diff --git "a/parse/train/6ugK-RQhIP5/6ugK-RQhIP5_middle.json" "b/parse/train/6ugK-RQhIP5/6ugK-RQhIP5_middle.json" new file mode 100644--- /dev/null +++ "b/parse/train/6ugK-RQhIP5/6ugK-RQhIP5_middle.json" @@ -0,0 +1,51190 @@ +{ + "pdf_info": [ + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 119, + 97, + 493, + 137 + ], + "lines": [ + { + "bbox": [ + 117, + 96, + 495, + 119 + ], + "spans": [ + { + "bbox": [ + 117, + 96, + 495, + 119 + ], + "score": 1.0, + "content": "A Probabilistic Representation for Deep Learning:", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 119, + 117, + 492, + 138 + ], + "spans": [ + { + "bbox": [ + 119, + 117, + 492, + 138 + ], + "score": 1.0, + "content": "Delving into The Information Bottleneck Principle", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 259, + 179, + 355, + 223 + ], + "lines": [ + { + "bbox": [ + 258, + 179, + 355, + 191 + ], + "spans": [ + { + "bbox": [ + 258, + 179, + 355, + 191 + ], + "score": 1.0, + "content": "Anonymous Author(s)", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 283, + 189, + 328, + 202 + ], + "spans": [ + { + "bbox": [ + 283, + 189, + 328, + 202 + ], + "score": 1.0, + "content": "Affiliation", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 286, + 200, + 324, + 213 + ], + "spans": [ + { + "bbox": [ + 286, + 200, + 324, + 213 + ], + "score": 1.0, + "content": "Address", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 290, + 213, + 320, + 222 + ], + "spans": [ + { + "bbox": [ + 290, + 213, + 320, + 222 + ], + "score": 1.0, + "content": "email", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 3.5 + }, + { + "type": "title", + "bbox": [ + 283, + 252, + 328, + 265 + ], + "lines": [ + { + "bbox": [ + 281, + 251, + 331, + 267 + ], + "spans": [ + { + "bbox": [ + 281, + 251, + 331, + 267 + ], + "score": 1.0, + "content": "Abstract", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 91, + 275, + 469, + 418 + ], + "lines": [ + { + "bbox": [ + 93, + 275, + 470, + 289 + ], + "spans": [ + { + "bbox": [ + 93, + 279, + 99, + 286 + ], + "score": 1.0, + "content": "1", + "type": "text" + }, + { + "bbox": [ + 141, + 275, + 470, + 289 + ], + "score": 1.0, + "content": "The Information Bottleneck (IB) principle has recently attracted great attention to", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 93, + 287, + 470, + 299 + ], + "spans": [ + { + "bbox": [ + 93, + 289, + 99, + 297 + ], + "score": 1.0, + "content": "2", + "type": "text" + }, + { + "bbox": [ + 142, + 287, + 470, + 299 + ], + "score": 1.0, + "content": "explaining Deep Neural Networks (DNNs), and the key is to accurately estimate the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 93, + 298, + 470, + 309 + ], + "spans": [ + { + "bbox": [ + 93, + 300, + 99, + 309 + ], + "score": 1.0, + "content": "3", + "type": "text" + }, + { + "bbox": [ + 141, + 298, + 470, + 309 + ], + "score": 1.0, + "content": "mutual information between a hidden layer and dataset. 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These", + "type": "text" + } + ], + "index": 40, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 705, + 439, + 717 + ], + "spans": [ + { + "bbox": [ + 89, + 707, + 100, + 716 + ], + "score": 1.0, + "content": "34", + "type": "text" + }, + { + "bbox": [ + 105, + 705, + 439, + 717 + ], + "score": 1.0, + "content": "unsettled limitations greatly weakens the validity of the IB explanations for DNNs.", + "type": "text" + } + ], + "index": 41, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 73, + 507, + 86 + ], + "spans": [ + { + "bbox": [ + 89, + 75, + 99, + 85 + ], + "score": 1.0, + "content": "35", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 73, + 507, + 86 + ], + "score": 1.0, + "content": "The key to examining the IB principle in DNNs is the accurate estimation of the mutual information.", + "type": "text", + "cross_page": true + } + ], + "index": 0, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 84, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 89, + 86, + 99, + 95 + ], + "score": 1.0, + "content": "36", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 84, + 505, + 96 + ], + "score": 1.0, + "content": "However, regarding DNNs as deterministic models hinders us from specifying the random variable", + "type": "text", + "cross_page": true + } + ], + "index": 1, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 94, + 506, + 108 + ], + "spans": [ + { + "bbox": [ + 89, + 96, + 99, + 106 + ], + "score": 1.0, + "content": "37", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 95, + 117, + 105 + ], + "score": 0.86, + "content": "T _ { i }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 117, + 94, + 198, + 108 + ], + "score": 1.0, + "content": "and the distribution", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 198, + 96, + 223, + 106 + ], + "score": 0.91, + "content": "P ( T _ { i } )", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 223, + 94, + 386, + 108 + ], + "score": 1.0, + "content": ", thus it is difficult to accurately estimate", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 387, + 95, + 423, + 106 + ], + "score": 0.92, + "content": "I ( X ; T _ { i } )", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 423, + 94, + 441, + 108 + ], + "score": 1.0, + "content": "and", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 441, + 95, + 476, + 106 + ], + "score": 0.91, + "content": "I ( Y ; T _ { i } )", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 477, + 94, + 506, + 108 + ], + "score": 1.0, + "content": ". 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Therefore, most previous works have to indirectly estimate", + "type": "text" + }, + { + "bbox": [ + 401, + 138, + 426, + 150 + ], + "score": 0.92, + "content": "P ( T _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 137, + 506, + 151 + ], + "score": 1.0, + "content": "via non-parametric", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 89, + 150, + 505, + 162 + ], + "spans": [ + { + "bbox": [ + 89, + 151, + 99, + 160 + ], + "score": 1.0, + "content": "42", + "type": "text" + }, + { + "bbox": [ + 105, + 150, + 505, + 162 + ], + "score": 1.0, + "content": "models [35], such as the empirical distribution [30], Kernel Density Estimation (KDE) [28], and", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 89, + 159, + 506, + 173 + ], + "spans": [ + { + "bbox": [ + 89, + 162, + 100, + 172 + ], + "score": 1.0, + "content": "43", + "type": "text" + }, + { + "bbox": [ + 105, + 159, + 506, + 173 + ], + "score": 1.0, + "content": "Gaussian convolution [10]. However, we experimentally confirm that classical non-parametric models", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 89, + 171, + 505, + 183 + ], + "spans": [ + { + "bbox": [ + 89, + 173, + 100, + 182 + ], + "score": 1.0, + "content": "44", + "type": "text" + }, + { + "bbox": [ + 105, + 171, + 505, + 183 + ], + "score": 1.0, + "content": "derives poor mutual information estimation [24, 22] in DNNs, and one reason is because activations", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 89, + 181, + 505, + 195 + ], + "spans": [ + { + "bbox": [ + 89, + 184, + 99, + 194 + ], + "score": 1.0, + "content": "45", + "type": "text" + }, + { + "bbox": [ + 105, + 181, + 505, + 195 + ], + "score": 1.0, + "content": "do not satisfy the i.i.d. prerequisite of non-parametric models (see Appendix G). In summary, the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 89, + 192, + 499, + 207 + ], + "spans": [ + { + "bbox": [ + 89, + 195, + 100, + 204 + ], + "score": 1.0, + "content": "46", + "type": "text" + }, + { + "bbox": [ + 104, + 192, + 499, + 207 + ], + "score": 1.0, + "content": "limitations mainly stem from the lack of an explicit probabilistic representation for deep learning.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 5.5 + }, + { + "type": "text", + "bbox": [ + 90, + 208, + 506, + 232 + ], + "lines": [ + { + "bbox": [ + 87, + 208, + 507, + 223 + ], + "spans": [ + { + "bbox": [ + 87, + 208, + 335, + 223 + ], + "score": 1.0, + "content": "47 The IB principle only formulates the information flow in", + "type": "text" + }, + { + "bbox": [ + 335, + 209, + 448, + 221 + ], + "score": 0.9, + "content": "\\mathbf { D N N s } = \\{ \\pmb { x } , \\pmb { t } _ { 1 } , \\cdot \\cdot \\cdot , \\pmb { t } _ { I } , \\hat { \\pmb { y } } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 208, + 507, + 223 + ], + "score": 1.0, + "content": "after training,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 87, + 219, + 333, + 232 + ], + "spans": [ + { + "bbox": [ + 87, + 219, + 333, + 232 + ], + "score": 1.0, + "content": "48 and the corresponding Markov chain (see Fig. 1 in [30])", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12.5 + }, + { + "type": "interline_equation", + "bbox": [ + 244, + 234, + 367, + 249 + ], + "lines": [ + { + "bbox": [ + 244, + 234, + 367, + 249 + ], + "spans": [ + { + "bbox": [ + 244, + 234, + 367, + 249 + ], + "score": 0.92, + "content": "Y X T _ { 1 } \\cdots T _ { I } { \\hat { Y } }", + "type": "interline_equation", + "image_path": "ead84a6b0118b658e9a1bf913fc61d857e2125f8496d78d794591ac5e0c3e24c.jpg" + } + ] + } + ], + "index": 14, + "virtual_lines": [ + { + "bbox": [ + 244, + 234, + 367, + 249 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 89, + 252, + 506, + 320 + ], + "lines": [ + { + "bbox": [ + 89, + 253, + 505, + 265 + ], + "spans": [ + { + "bbox": [ + 89, + 255, + 100, + 264 + ], + "score": 1.0, + "content": "49", + "type": "text" + }, + { + "bbox": [ + 106, + 253, + 245, + 265 + ], + "score": 1.0, + "content": "indicates that the information of", + "type": "text" + }, + { + "bbox": [ + 245, + 253, + 254, + 263 + ], + "score": 0.82, + "content": "Y", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 253, + 306, + 265 + ], + "score": 1.0, + "content": "transfers to", + "type": "text" + }, + { + "bbox": [ + 307, + 253, + 317, + 264 + ], + "score": 0.88, + "content": "T _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 253, + 441, + 265 + ], + "score": 1.0, + "content": "in the forward direction and", + "type": "text" + }, + { + "bbox": [ + 441, + 254, + 451, + 264 + ], + "score": 0.88, + "content": "T _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 253, + 505, + 265 + ], + "score": 1.0, + "content": "receives the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 89, + 265, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 89, + 266, + 100, + 276 + ], + "score": 1.0, + "content": "50", + "type": "text" + }, + { + "bbox": [ + 106, + 265, + 165, + 276 + ], + "score": 1.0, + "content": "information of", + "type": "text" + }, + { + "bbox": [ + 166, + 265, + 175, + 274 + ], + "score": 0.78, + "content": "Y", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 265, + 210, + 276 + ], + "score": 1.0, + "content": "only via", + "type": "text" + }, + { + "bbox": [ + 211, + 265, + 221, + 274 + ], + "score": 0.82, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 265, + 505, + 276 + ], + "score": 1.0, + "content": ". 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T _ { 1 } \\right. \\cdots \\right. T _ { I } \\right. \\hat { Y } } } \\\\ { { \\nonumber } } \\\\ { { T _ { 1 } \\left. \\cdot \\cdot \\cdot \\left. T _ { I } \\left. \\hat { Y } \\left. 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t _ { 1 } ; t _ { 2 } ; { \\hat { \\pmb y } } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 374, + 455, + 387 + ], + "score": 1.0, + "content": "forms a Markov chain", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5 + }, + { + "type": "interline_equation", + "bbox": [ + 261, + 391, + 350, + 405 + ], + "lines": [ + { + "bbox": [ + 261, + 391, + 350, + 405 + ], + "spans": [ + { + "bbox": [ + 261, + 391, + 350, + 405 + ], + "score": 0.91, + "content": "X T _ { 1 } T _ { 2 } { \\hat { Y } } .", + "type": "interline_equation", + "image_path": "4da2d70d9dacbaa625b7984616acefca401c5937c8e66798e413744240256425.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 261, + 391, + 350, + 405 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 86, + 411, + 505, + 447 + ], + "lines": [ + { + "bbox": [ + 86, + 411, + 506, + 425 + ], + "spans": [ + { + "bbox": [ + 86, + 414, + 101, + 424 + ], + "score": 1.0, + "content": "135", + "type": "text" + }, + { + "bbox": [ + 104, + 411, + 293, + 425 + ], + "score": 1.0, + "content": "Based on the corresponding joint distribution", + "type": "text" + }, + { + "bbox": [ + 293, + 411, + 486, + 425 + ], + "score": 0.91, + "content": "P ( \\hat { Y } , T _ { 2 } , T _ { 1 } | X ) = P ( T _ { 1 } | X ) P ( T _ { 2 } | T _ { 1 } ) P ( \\hat { Y } | T _ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 411, + 506, + 425 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 86, + 424, + 504, + 435 + ], + "spans": [ + { + "bbox": [ + 86, + 426, + 100, + 434 + ], + "score": 1.0, + "content": "136", + "type": "text" + }, + { + "bbox": [ + 105, + 424, + 504, + 435 + ], + "score": 1.0, + "content": "Definition 2, we derive a probabilistic explanation for the entire MLP, which is summarized in", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 86, + 434, + 363, + 447 + ], + "spans": [ + { + "bbox": [ + 86, + 436, + 100, + 446 + ], + "score": 1.0, + "content": "137", + "type": "text" + }, + { + "bbox": [ + 105, + 434, + 351, + 447 + ], + "score": 1.0, + "content": "Theorem 1. The detailed derivation is presented in Appendix", + "type": "text" + }, + { + "bbox": [ + 352, + 435, + 359, + 444 + ], + "score": 0.25, + "content": "\\mathbf { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 434, + 363, + 447 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 83, + 457, + 447, + 470 + ], + "lines": [ + { + "bbox": [ + 84, + 456, + 447, + 470 + ], + "spans": [ + { + "bbox": [ + 84, + 456, + 183, + 470 + ], + "score": 1.0, + "content": "138 Theorem 1. 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However,", + "type": "text" + }, + { + "bbox": [ + 291, + 348, + 342, + 360 + ], + "score": 0.92, + "content": "\\left( \\Omega _ { T } , \\mathcal { F } , P _ { T } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 347, + 402, + 360 + ], + "score": 1.0, + "content": "indicates that", + "type": "text" + }, + { + "bbox": [ + 403, + 349, + 414, + 359 + ], + "score": 0.87, + "content": "t _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 347, + 506, + 360 + ], + "score": 1.0, + "content": "actually is a variable", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 86, + 358, + 498, + 372 + ], + "spans": [ + { + "bbox": [ + 86, + 361, + 100, + 370 + ], + "score": 1.0, + "content": "160", + "type": "text" + }, + { + "bbox": [ + 105, + 358, + 270, + 372 + ], + "score": 1.0, + "content": "measuring the cross-correlation between", + "type": "text" + }, + { + "bbox": [ + 271, + 361, + 285, + 370 + ], + "score": 0.86, + "content": "{ \\pmb w } _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 358, + 303, + 372 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 304, + 361, + 311, + 369 + ], + "score": 0.78, + "content": "_ { z }", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 358, + 414, + 372 + ], + "score": 1.0, + "content": "rather than the sample of", + "type": "text" + }, + { + "bbox": [ + 414, + 359, + 422, + 369 + ], + "score": 0.82, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 358, + 457, + 372 + ], + "score": 1.0, + "content": ", namely", + "type": "text" + }, + { + "bbox": [ + 458, + 359, + 493, + 370 + ], + "score": 0.93, + "content": "t _ { n } \\notin E _ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 358, + 498, + 372 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 91, + 381, + 498, + 394 + ], + "lines": [ + { + "bbox": [ + 86, + 379, + 500, + 398 + ], + "spans": [ + { + "bbox": [ + 86, + 379, + 243, + 398 + ], + "score": 1.0, + "content": "161 Theorem 2. The information of", + "type": "text" + }, + { + "bbox": [ + 244, + 383, + 253, + 392 + ], + "score": 0.83, + "content": "Y", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 379, + 500, + 398 + ], + "score": 1.0, + "content": "flows into the MLP in the backward direction during training", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "interline_equation", + "bbox": [ + 262, + 400, + 349, + 414 + ], + "lines": [ + { + "bbox": [ + 262, + 400, + 349, + 414 + ], + "spans": [ + { + "bbox": [ + 262, + 400, + 349, + 414 + ], + "score": 0.92, + "content": "T _ { 1 } \\gets T _ { 2 } \\gets \\hat { Y } \\gets Y .", + "type": "interline_equation", + "image_path": "6e0bff2c4bdcdef0b2f4947e8e3d58ede655d10bd1f77754eb744a4829bcc63f.