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At a high level, there are two modeling paradigms which", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 430, + 505, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 505, + 442 + ], + "score": 1.0, + "content": "allow a model to deal with scale changes: models can be endowed with an internal notion of scale", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 441, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 505, + 453 + ], + "score": 1.0, + "content": "and transform their predictions accordingly, or instead, models can be designed to be specifically", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 452, + 505, + 463 + ], + "spans": [ + { + "bbox": [ + 106, + 452, + 505, + 463 + ], + "score": 1.0, + "content": "invariant to scale changes. In image classification, when scale changes are commonly a factor of 2,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 461, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 505, + 475 + ], + "score": 1.0, + "content": "it is often sufficient to make class prediction independent of scale. However, in tasks such as image", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 474, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 505, + 486 + ], + "score": 1.0, + "content": "segmentation, visual tracking, or object detection, scale changes can reach factors of 10 or more.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 484, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 505, + 497 + ], + "score": 1.0, + "content": "In these cases, it is intuitive that the ideal prediction should scale proportionally to the input. For", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 496, + 505, + 508 + ], + "spans": [ + { + "bbox": [ + 106, + 496, + 505, + 508 + ], + "score": 1.0, + "content": "example, the segmentation map of a nearby pedestrian should be easily converted to that of a distant", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 506, + 233, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 233, + 520 + ], + "score": 1.0, + "content": "person simply by downscaling.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 386, + 505, + 520 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 523, + 505, + 600 + ], + "lines": [ + { + "bbox": [ + 105, + 521, + 505, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 521, + 505, + 537 + ], + "score": 1.0, + "content": "Convolutional Neural Networks (CNNs) demonstrate state-of-the-art performance in a wide range", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 535, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 106, + 535, + 505, + 546 + ], + "score": 1.0, + "content": "of tasks. Yet, despite their built-in translation equivariance, they do not have a particular mechanism", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 546, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 106, + 546, + 505, + 557 + ], + "score": 1.0, + "content": "for dealing with scale changes. One way to make CNNs account for scale is to train them with data", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 555, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 104, + 555, + 505, + 569 + ], + "score": 1.0, + "content": "augmentation Barnard & Casasent (1991). This is, however, suitable only for global transformations.", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 567, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 505, + 579 + ], + "score": 1.0, + "content": "As an alternative, Henriques & Vedaldi (2017) and Tai et al. (2019) use the canonical coordinates", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 578, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 505, + 590 + ], + "score": 1.0, + "content": "of scale transformations to reduce scaling to well-studied translations. While these approaches do", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 589, + 422, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 422, + 602 + ], + "score": 1.0, + "content": "allow for scale equivariance, they consequently break translation equivariance.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 33, + "bbox_fs": [ + 104, + 521, + 505, + 602 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 605, + 505, + 694 + ], + "lines": [ + { + "bbox": [ + 106, + 606, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 106, + 606, + 505, + 618 + ], + "score": 1.0, + "content": "Several attempts have thus been made to extend CNNs to both scale and translation symmetry si-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 617, + 505, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 505, + 629 + ], + "score": 1.0, + "content": "multaneously. Some works use input or filter resizing to account for scaling in deep layers Xu et al.", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 627, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 506, + 640 + ], + "score": 1.0, + "content": "(2014); Kanazawa et al. (2014). Such methods are suboptimal due to the time complexity of tensor", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 639, + 506, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 506, + 651 + ], + "score": 1.0, + "content": "resizing and the need for interpolation. In Ghosh & Gupta (2019) the authors pre-calculate filters", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 649, + 506, + 663 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 506, + 663 + ], + "score": 1.0, + "content": "defined on several scales to build scale-invariant networks, while ignoring the important case of", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 659, + 505, + 674 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 505, + 674 + ], + "score": 1.0, + "content": "scale equivariance. In contrast, Worrall & Welling (2019) employ the theory of semigroup equiv-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 672, + 505, + 685 + ], + "spans": [ + { + "bbox": [ + 105, + 672, + 505, + 685 + ], + "score": 1.0, + "content": "ariant networks with scale-space as an example; however, this method is only suitable for integer", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 683, + 268, + 694 + ], + "spans": [ + { + "bbox": [ + 106, + 683, + 268, + 694 + ], + "score": 1.0, + "content": "downscale factors and therefore limited.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 40.5, + "bbox_fs": [ + 105, + 606, + 506, + 694 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 149 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "In this paper we develop a theory of scale-equivariant networks. We demonstrate the concept of", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "steerable filter parametrization which allows for scaling without the need for tensor resizing. Then", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 504, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 504, + 117 + ], + "score": 1.0, + "content": "we derive scale-equivariant convolution and demonstrate a fast algorithm for its implementation.", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "Furthermore, we experiment to determine to what degree the mathematical properties actually hold", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 138 + ], + "score": 1.0, + "content": "true. Finally, we conduct a set of experiments comparing our model with other methods for scale", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 138, + 267, + 149 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 267, + 149 + ], + "score": 1.0, + "content": "equivariance and local scale invariance.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 106, + 168, + 487, + 180 + ], + "lines": [ + { + "bbox": [ + 106, + 168, + 488, + 182 + ], + "spans": [ + { + "bbox": [ + 106, + 168, + 488, + 182 + ], + "score": 1.0, + "content": "The proposed model has the following advantages compared to other scale-equivariant models:", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 129, + 189, + 505, + 275 + ], + "lines": [ + { + "bbox": [ + 130, + 190, + 505, + 201 + ], + "spans": [ + { + "bbox": [ + 130, + 190, + 505, + 201 + ], + "score": 1.0, + "content": "1. It is equivariant to scale transformations with arbitrary discrete scale factors and is not", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 141, + 200, + 431, + 213 + ], + "spans": [ + { + "bbox": [ + 141, + 200, + 431, + 213 + ], + "score": 1.0, + "content": "limited to either integer scales or scales tailored by the image pixel grid.", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 129, + 215, + 505, + 228 + ], + "spans": [ + { + "bbox": [ + 129, + 215, + 505, + 228 + ], + "score": 1.0, + "content": "2. It does not rely on any image resampling techniques during training, and therefore, pro-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 141, + 226, + 457, + 239 + ], + "spans": [ + { + "bbox": [ + 141, + 226, + 457, + 239 + ], + "score": 1.0, + "content": "duces deep scale-equivariant representations free of any interpolation artifacts.", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 129, + 241, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 129, + 241, + 505, + 254 + ], + "score": 1.0, + "content": "3. The algorithm is based on the combination of tensor expansion and 2-dimensional convo-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 252, + 506, + 265 + ], + "spans": [ + { + "bbox": [ + 141, + 252, + 506, + 265 + ], + "score": 1.0, + "content": "lution, and demonstrates the same computation time as the general CNN with a comparable", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 142, + 263, + 188, + 275 + ], + "spans": [ + { + "bbox": [ + 142, + 263, + 188, + 275 + ], + "score": 1.0, + "content": "filter bank.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 10 + }, + { + "type": "title", + "bbox": [ + 107, + 290, + 208, + 303 + ], + "lines": [ + { + "bbox": [ + 104, + 289, + 209, + 306 + ], + "spans": [ + { + "bbox": [ + 104, + 289, + 209, + 306 + ], + "score": 1.0, + "content": "2 PRELIMINARIES", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 106, + 315, + 505, + 360 + ], + "lines": [ + { + "bbox": [ + 105, + 315, + 505, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 505, + 329 + ], + "score": 1.0, + "content": "Before we move into scale-equivariant mappings, we discuss some aspects of equivariance, scaling", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 327, + 504, + 339 + ], + "spans": [ + { + "bbox": [ + 106, + 327, + 504, + 339 + ], + "score": 1.0, + "content": "transformations, symmetry groups, and the functions defined on them. For simplicity, in this sec-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 338, + 505, + 350 + ], + "spans": [ + { + "bbox": [ + 106, + 338, + 505, + 350 + ], + "score": 1.0, + "content": "tion, we consider only 1-dimensional functions. The generalization to higher-dimensional cases is", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 348, + 173, + 361 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 173, + 361 + ], + "score": 1.0, + "content": "straightforward.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16.5 + }, + { + "type": "text", + "bbox": [ + 107, + 365, + 504, + 388 + ], + "lines": [ + { + "bbox": [ + 106, + 365, + 505, + 378 + ], + "spans": [ + { + "bbox": [ + 106, + 365, + 289, + 378 + ], + "score": 1.0, + "content": "Equivariance Let us consider some mapping", + "type": "text" + }, + { + "bbox": [ + 290, + 368, + 296, + 377 + ], + "score": 0.74, + "content": "g", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 365, + 390, + 378 + ], + "score": 1.0, + "content": ". It is equivariant under", + "type": "text" + }, + { + "bbox": [ + 390, + 366, + 402, + 376 + ], + "score": 0.88, + "content": "L _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 365, + 505, + 378 + ], + "score": 1.0, + "content": "if and only if there exists", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 107, + 375, + 477, + 390 + ], + "spans": [ + { + "bbox": [ + 107, + 377, + 119, + 389 + ], + "score": 0.89, + "content": "L _ { \\theta } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 119, + 375, + 158, + 390 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 159, + 376, + 225, + 389 + ], + "score": 0.93, + "content": "g \\circ L _ { \\theta } = L _ { \\theta } ^ { \\prime } \\circ g", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 375, + 259, + 390 + ], + "score": 1.0, + "content": ". In case", + "type": "text" + }, + { + "bbox": [ + 259, + 376, + 272, + 389 + ], + "score": 0.92, + "content": "L _ { \\theta } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 375, + 420, + 390 + ], + "score": 1.0, + "content": "is the identity mapping, the function", + "type": "text" + }, + { + "bbox": [ + 420, + 378, + 426, + 388 + ], + "score": 0.79, + "content": "g", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 375, + 477, + 390 + ], + "score": 1.0, + "content": "is invariant.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19.5 + }, + { + "type": "text", + "bbox": [ + 107, + 393, + 505, + 426 + ], + "lines": [ + { + "bbox": [ + 105, + 393, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 505, + 405 + ], + "score": 1.0, + "content": "In this paper we consider scaling transformations. In order to guarantee the equivariance of the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 405, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 505, + 416 + ], + "score": 1.0, + "content": "predictions to such transformations, and to improve the performance of the model, we seek to incor-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 416, + 276, + 427 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 276, + 427 + ], + "score": 1.0, + "content": "porate this property directly inside CNNs.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 106, + 432, + 435, + 444 + ], + "lines": [ + { + "bbox": [ + 105, + 431, + 435, + 445 + ], + "spans": [ + { + "bbox": [ + 105, + 431, + 209, + 445 + ], + "score": 1.0, + "content": "Scaling Given a function", + "type": "text" + }, + { + "bbox": [ + 209, + 432, + 254, + 444 + ], + "score": 0.92, + "content": "f : \\mathbb { R } \\to \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 431, + 435, + 445 + ], + "score": 1.0, + "content": ", a scale transformation is defined as follows:", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "interline_equation", + "bbox": [ + 240, + 448, + 371, + 463 + ], + "lines": [ + { + "bbox": [ + 240, + 448, + 371, + 463 + ], + "spans": [ + { + "bbox": [ + 240, + 448, + 371, + 463 + ], + "score": 0.91, + "content": "L _ { s } [ f ] ( x ) = f ( s ^ { - 1 } x ) , \\quad \\forall s > 0", + "type": "interline_equation", + "image_path": "13908d2afa72d06c5f750fd9545ff5fd44438e6667a9eef4ad001702836f5831.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 240, + 448, + 371, + 463 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 474, + 505, + 507 + ], + "lines": [ + { + "bbox": [ + 106, + 474, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 106, + 474, + 199, + 486 + ], + "score": 1.0, + "content": "We refer to cases with", + "type": "text" + }, + { + "bbox": [ + 199, + 475, + 224, + 484 + ], + "score": 0.9, + "content": "s > 1", + "type": "inline_equation" + }, + { + "bbox": [ + 224, + 474, + 341, + 486 + ], + "score": 1.0, + "content": "as upscale and to cases with", + "type": "text" + }, + { + "bbox": [ + 342, + 475, + 367, + 484 + ], + "score": 0.9, + "content": "s < 1", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 474, + 505, + 486 + ], + "score": 1.0, + "content": "as downscale. If we convolve the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 485, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 106, + 485, + 284, + 497 + ], + "score": 1.0, + "content": "downscaled function with an arbitrary filter", + "type": "text" + }, + { + "bbox": [ + 284, + 486, + 292, + 497 + ], + "score": 0.85, + "content": "\\psi", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 485, + 505, + 497 + ], + "score": 1.0, + "content": "and perform a simple change of variables inside the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 495, + 265, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 265, + 509 + ], + "score": 1.0, + "content": "integral, we get the following property:", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27 + }, + { + "type": "interline_equation", + "bbox": [ + 131, + 512, + 480, + 567 + ], + "lines": [ + { + "bbox": [ + 131, + 512, + 480, + 567 + ], + "spans": [ + { + "bbox": [ + 131, + 512, + 480, + 567 + ], + "score": 0.95, + "content": "\\begin{array} { l } { \\displaystyle [ L _ { s } [ f ] \\star \\psi ] ( x ) = \\int _ { \\mathbb R } L _ { s } [ f ] ( x ^ { \\prime } ) \\psi ( x ^ { \\prime } - x ) d x ^ { \\prime } = \\int _ { \\mathbb R } f ( s ^ { - 1 } x ^ { \\prime } ) \\psi ( x ^ { \\prime } - x ) d x ^ { \\prime } } \\\\ { \\displaystyle \\qquad = s \\int _ { \\mathbb R } f ( s ^ { - 1 } x ^ { \\prime } ) \\psi ( s ( s ^ { - 1 } x ^ { \\prime } - s ^ { - 1 } x ) ) d ( s ^ { - 1 } x ^ { \\prime } ) = s L _ { s } [ f \\star L _ { s ^ { - 1 } } [ \\psi ] ] ( x ) } \\end{array}", + "type": "interline_equation", + "image_path": "cecee878bf94f360d23a829c220e0eba59c655aa6e7d443c42052b97c09c1b7c.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 131, + 512, + 480, + 530.3333333333334 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 131, + 530.3333333333334, + 480, + 548.6666666666667 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 131, + 548.6666666666667, + 480, + 567.0000000000001 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 576, + 504, + 609 + ], + "lines": [ + { + "bbox": [ + 105, + 574, + 506, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 506, + 589 + ], + "score": 1.0, + "content": "In other words, convolution of the downscaled function with a filter can be expressed through a", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 587, + 505, + 599 + ], + "spans": [ + { + "bbox": [ + 106, + 587, + 505, + 599 + ], + "score": 1.0, + "content": "convolution of the function with the correspondingly upscaled filter where downscaling is performed", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 107, + 598, + 453, + 610 + ], + "spans": [ + { + "bbox": [ + 107, + 598, + 453, + 610 + ], + "score": 1.0, + "content": "afterwards. Equation 2 shows us that the standard convolution is not scale-equivariant.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 104, + 615, + 504, + 637 + ], + "lines": [ + { + "bbox": [ + 105, + 613, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 408, + 628 + ], + "score": 1.0, + "content": "Steerable Filters In order to make computations simpler, we reparametrize", + "type": "text" + }, + { + "bbox": [ + 408, + 614, + 501, + 627 + ], + "score": 0.93, + "content": "\\psi _ { \\sigma } ( x ) = \\sigma ^ { - 1 } \\psi ( \\sigma ^ { - 1 } x )", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 613, + 505, + 628 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 624, + 244, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 624, + 244, + 639 + ], + "score": 1.0, + "content": "which has the following property:", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35.5 + }, + { + "type": "interline_equation", + "bbox": [ + 223, + 642, + 388, + 657 + ], + "lines": [ + { + "bbox": [ + 223, + 642, + 388, + 657 + ], + "spans": [ + { + "bbox": [ + 223, + 642, + 388, + 657 + ], + "score": 0.91, + "content": "L _ { s ^ { - 1 } } [ \\psi _ { \\sigma } ] ( x ) = \\psi _ { \\sigma } ( s x ) = s ^ { - 1 } \\psi _ { s ^ { - 1 } \\sigma } ( x )", + "type": "interline_equation", + "image_path": "5c5e01cd9c1c7f93935e09fa08fbca4b21f2258a0937e7a59b257eb0bc5c51cc.jpg" + } + ] + } + ], + "index": 37, + "virtual_lines": [ + { + "bbox": [ + 223, + 642, + 388, + 657 + ], + "spans": [], + "index": 37 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 667, + 265, + 680 + ], + "lines": [ + { + "bbox": [ + 106, + 667, + 266, + 681 + ], + "spans": [ + { + "bbox": [ + 106, + 667, + 266, + 681 + ], + "score": 1.0, + "content": "It gives a shorter version of Equation 2:", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38 + }, + { + "type": "interline_equation", + "bbox": [ + 247, + 685, + 364, + 698 + ], + "lines": [ + { + "bbox": [ + 247, + 685, + 364, + 698 + ], + "spans": [ + { + "bbox": [ + 247, + 685, + 364, + 698 + ], + "score": 0.91, + "content": "L _ { s } [ f ] \\star \\psi _ { \\sigma } = L _ { s } [ f \\star \\psi _ { s ^ { - 1 } \\sigma } ]", + "type": "interline_equation", + "image_path": "7651f206035b5725275d06c6bc6580fff017e6d183066a68fe1bc3c8467c7de9.jpg" + } + ] + } + ], + "index": 39, + "virtual_lines": [ + { + "bbox": [ + 247, + 685, + 364, + 698 + ], + "spans": [], + "index": 39 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "We will refer to such a parameterization of filters as Steerable Filters because the scaling of these", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 720, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 734 + ], + "score": 1.0, + "content": "filters is the transformation of its parameters. Note that we may construct steerable filters from any", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 40.5 + } + ], + "page_idx": 1, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "score": 1.0, + "content": "2", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 149 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "In this paper we develop a theory of scale-equivariant networks. We demonstrate the concept of", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "steerable filter parametrization which allows for scaling without the need for tensor resizing. Then", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 504, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 504, + 117 + ], + "score": 1.0, + "content": "we derive scale-equivariant convolution and demonstrate a fast algorithm for its implementation.", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "Furthermore, we experiment to determine to what degree the mathematical properties actually hold", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 138 + ], + "score": 1.0, + "content": "true. Finally, we conduct a set of experiments comparing our model with other methods for scale", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 138, + 267, + 149 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 267, + 149 + ], + "score": 1.0, + "content": "equivariance and local scale invariance.