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The entire system, including the neural", "type": "text" } ], "index": 11 }, { "bbox": [ 106, 357, 437, 370 ], "spans": [ { "bbox": [ 106, 357, 437, 370 ], "score": 1.0, "content": "and probabilistic modules, are finally trained end-to-end via a single loss function.", "type": "text" } ], "index": 12 } ], "index": 7.5 } ], "index": 4.25 }, { "type": "text", "bbox": [ 107, 380, 505, 567 ], "lines": [ { "bbox": [ 106, 380, 505, 393 ], "spans": [ { "bbox": [ 106, 380, 505, 393 ], "score": 1.0, "content": "We propose SLASH – a novel DPPL that, similar to the punctuation symbol, can be used to efficiently", "type": "text" } ], "index": 13 }, { "bbox": [ 105, 390, 505, 404 ], "spans": [ { "bbox": [ 105, 390, 505, 404 ], "score": 1.0, "content": "combine several paradigms into one. 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SLASH consists of several key building blocks.", "type": "text" } ], "index": 17 }, { "bbox": [ 106, 434, 506, 447 ], "spans": [ { "bbox": [ 106, 434, 506, 447 ], "score": 1.0, "content": "Firstly, it makes use of Neural-Probabilistic Predicates (NPPs) for probability estimation. NPPs", "type": "text" } ], "index": 18 }, { "bbox": [ 104, 444, 506, 460 ], "spans": [ { "bbox": [ 104, 444, 506, 460 ], "score": 1.0, "content": "consist of neural and/or probabilistic circuit (PC) modules and act as a unifying term, encompassing", "type": "text" } ], "index": 19 }, { "bbox": [ 105, 455, 506, 470 ], "spans": [ { "bbox": [ 105, 455, 506, 470 ], "score": 1.0, "content": "the neural predicates of DeepProbLog and NeurASP, as well as purely probabilistic predicates. In", "type": "text" } ], "index": 20 }, { "bbox": [ 106, 467, 505, 480 ], "spans": [ { "bbox": [ 106, 467, 505, 480 ], "score": 1.0, "content": "this work, we introduce a much more powerful “flavor” of NPPs that consist jointly of neural and PC", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 478, 506, 492 ], "spans": [ { "bbox": [ 105, 478, 506, 492 ], "score": 1.0, "content": "modules, taking advantage of the power of neural computations together with true density estimation", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 487, 507, 504 ], "spans": [ { "bbox": [ 105, 487, 507, 504 ], "score": 1.0, "content": "of PCs. Depending on the underlying task one can thus ask a range of queries to the NPP, e.g.", "type": "text" } ], "index": 23 }, { "bbox": [ 105, 501, 505, 513 ], "spans": [ { "bbox": [ 105, 501, 505, 513 ], "score": 1.0, "content": "sample an unknown, desired variable, but also query for conditional class probabilities. Example", "type": "text" } ], "index": 24 }, { "bbox": [ 105, 511, 506, 524 ], "spans": [ { "bbox": [ 105, 511, 506, 524 ], "score": 1.0, "content": "NPPs consisting of a slot attention encoder and several PCs are depicted in Fig. 1 for the task of", "type": "text" } ], "index": 25 }, { "bbox": [ 105, 522, 506, 535 ], "spans": [ { "bbox": [ 105, 522, 506, 535 ], "score": 1.0, "content": "set prediction. The slot encoder is shared across all NPPs, whereas the PC of each NPP models a", "type": "text" } ], "index": 26 }, { "bbox": [ 105, 533, 505, 547 ], "spans": [ { "bbox": [ 105, 533, 505, 547 ], "score": 1.0, "content": "separate category of attributes. In this way, each NPP models the joint distribution over slot encodings", "type": "text" } ], "index": 27 }, { "bbox": [ 106, 545, 505, 558 ], "spans": [ { "bbox": [ 106, 545, 505, 558 ], "score": 1.0, "content": "and object attribute values, such as the color of an object. By querying the NPP, one can obtain", "type": "text" } ], "index": 28 }, { "bbox": [ 106, 555, 429, 568 ], "spans": [ { "bbox": [ 106, 555, 429, 568 ], "score": 1.0, "content": "task-related probability estimations, such as the conditional attribute probability.", "type": "text" } ], "index": 29 } ], "index": 21 }, { "type": "text", "bbox": [ 107, 572, 505, 672 ], "lines": [ { "bbox": [ 105, 572, 505, 585 ], "spans": [ { "bbox": [ 105, 572, 505, 585 ], "score": 1.0, "content": "The second component of SLASH is the logical program, which consists of a set of facts and logical", "type": "text" } ], "index": 30 }, { "bbox": [ 105, 583, 506, 595 ], "spans": [ { "bbox": [ 105, 583, 506, 595 ], "score": 1.0, "content": "statements defining the state of the world of the underlying task. For example, one can define the rules", "type": "text" } ], "index": 31 }, { "bbox": [ 105, 594, 506, 607 ], "spans": [ { "bbox": [ 105, 594, 506, 607 ], "score": 1.0, "content": "for