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 262, + 400, + 349, + 414 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 87, + 425, + 505, + 477 + ], + "lines": [ + { + "bbox": [ + 87, + 426, + 506, + 440 + ], + "spans": [ + { + "bbox": [ + 87, + 429, + 100, + 438 + ], + "score": 1.0, + "content": "162", + "type": "text" + }, + { + "bbox": [ + 104, + 426, + 185, + 440 + ], + "score": 1.0, + "content": "Proof: First, since", + "type": "text" + }, + { + "bbox": [ + 185, + 427, + 200, + 438 + ], + "score": 0.92, + "content": "\\Omega _ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 426, + 257, + 440 + ], + "score": 1.0, + "content": "is defined by", + "type": "text" + }, + { + "bbox": [ + 257, + 429, + 265, + 437 + ], + "score": 0.73, + "content": "\\omega", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 426, + 278, + 440 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 279, + 426, + 330, + 438 + ], + "score": 0.91, + "content": "\\left( \\Omega _ { T } , \\mathcal { F } , P _ { T } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 426, + 456, + 440 + ], + "score": 1.0, + "content": "and Equation (10) shows that", + "type": "text" + }, + { + "bbox": [ + 456, + 426, + 494, + 439 + ], + "score": 0.92, + "content": "\\omega ( s + 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 426, + 506, + 440 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 86, + 435, + 508, + 455 + ], + "spans": [ + { + "bbox": [ + 86, + 441, + 100, + 451 + ], + "score": 1.0, + "content": "163", + "type": "text" + }, + { + "bbox": [ + 105, + 439, + 275, + 452 + ], + "score": 1.0, + "content": "determined by all the previous gradients", + "type": "text" + }, + { + "bbox": [ + 275, + 439, + 322, + 454 + ], + "score": 0.92, + "content": "\\{ \\frac { \\partial \\ell _ { \\mathrm { C E } } } { \\partial \\omega ( s ) } \\} _ { s = 1 } ^ { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 435, + 508, + 455 + ], + "score": 1.0, + "content": "Es) }Ss=1 , and ω(0) is randomly initialized and α is a", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 86, + 451, + 506, + 469 + ], + "spans": [ + { + "bbox": [ + 86, + 457, + 100, + 465 + ], + "score": 1.0, + "content": "164", + "type": "text" + }, + { + "bbox": [ + 105, + 453, + 220, + 468 + ], + "score": 1.0, + "content": "constant, we can derive that", + "type": "text" + }, + { + "bbox": [ + 221, + 455, + 235, + 465 + ], + "score": 0.89, + "content": "\\Omega _ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 453, + 306, + 468 + ], + "score": 1.0, + "content": "is determined by", + "type": "text" + }, + { + "bbox": [ + 302, + 451, + 506, + 469 + ], + "score": 1.0, + "content": "∂`CE∂ω . Second, based on the back-propagation, the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 86, + 464, + 486, + 478 + ], + "spans": [ + { + "bbox": [ + 86, + 468, + 100, + 477 + ], + "score": 1.0, + "content": "165", + "type": "text" + }, + { + "bbox": [ + 104, + 464, + 329, + 478 + ], + "score": 1.0, + "content": "relation between gradients in two adjacent layers in the", + "type": "text" + }, + { + "bbox": [ + 329, + 465, + 417, + 477 + ], + "score": 0.92, + "content": "\\mathrm { M L P } = \\{ { \\pmb x } ; t _ { 1 } ; t _ { 2 } ; { \\hat { \\pmb y } } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 464, + 486, + 478 + ], + "score": 1.0, + "content": "is formulated as", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 19.5 + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 482, + 506, + 549 + ], + "lines": [ + { + "bbox": [ + 111, + 482, + 506, + 549 + ], + "spans": [ + { + "bbox": [ + 111, + 482, + 506, + 549 + ], + "score": 0.91, + "content": "\\begin{array} { r l } & { \\qquad \\displaystyle \\frac { \\partial \\ell _ { \\mathrm { C E } } ^ { \\star } } { \\partial \\omega _ { k l } ^ { ( 3 ) } } = [ P _ { \\hat { Y } | X } ( l | x ) - P _ { Y | X } ( l | x ) ] \\cdot t _ { 2 k } , } \\\\ { \\displaystyle \\partial \\ell _ { \\mathrm { C E } } ^ { ( \\zeta ) } = \\sum _ { l = 1 } ^ { L } \\displaystyle \\frac { \\partial \\ell _ { \\mathrm { C E } } ^ { \\star } } { \\partial \\omega _ { k l } ^ { ( 3 ) } } \\cdot \\omega _ { k l } ^ { ( 3 ) } \\cdot \\displaystyle \\frac { \\sigma _ { 2 } ^ { \\prime } \\big ( \\langle \\omega _ { k } ^ { ( 2 ) } , t _ { 1 } \\rangle \\big ) } { f _ { 2 k } } \\cdot t _ { 1 n } , ~ \\displaystyle \\frac { \\partial \\ell _ { \\mathrm { C E } } ^ { ( \\zeta ) } } { \\partial \\omega _ { m n } ^ { ( 1 ) } } = \\sum _ { k = 1 } ^ { K } \\displaystyle \\frac { \\partial \\ell _ { \\mathrm { C E } } ^ { ( \\zeta ) } } { \\partial \\omega _ { n k } ^ { ( 2 ) } } \\cdot \\omega _ { n k } ^ { ( 2 ) } \\cdot \\displaystyle \\frac { \\sigma _ { 1 } ^ { \\prime } \\big ( \\langle \\omega _ { n } ^ { ( 1 ) } , x \\rangle \\big ) } { t _ { 1 n } } \\cdot x _ { m } . } \\end{array}", + "type": "interline_equation", + "image_path": "209e76de1a5c3440575a539982ce200e9aab8f2258b0fece9e1c3f5ed6cb1a0b.jpg" + } + ] + } + ], + "index": 23, + "virtual_lines": [ + { + "bbox": [ + 111, + 482, + 506, + 504.3333333333333 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 111, + 504.3333333333333, + 506, + 526.6666666666666 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 111, + 526.6666666666666, + 506, + 549.0 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 86, + 565, + 505, + 618 + ], + "lines": [ + { + "bbox": [ + 85, + 563, + 506, + 581 + ], + "spans": [ + { + "bbox": [ + 85, + 569, + 101, + 579 + ], + "score": 1.0, + "content": "166", + "type": "text" + }, + { + "bbox": [ + 105, + 563, + 205, + 581 + ], + "score": 1.0, + "content": "Equation 13 shows that", + "type": "text" + }, + { + "bbox": [ + 205, + 566, + 227, + 581 + ], + "score": 0.93, + "content": "\\frac { \\partial \\ell _ { \\mathrm { C E } } } { \\partial \\omega ^ { ( 3 ) } }", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 563, + 293, + 581 + ], + "score": 1.0, + "content": "is a function of", + "type": "text" + }, + { + "bbox": [ + 293, + 567, + 336, + 580 + ], + "score": 0.94, + "content": "P _ { Y \\mid X } ( l \\mid x )", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 563, + 356, + 581 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 356, + 566, + 377, + 581 + ], + "score": 0.93, + "content": "\\frac { \\partial \\ell _ { \\mathrm { C E } } } { \\partial \\omega ^ { ( i ) } }", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 563, + 443, + 581 + ], + "score": 1.0, + "content": "is a function of", + "type": "text" + }, + { + "bbox": [ + 444, + 566, + 473, + 581 + ], + "score": 0.93, + "content": "\\frac { \\partial \\ell _ { \\mathrm { C E } } } { \\partial \\omega ^ { ( i + 1 ) } }", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 566, + 506, + 579 + ], + "score": 1.0, + "content": ", where", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 85, + 577, + 504, + 597 + ], + "spans": [ + { + "bbox": [ + 85, + 581, + 100, + 592 + ], + "score": 1.0, + "content": "167", + "type": "text" + }, + { + "bbox": [ + 106, + 579, + 124, + 591 + ], + "score": 0.87, + "content": "\\boldsymbol { \\omega } ^ { ( 3 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 125, + 577, + 212, + 597 + ], + "score": 1.0, + "content": "denotes the weight of", + "type": "text" + }, + { + "bbox": [ + 212, + 582, + 219, + 592 + ], + "score": 0.84, + "content": "\\hat { \\pmb { y } }", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 577, + 401, + 597 + ], + "score": 1.0, + "content": ". The two points above enable us to derive that", + "type": "text" + }, + { + "bbox": [ + 401, + 581, + 417, + 593 + ], + "score": 0.91, + "content": "\\Omega _ { T _ { i } }", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 577, + 479, + 597 + ], + "score": 1.0, + "content": "is a function of", + "type": "text" + }, + { + "bbox": [ + 480, + 581, + 504, + 594 + ], + "score": 0.91, + "content": "\\Omega _ { T _ { i + 1 } }", + "type": "inline_equation" + } + ], + "index": 26 + }, + { + "bbox": [ + 86, + 591, + 506, + 604 + ], + "spans": [ + { + "bbox": [ + 86, + 594, + 100, + 604 + ], + "score": 1.0, + "content": "168", + "type": "text" + }, + { + "bbox": [ + 105, + 591, + 123, + 604 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 592, + 138, + 604 + ], + "score": 0.9, + "content": "\\Omega _ { \\hat { Y } }", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 591, + 203, + 604 + ], + "score": 1.0, + "content": "is a function of", + "type": "text" + }, + { + "bbox": [ + 203, + 592, + 239, + 604 + ], + "score": 0.92, + "content": "P ( { Y \\vert } X )", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 591, + 441, + 604 + ], + "score": 1.0, + "content": ". Based on Definition 2, we can further derive that", + "type": "text" + }, + { + "bbox": [ + 441, + 592, + 452, + 603 + ], + "score": 0.87, + "content": "T _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 591, + 506, + 604 + ], + "score": 1.0, + "content": "is a function", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 86, + 603, + 507, + 618 + ], + "spans": [ + { + "bbox": [ + 86, + 607, + 100, + 617 + ], + "score": 1.0, + "content": "169", + "type": "text" + }, + { + "bbox": [ + 105, + 604, + 117, + 618 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 117, + 605, + 138, + 617 + ], + "score": 0.92, + "content": "T _ { i + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 604, + 155, + 618 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 156, + 603, + 165, + 615 + ], + "score": 0.86, + "content": "\\hat { Y }", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 604, + 228, + 618 + ], + "score": 1.0, + "content": "is a function of", + "type": "text" + }, + { + "bbox": [ + 229, + 605, + 238, + 615 + ], + "score": 0.74, + "content": "Y", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 604, + 258, + 618 + ], + "score": 1.0, + "content": ", i.e.,", + "type": "text" + }, + { + "bbox": [ + 259, + 603, + 343, + 617 + ], + "score": 0.93, + "content": "T _ { 1 } \\gets T _ { 2 } \\gets \\hat { Y } \\gets Y", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 604, + 507, + 618 + ], + "score": 1.0, + "content": ". (See the detailed proof in Appendix C).", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26.5 + }, + { + "type": "text", + "bbox": [ + 86, + 621, + 505, + 723 + ], + "lines": [ + { + "bbox": [ + 86, + 622, + 506, + 634 + ], + "spans": [ + { + "bbox": [ + 86, + 624, + 99, + 633 + ], + "score": 1.0, + "content": "170", + "type": "text" + }, + { + "bbox": [ + 105, + 622, + 506, + 634 + ], + "score": 1.0, + "content": "Theorem 2 is consistent with the prevailing explanation for deep learning. LeCunn et al. show that", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 86, + 633, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 86, + 635, + 99, + 644 + ], + "score": 1.0, + "content": "171", + "type": "text" + }, + { + "bbox": [ + 105, + 633, + 505, + 645 + ], + "score": 1.0, + "content": "deep learning exploits the hierarchical property of signals [18], i.e., the layers farther from output", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 86, + 644, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 86, + 645, + 100, + 656 + ], + "score": 1.0, + "content": "172", + "type": "text" + }, + { + "bbox": [ + 105, + 644, + 505, + 656 + ], + "score": 1.0, + "content": "learn lower-level features, such as edges, whereas the layers closer to output assemble lower-level", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 86, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 86, + 656, + 100, + 667 + ], + "score": 1.0, + "content": "173", + "type": "text" + }, + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "features into the higher-level features corresponding to labels (see Figure 2 in [37]). Notably, since", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 86, + 665, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 86, + 667, + 100, + 677 + ], + "score": 1.0, + "content": "174", + "type": "text" + }, + { + "bbox": [ + 104, + 665, + 506, + 678 + ], + "score": 1.0, + "content": "lower-level features commonly exist in signals with different labels (e.g., lower-level features, such", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 86, + 676, + 505, + 688 + ], + "spans": [ + { + "bbox": [ + 86, + 678, + 100, + 688 + ], + "score": 1.0, + "content": "175", + "type": "text" + }, + { + "bbox": [ + 105, + 676, + 505, + 688 + ], + "score": 1.0, + "content": "as the edges of the vehicle frame and the circular contour of wheels, exist in both the car and the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 86, + 687, + 505, + 699 + ], + "spans": [ + { + "bbox": [ + 86, + 689, + 100, + 699 + ], + "score": 1.0, + "content": "176", + "type": "text" + }, + { + "bbox": [ + 105, + 687, + 505, + 699 + ], + "score": 1.0, + "content": "truck classes in the CIFAR-10 dataset [16] in Figure 2), lower-level features do not contain much", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 86, + 698, + 505, + 710 + ], + "spans": [ + { + "bbox": [ + 86, + 700, + 100, + 709 + ], + "score": 1.0, + "content": "177", + "type": "text" + }, + { + "bbox": [ + 105, + 698, + 505, + 710 + ], + "score": 1.0, + "content": "information of labels. 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The mutual information between a fully connected layer and dataset is finite.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7, + "bbox_fs": [ + 86, + 244, + 484, + 258 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 274, + 262, + 336, + 276 + ], + "lines": [ + { + "bbox": [ + 274, + 262, + 336, + 276 + ], + "spans": [ + { + "bbox": [ + 274, + 262, + 336, + 276 + ], + "score": 0.92, + "content": "I ( X ; T ) < \\infty .", + "type": "interline_equation", + "image_path": "c2322c37754743cf88c1e39576148863252d3766f16220cff196fcfea7814eff.