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To shed light", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 345, + 469, + 358 + ], + "spans": [ + { + "bbox": [ + 141, + 345, + 309, + 358 + ], + "score": 1.0, + "content": "on the nature of the model learned by the", + "type": "text" + }, + { + "bbox": [ + 309, + 347, + 314, + 356 + ], + "score": 0.71, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 345, + 469, + 358 + ], + "score": 1.0, + "content": "-RevNet we reconstruct linear interpo-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 357, + 330, + 370 + ], + "spans": [ + { + "bbox": [ + 141, + 357, + 330, + 370 + ], + "score": 1.0, + "content": "lations between natural image representations.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 12, + "bbox_fs": [ + 141, + 203, + 470, + 370 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 390, + 205, + 403 + ], + "lines": [ + { + "bbox": [ + 105, + 389, + 208, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 208, + 406 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 106, + 415, + 505, + 537 + ], + "lines": [ + { + "bbox": [ + 105, + 415, + 506, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 506, + 429 + ], + "score": 1.0, + "content": "A CNN may be very effective in classifying images of all sorts (He et al., 2016; Krizhevsky et al.,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 427, + 506, + 439 + ], + "spans": [ + { + "bbox": [ + 106, + 427, + 506, + 439 + ], + "score": 1.0, + "content": "2012), but the cascade of linear and nonlinear operators reveals little about the contribution of the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 438, + 504, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 504, + 451 + ], + "score": 1.0, + "content": "internal representation to the classification. The learning process is characterized by a steady re-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 447, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 505, + 462 + ], + "score": 1.0, + "content": "duction of large amounts of uninformative variability in the images while simultaneously revealing", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 460, + 505, + 473 + ], + "spans": [ + { + "bbox": [ + 106, + 460, + 505, + 473 + ], + "score": 1.0, + "content": "the essence of the visual class. It is widely believed that this process is based on progressively dis-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 470, + 505, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 505, + 483 + ], + "score": 1.0, + "content": "carding uninformative variability about the input with respect to the problem at hand (Dosovitskiy", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 481, + 505, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 481, + 505, + 494 + ], + "score": 1.0, + "content": "& Brox, 2016; Mahendran & Vedaldi, 2016; Shwartz-Ziv & Tishby, 2017; Achille & Soatto, 2017).", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 492, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 505, + 505 + ], + "score": 1.0, + "content": "However, the extent to which information is discarded is lost somewhere in the intermediate non-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 503, + 506, + 517 + ], + "spans": [ + { + "bbox": [ + 105, + 503, + 506, + 517 + ], + "score": 1.0, + "content": "linear processing steps. In this paper, we aim to provide insight into the variability reduction process", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 514, + 506, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 506, + 528 + ], + "score": 1.0, + "content": "by proposing an invertible convolutional network, that does not discard any information about the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 525, + 133, + 540 + ], + "spans": [ + { + "bbox": [ + 104, + 525, + 133, + 540 + ], + "score": 1.0, + "content": "input.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 26, + "bbox_fs": [ + 104, + 415, + 506, + 540 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 542, + 504, + 597 + ], + "lines": [ + { + "bbox": [ + 106, + 542, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 106, + 542, + 505, + 555 + ], + "score": 1.0, + "content": "The difficulty to recover images from their hidden representations is found in many commonly used", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 553, + 506, + 566 + ], + "spans": [ + { + "bbox": [ + 105, + 553, + 506, + 566 + ], + "score": 1.0, + "content": "network architectures (Dosovitskiy & Brox, 2016; Mahendran & Vedaldi, 2016). This poses the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 564, + 506, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 506, + 577 + ], + "score": 1.0, + "content": "question if a substantial loss of information is necessary for successful classification. We show", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 575, + 506, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 575, + 506, + 588 + ], + "score": 1.0, + "content": "information does not have to be discarded. By using homeomorphic layers, the invariance can be", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 586, + 297, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 297, + 599 + ], + "score": 1.0, + "content": "built only at the very last layer via a projection.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34, + "bbox_fs": [ + 105, + 542, + 506, + 599 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 603, + 505, + 691 + ], + "lines": [ + { + "bbox": [ + 105, + 603, + 505, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 505, + 615 + ], + "score": 1.0, + "content": "In Shwartz-Ziv & Tishby (2017), minimal sufficient statistics are proposed as a candidate to explain", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 614, + 505, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 505, + 626 + ], + "score": 1.0, + "content": "the reduction of variability. Tishby & Zaslavsky (2015) introduces the information bottleneck princi-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 626, + 505, + 637 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 505, + 637 + ], + "score": 1.0, + "content": "ple which states that an optimal representation must reduce the mutual information between an input", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 636, + 505, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 636, + 505, + 648 + ], + "score": 1.0, + "content": "and its representation to reduce as much uninformative variability as possible. At the same time, the", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 647, + 505, + 659 + ], + "spans": [ + { + "bbox": [ + 105, + 647, + 505, + 659 + ], + "score": 1.0, + "content": "network should maximize the mutual information between the desired output and its representation", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 658, + 504, + 670 + ], + "spans": [ + { + "bbox": [ + 105, + 658, + 504, + 670 + ], + "score": 1.0, + "content": "to effectively preserve each class from collapsing onto other classes. The effect of the information", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 668, + 506, + 681 + ], + "spans": [ + { + "bbox": [ + 105, + 668, + 506, + 681 + ], + "score": 1.0, + "content": "bottleneck was demonstrated on small datasets in Shwartz-Ziv & Tishby (2017); Achille & Soatto", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 678, + 140, + 693 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 140, + 693 + ], + "score": 1.0, + "content": "(2017).", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 40.5, + "bbox_fs": [ + 105, + 603, + 506, + 693 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 148 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "However, in this work, we show it is not a necessary condition and we build a cascade of home-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "omorphic layers, which preserves the mutual information between input and hidden representation", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 505, + 116 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 505, + 116 + ], + "score": 1.0, + "content": "and shows that the loss of information can only occur at the final layer. This way we demonstrate", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 115, + 505, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 505, + 127 + ], + "score": 1.0, + "content": "that a loss of information can be avoided while maintaining discriminability, even for large-scale", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "score": 1.0, + "content": "problems like ImageNet. One way to reduce variability is progressive contraction with respect to a", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 136, + 336, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 154, + 150 + ], + "score": 1.0, + "content": "meaningful", + "type": "text" + }, + { + "bbox": [ + 155, + 137, + 164, + 147 + ], + "score": 0.83, + "content": "\\ell ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 136, + 336, + 150 + ], + "score": 1.0, + "content": "metric in the intermediate representations.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 107, + 154, + 505, + 242 + ], + "lines": [ + { + "bbox": [ + 105, + 153, + 505, + 168 + ], + "spans": [ + { + "bbox": [ + 105, + 153, + 505, + 168 + ], + "score": 1.0, + "content": "Several works (Oyallon, 2017; Zeiler & Fergus, 2014) observed a phenomenon of progressive sep-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "score": 1.0, + "content": "aration and contraction in non-invertible networks on limited datasets. Those progressive improve-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 177, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 177, + 505, + 189 + ], + "score": 1.0, + "content": "ments can be interpreted as the creation of progressively stronger invariants for classification. Ide-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 188, + 505, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 188, + 505, + 199 + ], + "score": 1.0, + "content": "ally, the contraction should not be too brutal to avoid removing important information from the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 505, + 210 + ], + "score": 1.0, + "content": "intermediate signal. This shows that a good trade-off between discriminability and invariance has", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 208, + 505, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 505, + 223 + ], + "score": 1.0, + "content": "to be progressively built. In this paper, we extend some findings of Zeiler & Fergus (2014); Oy-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 219, + 506, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 506, + 232 + ], + "score": 1.0, + "content": "allon (2017) to ImageNet (Russakovsky et al., 2015) and, most importantly, show that a loss of", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 230, + 382, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 382, + 243 + ], + "score": 1.0, + "content": "information is not necessary for observing a progressive contraction.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 9.5 + }, + { + "type": "text", + "bbox": [ + 107, + 248, + 505, + 369 + ], + "lines": [ + { + "bbox": [ + 106, + 247, + 506, + 260 + ], + "spans": [ + { + "bbox": [ + 106, + 247, + 506, + 260 + ], + "score": 1.0, + "content": "The duality between invariance and separation of the classes is discussed in Mallat (2016). Here,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 259, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 505, + 271 + ], + "score": 1.0, + "content": "intra-class variabilities are modeled as Lie groups that are processed by performing a parallel trans-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 270, + 506, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 506, + 282 + ], + "score": 1.0, + "content": "port along those symmetries. Filters are adapted through learning to the specific bias of the dataset", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 281, + 505, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 505, + 293 + ], + "score": 1.0, + "content": "and avoid to contract along discriminative directions. However, using groups beyond the Euclidean", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 291, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 505, + 304 + ], + "score": 1.0, + "content": "case for image classification is hard. Mainly because groups associated with abstract variabilities", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 304, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 106, + 304, + 505, + 315 + ], + "score": 1.0, + "content": "are difficult to estimate due to their high-dimensional nature, as well as the appropriate degree of", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 313, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 505, + 326 + ], + "score": 1.0, + "content": "invariance required. An illustration of this framework on the Euclidean group is given by the scatter-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 325, + 505, + 337 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 505, + 337 + ], + "score": 1.0, + "content": "ing transform (Mallat, 2012), which builds invariance to small translations while being recoverable", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 335, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 505, + 348 + ], + "score": 1.0, + "content": "to a certain extent. In this work, we introduce a network that cannot discard any information ex-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 347, + 506, + 359 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 506, + 359 + ], + "score": 1.0, + "content": "cept at the final classification stage, while we demonstrate numerically progressive contraction and", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 358, + 234, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 234, + 370 + ], + "score": 1.0, + "content": "separation of the signal classes.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 374, + 505, + 440 + ], + "lines": [ + { + "bbox": [ + 106, + 374, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 106, + 374, + 177, + 387 + ], + "score": 1.0, + "content": "We introduce the", + "type": "text" + }, + { + "bbox": [ + 177, + 375, + 182, + 385 + ], + "score": 0.7, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 374, + 334, + 387 + ], + "score": 1.0, + "content": "-RevNet, an invertible deep network.1", + "type": "text" + }, + { + "bbox": [ + 335, + 375, + 340, + 384 + ], + "score": 0.59, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 374, + 505, + 387 + ], + "score": 1.0, + "content": "-RevNets retain all information about the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 385, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 505, + 397 + ], + "score": 1.0, + "content": "input signal in any of their intermediate representations up until the last layer. Our architecture builds", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 396, + 506, + 409 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 506, + 409 + ], + "score": 1.0, + "content": "upon the recently introduced RevNet (Gomez et al., 2017), where we replace the non-invertible", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 407, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 327, + 420 + ], + "score": 1.0, + "content": "components of the original RevNets by invertible ones.", + "type": "text" + }, + { + "bbox": [ + 327, + 408, + 332, + 417 + ], + "score": 0.65, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 407, + 506, + 420 + ], + "score": 1.0, + "content": "-RevNets achieve the same performance on", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 419, + 506, + 431 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 506, + 431 + ], + "score": 1.0, + "content": "Imagenet compared to similar non-invertible RevNet and ResNet architectures (Gomez et al., 2017;", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 429, + 172, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 172, + 441 + ], + "score": 1.0, + "content": "He et al., 2016).", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 27.5 + }, + { + "type": "text", + "bbox": [ + 108, + 441, + 504, + 484 + ], + "lines": [ + { + "bbox": [ + 105, + 439, + 506, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 506, + 453 + ], + "score": 1.0, + "content": "To shed light on the mechanism underlying the generalization-ability of the learned representation,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 451, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 165, + 464 + ], + "score": 1.0, + "content": "we show that", + "type": "text" + }, + { + "bbox": [ + 165, + 452, + 170, + 461 + ], + "score": 0.63, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 170, + 451, + 506, + 464 + ], + "score": 1.0, + "content": "-RevNets progressively separate and contract signals with depth. Our results are", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 462, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 505, + 474 + ], + "score": 1.0, + "content": "evidence for an effective reduction of variability through a contraction with a recoverable input", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 473, + 297, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 297, + 486 + ], + "score": 1.0, + "content": "obtained from a series of one-to-one mappings.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 32.5 + }, + { + "type": "title", + "bbox": [ + 108, + 502, + 210, + 515 + ], + "lines": [ + { + "bbox": [ + 104, + 501, + 213, + 518 + ], + "spans": [ + { + "bbox": [ + 104, + 501, + 213, + 518 + ], + "score": 1.0, + "content": "2 RELATED WORK", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 107, + 528, + 505, + 649 + ], + "lines": [ + { + "bbox": [ + 106, + 529, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 106, + 529, + 505, + 540 + ], + "score": 1.0, + "content": "Several recent works show that significant information about the input images is lost with depth in", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 540, + 505, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 505, + 551 + ], + "score": 1.0, + "content": "successful Imagenet classification CNNs (Dosovitskiy & Brox, 2016; Mahendran & Vedaldi, 2016).", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "score": 1.0, + "content": "To understand the loss of information, the references propose to invert the representations by means", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 561, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 505, + 574 + ], + "score": 1.0, + "content": "of learned or hand-engineered priors. The approximate inversions indicate increased geometric", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 572, + 506, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 506, + 586 + ], + "score": 1.0, + "content": "and photometric invariance with depth. 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Parseval networks (Cisse et al., 2017) have been introduced to increase the robust-", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "score": 1.0, + "content": "ness of learned representations with respect to adversarial attacks. In this framework, the spectrum", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "of convolutional operators is constrained to norm 1 during learning. The linear operator is thus", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 699, + 145, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 145, + 711 + ], + "score": 1.0, + "content": "injective.", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 49, + "bbox_fs": [ + 105, + 655, + 506, + 711 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 182 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 94 + ], + "score": 1.0, + "content": "As a consequence, the input of Parseval networks can be recovered if but only if the built-in non-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 504, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 504, + 106 + ], + "score": 1.0, + "content": "linearities are invertible as well, which is typically not the case. Bruna et al. (2013) derive condi-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 505, + 118 + ], + "score": 1.0, + "content": "tions under which pooling representations are, but our method directly overcomes this issue. The", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 505, + 128 + ], + "score": 1.0, + "content": "Scattering transform (Mallat, 2012) is an example of predefined deep representation, approximately", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "score": 1.0, + "content": "invariant to translations, that can be reconstructed when the degree of invariance specified is small.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 137, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 106, + 137, + 505, + 150 + ], + "score": 1.0, + "content": "Yet, it requires a gradient descent optimization and no guarantee of convergences are known. In", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 148, + 505, + 160 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 505, + 160 + ], + "score": 1.0, + "content": "summary, the references make clear that invertibility requires special care in designing the architec-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "score": 1.0, + "content": "ture or special care in designing the optimization procedure. In this paper, we introduce a network,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 171, + 383, + 183 + ], + "spans": [ + { + "bbox": [ + 106, + 171, + 383, + 183 + ], + "score": 1.0, + "content": "that overcomes these issues and has an exact inverse by construction.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 107, + 187, + 505, + 297 + ], + "lines": [ + { + "bbox": [ + 106, + 187, + 505, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 505, + 199 + ], + "score": 1.0, + "content": "Our main inspiration for this work is the recent reversible residual network (RevNet), introduced", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 504, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 504, + 210 + ], + "score": 1.0, + "content": "in Gomez et al. (2017). RevNets are in turn closely related to NICE and Real-NVP architec-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 506, + 222 + ], + "score": 1.0, + "content": "tures (Dinh et al., 2016; 2014), which make use of constrained Jacobian determinants for generative", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 220, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 106, + 220, + 505, + 232 + ], + "score": 1.0, + "content": "modeling. All these architectures are similar to the lifting scheme (Sweldens, 1998) and Feistel", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 231, + 506, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 506, + 244 + ], + "score": 1.0, + "content": "cipher diagrams (Menezes et al., 1996), as we will show. RevNets illustrate how to build invert-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 241, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 505, + 255 + ], + "score": 1.0, + "content": "ible ResNet-type blocks that avoid storing intermediate activations necessary for the backward pass.", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 252, + 505, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 505, + 265 + ], + "score": 1.0, + "content": "However, RevNets still employ multiple non-invertible operators like max-pooling and downsam-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 264, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 505, + 276 + ], + "score": 1.0, + "content": "pling operators as part of the network. As such, RevNets are not invertible by construction. In this", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 274, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 505, + 288 + ], + "score": 1.0, + "content": "paper, we show how to build an invertible type of RevNet architecture that performs competitively", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 286, + 405, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 275, + 298 + ], + "score": 1.0, + "content": "with RevNets on Imagenet, which we call", + "type": "text" + }, + { + "bbox": [ + 276, + 286, + 280, + 296 + ], + "score": 0.62, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 286, + 405, + 298 + ], + "score": 1.0, + "content": "-RevNet for invertible RevNet.