when an object possesses a specific set of attributes (cf. Fig. 1). Thirdly, an ASP module is used to", "type": "text" } ], "index": 32 }, { "bbox": [ 105, 605, 505, 618 ], "spans": [ { "bbox": [ 105, 605, 505, 618 ], "score": 1.0, "content": "combine the first two components. Given a logical query about the input data, the logical program and", "type": "text" } ], "index": 33 }, { "bbox": [ 105, 615, 505, 629 ], "spans": [ { "bbox": [ 105, 615, 505, 629 ], "score": 1.0, "content": "the probability estimates obtained from the NPP(s), the ASP module produces a probability estimate", "type": "text" } ], "index": 34 }, { "bbox": [ 105, 625, 505, 641 ], "spans": [ { "bbox": [ 105, 625, 505, 641 ], "score": 1.0, "content": "about the truth value of the query, stating, e.g., how likely it is for a specific object in an image to be", "type": "text" } ], "index": 35 }, { "bbox": [ 105, 638, 506, 651 ], "spans": [ { "bbox": [ 105, 638, 506, 651 ], "score": 1.0, "content": "a large, dark red triangle. In contrast to query evaluation in Prolog (Colmerauer & Roussel, 1993;", "type": "text" } ], "index": 36 }, { "bbox": [ 105, 648, 506, 662 ], "spans": [ { "bbox": [ 105, 648, 506, 662 ], "score": 1.0, "content": "Clocksin & Mellish, 1981) which may lead to an infinite loop, many modern answer set solvers use", "type": "text" } ], "index": 37 }, { "bbox": [ 105, 659, 433, 673 ], "spans": [ { "bbox": [ 105, 659, 433, 673 ], "score": 1.0, "content": "Conflict-Driven-Clause-Learning (CDPL) which, in principle, always terminates.", "type": "text" } ], "index": 38 } ], "index": 34 }, { "type": "text", "bbox": [ 107, 677, 505, 732 ], "lines": [ { "bbox": [ 105, 676, 505, 690 ], "spans": [ { "bbox": [ 105, 676, 505, 690 ], "score": 1.0, "content": "Training in SLASH is performed efficiently in a batch-wise and end-to-end fashion, by integrating", "type": "text" } ], "index": 39 }, { "bbox": [ 105, 687, 506, 700 ], "spans": [ { "bbox": [ 105, 687, 506, 700 ], "score": 1.0, "content": "the parameters of all modules, neural and probabilistic, into a single loss term. SLASH thus allows a", "type": "text" } ], "index": 40 }, { "bbox": [ 105, 698, 506, 712 ], "spans": [ { "bbox": [ 105, 698, 506, 712 ], "score": 1.0, "content": "simple, fast and effective integration of sub-symbolic and symbolic computations. In our experiments,", "type": "text" } ], "index": 41 }, { "bbox": [ 105, 709, 506, 722 ], "spans": [ { "bbox": [ 105, 709, 506, 722 ], "score": 1.0, "content": "we investigate the advantages of SLASH in comparison to SOTA DPPLs on the benchmark task of", "type": "text" } ], "index": 42 }, { "bbox": [ 105, 720, 505, 734 ], "spans": [ { "bbox": [ 105, 720, 505, 734 ], "score": 1.0, "content": "MNIST-Addition (Manhaeve et al., 2018). We hereby show SLASH’s increased scalability regarding", "type": "text" } ], "index": 43 } ], "index": 41 } ], "page_idx": 1, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 107, 27, 308, 37 ], "lines": [ { "bbox": [ 106, 25, 309, 39 ], "spans": [ { "bbox": [ 106, 25, 309, 39 ], "score": 1.0, "content": "Under review as a conference paper at ICLR 2022", "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": "image", "bbox": [ 113, 83, 496, 246 ], "blocks": [ { "type": "image_body", "bbox": [ 113, 83, 496, 246 ], "group_id": 0, "lines": [ { "bbox": [ 113, 83, 496, 246 ], "spans": [ { "bbox": [ 113, 83, 496, 246 ], "score": 0.967, "type": "image", "image_path": "fbc474f94d8a2574e5362c4b2e3c3d61cc7e2bb6baf0aac66d0dc2840ab2de17.jpg" } ] } ], "index": 1, "virtual_lines": [ { "bbox": [ 113, 83, 496, 137.33333333333334 ], "spans": [], "index": 0 }, { "bbox": [ 113, 137.33333333333334, 496, 191.66666666666669 ], "spans": [], "index": 1 }, { "bbox": [ 113, 191.66666666666669, 496, 246.00000000000003 ], "spans": [], "index": 2 } ] }, { "type": "image_caption", "bbox": [ 106, 258, 505, 369 ], "group_id": 0, "lines": [ { "bbox": [ 106, 258, 505, 271 ], "spans": [ { "bbox": [ 106, 258, 505, 271 ], "score": 1.0, "content": "Figure 1: SLASH Attention illustrated for a visual reasoning task. SLASH with Neural-Probabilistic", "type": "text" } ], "index": 3 }, { "bbox": [ 105, 269, 506, 282 ], "spans": [ { "bbox": [ 105, 269, 506, 282 ], "score": 1.0, "content": "Predicates consisting of a slot attention encoder and Probabilistic Circuits (PCs) realised via EiNets.", "type": "text" } ], "index": 4 }, { "bbox": [ 105, 280, 506, 293 ], "spans": [ { "bbox": [ 105, 280, 506, 293 ], "score": 1.0, "content": "The slot encoder is shared over all