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 274, + 262, + 336, + 276 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "index", + "bbox": [ + 88, + 286, + 507, + 311 + ], + "lines": [ + { + "bbox": [ + 84, + 285, + 508, + 302 + ], + "spans": [ + { + "bbox": [ + 84, + 285, + 213, + 302 + ], + "score": 1.0, + "content": "154 Proof: Definition 2 shows", + "type": "text" + }, + { + "bbox": [ + 213, + 288, + 282, + 300 + ], + "score": 0.93, + "content": "E _ { T } = \\{ 1 , \\cdots N \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 283, + 285, + 309, + 302 + ], + "score": 1.0, + "content": ". Thus", + "type": "text" + }, + { + "bbox": [ + 310, + 288, + 318, + 298 + ], + "score": 0.84, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 285, + 454, + 302 + ], + "score": 1.0, + "content": "is a discrete random variable and", + "type": "text" + }, + { + "bbox": [ + 454, + 288, + 502, + 300 + ], + "score": 0.92, + "content": "H ( T ) < \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 285, + 508, + 302 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 9, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 298, + 239, + 311 + ], + "spans": [ + { + "bbox": [ + 85, + 298, + 139, + 311 + ], + "score": 1.0, + "content": "155 thereby", + "type": "text" + }, + { + "bbox": [ + 139, + 298, + 234, + 311 + ], + "score": 0.92, + "content": "I ( X ; T ) \\leq H ( T ) < \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 298, + 239, + 311 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 10, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 315, + 505, + 327 + ], + "spans": [ + { + "bbox": [ + 86, + 317, + 100, + 326 + ], + "score": 1.0, + "content": "156", + "type": "text" + }, + { + "bbox": [ + 106, + 315, + 505, + 327 + ], + "score": 1.0, + "content": "Proposition 1 circumvents the infinite mutual information problem. In the absence of a clear definition", + "type": "text" + } + ], + "index": 11, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 324, + 506, + 339 + ], + "spans": [ + { + "bbox": [ + 86, + 327, + 100, + 338 + ], + "score": 1.0, + "content": "157", + "type": "text" + }, + { + "bbox": [ + 107, + 326, + 166, + 337 + ], + "score": 0.91, + "content": "T : \\Omega _ { T } \\to E _ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 166, + 324, + 415, + 339 + ], + "score": 1.0, + "content": ", most previous works [28, 3, 1] simply viewing the activation", + "type": "text" + }, + { + "bbox": [ + 415, + 327, + 425, + 337 + ], + "score": 0.88, + "content": "t _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 324, + 494, + 339 + ], + "score": 1.0, + "content": "as the sample of", + "type": "text" + }, + { + "bbox": [ + 494, + 326, + 502, + 336 + ], + "score": 0.83, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 324, + 506, + 339 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 12, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 337, + 505, + 349 + ], + "spans": [ + { + "bbox": [ + 86, + 339, + 100, + 348 + ], + "score": 1.0, + "content": "158", + "type": "text" + }, + { + "bbox": [ + 106, + 337, + 138, + 349 + ], + "score": 1.0, + "content": "namely", + "type": "text" + }, + { + "bbox": [ + 139, + 337, + 195, + 348 + ], + "score": 0.92, + "content": "t _ { n } \\in E _ { T } = \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 337, + 230, + 349 + ], + "score": 1.0, + "content": ", implies", + "type": "text" + }, + { + "bbox": [ + 231, + 337, + 239, + 347 + ], + "score": 0.82, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 337, + 505, + 349 + ], + "score": 1.0, + "content": "being continuous and gives rise to the infinite mutual information", + "type": "text" + } + ], + "index": 13, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 347, + 506, + 360 + ], + "spans": [ + { + "bbox": [ + 86, + 349, + 100, + 360 + ], + "score": 1.0, + "content": "159", + "type": "text" + }, + { + "bbox": [ + 106, + 347, + 290, + 360 + ], + "score": 1.0, + "content": "problem in deterministic DNNs. However,", + "type": "text" + }, + { + "bbox": [ + 291, + 348, + 342, + 360 + ], + "score": 0.92, + "content": "\\left( \\Omega _ { T } , \\mathcal { F } , P _ { T } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 347, + 402, + 360 + ], + "score": 1.0, + "content": "indicates that", + "type": "text" + }, + { + "bbox": [ + 403, + 349, + 414, + 359 + ], + "score": 0.87, + "content": "t _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 347, + 506, + 360 + ], + "score": 1.0, + "content": "actually is a variable", + "type": "text" + } + ], + "index": 14, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 358, + 498, + 372 + ], + "spans": [ + { + "bbox": [ + 86, + 361, + 100, + 370 + ], + "score": 1.0, + "content": "160", + "type": "text" + }, + { + "bbox": [ + 105, + 358, + 270, + 372 + ], + "score": 1.0, + "content": "measuring the cross-correlation between", + "type": "text" + }, + { + "bbox": [ + 271, + 361, + 285, + 370 + ], + "score": 0.86, + "content": "{ \\pmb w } _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 358, + 303, + 372 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 304, + 361, + 311, + 369 + ], + "score": 0.78, + "content": "_ { z }", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 358, + 414, + 372 + ], + "score": 1.0, + "content": "rather than the sample of", + "type": "text" + }, + { + "bbox": [ + 414, + 359, + 422, + 369 + ], + "score": 0.82, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 358, + 457, + 372 + ], + "score": 1.0, + "content": ", namely", + "type": "text" + }, + { + "bbox": [ + 458, + 359, + 493, + 370 + ], + "score": 0.93, + "content": "t _ { n } \\notin E _ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 358, + 498, + 372 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 15, + "is_list_start_line": true + } + ], + "index": 9.5, + "bbox_fs": [ + 84, + 285, + 508, + 311 + ] + }, + { + "type": "index", + "bbox": [ + 87, + 315, + 505, + 371 + ], + "lines": [], + "index": 13, + "bbox_fs": [ + 86, + 315, + 506, + 372 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 91, + 381, + 498, + 394 + ], + "lines": [ + { + "bbox": [ + 86, + 379, + 500, + 398 + ], + "spans": [ + { + "bbox": [ + 86, + 379, + 243, + 398 + ], + "score": 1.0, + "content": "161 Theorem 2. 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P _ { Y | X } ( l | x ) ] \\cdot t _ { 2 k } , } \\\\ { \\displaystyle \\partial \\ell _ { \\mathrm { C E } } ^ { ( \\zeta ) } = \\sum _ { l = 1 } ^ { L } \\displaystyle \\frac { \\partial \\ell _ { \\mathrm { C E } } ^ { \\star } } { \\partial \\omega _ { k l } ^ { ( 3 ) } } \\cdot \\omega _ { k l } ^ { ( 3 ) } \\cdot \\displaystyle \\frac { \\sigma _ { 2 } ^ { \\prime } \\big ( \\langle \\omega _ { k } ^ { ( 2 ) } , t _ { 1 } \\rangle \\big ) } { f _ { 2 k } } \\cdot t _ { 1 n } , ~ \\displaystyle \\frac { \\partial \\ell _ { \\mathrm { C E } } ^ { ( \\zeta ) } } { \\partial \\omega _ { m n } ^ { ( 1 ) } } = \\sum _ { k = 1 } ^ { K } \\displaystyle \\frac { \\partial \\ell _ { \\mathrm { C E } } ^ { ( \\zeta ) } } { \\partial \\omega _ { n k } ^ { ( 2 ) } } \\cdot \\omega _ { n k } ^ { ( 2 ) } \\cdot \\displaystyle \\frac { \\sigma _ { 1 } ^ { \\prime } \\big ( \\langle \\omega _ { n } ^ { ( 1 ) } , x \\rangle \\big ) } { t _ { 1 n } } \\cdot x _ { m } . } \\end{array}", + "type": "interline_equation", + "image_path": "209e76de1a5c3440575a539982ce200e9aab8f2258b0fece9e1c3f5ed6cb1a0b.jpg" + } + ] + } + ], + "index": 23, + "virtual_lines": [ + { + "bbox": [ + 111, + 482, + 506, + 504.3333333333333 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 111, + 504.3333333333333, + 506, + 526.6666666666666 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 111, + 526.6666666666666, + 506, + 549.0 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "index", + "bbox": [ + 86, + 565, + 505, + 618 + ], + "lines": [ + { + "bbox": [ + 85, + 563, + 506, + 581 + ], + "spans": [ + { + "bbox": [ + 85, + 569, + 101, + 579 + ], + "score": 1.0, + "content": "166", + "type": "text" + }, + { + "bbox": [ + 105, + 563, + 205, + 581 + ], + "score": 1.0, + "content": "Equation 13 shows that", + "type": "text" + }, + { + "bbox": [ + 205, + 566, + 227, + 581 + ], + "score": 0.93, + "content": "\\frac { \\partial \\ell _ { \\mathrm { C E } } } { \\partial \\omega ^ { ( 3 ) } }", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 563, + 293, + 581 + ], + "score": 1.0, + "content": "is a function of", + "type": "text" + }, + { + "bbox": [ + 293, + 567, + 336, + 580 + ], + "score": 0.94, + "content": "P _ { Y \\mid X } ( l \\mid x )", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 563, + 356, + 581 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 356, + 566, + 377, + 581 + ], + "score": 0.93, + "content": "\\frac { \\partial \\ell _ { \\mathrm { C E } } } { \\partial \\omega ^ { ( i ) } }", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 563, + 443, + 581 + ], + "score": 1.0, + "content": "is a function of", + "type": "text" + }, + { + "bbox": [ + 444, + 566, + 473, + 581 + ], + "score": 0.93, + "content": "\\frac { \\partial \\ell _ { \\mathrm { C E } } } { \\partial \\omega ^ { ( i + 1 ) } }", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 566, + 506, + 579 + ], + "score": 1.0, + "content": ", where", + "type": "text" + } + ], + "index": 25, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 577, + 504, + 597 + ], + "spans": [ + { + "bbox": [ + 85, + 581, + 100, + 592 + ], + "score": 1.0, + "content": "167", + "type": "text" + }, + { + "bbox": [ + 106, + 579, + 124, + 591 + ], + "score": 0.87, + "content": "\\boldsymbol { \\omega } ^ { ( 3 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 125, + 577, + 212, + 597 + ], + "score": 1.0, + "content": "denotes the weight of", + "type": "text" + }, + { + "bbox": [ + 212, + 582, + 219, + 592 + ], + "score": 0.84, + "content": "\\hat { \\pmb { y } }", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 577, + 401, + 597 + ], + "score": 1.0, + "content": ". The two points above enable us to derive that", + "type": "text" + }, + { + "bbox": [ + 401, + 581, + 417, + 593 + ], + "score": 0.91, + "content": "\\Omega _ { T _ { i } }", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 577, + 479, + 597 + ], + "score": 1.0, + "content": "is a function of", + "type": "text" + }, + { + "bbox": [ + 480, + 581, + 504, + 594 + ], + "score": 0.91, + "content": "\\Omega _ { T _ { i + 1 } }", + "type": "inline_equation" + } + ], + "index": 26, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 591, + 506, + 604 + ], + "spans": [ + { + "bbox": [ + 86, + 594, + 100, + 604 + ], + "score": 1.0, + "content": "168", + "type": "text" + }, + { + "bbox": [ + 105, + 591, + 123, + 604 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 592, + 138, + 604 + ], + "score": 0.9, + "content": "\\Omega _ { \\hat { Y } }", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 591, + 203, + 604 + ], + "score": 1.0, + "content": "is a function of", + "type": "text" + }, + { + "bbox": [ + 203, + 592, + 239, + 604 + ], + "score": 0.92, + "content": "P ( { Y \\vert } X )", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 591, + 441, + 604 + ], + "score": 1.0, + "content": ". Based on Definition 2, we can further derive that", + "type": "text" + }, + { + "bbox": [ + 441, + 592, + 452, + 603 + ], + "score": 0.87, + "content": "T _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 591, + 506, + 604 + ], + "score": 1.0, + "content": "is a function", + "type": "text" + } + ], + "index": 27, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 603, + 507, + 618 + ], + "spans": [ + { + "bbox": [ + 86, + 607, + 100, + 617 + ], + "score": 1.0, + "content": "169", + "type": "text" + }, + { + "bbox": [ + 105, + 604, + 117, + 618 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 117, + 605, + 138, + 617 + ], + "score": 0.92, + "content": "T _ { i + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 604, + 155, + 618 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 156, + 603, + 165, + 615 + ], + "score": 0.86, + "content": "\\hat { Y }", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 604, + 228, + 618 + ], + "score": 1.0, + "content": "is a function of", + "type": "text" + }, + { + "bbox": [ + 229, + 605, + 238, + 615 + ], + "score": 0.74, + "content": "Y", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 604, + 258, + 618 + ], + "score": 1.0, + "content": ", i.e.,", + "type": "text" + }, + { + "bbox": [ + 259, + 603, + 343, + 617 + ], + "score": 0.93, + "content": "T _ { 1 } \\gets T _ { 2 } \\gets \\hat { Y } \\gets Y", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 604, + 507, + 618 + ], + "score": 1.0, + "content": ". (See the detailed proof in Appendix C).", + "type": "text" + } + ], + "index": 28, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 622, + 506, + 634 + ], + "spans": [ + { + "bbox": [ + 86, + 624, + 99, + 633 + ], + "score": 1.0, + "content": "170", + "type": "text" + }, + { + "bbox": [ + 105, + 622, + 506, + 634 + ], + "score": 1.0, + "content": "Theorem 2 is consistent with the prevailing explanation for deep learning. LeCunn et al. show that", + "type": "text" + } + ], + "index": 29, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 633, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 86, + 635, + 99, + 644 + ], + "score": 1.0, + "content": "171", + "type": "text" + }, + { + "bbox": [ + 105, + 633, + 505, + 645 + ], + "score": 1.0, + "content": "deep learning exploits the hierarchical property of signals [18], i.e., the layers farther from output", + "type": "text" + } + ], + "index": 30, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 644, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 86, + 645, + 100, + 656 + ], + "score": 1.0, + "content": "172", + "type": "text" + }, + { + "bbox": [ + 105, + 644, + 505, + 656 + ], + "score": 1.0, + "content": "learn lower-level features, such as edges, whereas the layers closer to output assemble lower-level", + "type": "text" + } + ], + "index": 31, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 86, + 656, + 100, + 667 + ], + "score": 1.0, + "content": "173", + "type": "text" + }, + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "features into the higher-level features corresponding to labels (see Figure 2 in [37]). Notably, since", + "type": "text" + } + ], + "index": 32, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 665, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 86, + 667, + 100, + 677 + ], + "score": 1.0, + "content": "174", + "type": "text" + }, + { + "bbox": [ + 104, + 665, + 506, + 678 + ], + "score": 1.0, + "content": "lower-level features commonly exist in signals with different labels (e.g., lower-level features, such", + "type": "text" + } + ], + "index": 33, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 676, + 505, + 688 + ], + "spans": [ + { + "bbox": [ + 86, + 678, + 100, + 688 + ], + "score": 1.0, + "content": "175", + "type": "text" + }, + { + "bbox": [ + 105, + 676, + 505, + 688 + ], + "score": 1.0, + "content": "as the edges of the vehicle frame and the circular contour of wheels, exist in both the car and the", + "type": "text" + } + ], + "index": 34, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 687, + 505, + 699 + ], + "spans": [ + { + "bbox": [ + 86, + 689, + 100, + 699 + ], + "score": 1.0, + "content": "176", + "type": "text" + }, + { + "bbox": [ + 105, + 687, + 505, + 699 + ], + "score": 1.0, + "content": "truck classes in the CIFAR-10 dataset [16] in Figure 2), lower-level features do not contain much", + "type": "text" + } + ], + "index": 35, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 698, + 505, + 710 + ], + "spans": [ + { + "bbox": [ + 86, + 700, + 100, + 709 + ], + "score": 1.0, + "content": "177", + "type": "text" + }, + { + "bbox": [ + 105, + 698, + 505, + 710 + ], + "score": 1.0, + "content": "information of labels. Therefore, the layers farther from output do not have much