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 13.5 + }, + { + "type": "title", + "bbox": [ + 108, + 314, + 204, + 326 + ], + "lines": [ + { + "bbox": [ + 104, + 312, + 206, + 329 + ], + "spans": [ + { + "bbox": [ + 104, + 312, + 150, + 329 + ], + "score": 1.0, + "content": "3 THE", + "type": "text" + }, + { + "bbox": [ + 150, + 315, + 155, + 325 + ], + "score": 0.69, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 312, + 206, + 329 + ], + "score": 1.0, + "content": "-REVNET", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 108, + 339, + 504, + 373 + ], + "lines": [ + { + "bbox": [ + 106, + 338, + 505, + 352 + ], + "spans": [ + { + "bbox": [ + 106, + 338, + 328, + 352 + ], + "score": 1.0, + "content": "This section introduces the general framework of the", + "type": "text" + }, + { + "bbox": [ + 328, + 340, + 333, + 350 + ], + "score": 0.67, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 338, + 505, + 352 + ], + "score": 1.0, + "content": "-RevNet architecture and explains how to", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 351, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 106, + 351, + 297, + 363 + ], + "score": 1.0, + "content": "explicitly build an inverse or a left-inverse to an", + "type": "text" + }, + { + "bbox": [ + 297, + 352, + 302, + 361 + ], + "score": 0.58, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 351, + 505, + 363 + ], + "score": 1.0, + "content": "-RevNet. 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The operator", + "type": "text" + }, + { + "bbox": [ + 344, + 656, + 352, + 667 + ], + "score": 0.84, + "content": "\\tilde { \\cal S }", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 657, + 505, + 670 + ], + "score": 1.0, + "content": "is linear, injective, reduces the spatial", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 667, + 505, + 681 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 505, + 681 + ], + "score": 1.0, + "content": "resolution of the coefficients and can potentially increase the layer size, as wider layers usually", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 678, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 323, + 691 + ], + "score": 1.0, + "content": "improve the classification performance (Zagoruyko", + "type": "text" + }, + { + "bbox": [ + 324, + 680, + 333, + 689 + ], + "score": 0.27, + "content": "\\&", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 678, + 506, + 691 + ], + "score": 1.0, + "content": "Komodakis, 2016). We can thus build a", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 689, + 505, + 703 + ], + "spans": [ + { + "bbox": [ + 104, + 689, + 167, + 703 + ], + "score": 1.0, + "content": "pseudo inverse", + "type": "text" + }, + { + "bbox": [ + 168, + 690, + 182, + 701 + ], + "score": 0.89, + "content": "{ \\tilde { S } } ^ { + }", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 689, + 375, + 703 + ], + "score": 1.0, + "content": "that will be used for the inversion. 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Bruna et al. (2013) derive condi-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 505, + 118 + ], + "score": 1.0, + "content": "tions under which pooling representations are, but our method directly overcomes this issue. The", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 505, + 128 + ], + "score": 1.0, + "content": "Scattering transform (Mallat, 2012) is an example of predefined deep representation, approximately", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "score": 1.0, + "content": "invariant to translations, that can be reconstructed when the degree of invariance specified is small.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 137, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 106, + 137, + 505, + 150 + ], + "score": 1.0, + "content": "Yet, it requires a gradient descent optimization and no guarantee of convergences are known. In", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 148, + 505, + 160 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 505, + 160 + ], + "score": 1.0, + "content": "summary, the references make clear that invertibility requires special care in designing the architec-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "score": 1.0, + "content": "ture or special care in designing the optimization procedure. In this paper, we introduce a network,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 171, + 383, + 183 + ], + "spans": [ + { + "bbox": [ + 106, + 171, + 383, + 183 + ], + "score": 1.0, + "content": "that overcomes these issues and has an exact inverse by construction.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 4, + "bbox_fs": [ + 105, + 82, + 505, + 183 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 187, + 505, + 297 + ], + "lines": [ + { + "bbox": [ + 106, + 187, + 505, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 505, + 199 + ], + "score": 1.0, + "content": "Our main inspiration for this work is the recent reversible residual network (RevNet), introduced", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 504, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 504, + 210 + ], + "score": 1.0, + "content": "in Gomez et al. (2017). RevNets are in turn closely related to NICE and Real-NVP architec-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 506, + 222 + ], + "score": 1.0, + "content": "tures (Dinh et al., 2016; 2014), which make use of constrained Jacobian determinants for generative", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 220, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 106, + 220, + 505, + 232 + ], + "score": 1.0, + "content": "modeling. All these architectures are similar to the lifting scheme (Sweldens, 1998) and Feistel", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 231, + 506, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 506, + 244 + ], + "score": 1.0, + "content": "cipher diagrams (Menezes et al., 1996), as we will show. RevNets illustrate how to build invert-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 241, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 505, + 255 + ], + "score": 1.0, + "content": "ible ResNet-type blocks that avoid storing intermediate activations necessary for the backward pass.", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 252, + 505, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 505, + 265 + ], + "score": 1.0, + "content": "However, RevNets still employ multiple non-invertible operators like max-pooling and downsam-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 264, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 505, + 276 + ], + "score": 1.0, + "content": "pling operators as part of the network. As such, RevNets are not invertible by construction. In this", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 274, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 505, + 288 + ], + "score": 1.0, + "content": "paper, we show how to build an invertible type of RevNet architecture that performs competitively", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 286, + 405, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 275, + 298 + ], + "score": 1.0, + "content": "with RevNets on Imagenet, which we call", + "type": "text" + }, + { + "bbox": [ + 276, + 286, + 280, + 296 + ], + "score": 0.62, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 286, + 405, + 298 + ], + "score": 1.0, + "content": "-RevNet for invertible RevNet.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 13.5, + "bbox_fs": [ + 105, + 187, + 506, + 298 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 314, + 204, + 326 + ], + "lines": [ + { + "bbox": [ + 104, + 312, + 206, + 329 + ], + "spans": [ + { + "bbox": [ + 104, + 312, + 150, + 329 + ], + "score": 1.0, + "content": "3 THE", + "type": "text" + }, + { + "bbox": [ + 150, + 315, + 155, + 325 + ], + "score": 0.69, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 312, + 206, + 329 + ], + "score": 1.0, + "content": "-REVNET", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 108, + 339, + 504, + 373 + ], + "lines": [ + { + "bbox": [ + 106, + 338, + 505, + 352 + ], + "spans": [ + { + "bbox": [ + 106, + 338, + 328, + 352 + ], + "score": 1.0, + "content": "This section introduces the general framework of the", + "type": "text" + }, + { + "bbox": [ + 328, + 340, + 333, + 350 + ], + "score": 0.67, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 338, + 505, + 352 + ], + "score": 1.0, + "content": "-RevNet architecture and explains how to", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 351, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 106, + 351, + 297, + 363 + ], + "score": 1.0, + "content": "explicitly build an inverse or a left-inverse to an", + "type": "text" + }, + { + "bbox": [ + 297, + 352, + 302, + 361 + ], + "score": 0.58, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 351, + 505, + 363 + ], + "score": 1.0, + "content": "-RevNet. Its practical implementation is discussed,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 362, + 310, + 373 + ], + "spans": [ + { + "bbox": [ + 106, + 362, + 310, + 373 + ], + "score": 1.0, + "content": "and we demonstrate competitive numerical results.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21, + "bbox_fs": [ + 106, + 338, + 505, + 373 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 388, + 268, + 398 + ], + "lines": [ + { + "bbox": [ + 105, + 387, + 271, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 271, + 400 + ], + "score": 1.0, + "content": "3.1 AN INVERTIBLE ARCHITECTURE", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "image", + "bbox": [ + 124, + 420, + 491, + 539 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 124, + 420, + 491, + 539 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 124, + 420, + 491, + 539 + ], + "spans": [ + { + "bbox": [ + 124, + 420, + 491, + 539 + ], + "score": 0.97, + "type": "image", + "image_path": "5cd194f32b37730890221ba1d7413ff2ff91a6729965b3a9be0fa94b8d97be5b.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 124, + 420, + 491, + 459.6666666666667 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 124, + 459.6666666666667, + 491, + 499.33333333333337 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 124, + 499.33333333333337, + 491, + 539.0 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 551, + 505, + 600 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 551, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 106, + 551, + 259, + 564 + ], + "score": 1.0, + "content": "Figure 1: The main component of the", + "type": "text" + }, + { + "bbox": [ + 259, + 553, + 264, + 562 + ], + "score": 0.63, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 551, + 505, + 564 + ], + "score": 1.0, + "content": "-RevNet and its inverse. RevNet blocks are interleaved with", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 563, + 505, + 576 + ], + "spans": [ + { + "bbox": [ + 106, + 563, + 210, + 576 + ], + "score": 1.0, + "content": "convolutional bottlenecks", + "type": "text" + }, + { + "bbox": [ + 211, + 563, + 223, + 576 + ], + "score": 0.89, + "content": "{ \\mathcal { F } } _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 563, + 330, + 576 + ], + "score": 1.0, + "content": "and reshuffling operations", + "type": "text" + }, + { + "bbox": [ + 330, + 563, + 342, + 576 + ], + "score": 0.89, + "content": "S _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 563, + 505, + 576 + ], + "score": 1.0, + "content": "to ensure invertibility of the architecture", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 574, + 505, + 588 + ], + "spans": [ + { + "bbox": [ + 106, + 575, + 437, + 588 + ], + "score": 1.0, + "content": "and computational efficiency. 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Observe that the inverse network is obtained with minimal adaptations.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28.5 + } + ], + "index": 26.75 + }, + { + "type": "text", + "bbox": [ + 107, + 620, + 505, + 703 + ], + "lines": [ + { + "bbox": [ + 106, + 620, + 505, + 633 + ], + "spans": [ + { + "bbox": [ + 106, + 620, + 157, + 633 + ], + "score": 1.0, + "content": "We describe", + "type": "text" + }, + { + "bbox": [ + 158, + 622, + 162, + 631 + ], + "score": 0.72, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 620, + 505, + 633 + ], + "score": 1.0, + "content": "-RevNets in their general setting. Their foundations are largely grounded in the recent", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 631, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 298, + 645 + ], + "score": 1.0, + "content": "RevNet architecture (Gomez et al., 2017). In an", + "type": "text" + }, + { + "bbox": [ + 299, + 633, + 303, + 642 + ], + "score": 0.66, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 631, + 505, + 645 + ], + "score": 1.0, + "content": "-RevNet, an initial input is split into two sublayers", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 107, + 643, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 107, + 644, + 140, + 657 + ], + "score": 0.92, + "content": "( x _ { 0 } , \\tilde { x } _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 643, + 315, + 658 + ], + "score": 1.0, + "content": "of equal size, thanks to a splitting operator", + "type": "text" + }, + { + "bbox": [ + 315, + 643, + 374, + 656 + ], + "score": 0.92, + "content": "\\tilde { S } x \\triangleq ( x _ { 0 } , \\tilde { x } _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 643, + 505, + 658 + ], + "score": 1.0, + "content": ", in this paper we choose to split", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 656, + 505, + 670 + ], + "spans": [ + { + "bbox": [ + 105, + 657, + 344, + 670 + ], + "score": 1.0, + "content": "the channel dimension as is done in RevNets. 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We can thus build a", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 689, + 505, + 703 + ], + "spans": [ + { + "bbox": [ + 104, + 689, + 167, + 703 + ], + "score": 1.0, + "content": "pseudo inverse", + "type": "text" + }, + { + "bbox": [ + 168, + 690, + 182, + 701 + ], + "score": 0.89, + "content": "{ \\tilde { S } } ^ { + }", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 689, + 375, + 703 + ], + "score": 1.0, + "content": "that will be used for the inversion. 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Figure 1 describes the blocks of an", + "type": "text" + }, + { + "bbox": [ + 316, + 130, + 320, + 138 + ], + "score": 0.67, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 128, + 505, + 141 + ], + "score": 1.0, + "content": "-RevNet. The design is similar to the Feistel", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 139, + 505, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 139, + 505, + 151 + ], + "score": 1.0, + "content": "cipher diagrams (Menezes et al., 1996) or a lifting scheme (Sweldens, 1998), which are invertible", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 150, + 451, + 163 + ], + "spans": [ + { + "bbox": [ + 106, + 150, + 451, + 163 + ], + "score": 1.0, + "content": "and efficient implementations of complex transforms like second generation wavelets.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 107, + 167, + 505, + 223 + ], + "lines": [ + { + "bbox": [ + 105, + 167, + 505, + 179 + ], + "spans": [ + { + "bbox": [ + 105, + 167, + 505, + 179 + ], + "score": 1.0, + "content": "In this way, we avoid the non-invertible modules of a RevNet (e.g. max-pooling or strides) which", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 178, + 505, + 191 + ], + "spans": [ + { + "bbox": [ + 105, + 178, + 505, + 191 + ], + "score": 1.0, + "content": "are necessary to train them in a reasonable time and are designed to build invariance w.r.t. translation", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 189, + 505, + 202 + ], + "spans": [ + { + "bbox": [ + 106, + 189, + 454, + 201 + ], + "score": 1.0, + "content": "variability. Our method shows we can replace them by linear and invertible modules", + "type": "text" + }, + { + "bbox": [ + 454, + 190, + 466, + 202 + ], + "score": 0.88, + "content": "S _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 189, + 505, + 201 + ], + "score": 1.0, + "content": ", that can", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 200, + 505, + 213 + ], + "spans": [ + { + "bbox": [ + 105, + 200, + 505, + 213 + ], + "score": 1.0, + "content": "reduce the spatial resolution (we refer to it as a spatial down-sampling for the sake of simplicity)", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 212, + 397, + 223 + ], + "spans": [ + { + "bbox": [ + 106, + 212, + 397, + 223 + ], + "score": 1.0, + "content": "while maintaining the layer’s size by increasing the number of channels.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 107, + 228, + 505, + 261 + ], + "lines": [ + { + "bbox": [ + 105, + 227, + 505, + 242 + ], + "spans": [ + { + "bbox": [ + 105, + 227, + 505, + 242 + ], + "score": 1.0, + "content": "We keep the computational cost manageable by tightly coupling downsampling and increase in", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 239, + 505, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 239, + 417, + 252 + ], + "score": 1.0, + "content": "width of the network. 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This leads to the following equations:", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13 + }, + { + "type": "interline_equation", + "bbox": [ + 181, + 275, + 420, + 305 + ], + "lines": [ + { + "bbox": [ + 181, + 275, + 420, + 305 + ], + "spans": [ + { + "bbox": [ + 181, + 275, + 420, + 305 + ], + "score": 0.92, + "content": "\\left\\{ \\begin{array} { l l } { x _ { j + 1 } = S _ { j + 1 } \\tilde { x } _ { j } } \\\\ { \\tilde { x } _ { j + 1 } = x _ { j } + \\mathcal { F } _ { j + 1 } \\tilde { x } _ { j } } \\end{array} \\right. \\iff \\quad \\left\\{ \\begin{array} { l l } { \\tilde { x } _ { j } = S _ { j + 1 } ^ { - 1 } x _ { j + 1 } } \\\\ { x _ { j } = \\tilde { x } _ { j + 1 } - \\mathcal { F } _ { j + 1 } \\tilde { x } _ { j } } \\end{array} \\right.", + "type": "interline_equation", + "image_path": "da0995a53b3995df67a5ec1405289067fd48910cb6ad5f7bd3f7226e7dcd5d2a.jpg" + } + ] + } + ], + "index": 15.5, + "virtual_lines": [ + { + "bbox": [ + 181, + 275, + 420, + 290.0 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 181, + 290.0, + 420, + 305.0 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 235, + 314, + 503, + 336 + ], + "lines": [ + { + "bbox": [ + 235, + 313, + 504, + 326 + ], + "spans": [ + { + "bbox": [ + 235, + 313, + 416, + 326 + ], + "score": 1.0, + "content": "Our downsampling layer can be written for", + "type": "text" + }, + { + "bbox": [ + 416, + 316, + 424, + 324 + ], + "score": 0.78, + "content": "u", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 313, + 504, + 326 + ], + "score": 1.0, + "content": "the spatial variable", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 234, + 325, + 336, + 336 + ], + "spans": [ + { + "bbox": [ + 234, + 325, + 252, + 336 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 252, + 326, + 259, + 335 + ], + "score": 0.8, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 325, + 336, + 336 + ], + "score": 1.0, + "content": "the channel index:", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17.5 + }, + { + "type": "interline_equation", + "bbox": [ + 320, + 342, + 420, + 356 + ], + "lines": [ + { + "bbox": [ + 320, + 342, + 420, + 356 + ], + "spans": [ + { + "bbox": [ + 320, + 342, + 420, + 356 + ], + "score": 0.9, + "content": "S _ { j } x ( u , \\lambda ) = x ( \\Psi ( u , \\lambda ) )", + "type": "interline_equation", + "image_path": "82df32856045c35be0eba3c2ee24262b514350416991477d0f1a642428812415.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 320, + 342, + 420, + 356 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "image", + "bbox": [ + 111, + 343, + 222, + 389 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 111, + 343, + 222, + 389 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 111, + 343, + 222, + 389 + ], + "spans": [ + { + "bbox": [ + 111, + 343, + 222, + 389 + ], + "score": 0.949, + "type": "image", + "image_path": "e760aaaf5fd2f19f2d2458df6118500adce3d86267e7acf371876f61f9b09078.jpg" + } + ] + } + ], + "index": 20.0, + "virtual_lines": [ + { + "bbox": [ + 111, + 343, + 222, + 366.0 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 111, + 366.0, + 222, + 389.0 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 415, + 227, + 438 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 415, + 227, + 427 + ], + "spans": [ + { + "bbox": [ + 106, + 415, + 227, + 427 + ], + "score": 1.0, + "content": "Figure 2: Illustration of the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 425, + 211, + 440 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 211, + 440 + ], + "score": 1.0, + "content": "invertible down-sampling", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26.5 + } + ], + "index": 23.25 + }, + { + "type": "text", + "bbox": [ + 236, + 362, + 504, + 452 + ], + "lines": [ + { + "bbox": [ + 235, + 362, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 235, + 362, + 264, + 375 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 264, + 362, + 273, + 372 + ], + "score": 0.81, + "content": "\\Psi", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 362, + 505, + 375 + ], + "score": 1.0, + "content": "is some invertible mapping. 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In consequence, its implementation is simple and specified by Equation (1). 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The hyper-parameters were selected to be either close to the", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 614, + 505, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 505, + 630 + ], + "score": 1.0, + "content": "ResNet and RevNet baselines in terms of the number of layers (a) or parameters (b) while keeping", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 627, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 505, + 640 + ], + "score": 1.0, + "content": "performance competitive. For the same reasons as in Gomez et al. 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Figure 1 describes the blocks of an", + "type": "text" + }, + { + "bbox": [ + 316, + 130, + 320, + 138 + ], + "score": 0.67, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 128, + 505, + 141 + ], + "score": 1.0, + "content": "-RevNet. The design is similar to the Feistel", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 139, + 505, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 139, + 505, + 151 + ], + "score": 1.0, + "content": "cipher diagrams (Menezes et al., 1996) or a lifting scheme (Sweldens, 1998), which are invertible", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 150, + 451, + 163 + ], + "spans": [ + { + "bbox": [ + 106, + 150, + 451, + 163 + ], + "score": 1.0, + "content": "and efficient implementations of complex transforms like second generation wavelets.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3, + "bbox_fs": [ + 104, + 82, + 506, + 163 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 167, + 505, + 223 + ], + "lines": [ + { + "bbox": [ + 105, + 167, + 505, + 179 + ], + "spans": [ + { + "bbox": [ + 105, + 167, + 505, + 179 + ], + "score": 1.0, + "content": "In this way, we avoid the non-invertible modules of a RevNet (e.g. max-pooling or strides) which", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 178, + 505, + 191 + ], + "spans": [ + { + "bbox": [ + 105, + 178, + 505, + 191 + ], + "score": 1.0, + "content": "are necessary to train them in a reasonable time and are designed to build invariance w.r.t. translation", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 189, + 505, + 202 + ], + "spans": [ + { + "bbox": [ + 106, + 189, + 454, + 201 + ], + "score": 1.0, + "content": "variability. 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In principle, any invert-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 235, + 373, + 505, + 385 + ], + "spans": [ + { + "bbox": [ + 235, + 373, + 505, + 385 + ], + "score": 1.0, + "content": "ible downsampling operation like e.g. dilated convolutions (Yu &", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 235, + 384, + 505, + 396 + ], + "spans": [ + { + "bbox": [ + 235, + 384, + 505, + 396 + ], + "score": 1.0, + "content": "Koltun, 2015) can be considered here. We use the inverse of the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 235, + 395, + 505, + 407 + ], + "spans": [ + { + "bbox": [ + 235, + 395, + 505, + 407 + ], + "score": 1.0, + "content": "operation described in Shi et al. 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In consequence, its implementation is simple and specified by Equation (1). 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The hyper-parameters were selected to be either close to the", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 614, + 505, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 505, + 630 + ], + "score": 1.0, + "content": "ResNet and RevNet baselines in terms of the number of layers (a) or parameters (b) while keeping", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 627, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 505, + 640 + ], + "score": 1.0, + "content": "performance competitive. For the same reasons as in Gomez et al. 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The second", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "layer has four times fewer channels than the other two, while their corresponding kernel sizes are", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 721, + 237, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 156, + 732 + ], + "score": 1.0, + "content": "respectively", + "type": "text" + }, + { + "bbox": [ + 157, + 721, + 233, + 732 + ], + "score": 0.92, + "content": "1 \\times 1 , 3 \\times 3 , 1 \\times 1", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 721, + 237, + 732 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 53 + } + ], + "index": 51.5, + "bbox_fs": [ + 105, + 687, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 162, + 79, + 449, + 144 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 162, + 79, + 449, + 144 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 162, + 79, + 449, + 144 + ], + "spans": [ + { + "bbox": [ + 162, + 79, + 449, + 144 + ], + "score": 0.963, + "html": "
