebb8d946eb97878711824e2f09632ab76746aa0e846c712.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 106, + 90, + 489, + 278.33333333333337 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 106, + 278.33333333333337, + 489, + 466.66666666666674 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 106, + 466.66666666666674, + 489, + 655.0000000000001 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 669, + 505, + 703 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 669, + 504, + 682 + ], + "spans": [ + { + "bbox": [ + 106, + 669, + 504, + 682 + ], + "score": 1.0, + "content": "Figure A.3: Embeddings of all considered datasets obtained using FDP (Fruchterman & Reingold,", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 679, + 505, + 693 + ], + "spans": [ + { + "bbox": [ + 106, + 679, + 271, + 693 + ], + "score": 1.0, + "content": "1991), DRGraph (Zhu et al., 2020a), and", + "type": "text" + }, + { + "bbox": [ + 271, + 681, + 276, + 691 + ], + "score": 0.76, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 679, + 440, + 693 + ], + "score": 1.0, + "content": "-FDP (Zhong et al., 2023), and our graph", + "type": "text" + }, + { + "bbox": [ + 440, + 681, + 445, + 690 + ], + "score": 0.52, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 679, + 505, + 693 + ], + "score": 1.0, + "content": "-SNE. Embed-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 691, + 337, + 704 + ], + "spans": [ + { + "bbox": [ + 106, + 691, + 337, + 704 + ], + "score": 1.0, + "content": "dings in each row were aligned using Procrustes rotation.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + } + ] + } + ], + "_backend": "pipeline", + "_version_name": "2.1.11" +} \ No newline at end of file