information of", + "type": "text" + } + ], + "index": 36, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 709, + 394, + 723 + ], + "spans": [ + { + "bbox": [ + 86, + 713, + 100, + 722 + ], + "score": 1.0, + "content": "178", + "type": "text" + }, + { + "bbox": [ + 104, + 710, + 306, + 723 + ], + "score": 1.0, + "content": "labels, which is consistent with the Markov chain", + "type": "text" + }, + { + "bbox": [ + 306, + 709, + 390, + 722 + ], + "score": 0.92, + "content": "T _ { 1 } \\gets T _ { 2 } \\gets \\hat { Y } \\gets Y", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 710, + 394, + 723 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 37, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 72, + 506, + 85 + ], + "spans": [ + { + "bbox": [ + 86, + 75, + 100, + 84 + ], + "score": 1.0, + "content": "179", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 72, + 226, + 85 + ], + "score": 1.0, + "content": "Since all the information of", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 227, + 73, + 236, + 83 + ], + "score": 0.79, + "content": "Y", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 236, + 72, + 288, + 85 + ], + "score": 1.0, + "content": "stems from", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 289, + 73, + 299, + 83 + ], + "score": 0.79, + "content": "X", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 299, + 72, + 323, + 85 + ], + "score": 1.0, + "content": "(i.e.,", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 324, + 73, + 401, + 85 + ], + "score": 0.92, + "content": "H ( Y ) = I ( X ; Y )", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 401, + 72, + 506, + 85 + ], + "score": 1.0, + "content": "proven in Appendix D),", + "type": "text", + "cross_page": true + } + ], + "index": 0, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 82, + 506, + 97 + ], + "spans": [ + { + "bbox": [ + 86, + 85, + 100, + 95 + ], + "score": 1.0, + "content": "180", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 82, + 286, + 97 + ], + "score": 1.0, + "content": "Theorem 2 implies that partial information of", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 286, + 84, + 296, + 93 + ], + "score": 0.85, + "content": "X", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 297, + 82, + 506, + 97 + ], + "score": 1.0, + "content": "flows into the MLP in the backward direction during", + "type": "text", + "cross_page": true + } + ], + "index": 1, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 94, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 86, + 97, + 99, + 106 + ], + "score": 1.0, + "content": "181", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 94, + 303, + 106 + ], + "score": 1.0, + "content": "training. Equation (2) shows the information of", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 303, + 95, + 313, + 104 + ], + "score": 0.83, + "content": "X", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 314, + 94, + 506, + 106 + ], + "score": 1.0, + "content": "flowing into the MLP in the forward direction", + "type": "text", + "cross_page": true + } + ], + "index": 2, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 104, + 506, + 119 + ], + "spans": [ + { + "bbox": [ + 86, + 108, + 100, + 117 + ], + "score": 1.0, + "content": "182", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 104, + 285, + 119 + ], + "score": 1.0, + "content": "during inference. Overall, the information of", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 285, + 106, + 295, + 115 + ], + "score": 0.82, + "content": "X", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 296, + 104, + 506, + 119 + ], + "score": 1.0, + "content": "flows in the backward and forward directions during", + "type": "text", + "cross_page": true + } + ], + "index": 3, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 116, + 506, + 129 + ], + "spans": [ + { + "bbox": [ + 86, + 118, + 100, + 128 + ], + "score": 1.0, + "content": "183", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 116, + 506, + 129 + ], + "score": 1.0, + "content": "training and inference, respectively. As a result, the Markov chain, Equation (2), proposed by recent", + "type": "text", + "cross_page": true + } + ], + "index": 4, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 128, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 86, + 129, + 100, + 138 + ], + "score": 1.0, + "content": "184", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 128, + 346, + 139 + ], + "score": 1.0, + "content": "works could not fully characterize the information flow of", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 347, + 128, + 357, + 137 + ], + "score": 0.84, + "content": "X", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 357, + 128, + 505, + 139 + ], + "score": 1.0, + "content": "in the MLP in each epoch. In other", + "type": "text", + "cross_page": true + } + ], + "index": 5, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 138, + 449, + 150 + ], + "spans": [ + { + "bbox": [ + 86, + 141, + 100, + 149 + ], + "score": 1.0, + "content": "185", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 138, + 135, + 150 + ], + "score": 1.0, + "content": "words,", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 136, + 138, + 172, + 150 + ], + "score": 0.93, + "content": "I ( X ; T _ { i } )", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 172, + 138, + 294, + 150 + ], + "score": 1.0, + "content": "is not necessarily greater than", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 295, + 138, + 341, + 150 + ], + "score": 0.93, + "content": "I ( X ; T _ { i + 1 } )", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 341, + 138, + 449, + 150 + ], + "score": 1.0, + "content": "in the MLP in each epoch.", + "type": "text", + "cross_page": true + } + ], + "index": 6, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 154, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 86, + 156, + 100, + 166 + ], + "score": 1.0, + "content": "186", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 154, + 203, + 167 + ], + "score": 1.0, + "content": "Equation (2) shows that", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 203, + 155, + 214, + 165 + ], + "score": 0.88, + "content": "T _ { i }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 214, + 154, + 324, + 167 + ], + "score": 1.0, + "content": "receives the information of", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 325, + 155, + 334, + 164 + ], + "score": 0.82, + "content": "Y", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 334, + 154, + 349, + 167 + ], + "score": 1.0, + "content": "via", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 350, + 155, + 360, + 164 + ], + "score": 0.83, + "content": "X", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 360, + 154, + 505, + 167 + ], + "score": 1.0, + "content": "during inference. 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Thus Equation (2) cannot fully characterize the information", + "type": "text", + "cross_page": true + } + ], + "index": 9, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 186, + 486, + 201 + ], + "spans": [ + { + "bbox": [ + 86, + 189, + 100, + 199 + ], + "score": 1.0, + "content": "189", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 186, + 137, + 201 + ], + "score": 1.0, + "content": "flow of", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 137, + 188, + 146, + 197 + ], + "score": 0.8, + "content": "Y", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 147, + 186, + 486, + 201 + ], + "score": 1.0, + "content": "in the MLP in each epoch, when we take into account the back-propagation training.", + "type": "text", + "cross_page": true + } + ], + "index": 10, + "is_list_start_line": true + }, + { + "bbox": [ + 87, + 204, + 487, + 216 + ], + "spans": [ + { + "bbox": [ + 87, + 205, + 100, + 214 + ], + "score": 1.0, + "content": "190", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 104, + 204, + 487, + 216 + ], + "score": 1.0, + "content": "To fully characterize the information flow in the MLP in each epoch, we introduce Corollary 1.", + "type": "text", + "cross_page": true + } + ], + "index": 11, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 214, + 483, + 227 + ], + "spans": [ + { + "bbox": [ + 86, + 216, + 100, + 227 + ], + "score": 1.0, + "content": "191", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 104, + 214, + 483, + 227 + ], + "score": 1.0, + "content": "Corollary 1. The information flow in the MLP can be characterized by two Markov chains as", + "type": "text", + "cross_page": true + } + ], + "index": 12, + "is_list_start_line": true + } + ], + "index": 26.5, + "bbox_fs": [ + 85, + 563, + 507, + 618 + ] + }, + { + "type": "index", + "bbox": [ + 86, + 621, + 505, + 723 + ], + "lines": [], + "index": 33, + "bbox_fs": [ + 86, + 622, + 506, + 723 + ], + "lines_deleted": true + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 86, + 72, + 505, + 150 + ], + "lines": [ + { + "bbox": [ + 86, + 72, + 506, + 85 + ], + "spans": [ + { + "bbox": [ + 86, + 75, + 100, + 84 + ], + "score": 1.0, + "content": "179", + "type": "text" + }, + { + "bbox": [ + 106, + 72, + 226, + 85 + ], + "score": 1.0, + "content": "Since all the information of", + "type": "text" + }, + { + "bbox": [ + 227, + 73, + 236, + 83 + ], + "score": 0.79, + "content": "Y", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 72, + 288, + 85 + ], + "score": 1.0, + "content": "stems from", + "type": "text" + }, + { + "bbox": [ + 289, + 73, + 299, + 83 + ], + "score": 0.79, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 72, + 323, + 85 + ], + "score": 1.0, + "content": "(i.e.,", + "type": "text" + }, + { + "bbox": [ + 324, + 73, + 401, + 85 + ], + "score": 0.92, + "content": "H ( Y ) = I ( X ; Y )", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 72, + 506, + 85 + ], + "score": 1.0, + "content": "proven in Appendix D),", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 86, + 82, + 506, + 97 + ], + "spans": [ + { + "bbox": [ + 86, + 85, + 100, + 95 + ], + "score": 1.0, + "content": "180", + "type": "text" + }, + { + "bbox": [ + 105, + 82, + 286, + 97 + ], + "score": 1.0, + "content": "Theorem 2 implies that partial information of", + "type": "text" + }, + { + "bbox": [ + 286, + 84, + 296, + 93 + ], + "score": 0.85, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 82, + 506, + 97 + ], + "score": 1.0, + "content": "flows into the MLP in the backward direction during", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 86, + 94, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 86, + 97, + 99, + 106 + ], + "score": 1.0, + "content": "181", + "type": "text" + }, + { + "bbox": [ + 105, + 94, + 303, + 106 + ], + "score": 1.0, + "content": "training. 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Y . } } \\end{array}", + "type": "interline_equation", + "image_path": "5009ee51df98562fce5a0f511b8df83d1af573616fc90fd96cb436e7cdb2eab5.jpg" + } + ] + } + ], + "index": 13.5, + "virtual_lines": [ + { + "bbox": [ + 250, + 228, + 360, + 244.5 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 250, + 244.5, + 360, + 261.0 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 90, + 264, + 504, + 277 + ], + "lines": [ + { + "bbox": [ + 86, + 261, + 501, + 280 + ], + "spans": [ + { + "bbox": [ + 86, + 261, + 220, + 280 + ], + "score": 1.0, + "content": "The virtual random variable 192", + "type": "text" + }, + { + "bbox": [ + 220, + 264, + 230, + 275 + ], + "score": 0.85, + "content": "\\bar { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 261, + 354, + 280 + ], + "score": 1.0, + "content": "contains all the information of", + "type": "text" + }, + { + "bbox": [ + 355, + 265, + 365, + 275 + ], + "score": 0.84, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 261, + 394, + 280 + ], + "score": 1.0, + "content": "except", + "type": "text" + }, + { + "bbox": [ + 394, + 265, + 403, + 275 + ], + "score": 0.76, + "content": "Y", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 261, + 424, + 280 + ], + "score": 1.0, + "content": ", i.e.,", + "type": "text" + }, + { + "bbox": [ + 425, + 264, + 501, + 277 + ], + "score": 0.93, + "content": "H ( { \\bar { X } } ) = H ( X | Y )", + "type": "inline_equation" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 86, + 280, + 506, + 328 + ], + "lines": [ + { + "bbox": [ + 86, + 281, + 506, + 294 + ], + "spans": [ + { + "bbox": [ + 86, + 283, + 100, + 293 + ], + "score": 1.0, + "content": "193", + "type": "text" + }, + { + "bbox": [ + 104, + 281, + 262, + 294 + ], + "score": 1.0, + "content": "Proof of the first Markov chain: Since", + "type": "text" + }, + { + "bbox": [ + 262, + 281, + 272, + 291 + ], + "score": 0.85, + "content": "\\bar { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 281, + 408, + 294 + ], + "score": 1.0, + "content": "does not have any information of", + "type": "text" + }, + { + "bbox": [ + 409, + 282, + 418, + 291 + ], + "score": 0.78, + "content": "Y", + "type": "inline_equation" + }, + { + "bbox": [ + 418, + 281, + 506, + 294 + ], + "score": 1.0, + "content": ", it can only flow into", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 86, + 291, + 506, + 304 + ], + "spans": [ + { + "bbox": [ + 86, + 294, + 100, + 304 + ], + "score": 1.0, + "content": "194", + "type": "text" + }, + { + "bbox": [ + 105, + 292, + 360, + 304 + ], + "score": 1.0, + "content": "the MLP in the forward direction during inference. 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Overall, the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 85, + 414, + 506, + 428 + ], + "spans": [ + { + "bbox": [ + 85, + 416, + 100, + 426 + ], + "score": 1.0, + "content": "204", + "type": "text" + }, + { + "bbox": [ + 105, + 414, + 185, + 428 + ], + "score": 1.0, + "content": "information flow of", + "type": "text" + }, + { + "bbox": [ + 185, + 415, + 195, + 424 + ], + "score": 0.77, + "content": "Y", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 414, + 506, + 428 + ], + "score": 1.0, + "content": "during inference will be the same as that during training. Based on Theorem 2,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 85, + 425, + 506, + 439 + ], + "spans": [ + { + "bbox": [ + 85, + 429, + 100, + 439 + ], + "score": 1.0, + "content": "205", + "type": "text" + }, + { + "bbox": [ + 105, + 426, + 177, + 439 + ], + "score": 1.0, + "content": "we conclude that", + "type": "text" + }, + { + "bbox": [ + 178, + 425, + 262, + 438 + ], + "score": 0.93, + "content": "T _ { 1 } \\gets T _ { 2 } \\gets \\hat { Y } \\gets Y", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 426, + 415, + 439 + ], + "score": 1.0, + "content": "characterizes the information flow of", + "type": "text" + }, + { + "bbox": [ + 415, + 427, + 424, + 437 + ], + "score": 0.79, + "content": "Y", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 426, + 506, + 439 + ], + "score": 1.0, + "content": "in the MLP in both", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 86, + 438, + 499, + 450 + ], + "spans": [ + { + "bbox": [ + 86, + 440, + 100, + 449 + ], + "score": 1.0, + "content": "206", + "type": "text" + }, + { + "bbox": [ + 105, + 438, + 499, + 450 + ], + "score": 1.0, + "content": "training and inference phases. Detailed derivations and explanations are presented in Appendix E.