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We now discuss how we progressively decrease the spatial resolution, while increasing the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 243, + 334, + 257 + ], + "spans": [ + { + "bbox": [ + 105, + 243, + 318, + 257 + ], + "score": 1.0, + "content": "number of channels per layer by use of the operators", + "type": "text" + }, + { + "bbox": [ + 318, + 243, + 330, + 255 + ], + "score": 0.89, + "content": "S _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 243, + 334, + 257 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 107, + 259, + 505, + 304 + ], + "lines": [ + { + "bbox": [ + 105, + 259, + 505, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 505, + 272 + ], + "score": 1.0, + "content": "We first describe the model (a), that consists of 56 layers which have been optimized to match", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 271, + 505, + 283 + ], + "spans": [ + { + "bbox": [ + 106, + 271, + 505, + 283 + ], + "score": 1.0, + "content": "the performances of a RevNet or a ResNet with approximatively the same number of layers. In", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 282, + 505, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 282, + 505, + 295 + ], + "score": 1.0, + "content": "particular, we explain how we progressively decrease the spatial resolution, while increasing the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 293, + 336, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 320, + 306 + ], + "score": 1.0, + "content": "number of channels per block by use of the operators", + "type": "text" + }, + { + "bbox": [ + 321, + 293, + 332, + 305 + ], + "score": 0.89, + "content": "S _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 332, + 293, + 336, + 306 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 9.5 + }, + { + "type": "text", + "bbox": [ + 106, + 311, + 505, + 443 + ], + "lines": [ + { + "bbox": [ + 105, + 310, + 506, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 311, + 196, + 325 + ], + "score": 1.0, + "content": "The splitting operator", + "type": "text" + }, + { + "bbox": [ + 196, + 310, + 205, + 322 + ], + "score": 0.82, + "content": "\\tilde { \\cal S }", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 311, + 506, + 325 + ], + "score": 1.0, + "content": "consists in a linear and injective embedding that downsamples by a factor", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 321, + 506, + 337 + ], + "spans": [ + { + "bbox": [ + 106, + 322, + 117, + 333 + ], + "score": 0.76, + "content": "4 ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 321, + 506, + 337 + ], + "score": 1.0, + "content": "the spatial resolution by increasing the number of output channels from 48 to 96 by simply adding", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 333, + 505, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 505, + 347 + ], + "score": 1.0, + "content": "0. The latter permits to increase the initial layer size, and consequently, the size of the next layers as", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 344, + 504, + 358 + ], + "spans": [ + { + "bbox": [ + 104, + 344, + 403, + 358 + ], + "score": 1.0, + "content": "performed in Gomez et al. (2017); it is thus not a bijective yet an injective", + "type": "text" + }, + { + "bbox": [ + 403, + 345, + 408, + 355 + ], + "score": 0.61, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 344, + 482, + 358 + ], + "score": 1.0, + "content": "-RevNet. At depth", + "type": "text" + }, + { + "bbox": [ + 482, + 345, + 504, + 357 + ], + "score": 0.53, + "content": "j , { \\mathcal { S } } _ { j }", + "type": "inline_equation" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 356, + 505, + 368 + ], + "spans": [ + { + "bbox": [ + 106, + 356, + 505, + 368 + ], + "score": 1.0, + "content": "allows us to reduce the number of computations while maintaining good classification performance.", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 365, + 506, + 380 + ], + "spans": [ + { + "bbox": [ + 104, + 365, + 397, + 380 + ], + "score": 1.0, + "content": "It will correspond to a downsampling operator respectively at the depth", + "type": "text" + }, + { + "bbox": [ + 397, + 367, + 457, + 378 + ], + "score": 0.55, + "content": "3 j = 1 5 , 2 7 , 4 ", + "type": "inline_equation" + }, + { + "bbox": [ + 458, + 365, + 506, + 380 + ], + "score": 1.0, + "content": "5 (3j as one", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 377, + 505, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 377, + 505, + 390 + ], + "score": 1.0, + "content": "block corresponds to three layers), similar to a normal RevNet. The spatial resolution of these layers", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 387, + 506, + 401 + ], + "spans": [ + { + "bbox": [ + 104, + 387, + 193, + 401 + ], + "score": 1.0, + "content": "is reduced by a factor", + "type": "text" + }, + { + "bbox": [ + 194, + 388, + 204, + 399 + ], + "score": 0.83, + "content": "2 ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 387, + 506, + 401 + ], + "score": 1.0, + "content": "while increasing the number of channels by a factor of 4 respectively to 48,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 399, + 506, + 412 + ], + "spans": [ + { + "bbox": [ + 106, + 399, + 506, + 412 + ], + "score": 1.0, + "content": "192, 768 and 3072. 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All the remaining blocks", + "type": "text" + }, + { + "bbox": [ + 280, + 422, + 292, + 433 + ], + "score": 0.88, + "content": "S _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 292, + 420, + 506, + 435 + ], + "score": 1.0, + "content": "are kept fix to the identity as explained in the section", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 431, + 137, + 445 + ], + "spans": [ + { + "bbox": [ + 105, + 431, + 137, + 445 + ], + "score": 1.0, + "content": "above.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 17.5 + }, + { + "type": "text", + "bbox": [ + 106, + 449, + 505, + 529 + ], + "lines": [ + { + "bbox": [ + 106, + 449, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 106, + 449, + 505, + 462 + ], + "score": 1.0, + "content": "Architecture (b) is bijective, it consists of 300 layers (100 blocks), whose total numbers of parame-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 460, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 493, + 474 + ], + "score": 1.0, + "content": "ters have been optimized to match those of a RevNet with 56 layers. Initially, the input is split via", + "type": "text" + }, + { + "bbox": [ + 493, + 460, + 501, + 472 + ], + "score": 0.82, + "content": "\\tilde { \\cal S }", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 461, + 505, + 474 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 471, + 505, + 485 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 350, + 485 + ], + "score": 1.0, + "content": "which corresponds to an invertible spatial downsampling of", + "type": "text" + }, + { + "bbox": [ + 351, + 471, + 361, + 483 + ], + "score": 0.85, + "content": "2 ^ { \\frac { 5 } { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 471, + 505, + 485 + ], + "score": 1.0, + "content": "that increases the number of chan-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 483, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 463, + 496 + ], + "score": 1.0, + "content": "nels from 3 to 12. It thus keeps the dimension constant and permits building a bijective", + "type": "text" + }, + { + "bbox": [ + 463, + 484, + 468, + 494 + ], + "score": 0.71, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 483, + 505, + 496 + ], + "score": 1.0, + "content": "-RevNet.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 493, + 506, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 168, + 507 + ], + "score": 1.0, + "content": "Then, at depth", + "type": "text" + }, + { + "bbox": [ + 168, + 495, + 213, + 506 + ], + "score": 0.5, + "content": "3 j = 3 , 2 1", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 493, + 396, + 507 + ], + "score": 1.0, + "content": ", 69, 285, the spatial resolution is reduced by", + "type": "text" + }, + { + "bbox": [ + 396, + 494, + 407, + 505 + ], + "score": 0.81, + "content": "2 ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 493, + 423, + 507 + ], + "score": 1.0, + "content": "via", + "type": "text" + }, + { + "bbox": [ + 424, + 495, + 435, + 507 + ], + "score": 0.87, + "content": "S _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 493, + 506, + 507 + ], + "score": 1.0, + "content": ". Contrary to the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 505, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 386, + 520 + ], + "score": 1.0, + "content": "architecture (a), the dimensionality of each layer is constantly equal to", + "type": "text" + }, + { + "bbox": [ + 387, + 506, + 422, + 517 + ], + "score": 0.92, + "content": "3 \\times 2 2 4 ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 505, + 505, + 520 + ], + "score": 1.0, + "content": ", until the final layer,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 518, + 267, + 528 + ], + "spans": [ + { + "bbox": [ + 106, + 518, + 267, + 528 + ], + "score": 1.0, + "content": "with channel sizes of 24, 96, 384, 1536.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 107, + 534, + 297, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 534, + 297, + 546 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 297, + 546 + ], + "score": 1.0, + "content": "For both networks, the training on Imagenet", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 545, + 297, + 558 + ], + "spans": [ + { + "bbox": [ + 106, + 545, + 297, + 558 + ], + "score": 1.0, + "content": "follows the same setup as Gomez et al. 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Contrary to the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 505, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 386, + 520 + ], + "score": 1.0, + "content": "architecture (a), the dimensionality of each layer is constantly equal to", + "type": "text" + }, + { + "bbox": [ + 387, + 506, + 422, + 517 + ], + "score": 0.92, + "content": "3 \\times 2 2 4 ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 505, + 505, + 520 + ], + "score": 1.0, + "content": ", until the final layer,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 518, + 267, + 528 + ], + "spans": [ + { + "bbox": [ + 106, + 518, + 267, + 528 + ], + "score": 1.0, + "content": "with channel sizes of 24, 96, 384, 1536.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 449, + 506, + 528 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 534, + 297, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 534, + 297, + 546 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 297, + 546 + ], + "score": 1.0, + "content": "For both networks, the training on Imagenet", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 545, + 297, + 558 + ], + "spans": [ + { + "bbox": [ + 106, + 545, + 297, + 558 + ], + "score": 1.0, + "content": "follows the same setup as Gomez et al. 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We", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 567, + 298, + 579 + ], + "spans": [ + { + "bbox": [ + 106, + 567, + 221, + 579 + ], + "score": 1.0, + "content": "regularized the model with a", + "type": "text" + }, + { + "bbox": [ + 221, + 567, + 231, + 577 + ], + "score": 0.83, + "content": "\\ell ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 567, + 298, + 579 + ], + "score": 1.0, + "content": "weight decay of", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 577, + 298, + 590 + ], + "spans": [ + { + "bbox": [ + 106, + 578, + 128, + 589 + ], + "score": 0.88, + "content": "1 0 ^ { - 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 129, + 577, + 298, + 590 + ], + "score": 1.0, + "content": "and batch normalization. The dataset is", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 589, + 298, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 162, + 600 + ], + "score": 1.0, + "content": "processed for", + "type": "text" + }, + { + "bbox": [ + 162, + 589, + 184, + 600 + ], + "score": 0.65, + "content": "6 0 0 \\mathrm { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 590, + 298, + 600 + ], + "score": 1.0, + "content": "iterations on a batch size of", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 599, + 298, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 298, + 613 + ], + "score": 1.0, + "content": "256, distributed on 4GPUs. The initial learning", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 611, + 297, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 275, + 622 + ], + "score": 1.0, + "content": "rate is 0.1, dropped by a factor of ten every", + "type": "text" + }, + { + "bbox": [ + 275, + 611, + 297, + 622 + ], + "score": 0.54, + "content": "1 6 0 \\mathrm { k }", + "type": "inline_equation" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 622, + 297, + 633 + ], + "spans": [ + { + "bbox": [ + 106, + 622, + 297, + 633 + ], + "score": 1.0, + "content": "iterations. The dataset was augmented accord-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 633, + 298, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 298, + 645 + ], + "score": 1.0, + "content": "ing to Gomez et al. (2017). The images values", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 644, + 297, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 297, + 656 + ], + "score": 1.0, + "content": "are mapped to [0, 1] while following geometric", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 654, + 297, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 297, + 668 + ], + "score": 1.0, + "content": "transformations were applied: random scaling,", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 666, + 297, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 297, + 678 + ], + "score": 1.0, + "content": "random horizontal flipping, random cropping", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 676, + 298, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 676, + 136, + 689 + ], + "score": 1.0, + "content": "of size", + "type": "text" + }, + { + "bbox": [ + 137, + 676, + 157, + 687 + ], + "score": 0.87, + "content": "2 2 4 ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 676, + 298, + 689 + ], + "score": 1.0, + "content": ", and finally color distortions. No", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 688, + 297, + 699 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 297, + 699 + ], + "score": 1.0, + "content": "other regularizations were incorporated into the", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 698, + 297, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 698, + 297, + 711 + ], + "score": 1.0, + "content": "classification pipeline. 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Observe that the decrease of both", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "training-losses are very similar which indicates that the constraint of invertibility does not interfere", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 506, + 128 + ], + "score": 1.0, + "content": "negatively with the learning process. However, we observed one third longer wall-clock times for", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 126, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 111, + 136 + ], + "score": 0.66, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 111, + 126, + 505, + 139 + ], + "score": 1.0, + "content": "-RevNets compared to plain RevNets because the channel size becomes larger. The Table 1 reports", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 137, + 505, + 149 + ], + "spans": [ + { + "bbox": [ + 106, + 137, + 207, + 149 + ], + "score": 1.0, + "content": "the performances of our", + "type": "text" + }, + { + "bbox": [ + 208, + 138, + 212, + 147 + ], + "score": 0.73, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 137, + 505, + 149 + ], + "score": 1.0, + "content": "-RevNets, with comparable RevNet and ResNet. First, we compare the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 147, + 506, + 162 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 111, + 158 + ], + "score": 0.69, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 111, + 147, + 506, + 162 + ], + "score": 1.0, + "content": "-RevNet (a) with the RevNet and ResNet. Indeed, those CNNs have the same number of layers,", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 158, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 158, + 138, + 172 + ], + "score": 1.0, + "content": "and the", + "type": "text" + }, + { + "bbox": [ + 138, + 160, + 143, + 169 + ], + "score": 0.53, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 143, + 158, + 505, + 172 + ], + "score": 1.0, + "content": "-RevNet (a) increases the channel width of the initial layer as done in Gomez et al. (2017).", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 169, + 482, + 184 + ], + "spans": [ + { + "bbox": [ + 105, + 169, + 482, + 184 + ], + "score": 1.0, + "content": "The drawback of this technique is that the kernel sizes will be larger for all subsequent layers.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 107, + 187, + 505, + 276 + ], + "lines": [ + { + "bbox": [ + 106, + 187, + 506, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 125, + 199 + ], + "score": 1.0, + "content": "The", + "type": "text" + }, + { + "bbox": [ + 125, + 188, + 130, + 197 + ], + "score": 0.66, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 131, + 187, + 506, + 199 + ], + "score": 1.0, + "content": "-RevNet (a) has about 6 times more parameters than a RevNet and a ResNet but leads to a", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 199, + 506, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 199, + 394, + 210 + ], + "score": 1.0, + "content": "similar accuracy on the validation set of ImageNet. On the contrary, the", + "type": "text" + }, + { + "bbox": [ + 394, + 199, + 399, + 208 + ], + "score": 0.59, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 199, + 506, + 210 + ], + "score": 1.0, + "content": "-RevNet (b) is designed to", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 505, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 505, + 221 + ], + "score": 1.0, + "content": "have roughly the same number of parameters as the RevNet and ResNet, while being bijective. Its", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 220, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 197, + 232 + ], + "score": 1.0, + "content": "accuracy decreases by", + "type": "text" + }, + { + "bbox": [ + 198, + 220, + 219, + 231 + ], + "score": 0.86, + "content": "1 . 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 220, + 505, + 232 + ], + "score": 1.0, + "content": "absolute percent on ImageNet compared to the RevNet baseline, which", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 232, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 505, + 243 + ], + "score": 1.0, + "content": "is not surprising because the number of channels was not drastically increased in the earlier layers", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 242, + 506, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 506, + 254 + ], + "score": 1.0, + "content": "as done in the baselines (Gomez et al., 2017; Krizhevsky et al., 2012; He et al., 2016); we did not", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 253, + 506, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 506, + 266 + ], + "score": 1.0, + "content": "explore wide ranges of hyper-parameters, thus the gap between (a) and (b) can likely be reduced", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 264, + 221, + 277 + ], + "spans": [ + { + "bbox": [ + 106, + 264, + 221, + 277 + ], + "score": 1.0, + "content": "with additional engineering.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 12.5 + }, + { + "type": "title", + "bbox": [ + 108, + 292, + 265, + 303 + ], + "lines": [ + { + "bbox": [ + 105, + 291, + 267, + 305 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 267, + 305 + ], + "score": 1.0, + "content": "4 ANALYSIS OF THE INVERSE", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 316, + 505, + 360 + ], + "lines": [ + { + "bbox": [ + 105, + 316, + 505, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 253, + 329 + ], + "score": 1.0, + "content": "We now analyze the representation", + "type": "text" + }, + { + "bbox": [ + 254, + 317, + 262, + 326 + ], + "score": 0.83, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 316, + 419, + 329 + ], + "score": 1.0, + "content": "built by our bijective neural network", + "type": "text" + }, + { + "bbox": [ + 419, + 317, + 424, + 326 + ], + "score": 0.76, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 316, + 505, + 329 + ], + "score": 1.0, + "content": "-RevNet (b) and its", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 326, + 505, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 137, + 340 + ], + "score": 1.0, + "content": "inverse", + "type": "text" + }, + { + "bbox": [ + 137, + 327, + 156, + 337 + ], + "score": 0.92, + "content": "\\Phi ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 326, + 403, + 340 + ], + "score": 1.0, + "content": ", as trained on ILSVRC-2012. We first explain why obtaining", + "type": "text" + }, + { + "bbox": [ + 403, + 327, + 422, + 338 + ], + "score": 0.9, + "content": "\\Phi ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 326, + 505, + 340 + ], + "score": 1.0, + "content": "is challenging, even", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 337, + 505, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 337, + 505, + 351 + ], + "score": 1.0, + "content": "locally. We then discuss the reconstruction, while displaying in the image space linear interpolations", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 349, + 207, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 207, + 362 + ], + "score": 1.0, + "content": "between representations.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 19.5 + }, + { + "type": "title", + "bbox": [ + 108, + 374, + 275, + 385 + ], + "lines": [ + { + "bbox": [ + 105, + 374, + 277, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 277, + 387 + ], + "score": 1.0, + "content": "4.1 AN ILL-CONDITIONED INVERSION", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 394, + 296, + 548 + ], + "lines": [ + { + "bbox": [ + 106, + 394, + 297, + 406 + ], + "spans": [ + { + "bbox": [ + 106, + 394, + 297, + 406 + ], + "score": 1.0, + "content": "In the previous section, we have described the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 404, + 297, + 418 + ], + "spans": [ + { + "bbox": [ + 106, + 406, + 111, + 415 + ], + "score": 0.75, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 111, + 404, + 297, + 418 + ], + "score": 1.0, + "content": "-RevNet architecture, that permits defining a", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 416, + 297, + 429 + ], + "spans": [ + { + "bbox": [ + 106, + 416, + 297, + 429 + ], + "score": 1.0, + "content": "deep network with an explicit inverse. We ex-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 426, + 297, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 297, + 441 + ], + "score": 1.0, + "content": "plain now why this is normally difficult, by", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 438, + 297, + 450 + ], + "spans": [ + { + "bbox": [ + 106, + 438, + 297, + 450 + ], + "score": 1.0, + "content": "studying its local inversion. 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Figure 4 corre-", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 622, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 466, + 635 + ], + "score": 1.0, + "content": "sponds to the singular values of the differential (i.e. the square roots of the eigen values of", + "type": "text" + }, + { + "bbox": [ + 466, + 622, + 498, + 632 + ], + "score": 0.9, + "content": "\\partial \\Phi ^ { * } \\partial \\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 499, + 622, + 505, + 635 + ], + "score": 1.0, + "content": "),", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 105, + 632, + 506, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 506, + 645 + ], + "score": 1.0, + "content": "in decreasing order, for a given natural image from ImageNet. The example we plot is typical of", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 106, + 643, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 169, + 656 + ], + "score": 1.0, + "content": "the behavior of", + "type": "text" + }, + { + "bbox": [ + 169, + 644, + 183, + 654 + ], + "score": 0.85, + "content": "\\partial \\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 644, + 392, + 656 + ], + "score": 1.0, + "content": ". 