NPPs. Each triangle in the figure represents a single EiNet that", "type": "text" } ], "index": 5 }, { "bbox": [ 105, 291, 506, 304 ], "spans": [ { "bbox": [ 105, 291, 506, 304 ], "score": 1.0, "content": "gives us a joint distribution at the root node. Thus, each PC learns the joint distribution over slot", "type": "text" } ], "index": 6 }, { "bbox": [ 105, 302, 506, 316 ], "spans": [ { "bbox": [ 105, 302, 153, 316 ], "score": 1.0, "content": "encodings,", "type": "text" }, { "bbox": [ 153, 302, 163, 313 ], "score": 0.84, "content": "z ^ { i }", "type": "inline_equation" }, { "bbox": [ 163, 302, 257, 316 ], "score": 1.0, "content": ", and object attributes,", "type": "text" }, { "bbox": [ 257, 303, 266, 313 ], "score": 0.76, "content": "C", "type": "inline_equation" }, { "bbox": [ 266, 302, 506, 316 ], "score": 1.0, "content": ", of a specific category, e.g. color attributes. Via targeted", "type": "text" } ], "index": 7 }, { "bbox": [ 105, 313, 506, 327 ], "spans": [ { "bbox": [ 105, 313, 506, 327 ], "score": 1.0, "content": "queries to the NPPs, one can obtain task-related probabilities, e.g. conditional probabilities for the", "type": "text" } ], "index": 8 }, { "bbox": [ 105, 323, 507, 338 ], "spans": [ { "bbox": [ 105, 323, 507, 338 ], "score": 1.0, "content": "task of set prediction. Given the probability estimates from the NPP(s) and a SLASH program,", "type": "text" } ], "index": 9 }, { "bbox": [ 105, 335, 506, 348 ], "spans": [ { "bbox": [ 105, 335, 506, 348 ], "score": 1.0, "content": "containing a set of facts and logical statements about the world, the probability of the truth value of a", "type": "text" } ], "index": 10 }, { "bbox": [ 106, 346, 506, 359 ], "spans": [ { "bbox": [ 106, 346, 506, 359 ], "score": 1.0, "content": "task-related query are computed via answer set programming. The entire system, including the neural", "type": "text" } ], "index": 11 }, { "bbox": [ 106, 357, 437, 370 ], "spans": [ { "bbox": [ 106, 357, 437, 370 ], "score": 1.0, "content": "and probabilistic modules, are finally trained end-to-end via a single loss function.", "type": "text" } ], "index": 12 } ], "index": 7.5 } ], "index": 4.25 }, { "type": "text", "bbox": [ 107, 380, 505, 567 ], "lines": [ { "bbox": [ 106, 380, 505, 393 ], "spans": [ { "bbox": [ 106, 380, 505, 393 ], "score": 1.0, "content": "We propose SLASH – a novel DPPL that, similar to the punctuation symbol, can be used to efficiently", "type": "text" } ], "index": 13 }, { "bbox": [ 105, 390, 505, 404 ], "spans": [ { "bbox": [ 105, 390, 505, 404 ], "score": 1.0, "content": "combine several paradigms into one. Specifically, SLASH represents a scalable programming", "type": "text" } ], "index": 14 }, { "bbox": [ 106, 402, 505, 415 ], "spans": [ { "bbox": [ 106, 402, 505, 415 ], "score": 1.0, "content": "language that seamlessly integrates probabilistic logical programming with neural representations", "type": "text" } ], "index": 15 }, { "bbox": [ 106, 412, 505, 425 ], "spans": [ { "bbox": [ 106, 412, 505, 425 ], "score": 1.0, "content": "and tractable probabilistic estimations. Fig. 1 shows an example instantiation of SLASH, termed", "type": "text" } ], "index": 16 }, { "bbox": [ 106, 424, 506, 437 ], "spans": [ { "bbox": [ 106, 424, 506, 437 ], "score": 1.0, "content": "SLASH Attention, for object-centric set prediction. SLASH consists of several key building blocks.", "type": "text" } ], "index": 17 }, { "bbox": [ 106, 434, 506, 447 ], "spans": [ { "bbox": [ 106, 434, 506, 447 ], "score": 1.0, "content": "Firstly, it makes use of Neural-Probabilistic Predicates (NPPs) for probability estimation. NPPs", "type": "text" } ], "index": 18 }, { "bbox": [ 104, 444, 506, 460 ], "spans": [ { "bbox": [ 104, 444, 506, 460 ], "score": 1.0, "content": "consist of neural and/or probabilistic circuit (PC) modules and act as a unifying term, encompassing", "type": "text" } ], "index": 19 }, { "bbox": [ 105, 455, 506, 470 ], "spans": [ { "bbox": [ 105, 455, 506, 470 ], "score": 1.0, "content": "the neural predicates of DeepProbLog and NeurASP, as well as purely probabilistic predicates. In", "type": "text" } ], "index": 20 }, { "bbox": [ 106, 467, 505, 480 ], "spans": [ { "bbox": [ 106, 467, 505, 480 ], "score": 1.0, "content": "this work, we introduce a much more powerful “flavor” of NPPs that consist jointly of neural and PC", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 478, 506, 492 ], "spans": [ { "bbox": [ 105, 478, 506, 492 ], "score": 1.0, "content": "modules, taking advantage of