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 24.5 + }, + { + "type": "text", + "bbox": [ + 95, + 453, + 506, + 477 + ], + "lines": [ + { + "bbox": [ + 92, + 454, + 507, + 467 + ], + "spans": [ + { + "bbox": [ + 92, + 454, + 263, + 467 + ], + "score": 1.0, + "content": "07 To quantify how much information of", + "type": "text" + }, + { + "bbox": [ + 264, + 455, + 274, + 464 + ], + "score": 0.85, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 454, + 293, + 467 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 293, + 455, + 302, + 464 + ], + "score": 0.81, + "content": "Y", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 454, + 507, + 467 + ], + "score": 1.0, + "content": "is learned by the MLP, we introduce Corollary 2.", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 93, + 465, + 484, + 477 + ], + "spans": [ + { + "bbox": [ + 93, + 465, + 484, + 477 + ], + "score": 1.0, + "content": "08 Corollary 2. The mutual information between dataset and the entire MLP can be expressed as", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30.5 + }, + { + "type": "interline_equation", + "bbox": [ + 232, + 479, + 378, + 512 + ], + "lines": [ + { + "bbox": [ + 232, + 479, + 378, + 512 + ], + "spans": [ + { + "bbox": [ + 232, + 479, + 378, + 512 + ], + "score": 0.74, + "content": "\\begin{array} { r l } & { I ( X ; T _ { \\mathrm { M L P } } ) = I ( \\bar { X } ; T _ { 1 } ) + I ( Y ; \\hat { Y } ) } \\\\ & { I ( Y ; T _ { \\mathrm { M L P } } ) = I ( Y ; \\hat { Y } ) } \\end{array}", + "type": "interline_equation", + "image_path": "d5b8ff95f0bad5682c7c714c8b67c980309ad74b0febef5e44e741a02bb75cf3.jpg" + } + ] + } + ], + "index": 32.5, + "virtual_lines": [ + { + "bbox": [ + 232, + 479, + 378, + 495.5 + ], + "spans": [], + "index": 32 + }, + { + "bbox": [ + 232, + 495.5, + 378, + 512.0 + ], + "spans": [], + "index": 33 + } + ] + }, + { + "type": "text", + "bbox": [ + 95, + 514, + 470, + 526 + ], + "lines": [ + { + "bbox": [ + 91, + 513, + 473, + 527 + ], + "spans": [ + { + "bbox": [ + 91, + 513, + 133, + 527 + ], + "score": 1.0, + "content": "09 where", + "type": "text" + }, + { + "bbox": [ + 133, + 515, + 155, + 526 + ], + "score": 0.87, + "content": "T _ { \\mathrm { M L P } }", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 513, + 473, + 527 + ], + "score": 1.0, + "content": "denotes a random variable corresponding to the entire architecture of the MLP.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 91, + 529, + 507, + 554 + ], + "lines": [ + { + "bbox": [ + 87, + 529, + 507, + 543 + ], + "spans": [ + { + "bbox": [ + 87, + 529, + 163, + 543 + ], + "score": 1.0, + "content": "Proof: Since 210", + "type": "text" + }, + { + "bbox": [ + 163, + 530, + 240, + 543 + ], + "score": 0.93, + "content": "H ( Y ) = I ( X ; Y )", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 529, + 304, + 543 + ], + "score": 1.0, + "content": "(Appendix D),", + "type": "text" + }, + { + "bbox": [ + 304, + 529, + 503, + 543 + ], + "score": 0.9, + "content": "H ( X ) = H ( { \\bar { X } } ) + I ( X ; Y ) = H ( { \\bar { X } } ) + H ( Y )", + "type": "inline_equation" + }, + { + "bbox": [ + 504, + 529, + 507, + 543 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 88, + 541, + 497, + 554 + ], + "spans": [ + { + "bbox": [ + 88, + 541, + 485, + 554 + ], + "score": 1.0, + "content": "211 Hence, Corollary 2 can be derived by Corollary 1 and the chain rule. 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Based on the definition of mutual information, we have", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 42 + }, + { + "type": "interline_equation", + "bbox": [ + 240, + 673, + 370, + 686 + ], + "lines": [ + { + "bbox": [ + 240, + 673, + 370, + 686 + ], + "spans": [ + { + "bbox": [ + 240, + 673, + 370, + 686 + ], + "score": 0.91, + "content": "I ( X ; T _ { i } ) = H ( T _ { i } ) - H ( T _ { i } | X ) .", + "type": "interline_equation", + "image_path": "970136fd3bee2b75a4bac9104899d20bdee6043ba65170ace25fc78acd861873.jpg" + } + ] + } + ], + "index": 43, + "virtual_lines": [ + { + "bbox": [ + 240, + 673, + 370, + 686 + ], + "spans": [], + "index": 43 + } + ] + }, + { + "type": "text", + "bbox": [ + 87, + 688, + 504, + 723 + ], + "lines": [ + { + "bbox": [ + 86, + 688, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 86, + 690, + 100, + 700 + ], + "score": 1.0, + "content": "218", + "type": "text" + }, + { + "bbox": [ + 104, + 688, + 235, + 702 + ], + "score": 1.0, + "content": "Previous works simply estimate", + "type": "text" + }, + { + "bbox": [ + 236, + 689, + 311, + 701 + ], + "score": 0.91, + "content": "I ( X ; T _ { i } ) = H ( T _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 688, + 349, + 702 + ], + "score": 1.0, + "content": ", because", + "type": "text" + }, + { + "bbox": [ + 349, + 689, + 359, + 700 + ], + "score": 0.86, + "content": "T _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 688, + 506, + 702 + ], + "score": 1.0, + "content": "is assumed to be entirely dependent", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 86, + 699, + 504, + 713 + ], + "spans": [ + { + "bbox": [ + 86, + 702, + 100, + 711 + ], + "score": 1.0, + "content": "219", + "type": "text" + }, + { + "bbox": [ + 104, + 699, + 118, + 713 + ], + "score": 1.0, + "content": "on", + "type": "text" + }, + { + "bbox": [ + 119, + 700, + 129, + 710 + ], + "score": 0.82, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 129, + 699, + 300, + 713 + ], + "score": 1.0, + "content": "in the Markov chain, Equation (2), thereby", + "type": "text" + }, + { + "bbox": [ + 301, + 700, + 358, + 712 + ], + "score": 0.92, + "content": "H ( T _ { i } | X ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 699, + 493, + 713 + ], + "score": 1.0, + "content": ". 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[], + "index": 12 + }, + { + "bbox": [ + 121, + 244.66666666666666, + 472, + 258.3333333333333 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 121, + 258.3333333333333, + 472, + 272.0 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 86, + 275, + 486, + 288 + ], + "lines": [ + { + "bbox": [ + 85, + 275, + 486, + 289 + ], + "spans": [ + { + "bbox": [ + 85, + 275, + 246, + 289 + ], + "score": 1.0, + "content": "To derive the marginal distribution 223", + "type": "text" + }, + { + "bbox": [ + 247, + 276, + 272, + 288 + ], + "score": 0.93, + "content": "P ( T _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 275, + 393, + 289 + ], + "score": 1.0, + "content": ", we sum the joint distribution", + "type": "text" + }, + { + "bbox": [ + 393, + 276, + 432, + 288 + ], + "score": 0.93, + "content": "P ( T _ { i } , X )", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 275, + 453, + 289 + ], + "score": 1.0, + "content": "over", + "type": "text" + }, + { + "bbox": [ + 454, + 277, + 482, + 286 + ], + "score": 0.89, + "content": "\\mathbf { \\boldsymbol { x } } \\in \\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 275, + 486, + 289 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 292, + 511, + 308 + ], + "lines": [ + { + "bbox": [ + 111, + 292, + 511, + 308 + ], + "spans": [ + { + "bbox": [ + 111, + 292, + 511, + 308 + ], + "score": 0.88, + "content": "\\begin{array} { r } { ^ { \\circ } ( T _ { i } = n ) = \\sum _ { \\mathbf { x } \\in \\mathcal { X } } P _ { X } ( \\mathbf { x } ) P _ { T _ { i } | X } ( n | \\mathbf { x } ) \\approx \\sum _ { \\mathbf { x } ^ { \\prime } \\in \\mathcal { D } } P _ { X } ( \\mathbf { x } ^ { j } ) P _ { T _ { i } | X } ( n | \\mathbf { x } ^ { j } ) = \\frac { 1 } { J } \\sum _ { \\mathbf { x } ^ { j } \\in \\mathcal { D } } P _ { T _ { i } | X } ( n | \\mathbf { x } ^ { j } ) , } \\end{array}", + "type": "interline_equation", + "image_path": "338336e0e6b08caad18fa6446b1c2ca4822cf1d36932b61a865fda6bba463246.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 111, + 292, + 511, + 308 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 84, + 324, + 504, + 347 + ], + "lines": [ + { + "bbox": [ + 84, + 324, + 505, + 338 + ], + "spans": [ + { + "bbox": [ + 84, + 324, + 132, + 338 + ], + "score": 1.0, + "content": "where 224", + "type": "text" + }, + { + "bbox": [ + 132, + 324, + 165, + 336 + ], + "score": 0.94, + "content": "P _ { X } ( \\pmb { x } ^ { j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 166, + 324, + 326, + 338 + ], + "score": 1.0, + "content": "is estimated by the empirical distribution", + "type": "text" + }, + { + "bbox": [ + 327, + 325, + 344, + 337 + ], + "score": 0.9, + "content": "1 / J", + "type": "inline_equation" + }, + { + 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Finally, we can derive", + "type": "text" + }, + { + "bbox": [ + 468, + 325, + 505, + 337 + ], + "score": 0.93, + "content": "I ( X ; T _ { i } )", + "type": "inline_equation" + } + ], + "index": 17 + }, + { + "bbox": [ + 84, + 335, + 483, + 348 + ], + "spans": [ + { + "bbox": [ + 84, + 335, + 483, + 348 + ], + "score": 1.0, + "content": "225 by Equation 16, 17, and 18. Similarly, based on the definition of mutual information, we have", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17.5 + }, + { + "type": "interline_equation", + "bbox": [ + 241, + 352, + 370, + 365 + ], + "lines": [ + { + "bbox": [ + 241, + 352, + 370, + 365 + ], + "spans": [ + { + "bbox": [ + 241, + 352, + 370, + 365 + ], + "score": 0.93, + "content": "I ( Y ; T _ { i } ) = H ( T _ { i } ) - H ( T _ { i } | Y ) .", + "type": "interline_equation", + "image_path": "08591322dff3a2a6bdf86ca7a5aa904e6540cd4ff726c9491295f0ad798bfdeb.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 241, + 352, + 370, + 365 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 86, + 369, + 308, + 382 + ], + "lines": [ + { + "bbox": [ + 85, + 369, + 308, + 383 + ], + "spans": [ + { + "bbox": [ + 85, + 369, + 155, + 383 + ], + "score": 1.0, + "content": "To estimate 226", + "type": "text" + }, + { + "bbox": [ + 155, + 370, + 192, + 382 + ], + "score": 0.93, + "content": "H ( T _ { i } | Y )", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 369, + 259, + 383 + ], + "score": 1.0, + "content": ", we reformulate", + "type": "text" + }, + { + "bbox": [ + 260, + 369, + 296, + 382 + ], + "score": 0.93, + "content": "P ( T _ { i } | Y )", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 369, + 308, + 383 + ], + "score": 1.0, + "content": "as", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "interline_equation", + "bbox": [ + 145, + 384, + 465, + 402 + ], + "lines": [ + { + "bbox": [ + 145, + 384, + 465, + 402 + ], + "spans": [ + { + "bbox": [ + 145, + 384, + 465, + 402 + ], + "score": 0.91, + "content": "\\begin{array} { r } { P _ { T _ { i } | Y } ( n | l ) = \\sum _ { \\pmb { x } \\in \\mathcal { X } } P _ { T _ { i } | X } ( n | \\pmb { x } ) P _ { X | Y } ( \\pmb { x } | l ) \\approx \\frac { 1 } { N ( l ) } \\sum _ { \\pmb { x } ^ { j } \\in \\mathcal { D } , y ^ { j } = l } P _ { T _ { i } | X } ( n | \\pmb { x } ^ { j } ) , } \\end{array}", + "type": "interline_equation", + "image_path": "be6ebf67fab350423a7426074d6cb414acb27a634668fffcee744eeafe6a6947.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 145, + 384, + 465, + 402 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 95, + 406, + 505, + 431 + ], + "lines": [ + { + "bbox": [ + 92, + 406, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 92, + 406, + 132, + 420 + ], + "score": 1.0, + "content": "where 27", + "type": "text" + }, + { + "bbox": [ + 133, + 406, + 181, + 420 + ], + "score": 0.94, + "content": "P _ { X | Y } ( \\pmb { x } ^ { j } | l )", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 406, + 344, + 420 + ], + "score": 1.0, + "content": "is estimated by the empirical distribution", + "type": "text" + }, + { + "bbox": [ + 345, + 407, + 375, + 419 + ], + "score": 0.92, + "content": "1 / N ( l )", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 406, + 393, + 420 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 393, + 407, + 414, + 419 + ], + "score": 0.91, + "content": "N ( l )", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 406, + 506, + 420 + ], + "score": 1.0, + "content": "denotes the number of", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 93, + 419, + 466, + 432 + ], + "spans": [ + { + "bbox": [ + 93, + 419, + 198, + 432 + ], + "score": 1.0, + "content": "28 samples with the label", + "type": "text" + }, + { + "bbox": [ + 198, + 420, + 202, + 429 + ], + "score": 0.76, + "content": "l", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 419, + 213, + 432 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 214, + 420, + 223, + 429 + ], + "score": 0.8, + "content": "\\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 419, + 317, + 432 + ], + "score": 1.0, + "content": ". Finally, we can derive", + "type": "text" + }, + { + "bbox": [ + 317, + 419, + 353, + 431 + ], + "score": 0.93, + "content": "I ( Y ; T _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 419, + 466, + 432 + ], + "score": 1.0, + "content": "by Equation 18, 19, and 20.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5 + }, + { + "type": "text", + "bbox": [ + 105, + 435, + 505, + 545 + ], + "lines": [ + { + "bbox": [ + 105, + 434, + 505, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 333, + 448 + ], + "score": 1.0, + "content": "Synthetic dataset. The dataset consists of 512 gray-scale", + "type": "text" + }, + { + "bbox": [ + 333, + 436, + 364, + 446 + ], + "score": 0.89, + "content": "3 2 \\times 3 2", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 434, + 505, + 448 + ], + "score": 1.0, + "content": "images, which are evenly generated", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 446, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 106, + 446, + 244, + 459 + ], + "score": 1.0, + "content": "by rotating a deterministic image", + "type": "text" + }, + { + "bbox": [ + 244, + 447, + 252, + 456 + ], + "score": 0.81, + "content": "\\hat { \\pmb x }", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 446, + 505, + 459 + ], + "score": 1.0, + "content": "in four different orientations and adding Gaussian noise with", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 456, + 506, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 156, + 470 + ], + "score": 1.0, + "content": "expectation", + "type": "text" + }, + { + "bbox": [ + 157, + 457, + 200, + 469 + ], + "score": 0.93, + "content": "\\boldsymbol { \\mu } = \\mathbb { E } ( \\hat { \\boldsymbol { x } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 456, + 257, + 470 + ], + "score": 1.0, + "content": "and variance", + "type": "text" + }, + { + "bbox": [ + 257, + 457, + 289, + 468 + ], + "score": 0.9, + "content": "\\sigma ^ { 2 } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 456, + 327, + 470 + ], + "score": 1.0, + "content": ", namely", + "type": "text" + }, + { + "bbox": [ + 327, + 457, + 420, + 469 + ], + "score": 0.91, + "content": "\\pmb { x } = r ( \\hat { \\pmb { x } } ) + \\mathcal { N } ( \\mu , \\sigma ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 456, + 453, + 470 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 453, + 457, + 469, + 469 + ], + "score": 0.91, + "content": "r ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 456, + 506, + 470 + ], + "score": 1.0, + "content": "denotes", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 468, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 106, + 468, + 505, + 480 + ], + "score": 1.0, + "content": "the rotation method shown in Figure 3. The reason for adding Gaussian noise is to avoid DNNs", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 479, + 505, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 505, + 492 + ], + "score": 1.0, + "content": "directly memorizing the deterministic image. In addition, the binary labels [1,0] and [0,1] evenly", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 489, + 506, + 503 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 506, + 503 + ], + "score": 1.0, + "content": "divide the synthetic dataset into two classes. As a result, the synthetic dataset has (approximately)", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 500, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 505, + 513 + ], + "score": 1.0, + "content": "2 bits information and the labels have 1 bit information. 