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This indicates", + "type": "text" + }, + { + "bbox": [ + 199, + 666, + 208, + 676 + ], + "score": 0.83, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "linearizes the space locally in a considerably smaller space in comparison", + "type": "text" + } + ], + "index": 58 + }, + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 442, + 689 + ], + "score": 1.0, + "content": "to the original input dimension. However, the dimensionality is still quite large (i.e.", + "type": "text" + }, + { + "bbox": [ + 442, + 677, + 465, + 688 + ], + "score": 0.83, + "content": "> 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 676, + 505, + 689 + ], + "score": 1.0, + "content": ") and thus", + "type": "text" + } + ], + "index": 59 + }, + { + "bbox": [ + 106, + 688, + 504, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 192, + 700 + ], + "score": 1.0, + "content": "we can not infer that", + "type": "text" + }, + { + "bbox": [ + 192, + 688, + 201, + 698 + ], + "score": 0.82, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 688, + 495, + 700 + ], + "score": 1.0, + "content": "lays locally in a low-dimensional manifold. It also proves that inversing", + "type": "text" + }, + { + "bbox": [ + 496, + 688, + 504, + 698 + ], + "score": 0.8, + "content": "\\Phi", + "type": "inline_equation" + } + ], + "index": 60 + }, + { + "bbox": [ + 105, + 699, + 505, + 710 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 710 + ], + "score": 1.0, + "content": "is difficult and is an ill-conditioned problem. 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Observe that the decrease of both", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "training-losses are very similar which indicates that the constraint of invertibility does not interfere", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 506, + 128 + ], + "score": 1.0, + "content": "negatively with the learning process. However, we observed one third longer wall-clock times for", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 126, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 111, + 136 + ], + "score": 0.66, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 111, + 126, + 505, + 139 + ], + "score": 1.0, + "content": "-RevNets compared to plain RevNets because the channel size becomes larger. The Table 1 reports", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 137, + 505, + 149 + ], + "spans": [ + { + "bbox": [ + 106, + 137, + 207, + 149 + ], + "score": 1.0, + "content": "the performances of our", + "type": "text" + }, + { + "bbox": [ + 208, + 138, + 212, + 147 + ], + "score": 0.73, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 137, + 505, + 149 + ], + "score": 1.0, + "content": "-RevNets, with comparable RevNet and ResNet. First, we compare the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 147, + 506, + 162 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 111, + 158 + ], + "score": 0.69, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 111, + 147, + 506, + 162 + ], + "score": 1.0, + "content": "-RevNet (a) with the RevNet and ResNet. Indeed, those CNNs have the same number of layers,", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 158, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 158, + 138, + 172 + ], + "score": 1.0, + "content": "and the", + "type": "text" + }, + { + "bbox": [ + 138, + 160, + 143, + 169 + ], + "score": 0.53, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 143, + 158, + 505, + 172 + ], + "score": 1.0, + "content": "-RevNet (a) increases the channel width of the initial layer as done in Gomez et al. 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On the contrary, the", + "type": "text" + }, + { + "bbox": [ + 394, + 199, + 399, + 208 + ], + "score": 0.59, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 199, + 506, + 210 + ], + "score": 1.0, + "content": "-RevNet (b) is designed to", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 505, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 505, + 221 + ], + "score": 1.0, + "content": "have roughly the same number of parameters as the RevNet and ResNet, while being bijective. Its", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 220, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 197, + 232 + ], + "score": 1.0, + "content": "accuracy decreases by", + "type": "text" + }, + { + "bbox": [ + 198, + 220, + 219, + 231 + ], + "score": 0.86, + "content": "1 . 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 220, + 505, + 232 + ], + "score": 1.0, + "content": "absolute percent on ImageNet compared to the RevNet baseline, which", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 232, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 505, + 243 + ], + "score": 1.0, + "content": "is not surprising because the number of channels was not drastically increased in the earlier layers", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 242, + 506, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 506, + 254 + ], + "score": 1.0, + "content": "as done in the baselines (Gomez et al., 2017; Krizhevsky et al., 2012; He et al., 2016); we did not", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 253, + 506, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 506, + 266 + ], + "score": 1.0, + "content": "explore wide ranges of hyper-parameters, thus the gap between (a) and (b) can likely be reduced", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 264, + 221, + 277 + ], + "spans": [ + { + "bbox": [ + 106, + 264, + 221, + 277 + ], + "score": 1.0, + "content": "with additional engineering.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 12.5, + "bbox_fs": [ + 105, + 187, + 506, + 277 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 292, + 265, + 303 + ], + "lines": [ + { + "bbox": [ + 105, + 291, + 267, + 305 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 267, + 305 + ], + "score": 1.0, + "content": "4 ANALYSIS OF THE INVERSE", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 316, + 505, + 360 + ], + "lines": [ + { + "bbox": [ + 105, + 316, + 505, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 253, + 329 + ], + "score": 1.0, + "content": "We now analyze the representation", + "type": "text" + }, + { + "bbox": [ + 254, + 317, + 262, + 326 + ], + "score": 0.83, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 316, + 419, + 329 + ], + "score": 1.0, + "content": "built by our bijective neural network", + "type": "text" + }, + { + "bbox": [ + 419, + 317, + 424, + 326 + ], + "score": 0.76, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 316, + 505, + 329 + ], + "score": 1.0, + "content": "-RevNet (b) and its", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 326, + 505, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 137, + 340 + ], + "score": 1.0, + "content": "inverse", + "type": "text" + }, + { + "bbox": [ + 137, + 327, + 156, + 337 + ], + "score": 0.92, + "content": "\\Phi ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 326, + 403, + 340 + ], + "score": 1.0, + "content": ", as trained on ILSVRC-2012. We first explain why obtaining", + "type": "text" + }, + { + "bbox": [ + 403, + 327, + 422, + 338 + ], + "score": 0.9, + "content": "\\Phi ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 326, + 505, + 340 + ], + "score": 1.0, + "content": "is challenging, even", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 337, + 505, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 337, + 505, + 351 + ], + "score": 1.0, + "content": "locally. We then discuss the reconstruction, while displaying in the image space linear interpolations", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 349, + 207, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 207, + 362 + ], + "score": 1.0, + "content": "between representations.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 19.5, + "bbox_fs": [ + 105, + 316, + 505, + 362 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 374, + 275, + 385 + ], + "lines": [ + { + "bbox": [ + 105, + 374, + 277, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 277, + 387 + ], + "score": 1.0, + "content": "4.1 AN ILL-CONDITIONED INVERSION", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 394, + 296, + 548 + ], + "lines": [ + { + "bbox": [ + 106, + 394, + 297, + 406 + ], + "spans": [ + { + "bbox": [ + 106, + 394, + 297, + 406 + ], + "score": 1.0, + "content": "In the previous section, we have described the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 404, + 297, + 418 + ], + "spans": [ + { + "bbox": [ + 106, + 406, + 111, + 415 + ], + "score": 0.75, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 111, + 404, + 297, + 418 + ], + "score": 1.0, + "content": "-RevNet architecture, that permits defining a", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 416, + 297, + 429 + ], + "spans": [ + { + "bbox": [ + 106, + 416, + 297, + 429 + ], + "score": 1.0, + "content": "deep network with an explicit inverse. We ex-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 426, + 297, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 297, + 441 + ], + "score": 1.0, + "content": "plain now why this is normally difficult, by", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 438, + 297, + 450 + ], + "spans": [ + { + "bbox": [ + 106, + 438, + 297, + 450 + ], + "score": 1.0, + "content": "studying its local inversion. We study the lo-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 449, + 295, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 208, + 461 + ], + "score": 1.0, + "content": "cal stability of a network", + "type": "text" + }, + { + "bbox": [ + 208, + 450, + 216, + 460 + ], + "score": 0.82, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 449, + 276, + 461 + ], + "score": 1.0, + "content": "and its inverse", + "type": "text" + }, + { + "bbox": [ + 277, + 449, + 295, + 460 + ], + "score": 0.89, + "content": "\\Phi ^ { - 1 }", + "type": "inline_equation" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 461, + 297, + 471 + ], + "spans": [ + { + "bbox": [ + 106, + 461, + 297, + 471 + ], + "score": 1.0, + "content": "w.r.t. to its input, which means that we will", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 472, + 297, + 483 + ], + "spans": [ + { + "bbox": [ + 106, + 472, + 297, + 483 + ], + "score": 1.0, + "content": "quantify locally the variations of the network", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 482, + 297, + 493 + ], + "spans": [ + { + "bbox": [ + 106, + 482, + 297, + 493 + ], + "score": 1.0, + "content": "and its inverse w.r.t. to small variations of an", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 493, + 298, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 147, + 505 + ], + "score": 1.0, + "content": "input. 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(2017).", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 50.5, + "bbox_fs": [ + 105, + 570, + 506, + 595 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 600, + 505, + 731 + ], + "lines": [ + { + "bbox": [ + 105, + 600, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 273, + 613 + ], + "score": 1.0, + "content": "In our numerical application to an image", + "type": "text" + }, + { + "bbox": [ + 274, + 602, + 281, + 610 + ], + "score": 0.45, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 600, + 285, + 613 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 285, + 600, + 304, + 611 + ], + "score": 0.75, + "content": "\\partial \\Phi _ { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 600, + 506, + 613 + ], + "score": 1.0, + "content": "corresponds to a very large matrix (square of the", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 611, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 505, + 623 + ], + "score": 1.0, + "content": "number of coefficients of the image at least) whose computations are expensive. Figure 4 corre-", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 622, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 466, + 635 + ], + "score": 1.0, + "content": "sponds to the singular values of the differential (i.e. the square roots of the eigen values of", + "type": "text" + }, + { + "bbox": [ + 466, + 622, + 498, + 632 + ], + "score": 0.9, + "content": "\\partial \\Phi ^ { * } \\partial \\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 499, + 622, + 505, + 635 + ], + "score": 1.0, + "content": "),", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 105, + 632, + 506, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 506, + 645 + ], + "score": 1.0, + "content": "in decreasing order, for a given natural image from ImageNet. The example we plot is typical of", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 106, + 643, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 169, + 656 + ], + "score": 1.0, + "content": "the behavior of", + "type": "text" + }, + { + "bbox": [ + 169, + 644, + 183, + 654 + ], + "score": 0.85, + "content": "\\partial \\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 644, + 392, + 656 + ], + "score": 1.0, + "content": ". Observe there is a fast decay: numerically, the first", + "type": "text" + }, + { + "bbox": [ + 392, + 643, + 408, + 654 + ], + "score": 0.9, + "content": "\\mathrm { 1 0 ^ { 3 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 644, + 425, + 656 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 426, + 644, + 442, + 654 + ], + "score": 0.88, + "content": "1 0 ^ { \\bar { 4 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 644, + 505, + 656 + ], + "score": 1.0, + "content": "singular values", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 105, + 654, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 232, + 668 + ], + "score": 1.0, + "content": "are responsible respectively for", + "type": "text" + }, + { + "bbox": [ + 233, + 655, + 253, + 666 + ], + "score": 0.89, + "content": "8 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 654, + 270, + 668 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 270, + 655, + 290, + 666 + ], + "score": 0.89, + "content": "9 7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 654, + 505, + 668 + ], + "score": 1.0, + "content": "of the cumulated energy (i.e. sum of squared singular", + "type": "text" + } + ], + "index": 57 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 199, + 678 + ], + "score": 1.0, + "content": "values). This indicates", + "type": "text" + }, + { + "bbox": [ + 199, + 666, + 208, + 676 + ], + "score": 0.83, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "linearizes the space locally in a considerably smaller space in comparison", + "type": "text" + } + ], + "index": 58 + }, + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 442, + 689 + ], + "score": 1.0, + "content": "to the original input dimension. However, the dimensionality is still quite large (i.e.", + "type": "text" + }, + { + "bbox": [ + 442, + 677, + 465, + 688 + ], + "score": 0.83, + "content": "> 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 676, + 505, + 689 + ], + "score": 1.0, + "content": ") and thus", + "type": "text" + } + ], + "index": 59 + }, + { + "bbox": [ + 106, + 688, + 504, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 192, + 700 + ], + "score": 1.0, + "content": "we can not infer that", + "type": "text" + }, + { + "bbox": [ + 192, + 688, + 201, + 698 + ], + "score": 0.82, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 688, + 495, + 700 + ], + "score": 1.0, + "content": "lays locally in a low-dimensional manifold. It also proves that inversing", + "type": "text" + }, + { + "bbox": [ + 496, + 688, + 504, + 698 + ], + "score": 0.8, + "content": "\\Phi", + "type": "inline_equation" + } + ], + "index": 60 + }, + { + "bbox": [ + 105, + 699, + 505, + 710 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 710 + ], + "score": 1.0, + "content": "is difficult and is an ill-conditioned problem. Thus obtaining implicitly this inverse would be a chal-", + "type": "text" + } + ], + "index": 61 + }, + { + "bbox": [ + 104, + 709, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 104, + 709, + 505, + 723 + ], + "score": 1.0, + "content": "lenging task that we avoided, thanks to the formal reconstruction algorithm provided by Subsection", + "type": "text" + } + ], + "index": 62 + }, + { + "bbox": [ + 105, + 720, + 126, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 126, + 732 + ], + "score": 1.0, + "content": "3.1.", + "type": "text" + } + ], + "index": 63 + } + ], + "index": 57.5, + "bbox_fs": [ + 104, + 600, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 106, + 64, + 505, + 288 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 106, + 64, + 505, + 288 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 64, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 106, + 64, + 505, + 288 + ], + "score": 0.97, + "type": "image", + "image_path": "a6fe6b5bd4ff7c42fa5cbd145f44da0d3e48b046022b4d75069d5e7eefe3144c.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 106, + 64, + 505, + 138.66666666666669 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 106, + 138.66666666666669, + 505, + 213.33333333333337 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 106, + 213.33333333333337, + 505, + 288.00000000000006 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 107, + 297, + 503, + 320 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 295, + 506, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 365, + 311 + ], + "score": 1.0, + "content": "Figure 5: This graphic displays several reconstructed sequences", + "type": "text" + }, + { + "bbox": [ + 365, + 297, + 388, + 309 + ], + "score": 0.92, + "content": "\\{ x ^ { t } \\} _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 295, + 506, + 311 + ], + "score": 1.0, + "content": ". 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One approach to reconstruct from an output layer consists in finding the input image that", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 396, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 505, + 410 + ], + "score": 1.0, + "content": "matches the activation through via gradient descent. However, this technique leads only to a partial", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 407, + 338, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 407, + 338, + 419 + ], + "score": 1.0, + "content": "or informal reconstruction (Mahendran & Vedaldi, 2015).", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 106, + 424, + 505, + 479 + ], + "lines": [ + { + "bbox": [ + 106, + 424, + 505, + 437 + ], + "spans": [ + { + "bbox": [ + 106, + 424, + 505, + 437 + ], + "score": 1.0, + "content": "Another method consists in embedding the representation in a lower dimensional space and com-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 435, + 505, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 505, + 448 + ], + "score": 1.0, + "content": "paring the common attributes of nearest neighbors (Szegedy et al., 2013). It is also possible to train", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 445, + 506, + 459 + ], + "spans": [ + { + "bbox": [ + 104, + 445, + 506, + 459 + ], + "score": 1.0, + "content": "a CNN to reconstruct the representation (Dosovitskiy & Brox, 2016). Yet these methods require a", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 456, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 505, + 471 + ], + "score": 1.0, + "content": "priori knowledge in order to find the appropriate embeddings or training sets. We now discuss the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 468, + 272, + 480 + ], + "spans": [ + { + "bbox": [ + 106, + 468, + 230, + 480 + ], + "score": 1.0, + "content": "improvements achieved by the", + "type": "text" + }, + { + "bbox": [ + 230, + 469, + 235, + 478 + ], + "score": 0.7, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 468, + 272, + 480 + ], + "score": 1.0, + "content": "-RevNet.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 106, + 484, + 505, + 551 + ], + "lines": [ + { + "bbox": [ + 106, + 484, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 106, + 484, + 427, + 497 + ], + "score": 1.0, + "content": "Our main claim is that while the local inversion is ill-conditioned, the inverse", + "type": "text" + }, + { + "bbox": [ + 428, + 484, + 447, + 495 + ], + "score": 0.9, + "content": "\\Phi ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 484, + 505, + 497 + ], + "score": 1.0, + "content": "computations", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 495, + 506, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 506, + 509 + ], + "score": 1.0, + "content": "do not involve significant round-off errors. The forward pass of the network does not seem to", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 506, + 506, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 473, + 519 + ], + "score": 1.0, + "content": "suffer from significant instabilities, thus it seems coherent to assume that this will hold for", + "type": "text" + }, + { + "bbox": [ + 474, + 506, + 492, + 517 + ], + "score": 0.9, + "content": "\\Phi ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 506, + 506, + 519 + ], + "score": 1.0, + "content": "as", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 518, + 505, + 530 + ], + "spans": [ + { + "bbox": [ + 106, + 518, + 505, + 530 + ], + "score": 1.0, + "content": "well. For example, adding constraints beyond vanishing moments in the case of a Lifting scheme", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 528, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 505, + 541 + ], + "score": 1.0, + "content": "is difficult (Sweldens, 1998; Mallat, 1999), and this is a weakness of this method. We validate our", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 540, + 415, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 371, + 551 + ], + "score": 1.0, + "content": "claim by computing the empirical relative error on several subsets", + "type": "text" + }, + { + "bbox": [ + 371, + 540, + 381, + 550 + ], + "score": 0.84, + "content": "\\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 541, + 415, + 551 + ], + "score": 1.0, + "content": "of data:", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 18.5 + }, + { + "type": "interline_equation", + "bbox": [ + 238, + 565, + 372, + 597 + ], + "lines": [ + { + "bbox": [ + 238, + 565, + 372, + 597 + ], + "spans": [ + { + "bbox": [ + 238, + 565, + 372, + 597 + ], + "score": 0.94, + "content": "\\epsilon ( \\mathcal { X } ) = \\frac { 1 } { | \\mathcal { X } | } \\sum _ { \\boldsymbol { x } \\in \\mathcal { X } } \\frac { \\| \\boldsymbol { x } - \\Phi ^ { - 1 } \\Phi \\boldsymbol { x } \\| } { \\| \\boldsymbol { x } \\| }", + "type": "interline_equation", + "image_path": "faf617bbc82767bf83b06069a10c7f59321cf2682e02e0004f4de34f1200046d.jpg" + } + ] + } + ], + "index": 22.5, + "virtual_lines": [ + { + "bbox": [ + 238, + 565, + 372, + 581.0 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 238, + 581.0, + 372, + 597.0 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 608, + 505, + 653 + ], + "lines": [ + { + "bbox": [ + 105, + 607, + 505, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 607, + 263, + 621 + ], + "score": 1.0, + "content": "We evaluate this measure on a subset", + "type": "text" + }, + { + "bbox": [ + 264, + 609, + 276, + 619 + ], + "score": 0.89, + "content": "\\mathcal { X } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 607, + 291, + 621 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 291, + 608, + 340, + 620 + ], + "score": 0.91, + "content": "| \\mathcal { X } _ { 1 } | = 1 0 ^ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 607, + 505, + 621 + ], + "score": 1.0, + "content": "independent uniform noises and on the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 618, + 506, + 632 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 163, + 632 + ], + "score": 1.0, + "content": "validation set", + "type": "text" + }, + { + "bbox": [ + 163, + 620, + 176, + 631 + ], + "score": 0.88, + "content": "\\mathcal { X } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 618, + 277, + 632 + ], + "score": 1.0, + "content": "of ImageNet. We report", + "type": "text" + }, + { + "bbox": [ + 277, + 620, + 354, + 632 + ], + "score": 0.9, + "content": "\\epsilon ( \\mathcal { X } _ { 1 } ) = 5 \\times 1 0 ^ { - 6 }", + "type": "inline_equation" + }, + { + "bbox": [ + 355, + 618, + 374, + 632 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 374, + 619, + 451, + 632 + ], + "score": 0.92, + "content": "\\epsilon ( \\mathcal { X } _ { 2 } ) = 3 \\times 1 0 ^ { - 6 }", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 618, + 506, + 632 + ], + "score": 1.0, + "content": "respectively,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 630, + 505, + 642 + ], + "spans": [ + { + "bbox": [ + 106, + 630, + 505, + 642 + ], + "score": 1.0, + "content": "which are close to the machine error and indicates that the inversion does not suffer from significant", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 641, + 175, + 653 + ], + "spans": [ + { + "bbox": [ + 106, + 641, + 175, + 653 + ], + "score": 1.0, + "content": "round-off errors.