the power of neural computations together with true density estimation", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 487, 507, 504 ], "spans": [ { "bbox": [ 105, 487, 507, 504 ], "score": 1.0, "content": "of PCs. Depending on the underlying task one can thus ask a range of queries to the NPP, e.g.", "type": "text" } ], "index": 23 }, { "bbox": [ 105, 501, 505, 513 ], "spans": [ { "bbox": [ 105, 501, 505, 513 ], "score": 1.0, "content": "sample an unknown, desired variable, but also query for conditional class probabilities. Example", "type": "text" } ], "index": 24 }, { "bbox": [ 105, 511, 506, 524 ], "spans": [ { "bbox": [ 105, 511, 506, 524 ], "score": 1.0, "content": "NPPs consisting of a slot attention encoder and several PCs are depicted in Fig. 1 for the task of", "type": "text" } ], "index": 25 }, { "bbox": [ 105, 522, 506, 535 ], "spans": [ { "bbox": [ 105, 522, 506, 535 ], "score": 1.0, "content": "set prediction. The slot encoder is shared across all NPPs, whereas the PC of each NPP models a", "type": "text" } ], "index": 26 }, { "bbox": [ 105, 533, 505, 547 ], "spans": [ { "bbox": [ 105, 533, 505, 547 ], "score": 1.0, "content": "separate category of attributes. In this way, each NPP models the joint distribution over slot encodings", "type": "text" } ], "index": 27 }, { "bbox": [ 106, 545, 505, 558 ], "spans": [ { "bbox": [ 106, 545, 505, 558 ], "score": 1.0, "content": "and object attribute values, such as the color of an object. By querying the NPP, one can obtain", "type": "text" } ], "index": 28 }, { "bbox": [ 106, 555, 429, 568 ], "spans": [ { "bbox": [ 106, 555, 429, 568 ], "score": 1.0, "content": "task-related probability estimations, such as the conditional attribute probability.", "type": "text" } ], "index": 29 } ], "index": 21, "bbox_fs": [ 104, 380, 507, 568 ] }, { "type": "text", "bbox": [ 107, 572, 505, 672 ], "lines": [ { "bbox": [ 105, 572, 505, 585 ], "spans": [ { "bbox": [ 105, 572, 505, 585 ], "score": 1.0, "content": "The second component of SLASH is the logical program, which consists of a set of facts and logical", "type": "text" } ], "index": 30 }, { "bbox": [ 105, 583, 506, 595 ], "spans": [ { "bbox": [ 105, 583, 506, 595 ], "score": 1.0, "content": "statements defining the state of the world of the underlying task. For example, one can define the rules", "type": "text" } ], "index": 31 }, { "bbox": [ 105, 594, 506, 607 ], "spans": [ { "bbox": [ 105, 594, 506, 607 ], "score": 1.0, "content": "for when an object possesses a specific set of attributes (cf. Fig. 1). Thirdly, an ASP module is used to", "type": "text" } ], "index": 32 }, { "bbox": [ 105, 605, 505, 618 ], "spans": [ { "bbox": [ 105, 605, 505, 618 ], "score": 1.0, "content": "combine the first two components. Given a logical query about the input data, the logical program and", "type": "text" } ], "index": 33 }, { "bbox": [ 105, 615, 505, 629 ], "spans": [ { "bbox": [ 105, 615, 505, 629 ], "score": 1.0, "content": "the probability estimates obtained from the NPP(s), the ASP module produces a probability estimate", "type": "text" } ], "index": 34 }, { "bbox": [ 105, 625, 505, 641 ], "spans": [ { "bbox": [ 105, 625, 505, 641 ], "score": 1.0, "content": "about the truth value of the query, stating, e.g., how likely it is for a specific object in an image to be", "type": "text" } ], "index": 35 }, { "bbox": [ 105, 638, 506, 651 ], "spans": [ { "bbox": [ 105, 638, 506, 651 ], "score": 1.0, "content": "a large, dark red triangle. In contrast to query evaluation in Prolog (Colmerauer & Roussel, 1993;", "type": "text" } ], "index": 36 }, { "bbox": [ 105, 648, 506, 662 ], "spans": [ { "bbox": [ 105, 648, 506, 662 ], "score": 1.0, "content": "Clocksin & Mellish, 1981) which may lead to an infinite loop, many modern answer set solvers use", "type": "text" } ], "index": 37 }, { "bbox": [ 105, 659, 433, 673 ], "spans": [ { "bbox": [ 105, 659, 433, 673 ], "score": 1.0, "content": "Conflict-Driven-Clause-Learning (CDPL) which, in principle, always terminates.", "type": "text" } ], "index": 38 } ], "index": 34, "bbox_fs": [ 105, 572, 506, 673 ] }, { "type": "text", "bbox": [ 107, 677, 505, 732 ], "lines": [ { "bbox": [ 105, 676, 505, 690 ], "spans": [ { "bbox": [ 105, 676, 505, 690 ], "score": 1.0, "content": "Training in SLASH is performed efficiently in a batch-wise and end-to-end fashion, by integrating", "type": "text" } ], "index": 39 }, { "bbox": [ 105, 687, 506, 700 ], "spans": [ { "bbox": [ 105, 687, 506, 700 ], "score": 1.0, "content": "the