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5, + "bbox_fs": [ + 105, + 82, + 506, + 149 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 168, + 487, + 180 + ], + "lines": [ + { + "bbox": [ + 106, + 168, + 488, + 182 + ], + "spans": [ + { + "bbox": [ + 106, + 168, + 488, + 182 + ], + "score": 1.0, + "content": "The proposed model has the following advantages compared to other scale-equivariant models:", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6, + "bbox_fs": [ + 106, + 168, + 488, + 182 + ] + }, + { + "type": "list", + "bbox": [ + 129, + 189, + 505, + 275 + ], + "lines": [ + { + "bbox": [ + 130, + 190, + 505, + 201 + ], + "spans": [ + { + "bbox": [ + 130, + 190, + 505, + 201 + ], + "score": 1.0, + "content": "1. It is equivariant to scale transformations with arbitrary discrete scale factors and is not", + "type": "text" + } + ], + "index": 7, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 200, + 431, + 213 + ], + "spans": [ + { + "bbox": [ + 141, + 200, + 431, + 213 + ], + "score": 1.0, + "content": "limited to either integer scales or scales tailored by the image pixel grid.", + "type": "text" + } + ], + "index": 8, + "is_list_end_line": true + }, + { + "bbox": [ + 129, + 215, + 505, + 228 + ], + "spans": [ + { + "bbox": [ + 129, + 215, + 505, + 228 + ], + "score": 1.0, + "content": "2. It does not rely on any image resampling techniques during training, and therefore, pro-", + "type": "text" + } + ], + "index": 9, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 226, + 457, + 239 + ], + "spans": [ + { + "bbox": [ + 141, + 226, + 457, + 239 + ], + "score": 1.0, + "content": "duces deep scale-equivariant representations free of any interpolation artifacts.", + "type": "text" + } + ], + "index": 10, + "is_list_end_line": true + }, + { + "bbox": [ + 129, + 241, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 129, + 241, + 505, + 254 + ], + "score": 1.0, + "content": "3. The algorithm is based on the combination of tensor expansion and 2-dimensional convo-", + "type": "text" + } + ], + "index": 11, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 252, + 506, + 265 + ], + "spans": [ + { + "bbox": [ + 141, + 252, + 506, + 265 + ], + "score": 1.0, + "content": "lution, and demonstrates the same computation time as the general CNN with a comparable", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 142, + 263, + 188, + 275 + ], + "spans": [ + { + "bbox": [ + 142, + 263, + 188, + 275 + ], + "score": 1.0, + "content": "filter bank.", + "type": "text" + } + ], + "index": 13, + "is_list_end_line": true + } + ], + "index": 10, + "bbox_fs": [ + 129, + 190, + 506, + 275 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 290, + 208, + 303 + ], + "lines": [ + { + "bbox": [ + 104, + 289, + 209, + 306 + ], + "spans": [ + { + "bbox": [ + 104, + 289, + 209, + 306 + ], + "score": 1.0, + "content": "2 PRELIMINARIES", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 106, + 315, + 505, + 360 + ], + "lines": [ + { + "bbox": [ + 105, + 315, + 505, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 505, + 329 + ], + "score": 1.0, + "content": "Before we move into scale-equivariant mappings, we discuss some aspects of equivariance, scaling", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 327, + 504, + 339 + ], + "spans": [ + { + "bbox": [ + 106, + 327, + 504, + 339 + ], + "score": 1.0, + "content": "transformations, symmetry groups, and the functions defined on them. For simplicity, in this sec-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 338, + 505, + 350 + ], + "spans": [ + { + "bbox": [ + 106, + 338, + 505, + 350 + ], + "score": 1.0, + "content": "tion, we consider only 1-dimensional functions. The generalization to higher-dimensional cases is", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 348, + 173, + 361 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 173, + 361 + ], + "score": 1.0, + "content": "straightforward.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16.5, + "bbox_fs": [ + 105, + 315, + 505, + 361 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 365, + 504, + 388 + ], + "lines": [ + { + "bbox": [ + 106, + 365, + 505, + 378 + ], + "spans": [ + { + "bbox": [ + 106, + 365, + 289, + 378 + ], + "score": 1.0, + "content": "Equivariance Let us consider some mapping", + "type": "text" + }, + { + "bbox": [ + 290, + 368, + 296, + 377 + ], + "score": 0.74, + "content": "g", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 365, + 390, + 378 + ], + "score": 1.0, + "content": ". It is equivariant under", + "type": "text" + }, + { + "bbox": [ + 390, + 366, + 402, + 376 + ], + "score": 0.88, + "content": "L _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 365, + 505, + 378 + ], + "score": 1.0, + "content": "if and only if there exists", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 107, + 375, + 477, + 390 + ], + "spans": [ + { + "bbox": [ + 107, + 377, + 119, + 389 + ], + "score": 0.89, + "content": "L _ { \\theta } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 119, + 375, + 158, + 390 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 159, + 376, + 225, + 389 + ], + "score": 0.93, + "content": "g \\circ L _ { \\theta } = L _ { \\theta } ^ { \\prime } \\circ g", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 375, + 259, + 390 + ], + "score": 1.0, + "content": ". In case", + "type": "text" + }, + { + "bbox": [ + 259, + 376, + 272, + 389 + ], + "score": 0.92, + "content": "L _ { \\theta } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 375, + 420, + 390 + ], + "score": 1.0, + "content": "is the identity mapping, the function", + "type": "text" + }, + { + "bbox": [ + 420, + 378, + 426, + 388 + ], + "score": 0.79, + "content": "g", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 375, + 477, + 390 + ], + "score": 1.0, + "content": "is invariant.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19.5, + "bbox_fs": [ + 106, + 365, + 505, + 390 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 393, + 505, + 426 + ], + "lines": [ + { + "bbox": [ + 105, + 393, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 505, + 405 + ], + "score": 1.0, + "content": "In this paper we consider scaling transformations. In order to guarantee the equivariance of the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 405, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 505, + 416 + ], + "score": 1.0, + "content": "predictions to such transformations, and to improve the performance of the model, we seek to incor-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 416, + 276, + 427 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 276, + 427 + ], + "score": 1.0, + "content": "porate this property directly inside CNNs.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22, + "bbox_fs": [ + 105, + 393, + 505, + 427 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 432, + 435, + 444 + ], + "lines": [ + { + "bbox": [ + 105, + 431, + 435, + 445 + ], + "spans": [ + { + "bbox": [ + 105, + 431, + 209, + 445 + ], + "score": 1.0, + "content": "Scaling Given a function", + "type": "text" + }, + { + "bbox": [ + 209, + 432, + 254, + 444 + ], + "score": 0.92, + "content": "f : \\mathbb { R } \\to \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 431, + 435, + 445 + ], + "score": 1.0, + "content": ", a scale transformation is defined as follows:", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 431, + 435, + 445 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 240, + 448, + 371, + 463 + ], + "lines": [ + { + "bbox": [ + 240, + 448, + 371, + 463 + ], + "spans": [ + { + "bbox": [ + 240, + 448, + 371, + 463 + ], + "score": 0.91, + "content": "L _ { s } [ f ] ( x ) = f ( s ^ { - 1 } x ) , \\quad \\forall s > 0", + "type": "interline_equation", + "image_path": "13908d2afa72d06c5f750fd9545ff5fd44438e6667a9eef4ad001702836f5831.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 240, + 448, + 371, + 463 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 474, + 505, + 507 + ], + "lines": [ + { + "bbox": [ + 106, + 474, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 106, + 474, + 199, + 486 + ], + "score": 1.0, + "content": "We refer to cases with", + "type": "text" + }, + { + "bbox": [ + 199, + 475, + 224, + 484 + ], + "score": 0.9, + "content": "s > 1", + "type": "inline_equation" + }, + { + "bbox": [ + 224, + 474, + 341, + 486 + ], + "score": 1.0, + "content": "as upscale and to cases with", + "type": "text" + }, + { + "bbox": [ + 342, + 475, + 367, + 484 + ], + "score": 0.9, + "content": "s < 1", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 474, + 505, + 486 + ], + "score": 1.0, + "content": "as downscale. If we convolve the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 485, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 106, + 485, + 284, + 497 + ], + "score": 1.0, + "content": "downscaled function with an arbitrary filter", + "type": "text" + }, + { + "bbox": [ + 284, + 486, + 292, + 497 + ], + "score": 0.85, + "content": "\\psi", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 485, + 505, + 497 + ], + "score": 1.0, + "content": "and perform a simple change of variables inside the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 495, + 265, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 265, + 509 + ], + "score": 1.0, + "content": "integral, we get the following property:", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 474, + 505, + 509 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 131, + 512, + 480, + 567 + ], + "lines": [ + { + "bbox": [ + 131, + 512, + 480, + 567 + ], + "spans": [ + { + "bbox": [ + 131, + 512, + 480, + 567 + ], + "score": 0.95, + "content": "\\begin{array} { l } { \\displaystyle [ L _ { s } [ f ] \\star \\psi ] ( x ) = \\int _ { \\mathbb R } L _ { s } [ f ] ( x ^ { \\prime } ) \\psi ( x ^ { \\prime } - x ) d x ^ { \\prime } = \\int _ { \\mathbb R } f ( s ^ { - 1 } x ^ { \\prime } ) \\psi ( x ^ { \\prime } - x ) d x ^ { \\prime } } \\\\ { \\displaystyle \\qquad = s \\int _ { \\mathbb R } f ( s ^ { - 1 } x ^ { \\prime } ) \\psi ( s ( s ^ { - 1 } x ^ { \\prime } - s ^ { - 1 } x ) ) d ( s ^ { - 1 } x ^ { \\prime } ) = s L _ { s } [ f \\star L _ { s ^ { - 1 } } [ \\psi ] ] ( x ) } \\end{array}", + "type": "interline_equation", + "image_path": "cecee878bf94f360d23a829c220e0eba59c655aa6e7d443c42052b97c09c1b7c.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 131, + 512, + 480, + 530.3333333333334 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 131, + 530.3333333333334, + 480, + 548.6666666666667 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 131, + 548.6666666666667, + 480, + 567.0000000000001 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 576, + 504, + 609 + ], + "lines": [ + { + "bbox": [ + 105, + 574, + 506, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 506, + 589 + ], + "score": 1.0, + "content": "In other words, convolution of the downscaled function with a filter can be expressed through a", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 587, + 505, + 599 + ], + "spans": [ + { + "bbox": [ + 106, + 587, + 505, + 599 + ], + "score": 1.0, + "content": "convolution of the function with the correspondingly upscaled filter where downscaling is performed", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 107, + 598, + 453, + 610 + ], + "spans": [ + { + "bbox": [ + 107, + 598, + 453, + 610 + ], + "score": 1.0, + "content": "afterwards. Equation 2 shows us that the standard convolution is not scale-equivariant.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 574, + 506, + 610 + ] + }, + { + "type": "text", + "bbox": [ + 104, + 615, + 504, + 637 + ], + "lines": [ + { + "bbox": [ + 105, + 613, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 408, + 628 + ], + "score": 1.0, + "content": "Steerable Filters In order to make computations simpler, we reparametrize", + "type": "text" + }, + { + "bbox": [ + 408, + 614, + 501, + 627 + ], + "score": 0.93, + "content": "\\psi _ { \\sigma } ( x ) = \\sigma ^ { - 1 } \\psi ( \\sigma ^ { - 1 } x )", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 613, + 505, + 628 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 624, + 244, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 624, + 244, + 639 + ], + "score": 1.0, + "content": "which has the following property:", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35.5, + "bbox_fs": [ + 105, + 613, + 505, + 639 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 223, + 642, + 388, + 657 + ], + "lines": [ + { + "bbox": [ + 223, + 642, + 388, + 657 + ], + "spans": [ + { + "bbox": [ + 223, + 642, + 388, + 657 + ], + "score": 0.91, + "content": "L _ { s ^ { - 1 } } [ \\psi _ { \\sigma } ] ( x ) = \\psi _ { \\sigma } ( s x ) = s ^ { - 1 } \\psi _ { s ^ { - 1 } \\sigma } ( x )", + "type": "interline_equation", + "image_path": "5c5e01cd9c1c7f93935e09fa08fbca4b21f2258a0937e7a59b257eb0bc5c51cc.jpg" + } + ] + } + ], + "index": 37, + "virtual_lines": [ + { + "bbox": [ + 223, + 642, + 388, + 657 + ], + "spans": [], + "index": 37 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 667, + 265, + 680 + ], + "lines": [ + { + "bbox": [ + 106, + 667, + 266, + 681 + ], + "spans": [ + { + "bbox": [ + 106, + 667, + 266, + 681 + ], + "score": 1.0, + "content": "It gives a shorter version of Equation 2:", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38, + "bbox_fs": [ + 106, + 667, + 266, + 681 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 247, + 685, + 364, + 698 + ], + "lines": [ + { + "bbox": [ + 247, + 685, + 364, + 698 + ], + "spans": [ + { + "bbox": [ + 247, + 685, + 364, + 698 + ], + "score": 0.91, + "content": "L _ { s } [ f ] \\star \\psi _ { \\sigma } = L _ { s } [ f \\star \\psi _ { s ^ { - 1 } \\sigma } ]", + "type": "interline_equation", + "image_path": "7651f206035b5725275d06c6bc6580fff017e6d183066a68fe1bc3c8467c7de9.jpg" + } + ] + } + ], + "index": 39, + "virtual_lines": [ + { + "bbox": [ + 247, + 685, + 364, + 698 + ], + "spans": [], + "index": 39 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "We will refer to such a parameterization of filters as Steerable Filters because the scaling of these", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 720, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 734 + ], + "score": 1.0, + "content": "filters is the transformation of its parameters. Note that we may construct steerable filters from any", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 82, + 505, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 94 + ], + "score": 1.0, + "content": "function. This has the important consequence that it does not restrict our approach. Rather it will", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "make the analysis easier for discrete data. Moreover, note that any linear combination of steerable", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 199, + 116 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 199, + 116 + ], + "score": 1.0, + "content": "filters is still steerable.", + "type": "text", + "cross_page": true + } + ], + "index": 2 + } + ], + "index": 40.5, + "bbox_fs": [ + 105, + 709, + 505, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 108, + 82, + 504, + 115 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 94 + ], + "score": 1.0, + "content": "function. This has the important consequence that it does not restrict our approach. Rather it will", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "make the analysis easier for discrete data. Moreover, note that any linear combination of steerable", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 199, + 116 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 199, + 116 + ], + "score": 1.0, + "content": "filters is still steerable.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 106, + 121, + 505, + 222 + ], + "lines": [ + { + "bbox": [ + 106, + 121, + 506, + 135 + ], + "spans": [ + { + "bbox": [ + 106, + 121, + 397, + 135 + ], + "score": 1.0, + "content": "Scale-Translation Group All possible scales form the scaling group", + "type": "text" + }, + { + "bbox": [ + 397, + 122, + 405, + 131 + ], + "score": 0.78, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 121, + 506, + 135 + ], + "score": 1.0, + "content": ". Here we consider the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 132, + 506, + 145 + ], + "spans": [ + { + "bbox": [ + 106, + 132, + 282, + 145 + ], + "score": 1.0, + "content": "discrete scale group, i.e. scales of the form", + "type": "text" + }, + { + "bbox": [ + 282, + 132, + 381, + 144 + ], + "score": 0.91, + "content": "\\dots a ^ { - 1 } , a ^ { - 1 } , 1 , a , a ^ { \\tilde { 2 } } , \\tilde { \\dots } .", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 132, + 425, + 145 + ], + "score": 1.0, + "content": "with base", + "type": "text" + }, + { + "bbox": [ + 426, + 135, + 432, + 142 + ], + "score": 0.69, + "content": "a", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 132, + 506, + 145 + ], + "score": 1.0, + "content": "as a parameter of", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 143, + 506, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 506, + 156 + ], + "score": 1.0, + "content": "our method. Analysis of this group by itself breaks the translation equivariance of CNNs. Thus we", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "score": 1.0, + "content": "seek to incorporate scale and translation symmetries into CNNs, and, therefore consider the Scale-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 164, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 182, + 178 + ], + "score": 1.0, + "content": "Translation Group", + "type": "text" + }, + { + "bbox": [ + 182, + 166, + 192, + 175 + ], + "score": 0.72, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 164, + 379, + 178 + ], + "score": 1.0, + "content": ". It is a semidirect product of the scaling group", + "type": "text" + }, + { + "bbox": [ + 380, + 166, + 388, + 175 + ], + "score": 0.8, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 164, + 505, + 178 + ], + "score": 1.0, + "content": "and the group of translations", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 176, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 106, + 176, + 136, + 186 + ], + "score": 0.88, + "content": "T \\cong \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 136, + 176, + 205, + 189 + ], + "score": 1.0, + "content": ". In other words:", + "type": "text" + }, + { + "bbox": [ + 205, + 176, + 313, + 188 + ], + "score": 0.91, + "content": "H = \\{ ( s , t ) | s \\in S , t \\in T \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 176, + 505, + 189 + ], + "score": 1.0, + "content": ". For multiplication of group elements, we have", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 107, + 187, + 505, + 202 + ], + "spans": [ + { + "bbox": [ + 107, + 188, + 250, + 200 + ], + "score": 0.9, + "content": "( s _ { 2 } , t _ { 2 } ) \\cdot ( s _ { 1 } , t _ { 1 } ) = ( s _ { 2 } s _ { 1 } , s _ { 2 } t _ { 1 } + t _ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 187, + 327, + 202 + ], + "score": 1.0, + "content": "and for the inverse", + "type": "text" + }, + { + "bbox": [ + 327, + 187, + 500, + 201 + ], + "score": 0.91, + "content": "( s _ { 2 } , t _ { 2 } ) ^ { - 1 } \\cdot ( s _ { 1 } , t _ { 1 } ) = ( s _ { 2 } ^ { - 1 } s _ { 1 } , s _ { 2 } ^ { - 1 } ( t _ { 1 } - t _ { 2 } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 187, + 505, + 202 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 104, + 198, + 505, + 213 + ], + "spans": [ + { + "bbox": [ + 104, + 198, + 443, + 213 + ], + "score": 1.0, + "content": "Additionally, for the corresponding scaling and translation transformations, we have", + "type": "text" + }, + { + "bbox": [ + 443, + 200, + 505, + 211 + ], + "score": 0.88, + "content": "L _ { s t } = L _ { s } L _ { t } \\neq", + "type": "inline_equation" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 210, + 347, + 223 + ], + "spans": [ + { + "bbox": [ + 106, + 210, + 129, + 222 + ], + "score": 0.9, + "content": "L _ { t } L _ { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 129, + 210, + 347, + 223 + ], + "score": 1.0, + "content": ", which means that the order of the operations matters.