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25.5 + }, + { + "type": "text", + "bbox": [ + 106, + 658, + 505, + 702 + ], + "lines": [ + { + "bbox": [ + 105, + 656, + 507, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 656, + 206, + 672 + ], + "score": 1.0, + "content": "Given a pair of images", + "type": "text" + }, + { + "bbox": [ + 207, + 658, + 242, + 670 + ], + "score": 0.94, + "content": "\\{ x ^ { 0 } , x ^ { 1 } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 656, + 507, + 672 + ], + "score": 1.0, + "content": ", we propose to study linear interpolations between the pair of", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 104, + 668, + 506, + 683 + ], + "spans": [ + { + "bbox": [ + 104, + 668, + 171, + 683 + ], + "score": 1.0, + "content": "representations", + "type": "text" + }, + { + "bbox": [ + 171, + 669, + 221, + 681 + ], + "score": 0.91, + "content": "\\{ \\Phi x ^ { \\bar { 0 } } , \\Phi x ^ { \\mathrm { { 1 } } } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 668, + 506, + 683 + ], + "score": 1.0, + "content": ", in the feature domain. Those interpolations correspond to existing", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 680, + 506, + 693 + ], + "spans": [ + { + "bbox": [ + 105, + 680, + 147, + 693 + ], + "score": 1.0, + "content": "images as", + "type": "text" + }, + { + "bbox": [ + 148, + 680, + 167, + 690 + ], + "score": 0.91, + "content": "\\Phi ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 680, + 506, + 693 + ], + "score": 1.0, + "content": "is an exact inverse. 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The left image corresponds", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 307, + 235, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 117, + 321 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 117, + 308, + 128, + 318 + ], + "score": 0.87, + "content": "{ \\bar { \\mathbf { \\Gamma } } } _ { x } 0", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 307, + 219, + 321 + ], + "score": 1.0, + "content": "and the right image to", + "type": "text" + }, + { + "bbox": [ + 219, + 308, + 230, + 318 + ], + "score": 0.87, + "content": "x ^ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 307, + 235, + 321 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "title", + "bbox": [ + 106, + 343, + 339, + 353 + ], + "lines": [ + { + "bbox": [ + 106, + 342, + 340, + 354 + ], + "spans": [ + { + "bbox": [ + 106, + 342, + 340, + 354 + ], + "score": 1.0, + "content": "4.2 LINEAR INTERPOLATION AND RECONSTRUCTION", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 106, + 363, + 505, + 419 + ], + "lines": [ + { + "bbox": [ + 107, + 364, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 107, + 364, + 505, + 376 + ], + "score": 1.0, + "content": "Visualizing or understanding the important directions in the representation of inner layers of a CNN,", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 375, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 106, + 375, + 505, + 387 + ], + "score": 1.0, + "content": "and in particular, the final layer is complex because typically the cascade is either not invertible or", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 385, + 506, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 506, + 398 + ], + "score": 1.0, + "content": "unstable. One approach to reconstruct from an output layer consists in finding the input image that", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 396, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 505, + 410 + ], + "score": 1.0, + "content": "matches the activation through via gradient descent. However, this technique leads only to a partial", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 407, + 338, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 407, + 338, + 419 + ], + "score": 1.0, + "content": "or informal reconstruction (Mahendran & Vedaldi, 2015).", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 8, + "bbox_fs": [ + 105, + 364, + 506, + 419 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 424, + 505, + 479 + ], + "lines": [ + { + "bbox": [ + 106, + 424, + 505, + 437 + ], + "spans": [ + { + "bbox": [ + 106, + 424, + 505, + 437 + ], + "score": 1.0, + "content": "Another method consists in embedding the representation in a lower dimensional space and com-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 435, + 505, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 505, + 448 + ], + "score": 1.0, + "content": "paring the common attributes of nearest neighbors (Szegedy et al., 2013). It is also possible to train", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 445, + 506, + 459 + ], + "spans": [ + { + "bbox": [ + 104, + 445, + 506, + 459 + ], + "score": 1.0, + "content": "a CNN to reconstruct the representation (Dosovitskiy & Brox, 2016). Yet these methods require a", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 456, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 505, + 471 + ], + "score": 1.0, + "content": "priori knowledge in order to find the appropriate embeddings or training sets. We now discuss the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 468, + 272, + 480 + ], + "spans": [ + { + "bbox": [ + 106, + 468, + 230, + 480 + ], + "score": 1.0, + "content": "improvements achieved by the", + "type": "text" + }, + { + "bbox": [ + 230, + 469, + 235, + 478 + ], + "score": 0.7, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 468, + 272, + 480 + ], + "score": 1.0, + "content": "-RevNet.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13, + "bbox_fs": [ + 104, + 424, + 506, + 480 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 484, + 505, + 551 + ], + "lines": [ + { + "bbox": [ + 106, + 484, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 106, + 484, + 427, + 497 + ], + "score": 1.0, + "content": "Our main claim is that while the local inversion is ill-conditioned, the inverse", + "type": "text" + }, + { + "bbox": [ + 428, + 484, + 447, + 495 + ], + "score": 0.9, + "content": "\\Phi ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 484, + 505, + 497 + ], + "score": 1.0, + "content": "computations", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 495, + 506, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 506, + 509 + ], + "score": 1.0, + "content": "do not involve significant round-off errors. The forward pass of the network does not seem to", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 506, + 506, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 473, + 519 + ], + "score": 1.0, + "content": "suffer from significant instabilities, thus it seems coherent to assume that this will hold for", + "type": "text" + }, + { + "bbox": [ + 474, + 506, + 492, + 517 + ], + "score": 0.9, + "content": "\\Phi ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 506, + 506, + 519 + ], + "score": 1.0, + "content": "as", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 518, + 505, + 530 + ], + "spans": [ + { + "bbox": [ + 106, + 518, + 505, + 530 + ], + "score": 1.0, + "content": "well. For example, adding constraints beyond vanishing moments in the case of a Lifting scheme", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 528, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 505, + 541 + ], + "score": 1.0, + "content": "is difficult (Sweldens, 1998; Mallat, 1999), and this is a weakness of this method. We validate our", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 540, + 415, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 371, + 551 + ], + "score": 1.0, + "content": "claim by computing the empirical relative error on several subsets", + "type": "text" + }, + { + "bbox": [ + 371, + 540, + 381, + 550 + ], + "score": 0.84, + "content": "\\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 541, + 415, + 551 + ], + "score": 1.0, + "content": "of data:", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 18.5, + "bbox_fs": [ + 105, + 484, + 506, + 551 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 238, + 565, + 372, + 597 + ], + "lines": [ + { + "bbox": [ + 238, + 565, + 372, + 597 + ], + "spans": [ + { + "bbox": [ + 238, + 565, + 372, + 597 + ], + "score": 0.94, + "content": "\\epsilon ( \\mathcal { X } ) = \\frac { 1 } { | \\mathcal { X } | } \\sum _ { \\boldsymbol { x } \\in \\mathcal { X } } \\frac { \\| \\boldsymbol { x } - \\Phi ^ { - 1 } \\Phi \\boldsymbol { x } \\| } { \\| \\boldsymbol { x } \\| }", + "type": "interline_equation", + "image_path": "faf617bbc82767bf83b06069a10c7f59321cf2682e02e0004f4de34f1200046d.jpg" + } + ] + } + ], + "index": 22.5, + "virtual_lines": [ + { + "bbox": [ + 238, + 565, + 372, + 581.0 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 238, + 581.0, + 372, + 597.0 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 608, + 505, + 653 + ], + "lines": [ + { + "bbox": [ + 105, + 607, + 505, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 607, + 263, + 621 + ], + "score": 1.0, + "content": "We evaluate this measure on a subset", + "type": "text" + }, + { + "bbox": [ + 264, + 609, + 276, + 619 + ], + "score": 0.89, + "content": "\\mathcal { X } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 607, + 291, + 621 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 291, + 608, + 340, + 620 + ], + "score": 0.91, + "content": "| \\mathcal { X } _ { 1 } | = 1 0 ^ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 607, + 505, + 621 + ], + "score": 1.0, + "content": "independent uniform noises and on the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 618, + 506, + 632 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 163, + 632 + ], + "score": 1.0, + "content": "validation set", + "type": "text" + }, + { + "bbox": [ + 163, + 620, + 176, + 631 + ], + "score": 0.88, + "content": "\\mathcal { X } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 618, + 277, + 632 + ], + "score": 1.0, + "content": "of ImageNet. We report", + "type": "text" + }, + { + "bbox": [ + 277, + 620, + 354, + 632 + ], + "score": 0.9, + "content": "\\epsilon ( \\mathcal { X } _ { 1 } ) = 5 \\times 1 0 ^ { - 6 }", + "type": "inline_equation" + }, + { + "bbox": [ + 355, + 618, + 374, + 632 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 374, + 619, + 451, + 632 + ], + "score": 0.92, + "content": "\\epsilon ( \\mathcal { X } _ { 2 } ) = 3 \\times 1 0 ^ { - 6 }", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 618, + 506, + 632 + ], + "score": 1.0, + "content": "respectively,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 630, + 505, + 642 + ], + "spans": [ + { + "bbox": [ + 106, + 630, + 505, + 642 + ], + "score": 1.0, + "content": "which are close to the machine error and indicates that the inversion does not suffer from significant", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 641, + 175, + 653 + ], + "spans": [ + { + "bbox": [ + 106, + 641, + 175, + 653 + ], + "score": 1.0, + "content": "round-off errors.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25.5, + "bbox_fs": [ + 105, + 607, + 506, + 653 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 658, + 505, + 702 + ], + "lines": [ + { + "bbox": [ + 105, + 656, + 507, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 656, + 206, + 672 + ], + "score": 1.0, + "content": "Given a pair of images", + "type": "text" + }, + { + "bbox": [ + 207, + 658, + 242, + 670 + ], + "score": 0.94, + "content": "\\{ x ^ { 0 } , x ^ { 1 } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 656, + 507, + 672 + ], + "score": 1.0, + "content": ", we propose to study linear interpolations between the pair of", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 104, + 668, + 506, + 683 + ], + "spans": [ + { + "bbox": [ + 104, + 668, + 171, + 683 + ], + "score": 1.0, + "content": "representations", + "type": "text" + }, + { + "bbox": [ + 171, + 669, + 221, + 681 + ], + "score": 0.91, + "content": "\\{ \\Phi x ^ { \\bar { 0 } } , \\Phi x ^ { \\mathrm { { 1 } } } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 668, + 506, + 683 + ], + "score": 1.0, + "content": ", in the feature domain. Those interpolations correspond to existing", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 680, + 506, + 693 + ], + "spans": [ + { + "bbox": [ + 105, + 680, + 147, + 693 + ], + "score": 1.0, + "content": "images as", + "type": "text" + }, + { + "bbox": [ + 148, + 680, + 167, + 690 + ], + "score": 0.91, + "content": "\\Phi ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 680, + 506, + 693 + ], + "score": 1.0, + "content": "is an exact inverse. We reconstruct a convex path between two input points; it means", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 690, + 136, + 703 + ], + "spans": [ + { + "bbox": [ + 105, + 690, + 136, + 703 + ], + "score": 1.0, + "content": "that if:", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29.5, + "bbox_fs": [ + 104, + 656, + 507, + 703 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 252, + 702, + 358, + 716 + ], + "lines": [ + { + "bbox": [ + 252, + 702, + 358, + 716 + ], + "spans": [ + { + "bbox": [ + 252, + 702, + 358, + 716 + ], + "score": 0.91, + "content": "\\phi ^ { t } = t \\Phi x ^ { 0 } + ( 1 - t ) \\Phi x ^ { 1 } ,", + "type": "interline_equation", + "image_path": "b0fc9ca852c1841fc195fdab2bbbe01c79b13bcab5247de6e4056b08676f6395.jpg" + } + ] + } + ], + "index": 32, + "virtual_lines": [ + { + "bbox": [ + 252, + 702, + 358, + 716 + ], + "spans": [], + "index": 32 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 720, + 343, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 718, + 344, + 735 + ], + "spans": [ + { + "bbox": [ + 105, + 718, + 129, + 735 + ], + "score": 1.0, + "content": "then:", + "type": "text" + }, + { + "bbox": [ + 129, + 720, + 181, + 732 + ], + "score": 0.93, + "content": "x ^ { t } = \\Phi ^ { - 1 } \\phi ^ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 718, + 344, + 735 + ], + "score": 1.0, + "content": "is a signal that corresponds to an image.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 718, + 344, + 735 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 119, + 62, + 492, + 218 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 119, + 62, + 492, + 218 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 119, + 62, + 492, + 218 + ], + "spans": [ + { + "bbox": [ + 119, + 62, + 492, + 218 + ], + "score": 0.971, + "type": "image", + "image_path": "293bac6aae019b36fb6785db214dbbee593d7a2ee2f0f04de109fb47c7cb0a9a.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 119, + 62, + 492, + 114.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 119, + 114.0, + 492, + 166.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 119, + 166.0, + 492, + 218.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 227, + 503, + 250 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 226, + 505, + 240 + ], + "spans": [ + { + "bbox": [ + 106, + 226, + 224, + 240 + ], + "score": 1.0, + "content": "Figure 6: Accuracy at depth", + "type": "text" + }, + { + "bbox": [ + 225, + 228, + 231, + 239 + ], + "score": 0.83, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 226, + 505, + 240 + ], + "score": 1.0, + "content": "for a linear SVM and a 1-nearest neighbor classifier applied to the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 237, + 198, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 181, + 252 + ], + "score": 1.0, + "content": "spatially averaged", + "type": "text" + }, + { + "bbox": [ + 181, + 239, + 193, + 251 + ], + "score": 0.89, + "content": "\\Phi _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 237, + 198, + 252 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "text", + "bbox": [ + 107, + 275, + 505, + 308 + ], + "lines": [ + { + "bbox": [ + 105, + 273, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 273, + 170, + 288 + ], + "score": 1.0, + "content": "We discretized", + "type": "text" + }, + { + "bbox": [ + 171, + 275, + 191, + 287 + ], + "score": 0.48, + "content": "[ 0 , 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 273, + 213, + 288 + ], + "score": 1.0, + "content": "into", + "type": "text" + }, + { + "bbox": [ + 213, + 275, + 258, + 287 + ], + "score": 0.94, + "content": "\\{ t _ { 1 } , . . . , t _ { k } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 273, + 505, + 288 + ], + "score": 1.0, + "content": ", adapt the step size manually and reconstruct the sequence", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 284, + 506, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 118, + 299 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 119, + 286, + 173, + 298 + ], + "score": 0.93, + "content": "\\{ x ^ { t _ { 1 } } , . . . , x ^ { t _ { k } } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 284, + 506, + 299 + ], + "score": 1.0, + "content": ". Results are displayed in the Figure 5. We selected images from the basel face", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 296, + 475, + 310 + ], + "spans": [ + { + "bbox": [ + 105, + 296, + 475, + 310 + ], + "score": 1.0, + "content": "dataset (Paysan et al., 2009), describable texture dataset (Cimpoi et al., 2014) and imagenet.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 107, + 314, + 505, + 369 + ], + "lines": [ + { + "bbox": [ + 106, + 314, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 106, + 314, + 505, + 326 + ], + "score": 1.0, + "content": "We now interpret the results. First, observe that a linear interpolation in the feature space is not", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 325, + 505, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 505, + 338 + ], + "score": 1.0, + "content": "a linear interpolation in the image space and that intermediary images are noisy, even for small", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 336, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 106, + 336, + 505, + 348 + ], + "score": 1.0, + "content": "deformations, yet they mostly remain recognizable. However, some geometric transformations such", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 347, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 505, + 360 + ], + "score": 1.0, + "content": "as a 3D-rotation seem to have been linearized, as suggested in Aubry & Russell (2015). In the next", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 357, + 411, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 411, + 371 + ], + "score": 1.0, + "content": "section, we thus investigate how the linear separation progresses with depth.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10 + }, + { + "type": "title", + "bbox": [ + 108, + 389, + 213, + 402 + ], + "lines": [ + { + "bbox": [ + 105, + 388, + 214, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 214, + 403 + ], + "score": 1.0, + "content": "5 A CONTRACTION", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 107, + 416, + 505, + 460 + ], + "lines": [ + { + "bbox": [ + 105, + 416, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 292, + 429 + ], + "score": 1.0, + "content": "In this section, we study again the bijective", + "type": "text" + }, + { + "bbox": [ + 292, + 417, + 297, + 426 + ], + "score": 0.69, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 416, + 505, + 429 + ], + "score": 1.0, + "content": "-RevNet. We first show that a localized or linear", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 427, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 505, + 439 + ], + "score": 1.0, + "content": "classifier progressively improves with depth. Then, we describe the linear subspace spanned by", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 438, + 506, + 451 + ], + "spans": [ + { + "bbox": [ + 106, + 439, + 115, + 449 + ], + "score": 0.76, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 115, + 438, + 506, + 451 + ], + "score": 1.0, + "content": ", namely the feature space, showing that the classification can be performed on a much smaller", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 450, + 269, + 461 + ], + "spans": [ + { + "bbox": [ + 106, + 450, + 269, + 461 + ], + "score": 1.0, + "content": "subspace, which can be built via a PCA.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15.5 + }, + { + "type": "title", + "bbox": [ + 112, + 479, + 365, + 489 + ], + "lines": [ + { + "bbox": [ + 110, + 478, + 367, + 491 + ], + "spans": [ + { + "bbox": [ + 110, + 478, + 367, + 491 + ], + "score": 1.0, + "content": "5.1 PROGRESSIVE LINEAR SEPARATION AND CONTRACTION", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 500, + 505, + 533 + ], + "lines": [ + { + "bbox": [ + 106, + 500, + 505, + 512 + ], + "spans": [ + { + "bbox": [ + 106, + 500, + 259, + 512 + ], + "score": 1.0, + "content": "We show that both a ResNet and an", + "type": "text" + }, + { + "bbox": [ + 259, + 501, + 264, + 510 + ], + "score": 0.64, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 500, + 505, + 512 + ], + "score": 1.0, + "content": "-RevNet build a progressively more linearly separable and", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 511, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 496, + 523 + ], + "score": 1.0, + "content": "contracted representation as measured in Oyallon (2017). Observe this property holds for the", + "type": "text" + }, + { + "bbox": [ + 496, + 512, + 500, + 521 + ], + "score": 0.71, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 511, + 505, + 523 + ], + "score": 1.0, + "content": "-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 522, + 359, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 359, + 535 + ], + "score": 1.0, + "content": "RevNet despite the fact that it can not discard any information.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 106, + 539, + 505, + 649 + ], + "lines": [ + { + "bbox": [ + 106, + 538, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 106, + 538, + 505, + 552 + ], + "score": 1.0, + "content": "We investigate these properties in each block, with the following experimental protocol. To reduce", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 550, + 505, + 562 + ], + "spans": [ + { + "bbox": [ + 106, + 550, + 505, + 562 + ], + "score": 1.0, + "content": "the computational burden we used a subset of 100 randomly selected imagenet classes, that consist", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 560, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 117, + 574 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 117, + 561, + 161, + 572 + ], + "score": 0.91, + "content": "N = 1 2 0 k", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 560, + 505, + 574 + ], + "score": 1.0, + "content": "images, and keep the same subset during all our following experiments. At each depth", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 572, + 506, + 586 + ], + "spans": [ + { + "bbox": [ + 106, + 573, + 112, + 584 + ], + "score": 0.72, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 113, + 572, + 207, + 586 + ], + "score": 1.0, + "content": ", we extract the features", + "type": "text" + }, + { + "bbox": [ + 208, + 572, + 259, + 585 + ], + "score": 0.93, + "content": "\\{ \\Phi _ { j } x ^ { n } \\} _ { n \\leq N }", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 572, + 506, + 586 + ], + "score": 1.0, + "content": "of the training set, we average them along the spatial variable", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 583, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 505, + 595 + ], + "score": 1.0, + "content": "and standardize them in order to avoid any ill-conditioning effects. We used both a nearest neighbor", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 593, + 506, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 455, + 606 + ], + "score": 1.0, + "content": "classifier and a linear SVM. The former is a localized classifier that indicates that the", + "type": "text" + }, + { + "bbox": [ + 456, + 594, + 465, + 604 + ], + "score": 0.82, + "content": "\\ell ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 593, + 506, + 606 + ], + "score": 1.0, + "content": "metric is", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 605, + 506, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 506, + 617 + ], + "score": 1.0, + "content": "progressively more important for classification, while a linear SVM measures the linear separation", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 616, + 506, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 506, + 628 + ], + "score": 1.0, + "content": "of the different classes. The parameters of the linear SVM are cross-validated on a small subset of", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 627, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 506, + 640 + ], + "score": 1.0, + "content": "the training set, prior to training on the 100 classes. We evaluate both classifiers for each model on", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 637, + 399, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 637, + 399, + 650 + ], + "score": 1.0, + "content": "the validation set of ImageNet and report the Top-1 accuracy in Figure 6.