parameters of all modules, neural and probabilistic, into a single loss term. SLASH thus allows a", "type": "text" } ], "index": 40 }, { "bbox": [ 105, 698, 506, 712 ], "spans": [ { "bbox": [ 105, 698, 506, 712 ], "score": 1.0, "content": "simple, fast and effective integration of sub-symbolic and symbolic computations. In our experiments,", "type": "text" } ], "index": 41 }, { "bbox": [ 105, 709, 506, 722 ], "spans": [ { "bbox": [ 105, 709, 506, 722 ], "score": 1.0, "content": "we investigate the advantages of SLASH in comparison to SOTA DPPLs on the benchmark task of", "type": "text" } ], "index": 42 }, { "bbox": [ 105, 720, 505, 734 ], "spans": [ { "bbox": [ 105, 720, 505, 734 ], "score": 1.0, "content": "MNIST-Addition (Manhaeve et al., 2018). We hereby show SLASH’s increased scalability regarding", "type": "text" } ], "index": 43 }, { "bbox": [ 105, 82, 505, 95 ], "spans": [ { "bbox": [ 105, 82, 505, 95 ], "score": 1.0, "content": "computation time, as well as SLASH’s ability to handle incomplete data via true probabilistic density", "type": "text", "cross_page": true } ], "index": 0 }, { "bbox": [ 106, 94, 505, 106 ], "spans": [ { "bbox": [ 106, 94, 505, 106 ], "score": 1.0, "content": "modelling. Next, we show that SLASH Attention provides superior results for set prediction in terms", "type": "text", "cross_page": true } ], "index": 1 }, { "bbox": [ 105, 105, 505, 117 ], "spans": [ { "bbox": [ 105, 105, 505, 117 ], "score": 1.0, "content": "of accuracy and generalization abilities compared to a baseline slot attention encoder. With our", "type": "text", "cross_page": true } ], "index": 2 }, { "bbox": [ 105, 114, 506, 129 ], "spans": [ { "bbox": [ 105, 114, 506, 129 ], "score": 1.0, "content": "experiments, we thus show that SLASH is a realization of “one system – two approaches” (Bengio,", "type": "text", "cross_page": true } ], "index": 3 }, { "bbox": [ 105, 126, 489, 140 ], "spans": [ { "bbox": [ 105, 126, 489, 140 ], "score": 1.0, "content": "2019), that can successfully be used for performing various tasks and on a variety of data types.", "type": "text", "cross_page": true } ], "index": 4 } ], "index": 41, "bbox_fs": [ 105, 676, 506, 734 ] } ] }, { "preproc_blocks": [ { "type": "text", "bbox": [ 107, 82, 505, 138 ], "lines": [ { "bbox": [ 105, 82, 505, 95 ], "spans": [ { "bbox": [ 105, 82, 505, 95 ], "score": 1.0, "content": "computation time, as well as SLASH’s ability to handle incomplete data via true probabilistic density", "type": "text" } ], "index": 0 }, { "bbox": [ 106, 94, 505, 106 ], "spans": [ { "bbox": [ 106, 94, 505, 106 ], "score": 1.0, "content": "modelling. Next, we show that SLASH Attention provides superior results for set prediction in terms", "type": "text" } ], "index": 1 }, { "bbox": [ 105, 105, 505, 117 ], "spans": [ { "bbox": [ 105, 105, 505, 117 ], "score": 1.0, "content": "of accuracy and generalization abilities compared to a baseline slot attention encoder. With our", "type": "text" } ], "index": 2 }, { "bbox": [ 105, 114, 506, 129 ], "spans": [ { "bbox": [ 105, 114, 506, 129 ], "score": 1.0, "content": "experiments, we thus show that SLASH is a realization of “one system – two approaches” (Bengio,", "type": "text" } ], "index": 3 }, { "bbox": [ 105, 126, 489, 140 ], "spans": [ { "bbox": [ 105, 126, 489, 140 ], "score": 1.0, "content": "2019), that can successfully be used for performing various tasks and on a variety of data types.", "type": "text" } ], "index": 4 } ], "index": 2 }, { "type": "text", "bbox": [ 107, 143, 505, 198 ], "lines": [ { "bbox": [ 106, 144, 505, 155 ], "spans": [ { "bbox": [ 106, 144, 505, 155 ], "score": 1.0, "content": "We make the following contributions: (1) We introduce neural-probabilistic predicates, efficiently", "type": "text" } ], "index": 5 }, { "bbox": [ 106, 155, 505, 167 ], "spans": [ { "bbox": [ 106, 155, 505, 167 ], "score": 1.0, "content": "integrating answer set programming with probabilistic inference via our novel DPPL, SLASH. (2) We", "type": "text" } ], "index": 6 }, { "bbox": [ 105, 165, 506, 178 ], "spans": [ { "bbox": [ 105, 165, 506, 178 ], "score": 1.0, "content": "successfully train neural, probabilistic and logic modules within SLASH for complex data structures", "type": "text" } ], "index": 7 }, { "bbox": [ 105, 175, 505, 190 ], "spans": [ { "bbox": [ 105, 175, 505, 190 ], "score": 1.0, "content": "end-to-end via a simple, single loss term. (3) We show that SLASH provides various advantages", "type": "text" } ], "index": 8 }, { "bbox": [ 105, 187, 