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 106, + 227, + 505, + 284 + ], + "lines": [ + { + "bbox": [ + 106, + 227, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 106, + 227, + 421, + 239 + ], + "score": 1.0, + "content": "From now on, we will work with functions defined on groups, i.e. mappings", + "type": "text" + }, + { + "bbox": [ + 421, + 227, + 456, + 237 + ], + "score": 0.9, + "content": "H \\to \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 227, + 505, + 239 + ], + "score": 1.0, + "content": ". Note, that", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 238, + 505, + 250 + ], + "spans": [ + { + "bbox": [ + 106, + 238, + 172, + 250 + ], + "score": 1.0, + "content": "simple function", + "type": "text" + }, + { + "bbox": [ + 173, + 239, + 222, + 250 + ], + "score": 0.93, + "content": "f : \\mathbb { R } \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 238, + 371, + 250 + ], + "score": 1.0, + "content": "may be considered as a function on", + "type": "text" + }, + { + "bbox": [ + 372, + 239, + 382, + 248 + ], + "score": 0.8, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 238, + 505, + 250 + ], + "score": 1.0, + "content": "with constant value along the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 107, + 248, + 505, + 262 + ], + "spans": [ + { + "bbox": [ + 107, + 250, + 114, + 259 + ], + "score": 0.81, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 115, + 248, + 340, + 262 + ], + "score": 1.0, + "content": "axis. Therefore, Equation 4 holds true for functions on", + "type": "text" + }, + { + "bbox": [ + 340, + 250, + 350, + 259 + ], + "score": 0.83, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 248, + 505, + 262 + ], + "score": 1.0, + "content": "as well. One thing we should keep in", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 260, + 505, + 272 + ], + "spans": [ + { + "bbox": [ + 106, + 260, + 222, + 272 + ], + "score": 1.0, + "content": "mind is that when we apply", + "type": "text" + }, + { + "bbox": [ + 222, + 261, + 234, + 271 + ], + "score": 0.88, + "content": "L _ { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 260, + 300, + 272 + ], + "score": 1.0, + "content": "to functions on", + "type": "text" + }, + { + "bbox": [ + 300, + 261, + 311, + 270 + ], + "score": 0.83, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 260, + 330, + 272 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 330, + 261, + 338, + 270 + ], + "score": 0.83, + "content": "\\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 260, + 505, + 272 + ], + "score": 1.0, + "content": "we use different notations. For example", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 270, + 427, + 284 + ], + "spans": [ + { + "bbox": [ + 106, + 271, + 197, + 284 + ], + "score": 0.92, + "content": "L _ { s } [ f ] ( x ^ { \\prime } ) = f ( s ^ { - 1 } x ^ { \\prime } )", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 270, + 215, + 284 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 216, + 271, + 427, + 284 + ], + "score": 0.83, + "content": "\\begin{array} { r } { \\bar { L } _ { s } [ f ] ( s ^ { \\prime } , t ^ { \\prime } ) = f ( ( s , 0 ) ^ { - 1 } ( s ^ { \\prime } , t ^ { \\prime } ) ) = f ( s ^ { - 1 } s ^ { \\prime } , s ^ { - 1 } t ^ { \\prime } ) } \\end{array}", + "type": "inline_equation" + } + ], + "index": 16 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 105, + 287, + 505, + 311 + ], + "lines": [ + { + "bbox": [ + 105, + 287, + 505, + 301 + ], + "spans": [ + { + "bbox": [ + 105, + 287, + 304, + 301 + ], + "score": 1.0, + "content": "Group-Equivariant Convolution Given group", + "type": "text" + }, + { + "bbox": [ + 304, + 288, + 313, + 298 + ], + "score": 0.82, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 287, + 393, + 301 + ], + "score": 1.0, + "content": "and two functions", + "type": "text" + }, + { + "bbox": [ + 393, + 289, + 401, + 300 + ], + "score": 0.86, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 287, + 421, + 301 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 421, + 289, + 430, + 299 + ], + "score": 0.85, + "content": "\\psi", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 287, + 491, + 301 + ], + "score": 1.0, + "content": "defined on it,", + "type": "text" + }, + { + "bbox": [ + 492, + 288, + 501, + 298 + ], + "score": 0.77, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 287, + 505, + 301 + ], + "score": 1.0, + "content": "-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 299, + 248, + 312 + ], + "spans": [ + { + "bbox": [ + 106, + 299, + 248, + 312 + ], + "score": 1.0, + "content": "equivariant convolution is given by", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17.5 + }, + { + "type": "interline_equation", + "bbox": [ + 165, + 313, + 445, + 340 + ], + "lines": [ + { + "bbox": [ + 165, + 313, + 445, + 340 + ], + "spans": [ + { + "bbox": [ + 165, + 313, + 445, + 340 + ], + "score": 0.94, + "content": "[ f \\star _ { G } \\psi ] ( g ) = \\int _ { G } f ( g ^ { \\prime } ) L _ { g } [ \\psi ] ( g ^ { \\prime } ) d \\mu ( g ^ { \\prime } ) = \\int _ { G } f ( g ^ { \\prime } ) \\psi ( g ^ { - 1 } g ^ { \\prime } ) d \\mu ( g ^ { \\prime } )", + "type": "interline_equation", + "image_path": "5988433815b304c1229351695466f4b8fe3fc481fadcd0844c1eef82a7d8bd97.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 165, + 313, + 445, + 340 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 344, + 505, + 389 + ], + "lines": [ + { + "bbox": [ + 105, + 343, + 506, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 129, + 357 + ], + "score": 1.0, + "content": "Here", + "type": "text" + }, + { + "bbox": [ + 129, + 344, + 151, + 356 + ], + "score": 0.91, + "content": "\\mu ( g ^ { \\prime } )", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 343, + 457, + 357 + ], + "score": 1.0, + "content": "is the Haar measure also known as invariant measure Folland (2016). For", + "type": "text" + }, + { + "bbox": [ + 457, + 344, + 489, + 354 + ], + "score": 0.91, + "content": "T \\cong \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 343, + 506, + 357 + ], + "score": 1.0, + "content": "we", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 354, + 506, + 368 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 127, + 368 + ], + "score": 1.0, + "content": "have", + "type": "text" + }, + { + "bbox": [ + 128, + 355, + 182, + 367 + ], + "score": 0.92, + "content": "d \\mu ( g ^ { \\prime } ) = d g ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 354, + 506, + 368 + ], + "score": 1.0, + "content": ". For discrete groups, the Haar measure is the counting measure, and integration", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 366, + 504, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 497, + 379 + ], + "score": 1.0, + "content": "becomes a discrete sum. This formula tells us that the output of the convolution evaluated at point", + "type": "text" + }, + { + "bbox": [ + 498, + 370, + 504, + 378 + ], + "score": 0.73, + "content": "g", + "type": "inline_equation" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 376, + 413, + 391 + ], + "spans": [ + { + "bbox": [ + 104, + 376, + 272, + 391 + ], + "score": 1.0, + "content": "is the inner product between the function", + "type": "text" + }, + { + "bbox": [ + 272, + 378, + 279, + 389 + ], + "score": 0.86, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 376, + 384, + 391 + ], + "score": 1.0, + "content": "and the transformed filter", + "type": "text" + }, + { + "bbox": [ + 384, + 377, + 409, + 390 + ], + "score": 0.92, + "content": "L _ { g } [ \\psi ]", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 376, + 413, + 391 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21.5 + }, + { + "type": "title", + "bbox": [ + 108, + 404, + 296, + 417 + ], + "lines": [ + { + "bbox": [ + 104, + 402, + 298, + 420 + ], + "spans": [ + { + "bbox": [ + 104, + 402, + 298, + 420 + ], + "score": 1.0, + "content": "3 SCALE-EQUIVARIANT MAPPINGS", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 106, + 428, + 384, + 441 + ], + "lines": [ + { + "bbox": [ + 105, + 426, + 385, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 385, + 443 + ], + "score": 1.0, + "content": "Now we define the main building blocks of scale-equivariant models.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 108, + 446, + 505, + 491 + ], + "lines": [ + { + "bbox": [ + 106, + 446, + 505, + 458 + ], + "spans": [ + { + "bbox": [ + 106, + 446, + 505, + 458 + ], + "score": 1.0, + "content": "Scale Convolution In order to derive scale convolution, we start from group equivariant convolution", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 457, + 505, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 127, + 470 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 127, + 457, + 161, + 467 + ], + "score": 0.89, + "content": "G = H", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 457, + 505, + 470 + ], + "score": 1.0, + "content": ". 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Given", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 478, + 489, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 157, + 492 + ], + "score": 1.0, + "content": "the function", + "type": "text" + }, + { + "bbox": [ + 157, + 479, + 184, + 491 + ], + "score": 0.92, + "content": "f ( s , t )", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 478, + 269, + 492 + ], + "score": 1.0, + "content": "and a steerable filter", + "type": "text" + }, + { + "bbox": [ + 269, + 479, + 302, + 491 + ], + "score": 0.93, + "content": "\\psi _ { \\sigma } ( s , t )", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 478, + 348, + 492 + ], + "score": 1.0, + "content": "defined on", + "type": "text" + }, + { + "bbox": [ + 348, + 479, + 358, + 489 + ], + "score": 0.82, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 478, + 489, + 492 + ], + "score": 1.0, + "content": ", a scale convolution is given by:", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 27.5 + }, + { + "type": "interline_equation", + "bbox": [ + 119, + 495, + 479, + 551 + ], + "lines": [ + { + "bbox": [ + 119, + 495, + 479, + 551 + ], + "spans": [ + { + "bbox": [ + 119, + 495, + 479, + 551 + ], + "score": 0.93, + "content": "\\begin{array} { l } { { [ f \\star _ { \\cal H } \\psi _ { \\sigma } ] ( s , t ) = \\displaystyle \\int _ { \\cal S } \\int _ { \\cal T } f ( s ^ { \\prime } , t ^ { \\prime } ) { \\cal L } _ { s t } [ \\psi _ { \\sigma } ] ( s ^ { \\prime } , t ^ { \\prime } ) d \\mu ( s ^ { \\prime } ) d \\mu ( t ^ { \\prime } ) } } \\\\ { { { } } } \\\\ { { { } = \\displaystyle \\sum _ { s ^ { \\prime } } \\displaystyle \\int _ { \\cal T } f ( s ^ { \\prime } , t ^ { \\prime } ) \\psi _ { s \\sigma } ( s ^ { - 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We will refer to models using scale-equivariant layers", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 673, + 426, + 687 + ], + "spans": [ + { + "bbox": [ + 105, + 673, + 398, + 687 + ], + "score": 1.0, + "content": "with steerable filters as Scale-Equivariant Steerable Networks, or shortly", + "type": "text" + }, + { + "bbox": [ + 398, + 674, + 426, + 685 + ], + "score": 0.33, + "content": "S \\bar { E } S N ^ { 1 }", + "type": "inline_equation" + } + ], + "index": 42 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 107, + 691, + 504, + 714 + ], + "lines": [ + { + "bbox": [ + 105, + 690, + 506, + 704 + ], + "spans": [ + { + "bbox": [ + 105, + 690, + 506, + 704 + ], + "score": 1.0, + "content": "Nonlinearities In order to guarantee the equivariance of the network to scale transformations, we", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 702, + 506, + 714 + ], + "spans": [ + { + "bbox": [ + 105, + 702, + 506, + 714 + ], + "score": 1.0, + "content": "use scale equivariant nonlinearities. 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Here we consider the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 132, + 506, + 145 + ], + "spans": [ + { + "bbox": [ + 106, + 132, + 282, + 145 + ], + "score": 1.0, + "content": "discrete scale group, i.e. scales of the form", + "type": "text" + }, + { + "bbox": [ + 282, + 132, + 381, + 144 + ], + "score": 0.91, + "content": "\\dots a ^ { - 1 } , a ^ { - 1 } , 1 , a , a ^ { \\tilde { 2 } } , \\tilde { \\dots } .", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 132, + 425, + 145 + ], + "score": 1.0, + "content": "with base", + "type": "text" + }, + { + "bbox": [ + 426, + 135, + 432, + 142 + ], + "score": 0.69, + "content": "a", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 132, + 506, + 145 + ], + "score": 1.0, + "content": "as a parameter of", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 143, + 506, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 506, + 156 + ], + "score": 1.0, + "content": "our method. Analysis of this group by itself breaks the translation equivariance of CNNs. Thus we", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "score": 1.0, + "content": "seek to incorporate scale and translation symmetries into CNNs, and, therefore consider the Scale-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 164, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 182, + 178 + ], + "score": 1.0, + "content": "Translation Group", + "type": "text" + }, + { + "bbox": [ + 182, + 166, + 192, + 175 + ], + "score": 0.72, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 164, + 379, + 178 + ], + "score": 1.0, + "content": ". It is a semidirect product of the scaling group", + "type": "text" + }, + { + "bbox": [ + 380, + 166, + 388, + 175 + ], + "score": 0.8, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 164, + 505, + 178 + ], + "score": 1.0, + "content": "and the group of translations", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 176, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 106, + 176, + 136, + 186 + ], + "score": 0.88, + "content": "T \\cong \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 136, + 176, + 205, + 189 + ], + "score": 1.0, + "content": ". In other words:", + "type": "text" + }, + { + "bbox": [ + 205, + 176, + 313, + 188 + ], + "score": 0.91, + "content": "H = \\{ ( s , t ) | s \\in S , t \\in T \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 176, + 505, + 189 + ], + "score": 1.0, + "content": ". For multiplication of group elements, we have", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 107, + 187, + 505, + 202 + ], + "spans": [ + { + "bbox": [ + 107, + 188, + 250, + 200 + ], + "score": 0.9, + "content": "( s _ { 2 } , t _ { 2 } ) \\cdot ( s _ { 1 } , t _ { 1 } ) = ( s _ { 2 } s _ { 1 } , s _ { 2 } t _ { 1 } + t _ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 187, + 327, + 202 + ], + "score": 1.0, + "content": "and for the inverse", + "type": "text" + }, + { + "bbox": [ + 327, + 187, + 500, + 201 + ], + "score": 0.91, + "content": "( s _ { 2 } , t _ { 2 } ) ^ { - 1 } \\cdot ( s _ { 1 } , t _ { 1 } ) = ( s _ { 2 } ^ { - 1 } s _ { 1 } , s _ { 2 } ^ { - 1 } ( t _ { 1 } - t _ { 2 } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 187, + 505, + 202 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 104, + 198, + 505, + 213 + ], + "spans": [ + { + "bbox": [ + 104, + 198, + 443, + 213 + ], + "score": 1.0, + "content": "Additionally, for the corresponding scaling and translation transformations, we have", + "type": "text" + }, + { + "bbox": [ + 443, + 200, + 505, + 211 + ], + "score": 0.88, + "content": "L _ { s t } = L _ { s } L _ { t } \\neq", + "type": "inline_equation" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 210, + 347, + 223 + ], + "spans": [ + { + "bbox": [ + 106, + 210, + 129, + 222 + ], + "score": 0.9, + "content": "L _ { t } L _ { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 129, + 210, + 347, + 223 + ], + "score": 1.0, + "content": ", which means that the order of the operations matters.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 7, + "bbox_fs": [ + 104, + 121, + 506, + 223 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 227, + 505, + 284 + ], + "lines": [ + { + "bbox": [ + 106, + 227, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 106, + 227, + 421, + 239 + ], + "score": 1.0, + "content": "From now on, we will work with functions defined on groups, i.e. mappings", + "type": "text" + }, + { + "bbox": [ + 421, + 227, + 456, + 237 + ], + "score": 0.9, + "content": "H \\to \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 227, + 505, + 239 + ], + "score": 1.0, + "content": ". Note, that", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 238, + 505, + 250 + ], + "spans": [ + { + "bbox": [ + 106, + 238, + 172, + 250 + ], + "score": 1.0, + "content": "simple function", + "type": "text" + }, + { + "bbox": [ + 173, + 239, + 222, + 250 + ], + "score": 0.93, + "content": "f : \\mathbb { R } \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 238, + 371, + 250 + ], + "score": 1.0, + "content": "may be considered as a function on", + "type": "text" + }, + { + "bbox": [ + 372, + 239, + 382, + 248 + ], + "score": 0.8, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 238, + 505, + 250 + ], + "score": 1.0, + "content": "with constant value along the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 107, + 248, + 505, + 262 + ], + "spans": [ + { + "bbox": [ + 107, + 250, + 114, + 259 + ], + "score": 0.81, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 115, + 248, + 340, + 262 + ], + "score": 1.0, + "content": "axis. 