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 26.5 + }, + { + "type": "text", + "bbox": [ + 107, + 654, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "score": 1.0, + "content": "We observe that both classifiers progressively improve similarly with depth for each model, the lin-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 665, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 505, + 677 + ], + "score": 1.0, + "content": "ear SVM performing slightly better than the nearest neighbor classifier because it is the more robust", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 337, + 690 + ], + "score": 1.0, + "content": "and discriminative classifier of the two. In the case of the", + "type": "text" + }, + { + "bbox": [ + 338, + 678, + 342, + 687 + ], + "score": 0.75, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "-RevNet, the classification performed by", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 180, + 700 + ], + "score": 1.0, + "content": "the CNN leads to", + "type": "text" + }, + { + "bbox": [ + 181, + 687, + 200, + 699 + ], + "score": 0.88, + "content": "7 7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 687, + 506, + 700 + ], + "score": 1.0, + "content": ", and the linear SVM performs slightly better because we did not fine-tune", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "the model to 100 classes. Observe that there is a more intense jump of performance on the 3 last", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "score": 1.0, + "content": "layers, which seems to indicate that the former layers have prepared the representation to be more", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 720, + 318, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 318, + 733 + ], + "score": 1.0, + "content": "contracted and linearly separated for the final layers.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 35 + } + ], + "page_idx": 7, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 761 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 761 + ], + "score": 1.0, + "content": "8", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 119, + 62, + 492, + 218 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 119, + 62, + 492, + 218 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 119, + 62, + 492, + 218 + ], + "spans": [ + { + "bbox": [ + 119, + 62, + 492, + 218 + ], + "score": 0.971, + "type": "image", + "image_path": "293bac6aae019b36fb6785db214dbbee593d7a2ee2f0f04de109fb47c7cb0a9a.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 119, + 62, + 492, + 114.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 119, + 114.0, + 492, + 166.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 119, + 166.0, + 492, + 218.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 227, + 503, + 250 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 226, + 505, + 240 + ], + "spans": [ + { + "bbox": [ + 106, + 226, + 224, + 240 + ], + "score": 1.0, + "content": "Figure 6: Accuracy at depth", + "type": "text" + }, + { + "bbox": [ + 225, + 228, + 231, + 239 + ], + "score": 0.83, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 226, + 505, + 240 + ], + "score": 1.0, + "content": "for a linear SVM and a 1-nearest neighbor classifier applied to the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 237, + 198, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 181, + 252 + ], + "score": 1.0, + "content": "spatially averaged", + "type": "text" + }, + { + "bbox": [ + 181, + 239, + 193, + 251 + ], + "score": 0.89, + "content": "\\Phi _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 237, + 198, + 252 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "text", + "bbox": [ + 107, + 275, + 505, + 308 + ], + "lines": [ + { + "bbox": [ + 105, + 273, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 273, + 170, + 288 + ], + "score": 1.0, + "content": "We discretized", + "type": "text" + }, + { + "bbox": [ + 171, + 275, + 191, + 287 + ], + "score": 0.48, + "content": "[ 0 , 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 273, + 213, + 288 + ], + "score": 1.0, + "content": "into", + "type": "text" + }, + { + "bbox": [ + 213, + 275, + 258, + 287 + ], + "score": 0.94, + "content": "\\{ t _ { 1 } , . . . , t _ { k } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 273, + 505, + 288 + ], + "score": 1.0, + "content": ", adapt the step size manually and reconstruct the sequence", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 284, + 506, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 118, + 299 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 119, + 286, + 173, + 298 + ], + "score": 0.93, + "content": "\\{ x ^ { t _ { 1 } } , . . . , x ^ { t _ { k } } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 284, + 506, + 299 + ], + "score": 1.0, + "content": ". Results are displayed in the Figure 5. We selected images from the basel face", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 296, + 475, + 310 + ], + "spans": [ + { + "bbox": [ + 105, + 296, + 475, + 310 + ], + "score": 1.0, + "content": "dataset (Paysan et al., 2009), describable texture dataset (Cimpoi et al., 2014) and imagenet.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6, + "bbox_fs": [ + 105, + 273, + 506, + 310 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 314, + 505, + 369 + ], + "lines": [ + { + "bbox": [ + 106, + 314, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 106, + 314, + 505, + 326 + ], + "score": 1.0, + "content": "We now interpret the results. First, observe that a linear interpolation in the feature space is not", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 325, + 505, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 505, + 338 + ], + "score": 1.0, + "content": "a linear interpolation in the image space and that intermediary images are noisy, even for small", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 336, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 106, + 336, + 505, + 348 + ], + "score": 1.0, + "content": "deformations, yet they mostly remain recognizable. However, some geometric transformations such", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 347, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 505, + 360 + ], + "score": 1.0, + "content": "as a 3D-rotation seem to have been linearized, as suggested in Aubry & Russell (2015). In the next", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 357, + 411, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 411, + 371 + ], + "score": 1.0, + "content": "section, we thus investigate how the linear separation progresses with depth.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10, + "bbox_fs": [ + 105, + 314, + 505, + 371 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 389, + 213, + 402 + ], + "lines": [ + { + "bbox": [ + 105, + 388, + 214, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 214, + 403 + ], + "score": 1.0, + "content": "5 A CONTRACTION", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 107, + 416, + 505, + 460 + ], + "lines": [ + { + "bbox": [ + 105, + 416, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 292, + 429 + ], + "score": 1.0, + "content": "In this section, we study again the bijective", + "type": "text" + }, + { + "bbox": [ + 292, + 417, + 297, + 426 + ], + "score": 0.69, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 416, + 505, + 429 + ], + "score": 1.0, + "content": "-RevNet. We first show that a localized or linear", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 427, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 505, + 439 + ], + "score": 1.0, + "content": "classifier progressively improves with depth. Then, we describe the linear subspace spanned by", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 438, + 506, + 451 + ], + "spans": [ + { + "bbox": [ + 106, + 439, + 115, + 449 + ], + "score": 0.76, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 115, + 438, + 506, + 451 + ], + "score": 1.0, + "content": ", namely the feature space, showing that the classification can be performed on a much smaller", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 450, + 269, + 461 + ], + "spans": [ + { + "bbox": [ + 106, + 450, + 269, + 461 + ], + "score": 1.0, + "content": "subspace, which can be built via a PCA.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15.5, + "bbox_fs": [ + 105, + 416, + 506, + 461 + ] + }, + { + "type": "title", + "bbox": [ + 112, + 479, + 365, + 489 + ], + "lines": [ + { + "bbox": [ + 110, + 478, + 367, + 491 + ], + "spans": [ + { + "bbox": [ + 110, + 478, + 367, + 491 + ], + "score": 1.0, + "content": "5.1 PROGRESSIVE LINEAR SEPARATION AND CONTRACTION", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 500, + 505, + 533 + ], + "lines": [ + { + "bbox": [ + 106, + 500, + 505, + 512 + ], + "spans": [ + { + "bbox": [ + 106, + 500, + 259, + 512 + ], + "score": 1.0, + "content": "We show that both a ResNet and an", + "type": "text" + }, + { + "bbox": [ + 259, + 501, + 264, + 510 + ], + "score": 0.64, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 500, + 505, + 512 + ], + "score": 1.0, + "content": "-RevNet build a progressively more linearly separable and", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 511, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 496, + 523 + ], + "score": 1.0, + "content": "contracted representation as measured in Oyallon (2017). Observe this property holds for the", + "type": "text" + }, + { + "bbox": [ + 496, + 512, + 500, + 521 + ], + "score": 0.71, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 511, + 505, + 523 + ], + "score": 1.0, + "content": "-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 522, + 359, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 359, + 535 + ], + "score": 1.0, + "content": "RevNet despite the fact that it can not discard any information.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 500, + 505, + 535 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 539, + 505, + 649 + ], + "lines": [ + { + "bbox": [ + 106, + 538, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 106, + 538, + 505, + 552 + ], + "score": 1.0, + "content": "We investigate these properties in each block, with the following experimental protocol. To reduce", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 550, + 505, + 562 + ], + "spans": [ + { + "bbox": [ + 106, + 550, + 505, + 562 + ], + "score": 1.0, + "content": "the computational burden we used a subset of 100 randomly selected imagenet classes, that consist", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 560, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 117, + 574 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 117, + 561, + 161, + 572 + ], + "score": 0.91, + "content": "N = 1 2 0 k", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 560, + 505, + 574 + ], + "score": 1.0, + "content": "images, and keep the same subset during all our following experiments. At each depth", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 572, + 506, + 586 + ], + "spans": [ + { + "bbox": [ + 106, + 573, + 112, + 584 + ], + "score": 0.72, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 113, + 572, + 207, + 586 + ], + "score": 1.0, + "content": ", we extract the features", + "type": "text" + }, + { + "bbox": [ + 208, + 572, + 259, + 585 + ], + "score": 0.93, + "content": "\\{ \\Phi _ { j } x ^ { n } \\} _ { n \\leq N }", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 572, + 506, + 586 + ], + "score": 1.0, + "content": "of the training set, we average them along the spatial variable", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 583, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 505, + 595 + ], + "score": 1.0, + "content": "and standardize them in order to avoid any ill-conditioning effects. We used both a nearest neighbor", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 593, + 506, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 455, + 606 + ], + "score": 1.0, + "content": "classifier and a linear SVM. The former is a localized classifier that indicates that the", + "type": "text" + }, + { + "bbox": [ + 456, + 594, + 465, + 604 + ], + "score": 0.82, + "content": "\\ell ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 593, + 506, + 606 + ], + "score": 1.0, + "content": "metric is", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 605, + 506, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 506, + 617 + ], + "score": 1.0, + "content": "progressively more important for classification, while a linear SVM measures the linear separation", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 616, + 506, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 506, + 628 + ], + "score": 1.0, + "content": "of the different classes. The parameters of the linear SVM are cross-validated on a small subset of", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 627, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 506, + 640 + ], + "score": 1.0, + "content": "the training set, prior to training on the 100 classes. We evaluate both classifiers for each model on", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 637, + 399, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 637, + 399, + 650 + ], + "score": 1.0, + "content": "the validation set of ImageNet and report the Top-1 accuracy in Figure 6.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 26.5, + "bbox_fs": [ + 105, + 538, + 506, + 650 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 654, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "score": 1.0, + "content": "We observe that both classifiers progressively improve similarly with depth for each model, the lin-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 665, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 505, + 677 + ], + "score": 1.0, + "content": "ear SVM performing slightly better than the nearest neighbor classifier because it is the more robust", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 337, + 690 + ], + "score": 1.0, + "content": "and discriminative classifier of the two. In the case of the", + "type": "text" + }, + { + "bbox": [ + 338, + 678, + 342, + 687 + ], + "score": 0.75, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "-RevNet, the classification performed by", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 180, + 700 + ], + "score": 1.0, + "content": "the CNN leads to", + "type": "text" + }, + { + "bbox": [ + 181, + 687, + 200, + 699 + ], + "score": 0.88, + "content": "7 7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 687, + 506, + 700 + ], + "score": 1.0, + "content": ", and the linear SVM performs slightly better because we did not fine-tune", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "the model to 100 classes. Observe that there is a more intense jump of performance on the 3 last", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "score": 1.0, + "content": "layers, which seems to indicate that the former layers have prepared the representation to be more", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 720, + 318, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 318, + 733 + ], + "score": 1.0, + "content": "contracted and linearly separated for the final layers.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 35, + "bbox_fs": [ + 105, + 654, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 108, + 82, + 505, + 116 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "The results suggest a low-dimensional embedding of the data, but this is difficult to validate as", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "estimating local dimensionality in high dimensions is an open problem. However, in the next section,", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 104, + 463, + 117 + ], + "score": 1.0, + "content": "we try to compute the dimension of the discriminative part of the representation built by an", + "type": "text" + }, + { + "bbox": [ + 463, + 105, + 468, + 114 + ], + "score": 0.68, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "-RevNet.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "title", + "bbox": [ + 107, + 130, + 356, + 141 + ], + "lines": [ + { + "bbox": [ + 106, + 129, + 357, + 142 + ], + "spans": [ + { + "bbox": [ + 106, + 129, + 357, + 142 + ], + "score": 1.0, + "content": "5.2 DIMENSIONALITY ANALYSIS OF THE FEATURE SPACE", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 107, + 150, + 505, + 194 + ], + "lines": [ + { + "bbox": [ + 105, + 151, + 505, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 151, + 505, + 162 + ], + "score": 1.0, + "content": "In this section, we investigate if we can refine the dimensionality of informative variabilities in the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 162, + 505, + 173 + ], + "spans": [ + { + "bbox": [ + 106, + 162, + 173, + 173 + ], + "score": 1.0, + "content": "final layer of an", + "type": "text" + }, + { + "bbox": [ + 174, + 162, + 178, + 172 + ], + "score": 0.71, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 162, + 505, + 173 + ], + "score": 1.0, + "content": "-RevNet. Indeed, the cascade of convolutional operators has been trained on the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 172, + 506, + 186 + ], + "spans": [ + { + "bbox": [ + 105, + 172, + 506, + 186 + ], + "score": 1.0, + "content": "training set to separate the 1000 different classes while being a homeomorphism on its feature space.", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 182, + 367, + 197 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 367, + 197 + ], + "score": 1.0, + "content": "Thus, the dimensionality of the feature space is potentially large.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5.5 + }, + { + "type": "text", + "bbox": [ + 107, + 200, + 505, + 266 + ], + "lines": [ + { + "bbox": [ + 106, + 199, + 505, + 213 + ], + "spans": [ + { + "bbox": [ + 106, + 199, + 505, + 213 + ], + "score": 1.0, + "content": "As shown in the previous subsection, the final layer is progressively prepared to be projected on", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 211, + 506, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 211, + 506, + 223 + ], + "score": 1.0, + "content": "the final probes corresponding to the classes. This indicates that the non-informative variabilities", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 221, + 506, + 235 + ], + "spans": [ + { + "bbox": [ + 105, + 221, + 408, + 235 + ], + "score": 1.0, + "content": "for classification can be removed via a linear projection on the final layer", + "type": "text" + }, + { + "bbox": [ + 408, + 223, + 417, + 232 + ], + "score": 0.77, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 221, + 506, + 235 + ], + "score": 1.0, + "content": ", which lie in a space", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 232, + 505, + 245 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 505, + 245 + ], + "score": 1.0, + "content": "of dimension 1000, at most. However, this projection has been built via supervision, which can", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 245, + 505, + 256 + ], + "spans": [ + { + "bbox": [ + 106, + 245, + 505, + 256 + ], + "score": 1.0, + "content": "still retain directions that have been contracted and thus will not be selected by an algorithm such as", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 254, + 505, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 505, + 268 + ], + "score": 1.0, + "content": "PCA. We show in fact a PCA retains the necessary information for classification in a small subspace.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 10.5 + }, + { + "type": "text", + "bbox": [ + 107, + 271, + 296, + 469 + ], + "lines": [ + { + "bbox": [ + 106, + 271, + 297, + 285 + ], + "spans": [ + { + "bbox": [ + 106, + 271, + 271, + 285 + ], + "score": 1.0, + "content": "To do so, we build the linear projectors", + "type": "text" + }, + { + "bbox": [ + 271, + 274, + 283, + 283 + ], + "score": 0.85, + "content": "\\pi _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 283, + 271, + 297, + 285 + ], + "score": 1.0, + "content": "on", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 283, + 296, + 295 + ], + "spans": [ + { + "bbox": [ + 106, + 283, + 194, + 295 + ], + "score": 1.0, + "content": "the subspace of the", + "type": "text" + }, + { + "bbox": [ + 194, + 284, + 201, + 293 + ], + "score": 0.7, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 283, + 296, + 295 + ], + "score": 1.0, + "content": "first principal compo-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 294, + 297, + 305 + ], + "spans": [ + { + "bbox": [ + 105, + 294, + 297, + 305 + ], + "score": 1.0, + "content": "nents, and we propose to measure the classifica-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 305, + 297, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 305, + 297, + 317 + ], + "score": 1.0, + "content": "tion power of the projected representation with", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 316, + 297, + 328 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 297, + 328 + ], + "score": 1.0, + "content": "a supervised classifier, e.g. nearest neighbor or", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 326, + 297, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 297, + 339 + ], + "score": 1.0, + "content": "a linear SVM, on the previous 100 class task.", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 335, + 295, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 248, + 352 + ], + "score": 1.0, + "content": "Again, the feature representation", + "type": "text" + }, + { + "bbox": [ + 248, + 338, + 295, + 350 + ], + "score": 0.92, + "content": "\\{ \\Phi x ^ { n } \\} _ { n \\leq N }", + "type": "inline_equation" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 348, + 298, + 361 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 298, + 361 + ], + "score": 1.0, + "content": "are spatially averaged to remove the translation", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 360, + 297, + 371 + ], + "spans": [ + { + "bbox": [ + 106, + 360, + 297, + 371 + ], + "score": 1.0, + "content": "variability, and standardized on the training set.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 370, + 298, + 383 + ], + "spans": [ + { + "bbox": [ + 106, + 370, + 298, + 383 + ], + "score": 1.0, + "content": "We apply both classifiers, and we report the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 379, + 298, + 395 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 214, + 395 + ], + "score": 1.0, + "content": "classification accuracy of", + "type": "text" + }, + { + "bbox": [ + 214, + 381, + 272, + 394 + ], + "score": 0.93, + "content": "\\{ \\pi _ { d } \\Phi x ^ { n } \\} _ { n \\leq N }", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 379, + 298, + 395 + ], + "score": 1.0, + "content": "w.r.t.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 392, + 297, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 118, + 405 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 118, + 393, + 125, + 403 + ], + "score": 0.77, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 126, + 392, + 297, + 405 + ], + "score": 1.0, + "content": "on the Figure 7. A linear projection re-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 404, + 297, + 415 + ], + "spans": [ + { + "bbox": [ + 106, + 404, + 297, + 415 + ], + "score": 1.0, + "content": "moves some information that can not be recov-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 414, + 297, + 426 + ], + "spans": [ + { + "bbox": [ + 106, + 414, + 297, + 426 + ], + "score": 1.0, + "content": "ered by a linear classifier, therefore we observe", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 425, + 297, + 437 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 297, + 437 + ], + "score": 1.0, + "content": "that the classification accuracy only decreases", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 437, + 297, + 447 + ], + "spans": [ + { + "bbox": [ + 106, + 437, + 173, + 447 + ], + "score": 1.0, + "content": "significantly for", + "type": "text" + }, + { + "bbox": [ + 174, + 437, + 210, + 447 + ], + "score": 0.91, + "content": "d \\leq 2 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 437, + 297, + 447 + ], + "score": 1.0, + "content": ". 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However, this projection has been built via supervision, which can", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 245, + 505, + 256 + ], + "spans": [ + { + "bbox": [ + 106, + 245, + 505, + 256 + ], + "score": 1.0, + "content": "still retain directions that have been contracted and thus will not be selected by an algorithm such as", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 254, + 505, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 505, + 268 + ], + "score": 1.0, + "content": "PCA. We show in fact a PCA retains the necessary information for classification in a small subspace.