480, 200 ], "spans": [ { "bbox": [ 105, 187, 480, 200 ], "score": 1.0, "content": "across a variety of tasks and data sets compared to state-of-the-art DPPLs and neural models.", "type": "text" } ], "index": 9 } ], "index": 7 }, { "type": "title", "bbox": [ 107, 214, 344, 227 ], "lines": [ { "bbox": [ 104, 213, 345, 229 ], "spans": [ { "bbox": [ 104, 213, 345, 229 ], "score": 1.0, "content": "2 NEURO-SYMBOLIC LOGIC PROGRAMMING", "type": "text" } ], "index": 10 } ], "index": 10 }, { "type": "text", "bbox": [ 107, 239, 506, 273 ], "lines": [ { "bbox": [ 105, 237, 506, 252 ], "spans": [ { "bbox": [ 105, 237, 506, 252 ], "score": 1.0, "content": "Neuro-Symbolic AI can be divided into two lines of research, depending on the starting point. Both,", "type": "text" } ], "index": 11 }, { "bbox": [ 105, 250, 505, 263 ], "spans": [ { "bbox": [ 105, 250, 505, 263 ], "score": 1.0, "content": "however, have the same final goal: to combine low-level perception with logical constraints and", "type": "text" } ], "index": 12 }, { "bbox": [ 105, 260, 150, 275 ], "spans": [ { "bbox": [ 105, 260, 150, 275 ], "score": 1.0, "content": "reasoning.", "type": "text" } ], "index": 13 } ], "index": 12 }, { "type": "text", "bbox": [ 107, 277, 505, 399 ], "lines": [ { "bbox": [ 106, 278, 506, 290 ], "spans": [ { "bbox": [ 106, 278, 506, 290 ], "score": 1.0, "content": "A key motivation of Neuro-Symbolic AI (d’Avila Garcez et al., 2009; Mao et al., 2019; Hudson &", "type": "text" } ], "index": 14 }, { "bbox": [ 106, 289, 505, 302 ], "spans": [ { "bbox": [ 106, 289, 505, 302 ], "score": 1.0, "content": "Manning, 2019; d’Avila Garcez et al., 2019; Jiang & Ahn, 2020; d’Avila Garcez & Lamb, 2020) is", "type": "text" } ], "index": 15 }, { "bbox": [ 106, 300, 505, 313 ], "spans": [ { "bbox": [ 106, 300, 505, 313 ], "score": 1.0, "content": "to combine the advantages of symbolic and neural representations into a joint system. This is often", "type": "text" } ], "index": 16 }, { "bbox": [ 106, 311, 506, 324 ], "spans": [ { "bbox": [ 106, 311, 506, 324 ], "score": 1.0, "content": "done in a hybrid approach where a neural network acts as a perception module that interfaces with a", "type": "text" } ], "index": 17 }, { "bbox": [ 105, 322, 506, 334 ], "spans": [ { "bbox": [ 105, 322, 506, 334 ], "score": 1.0, "content": "symbolic reasoning system, e.g. (Mao et al., 2019; Yi et al., 2018). The goal of such an approach is", "type": "text" } ], "index": 18 }, { "bbox": [ 105, 332, 506, 346 ], "spans": [ { "bbox": [ 105, 332, 506, 346 ], "score": 1.0, "content": "to mitigate the issues of one type of representation by the other, e.g. using the power of symbolic", "type": "text" } ], "index": 19 }, { "bbox": [ 105, 344, 506, 356 ], "spans": [ { "bbox": [ 105, 344, 506, 356 ], "score": 1.0, "content": "reasoning systems to handle the generalizability issues of neural networks and on the other hand", "type": "text" } ], "index": 20 }, { "bbox": [ 105, 354, 506, 368 ], "spans": [ { "bbox": [ 105, 354, 506, 368 ], "score": 1.0, "content": "handle the difficulty of noisy data for symbolic systems via neural networks. Recent work has also", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 365, 505, 379 ], "spans": [ { "bbox": [ 105, 365, 505, 379 ], "score": 1.0, "content": "shown the advantage of Neuro-Symbolic approaches for explaining and revising incorrect decisions", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 376, 506, 390 ], "spans": [ { "bbox": [ 105, 376, 506, 390 ], "score": 1.0, "content": "(Ciravegna et al., 2020; Stammer et al., 2021). Many of these previous works, however, train the", "type": "text" } ], "index": 23 }, { "bbox": [ 106, 388, 299, 400 ], "spans": [ { "bbox": [ 106, 388, 299, 400 ], "score": 1.0, "content": "sub-symbolic and symbolic modules separately.", "type": "text" } ], "index": 24 } ], "index": 19 }, { "type": "text", "bbox": [ 107, 405, 505, 493 ], "lines": [ { "bbox": [ 105, 403, 506, 419 ], "spans": [ { "bbox": [ 105, 403, 506, 419 ], "score": 1.0, "content": "Deep Probabilistic Programming Languages (DPPLs) are programming languages that combine deep", "type": "text" } ], "index": 25 }, { "bbox": [ 105, 416, 506, 428 ], "spans": [ { "bbox": [ 105, 416, 506, 428 ], "score": 1.0, "content": "neural networks with probabilistic models and allow a user to express a probabilistic model via a", "type": "text" } ], "index": 26 }, { "bbox": [ 105, 426, 506, 439 ], "spans": [ { "bbox": [ 105, 426, 506, 439 ], "score": 1.0, "content": "logical