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One thing we should keep in", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 260, + 505, + 272 + ], + "spans": [ + { + "bbox": [ + 106, + 260, + 222, + 272 + ], + "score": 1.0, + "content": "mind is that when we apply", + "type": "text" + }, + { + "bbox": [ + 222, + 261, + 234, + 271 + ], + "score": 0.88, + "content": "L _ { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 260, + 300, + 272 + ], + "score": 1.0, + "content": "to functions on", + "type": "text" + }, + { + "bbox": [ + 300, + 261, + 311, + 270 + ], + "score": 0.83, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 260, + 330, + 272 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 330, + 261, + 338, + 270 + ], + "score": 0.83, + "content": "\\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 260, + 505, + 272 + ], + "score": 1.0, + "content": "we use different notations. For example", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 270, + 427, + 284 + ], + "spans": [ + { + "bbox": [ + 106, + 271, + 197, + 284 + ], + "score": 0.92, + "content": "L _ { s } [ f ] ( x ^ { \\prime } ) = f ( s ^ { - 1 } x ^ { \\prime } )", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 270, + 215, + 284 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 216, + 271, + 427, + 284 + ], + "score": 0.83, + "content": "\\begin{array} { r } { \\bar { L } _ { s } [ f ] ( s ^ { \\prime } , t ^ { \\prime } ) = f ( ( s , 0 ) ^ { - 1 } ( s ^ { \\prime } , t ^ { \\prime } ) ) = f ( s ^ { - 1 } s ^ { \\prime } , s ^ { - 1 } t ^ { \\prime } ) } \\end{array}", + "type": "inline_equation" + } + ], + "index": 16 + } + ], + "index": 14, + "bbox_fs": [ + 106, + 227, + 505, + 284 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 287, + 505, + 311 + ], + "lines": [ + { + "bbox": [ + 105, + 287, + 505, + 301 + ], + "spans": [ + { + "bbox": [ + 105, + 287, + 304, + 301 + ], + "score": 1.0, + "content": "Group-Equivariant Convolution Given group", + "type": "text" + }, + { + "bbox": [ + 304, + 288, + 313, + 298 + ], + "score": 0.82, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 287, + 393, + 301 + ], + "score": 1.0, + "content": "and two functions", + "type": "text" + }, + { + "bbox": [ + 393, + 289, + 401, + 300 + ], + "score": 0.86, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 287, + 421, + 301 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 421, + 289, + 430, + 299 + ], + "score": 0.85, + "content": "\\psi", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 287, + 491, + 301 + ], + "score": 1.0, + "content": "defined on it,", + "type": "text" + }, + { + "bbox": [ + 492, + 288, + 501, + 298 + ], + "score": 0.77, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 287, + 505, + 301 + ], + "score": 1.0, + "content": "-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 299, + 248, + 312 + ], + "spans": [ + { + "bbox": [ + 106, + 299, + 248, + 312 + ], + "score": 1.0, + "content": "equivariant convolution is given by", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17.5, + "bbox_fs": [ + 105, + 287, + 505, + 312 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 165, + 313, + 445, + 340 + ], + "lines": [ + { + "bbox": [ + 165, + 313, + 445, + 340 + ], + "spans": [ + { + "bbox": [ + 165, + 313, + 445, + 340 + ], + "score": 0.94, + "content": "[ f \\star _ { G } \\psi ] ( g ) = \\int _ { G } f ( g ^ { \\prime } ) L _ { g } [ \\psi ] ( g ^ { \\prime } ) d \\mu ( g ^ { \\prime } ) = \\int _ { G } f ( g ^ { \\prime } ) \\psi ( g ^ { - 1 } g ^ { \\prime } ) d \\mu ( g ^ { \\prime } )", + "type": "interline_equation", + "image_path": "5988433815b304c1229351695466f4b8fe3fc481fadcd0844c1eef82a7d8bd97.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 165, + 313, + 445, + 340 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 344, + 505, + 389 + ], + "lines": [ + { + "bbox": [ + 105, + 343, + 506, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 129, + 357 + ], + "score": 1.0, + "content": "Here", + "type": "text" + }, + { + "bbox": [ + 129, + 344, + 151, + 356 + ], + "score": 0.91, + "content": "\\mu ( g ^ { \\prime } )", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 343, + 457, + 357 + ], + "score": 1.0, + "content": "is the Haar measure also known as invariant measure Folland (2016). For", + "type": "text" + }, + { + "bbox": [ + 457, + 344, + 489, + 354 + ], + "score": 0.91, + "content": "T \\cong \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 343, + 506, + 357 + ], + "score": 1.0, + "content": "we", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 354, + 506, + 368 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 127, + 368 + ], + "score": 1.0, + "content": "have", + "type": "text" + }, + { + "bbox": [ + 128, + 355, + 182, + 367 + ], + "score": 0.92, + "content": "d \\mu ( g ^ { \\prime } ) = d g ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 354, + 506, + 368 + ], + "score": 1.0, + "content": ". For discrete groups, the Haar measure is the counting measure, and integration", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 366, + 504, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 497, + 379 + ], + "score": 1.0, + "content": "becomes a discrete sum. This formula tells us that the output of the convolution evaluated at point", + "type": "text" + }, + { + "bbox": [ + 498, + 370, + 504, + 378 + ], + "score": 0.73, + "content": "g", + "type": "inline_equation" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 376, + 413, + 391 + ], + "spans": [ + { + "bbox": [ + 104, + 376, + 272, + 391 + ], + "score": 1.0, + "content": "is the inner product between the function", + "type": "text" + }, + { + "bbox": [ + 272, + 378, + 279, + 389 + ], + "score": 0.86, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 376, + 384, + 391 + ], + "score": 1.0, + "content": "and the transformed filter", + "type": "text" + }, + { + "bbox": [ + 384, + 377, + 409, + 390 + ], + "score": 0.92, + "content": "L _ { g } [ \\psi ]", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 376, + 413, + 391 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21.5, + "bbox_fs": [ + 104, + 343, + 506, + 391 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 404, + 296, + 417 + ], + "lines": [ + { + "bbox": [ + 104, + 402, + 298, + 420 + ], + "spans": [ + { + "bbox": [ + 104, + 402, + 298, + 420 + ], + "score": 1.0, + "content": "3 SCALE-EQUIVARIANT MAPPINGS", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 106, + 428, + 384, + 441 + ], + "lines": [ + { + "bbox": [ + 105, + 426, + 385, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 385, + 443 + ], + "score": 1.0, + "content": "Now we define the main building blocks of scale-equivariant models.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 426, + 385, + 443 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 446, + 505, + 491 + ], + "lines": [ + { + "bbox": [ + 106, + 446, + 505, + 458 + ], + "spans": [ + { + "bbox": [ + 106, + 446, + 505, + 458 + ], + "score": 1.0, + "content": "Scale Convolution In order to derive scale convolution, we start from group equivariant convolution", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 457, + 505, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 127, + 470 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 127, + 457, + 161, + 467 + ], + "score": 0.89, + "content": "G = H", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 457, + 505, + 470 + ], + "score": 1.0, + "content": ". We first use the property of semidirect product of groups which splits the integral,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 467, + 506, + 481 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 506, + 481 + ], + "score": 1.0, + "content": "then choose the appropriate Haar measures, and finally use the properties of steerable filters. 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Thus Equation 7", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 653, + 505, + 665 + ], + "spans": [ + { + "bbox": [ + 106, + 653, + 505, + 665 + ], + "score": 1.0, + "content": "shows the most general form of scale-equivariant layers which allows for building scale-equivariant", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 662, + 506, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 662, + 280, + 677 + ], + "score": 1.0, + "content": "convolutional networks with such choice of", + "type": "text" + }, + { + "bbox": [ + 280, + 664, + 288, + 673 + ], + "score": 0.75, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 662, + 506, + 677 + ], + "score": 1.0, + "content": ". 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It is useful to utilize this transformation closer to the end of", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 279, + 505, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 279, + 505, + 293 + ], + "score": 1.0, + "content": "the network, when the deep representation must be invariant to nuisance input variations, but already", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 290, + 237, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 237, + 303 + ], + "score": 1.0, + "content": "has very rich semantic meaning.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14.5 + }, + { + "type": "title", + "bbox": [ + 108, + 318, + 219, + 330 + ], + "lines": [ + { + "bbox": [ + 105, + 317, + 221, + 332 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 221, + 332 + ], + "score": 1.0, + "content": "4 IMPLEMENTATION", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 342, + 505, + 376 + ], + "lines": [ + { + "bbox": [ + 105, + 343, + 504, + 355 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 504, + 355 + ], + "score": 1.0, + "content": "In this paragraph we discuss an efficient implementation of Scale-Equivariant Steerable Networks.", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 353, + 505, + 366 + ], + "spans": [ + { + "bbox": [ + 106, + 353, + 505, + 366 + ], + "score": 1.0, + "content": "We illustrate all algorithms in Figure 1. For simplicity we assume that zero padding is applied when", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 364, + 327, + 377 + ], + "spans": [ + { + "bbox": [ + 105, + 364, + 327, + 377 + ], + "score": 1.0, + "content": "it is needed for both the spatial axes and the scale axis.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 106, + 381, + 505, + 475 + ], + "lines": [ + { + "bbox": [ + 106, + 381, + 505, + 394 + ], + "spans": [ + { + "bbox": [ + 106, + 381, + 505, + 394 + ], + "score": 1.0, + "content": "Filter Basis A direct implementation of Equation 7 is impossible due to several limitations. First,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 391, + 505, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 391, + 236, + 406 + ], + "score": 1.0, + "content": "the infinite number of scales in", + "type": "text" + }, + { + "bbox": [ + 236, + 393, + 244, + 402 + ], + "score": 0.81, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 391, + 505, + 406 + ], + "score": 1.0, + "content": "calls for a discrete approximation. 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In other words, we do the following substitution in Equation 7:", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 462, + 191, + 475 + ], + "spans": [ + { + "bbox": [ + 106, + 462, + 191, + 475 + ], + "score": 0.88, + "content": "\\begin{array} { r } { \\psi _ { \\sigma } \\kappa \\stackrel { - } { = } \\sum _ { i } w _ { i } \\Psi _ { i } } \\end{array}", + "type": "inline_equation" + } + ], + "index": 28 + } + ], + "index": 24.5 + }, + { + "type": "text", + "bbox": [ + 107, + 479, + 505, + 513 + ], + "lines": [ + { + "bbox": [ + 105, + 478, + 506, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 506, + 492 + ], + "score": 1.0, + "content": "In our experiments we use a basis of 2D Hermite polynomials with 2D Gaussian envelope, as it", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 490, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 490, + 505, + 502 + ], + "score": 1.0, + "content": "demonstrates good results. The basis is pre-calculated for all scales and fixed. For filters of size", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 107, + 499, + 488, + 515 + ], + "spans": [ + { + "bbox": [ + 107, + 501, + 136, + 512 + ], + "score": 0.89, + "content": "V \\times V", + "type": "inline_equation" + }, + { + "bbox": [ + 136, + 499, + 294, + 515 + ], + "score": 1.0, + "content": ", the basis is stored as an array of shape", + "type": "text" + }, + { + "bbox": [ + 295, + 501, + 347, + 513 + ], + "score": 0.92, + "content": "[ N _ { b } , S , V , V ]", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 499, + 488, + 515 + ], + "score": 1.0, + "content": ". See Appendix C for more details.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 106, + 518, + 504, + 552 + ], + "lines": [ + { + "bbox": [ + 106, + 516, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 106, + 516, + 131, + 532 + ], + "score": 1.0, + "content": "Conv", + "type": "text" + }, + { + "bbox": [ + 131, + 518, + 168, + 528 + ], + "score": 0.89, + "content": "T \\to H", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 516, + 323, + 532 + ], + "score": 1.0, + "content": "If the input signal is just a function on", + "type": "text" + }, + { + "bbox": [ + 323, + 519, + 332, + 528 + ], + "score": 0.8, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 332, + 516, + 399, + 532 + ], + "score": 1.0, + "content": "with spatial size", + "type": "text" + }, + { + "bbox": [ + 400, + 519, + 429, + 528 + ], + "score": 0.9, + "content": "U \\times U", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 516, + 505, + 532 + ], + "score": 1.0, + "content": ", stored as an array", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 529, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 106, + 529, + 142, + 541 + ], + "score": 1.0, + "content": "of shape", + "type": "text" + }, + { + "bbox": [ + 143, + 529, + 185, + 541 + ], + "score": 0.92, + "content": "[ C _ { \\mathrm { i n } } , U , U ]", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 529, + 412, + 541 + ], + "score": 1.0, + "content": ", then Equation 7 can be simplified. The summation over", + "type": "text" + }, + { + "bbox": [ + 413, + 530, + 421, + 539 + ], + "score": 0.81, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 529, + 505, + 541 + ], + "score": 1.0, + "content": "degenerates, and the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 540, + 301, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 301, + 552 + ], + "score": 1.0, + "content": "final result can be written in the following form:", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33 + }, + { + "type": "interline_equation", + "bbox": [ + 172, + 556, + 438, + 570 + ], + "lines": [ + { + "bbox": [ + 172, + 556, + 438, + 570 + ], + "spans": [ + { + "bbox": [ + 172, + 556, + 438, + 570 + ], + "score": 0.84, + "content": "\\mathtt { c o n v T H } ( f , w , \\Psi ) = \\mathtt { s q u e e } z \\in \\left( \\mathtt { c o n v } 2 \\mathrm { d } ( f , \\mathtt { e x p a n d } ( w \\times \\Psi ) ) \\right)", + "type": "interline_equation", + "image_path": "128a59139f7def3ef1c6918aed47f7ea431e6bba05ddfd59792f60d079b3c1ca.jpg" + } + ] + } + ], + "index": 35, + "virtual_lines": [ + { + "bbox": [ + 172, + 556, + 438, + 570 + ], + "spans": [], + "index": 35 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 580, + 505, + 636 + ], + "lines": [ + { + "bbox": [ + 105, + 581, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 128, + 595 + ], + "score": 1.0, + "content": "Here", + "type": "text" + }, + { + "bbox": [ + 128, + 583, + 137, + 591 + ], + "score": 0.78, + "content": "w", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 581, + 216, + 595 + ], + "score": 1.0, + "content": "is an array of shape", + "type": "text" + }, + { + "bbox": [ + 217, + 581, + 273, + 593 + ], + "score": 0.92, + "content": "[ C _ { \\mathrm { o u t } } , C _ { \\mathrm { i n } } , N _ { b } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 581, + 350, + 595 + ], + "score": 1.0, + "content": ". We compute filter", + "type": "text" + }, + { + "bbox": [ + 350, + 582, + 376, + 592 + ], + "score": 0.91, + "content": "w \\times \\Psi", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 581, + 413, + 595 + ], + "score": 1.0, + "content": "of shape", + "type": "text" + }, + { + "bbox": [ + 413, + 581, + 487, + 594 + ], + "score": 0.9, + "content": "[ C _ { \\mathrm { o u t } } , C _ { \\mathrm { i n } } , S , V , V ]", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 581, + 506, + 595 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 591, + 506, + 605 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 183, + 605 + ], + "score": 1.0, + "content": "expand it to shape", + "type": "text" + }, + { + "bbox": [ + 184, + 593, + 253, + 604 + ], + "score": 0.91, + "content": "[ C _ { \\mathrm { o u t } } , C _ { \\mathrm { i n } } S , V , V ]", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 591, + 506, + 605 + ], + "score": 1.0, + "content": ". 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Note that the output can be viewed as", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 614, + 505, + 627 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 445, + 627 + ], + "score": 1.0, + "content": "a stack of feature maps, where all the features in each spatial position are vectors of", + "type": "text" + }, + { + "bbox": [ + 445, + 615, + 453, + 624 + ], + "score": 0.81, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 614, + 505, + 627 + ], + "score": 1.0, + "content": "components", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 624, + 289, + 638 + ], + "spans": [ + { + "bbox": [ + 106, + 624, + 289, + 638 + ], + "score": 1.0, + "content": "instead of being scalars as in standard CNNs.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 106, + 641, + 505, + 698 + ], + "lines": [ + { + "bbox": [ + 106, + 640, + 505, + 655 + ], + "spans": [ + { + "bbox": [ + 106, + 640, + 131, + 655 + ], + "score": 1.0, + "content": "Conv", + "type": "text" + }, + { + "bbox": [ + 132, + 642, + 171, + 652 + ], + "score": 0.9, + "content": "H H", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 640, + 237, + 655 + ], + "score": 1.0, + "content": "The function on", + "type": "text" + }, + { + "bbox": [ + 237, + 642, + 248, + 652 + ], + "score": 0.79, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 640, + 505, + 655 + ], + "score": 1.0, + "content": "has a scale axis and therefore there are two options for choosing", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 652, + 505, + 666 + ], + "spans": [ + { + "bbox": [ + 105, + 652, + 505, + 666 + ], + "score": 1.0, + "content": "weights of the convolutional filter. The filter may have just one scale and, therefore, does not capture", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 664, + 505, + 676 + ], + "spans": [ + { + "bbox": [ + 106, + 664, + 505, + 676 + ], + "score": 1.0, + "content": "the correlations between different scales of the input function; or, it may have a non-unitary extent", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 107, + 675, + 505, + 687 + ], + "spans": [ + { + "bbox": [ + 107, + 675, + 122, + 686 + ], + "score": 0.9, + "content": "K _ { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 122, + 675, + 341, + 687 + ], + "score": 1.0, + "content": "in the scale axis and capture the correlation between", + "type": "text" + }, + { + "bbox": [ + 341, + 675, + 356, + 686 + ], + "score": 0.89, + "content": "K _ { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 675, + 505, + 687 + ], + "score": 1.0, + "content": "neighboring scales. 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One", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 152, + 506, + 164 + ], + "spans": [ + { + "bbox": [ + 106, + 152, + 506, + 164 + ], + "score": 1.0, + "content": "way to do this is to calculate the invariant measure of the signal. In case of translation, such a", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 163, + 312, + 175 + ], + "spans": [ + { + "bbox": [ + 105, + 163, + 312, + 175 + ], + "score": 1.0, + "content": "measure could be the maximum value for example.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5, + "bbox_fs": [ + 105, + 141, + 506, + 175 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 179, + 503, + 213 + ], + "lines": [ + { + "bbox": [ + 105, + 178, + 505, + 192 + ], + "spans": [ + { + "bbox": [ + 105, + 178, + 351, + 192 + ], + "score": 1.0, + "content": "First, we propose the maximum scale projection defined as", + "type": "text" + }, + { + "bbox": [ + 351, + 180, + 452, + 192 + ], + "score": 0.92, + "content": "f ( s , x ) \\operatorname* { m a x } _ { s } f ( s , x )", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 178, + 505, + 192 + ], + "score": 1.0, + "content": ". This trans-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 190, + 505, + 203 + ], + "spans": [ + { + "bbox": [ + 106, + 190, + 235, + 203 + ], + "score": 1.0, + "content": "formation projects the function", + "type": "text" + }, + { + "bbox": [ + 235, + 191, + 243, + 202 + ], + "score": 0.86, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 190, + 267, + 203 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 267, + 191, + 277, + 200 + ], + "score": 0.82, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 190, + 289, + 203 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 290, + 191, + 298, + 200 + ], + "score": 0.81, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 190, + 505, + 203 + ], + "score": 1.0, + "content": ". 