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 10.5, + "bbox_fs": [ + 105, + 199, + 506, + 268 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 271, + 296, + 469 + ], + "lines": [ + { + "bbox": [ + 106, + 271, + 297, + 285 + ], + "spans": [ + { + "bbox": [ + 106, + 271, + 271, + 285 + ], + "score": 1.0, + "content": "To do so, we build the linear projectors", + "type": "text" + }, + { + "bbox": [ + 271, + 274, + 283, + 283 + ], + "score": 0.85, + "content": "\\pi _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 283, + 271, + 297, + 285 + ], + "score": 1.0, + "content": "on", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 283, + 296, + 295 + ], + "spans": [ + { + "bbox": [ + 106, + 283, + 194, + 295 + ], + "score": 1.0, + "content": "the subspace of the", + "type": "text" + }, + { + "bbox": [ + 194, + 284, + 201, + 293 + ], + "score": 0.7, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 283, + 296, + 295 + ], + "score": 1.0, + "content": "first principal compo-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 294, + 297, + 305 + ], + "spans": [ + { + "bbox": [ + 105, + 294, + 297, + 305 + ], + "score": 1.0, + "content": "nents, and we propose to measure the classifica-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 305, + 297, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 305, + 297, + 317 + ], + "score": 1.0, + "content": "tion power of the projected representation with", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 316, + 297, + 328 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 297, + 328 + ], + "score": 1.0, + "content": "a supervised classifier, e.g. nearest neighbor or", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 326, + 297, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 297, + 339 + ], + "score": 1.0, + "content": "a linear SVM, on the previous 100 class task.", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 335, + 295, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 248, + 352 + ], + "score": 1.0, + "content": "Again, the feature representation", + "type": "text" + }, + { + "bbox": [ + 248, + 338, + 295, + 350 + ], + "score": 0.92, + "content": "\\{ \\Phi x ^ { n } \\} _ { n \\leq N }", + "type": "inline_equation" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 348, + 298, + 361 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 298, + 361 + ], + "score": 1.0, + "content": "are spatially averaged to remove the translation", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 360, + 297, + 371 + ], + "spans": [ + { + "bbox": [ + 106, + 360, + 297, + 371 + ], + "score": 1.0, + "content": "variability, and standardized on the training set.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 370, + 298, + 383 + ], + "spans": [ + { + "bbox": [ + 106, + 370, + 298, + 383 + ], + "score": 1.0, + "content": "We apply both classifiers, and we report the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 379, + 298, + 395 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 214, + 395 + ], + "score": 1.0, + "content": "classification accuracy of", + "type": "text" + }, + { + "bbox": [ + 214, + 381, + 272, + 394 + ], + "score": 0.93, + "content": "\\{ \\pi _ { d } \\Phi x ^ { n } \\} _ { n \\leq N }", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 379, + 298, + 395 + ], + "score": 1.0, + "content": "w.r.t.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 392, + 297, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 118, + 405 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 118, + 393, + 125, + 403 + ], + "score": 0.77, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 126, + 392, + 297, + 405 + ], + "score": 1.0, + "content": "on the Figure 7. A linear projection re-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 404, + 297, + 415 + ], + "spans": [ + { + "bbox": [ + 106, + 404, + 297, + 415 + ], + "score": 1.0, + "content": "moves some information that can not be recov-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 414, + 297, + 426 + ], + "spans": [ + { + "bbox": [ + 106, + 414, + 297, + 426 + ], + "score": 1.0, + "content": "ered by a linear classifier, therefore we observe", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 425, + 297, + 437 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 297, + 437 + ], + "score": 1.0, + "content": "that the classification accuracy only decreases", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 437, + 297, + 447 + ], + "spans": [ + { + "bbox": [ + 106, + 437, + 173, + 447 + ], + "score": 1.0, + "content": "significantly for", + "type": "text" + }, + { + "bbox": [ + 174, + 437, + 210, + 447 + ], + "score": 0.91, + "content": "d \\leq 2 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 437, + 297, + 447 + ], + "score": 1.0, + "content": ". 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We show that this is not", + "type": "text" + } + ], + "index": 62 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "the case and propose to explain the generalization property with empirical evidence of progressive", + "type": "text" + } + ], + "index": 63 + }, + { + "bbox": [ + 105, + 721, + 318, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 318, + 732 + ], + "score": 1.0, + "content": "separation and contraction with depth, on ImageNet.", + "type": "text" + } + ], + "index": 64 + } + ], + "index": 62.5, + "bbox_fs": [ + 105, + 687, + 505, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 82, + 225, + 93 + ], + "lines": [ + { + "bbox": [ + 107, + 80, + 226, + 96 + ], + "spans": [ + { + "bbox": [ + 107, + 80, + 226, + 96 + ], + "score": 1.0, + "content": "ACKNOWLEDGEMENTS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 106, + 505, + 172 + ], + "lines": [ + { + "bbox": [ + 105, + 105, + 506, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 506, + 119 + ], + "score": 1.0, + "content": "Jorn-Henrik Jacobsen was partially funded by the STW perspective program ImaGene. Edouard ¨", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 117, + 505, + 130 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 505, + 130 + ], + "score": 1.0, + "content": "Oyallon was partially funded by the ERC grant InvariantClass 320959, via a grant for PhD Students", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 127, + 505, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 505, + 141 + ], + "score": 1.0, + "content": "of the Conseil regional dIle-de-France (RDM-IdF), and a postdoctoral grant from the from DPEI ´", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 139, + 505, + 152 + ], + "spans": [ + { + "bbox": [ + 105, + 139, + 505, + 152 + ], + "score": 1.0, + "content": "of Inria (AAR 2017POD057) for the collaboration with CWI. We thank Berkay Kicanaoglu for the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 149, + 505, + 163 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 505, + 163 + ], + "score": 1.0, + "content": "Basel Face data, Mathieu Andreux, Eugene Belilovsky, Amal Rannen, Patrick Putzky and Kyriacos", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 160, + 281, + 174 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 281, + 174 + ], + "score": 1.0, + "content": "Shiarlis for feedback on drafts of the paper.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3.5 + }, + { + "type": "title", + "bbox": [ + 107, + 189, + 175, + 201 + ], + "lines": [ + { + "bbox": [ + 106, + 189, + 176, + 202 + ], + "spans": [ + { + "bbox": [ + 106, + 189, + 176, + 202 + ], + "score": 1.0, + "content": "REFERENCES", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 106, + 208, + 504, + 230 + ], + "lines": [ + { + "bbox": [ + 106, + 207, + 505, + 221 + ], + "spans": [ + { + "bbox": [ + 106, + 207, + 505, + 221 + ], + "score": 1.0, + "content": "Alessandro Achille and Stefano Soatto. On the emergence of invariance and disentangling in deep", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 115, + 219, + 345, + 231 + ], + "spans": [ + { + "bbox": [ + 115, + 219, + 345, + 231 + ], + "score": 1.0, + "content": "representations. arXiv preprint arXiv:1706.01350, 2017.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8.5 + }, + { + "type": "text", + "bbox": [ + 107, + 237, + 504, + 271 + ], + "lines": [ + { + "bbox": [ + 105, + 237, + 504, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 504, + 252 + ], + "score": 1.0, + "content": "Mathieu Aubry and Bryan C Russell. Understanding deep features with computer-generated im-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 114, + 249, + 505, + 262 + ], + "spans": [ + { + "bbox": [ + 114, + 249, + 505, + 262 + ], + "score": 1.0, + "content": "agery. In Proceedings of the IEEE International Conference on Computer Vision, pp. 2875–2883,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 115, + 260, + 142, + 272 + ], + "spans": [ + { + "bbox": [ + 115, + 260, + 142, + 272 + ], + "score": 1.0, + "content": "2015.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 104, + 279, + 505, + 302 + ], + "lines": [ + { + "bbox": [ + 105, + 278, + 505, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 278, + 505, + 293 + ], + "score": 1.0, + "content": "Joan Bruna, Arthur Szlam, and Yann LeCun. Signal recovery from pooling representations. arXiv", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 115, + 291, + 249, + 302 + ], + "spans": [ + { + "bbox": [ + 115, + 291, + 249, + 302 + ], + "score": 1.0, + "content": "preprint arXiv:1311.4025, 2013.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13.5 + }, + { + "type": "text", + "bbox": [ + 106, + 309, + 505, + 344 + ], + "lines": [ + { + "bbox": [ + 106, + 309, + 505, + 322 + ], + "spans": [ + { + "bbox": [ + 106, + 309, + 505, + 322 + ], + "score": 1.0, + "content": "Mircea Cimpoi, Subhransu Maji, Iasonas Kokkinos, Sammy Mohamed, and Andrea Vedaldi. De-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 116, + 321, + 505, + 334 + ], + "spans": [ + { + "bbox": [ + 116, + 321, + 505, + 334 + ], + "score": 1.0, + "content": "scribing textures in the wild. In Proceedings of the IEEE Conference on Computer Vision and", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 116, + 332, + 291, + 344 + ], + "spans": [ + { + "bbox": [ + 116, + 332, + 291, + 344 + ], + "score": 1.0, + "content": "Pattern Recognition, pp. 3606–3613, 2014.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 106, + 351, + 504, + 385 + ], + "lines": [ + { + "bbox": [ + 106, + 351, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 106, + 351, + 505, + 363 + ], + "score": 1.0, + "content": "Moustapha Cisse, Piotr Bojanowski, Edouard Grave, Yann Dauphin, and Nicolas Usunier. Parseval", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 115, + 362, + 506, + 375 + ], + "spans": [ + { + "bbox": [ + 115, + 362, + 506, + 375 + ], + "score": 1.0, + "content": "networks: Improving robustness to adversarial examples. In International Conference on Machine", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 115, + 373, + 238, + 385 + ], + "spans": [ + { + "bbox": [ + 115, + 373, + 238, + 385 + ], + "score": 1.0, + "content": "Learning, pp. 854–863, 2017.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 105, + 392, + 502, + 415 + ], + "lines": [ + { + "bbox": [ + 105, + 391, + 503, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 391, + 503, + 406 + ], + "score": 1.0, + "content": "Laurent Dinh, David Krueger, and Yoshua Bengio. Nice: Non-linear independent components esti-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 115, + 404, + 308, + 415 + ], + "spans": [ + { + "bbox": [ + 115, + 404, + 308, + 415 + ], + "score": 1.0, + "content": "mation. arXiv preprint arXiv:1410.8516, 2014.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21.5 + }, + { + "type": "text", + "bbox": [ + 106, + 422, + 503, + 445 + ], + "lines": [ + { + "bbox": [ + 104, + 421, + 505, + 437 + ], + "spans": [ + { + "bbox": [ + 104, + 421, + 505, + 437 + ], + "score": 1.0, + "content": "Laurent Dinh, Jascha Sohl-Dickstein, and Samy Bengio. Density estimation using real nvp. arXiv", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 115, + 434, + 254, + 445 + ], + "spans": [ + { + "bbox": [ + 115, + 434, + 254, + 445 + ], + "score": 1.0, + "content": "preprint arXiv:1605.08803, 2016.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5 + }, + { + "type": "text", + "bbox": [ + 106, + 452, + 506, + 487 + ], + "lines": [ + { + "bbox": [ + 106, + 452, + 504, + 466 + ], + "spans": [ + { + "bbox": [ + 106, + 452, + 504, + 466 + ], + "score": 1.0, + "content": "Alexey Dosovitskiy and Thomas Brox. Inverting visual representations with convolutional networks.", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 115, + 463, + 505, + 477 + ], + "spans": [ + { + "bbox": [ + 115, + 463, + 505, + 477 + ], + "score": 1.0, + "content": "In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 4829–", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 115, + 474, + 167, + 488 + ], + "spans": [ + { + "bbox": [ + 115, + 474, + 167, + 488 + ], + "score": 1.0, + "content": "4837, 2016.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 105, + 494, + 504, + 517 + ], + "lines": [ + { + "bbox": [ + 106, + 495, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 106, + 495, + 505, + 507 + ], + "score": 1.0, + "content": "Aidan N Gomez, Mengye Ren, Raquel Urtasun, and Roger B Grosse. The reversible residual net-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 116, + 506, + 487, + 518 + ], + "spans": [ + { + "bbox": [ + 116, + 506, + 487, + 518 + ], + "score": 1.0, + "content": "work: Backpropagation without storing activations. arXiv preprint arXiv:1707.04585, 2017.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28.5 + }, + { + "type": "text", + "bbox": [ + 106, + 524, + 505, + 559 + ], + "lines": [ + { + "bbox": [ + 105, + 523, + 504, + 539 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 504, + 539 + ], + "score": 1.0, + "content": "Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recog-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 115, + 534, + 505, + 550 + ], + "spans": [ + { + "bbox": [ + 115, + 534, + 505, + 550 + ], + "score": 1.0, + "content": "nition. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 115, + 545, + 182, + 559 + ], + "spans": [ + { + "bbox": [ + 115, + 545, + 182, + 559 + ], + "score": 1.0, + "content": "770–778, 2016.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 107, + 565, + 504, + 599 + ], + "lines": [ + { + "bbox": [ + 105, + 565, + 505, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 505, + 580 + ], + "score": 1.0, + "content": "Sergey Ioffe and Christian Szegedy. Batch normalization: Accelerating deep network training by", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 115, + 577, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 115, + 577, + 505, + 590 + ], + "score": 1.0, + "content": "reducing internal covariate shift. In International Conference on Machine Learning, pp. 448–456,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 115, + 588, + 142, + 600 + ], + "spans": [ + { + "bbox": [ + 115, + 588, + 142, + 600 + ], + "score": 1.0, + "content": "2015.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 607, + 504, + 641 + ], + "lines": [ + { + "bbox": [ + 105, + 606, + 504, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 504, + 621 + ], + "score": 1.0, + "content": "Alex Krizhevsky, Ilya Sutskever, and Geoffrey E Hinton. Imagenet classification with deep convo-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 115, + 618, + 506, + 632 + ], + "spans": [ + { + "bbox": [ + 115, + 618, + 506, + 632 + ], + "score": 1.0, + "content": "lutional neural networks. In Advances in neural information processing systems, pp. 1097–1105,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 114, + 627, + 143, + 642 + ], + "spans": [ + { + "bbox": [ + 114, + 627, + 143, + 642 + ], + "score": 1.0, + "content": "2012.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 106, + 648, + 505, + 682 + ], + "lines": [ + { + "bbox": [ + 105, + 647, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 647, + 505, + 662 + ], + "score": 1.0, + "content": "Aravindh Mahendran and Andrea Vedaldi. Understanding deep image representations by inverting", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 115, + 659, + 506, + 674 + ], + "spans": [ + { + "bbox": [ + 115, + 659, + 506, + 674 + ], + "score": 1.0, + "content": "them. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp.", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 116, + 671, + 191, + 682 + ], + "spans": [ + { + "bbox": [ + 116, + 671, + 191, + 682 + ], + "score": 1.0, + "content": "5188–5196, 2015.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 106, + 690, + 505, + 713 + ], + "lines": [ + { + "bbox": [ + 106, + 688, + 505, + 704 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 505, + 704 + ], + "score": 1.0, + "content": "Aravindh Mahendran and Andrea Vedaldi. Visualizing deep convolutional neural networks using", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 115, + 702, + 462, + 714 + ], + "spans": [ + { + "bbox": [ + 115, + 702, + 462, + 714 + ], + "score": 1.0, + "content": "natural pre-images. International Journal of Computer Vision, 120(3):233–255, 2016.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 42.5 + }, + { + "type": "text", + "bbox": [ + 107, + 720, + 417, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 719, + 417, + 734 + ], + "spans": [ + { + "bbox": [ + 106, + 719, + 417, + 734 + ], + "score": 1.0, + "content": "Stephane Mallat. ´ A wavelet tour of signal processing. Academic press, 1999.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 44 + } + ], + "page_idx": 9, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 301, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 313, + 765 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 313, + 765 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 15, + "width": 14 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 82, + 225, + 93 + ], + "lines": [ + { + "bbox": [ + 107, + 80, + 226, + 96 + ], + "spans": [ + { + "bbox": [ + 107, + 80, + 226, + 96 + ], + "score": 1.0, + "content": "ACKNOWLEDGEMENTS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 106, + 505, + 172 + ], + "lines": [ + { + "bbox": [ + 105, + 105, + 506, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 506, + 119 + ], + "score": 1.0, + "content": "Jorn-Henrik Jacobsen was partially funded by the STW perspective program ImaGene. 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On the emergence of invariance and disentangling in deep", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 115, + 219, + 345, + 231 + ], + "spans": [ + { + "bbox": [ + 115, + 219, + 345, + 231 + ], + "score": 1.0, + "content": "representations. arXiv preprint arXiv:1706.01350, 2017.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8.5, + "bbox_fs": [ + 106, + 207, + 505, + 231 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 237, + 504, + 271 + ], + "lines": [ + { + "bbox": [ + 105, + 237, + 504, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 504, + 252 + ], + "score": 1.0, + "content": "Mathieu Aubry and Bryan C Russell. Understanding deep features with computer-generated im-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 114, + 249, + 505, + 262 + ], + "spans": [ + { + "bbox": [ + 114, + 249, + 505, + 262 + ], + "score": 1.0, + "content": "agery. In Proceedings of the IEEE International Conference on Computer Vision, pp. 2875–2883,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 115, + 260, + 142, + 272 + ], + "spans": [ + { + "bbox": [ + 115, + 260, + 142, + 272 + ], + "score": 1.0, + "content": "2015.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11, + "bbox_fs": [ + 105, + 237, + 505, + 272 + ] + }, + { + "type": "text", + "bbox": [ + 104, + 279, + 505, + 302 + ], + "lines": [ + { + "bbox": [ + 105, + 278, + 505, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 278, + 505, + 293 + ], + "score": 1.0, + "content": "Joan Bruna, Arthur Szlam, and Yann LeCun. Signal recovery from pooling representations. arXiv", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 115, + 291, + 249, + 302 + ], + "spans": [ + { + "bbox": [ + 115, + 291, + 249, + 302 + ], + "score": 1.0, + "content": "preprint arXiv:1311.4025, 2013.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13.5, + "bbox_fs": [ + 105, + 278, + 505, + 302 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 309, + 505, + 344 + ], + "lines": [ + { + "bbox": [ + 106, + 309, + 505, + 322 + ], + "spans": [ + { + "bbox": [ + 106, + 309, + 505, + 322 + ], + "score": 1.0, + "content": "Mircea Cimpoi, Subhransu Maji, Iasonas Kokkinos, Sammy Mohamed, and Andrea Vedaldi. De-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 116, + 321, + 505, + 334 + ], + "spans": [ + { + "bbox": [ + 116, + 321, + 505, + 334 + ], + "score": 1.0, + "content": "scribing textures in the wild. In Proceedings of the IEEE Conference on Computer Vision and", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 116, + 332, + 291, + 344 + ], + "spans": [ + { + "bbox": [ + 116, + 332, + 291, + 344 + ], + "score": 1.0, + "content": "Pattern Recognition, pp. 3606–3613, 2014.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16, + "bbox_fs": [ + 106, + 309, + 505, + 344 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 351, + 504, + 385 + ], + "lines": [ + { + "bbox": [ + 106, + 351, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 106, + 351, + 505, + 363 + ], + "score": 1.0, + "content": "Moustapha Cisse, Piotr Bojanowski, Edouard Grave, Yann Dauphin, and Nicolas Usunier. Parseval", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 115, + 362, + 506, + 375 + ], + "spans": [ + { + "bbox": [ + 115, + 362, + 506, + 375 + ], + "score": 1.0, + "content": "networks: Improving robustness to adversarial examples. In International Conference on Machine", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 115, + 373, + 238, + 385 + ], + "spans": [ + { + "bbox": [ + 115, + 373, + 238, + 385 + ], + "score": 1.0, + "content": "Learning, pp. 854–863, 2017.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19, + "bbox_fs": [ + 106, + 351, + 506, + 385 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 392, + 502, + 415 + ], + "lines": [ + { + "bbox": [ + 105, + 391, + 503, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 391, + 503, + 406 + ], + "score": 1.0, + "content": "Laurent Dinh, David Krueger, and Yoshua Bengio. Nice: Non-linear independent components esti-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 115, + 404, + 308, + 415 + ], + "spans": [ + { + "bbox": [ + 115, + 404, + 308, + 415 + ], + "score": 1.0, + "content": "mation. arXiv preprint arXiv:1410.8516, 2014.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21.5, + "bbox_fs": [ + 105, + 391, + 503, + 415 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 422, + 503, + 445 + ], + "lines": [ + { + "bbox": [ + 104, + 421, + 505, + 437 + ], + "spans": [ + { + "bbox": [ + 104, + 421, + 505, + 437 + ], + "score": 1.0, + "content": "Laurent Dinh, Jascha Sohl-Dickstein, and Samy Bengio. Density estimation using real nvp. arXiv", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 115, + 434, + 254, + 445 + ], + "spans": [ + { + "bbox": [ + 115, + 434, + 254, + 445 + ], + "score": 1.0, + "content": "preprint arXiv:1605.08803, 2016.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5, + "bbox_fs": [ + 104, + 421, + 505, + 445 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 452, + 506, + 487 + ], + "lines": [ + { + "bbox": [ + 106, + 452, + 504, + 466 + ], + "spans": [ + { + "bbox": [ + 106, + 452, + 504, + 466 + ], + "score": 1.0, + "content": "Alexey Dosovitskiy and Thomas Brox. Inverting visual representations with convolutional networks.", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 115, + 463, + 505, + 477 + ], + "spans": [ + { + "bbox": [ + 115, + 463, + 505, + 477 + ], + "score": 1.0, + "content": "In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 4829–", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 115, + 474, + 167, + 488 + ], + "spans": [ + { + "bbox": [ + 115, + 474, + 167, + 488 + ], + "score": 1.0, + "content": "4837, 2016.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26, + "bbox_fs": [ + 106, + 452, + 505, + 488 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 494, + 504, + 517 + ], + "lines": [ + { + "bbox": [ + 106, + 495, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 106, + 495, + 505, + 507 + ], + "score": 1.0, + "content": "Aidan N Gomez, Mengye Ren, Raquel Urtasun, and Roger B Grosse. The reversible residual net-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 116, + 506, + 487, + 518 + ], + "spans": [ + { + "bbox": [ + 116, + 506, + 487, + 518 + ], + "score": 1.0, + "content": "work: Backpropagation without storing activations. arXiv preprint arXiv:1707.04585, 2017.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28.5, + "bbox_fs": [ + 106, + 495, + 505, + 518 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 524, + 505, + 559 + ], + "lines": [ + { + "bbox": [ + 105, + 523, + 504, + 539 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 504, + 539 + ], + "score": 1.0, + "content": "Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recog-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 115, + 534, + 505, + 550 + ], + "spans": [ + { + "bbox": [ + 115, + 534, + 505, + 550 + ], + "score": 1.0, + "content": "nition. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 115, + 545, + 182, + 559 + ], + "spans": [ + { + "bbox": [ + 115, + 545, + 182, + 559 + ], + "score": 1.0, + "content": "770–778, 2016.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31, + "bbox_fs": [ + 105, + 523, + 505, + 559 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 565, + 504, + 599 + ], + "lines": [ + { + "bbox": [ + 105, + 565, + 505, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 505, + 580 + ], + "score": 1.0, + "content": "Sergey Ioffe and Christian Szegedy. Batch normalization: Accelerating deep network training by", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 115, + 577, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 115, + 577, + 505, + 590 + ], + "score": 1.0, + "content": "reducing internal covariate shift. In International Conference on Machine Learning, pp. 448–456,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 115, + 588, + 142, + 600 + ], + "spans": [ + { + "bbox": [ + 115, + 588, + 142, + 600 + ], + "score": 1.0, + "content": "2015.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34, + "bbox_fs": [ + 105, + 565, + 505, + 600 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 607, + 504, + 641 + ], + "lines": [ + { + "bbox": [ + 105, + 606, + 504, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 504, + 621 + ], + "score": 1.0, + "content": "Alex Krizhevsky, Ilya Sutskever, and Geoffrey E Hinton. Imagenet classification with deep convo-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 115, + 618, + 506, + 632 + ], + "spans": [ + { + "bbox": [ + 115, + 618, + 506, + 632 + ], + "score": 1.0, + "content": "lutional neural networks. In Advances in neural information processing systems, pp. 1097–1105,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 114, + 627, + 143, + 642 + ], + "spans": [ + { + "bbox": [ + 114, + 627, + 143, + 642 + ], + "score": 1.0, + "content": "2012.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37, + "bbox_fs": [ + 105, + 606, + 506, + 642 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 648, + 505, + 682 + ], + "lines": [ + { + "bbox": [ + 105, + 647, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 647, + 505, + 662 + ], + "score": 1.0, + "content": "Aravindh Mahendran and Andrea Vedaldi. Understanding deep image representations by inverting", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 115, + 659, + 506, + 674 + ], + "spans": [ + { + "bbox": [ + 115, + 659, + 506, + 674 + ], + "score": 1.0, + "content": "them. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp.", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 116, + 671, + 191, + 682 + ], + "spans": [ + { + "bbox": [ + 116, + 671, + 191, + 682 + ], + "score": 1.0, + "content": "5188–5196, 2015.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 40, + "bbox_fs": [ + 105, + 647, + 506, + 682 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 690, + 505, + 713 + ], + "lines": [ + { + "bbox": [ + 106, + 688, + 505, + 704 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 505, + 704 + ], + "score": 1.0, + "content": "Aravindh Mahendran and Andrea Vedaldi. Visualizing deep convolutional neural networks using", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 115, + 702, + 462, + 714 + ], + "spans": [ + { + "bbox": [ + 115, + 702, + 462, + 714 + ], + "score": 1.0, + "content": "natural pre-images. International Journal of Computer Vision, 120(3):233–255, 2016.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 42.5, + "bbox_fs": [ + 106, + 688, + 505, + 714 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 720, + 417, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 719, + 417, + 734 + ], + "spans": [ + { + "bbox": [ + 106, + 719, + 417, + 734 + ], + "score": 1.0, + "content": "Stephane Mallat. ´ A wavelet tour of signal processing. Academic press, 1999.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 44, + "bbox_fs": [ + 106, + 719, + 417, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 504, + 105 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 506, + 96 + ], + "score": 1.0, + "content": "Stephane Mallat. Group invariant scattering. ´ Communications on Pure and Applied Mathematics,", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 116, + 93, + 220, + 104 + ], + "spans": [ + { + "bbox": [ + 116, + 93, + 220, + 104 + ], + "score": 1.0, + "content": "65(10):1331–1398, 2012.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 106, + 111, + 504, + 135 + ], + "lines": [ + { + "bbox": [ + 106, + 112, + 505, + 125 + ], + "spans": [ + { + "bbox": [ + 106, + 112, + 505, + 125 + ], + "score": 1.0, + "content": "Stephane Mallat. Understanding deep convolutional networks. ´ Phil. Trans. R. Soc. A, 374(2065):", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 116, + 123, + 186, + 135 + ], + "spans": [ + { + "bbox": [ + 116, + 123, + 186, + 135 + ], + "score": 1.0, + "content": "20150203, 2016.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 106, + 141, + 504, + 165 + ], + "lines": [ + { + "bbox": [ + 105, + 140, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 140, + 505, + 156 + ], + "score": 1.0, + "content": "Alfred J Menezes, Paul C Van Oorschot, and Scott A Vanstone. Handbook of applied cryptography.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 116, + 153, + 189, + 165 + ], + "spans": [ + { + "bbox": [ + 116, + 153, + 189, + 165 + ], + "score": 1.0, + "content": "CRC press, 1996.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4.5 + }, + { + "type": "text", + "bbox": [ + 106, + 171, + 505, + 195 + ], + "lines": [ + { + "bbox": [ + 105, + 170, + 505, + 186 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 505, + 186 + ], + "score": 1.0, + "content": "Edouard Oyallon. Building a regular decision boundary with deep networks. arXiv preprint", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 116, + 183, + 220, + 195 + ], + "spans": [ + { + "bbox": [ + 116, + 183, + 220, + 195 + ], + "score": 1.0, + "content": "arXiv:1703.01775, 2017.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6.5 + }, + { + "type": "text", + "bbox": [ + 107, + 201, + 504, + 236 + ], + "lines": [ + { + "bbox": [ + 105, + 201, + 506, + 215 + ], + "spans": [ + { + "bbox": [ + 105, + 201, + 506, + 215 + ], + "score": 1.0, + "content": "Pascal Paysan, Reinhard Knothe, Brian Amberg, Sami Romdhani, and Thomas Vetter. A 3d face", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 115, + 213, + 506, + 226 + ], + "spans": [ + { + "bbox": [ + 115, + 213, + 506, + 226 + ], + "score": 1.0, + "content": "model for pose and illumination invariant face recognition. In Advanced Video and Signal Based", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 115, + 223, + 502, + 236 + ], + "spans": [ + { + "bbox": [ + 115, + 223, + 502, + 236 + ], + "score": 1.0, + "content": "Surveillance, 2009. AVSS’09. Sixth IEEE International Conference on, pp. 296–301. Ieee, 2009.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 105, + 242, + 504, + 266 + ], + "lines": [ + { + "bbox": [ + 106, + 241, + 505, + 257 + ], + "spans": [ + { + "bbox": [ + 106, + 241, + 505, + 257 + ], + "score": 1.0, + "content": "Alec Radford, Luke Metz, and Soumith Chintala. Unsupervised representation learning with deep", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 115, + 254, + 468, + 266 + ], + "spans": [ + { + "bbox": [ + 115, + 254, + 468, + 266 + ], + "score": 1.0, + "content": "convolutional generative adversarial networks. arXiv preprint arXiv:1511.06434, 2015.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5 + }, + { + "type": "text", + "bbox": [ + 108, + 272, + 504, + 307 + ], + "lines": [ + { + "bbox": [ + 106, + 271, + 505, + 286 + ], + "spans": [ + { + "bbox": [ + 106, + 271, + 505, + 286 + ], + "score": 1.0, + "content": "Olga Russakovsky, Jia Deng, Hao Su, Jonathan Krause, Sanjeev Satheesh, Sean Ma, Zhiheng", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 116, + 284, + 505, + 296 + ], + "spans": [ + { + "bbox": [ + 116, + 284, + 505, + 296 + ], + "score": 1.0, + "content": "Huang, Andrej Karpathy, Aditya Khosla, Michael Bernstein, et al. Imagenet large scale visual", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 115, + 295, + 474, + 308 + ], + "spans": [ + { + "bbox": [ + 115, + 295, + 474, + 308 + ], + "score": 1.0, + "content": "recognition challenge. International Journal of Computer Vision, 115(3):211–252, 2015.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 106, + 313, + 505, + 358 + ], + "lines": [ + { + "bbox": [ + 106, + 313, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 106, + 313, + 505, + 325 + ], + "score": 1.0, + "content": "Wenzhe Shi, Jose Caballero, Ferenc Huszar, Johannes Totz, Andrew P Aitken, Rob Bishop, Daniel ´", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 115, + 324, + 505, + 337 + ], + "spans": [ + { + "bbox": [ + 115, + 324, + 505, + 337 + ], + "score": 1.0, + "content": "Rueckert, and Zehan Wang. Real-time single image and video super-resolution using an efficient", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 116, + 335, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 116, + 335, + 505, + 348 + ], + "score": 1.0, + "content": "sub-pixel convolutional neural network. In Proceedings of the IEEE Conference on Computer", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 116, + 346, + 336, + 359 + ], + "spans": [ + { + "bbox": [ + 116, + 346, + 336, + 359 + ], + "score": 1.0, + "content": "Vision and Pattern Recognition, pp. 1874–1883, 2016.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17.5 + }, + { + "type": "text", + "bbox": [ + 106, + 365, + 503, + 388 + ], + "lines": [ + { + "bbox": [ + 106, + 365, + 505, + 378 + ], + "spans": [ + { + "bbox": [ + 106, + 365, + 505, + 378 + ], + "score": 1.0, + "content": "Ravid Shwartz-Ziv and Naftali Tishby. Opening the black box of deep neural networks via informa-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 116, + 377, + 300, + 387 + ], + "spans": [ + { + "bbox": [ + 116, + 377, + 300, + 387 + ], + "score": 1.0, + "content": "tion. arXiv preprint arXiv:1703.00810, 2017.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5 + }, + { + "type": "text", + "bbox": [ + 106, + 395, + 504, + 418 + ], + "lines": [ + { + "bbox": [ + 106, + 394, + 505, + 407 + ], + "spans": [ + { + "bbox": [ + 106, + 394, + 505, + 407 + ], + "score": 1.0, + "content": "Wim Sweldens. The lifting scheme: A construction of second generation wavelets. SIAM journal", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 115, + 406, + 312, + 418 + ], + "spans": [ + { + "bbox": [ + 115, + 406, + 312, + 418 + ], + "score": 1.0, + "content": "on mathematical analysis, 29(2):511–546, 1998.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5 + }, + { + "type": "text", + "bbox": [ + 105, + 425, + 505, + 448 + ], + "lines": [ + { + "bbox": [ + 106, + 425, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 505, + 438 + ], + "score": 1.0, + "content": "Christian Szegedy, Wojciech Zaremba, Ilya Sutskever, Joan Bruna, Dumitru Erhan, Ian Goodfellow,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 115, + 437, + 505, + 448 + ], + "spans": [ + { + "bbox": [ + 115, + 437, + 505, + 448 + ], + "score": 1.0, + "content": "and Rob Fergus. Intriguing properties of neural networks. arXiv preprint arXiv:1312.6199, 2013.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5 + }, + { + "type": "text", + "bbox": [ + 106, + 455, + 505, + 478 + ], + "lines": [ + { + "bbox": [ + 106, + 455, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 106, + 455, + 505, + 468 + ], + "score": 1.0, + "content": "Naftali Tishby and Noga Zaslavsky. Deep learning and the information bottleneck principle. In", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 116, + 466, + 403, + 478 + ], + "spans": [ + { + "bbox": [ + 116, + 466, + 403, + 478 + ], + "score": 1.0, + "content": "Information Theory Workshop (ITW), 2015 IEEE, pp. 1–5. IEEE, 2015.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26.5 + }, + { + "type": "text", + "bbox": [ + 107, + 484, + 504, + 508 + ], + "lines": [ + { + "bbox": [ + 105, + 484, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 505, + 497 + ], + "score": 1.0, + "content": "Fisher Yu and Vladlen Koltun. Multi-scale context aggregation by dilated convolutions. arXiv", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 114, + 496, + 254, + 508 + ], + "spans": [ + { + "bbox": [ + 114, + 496, + 254, + 508 + ], + "score": 1.0, + "content": "preprint arXiv:1511.07122, 2015.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28.5 + }, + { + "type": "text", + "bbox": [ + 106, + 514, + 505, + 538 + ], + "lines": [ + { + "bbox": [ + 106, + 513, + 505, + 527 + ], + "spans": [ + { + "bbox": [ + 106, + 514, + 297, + 527 + ], + "score": 1.0, + "content": "Sergey Zagoruyko and Nikos Komodakis.", + "type": "text" + }, + { + "bbox": [ + 312, + 513, + 426, + 527 + ], + "score": 1.0, + "content": "Wide residual networks.", + "type": "text" + }, + { + "bbox": [ + 438, + 514, + 505, + 527 + ], + "score": 1.0, + "content": "arXiv preprint", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 115, + 525, + 219, + 537 + ], + "spans": [ + { + "bbox": [ + 115, + 525, + 219, + 537 + ], + "score": 1.0, + "content": "arXiv:1605.07146, 2016.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30.5 + }, + { + "type": "text", + "bbox": [ + 105, + 544, + 505, + 568 + ], + "lines": [ + { + "bbox": [ + 105, + 544, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 505, + 557 + ], + "score": 1.0, + "content": "Matthew D Zeiler and Rob Fergus. Visualizing and understanding convolutional networks. In", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 116, + 555, + 403, + 568 + ], + "spans": [ + { + "bbox": [ + 116, + 555, + 403, + 568 + ], + "score": 1.0, + "content": "European conference on computer vision, pp. 818–833. Springer, 2014.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32.5 + } + ], + "page_idx": 10, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 293, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 39 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 39 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 310, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 765 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 765 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 15, + "width": 13 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 504, + 105 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 506, + 96 + ], + "score": 1.0, + "content": "Stephane Mallat. Group invariant scattering. ´ Communications on Pure and Applied Mathematics,", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 116, + 93, + 220, + 104 + ], + "spans": [ + { + "bbox": [ + 116, + 93, + 220, + 104 + ], + "score": 1.0, + "content": "65(10):1331–1398, 2012.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5, + "bbox_fs": [ + 105, + 81, + 506, + 104 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 111, + 504, + 135 + ], + "lines": [ + { + "bbox": [ + 106, + 112, + 505, + 125 + ], + "spans": [ + { + "bbox": [ + 106, + 112, + 505, + 125 + ], + "score": 1.0, + "content": "Stephane Mallat. Understanding deep convolutional networks. ´ Phil. Trans. R. Soc. A, 374(2065):", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 116, + 123, + 186, + 135 + ], + "spans": [ + { + "bbox": [ + 116, + 123, + 186, + 135 + ], + "score": 1.0, + "content": "20150203, 2016.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2.5, + "bbox_fs": [ + 106, + 112, + 505, + 135 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 141, + 504, + 165 + ], + "lines": [ + { + "bbox": [ + 105, + 140, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 140, + 505, + 156 + ], + "score": 1.0, + "content": "Alfred J Menezes, Paul C Van Oorschot, and Scott A Vanstone. Handbook of applied cryptography.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 116, + 153, + 189, + 165 + ], + "spans": [ + { + "bbox": [ + 116, + 153, + 189, + 165 + ], + "score": 1.0, + "content": "CRC press, 1996.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4.5, + "bbox_fs": [ + 105, + 140, + 505, + 165 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 171, + 505, + 195 + ], + "lines": [ + { + "bbox": [ + 105, + 170, + 505, + 186 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 505, + 186 + ], + "score": 1.0, + "content": "Edouard Oyallon. Building a regular decision boundary with deep networks. arXiv preprint", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 116, + 183, + 220, + 195 + ], + "spans": [ + { + "bbox": [ + 116, + 183, + 220, + 195 + ], + "score": 1.0, + "content": "arXiv:1703.01775, 2017.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6.5, + "bbox_fs": [ + 105, + 170, + 505, + 195 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 201, + 504, + 236 + ], + "lines": [ + { + "bbox": [ + 105, + 201, + 506, + 215 + ], + "spans": [ + { + "bbox": [ + 105, + 201, + 506, + 215 + ], + "score": 1.0, + "content": "Pascal Paysan, Reinhard Knothe, Brian Amberg, Sami Romdhani, and Thomas Vetter. A 3d face", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 115, + 213, + 506, + 226 + ], + "spans": [ + { + "bbox": [ + 115, + 213, + 506, + 226 + ], + "score": 1.0, + "content": "model for pose and illumination invariant face recognition. In Advanced Video and Signal Based", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 115, + 223, + 502, + 236 + ], + "spans": [ + { + "bbox": [ + 115, + 223, + 502, + 236 + ], + "score": 1.0, + "content": "Surveillance, 2009. AVSS’09. Sixth IEEE International Conference on, pp. 296–301. Ieee, 2009.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9, + "bbox_fs": [ + 105, + 201, + 506, + 236 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 242, + 504, + 266 + ], + "lines": [ + { + "bbox": [ + 106, + 241, + 505, + 257 + ], + "spans": [ + { + "bbox": [ + 106, + 241, + 505, + 257 + ], + "score": 1.0, + "content": "Alec Radford, Luke Metz, and Soumith Chintala. Unsupervised representation learning with deep", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 115, + 254, + 468, + 266 + ], + "spans": [ + { + "bbox": [ + 115, + 254, + 468, + 266 + ], + "score": 1.0, + "content": "convolutional generative adversarial networks. arXiv preprint arXiv:1511.06434, 2015.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5, + "bbox_fs": [ + 106, + 241, + 505, + 266 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 272, + 504, + 307 + ], + "lines": [ + { + "bbox": [ + 106, + 271, + 505, + 286 + ], + "spans": [ + { + "bbox": [ + 106, + 271, + 505, + 286 + ], + "score": 1.0, + "content": "Olga Russakovsky, Jia Deng, Hao Su, Jonathan Krause, Sanjeev Satheesh, Sean Ma, Zhiheng", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 116, + 284, + 505, + 296 + ], + "spans": [ + { + "bbox": [ + 116, + 284, + 505, + 296 + ], + "score": 1.0, + "content": "Huang, Andrej Karpathy, Aditya Khosla, Michael Bernstein, et al. Imagenet large scale visual", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 115, + 295, + 474, + 308 + ], + "spans": [ + { + "bbox": [ + 115, + 295, + 474, + 308 + ], + "score": 1.0, + "content": "recognition challenge. International Journal of Computer Vision, 115(3):211–252, 2015.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14, + "bbox_fs": [ + 106, + 271, + 505, + 308 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 313, + 505, + 358 + ], + "lines": [ + { + "bbox": [ + 106, + 313, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 106, + 313, + 505, + 325 + ], + "score": 1.0, + "content": "Wenzhe Shi, Jose Caballero, Ferenc Huszar, Johannes Totz, Andrew P Aitken, Rob Bishop, Daniel ´", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 115, + 324, + 505, + 337 + ], + "spans": [ + { + "bbox": [ + 115, + 324, + 505, + 337 + ], + "score": 1.0, + "content": "Rueckert, and Zehan Wang. Real-time single image and video super-resolution using an efficient", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 116, + 335, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 116, + 335, + 505, + 348 + ], + "score": 1.0, + "content": "sub-pixel convolutional neural network. In Proceedings of the IEEE Conference on Computer", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 116, + 346, + 336, + 359 + ], + "spans": [ + { + "bbox": [ + 116, + 346, + 336, + 359 + ], + "score": 1.0, + "content": "Vision and Pattern Recognition, pp. 1874–1883, 2016.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17.5, + "bbox_fs": [ + 106, + 313, + 505, + 359 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 365, + 503, + 388 + ], + "lines": [ + { + "bbox": [ + 106, + 365, + 505, + 378 + ], + "spans": [ + { + "bbox": [ + 106, + 365, + 505, + 378 + ], + "score": 1.0, + "content": "Ravid Shwartz-Ziv and Naftali Tishby. Opening the black box of deep neural networks via informa-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 116, + 377, + 300, + 387 + ], + "spans": [ + { + "bbox": [ + 116, + 377, + 300, + 387 + ], + "score": 1.0, + "content": "tion. arXiv preprint arXiv:1703.00810, 2017.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5, + "bbox_fs": [ + 106, + 365, + 505, + 387 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 395, + 504, + 418 + ], + "lines": [ + { + "bbox": [ + 106, + 394, + 505, + 407 + ], + "spans": [ + { + "bbox": [ + 106, + 394, + 505, + 407 + ], + "score": 1.0, + "content": "Wim Sweldens. The lifting scheme: A construction of second generation wavelets. SIAM journal", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 115, + 406, + 312, + 418 + ], + "spans": [ + { + "bbox": [ + 115, + 406, + 312, + 418 + ], + "score": 1.0, + "content": "on mathematical analysis, 29(2):511–546, 1998.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5, + "bbox_fs": [ + 106, + 394, + 505, + 418 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 425, + 505, + 448 + ], + "lines": [ + { + "bbox": [ + 106, + 425, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 505, + 438 + ], + "score": 1.0, + "content": "Christian Szegedy, Wojciech Zaremba, Ilya Sutskever, Joan Bruna, Dumitru Erhan, Ian Goodfellow,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 115, + 437, + 505, + 448 + ], + "spans": [ + { + "bbox": [ + 115, + 437, + 505, + 448 + ], + "score": 1.0, + "content": "and Rob Fergus. Intriguing properties of neural networks. arXiv preprint arXiv:1312.6199, 2013.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5, + "bbox_fs": [ + 106, + 425, + 505, + 448 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 455, + 505, + 478 + ], + "lines": [ + { + "bbox": [ + 106, + 455, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 106, + 455, + 505, + 468 + ], + "score": 1.0, + "content": "Naftali Tishby and Noga Zaslavsky. Deep learning and the information bottleneck principle. In", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 116, + 466, + 403, + 478 + ], + "spans": [ + { + "bbox": [ + 116, + 466, + 403, + 478 + ], + "score": 1.0, + "content": "Information Theory Workshop (ITW), 2015 IEEE, pp. 1–5. IEEE, 2015.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26.5, + "bbox_fs": [ + 106, + 455, + 505, + 478 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 484, + 504, + 508 + ], + "lines": [ + { + "bbox": [ + 105, + 484, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 505, + 497 + ], + "score": 1.0, + "content": "Fisher Yu and Vladlen Koltun. Multi-scale context aggregation by dilated convolutions. arXiv", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 114, + 496, + 254, + 508 + ], + "spans": [ + { + "bbox": [ + 114, + 496, + 254, + 508 + ], + "score": 1.0, + "content": "preprint arXiv:1511.07122, 2015.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 484, + 505, + 508 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 514, + 505, + 538 + ], + "lines": [ + { + "bbox": [ + 106, + 513, + 505, + 527 + ], + "spans": [ + { + "bbox": [ + 106, + 514, + 297, + 527 + ], + "score": 1.0, + "content": "Sergey Zagoruyko and Nikos Komodakis.", + "type": "text" + }, + { + "bbox": [ + 312, + 513, + 426, + 527 + ], + "score": 1.0, + "content": "Wide residual networks.", + "type": "text" + }, + { + "bbox": [ + 438, + 514, + 505, + 527 + ], + "score": 1.0, + "content": "arXiv preprint", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 115, + 525, + 219, + 537 + ], + "spans": [ + { + "bbox": [ + 115, + 525, + 219, + 537 + ], + "score": 1.0, + "content": "arXiv:1605.07146, 2016.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30.5, + "bbox_fs": [ + 106, + 513, + 505, + 537 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 544, + 505, + 568 + ], + "lines": [ + { + "bbox": [ + 105, + 544, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 505, + 557 + ], + "score": 1.0, + "content": "Matthew D Zeiler and Rob Fergus. Visualizing and understanding convolutional networks. In", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 116, + 555, + 403, + 568 + ], + "spans": [ + { + "bbox": [ + 116, + 555, + 403, + 568 + ], + "score": 1.0, + "content": "European conference on computer vision, pp. 818–833. Springer, 2014.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32.5, + "bbox_fs": [ + 105, + 544, + 505, + 568 + ] + } + ] + } + ], + "_backend": "pipeline", + "_version_name": "2.2.2" +} \ No newline at end of file