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Table 1 summarizes the architecture of the three MLPs.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35 + }, + { + "type": "title", + "bbox": [ + 106, + 595, + 429, + 608 + ], + "lines": [ + { + "bbox": [ + 105, + 596, + 430, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 430, + 609 + ], + "score": 1.0, + "content": "4.2 Validating the probability space and the mutual information estimator", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 106, + 615, + 506, + 694 + ], + "lines": [ + { + "bbox": [ + 104, + 614, + 507, + 629 + ], + "spans": [ + { + "bbox": [ + 104, + 614, + 247, + 629 + ], + "score": 1.0, + "content": "We demonstrate the sample space", + "type": "text" + }, + { + "bbox": [ + 248, + 616, + 262, + 627 + ], + "score": 0.88, + "content": "\\Omega _ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 614, + 474, + 629 + ], + "score": 1.0, + "content": "by visualizing the weights4 of the eight neurons in", + "type": "text" + }, + { + "bbox": [ + 474, + 617, + 484, + 627 + ], + "score": 0.83, + "content": "\\mathbf { t } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 614, + 507, + 629 + ], + "score": 1.0, + "content": ", i.e.,", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 107, + 622, + 510, + 649 + ], + "spans": [ + { + "bbox": [ + 107, + 627, + 189, + 642 + ], + "score": 0.93, + "content": "\\omega _ { n } ^ { ( 1 ) } = \\{ \\omega _ { m n } ^ { ( 1 ) } \\} _ { m = 1 } ^ { 1 0 2 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 622, + 510, + 649 + ], + "score": 1.0, + "content": ", in 5 different epochs (i.e., 0,1,4,128,1000) in Figure 4 (Left). 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472, + 244.66666666666666 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 121, + 244.66666666666666, + 472, + 258.3333333333333 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 121, + 258.3333333333333, + 472, + 272.0 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 86, + 275, + 486, + 288 + ], + "lines": [ + { + "bbox": [ + 85, + 275, + 486, + 289 + ], + "spans": [ + { + "bbox": [ + 85, + 275, + 246, + 289 + ], + "score": 1.0, + "content": "To derive the marginal distribution 223", + "type": "text" + }, + { + "bbox": [ + 247, + 276, + 272, + 288 + ], + "score": 0.93, + "content": "P ( T _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 275, + 393, + 289 + ], + "score": 1.0, + "content": ", we sum the joint distribution", + "type": "text" + }, + { + "bbox": [ + 393, + 276, + 432, + 288 + ], + "score": 0.93, + "content": "P ( T _ { i } , X )", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 275, + 453, + 289 + ], + "score": 1.0, + "content": "over", + "type": "text" + }, + { + "bbox": [ + 454, + 277, + 482, + 286 + ], + "score": 0.89, + "content": "\\mathbf { \\boldsymbol { x } } \\in \\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 275, + 486, + 289 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15, + "bbox_fs": [ + 85, + 275, + 486, + 289 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 292, + 511, + 308 + ], + "lines": [ + { + "bbox": [ + 111, + 292, + 511, + 308 + ], + "spans": [ + { + "bbox": [ + 111, + 292, + 511, + 308 + ], + "score": 0.88, + "content": "\\begin{array} { r } { ^ { \\circ } ( T _ { i } = n ) = \\sum _ { \\mathbf { x } \\in \\mathcal { X } } P _ { X } ( \\mathbf { x } ) P _ { T _ { i } | X } ( n | \\mathbf { x } ) \\approx \\sum _ { \\mathbf { x } ^ { \\prime } \\in \\mathcal { D } } P _ { X } ( \\mathbf { x } ^ { j } ) P _ { T _ { i } | X } ( n | \\mathbf { x } ^ { j } ) = \\frac { 1 } { J } \\sum _ { \\mathbf { x } ^ { j } \\in \\mathcal { D } } P _ { T _ { i } | X } ( n | \\mathbf { x } ^ { j } ) , } \\end{array}", + "type": "interline_equation", + "image_path": "338336e0e6b08caad18fa6446b1c2ca4822cf1d36932b61a865fda6bba463246.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 111, + 292, + 511, + 308 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 84, + 324, + 504, + 347 + ], + "lines": [ + { + "bbox": [ + 84, + 324, + 505, + 338 + ], + "spans": [ + { + "bbox": [ + 84, + 324, + 132, + 338 + ], + "score": 1.0, + "content": "where 224", + "type": "text" + }, + { + "bbox": [ + 132, + 324, + 165, + 336 + ], + "score": 0.94, + "content": "P _ { X } ( \\pmb { x } ^ { j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 166, + 324, + 326, + 338 + ], + "score": 1.0, + "content": "is estimated by the empirical distribution", + "type": "text" + }, + { + "bbox": [ + 327, + 325, + 344, + 337 + 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Finally, we can derive", + "type": "text" + }, + { + "bbox": [ + 468, + 325, + 505, + 337 + ], + "score": 0.93, + "content": "I ( X ; T _ { i } )", + "type": "inline_equation" + } + ], + "index": 17 + }, + { + "bbox": [ + 84, + 335, + 483, + 348 + ], + "spans": [ + { + "bbox": [ + 84, + 335, + 483, + 348 + ], + "score": 1.0, + "content": "225 by Equation 16, 17, and 18. Similarly, based on the definition of mutual information, we have", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17.5, + "bbox_fs": [ + 84, + 324, + 505, + 348 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 241, + 352, + 370, + 365 + ], + "lines": [ + { + "bbox": [ + 241, + 352, + 370, + 365 + ], + "spans": [ + { + "bbox": [ + 241, + 352, + 370, + 365 + ], + "score": 0.93, + "content": "I ( Y ; T _ { i } ) = H ( T _ { i } ) - H ( T _ { i } | Y ) .", + "type": "interline_equation", + "image_path": "08591322dff3a2a6bdf86ca7a5aa904e6540cd4ff726c9491295f0ad798bfdeb.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 241, + 352, + 370, + 365 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 86, + 369, + 308, + 382 + ], + "lines": [ + { + "bbox": [ + 85, + 369, + 308, + 383 + ], + "spans": [ + { + "bbox": [ + 85, + 369, + 155, + 383 + ], + "score": 1.0, + "content": "To estimate 226", + "type": "text" + }, + { + "bbox": [ + 155, + 370, + 192, + 382 + ], + "score": 0.93, + "content": "H ( T _ { i } | Y )", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 369, + 259, + 383 + ], + "score": 1.0, + "content": ", we reformulate", + "type": "text" + }, + { + "bbox": [ + 260, + 369, + 296, + 382 + ], + "score": 0.93, + "content": "P ( T _ { i } | Y )", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 369, + 308, + 383 + ], + "score": 1.0, + "content": "as", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20, + "bbox_fs": [ + 85, + 369, + 308, + 383 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 145, + 384, + 465, + 402 + ], + "lines": [ + { + "bbox": [ + 145, + 384, + 465, + 402 + ], + "spans": [ + { + "bbox": [ + 145, + 384, + 465, + 402 + ], + "score": 0.91, + "content": "\\begin{array} { r } { P _ { T _ { i } | Y } ( n | l ) = \\sum _ { \\pmb { x } \\in \\mathcal { X } } P _ { T _ { i } | X } ( n | \\pmb { x } ) P _ { X | Y } ( \\pmb { x } | l ) \\approx \\frac { 1 } { N ( l ) } \\sum _ { \\pmb { x } ^ { j } \\in \\mathcal { D } , y ^ { j } = l } P _ { T _ { i } | X } ( n | \\pmb { x } ^ { j } ) , } \\end{array}", + "type": "interline_equation", + "image_path": "be6ebf67fab350423a7426074d6cb414acb27a634668fffcee744eeafe6a6947.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 145, + 384, + 465, + 402 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 95, + 406, + 505, + 431 + ], + "lines": [ + { + "bbox": [ + 92, + 406, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 92, + 406, + 132, + 420 + ], + "score": 1.0, + "content": "where 27", + "type": "text" + }, + { + "bbox": [ + 133, + 406, + 181, + 420 + ], + "score": 0.94, + "content": "P _ { X | Y } ( \\pmb { x } ^ { j } | l )", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 406, + 344, + 420 + ], + "score": 1.0, + "content": "is estimated by the empirical distribution", + "type": "text" + }, + { + "bbox": [ + 345, + 407, + 375, + 419 + ], + "score": 0.92, + "content": "1 / N ( l )", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 406, + 393, + 420 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 393, + 407, + 414, + 419 + ], + "score": 0.91, + "content": "N ( l )", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 406, + 506, + 420 + ], + "score": 1.0, + "content": "denotes the number of", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 93, + 419, + 466, + 432 + ], + "spans": [ + { + "bbox": [ + 93, + 419, + 198, + 432 + ], + "score": 1.0, + "content": "28 samples with the label", + "type": "text" + }, + { + "bbox": [ + 198, + 420, + 202, + 429 + ], + "score": 0.76, + "content": "l", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 419, + 213, + 432 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 214, + 420, + 223, + 429 + ], + "score": 0.8, + "content": "\\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 419, + 317, + 432 + ], + "score": 1.0, + "content": ". Finally, we can derive", + "type": "text" + }, + { + "bbox": [ + 317, + 419, + 353, + 431 + ], + "score": 0.93, + "content": "I ( Y ; T _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 419, + 466, + 432 + ], + "score": 1.0, + "content": "by Equation 18, 19, and 20.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5, + "bbox_fs": [ + 92, + 406, + 506, + 432 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 435, + 505, + 545 + ], + "lines": [ + { + "bbox": [ + 105, + 434, + 505, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 333, + 448 + ], + "score": 1.0, + "content": "Synthetic dataset. 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The reason for adding Gaussian noise is to avoid DNNs", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 479, + 505, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 505, + 492 + ], + "score": 1.0, + "content": "directly memorizing the deterministic image. In addition, the binary labels [1,0] and [0,1] evenly", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 489, + 506, + 503 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 506, + 503 + ], + "score": 1.0, + "content": "divide the synthetic dataset into two classes. As a result, the synthetic dataset has (approximately)", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 500, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 505, + 513 + ], + "score": 1.0, + "content": "2 bits information and the labels have 1 bit information. 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We train three MLPs, namely MLP1, MLP2 and MLP3, on the synthetic dataset", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 561, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 505, + 573 + ], + "score": 1.0, + "content": "by a variant of Stochastic Gradient Descent (SGD) method, namely Adam [13], over 1000 epochs", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 572, + 460, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 193, + 584 + ], + "score": 1.0, + "content": "with the learning rate", + "type": "text" + }, + { + "bbox": [ + 194, + 572, + 232, + 582 + ], + "score": 0.87, + "content": "\\alpha = 0 . 0 3", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 572, + 460, + 584 + ], + "score": 1.0, + "content": ". Table 1 summarizes the architecture of the three MLPs.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35, + "bbox_fs": [ + 105, + 549, + 505, + 584 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 595, + 429, + 608 + ], + "lines": [ + { + "bbox": [ + 105, + 596, + 430, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 430, + 609 + ], + "score": 1.0, + "content": "4.2 Validating the probability space and the mutual information estimator", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 106, + 615, + 506, + 694 + ], + "lines": [ + { + "bbox": [ + 104, + 614, + 507, + 629 + ], + "spans": [ + { + "bbox": [ + 104, + 614, + 247, + 629 + ], + "score": 1.0, + "content": "We demonstrate the sample space", + "type": "text" + }, + { + "bbox": [ + 248, + 616, + 262, + 627 + ], + "score": 0.88, + "content": "\\Omega _ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 614, + 474, + 629 + ], + "score": 1.0, + "content": "by visualizing the weights4 of the eight neurons in", + "type": "text" + }, + { + "bbox": [ + 474, + 617, + 484, + 627 + ], + "score": 0.83, + "content": "\\mathbf { t } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 614, + 507, + 629 + ], + "score": 1.0, + "content": ", i.e.,", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 107, + 622, + 510, + 649 + ], + "spans": [ + { + "bbox": [ + 107, + 627, + 189, + 642 + ], + "score": 0.93, + "content": "\\omega _ { n } ^ { ( 1 ) } = \\{ \\omega _ { m n } ^ { ( 1 ) } \\} _ { m = 1 } ^ { 1 0 2 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 622, + 510, + 649 + ], + "score": 1.0, + "content": ", in 5 different epochs (i.e., 0,1,4,128,1000) in Figure 4 (Left). 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For insta", + "type": "text" + }, + { + "bbox": [ + 375, + 294, + 420, + 309 + ], + "score": 0.93, + "content": "\\{ \\omega _ { n } ^ { ( 1 ) } \\} _ { n = 1 } ^ { 8 }", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 309, + 410, + 322 + ], + "score": 0.9, + "content": "{ \\boldsymbol \\omega } _ { 2 } ^ { ( 1 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 291, + 506, + 328 + ], + "score": 1.0, + "content": "being recognized theorrectly characterizes", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 85, + 321, + 507, + 340 + ], + "spans": [ + { + "bbox": [ + 85, + 325, + 99, + 336 + ], + "score": 1.0, + "content": "251", + "type": "text" + }, + { + "bbox": [ + 103, + 321, + 341, + 340 + ], + "score": 1.0, + "content": "the feature of Image0 and has the largest cross-correlation", + "type": "text" + }, + { + "bbox": [ + 341, + 322, + 415, + 337 + ], + "score": 0.93, + "content": "\\langle \\omega _ { 2 } ^ { ( 1 ) } , \\pmb { x } \\rangle = 1 9 0 . 8", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 321, + 507, + 340 + ], + "score": 1.0, + "content": ", thus it has the largest", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 86, + 331, + 510, + 359 + ], + "spans": [ + { + "bbox": [ + 86, + 340, + 100, + 351 + ], + "score": 1.0, + "content": "252", + "type": "text" + }, + { + "bbox": [ + 100, + 331, + 152, + 359 + ], + "score": 1.0, + "content": "probability", + "type": "text" + }, + { + "bbox": [ + 152, + 336, + 263, + 353 + ], + "score": 0.94, + "content": "P _ { T _ { 1 } | X } ^ { \\mathrm { R e L U } } ( \\omega _ { 2 } ^ { ( 1 ) } | \\mathrm { I m a g e 0 } ) = 1 . 0", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 331, + 510, + 359 + ], + "score": 1.0, + "content": "being recognized as the feature with largest cross-correlation", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 85, + 350, + 508, + 370 + ], + "spans": [ + { + "bbox": [ + 85, + 357, + 100, + 367 + ], + "score": 1.0, + "content": "253", + "type": "text" + }, + { + "bbox": [ + 102, + 350, + 222, + 370 + ], + "score": 1.0, + "content": "to Image0. 