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Trans-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 229, + 506, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 229, + 147, + 243 + ], + "score": 1.0, + "content": "formation", + "type": "text" + }, + { + "bbox": [ + 148, + 229, + 246, + 241 + ], + "score": 0.91, + "content": "f ( s , x ) \\to \\operatorname* { m a x } _ { x } { \\overline { { f } } } ( s , x )", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 229, + 330, + 243 + ], + "score": 1.0, + "content": "projects the function", + "type": "text" + }, + { + "bbox": [ + 330, + 230, + 338, + 241 + ], + "score": 0.86, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 229, + 359, + 243 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 360, + 230, + 370, + 239 + ], + "score": 0.79, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 229, + 380, + 243 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 380, + 230, + 388, + 239 + ], + "score": 0.76, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 229, + 506, + 243 + ], + "score": 1.0, + "content": ". The obtained representation", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 241, + 453, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 453, + 252 + ], + "score": 1.0, + "content": "is invariant to scaling in spatial domain, however, it stores the information about scale.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11, + "bbox_fs": [ + 105, + 218, + 506, + 252 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 257, + 505, + 302 + ], + "lines": [ + { + "bbox": [ + 105, + 258, + 505, + 269 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 505, + 269 + ], + "score": 1.0, + "content": "Finally, we can combine both of these pooling mechanisms in any order. The obtained transforma-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 268, + 505, + 280 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 505, + 280 + ], + "score": 1.0, + "content": "tion produces a scale invariant function. It is useful to utilize this transformation closer to the end of", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 279, + 505, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 279, + 505, + 293 + ], + "score": 1.0, + "content": "the network, when the deep representation must be invariant to nuisance input variations, but already", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 290, + 237, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 237, + 303 + ], + "score": 1.0, + "content": "has very rich semantic meaning.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14.5, + "bbox_fs": [ + 105, + 258, + 505, + 303 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 318, + 219, + 330 + ], + "lines": [ + { + "bbox": [ + 105, + 317, + 221, + 332 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 221, + 332 + ], + "score": 1.0, + "content": "4 IMPLEMENTATION", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 342, + 505, + 376 + ], + "lines": [ + { + "bbox": [ + 105, + 343, + 504, + 355 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 504, + 355 + ], + "score": 1.0, + "content": "In this paragraph we discuss an efficient implementation of Scale-Equivariant Steerable Networks.", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 353, + 505, + 366 + ], + "spans": [ + { + "bbox": [ + 106, + 353, + 505, + 366 + ], + "score": 1.0, + "content": "We illustrate all algorithms in Figure 1. For simplicity we assume that zero padding is applied when", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 364, + 327, + 377 + ], + "spans": [ + { + "bbox": [ + 105, + 364, + 327, + 377 + ], + "score": 1.0, + "content": "it is needed for both the spatial axes and the scale axis.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 343, + 505, + 377 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 381, + 505, + 475 + ], + "lines": [ + { + "bbox": [ + 106, + 381, + 505, + 394 + ], + "spans": [ + { + "bbox": [ + 106, + 381, + 505, + 394 + ], + "score": 1.0, + "content": "Filter Basis A direct implementation of Equation 7 is impossible due to several limitations. First,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 391, + 505, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 391, + 236, + 406 + ], + "score": 1.0, + "content": "the infinite number of scales in", + "type": "text" + }, + { + "bbox": [ + 236, + 393, + 244, + 402 + ], + "score": 0.81, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 391, + 505, + 406 + ], + "score": 1.0, + "content": "calls for a discrete approximation. We truncate the scale group", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 403, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 198, + 416 + ], + "score": 1.0, + "content": "and limit ourselves to", + "type": "text" + }, + { + "bbox": [ + 199, + 404, + 213, + 415 + ], + "score": 0.89, + "content": "N _ { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 403, + 505, + 416 + ], + "score": 1.0, + "content": "scales and use discrete translations instead of continuous ones. Train-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 414, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 104, + 414, + 505, + 426 + ], + "score": 1.0, + "content": "ing of SESN involves searching for the optimal filter in functional space which is a problem", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 425, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 410, + 438 + ], + "score": 1.0, + "content": "by itself. Rather than solving it directly, we choose a complete basis of", + "type": "text" + }, + { + "bbox": [ + 411, + 426, + 424, + 437 + ], + "score": 0.89, + "content": "N _ { b }", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 425, + 505, + 438 + ], + "score": 1.0, + "content": "steerable functions", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 107, + 433, + 507, + 454 + ], + "spans": [ + { + "bbox": [ + 107, + 436, + 181, + 451 + ], + "score": 0.93, + "content": "\\dot { \\Psi } = \\{ \\psi _ { s ^ { - 1 } \\sigma , i } \\} _ { i = 1 } ^ { N _ { b } }", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 433, + 507, + 454 + ], + "score": 1.0, + "content": "and represent convolutional filter as a linear combination of basis functions with", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 447, + 507, + 467 + ], + "spans": [ + { + "bbox": [ + 104, + 447, + 190, + 467 + ], + "score": 1.0, + "content": "trainable parameters", + "type": "text" + }, + { + "bbox": [ + 190, + 449, + 246, + 464 + ], + "score": 0.93, + "content": "w = \\{ w _ { i } \\} _ { i = 1 } ^ { N _ { b } }", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 447, + 507, + 467 + ], + "score": 1.0, + "content": ". In other words, we do the following substitution in Equation 7:", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 462, + 191, + 475 + ], + "spans": [ + { + "bbox": [ + 106, + 462, + 191, + 475 + ], + "score": 0.88, + "content": "\\begin{array} { r } { \\psi _ { \\sigma } \\kappa \\stackrel { - } { = } \\sum _ { i } w _ { i } \\Psi _ { i } } \\end{array}", + "type": "inline_equation" + } + ], + "index": 28 + } + ], + "index": 24.5, + "bbox_fs": [ + 104, + 381, + 507, + 475 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 479, + 505, + 513 + ], + "lines": [ + { + "bbox": [ + 105, + 478, + 506, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 506, + 492 + ], + "score": 1.0, + "content": "In our experiments we use a basis of 2D Hermite polynomials with 2D Gaussian envelope, as it", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 490, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 490, + 505, + 502 + ], + "score": 1.0, + "content": "demonstrates good results. The basis is pre-calculated for all scales and fixed. For filters of size", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 107, + 499, + 488, + 515 + ], + "spans": [ + { + "bbox": [ + 107, + 501, + 136, + 512 + ], + "score": 0.89, + "content": "V \\times V", + "type": "inline_equation" + }, + { + "bbox": [ + 136, + 499, + 294, + 515 + ], + "score": 1.0, + "content": ", the basis is stored as an array of shape", + "type": "text" + }, + { + "bbox": [ + 295, + 501, + 347, + 513 + ], + "score": 0.92, + "content": "[ N _ { b } , S , V , V ]", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 499, + 488, + 515 + ], + "score": 1.0, + "content": ". See Appendix C for more details.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30, + "bbox_fs": [ + 105, + 478, + 506, + 515 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 518, + 504, + 552 + ], + "lines": [ + { + "bbox": [ + 106, + 516, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 106, + 516, + 131, + 532 + ], + "score": 1.0, + "content": "Conv", + "type": "text" + }, + { + "bbox": [ + 131, + 518, + 168, + 528 + ], + "score": 0.89, + "content": "T \\to H", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 516, + 323, + 532 + ], + "score": 1.0, + "content": "If the input signal is just a function on", + "type": "text" + }, + { + "bbox": [ + 323, + 519, + 332, + 528 + ], + "score": 0.8, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 332, + 516, + 399, + 532 + ], + "score": 1.0, + "content": "with spatial size", + "type": "text" + }, + { + "bbox": [ + 400, + 519, + 429, + 528 + ], + "score": 0.9, + "content": "U \\times U", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 516, + 505, + 532 + ], + "score": 1.0, + "content": ", stored as an array", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 529, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 106, + 529, + 142, + 541 + ], + "score": 1.0, + "content": "of shape", + "type": "text" + }, + { + "bbox": [ + 143, + 529, + 185, + 541 + ], + "score": 0.92, + "content": "[ C _ { \\mathrm { i n } } , U , U ]", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 529, + 412, + 541 + ], + "score": 1.0, + "content": ", then Equation 7 can be simplified. The summation over", + "type": "text" + }, + { + "bbox": [ + 413, + 530, + 421, + 539 + ], + "score": 0.81, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 529, + 505, + 541 + ], + "score": 1.0, + "content": "degenerates, and the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 540, + 301, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 301, + 552 + ], + "score": 1.0, + "content": "final result can be written in the following form:", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 516, + 505, + 552 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 172, + 556, + 438, + 570 + ], + "lines": [ + { + "bbox": [ + 172, + 556, + 438, + 570 + ], + "spans": [ + { + "bbox": [ + 172, + 556, + 438, + 570 + ], + "score": 0.84, + "content": "\\mathtt { c o n v T H } ( f , w , \\Psi ) = \\mathtt { s q u e e } z \\in \\left( \\mathtt { c o n v } 2 \\mathrm { d } ( f , \\mathtt { e x p a n d } ( w \\times \\Psi ) ) \\right)", + "type": "interline_equation", + "image_path": "128a59139f7def3ef1c6918aed47f7ea431e6bba05ddfd59792f60d079b3c1ca.jpg" + } + ] + } + ], + "index": 35, + "virtual_lines": [ + { + "bbox": [ + 172, + 556, + 438, + 570 + ], + "spans": [], + "index": 35 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 580, + 505, + 636 + ], + "lines": [ + { + "bbox": [ + 105, + 581, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 128, + 595 + ], + "score": 1.0, + "content": "Here", + "type": "text" + }, + { + "bbox": [ + 128, + 583, + 137, + 591 + ], + "score": 0.78, + "content": "w", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 581, + 216, + 595 + ], + "score": 1.0, + "content": "is an array of shape", + "type": "text" + }, + { + "bbox": [ + 217, + 581, + 273, + 593 + ], + "score": 0.92, + "content": "[ C _ { \\mathrm { o u t } } , C _ { \\mathrm { i n } } , N _ { b } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 581, + 350, + 595 + ], + "score": 1.0, + "content": ". 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Note that the output can be viewed as", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 614, + 505, + 627 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 445, + 627 + ], + "score": 1.0, + "content": "a stack of feature maps, where all the features in each spatial position are vectors of", + "type": "text" + }, + { + "bbox": [ + 445, + 615, + 453, + 624 + ], + "score": 0.81, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 614, + 505, + 627 + ], + "score": 1.0, + "content": "components", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 624, + 289, + 638 + ], + "spans": [ + { + "bbox": [ + 106, + 624, + 289, + 638 + ], + "score": 1.0, + "content": "instead of being scalars as in standard CNNs.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 38, + "bbox_fs": [ + 105, + 581, + 506, + 638 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 641, + 505, + 698 + ], + "lines": [ + { + "bbox": [ + 106, + 640, + 505, + 655 + ], + "spans": [ + { + "bbox": [ + 106, + 640, + 131, + 655 + ], + "score": 1.0, + "content": "Conv", + "type": "text" + }, + { + "bbox": [ + 132, + 642, + 171, + 652 + ], + "score": 0.9, + "content": "H H", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 640, + 237, + 655 + ], + "score": 1.0, + "content": "The function on", + "type": "text" + }, + { + "bbox": [ + 237, + 642, + 248, + 652 + ], + "score": 0.79, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 640, + 505, + 655 + ], + "score": 1.0, + "content": "has a scale axis and therefore there are two options for choosing", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 652, + 505, + 666 + ], + "spans": [ + { + "bbox": [ + 105, + 652, + 505, + 666 + ], + "score": 1.0, + "content": "weights of the convolutional filter. The filter may have just one scale and, therefore, does not capture", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 664, + 505, + 676 + ], + "spans": [ + { + "bbox": [ + 106, + 664, + 505, + 676 + ], + "score": 1.0, + "content": "the correlations between different scales of the input function; or, it may have a non-unitary extent", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 107, + 675, + 505, + 687 + ], + "spans": [ + { + "bbox": [ + 107, + 675, + 122, + 686 + ], + "score": 0.9, + "content": "K _ { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 122, + 675, + 341, + 687 + ], + "score": 1.0, + "content": "in the scale axis and capture the correlation between", + "type": "text" + }, + { + "bbox": [ + 341, + 675, + 356, + 686 + ], + "score": 0.89, + "content": "K _ { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 675, + 505, + 687 + ], + "score": 1.0, + "content": "neighboring scales. 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Middle and right: a", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 336, + 506, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 506, + 351 + ], + "score": 1.0, + "content": "representation of scale-convolution using Equation 9 and Equation 10. As an example we use input", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 347, + 504, + 361 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 132, + 361 + ], + "score": 1.0, + "content": "signal", + "type": "text" + }, + { + "bbox": [ + 133, + 349, + 140, + 360 + ], + "score": 0.86, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 347, + 272, + 361 + ], + "score": 1.0, + "content": "with 3 channels. It has 1 scale on", + "type": "text" + }, + { + "bbox": [ + 272, + 349, + 281, + 358 + ], + "score": 0.75, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 347, + 343, + 361 + ], + "score": 1.0, + "content": "and 4 scales on", + "type": "text" + }, + { + "bbox": [ + 344, + 349, + 354, + 358 + ], + "score": 0.74, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 347, + 459, + 361 + ], + "score": 1.0, + "content": ". It is convolved with filter", + "type": "text" + }, + { + "bbox": [ + 459, + 349, + 504, + 359 + ], + "score": 0.9, + "content": "\\kappa = w \\times \\Psi", + "type": "inline_equation" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 360, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 106, + 360, + 505, + 371 + ], + "score": 1.0, + "content": "without scale interaction, which produces the output with 2 channels and 4 scales as well. Here we", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 370, + 497, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 497, + 383 + ], + "score": 1.0, + "content": "represent only channels of the signals and the filter. Spatial components are hidden for simplicity.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10 + } + ], + "index": 8.0 + }, + { + "type": "title", + "bbox": [ + 108, + 408, + 211, + 421 + ], + "lines": [ + { + "bbox": [ + 105, + 407, + 213, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 213, + 423 + ], + "score": 1.0, + "content": "5 RELATED WORK", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 106, + 434, + 505, + 566 + ], + "lines": [ + { + "bbox": [ + 106, + 434, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 106, + 434, + 506, + 448 + ], + "score": 1.0, + "content": "Various works on group-equivariant convolutional networks have been published recently. These", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 446, + 505, + 458 + ], + "spans": [ + { + "bbox": [ + 106, + 446, + 505, + 458 + ], + "score": 1.0, + "content": "works have considered roto-translation groups in 2D Cohen & Welling (2016a); Hoogeboom et al.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 457, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 106, + 457, + 505, + 468 + ], + "score": 1.0, + "content": "(2018); Worrall et al. (2017); Weiler & Cesa (2019) and 3D Worrall & Brostow (2018); Kondor", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 466, + 506, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 506, + 480 + ], + "score": 1.0, + "content": "(2018); Thomas et al. (2018) and rotation equivariant networks in 3D Cohen et al. (2017); Esteves", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 478, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 505, + 491 + ], + "score": 1.0, + "content": "et al. (2018); Cohen et al. (2019). In Freeman & Adelson (1991) authors describe the algorithm", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 488, + 506, + 503 + ], + "spans": [ + { + "bbox": [ + 104, + 488, + 506, + 503 + ], + "score": 1.0, + "content": "for designing steerable filters for rotations. Rotation steerable filters are used in Cohen & Welling", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 501, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 106, + 501, + 505, + 513 + ], + "score": 1.0, + "content": "(2016b); Weiler et al. (2018a;b) for building equivariant networks. In Jacobsen et al. (2017) the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 510, + 506, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 369, + 525 + ], + "score": 1.0, + "content": "authors build convolutional blocks locally equivariant to arbitrary", + "type": "text" + }, + { + "bbox": [ + 369, + 512, + 376, + 521 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 510, + 506, + 525 + ], + "score": 1.0, + "content": "-parameter Lie group by using a", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 523, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 505, + 535 + ], + "score": 1.0, + "content": "steerable basis. And in Murugan et al. the authors discuss the approach for learning steerable filters", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 532, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 104, + 532, + 506, + 546 + ], + "score": 1.0, + "content": "from data. To date, the majority of papers on group equivariant networks have considered rotations", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 543, + 506, + 558 + ], + "spans": [ + { + "bbox": [ + 104, + 543, + 506, + 558 + ], + "score": 1.0, + "content": "in 2D and 3D, but have not payed attention to scale symmetry. As we have argued above, it is a", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 555, + 225, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 225, + 568 + ], + "score": 1.0, + "content": "fundamentally different case.