program. Similar to Neuro-Symbolic architectures, DPPLs thereby unite the advantages of", "type": "text" } ], "index": 27 }, { "bbox": [ 105, 437, 506, 450 ], "spans": [ { "bbox": [ 105, 437, 506, 450 ], "score": 1.0, "content": "different paradigms. DPPLs are related to earlier works such as Markov Logic Networks (MLNs)", "type": "text" } ], "index": 28 }, { "bbox": [ 105, 448, 506, 461 ], "spans": [ { "bbox": [ 105, 448, 506, 461 ], "score": 1.0, "content": "(Richardson & Domingos, 2006). Thereby, the binding link is the Weighted Model Counting (WMC)", "type": "text" } ], "index": 29 }, { "bbox": [ 104, 456, 507, 474 ], "spans": [ { "bbox": [ 104, 456, 160, 474 ], "score": 1.0, "content": "introduced in", "type": "text" }, { "bbox": [ 160, 459, 189, 470 ], "score": 0.41, "content": "\\mathrm { L P ^ { M L N } }", "type": "inline_equation" }, { "bbox": [ 190, 456, 507, 474 ], "score": 1.0, "content": "(Lee & Wang, 2016). Several DPPLs have been proposed by now, among which", "type": "text" } ], "index": 30 }, { "bbox": [ 105, 470, 507, 484 ], "spans": [ { "bbox": [ 105, 470, 507, 484 ], "score": 1.0, "content": "are Pyro (Bingham et al., 2019), Edward (Tran et al., 2017), DeepProbLog (Manhaeve et al., 2018),", "type": "text" } ], "index": 31 }, { "bbox": [ 105, 482, 243, 494 ], "spans": [ { "bbox": [ 105, 482, 243, 494 ], "score": 1.0, "content": "and NeurASP (Yang et al., 2020).", "type": "text" } ], "index": 32 } ], "index": 28.5 }, { "type": "text", "bbox": [ 107, 498, 506, 597 ], "lines": [ { "bbox": [ 106, 498, 506, 511 ], "spans": [ { "bbox": [ 106, 498, 506, 511 ], "score": 1.0, "content": "To resolve the scalability issues of DeepProbLog, which use Sentential Decision Diagrams (SDDs)", "type": "text" } ], "index": 33 }, { "bbox": [ 106, 509, 506, 521 ], "spans": [ { "bbox": [ 106, 509, 506, 521 ], "score": 1.0, "content": "(Darwiche, 2011) as the underlying data structure to evaluate queries, NeurASP (Yang et al., 2020),", "type": "text" } ], "index": 34 }, { "bbox": [ 105, 520, 506, 533 ], "spans": [ { "bbox": [ 105, 520, 506, 533 ], "score": 1.0, "content": "offers a solution by utilizing Answer Set Programming (ASP) (Dimopoulos et al., 1997; Soininen &", "type": "text" } ], "index": 35 }, { "bbox": [ 105, 529, 506, 545 ], "spans": [ { "bbox": [ 105, 529, 506, 545 ], "score": 1.0, "content": "Niemelä, 1999; Marek & Truszczynski, 1999; Calimeri et al., 2020). In this way, NeurASP changes", "type": "text" } ], "index": 36 }, { "bbox": [ 105, 541, 506, 556 ], "spans": [ { "bbox": [ 105, 541, 506, 556 ], "score": 1.0, "content": "the paradigm from query evaluation to model generation, i.e. instead of constructing an SDD or", "type": "text" } ], "index": 37 }, { "bbox": [ 106, 554, 505, 565 ], "spans": [ { "bbox": [ 106, 554, 505, 565 ], "score": 1.0, "content": "similar knowledge representation system, NeurASP generates a set of all possible solutions (one", "type": "text" } ], "index": 38 }, { "bbox": [ 106, 564, 505, 576 ], "spans": [ { "bbox": [ 106, 564, 505, 576 ], "score": 1.0, "content": "model per solution) and estimates the probability for the truth value of each of these solutions. Of", "type": "text" } ], "index": 39 }, { "bbox": [ 106, 575, 506, 588 ], "spans": [ { "bbox": [ 106, 575, 506, 588 ], "score": 1.0, "content": "those DPPLs that handle learning in a relational, probabilistic setting and in an end-to-end fashion,", "type": "text" } ], "index": 40 }, { "bbox": [ 105, 585, 396, 599 ], "spans": [ { "bbox": [ 105, 585, 396, 599 ], "score": 1.0, "content": "all of these are limited to estimating only conditional class probabilities.", "type": "text" } ], "index": 41 } ], "index": 37 }, { "type": "title", "bbox": [ 108, 613, 263, 626 ], "lines": [ { "bbox": [ 105, 612, 265, 628 ], "spans": [ { "bbox": [ 105, 612, 265, 628 ], "score": 1.0, "content": "3 THE SLASH FRAMEWORK", "type": "text" } ], "index": 42 } ], "index": 42 }, { "type": "text", "bbox": [ 108, 637, 504, 671 ], "lines": [ { "bbox": [ 105, 638, 505, 650 ], "spans": [ { "bbox": [ 105, 638, 505, 650 ], "score": 1.0, "content": "In this section, we introduce our novel DPPL, SLASH. Before we dive into the details of this, it is", "type": "text" } ], "index": 43 }, { "bbox": [ 105, 649, 506, 661 ], "spans": [ { "bbox": [ 105, 649, 506, 661 ], "score": 1.0, "content": "necessary to first introduce Neural-Probabilistic Predicates, for which we require an understanding of", "type": "text" } ], "index": 44 }, { "bbox": [ 105, 660, 430, 672 ], "spans": [ { "bbox": [ 105, 