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For insta", + "type": "text" + }, + { + "bbox": [ + 375, + 294, + 420, + 309 + ], + "score": 0.93, + "content": "\\{ \\omega _ { n } ^ { ( 1 ) } \\} _ { n = 1 } ^ { 8 }", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 309, + 410, + 322 + ], + "score": 0.9, + "content": "{ \\boldsymbol \\omega } _ { 2 } ^ { ( 1 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 291, + 506, + 328 + ], + "score": 1.0, + "content": "being recognized theorrectly characterizes", + "type": "text" + } + ], + "index": 11, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 321, + 507, + 340 + ], + "spans": [ + { + "bbox": [ + 85, + 325, + 99, + 336 + ], + "score": 1.0, + "content": "251", + "type": "text" + }, + { + "bbox": [ + 103, + 321, + 341, + 340 + ], + "score": 1.0, + "content": "the feature of Image0 and has the largest cross-correlation", + "type": "text" + }, + { + "bbox": [ + 341, + 322, + 415, + 337 + ], + "score": 0.93, + "content": "\\langle \\omega _ { 2 } ^ { ( 1 ) } , \\pmb { x } \\rangle = 1 9 0 . 8", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 321, + 507, + 340 + ], + "score": 1.0, + "content": ", thus it has the largest", + "type": "text" + } + ], + "index": 12, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 331, + 510, + 359 + ], + "spans": [ + { + "bbox": [ + 86, + 340, + 100, + 351 + ], + "score": 1.0, + "content": "252", + "type": "text" + }, + { + "bbox": [ + 100, + 331, + 152, + 359 + ], + "score": 1.0, + "content": "probability", + "type": "text" + }, + { + "bbox": [ + 152, + 336, + 263, + 353 + ], + "score": 0.94, + "content": "P _ { T _ { 1 } | X } ^ { \\mathrm { R e L U } } ( \\omega _ { 2 } ^ { ( 1 ) } | \\mathrm { I m a g e 0 } ) = 1 . 0", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 331, + 510, + 359 + ], + "score": 1.0, + "content": "being recognized as the feature with largest cross-correlation", + "type": "text" + } + ], + "index": 13, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 350, + 508, + 370 + ], + "spans": [ + { + "bbox": [ + 85, + 357, + 100, + 367 + ], + "score": 1.0, + "content": "253", + "type": "text" + }, + { + "bbox": [ + 102, + 350, + 222, + 370 + ], + "score": 1.0, + "content": "to Image0. In contrast, since", + "type": "text" + }, + { + "bbox": [ + 222, + 352, + 241, + 367 + ], + "score": 0.88, + "content": "{ \\omega } _ { 7 } ^ { ( 1 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 350, + 508, + 370 + ], + "score": 1.0, + "content": "incorrectly characterizes the feature of Image0 and has the lowest", + "type": "text" + } + ], + "index": 14, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 362, + 505, + 390 + ], + "spans": [ + { + "bbox": [ + 86, + 372, + 99, + 380 + ], + "score": 1.0, + "content": "254", + "type": "text" + }, + { + "bbox": [ + 100, + 362, + 178, + 390 + ], + "score": 1.0, + "content": "cross-correlation", + "type": "text" + }, + { + "bbox": [ + 178, + 367, + 261, + 381 + ], + "score": 0.93, + "content": "\\langle \\omega _ { 7 } ^ { ( 1 ) } , \\pmb { x } \\rangle = - 2 1 0 . 0", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 362, + 392, + 390 + ], + "score": 1.0, + "content": ", so it has the lowest probability", + "type": "text" + }, + { + "bbox": [ + 392, + 366, + 505, + 384 + ], + "score": 0.93, + "content": "P _ { T _ { 1 } | X } ^ { \\mathrm { R e L U } } ( \\omega _ { 7 } ^ { ( 1 ) } | \\mathrm { I m a g e 0 } ) = 0 . 0", + "type": "inline_equation" + } + ], + "index": 15, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 381, + 397, + 394 + ], + "spans": [ + { + "bbox": [ + 86, + 384, + 99, + 393 + ], + "score": 1.0, + "content": "255", + "type": "text" + }, + { + "bbox": [ + 105, + 381, + 397, + 394 + ], + "score": 1.0, + "content": "being recognized as the feature with largest cross-correlation to Image0.", + "type": "text" + } + ], + "index": 16, + "is_list_start_line": true + } + ], + "index": 13.5, + "bbox_fs": [ + 85, + 291, + 510, + 394 + ] + }, + { + "type": "text", + "bbox": [ + 96, + 398, + 506, + 515 + ], + "lines": [ + { + "bbox": [ + 94, + 397, + 506, + 411 + ], + "spans": [ + { + "bbox": [ + 94, + 397, + 506, + 411 + ], + "score": 1.0, + "content": "6 We observe that an activation function (abbr. ACT) plays an important role in the distribution.", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 104, + 409, + 505, + 422 + ], + "spans": [ + { + "bbox": [ + 104, + 409, + 505, + 422 + ], + "score": 1.0, + "content": "Specifically, ReLU, a non-saturating (unbounded) ACT [9], preserves the positive cross-correlations", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 99, + 416, + 506, + 443 + ], + "spans": [ + { + "bbox": [ + 99, + 416, + 274, + 443 + ], + "score": 1.0, + "content": "while resets all the negative ones as zero.", + "type": "text" + }, + { + "bbox": [ + 275, + 420, + 386, + 438 + ], + "score": 0.93, + "content": "P _ { T _ { 1 } | X } ^ { \\mathrm { R e L U } } ( \\omega _ { 2 } ^ { ( 1 ) } | \\mathrm { I m a g e 0 } ) = 1 . 0", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 416, + 506, + 443 + ], + "score": 1.0, + "content": "shows that ReLU derives the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 101, + 436, + 507, + 456 + ], + "spans": [ + { + "bbox": [ + 101, + 436, + 193, + 456 + ], + "score": 1.0, + "content": "correct probability of", + "type": "text" + }, + { + "bbox": [ + 193, + 436, + 212, + 452 + ], + "score": 0.93, + "content": "{ \\boldsymbol \\omega } _ { 2 } ^ { ( 1 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 436, + 507, + 456 + ], + "score": 1.0, + "content": "being recognized as the feature with largest cross-correlation. In contrast,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 100, + 445, + 510, + 472 + ], + "spans": [ + { + "bbox": [ + 100, + 445, + 137, + 472 + ], + "score": 1.0, + "content": "though", + "type": "text" + }, + { + "bbox": [ + 137, + 451, + 156, + 466 + ], + "score": 0.92, + "content": "{ \\boldsymbol \\omega } _ { 2 } ^ { ( 1 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 445, + 340, + 472 + ], + "score": 1.0, + "content": "has stronger cross-correlation to Image0 than", + "type": "text" + }, + { + "bbox": [ + 341, + 451, + 360, + 466 + ], + "score": 0.91, + "content": "{ \\omega } _ { 4 } ^ { ( 1 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 445, + 381, + 472 + ], + "score": 1.0, + "content": ", i.e.,", + "type": "text" + }, + { + "bbox": [ + 382, + 450, + 469, + 466 + ], + "score": 0.93, + "content": "\\langle \\omega _ { 2 } ^ { ( 1 ) } , \\pmb { x } \\rangle > \\langle \\pmb { \\omega } _ { 4 } ^ { ( 1 ) } , \\pmb { x } \\rangle", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 445, + 510, + 472 + ], + "score": 1.0, + "content": ", Tanh, a", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 100, + 462, + 506, + 485 + ], + "spans": [ + { + "bbox": [ + 100, + 462, + 248, + 485 + ], + "score": 1.0, + "content": "saturating (bounded) ACT, derives", + "type": "text" + }, + { + "bbox": [ + 248, + 466, + 358, + 480 + ], + "score": 0.93, + "content": "f _ { 1 2 } ^ { \\mathrm { T a n h } } ( { \\pmb x } ) = f _ { 1 4 } ^ { \\mathrm { T a n h } } ( { \\pmb x } ) = 1 . 0", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 462, + 406, + 485 + ], + "score": 1.0, + "content": ", and makes", + "type": "text" + }, + { + "bbox": [ + 407, + 465, + 426, + 480 + ], + "score": 0.92, + "content": "{ \\omega } _ { 4 } ^ { ( 1 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 462, + 506, + 485 + ], + "score": 1.0, + "content": "to incorrectly have", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 103, + 477, + 507, + 496 + ], + "spans": [ + { + "bbox": [ + 103, + 477, + 225, + 496 + ], + "score": 1.0, + "content": "the same probability 0.272 to", + "type": "text" + }, + { + "bbox": [ + 226, + 479, + 245, + 494 + ], + "score": 0.92, + "content": "{ \\boldsymbol \\omega } _ { 2 } ^ { ( 1 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 477, + 507, + 496 + ], + "score": 1.0, + "content": "being recognized as the feature with the largest cross-correlation", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 94, + 492, + 506, + 505 + ], + "spans": [ + { + "bbox": [ + 94, + 497, + 99, + 502 + ], + "score": 1.0, + "content": "3", + "type": "text" + }, + { + "bbox": [ + 104, + 492, + 221, + 505 + ], + "score": 1.0, + "content": "to Image0, i.e., Tanh hinders", + "type": "text" + }, + { + "bbox": [ + 221, + 494, + 231, + 504 + ], + "score": 0.86, + "content": "\\mathbf { t } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 492, + 506, + 505 + ], + "score": 1.0, + "content": "from correctly recognizing the features of input. The simulations for", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 99, + 503, + 484, + 516 + ], + "spans": [ + { + "bbox": [ + 99, + 503, + 484, + 516 + ], + "score": 1.0, + "content": "validating the probability space based on other synthetic images are presented in Appendix G.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 21, + "bbox_fs": [ + 94, + 397, + 510, + 516 + ] + }, + { + "type": "index", + "bbox": [ + 85, + 519, + 505, + 630 + ], + "lines": [ + { + "bbox": [ + 86, + 520, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 86, + 522, + 100, + 532 + ], + "score": 1.0, + "content": "265", + "type": "text" + }, + { + "bbox": [ + 105, + 520, + 505, + 532 + ], + "score": 1.0, + "content": "To validate the mutual information estimator, we follow recent works [30, 28] to train the three", + "type": "text" + } + ], + "index": 26, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 530, + 505, + 542 + ], + "spans": [ + { + "bbox": [ + 86, + 533, + 99, + 542 + ], + "score": 1.0, + "content": "266", + "type": "text" + }, + { + "bbox": [ + 105, + 530, + 505, + 542 + ], + "score": 1.0, + "content": "MLPs with 50 different random initialization and study the average mutual information. Figure 4", + "type": "text" + } + ], + "index": 27, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 541, + 506, + 554 + ], + "spans": [ + { + "bbox": [ + 86, + 544, + 99, + 552 + ], + "score": 1.0, + "content": "267", + "type": "text" + }, + { + "bbox": [ + 105, + 541, + 183, + 554 + ], + "score": 1.0, + "content": "(Right) shows that", + "type": "text" + }, + { + "bbox": [ + 183, + 542, + 221, + 554 + ], + "score": 0.93, + "content": "I ( X ; T _ { 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 541, + 467, + 554 + ], + "score": 1.0, + "content": "quickly increases to 1.81 and keeps stable in the MLP1, i.e.,", + "type": "text" + }, + { + "bbox": [ + 467, + 542, + 477, + 552 + ], + "score": 0.84, + "content": "\\mathbf { t } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 541, + 506, + 554 + ], + "score": 1.0, + "content": "learns", + "type": "text" + } + ], + "index": 28, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 552, + 505, + 565 + ], + "spans": [ + { + "bbox": [ + 86, + 555, + 99, + 564 + ], + "score": 1.0, + "content": "268", + "type": "text" + }, + { + "bbox": [ + 105, + 553, + 245, + 564 + ], + "score": 1.0, + "content": "most information of the dataset as", + "type": "text" + }, + { + "bbox": [ + 246, + 552, + 299, + 565 + ], + "score": 0.92, + "content": "H ( X ) = 2 . 0", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 553, + 505, + 564 + ], + "score": 1.0, + "content": ". Notably, the result is consistent with the variation", + "type": "text" + } + ], + "index": 29, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 563, + 506, + 576 + ], + "spans": [ + { + "bbox": [ + 86, + 566, + 99, + 574 + ], + "score": 1.0, + "content": "269", + "type": "text" + }, + { + "bbox": [ + 105, + 563, + 506, + 576 + ], + "score": 1.0, + "content": "of the weights in Figure 4 (Left), which shows that the weights correctly characterize the features", + "type": "text" + } + ], + "index": 30, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 574, + 505, + 587 + ], + "spans": [ + { + "bbox": [ + 86, + 576, + 100, + 586 + ], + "score": 1.0, + "content": "270", + "type": "text" + }, + { + "bbox": [ + 105, + 574, + 467, + 587 + ], + "score": 1.0, + "content": "of the dataset and keeps stable after the fourth epoch. As a comparison, we observe that", + "type": "text" + }, + { + "bbox": [ + 468, + 574, + 505, + 586 + ], + "score": 0.92, + "content": "I ( X ; T _ { 1 } )", + "type": "inline_equation" + } + ], + "index": 31, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 585, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 86, + 587, + 99, + 597 + ], + "score": 1.0, + "content": "271", + "type": "text" + }, + { + "bbox": [ + 106, + 585, + 434, + 597 + ], + "score": 1.0, + "content": "keeps stable at 0.44 in the MLP2, which confirms the statement that Tanh hinders", + "type": "text" + }, + { + "bbox": [ + 434, + 586, + 444, + 596 + ], + "score": 0.85, + "content": "\\mathbf { \\delta t } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 585, + 505, + 597 + ], + "score": 1.0, + "content": "from correctly", + "type": "text" + } + ], + "index": 32, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 595, + 506, + 609 + ], + "spans": [ + { + "bbox": [ + 86, + 598, + 100, + 608 + ], + "score": 1.0, + "content": "272", + "type": "text" + }, + { + "bbox": [ + 104, + 595, + 397, + 609 + ], + "score": 1.0, + "content": "recognizing the features of input. In addition, Figure 4 (Right) shows that", + "type": "text" + }, + { + "bbox": [ + 398, + 596, + 466, + 608 + ], + "score": 0.9, + "content": "I ( X ; T _ { 1 } ) \\approx 0 . 7 9", + "type": "inline_equation" + }, + { + "bbox": [ + 467, + 595, + 506, + 609 + ], + "score": 1.0, + "content": "in MLP3", + "type": "text" + } + ], + "index": 33, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 607, + 505, + 619 + ], + "spans": [ + { + "bbox": [ + 86, + 609, + 100, + 619 + ], + "score": 1.0, + "content": "273", + "type": "text" + }, + { + "bbox": [ + 105, + 607, + 169, + 619 + ], + "score": 1.0, + "content": "is smaller than", + "type": "text" + }, + { + "bbox": [ + 169, + 607, + 238, + 619 + ], + "score": 0.92, + "content": "I ( X ; T _ { 1 } ) \\approx \\bar { 1 } . 8 1", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 607, + 505, + 619 + ], + "score": 1.0, + "content": "in MLP1, which is consistent with Definition 1, i.e., a layer with", + "type": "text" + } + ], + "index": 34, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 618, + 460, + 630 + ], + "spans": [ + { + "bbox": [ + 86, + 620, + 99, + 629 + ], + "score": 1.0, + "content": "274", + "type": "text" + }, + { + "bbox": [ + 105, + 618, + 460, + 630 + ], + "score": 1.0, + "content": "fewer neurons would represent fewer possible features, thus it contains less information.", + "type": "text" + } + ], + "index": 35, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 634, + 506, + 647 + ], + "spans": [ + { + "bbox": [ + 86, + 636, + 99, + 646 + ], + "score": 1.0, + "content": "275", + "type": "text" + }, + { + "bbox": [ + 105, + 635, + 317, + 647 + ], + "score": 1.0, + "content": "In summary, we demonstrate the probability space", + "type": "text" + }, + { + "bbox": [ + 317, + 634, + 369, + 646 + ], + "score": 0.93, + "content": "\\left( \\Omega _ { T } , \\mathcal { F } , P _ { T } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 635, + 506, + 647 + ], + "score": 1.0, + "content": "and show that if an ACT cannot", + "type": "text" + } + ], + "index": 36, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 645, + 506, + 658 + ], + "spans": [ + { + "bbox": [ + 85, + 647, + 100, + 658 + ], + "score": 1.0, + "content": "276", + "type": "text" + }, + { + "bbox": [ + 105, + 645, + 506, + 658 + ], + "score": 1.0, + "content": "preserve the cross-correlation between weights(features) and input, it would distort the distribution", + "type": "text" + } + ], + "index": 37, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 656, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 86, + 658, + 99, + 667 + ], + "score": 1.0, + "content": "277", + "type": "text" + }, + { + "bbox": [ + 106, + 656, + 506, + 668 + ], + "score": 1.0, + "content": "of a layer, thereby affecting the mutual information between the layer and data/labels. 