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 19.5 + }, + { + "type": "text", + "bbox": [ + 106, + 572, + 505, + 649 + ], + "lines": [ + { + "bbox": [ + 106, + 572, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 505, + 585 + ], + "score": 1.0, + "content": "Many papers and even conferences have been dedicated to image scale-space — a concept where", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 583, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 505, + 595 + ], + "score": 1.0, + "content": "the image is analyzed together with all its downscaled versions. Initially introduced in Iijima (1959)", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 594, + 506, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 506, + 607 + ], + "score": 1.0, + "content": "and later developed by Witkin (1987); Perona & Malik (1990); Lindeberg (2013) scale space relies", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 605, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 505, + 618 + ], + "score": 1.0, + "content": "on the scale symmetry of images. The differential structure of the image Koenderink (1984) allows", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 616, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 505, + 628 + ], + "score": 1.0, + "content": "one to make a connection between image formation mechanisms and the space of solutions of the 2-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 626, + 505, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 505, + 641 + ], + "score": 1.0, + "content": "dimensional heat equation, which significantly improved the image analysis models in the pre-deep", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 637, + 159, + 652 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 159, + 652 + ], + "score": 1.0, + "content": "learning era.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 106, + 654, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 654, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 654, + 505, + 667 + ], + "score": 1.0, + "content": "One of the first works on scale equivariance and local scale invariance in the framework of CNNs was", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 666, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 677 + ], + "score": 1.0, + "content": "proposed by Xu et al. 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Middle and right: a", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 336, + 506, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 506, + 351 + ], + "score": 1.0, + "content": "representation of scale-convolution using Equation 9 and Equation 10. As an example we use input", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 347, + 504, + 361 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 132, + 361 + ], + "score": 1.0, + "content": "signal", + "type": "text" + }, + { + "bbox": [ + 133, + 349, + 140, + 360 + ], + "score": 0.86, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 347, + 272, + 361 + ], + "score": 1.0, + "content": "with 3 channels. 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Here we", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 370, + 497, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 497, + 383 + ], + "score": 1.0, + "content": "represent only channels of the signals and the filter. Spatial components are hidden for simplicity.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10 + } + ], + "index": 8.0 + }, + { + "type": "title", + "bbox": [ + 108, + 408, + 211, + 421 + ], + "lines": [ + { + "bbox": [ + 105, + 407, + 213, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 213, + 423 + ], + "score": 1.0, + "content": "5 RELATED WORK", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 106, + 434, + 505, + 566 + ], + "lines": [ + { + "bbox": [ + 106, + 434, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 106, + 434, + 506, + 448 + ], + "score": 1.0, + "content": "Various works on group-equivariant convolutional networks have been published recently. These", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 446, + 505, + 458 + ], + "spans": [ + { + "bbox": [ + 106, + 446, + 505, + 458 + ], + "score": 1.0, + "content": "works have considered roto-translation groups in 2D Cohen & Welling (2016a); Hoogeboom et al.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 457, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 106, + 457, + 505, + 468 + ], + "score": 1.0, + "content": "(2018); Worrall et al. (2017); Weiler & Cesa (2019) and 3D Worrall & Brostow (2018); Kondor", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 466, + 506, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 506, + 480 + ], + "score": 1.0, + "content": "(2018); Thomas et al. (2018) and rotation equivariant networks in 3D Cohen et al. 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MethodEquivarianceAdmissible ScalesApproachInterscale
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MethodEquivarianceAdmissible ScalesApproachInterscale
SiCNNGridFilter Rescaling
SI-ConvNetGridInput Rescaling
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DSS<x</IntegerFilter Dilationxx<>
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Method(28×28)(28 × 28) +(56 × 56)(56 × 56) +# Params
CNN2.56 ± 0.041.96 ± 0.072.02 ± 0.071.60 ± 0.09495K
SiCNN2.40 ± 0.031.86 ± 0.102.02 ± 0.141.59 ± 0.03497K
SI-ConvNet2.40± 0.121.94 ± 0.071.82 ± 0.111.59 ± 0.10495K
SEVF Scalar2.30 ± 0.061.96 ± 0.071.87 ± 0.091.62 ± 0.07494 K
SEVF Vector2.63 ± 0.092.23 ± 0.092.12 ± 0.131.81 ± 0.09475 K
DSS Scalar2.53 ± 0.102.04± 0.081.92 ± 0.081.57 ± 0.08494 K
DSS Vector2.58 ± 0.111.95 ± 0.071.97 ± 0.081.57 ± 0.09494 K
SS-CNN2.32 ± 0.152.10 ±0.151.84 ± 0.101.76 ± 0.07494 K
SESN Scalar2.10 ±0.101.79 ± 0.091.74 ± 0.091.50 ± 0.07495K
SESN Vector2.08 ±0.091.76 ± 0.081.68 ± 0.061.42 ± 0.07495 K
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In ex-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 104, + 243, + 506, + 255 + ], + "spans": [ + { + "bbox": [ + 104, + 243, + 258, + 255 + ], + "score": 1.0, + "content": "periment we use image resolution of", + "type": "text" + }, + { + "bbox": [ + 258, + 243, + 293, + 253 + ], + "score": 0.91, + "content": "2 8 \\times 2 8", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 243, + 312, + 255 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 312, + 243, + 346, + 254 + ], + "score": 0.9, + "content": "5 6 \\times 5 6", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 243, + 506, + 255 + ], + "score": 1.0, + "content": ". We test both the regime without data", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 254, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 421, + 266 + ], + "score": 1.0, + "content": "augmentation, and the regime with scaling data augmentation, denoted with", + "type": "text" + }, + { + "bbox": [ + 421, + 254, + 438, + 265 + ], + "score": 0.56, + "content": "\" + \"", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 254, + 505, + 266 + ], + "score": 1.0, + "content": ". All results are", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 265, + 498, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 265, + 176, + 276 + ], + "score": 1.0, + "content": "reported as mean", + "type": "text" + }, + { + "bbox": [ + 176, + 266, + 187, + 275 + ], + "score": 0.74, + "content": "\\pm", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 265, + 498, + 276 + ], + "score": 1.0, + "content": "std over 6 different fixed realizations of the dataset. The best results are bold.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + }, + { + "type": "title", + "bbox": [ + 107, + 301, + 229, + 312 + ], + "lines": [ + { + "bbox": [ + 106, + 300, + 231, + 313 + ], + "spans": [ + { + "bbox": [ + 106, + 300, + 231, + 313 + ], + "score": 1.0, + "content": "6.1 EQUIVARIANCE ERROR", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 107, + 323, + 505, + 389 + ], + "lines": [ + { + "bbox": [ + 106, + 324, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 106, + 324, + 505, + 335 + ], + "score": 1.0, + "content": "We have presented scale-convolution which is equivariant to scale transformation and translation for", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 334, + 505, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 505, + 347 + ], + "score": 1.0, + "content": "continuous signals. While translation equivariance holds true even for discretized signals and filters,", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 345, + 505, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 505, + 357 + ], + "score": 1.0, + "content": "scale equivariance may not be exact. Therefore, before starting any experiments, we check to which", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 356, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 505, + 369 + ], + "score": 1.0, + "content": "degree the predicted properties of scale-convolution hold true. We do so by measuring the difference", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 365, + 506, + 380 + ], + "spans": [ + { + "bbox": [ + 106, + 366, + 276, + 379 + ], + "score": 0.93, + "content": "\\dot { \\Delta } = \\| [ L _ { s } \\mathbf { \\hat { \\Phi } } ( f ) - \\Phi \\mathbf { \\hat { L } } _ { s } ( f ) \\| _ { 2 } ^ { 2 } / \\| L _ { s } \\Phi ( f ) \\| _ { 2 } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 365, + 308, + 380 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 308, + 367, + 317, + 377 + ], + "score": 0.84, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 365, + 506, + 380 + ], + "score": 1.0, + "content": "is scale-convolution with randomly initialized", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 378, + 143, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 143, + 391 + ], + "score": 1.0, + "content": "weights.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 10.5 + }, + { + "type": "text", + "bbox": [ + 106, + 394, + 505, + 526 + ], + "lines": [ + { + "bbox": [ + 106, + 395, + 505, + 407 + ], + "spans": [ + { + "bbox": [ + 106, + 395, + 505, + 407 + ], + "score": 1.0, + "content": "In case of perfect equivariance the difference is equal to zero. We calculate the error on randomly", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 405, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 505, + 419 + ], + "score": 1.0, + "content": "sampled images from the STL-10 dataset Coates et al. (2011). The results are represented in Fig-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 417, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 505, + 429 + ], + "score": 1.0, + "content": "ure 2. The networks on the left and on the middle plots do not have interscale interactions. The", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 428, + 505, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 420, + 440 + ], + "score": 1.0, + "content": "networks on the middle and on the right plots consist of just one layer. We use", + "type": "text" + }, + { + "bbox": [ + 420, + 428, + 478, + 439 + ], + "score": 0.79, + "content": "N _ { S } = 5 , 1 3 , 5", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 428, + 505, + 440 + ], + "score": 1.0, + "content": "scales", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 439, + 505, + 451 + ], + "spans": [ + { + "bbox": [ + 106, + 439, + 505, + 451 + ], + "score": 1.0, + "content": "for the networks on the left, the middle, and the right plots respectively. While discretization intro-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 450, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 106, + 450, + 360, + 462 + ], + "score": 1.0, + "content": "duces some error, it stays very low, and is not much higher than", + "type": "text" + }, + { + "bbox": [ + 361, + 450, + 375, + 460 + ], + "score": 0.85, + "content": "6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 450, + 505, + 462 + ], + "score": 1.0, + "content": "for the networks with 50 layers.", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 460, + 505, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 505, + 473 + ], + "score": 1.0, + "content": "The difference, however, increases if the input image is downscaled more than 16 times. There-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 471, + 505, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 505, + 484 + ], + "score": 1.0, + "content": "fore, we are free to use deep networks. However, we should pay extra attention to extreme cases", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 482, + 505, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 505, + 495 + ], + "score": 1.0, + "content": "where scale changes are of very big magnitude. These are quite rare but still appear in practice.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 493, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 106, + 493, + 505, + 505 + ], + "score": 1.0, + "content": "Finally, we see that using SESN with interscale interaction introduces extra equivariance error due", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 505, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 106, + 505, + 185, + 516 + ], + "score": 1.0, + "content": "to the truncation of", + "type": "text" + }, + { + "bbox": [ + 185, + 505, + 192, + 514 + ], + "score": 0.76, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 193, + 505, + 505, + 516 + ], + "score": 1.0, + "content": ". We will build the networks with either no scale interaction or interaction of 2", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 516, + 136, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 136, + 527 + ], + "score": 1.0, + "content": "scales.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 19.5 + }, + { + "type": "title", + "bbox": [ + 107, + 543, + 196, + 555 + ], + "lines": [ + { + "bbox": [ + 105, + 542, + 198, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 198, + 558 + ], + "score": 1.0, + "content": "6.2 MNIST-SCALE", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 565, + 505, + 632 + ], + "lines": [ + { + "bbox": [ + 106, + 566, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 566, + 505, + 578 + ], + "score": 1.0, + "content": "Following Kanazawa et al. (2014); Marcos et al. (2018); Ghosh & Gupta (2019) we conduct experi-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 577, + 505, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 505, + 589 + ], + "score": 1.0, + "content": "ments on the MNIST-scale dataset. We rescale the images of the MNIST dataset LeCun et al. (1998)", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 588, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 106, + 588, + 117, + 600 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 117, + 588, + 157, + 599 + ], + "score": 0.78, + "content": "0 . 3 - 1 . 0", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 588, + 505, + 600 + ], + "score": 1.0, + "content": "of the original size and pad them with zeros to retain the initial resolution. The scal-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "score": 1.0, + "content": "ing factors are sampled uniformly and independently for each image. The obtained dataset is then", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 609, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 505, + 622 + ], + "score": 1.0, + "content": "split into 10,000 for training, 2,000 for evaluation and 50,000 for testing. We generate 6 different", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 621, + 287, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 287, + 633 + ], + "score": 1.0, + "content": "realizations and fix them for all experiments.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 29.5 + }, + { + "type": "text", + "bbox": [ + 107, + 637, + 505, + 693 + ], + "lines": [ + { + "bbox": [ + 106, + 638, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 505, + 650 + ], + "score": 1.0, + "content": "As a baseline model we use the model described in Ghosh & Gupta (2019), which currently holds", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 648, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 505, + 662 + ], + "score": 1.0, + "content": "the state-of-the-art result on this dataset. It consists of 3 convolutional and 2 fully-connected layers.", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 659, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 224, + 673 + ], + "score": 1.0, + "content": "Each layer has filters of size", + "type": "text" + }, + { + "bbox": [ + 225, + 660, + 249, + 671 + ], + "score": 0.88, + "content": "7 \\times 7", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 659, + 505, + 673 + ], + "score": 1.0, + "content": ". We keep the number of trainable parameters almost the same", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 671, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 106, + 671, + 505, + 683 + ], + "score": 1.0, + "content": "for all tested methods. This is achieved by varying the number of channels. For scale equivariant", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 681, + 384, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 384, + 694 + ], + "score": 1.0, + "content": "models we add scale projection at the end of the convolutional block.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 108, + 699, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "score": 1.0, + "content": "For SiCNN, DSS, SEVF and our model, we additionally train counterparts where after each con-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 709, + 506, + 721 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 506, + 721 + ], + "score": 1.0, + "content": "volution, an extra projection layer is inserted. Projection layers transform vector features in each", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "score": 1.0, + "content": "spatial position of each channel into scalar ones. 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Method(28×28)(28 × 28) +(56 × 56)(56 × 56) +# Params
CNN2.56 ± 0.041.96 ± 0.072.02 ± 0.071.60 ± 0.09495K
SiCNN2.40 ± 0.031.86 ± 0.102.02 ± 0.141.59 ± 0.03497K
SI-ConvNet2.40± 0.121.94 ± 0.071.82 ± 0.111.59 ± 0.10495K
SEVF Scalar2.30 ± 0.061.96 ± 0.071.87 ± 0.091.62 ± 0.07494 K
SEVF Vector2.63 ± 0.092.23 ± 0.092.12 ± 0.131.81 ± 0.09475 K
DSS Scalar2.53 ± 0.102.04± 0.081.92 ± 0.081.57 ± 0.08494 K
DSS Vector2.58 ± 0.111.95 ± 0.071.97 ± 0.081.57 ± 0.09494 K
SS-CNN2.32 ± 0.152.10 ±0.151.84 ± 0.101.76 ± 0.07494 K
SESN Scalar2.10 ±0.101.79 ± 0.091.74 ± 0.091.50 ± 0.07495K
SESN Vector2.08 ±0.091.76 ± 0.081.68 ± 0.061.42 ± 0.07495 K
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In ex-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 104, + 243, + 506, + 255 + ], + "spans": [ + { + "bbox": [ + 104, + 243, + 258, + 255 + ], + "score": 1.0, + "content": "periment we use image resolution of", + "type": "text" + }, + { + "bbox": [ + 258, + 243, + 293, + 253 + ], + "score": 0.91, + "content": "2 8 \\times 2 8", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 243, + 312, + 255 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 312, + 243, + 346, + 254 + ], + "score": 0.9, + "content": "5 6 \\times 5 6", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 243, + 506, + 255 + ], + "score": 1.0, + "content": ". We test both the regime without data", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 254, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 421, + 266 + ], + "score": 1.0, + "content": "augmentation, and the regime with scaling data augmentation, denoted with", + "type": "text" + }, + { + "bbox": [ + 421, + 254, + 438, + 265 + ], + "score": 0.56, + "content": "\" + \"", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 254, + 505, + 266 + ], + "score": 1.0, + "content": ". All results are", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 265, + 498, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 265, + 176, + 276 + ], + "score": 1.0, + "content": "reported as mean", + "type": "text" + }, + { + "bbox": [ + 176, + 266, + 187, + 275 + ], + "score": 0.74, + "content": "\\pm", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 265, + 498, + 276 + ], + "score": 1.0, + "content": "std over 6 different fixed realizations of the dataset. The best results are bold.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5, + "bbox_fs": [ + 104, + 231, + 506, + 276 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 301, + 229, + 312 + ], + "lines": [ + { + "bbox": [ + 106, + 300, + 231, + 313 + ], + "spans": [ + { + "bbox": [ + 106, + 300, + 231, + 313 + ], + "score": 1.0, + "content": "6.1 EQUIVARIANCE ERROR", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 107, + 323, + 505, + 389 + ], + "lines": [ + { + "bbox": [ + 106, + 324, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 106, + 324, + 505, + 335 + ], + "score": 1.0, + "content": "We have presented scale-convolution which is equivariant to scale transformation and translation for", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 334, + 505, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 505, + 347 + ], + "score": 1.0, + "content": "continuous signals. While translation equivariance holds true even for discretized signals and filters,", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 345, + 505, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 505, + 357 + ], + "score": 1.0, + "content": "scale equivariance may not be exact. Therefore, before starting any experiments, we check to which", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 356, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 505, + 369 + ], + "score": 1.0, + "content": "degree the predicted properties of scale-convolution hold true. We do so by measuring the difference", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 365, + 506, + 380 + ], + "spans": [ + { + "bbox": [ + 106, + 366, + 276, + 379 + ], + "score": 0.93, + "content": "\\dot { \\Delta } = \\| [ L _ { s } \\mathbf { \\hat { \\Phi } } ( f ) - \\Phi \\mathbf { \\hat { L } } _ { s } ( f ) \\| _ { 2 } ^ { 2 } / \\| L _ { s } \\Phi ( f ) \\| _ { 2 } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 365, + 308, + 380 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 308, + 367, + 317, + 377 + ], + "score": 0.84, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 365, + 506, + 380 + ], + "score": 1.0, + "content": "is scale-convolution with randomly initialized", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 378, + 143, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 143, + 391 + ], + "score": 1.0, + "content": "weights.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 10.5, + "bbox_fs": [ + 105, + 324, + 506, + 391 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 394, + 505, + 526 + ], + "lines": [ + { + "bbox": [ + 106, + 395, + 505, + 407 + ], + "spans": [ + { + "bbox": [ + 106, + 395, + 505, + 407 + ], + "score": 1.0, + "content": "In case of perfect equivariance the difference is equal to zero. We calculate the error on randomly", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 405, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 505, + 419 + ], + "score": 1.0, + "content": "sampled images from the STL-10 dataset Coates et al. (2011). The results are represented in Fig-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 417, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 505, + 429 + ], + "score": 1.0, + "content": "ure 2. The networks on the left and on the middle plots do not have interscale interactions. The", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 428, + 505, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 420, + 440 + ], + "score": 1.0, + "content": "networks on the middle and on the right plots consist of just one layer. We use", + "type": "text" + }, + { + "bbox": [ + 420, + 428, + 478, + 439 + ], + "score": 0.79, + "content": "N _ { S } = 5 , 1 3 , 5", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 428, + 505, + 440 + ], + "score": 1.0, + "content": "scales", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 439, + 505, + 451 + ], + "spans": [ + { + "bbox": [ + 106, + 439, + 505, + 451 + ], + "score": 1.0, + "content": "for the networks on the left, the middle, and the right plots respectively. While discretization intro-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 450, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 106, + 450, + 360, + 462 + ], + "score": 1.0, + "content": "duces some error, it stays very low, and is not much higher than", + "type": "text" + }, + { + "bbox": [ + 361, + 450, + 375, + 460 + ], + "score": 0.85, + "content": "6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 450, + 505, + 462 + ], + "score": 1.0, + "content": "for the networks with 50 layers.", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 460, + 505, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 505, + 473 + ], + "score": 1.0, + "content": "The difference, however, increases if the input image is downscaled more than 16 times. There-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 471, + 505, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 505, + 484 + ], + "score": 1.0, + "content": "fore, we are free to use deep networks. However, we should pay extra attention to extreme cases", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 482, + 505, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 505, + 495 + ], + "score": 1.0, + "content": "where scale changes are of very big magnitude. These are quite rare but still appear in practice.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 493, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 106, + 493, + 505, + 505 + ], + "score": 1.0, + "content": "Finally, we see that using SESN with interscale interaction introduces extra equivariance error due", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 505, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 106, + 505, + 185, + 516 + ], + "score": 1.0, + "content": "to the truncation of", + "type": "text" + }, + { + "bbox": [ + 185, + 505, + 192, + 514 + ], + "score": 0.76, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 193, + 505, + 505, + 516 + ], + "score": 1.0, + "content": ". We will build the networks with either no scale interaction or interaction of 2", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 516, + 136, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 136, + 527 + ], + "score": 1.0, + "content": "scales.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 19.5, + "bbox_fs": [ + 105, + 395, + 505, + 527 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 543, + 196, + 555 + ], + "lines": [ + { + "bbox": [ + 105, + 542, + 198, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 198, + 558 + ], + "score": 1.0, + "content": "6.2 MNIST-SCALE", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 565, + 505, + 632 + ], + "lines": [ + { + "bbox": [ + 106, + 566, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 566, + 505, + 578 + ], + "score": 1.0, + "content": "Following Kanazawa et al. (2014); Marcos et al. (2018); Ghosh & Gupta (2019) we conduct experi-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 577, + 505, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 505, + 589 + ], + "score": 1.0, + "content": "ments on the MNIST-scale dataset. We rescale the images of the MNIST dataset LeCun et al. (1998)", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 588, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 106, + 588, + 117, + 600 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 117, + 588, + 157, + 599 + ], + "score": 0.78, + "content": "0 . 3 - 1 . 0", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 588, + 505, + 600 + ], + "score": 1.0, + "content": "of the original size and pad them with zeros to retain the initial resolution. The scal-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "score": 1.0, + "content": "ing factors are sampled uniformly and independently for each image. The obtained dataset is then", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 609, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 505, + 622 + ], + "score": 1.0, + "content": "split into 10,000 for training, 2,000 for evaluation and 50,000 for testing. We generate 6 different", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 621, + 287, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 287, + 633 + ], + "score": 1.0, + "content": "realizations and fix them for all experiments.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 29.5, + "bbox_fs": [ + 105, + 566, + 505, + 633 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 637, + 505, + 693 + ], + "lines": [ + { + "bbox": [ + 106, + 638, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 505, + 650 + ], + "score": 1.0, + "content": "As a baseline model we use the model described in Ghosh & Gupta (2019), which currently holds", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 648, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 505, + 662 + ], + "score": 1.0, + "content": "the state-of-the-art result on this dataset. It consists of 3 convolutional and 2 fully-connected layers.", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 659, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 224, + 673 + ], + "score": 1.0, + "content": "Each layer has filters of size", + "type": "text" + }, + { + "bbox": [ + 225, + 660, + 249, + 671 + ], + "score": 0.88, + "content": "7 \\times 7", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 659, + 505, + 673 + ], + "score": 1.0, + "content": ". We keep the number of trainable parameters almost the same", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 671, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 106, + 671, + 505, + 683 + ], + "score": 1.0, + "content": "for all tested methods. This is achieved by varying the number of channels. For scale equivariant", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 681, + 384, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 384, + 694 + ], + "score": 1.0, + "content": "models we add scale projection at the end of the convolutional block.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 35, + "bbox_fs": [ + 105, + 638, + 505, + 694 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 699, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "score": 1.0, + "content": "For SiCNN, DSS, SEVF and our model, we additionally train counterparts where after each con-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 709, + 506, + 721 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 506, + 721 + ], + "score": 1.0, + "content": "volution, an extra projection layer is inserted. Projection layers transform vector features in each", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "score": 1.0, + "content": "spatial position of each channel into scalar ones. All of the layers have now scalar inputs instead of", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "vector inputs. Therefore, we denote these models with “Scalar”. The original models are denoted as", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 92, + 506, + 108 + ], + "spans": [ + { + "bbox": [ + 105, + 92, + 506, + 108 + ], + "score": 1.0, + "content": "“Vector”. The exact type of projection depends on the way the vector features are constructed. For", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "score": 1.0, + "content": "SiCNN, DSS, and SESN, we use maximum pooling along the scale dimension, while for SEVF, it", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 287, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 194, + 127 + ], + "score": 1.0, + "content": "is a calculation of the", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 194, + 116, + 206, + 127 + ], + "score": 0.88, + "content": "L _ { 2 }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 207, + 115, + 287, + 127 + ], + "score": 1.0, + "content": "-norm of the vector.", + "type": "text", + "cross_page": true + } + ], + "index": 3 + } + ], + "index": 39, + "bbox_fs": [ + 105, + 698, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 126 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "vector inputs. Therefore, we denote these models with “Scalar”. The original models are denoted as", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 92, + 506, + 108 + ], + "spans": [ + { + "bbox": [ + 105, + 92, + 506, + 108 + ], + "score": 1.0, + "content": "“Vector”. The exact type of projection depends on the way the vector features are constructed. For", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "score": 1.0, + "content": "SiCNN, DSS, and SESN, we use maximum pooling along the scale dimension, while for SEVF, it", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 287, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 194, + 127 + ], + "score": 1.0, + "content": "is a calculation of the", + "type": "text" + }, + { + "bbox": [ + 194, + 116, + 206, + 127 + ], + "score": 0.88, + "content": "L _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 115, + 287, + 127 + ], + "score": 1.0, + "content": "-norm of the vector.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "text", + "bbox": [ + 106, + 132, + 505, + 231 + ], + "lines": [ + { + "bbox": [ + 105, + 131, + 506, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 131, + 506, + 145 + ], + "score": 1.0, + "content": "All models are trained with the Adam optimizer Kingma & Ba (2014) for 60 epochs with a batch", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 143, + 505, + 155 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 505, + 155 + ], + "score": 1.0, + "content": "size of 128. Initial learning rate is set to 0.01 and divided by 10 after 20 and 40 epochs. We conduct", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 506, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 506, + 167 + ], + "score": 1.0, + "content": "the experiments with 4 different settings. Following the idea discussed in Ghosh & Gupta (2019),", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 504, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 468, + 178 + ], + "score": 1.0, + "content": "in addition to the standard setting we train the networks with input images upscaled to", + "type": "text" + }, + { + "bbox": [ + 469, + 165, + 504, + 176 + ], + "score": 0.89, + "content": "5 6 \\times 5 6", + "type": "inline_equation" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 177, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 106, + 177, + 505, + 189 + ], + "score": 1.0, + "content": "using bilinear interpolation. This results in all image transformations performed by the network", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 187, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 506, + 200 + ], + "score": 1.0, + "content": "becoming more stable, which produces less interpolation artifacts. For both input sizes we conduct", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 198, + 506, + 211 + ], + "spans": [ + { + "bbox": [ + 106, + 198, + 506, + 211 + ], + "score": 1.0, + "content": "the experiments without data augmentation and with scaling augmentation, which results in 4 setups", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 209, + 505, + 221 + ], + "spans": [ + { + "bbox": [ + 106, + 209, + 479, + 221 + ], + "score": 1.0, + "content": "in total. We run the experiments on 6 different realizations of MNIST-scale and report mean", + "type": "text" + }, + { + "bbox": [ + 480, + 210, + 490, + 220 + ], + "score": 0.72, + "content": "\\pm", + "type": "inline_equation" + }, + { + "bbox": [ + 491, + 209, + 505, + 221 + ], + "score": 1.0, + "content": "std", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 220, + 214, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 214, + 232 + ], + "score": 1.0, + "content": "calculated over these runs.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 107, + 236, + 505, + 303 + ], + "lines": [ + { + "bbox": [ + 105, + 236, + 505, + 250 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 505, + 250 + ], + "score": 1.0, + "content": "The obtained results are summarized in Table 2. The reported errors may differ a bit from the ones", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 246, + 506, + 262 + ], + "spans": [ + { + "bbox": [ + 105, + 246, + 506, + 262 + ], + "score": 1.0, + "content": "in the original paper because of the variations in generated datasets and slightly different training", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 258, + 506, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 506, + 272 + ], + "score": 1.0, + "content": "procedure. Nevertheless, we try to keep our configuration as close as possible to Ghosh & Gupta", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 269, + 505, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 269, + 505, + 282 + ], + "score": 1.0, + "content": "(2019) which currently demonstrated the best classification accuracy on MNIST-scale. For example,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 281, + 506, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 211, + 294 + ], + "score": 1.0, + "content": "SS-CNN reports error of", + "type": "text" + }, + { + "bbox": [ + 212, + 281, + 262, + 292 + ], + "score": 0.88, + "content": "1 . 9 1 \\pm 0 . 0 4", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 281, + 425, + 294 + ], + "score": 1.0, + "content": "in Ghosh & Gupta (2019) while it has", + "type": "text" + }, + { + "bbox": [ + 425, + 281, + 476, + 292 + ], + "score": 0.88, + "content": "1 . 8 4 \\pm 0 . 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 281, + 506, + 294 + ], + "score": 1.0, + "content": "in our", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 292, + 160, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 292, + 160, + 306 + ], + "score": 1.0, + "content": "experiments.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 15.5 + }, + { + "type": "text", + "bbox": [ + 107, + 308, + 504, + 353 + ], + "lines": [ + { + "bbox": [ + 105, + 308, + 505, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 505, + 321 + ], + "score": 1.0, + "content": "SESN significantly outperforms other methods in all 4 regimes. “Scalar” versions of it already", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 320, + 506, + 332 + ], + "spans": [ + { + "bbox": [ + 106, + 320, + 506, + 332 + ], + "score": 1.0, + "content": "outperform all previous methods, and “Vector” versions make the gain even more significant. The", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 331, + 506, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 506, + 343 + ], + "score": 1.0, + "content": "global architectures of all models are the same for all rows, which indicates that the way scale", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 342, + 284, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 284, + 353 + ], + "score": 1.0, + "content": "convolution is done plays an important role.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20.5 + }, + { + "type": "title", + "bbox": [ + 107, + 368, + 165, + 379 + ], + "lines": [ + { + "bbox": [ + 105, + 366, + 168, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 168, + 381 + ], + "score": 1.0, + "content": "6.3 STL-10", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 389, + 338, + 489 + ], + "lines": [ + { + "bbox": [ + 106, + 390, + 339, + 401 + ], + "spans": [ + { + "bbox": [ + 106, + 390, + 339, + 401 + ], + "score": 1.0, + "content": "In order to evaluate the role of scale equivariance in nat-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 401, + 339, + 413 + ], + "spans": [ + { + "bbox": [ + 106, + 401, + 339, + 413 + ], + "score": 1.0, + "content": "ural image classification, we conduct the experiments on", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 411, + 340, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 340, + 423 + ], + "score": 1.0, + "content": "STL-10 dataset Coates et al. (2011). This dataset consists", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 423, + 339, + 434 + ], + "spans": [ + { + "bbox": [ + 106, + 423, + 339, + 434 + ], + "score": 1.0, + "content": "of 8,000 training and 5,000 testing labeled images. Ad-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 434, + 339, + 445 + ], + "spans": [ + { + "bbox": [ + 106, + 434, + 339, + 445 + ], + "score": 1.0, + "content": "ditionally, it includes 100,000 unlabeled images. The im-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 444, + 339, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 205, + 456 + ], + "score": 1.0, + "content": "ages have a resolution of", + "type": "text" + }, + { + "bbox": [ + 205, + 445, + 235, + 455 + ], + "score": 0.91, + "content": "9 6 \\times 9 6", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 444, + 339, + 456 + ], + "score": 1.0, + "content": "pixels and RGB channels.", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 455, + 340, + 467 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 340, + 467 + ], + "score": 1.0, + "content": "Labeled images belong to 10 classes such as bird, horse or", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 467, + 339, + 478 + ], + "spans": [ + { + "bbox": [ + 106, + 467, + 339, + 478 + ], + "score": 1.0, + "content": "car. We use only the labeled subset to demonstrate the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 478, + 310, + 490 + ], + "spans": [ + { + "bbox": [ + 106, + 478, + 310, + 490 + ], + "score": 1.0, + "content": "performance of the models in the low data regime.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 107, + 494, + 338, + 572 + ], + "lines": [ + { + "bbox": [ + 106, + 493, + 339, + 507 + ], + "spans": [ + { + "bbox": [ + 106, + 493, + 339, + 507 + ], + "score": 1.0, + "content": "The dataset is normalized by subtracting the per-channel", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 505, + 339, + 517 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 339, + 517 + ], + "score": 1.0, + "content": "mean and dividing by the per-channel standard deviation.", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 516, + 339, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 339, + 529 + ], + "score": 1.0, + "content": "During training, we augment the dataset by applying 12", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 528, + 339, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 339, + 540 + ], + "score": 1.0, + "content": "pixel zero padding and randomly cropping the images to", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 538, + 339, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 125, + 550 + ], + "score": 1.0, + "content": "size", + "type": "text" + }, + { + "bbox": [ + 125, + 538, + 159, + 549 + ], + "score": 0.9, + "content": "9 6 \\times 9 6", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 538, + 339, + 550 + ], + "score": 1.0, + "content": ". 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MethodError, %#Params