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The forth figure shows", + "type": "text" + }, + { + "bbox": [ + 285, + 318, + 329, + 329 + ], + "score": 0.92, + "content": "I ( X ; T _ { \\mathrm { M L P } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 317, + 345, + 330 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 345, + 318, + 387, + 329 + ], + "score": 0.94, + "content": "I ( Y ; T _ { \\mathrm { M L P } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 317, + 506, + 330 + ], + "score": 1.0, + "content": "in a MLP. 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Figure 5 visualizes", + "type": "text" + }, + { + "bbox": [ + 199, + 523, + 246, + 535 + ], + "score": 0.93, + "content": "I ( X ; T _ { \\mathrm { M L P } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 523, + 265, + 536 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 265, + 523, + 312, + 534 + ], + "score": 0.9, + "content": "I ( Y ; { \\bar { T } } _ { \\mathrm { M L P } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 523, + 505, + 536 + ], + "score": 1.0, + "content": "based on Corollary 2. It shows that all of three", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 533, + 502, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 267, + 547 + ], + "score": 1.0, + "content": "MLPs satisfy the IB principle, namely", + "type": "text" + }, + { + "bbox": [ + 267, + 534, + 376, + 546 + ], + "score": 0.91, + "content": "I ( X ; T _ { \\mathrm { M L P } } ) < H ( X ) = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 533, + 395, + 547 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 396, + 534, + 502, + 546 + ], + "score": 0.92, + "content": "I ( Y ; T _ { \\mathrm { M L P } } ) = H ( Y ) = 1", + "type": "inline_equation" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 545, + 506, + 558 + ], + "spans": [ + { + "bbox": [ + 106, + 545, + 506, + 558 + ], + "score": 1.0, + "content": "though they have different architectures. Importantly, in contrast to previous work [28] claiming that", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 555, + 506, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 506, + 569 + ], + "score": 1.0, + "content": "the compression not exists in DNNs with non-saturating ACT, such as ReLU, Figure 5 clearly shows", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 567, + 455, + 579 + ], + "spans": [ + { + "bbox": [ + 106, + 567, + 455, + 579 + ], + "score": 1.0, + "content": "that the compression exists in all the MLPs, no matter the activation function of MLPs.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21.5 + }, + { + "type": "text", + "bbox": [ + 101, + 583, + 504, + 606 + ], + "lines": [ + { + "bbox": [ + 105, + 582, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 505, + 595 + ], + "score": 1.0, + "content": "We further demonstrate the information theoretic explanations for DNNs on the benchmark MNIST", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 594, + 417, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 405, + 606 + ], + "score": 1.0, + "content": "and Fashion-MNIST datasets. The experiments are presented in Appendix", + "type": "text" + }, + { + "bbox": [ + 405, + 595, + 414, + 604 + ], + "score": 0.27, + "content": "_ \\mathrm { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 594, + 417, + 606 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25.5 + }, + { + "type": "title", + "bbox": [ + 96, + 621, + 270, + 634 + ], + "lines": [ + { + "bbox": [ + 93, + 620, + 270, + 636 + ], + "spans": [ + { + "bbox": [ + 93, + 620, + 270, + 636 + ], + "score": 1.0, + "content": "04 5 Conclusion and future work", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 105, + 645, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 644, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 505, + 658 + ], + "score": 1.0, + "content": "In this work, we (1) specify the probability space for a hidden layer for (2) accurately estimating the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 656, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 656, + 505, + 668 + ], + "score": 1.0, + "content": "mutual information and (3) clearly explaining how the components of the layer affect the mutual", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 667, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 106, + 667, + 505, + 680 + ], + "score": 1.0, + "content": "information. We take into account the back-propagation training and derive two novel Markov chains", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 678, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 505, + 689 + ], + "score": 1.0, + "content": "to characterize the information flow in DNNs. Furthermore, we demonstrate that a DNN satisfies the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 689, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 505, + 701 + ], + "score": 1.0, + "content": "IB principle no matter the architecture of the DNN. In contrast, different hidden layers show different", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 700, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 505, + 712 + ], + "score": 1.0, + "content": "IB trade-offs depending on the architecture and the position of the layers in DNNs. 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The forth figure shows", + "type": "text" + }, + { + "bbox": [ + 285, + 318, + 329, + 329 + ], + "score": 0.92, + "content": "I ( X ; T _ { \\mathrm { M L P } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 317, + 345, + 330 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 345, + 318, + 387, + 329 + ], + "score": 0.94, + "content": "I ( Y ; T _ { \\mathrm { M L P } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 317, + 506, + 330 + ], + "score": 1.0, + "content": "in a MLP. The pink line denotes", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 107, + 329, + 308, + 339 + ], + "spans": [ + { + "bbox": [ + 107, + 329, + 155, + 339 + ], + "score": 0.9, + "content": "H ( Y ) = 1 . 0", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 329, + 256, + 339 + ], + "score": 1.0, + "content": "and the orange line denotes", + "type": "text" + }, + { + "bbox": [ + 256, + 329, + 305, + 339 + ], + "score": 0.91, + "content": "H ( X ) = 2 . 0", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 329, + 308, + 339 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "title", + "bbox": [ + 91, + 345, + 383, + 357 + ], + "lines": [ + { + "bbox": [ + 87, + 344, + 384, + 358 + ], + "spans": [ + { + "bbox": [ + 87, + 344, + 384, + 358 + ], + "score": 1.0, + "content": "283 4.3 Validating the information theoretic explanations for DNNs", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 107, + 365, + 505, + 424 + ], + "lines": [ + { + "bbox": [ + 106, + 365, + 506, + 380 + ], + "spans": [ + { + "bbox": [ + 106, + 366, + 203, + 380 + ], + "score": 1.0, + "content": "In Figure 5, we observe", + "type": "text" + }, + { + "bbox": [ + 204, + 365, + 288, + 379 + ], + "score": 0.92, + "content": "I ( X ; T _ { i } ) \\leq I ( X ; { \\hat { Y } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 366, + 506, + 380 + ], + "score": 1.0, + "content": "in MLP2 and MLP3, which confirms that the Markov", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 377, + 506, + 389 + ], + "spans": [ + { + "bbox": [ + 106, + 377, + 506, + 389 + ], + "score": 1.0, + "content": "chain proposed by previous works, Equation (2), cannot fully explain the information flow in MLPs,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 389, + 506, + 401 + ], + "spans": [ + { + "bbox": [ + 106, + 389, + 506, + 401 + ], + "score": 1.0, + "content": "if taking into account the back-propagation training. As a comparison, the second and third row", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 399, + 507, + 414 + ], + "spans": [ + { + "bbox": [ + 106, + 400, + 130, + 414 + ], + "score": 1.0, + "content": "show", + "type": "text" + }, + { + "bbox": [ + 131, + 400, + 265, + 413 + ], + "score": 0.92, + "content": "I ( \\bar { X } ; T _ { 1 } ) \\ge I ( \\bar { X } ; T _ { 2 } ) \\ge I ( \\bar { X } ; \\hat { Y } )", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 400, + 284, + 414 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 284, + 399, + 416, + 413 + ], + "score": 0.92, + "content": "I ( Y ; T _ { 1 } ) \\leq I ( Y ; T _ { 2 } ) \\geq I ( Y ; \\hat { Y } )", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 400, + 507, + 414 + ], + "score": 1.0, + "content": "in all the three MLPs,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 412, + 492, + 424 + ], + "spans": [ + { + "bbox": [ + 106, + 412, + 492, + 424 + ], + "score": 1.0, + "content": "which validates that Corollary 1, i.e., Equation (14) characterizes the information flow in MLPs.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 9, + "bbox_fs": [ + 106, + 365, + 507, + 424 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 428, + 505, + 507 + ], + "lines": [ + { + "bbox": [ + 105, + 428, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 505, + 441 + ], + "score": 1.0, + "content": "Figure 5 demonstrates that different hidden layers achieve different IB trade-offs depending on", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 439, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 106, + 439, + 440, + 452 + ], + "score": 1.0, + "content": "the architecture and the position of the layers in MLPs. 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In terms of position,", + "type": "text" + }, + { + "bbox": [ + 380, + 484, + 433, + 497 + ], + "score": 0.93, + "content": "I ( Y ; { \\hat { Y } } ) = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 484, + 450, + 498 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 450, + 483, + 504, + 497 + ], + "score": 0.93, + "content": "I ( { \\bar { X } } ; { \\hat { Y } } ) = 0", + "type": "inline_equation" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 496, + 371, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 190, + 508 + ], + "score": 1.0, + "content": "in MLP1 means that", + "type": "text" + }, + { + "bbox": [ + 190, + 496, + 198, + 507 + ], + "score": 0.84, + "content": "\\hat { \\pmb { y } }", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 496, + 317, + 508 + ], + "score": 1.0, + "content": "has a different IB trade-off to", + "type": "text" + }, + { + "bbox": [ + 318, + 497, + 327, + 507 + ], + "score": 0.87, + "content": "\\mathbf { \\delta t } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 496, + 371, + 508 + ], + "score": 1.0, + "content": "in MLP1.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 15, + "bbox_fs": [ + 105, + 428, + 506, + 508 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 512, + 505, + 578 + ], + "lines": [ + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "score": 1.0, + "content": "We demonstrate that a MLP satisfies the IB principle no matter what the architecture of the MLP", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 523, + 505, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 198, + 536 + ], + "score": 1.0, + "content": "is. 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It shows that all of three", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 533, + 502, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 267, + 547 + ], + "score": 1.0, + "content": "MLPs satisfy the IB principle, namely", + "type": "text" + }, + { + "bbox": [ + 267, + 534, + 376, + 546 + ], + "score": 0.91, + "content": "I ( X ; T _ { \\mathrm { M L P } } ) < H ( X ) = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 533, + 395, + 547 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 396, + 534, + 502, + 546 + ], + "score": 0.92, + "content": "I ( Y ; T _ { \\mathrm { M L P } } ) = H ( Y ) = 1", + "type": "inline_equation" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 545, + 506, + 558 + ], + "spans": [ + { + "bbox": [ + 106, + 545, + 506, + 558 + ], + "score": 1.0, + "content": "though they have different architectures. Importantly, in contrast to previous work [28] claiming that", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 555, + 506, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 506, + 569 + ], + "score": 1.0, + "content": "the compression not exists in DNNs with non-saturating ACT, such as ReLU, Figure 5 clearly shows", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 567, + 455, + 579 + ], + "spans": [ + { + "bbox": [ + 106, + 567, + 455, + 579 + ], + "score": 1.0, + "content": "that the compression exists in all the MLPs, no matter the activation function of MLPs.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21.5, + "bbox_fs": [ + 105, + 511, + 506, + 579 + ] + }, + { + "type": "text", + "bbox": [ + 101, + 583, + 504, + 606 + ], + "lines": [ + { + "bbox": [ + 105, + 582, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 505, + 595 + ], + "score": 1.0, + "content": "We further demonstrate the information theoretic explanations for DNNs on the benchmark MNIST", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 594, + 417, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 405, + 606 + ], + "score": 1.0, + "content": "and Fashion-MNIST datasets. The experiments are presented in Appendix", + "type": "text" + }, + { + "bbox": [ + 405, + 595, + 414, + 604 + ], + "score": 0.27, + "content": "_ \\mathrm { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 594, + 417, + 606 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25.5, + "bbox_fs": [ + 105, + 582, + 505, + 606 + ] + }, + { + "type": "title", + "bbox": [ + 96, + 621, + 270, + 634 + ], + "lines": [ + { + "bbox": [ + 93, + 620, + 270, + 636 + ], + "spans": [ + { + "bbox": [ + 93, + 620, + 270, + 636 + ], + "score": 1.0, + "content": "04 5 Conclusion and future work", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 105, + 645, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 644, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 505, + 658 + ], + "score": 1.0, + "content": "In this work, we (1) specify the probability space for a hidden layer for (2) accurately estimating the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 656, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 656, + 505, + 668 + ], + "score": 1.0, + "content": "mutual information and (3) clearly explaining how the components of the layer affect the mutual", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 667, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 106, + 667, + 505, + 680 + ], + "score": 1.0, + "content": "information. 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