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SiCNN11.6211.0 M
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DSS11.2811.0 M
SS-CNN25.4710.8 M
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SESN-C14.0811.0 M
Harm WRN9.5511.0 M
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The best results are bold.", + "type": "text" + } + ], + "index": 57 + }, + { + "bbox": [ + 346, + 575, + 505, + 587 + ], + "spans": [ + { + "bbox": [ + 346, + 575, + 505, + 587 + ], + "score": 1.0, + "content": "We additionally report the current best", + "type": "text" + } + ], + "index": 58 + }, + { + "bbox": [ + 345, + 586, + 505, + 598 + ], + "spans": [ + { + "bbox": [ + 345, + 586, + 505, + 598 + ], + "score": 1.0, + "content": "result achieved by Harm WRN from", + "type": "text" + } + ], + "index": 59 + }, + { + "bbox": [ + 346, + 597, + 429, + 610 + ], + "spans": [ + { + "bbox": [ + 346, + 597, + 429, + 610 + ], + "score": 1.0, + "content": "Ulicny et al. 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The initial learning rate is set to 0.1 and", + "type": "text" + } + ], + "index": 62 + }, + { + "bbox": [ + 106, + 671, + 303, + 683 + ], + "spans": [ + { + "bbox": [ + 106, + 671, + 303, + 683 + ], + "score": 1.0, + "content": "divided by 5 after 300, 400, 600 and 800 epochs.", + "type": "text" + } + ], + "index": 63 + } + ], + "index": 62 + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "The results are summarized in Table 3. We found SEVF training unstable and therefore do not", + "type": "text" + } + ], + "index": 64 + }, + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "score": 1.0, + "content": "include it in the table. Pure scale-invariant SI-ConvNet and SS-CNN demonstrate significantly worse", + "type": "text" + } + ], + "index": 65 + }, + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "results than the baseline. We note the importance of equivariance for deep networks. We also find", + "type": "text" + } + ], + "index": 66 + }, + { + "bbox": [ + 105, + 720, + 506, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 506, + 734 + ], + "score": 1.0, + "content": "that SESN-C performs significantly worse than SESN-A and SESN-B due to high equivariance", + "type": "text" + } + ], + "index": 67 + } + ], + "index": 65.5 + } + ], + "page_idx": 7, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 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": "text", + "bbox": [ + 107, + 82, + 505, + 126 + ], + "lines": [], + "index": 1.5, + "bbox_fs": [ + 105, + 82, + 506, + 127 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 106, + 132, + 505, + 231 + ], + "lines": [ + { + "bbox": [ + 105, + 131, + 506, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 131, + 506, + 145 + ], + "score": 1.0, + "content": "All models are trained with the Adam optimizer Kingma & Ba (2014) for 60 epochs with a batch", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 143, + 505, + 155 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 505, + 155 + ], + "score": 1.0, + "content": "size of 128. Initial learning rate is set to 0.01 and divided by 10 after 20 and 40 epochs. We conduct", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 506, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 506, + 167 + ], + "score": 1.0, + "content": "the experiments with 4 different settings. Following the idea discussed in Ghosh & Gupta (2019),", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 504, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 468, + 178 + ], + "score": 1.0, + "content": "in addition to the standard setting we train the networks with input images upscaled to", + "type": "text" + }, + { + "bbox": [ + 469, + 165, + 504, + 176 + ], + "score": 0.89, + "content": "5 6 \\times 5 6", + "type": "inline_equation" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 177, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 106, + 177, + 505, + 189 + ], + "score": 1.0, + "content": "using bilinear interpolation. This results in all image transformations performed by the network", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 187, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 506, + 200 + ], + "score": 1.0, + "content": "becoming more stable, which produces less interpolation artifacts. For both input sizes we conduct", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 198, + 506, + 211 + ], + "spans": [ + { + "bbox": [ + 106, + 198, + 506, + 211 + ], + "score": 1.0, + "content": "the experiments without data augmentation and with scaling augmentation, which results in 4 setups", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 209, + 505, + 221 + ], + "spans": [ + { + "bbox": [ + 106, + 209, + 479, + 221 + ], + "score": 1.0, + "content": "in total. We run the experiments on 6 different realizations of MNIST-scale and report mean", + "type": "text" + }, + { + "bbox": [ + 480, + 210, + 490, + 220 + ], + "score": 0.72, + "content": "\\pm", + "type": "inline_equation" + }, + { + "bbox": [ + 491, + 209, + 505, + 221 + ], + "score": 1.0, + "content": "std", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 220, + 214, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 214, + 232 + ], + "score": 1.0, + "content": "calculated over these runs.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 8, + "bbox_fs": [ + 105, + 131, + 506, + 232 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 236, + 505, + 303 + ], + "lines": [ + { + "bbox": [ + 105, + 236, + 505, + 250 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 505, + 250 + ], + "score": 1.0, + "content": "The obtained results are summarized in Table 2. The reported errors may differ a bit from the ones", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 246, + 506, + 262 + ], + "spans": [ + { + "bbox": [ + 105, + 246, + 506, + 262 + ], + "score": 1.0, + "content": "in the original paper because of the variations in generated datasets and slightly different training", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 258, + 506, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 506, + 272 + ], + "score": 1.0, + "content": "procedure. Nevertheless, we try to keep our configuration as close as possible to Ghosh & Gupta", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 269, + 505, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 269, + 505, + 282 + ], + "score": 1.0, + "content": "(2019) which currently demonstrated the best classification accuracy on MNIST-scale. For example,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 281, + 506, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 211, + 294 + ], + "score": 1.0, + "content": "SS-CNN reports error of", + "type": "text" + }, + { + "bbox": [ + 212, + 281, + 262, + 292 + ], + "score": 0.88, + "content": "1 . 9 1 \\pm 0 . 0 4", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 281, + 425, + 294 + ], + "score": 1.0, + "content": "in Ghosh & Gupta (2019) while it has", + "type": "text" + }, + { + "bbox": [ + 425, + 281, + 476, + 292 + ], + "score": 0.88, + "content": "1 . 8 4 \\pm 0 . 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 281, + 506, + 294 + ], + "score": 1.0, + "content": "in our", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 292, + 160, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 292, + 160, + 306 + ], + "score": 1.0, + "content": "experiments.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 15.5, + "bbox_fs": [ + 105, + 236, + 506, + 306 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 308, + 504, + 353 + ], + "lines": [ + { + "bbox": [ + 105, + 308, + 505, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 505, + 321 + ], + "score": 1.0, + "content": "SESN significantly outperforms other methods in all 4 regimes. “Scalar” versions of it already", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 320, + 506, + 332 + ], + "spans": [ + { + "bbox": [ + 106, + 320, + 506, + 332 + ], + "score": 1.0, + "content": "outperform all previous methods, and “Vector” versions make the gain even more significant. The", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 331, + 506, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 506, + 343 + ], + "score": 1.0, + "content": "global architectures of all models are the same for all rows, which indicates that the way scale", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 342, + 284, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 284, + 353 + ], + "score": 1.0, + "content": "convolution is done plays an important role.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 308, + 506, + 353 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 368, + 165, + 379 + ], + "lines": [ + { + "bbox": [ + 105, + 366, + 168, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 168, + 381 + ], + "score": 1.0, + "content": "6.3 STL-10", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 389, + 338, + 489 + ], + "lines": [ + { + "bbox": [ + 106, + 390, + 339, + 401 + ], + "spans": [ + { + "bbox": [ + 106, + 390, + 339, + 401 + ], + "score": 1.0, + "content": "In order to evaluate the role of scale equivariance in nat-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 401, + 339, + 413 + ], + "spans": [ + { + "bbox": [ + 106, + 401, + 339, + 413 + ], + "score": 1.0, + "content": "ural image classification, we conduct the experiments on", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 411, + 340, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 340, + 423 + ], + "score": 1.0, + "content": "STL-10 dataset Coates et al. (2011). This dataset consists", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 423, + 339, + 434 + ], + "spans": [ + { + "bbox": [ + 106, + 423, + 339, + 434 + ], + "score": 1.0, + "content": "of 8,000 training and 5,000 testing labeled images. Ad-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 434, + 339, + 445 + ], + "spans": [ + { + "bbox": [ + 106, + 434, + 339, + 445 + ], + "score": 1.0, + "content": "ditionally, it includes 100,000 unlabeled images. The im-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 444, + 339, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 205, + 456 + ], + "score": 1.0, + "content": "ages have a resolution of", + "type": "text" + }, + { + "bbox": [ + 205, + 445, + 235, + 455 + ], + "score": 0.91, + "content": "9 6 \\times 9 6", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 444, + 339, + 456 + ], + "score": 1.0, + "content": "pixels and RGB channels.", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 455, + 340, + 467 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 340, + 467 + ], + "score": 1.0, + "content": "Labeled images belong to 10 classes such as bird, horse or", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 467, + 339, + 478 + ], + "spans": [ + { + "bbox": [ + 106, + 467, + 339, + 478 + ], + "score": 1.0, + "content": "car. We use only the labeled subset to demonstrate the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 478, + 310, + 490 + ], + "spans": [ + { + "bbox": [ + 106, + 478, + 310, + 490 + ], + "score": 1.0, + "content": "performance of the models in the low data regime.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 390, + 340, + 490 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 494, + 338, + 572 + ], + "lines": [ + { + "bbox": [ + 106, + 493, + 339, + 507 + ], + "spans": [ + { + "bbox": [ + 106, + 493, + 339, + 507 + ], + "score": 1.0, + "content": "The dataset is normalized by subtracting the per-channel", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 505, + 339, + 517 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 339, + 517 + ], + "score": 1.0, + "content": "mean and dividing by the per-channel standard deviation.", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 516, + 339, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 339, + 529 + ], + "score": 1.0, + "content": "During training, we augment the dataset by applying 12", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 528, + 339, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 339, + 540 + ], + "score": 1.0, + "content": "pixel zero padding and randomly cropping the images to", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 538, + 339, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 125, + 550 + ], + "score": 1.0, + "content": "size", + "type": "text" + }, + { + "bbox": [ + 125, + 538, + 159, + 549 + ], + "score": 0.9, + "content": "9 6 \\times 9 6", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 538, + 339, + 550 + ], + "score": 1.0, + "content": ". Additionally, random horizontal flips with", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 548, + 340, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 548, + 152, + 562 + ], + "score": 1.0, + "content": "probability", + "type": "text" + }, + { + "bbox": [ + 152, + 549, + 172, + 560 + ], + "score": 0.89, + "content": "5 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 548, + 340, + 562 + ], + "score": 1.0, + "content": "and Cutout DeVries & Taylor (2017) with", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 559, + 221, + 573 + ], + "spans": [ + { + "bbox": [ + 106, + 559, + 221, + 573 + ], + "score": 1.0, + "content": "1 hole of 32 pixels are used.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 36, + "bbox_fs": [ + 105, + 493, + 340, + 573 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 577, + 340, + 643 + ], + "lines": [ + { + "bbox": [ + 106, + 576, + 339, + 589 + ], + "spans": [ + { + "bbox": [ + 106, + 576, + 339, + 589 + ], + "score": 1.0, + "content": "As a baseline we choose WideResNet Zagoruyko & Ko-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 588, + 339, + 600 + ], + "spans": [ + { + "bbox": [ + 106, + 588, + 339, + 600 + ], + "score": 1.0, + "content": "modakis (2016) with 16 layers and a widening factor of 8.", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 599, + 339, + 611 + ], + "spans": [ + { + "bbox": [ + 106, + 599, + 339, + 611 + ], + "score": 1.0, + "content": "We set dropout probability to 0.3 in all blocks. 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MethodError, %#Params
WRN11.4811.0 M
SiCNN11.6211.0 M
SI-ConvNet12.4811.0 M
DSS11.2811.0 M
SS-CNN25.4710.8 M
SESN-A10.8311.0 M
SESN-B8.5111.0 M
SESN-C14.0811.0 M
Harm WRN9.5511.0 M
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The best results are bold.", + "type": "text" + } + ], + "index": 57 + }, + { + "bbox": [ + 346, + 575, + 505, + 587 + ], + "spans": [ + { + "bbox": [ + 346, + 575, + 505, + 587 + ], + "score": 1.0, + "content": "We additionally report the current best", + "type": "text" + } + ], + "index": 58 + }, + { + "bbox": [ + 345, + 586, + 505, + 598 + ], + "spans": [ + { + "bbox": [ + 345, + 586, + 505, + 598 + ], + "score": 1.0, + "content": "result achieved by Harm WRN from", + "type": "text" + } + ], + "index": 59 + }, + { + "bbox": [ + 346, + 597, + 429, + 610 + ], + "spans": [ + { + "bbox": [ + 346, + 597, + 429, + 610 + ], + "score": 1.0, + "content": "Ulicny et al. 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The initial learning rate is set to 0.1 and", + "type": "text" + } + ], + "index": 62 + }, + { + "bbox": [ + 106, + 671, + 303, + 683 + ], + "spans": [ + { + "bbox": [ + 106, + 671, + 303, + 683 + ], + "score": 1.0, + "content": "divided by 5 after 300, 400, 600 and 800 epochs.", + "type": "text" + } + ], + "index": 63 + } + ], + "index": 62, + "bbox_fs": [ + 105, + 647, + 502, + 683 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "The results are summarized in Table 3. We found SEVF training unstable and therefore do not", + "type": "text" + } + ], + "index": 64 + }, + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "score": 1.0, + "content": "include it in the table. Pure scale-invariant SI-ConvNet and SS-CNN demonstrate significantly worse", + "type": "text" + } + ], + "index": 65 + }, + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "results than the baseline. We note the importance of equivariance for deep networks. We also find", + "type": "text" + } + ], + "index": 66 + }, + { + "bbox": [ + 105, + 720, + 506, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 506, + 734 + ], + "score": 1.0, + "content": "that SESN-C performs significantly worse than SESN-A and SESN-B due to high equivariance", + "type": "text" + } + ], + "index": 67 + }, + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "error caused by interscale interaction. SESN-B significantly improves the results of both WRN", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "and DSS due to the projection between scales. The maximum scale projection makes the weights", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "score": 1.0, + "content": "of the next layer to have a maximum receptive field in the space of scales. This is an easy yet", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 506, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 506, + 127 + ], + "score": 1.0, + "content": "effective method for capturing the correlations between different scales. This experiment shows that", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 127, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 505, + 138 + ], + "score": 1.0, + "content": "scale-equivariance is a very useful inductive bias for natural image classification with deep neural", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 148, + 149 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 148, + 149 + ], + "score": 1.0, + "content": "networks.", + "type": "text", + "cross_page": true + } + ], + "index": 5 + } + ], + "index": 65.5, + "bbox_fs": [ + 105, + 687, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 148 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "error caused by interscale interaction. SESN-B significantly improves the results of both WRN", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "and DSS due to the projection between scales. The maximum scale projection makes the weights", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "score": 1.0, + "content": "of the next layer to have a maximum receptive field in the space of scales. This is an easy yet", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 506, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 506, + 127 + ], + "score": 1.0, + "content": "effective method for capturing the correlations between different scales. 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Method28×28,s56 × 56,s
CNN3.83.8
SiCNN Scalar13.518.9
SiCNN Vector15.322.8
SI-ConvNet18.433.1
SEVF Scalar21.038.4
SEVF Vector25.446.0
DSS Scalar3.95.0
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CNN3263951
SiCNN3263957
SI-ConvNet3263957
SEVF Scalar3263952568
SEVF Vector2345688
DSS3263954
SS-CNN3060906
SESN3263954
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MethodBlock 1Block 2Block 3# Scales
CNN1632641
SiCNN1632643
SI-ConvNet1632643
SEVF1123453
DSS1632644
SS-CNN1122443
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MethodConv 1Conv 2Conv 3FC1# Scales
CNN3263951
SiCNN3263957
SI-ConvNet3263957
SEVF Scalar3263952568
SEVF Vector2345688
DSS3263954
SS-CNN3060906
SESN3263954
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MethodBlock 1Block 2Block 3# Scales
CNN1632641
SiCNN1632643
SI-ConvNet1632643
SEVF1123453
DSS1632644
SS-CNN1122443
SESN1632643
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