660, 430, 672 ], "score": 1.0, "content": "Probabilistic Circuits. Finally, we will present the learning paradigm of SLASH.", "type": "text" } ], "index": 45 } ], "index": 44 }, { "type": "text", "bbox": [ 107, 676, 504, 731 ], "lines": [ { "bbox": [ 106, 677, 506, 689 ], "spans": [ { "bbox": [ 106, 677, 506, 689 ], "score": 1.0, "content": "The term probabilistic circuit (PC) (Choi et al., 2020) represents a unifying framework that en-", "type": "text" } ], "index": 46 }, { "bbox": [ 105, 687, 506, 700 ], "spans": [ { "bbox": [ 105, 687, 506, 700 ], "score": 1.0, "content": "compasses all computational graphs which encode probability distributions and guarantee tractable", "type": "text" } ], "index": 47 }, { "bbox": [ 105, 698, 506, 712 ], "spans": [ { "bbox": [ 105, 698, 506, 712 ], "score": 1.0, "content": "probabilistic modelling. These include Sum-Product Networks (SPNs) (Poon & Domingos, 2011)", "type": "text" } ], "index": 48 }, { "bbox": [ 105, 709, 505, 723 ], "spans": [ { "bbox": [ 105, 709, 505, 723 ], "score": 1.0, "content": "which are deep mixture models represented via a rooted directed acyclic graphs with a recursively", "type": "text" } ], "index": 49 }, { "bbox": [ 105, 720, 178, 733 ], "spans": [ { "bbox": [ 105, 720, 178, 733 ], "score": 1.0, "content": "defined structure.", "type": "text" } ], "index": 50 } ], "index": 48 } ], "page_idx": 2, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 107, 27, 308, 37 ], "lines": [ { "bbox": [ 107, 26, 308, 38 ], "spans": [ { "bbox": [ 107, 26, 308, 38 ], "score": 1.0, "content": "Under review as a conference paper at ICLR 2022", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 303, 751, 309, 759 ], "lines": [ { "bbox": [ 302, 750, 309, 762 ], "spans": [ { "bbox": [ 302, 750, 309, 762 ], "score": 1.0, "content": "3", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "text", "bbox": [ 107, 82, 505, 138 ], "lines": [], "index": 2, "bbox_fs": [ 105, 82, 506, 140 ], "lines_deleted": true }, { "type": "text", "bbox": [ 107, 143, 505, 198 ], "lines": [ { "bbox": [ 106, 144, 505, 155 ], "spans": [ { "bbox": [ 106, 144, 505, 155 ], "score": 1.0, "content": "We make the following contributions: (1) We introduce neural-probabilistic predicates, efficiently", "type": "text" } ], "index": 5 }, { "bbox": [ 106, 155, 505, 167 ], "spans": [ { "bbox": [ 106, 155, 505, 167 ], "score": 1.0, "content": "integrating answer set programming with probabilistic inference via our novel DPPL, SLASH. (2) We", "type": "text" } ], "index": 6 }, { "bbox": [ 105, 165, 506, 178 ], "spans": [ { "bbox": [ 105, 165, 506, 178 ], "score": 1.0, "content": "successfully train neural, probabilistic and logic modules within SLASH for complex data structures", "type": "text" } ], "index": 7 }, { "bbox": [ 105, 175, 505, 190 ], "spans": [ { "bbox": [ 105, 175, 505, 190 ], "score": 1.0, "content": "end-to-end via a simple, single loss term. (3) We show that SLASH provides various advantages", "type": "text" } ], "index": 8 }, { "bbox": [ 105, 187, 480, 200 ], "spans": [ { "bbox": [ 105, 187, 480, 200 ], "score": 1.0, "content": "across a variety of tasks and data sets compared to state-of-the-art DPPLs and neural models.", "type": "text" } ], "index": 9 } ], "index": 7, "bbox_fs": [ 105, 144, 506, 200 ] }, { "type": "title", "bbox": [ 107, 214, 344, 227 ], "lines": [ { "bbox": [ 104, 213, 345, 229 ], "spans": [ { "bbox": [ 104, 213, 345, 229 ], "score": 1.0, "content": "2 NEURO-SYMBOLIC LOGIC PROGRAMMING", "type": "text" } ], "index": 10 } ], "index": 10 }, { "type": "text", "bbox": [ 107, 239, 506, 273 ], "lines": [ { "bbox": [ 105, 237, 506, 252 ], "spans": [ { "bbox": [ 105, 237, 506, 252 ], "score": 1.0, "content": "Neuro-Symbolic AI can be divided into two lines of research, depending on the starting point. Both,", "type": "text" } ], "index": 11 }, { "bbox": [ 105, 250, 505, 263 ], "spans": [ { "bbox": [ 105, 250, 505, 263 ], "score": 1.0, "content": "however, have the same final goal: to combine low-level perception with logical constraints and", "type": "text" } ], "index": 12 }, { "bbox": [ 105, 260, 150, 275 ], "spans": [ { "bbox": [ 105, 260, 150, 275 ], "score": 1.0, "content": "reasoning.", "type": "text" } ], "index": 13 } ], "index": 12, "bbox_fs": [ 105, 237, 506, 275 ] }, { "type": "text", "bbox": [ 107, 277, 505, 399 ], "lines": [ { "bbox": [ 106, 278, 506, 290 ], "spans": [ { "bbox": [ 106, 278, 506, 290 ], "score": 1.0, "content": "A key motivation of Neuro-Symbolic AI (d’Avila Garcez et al., 2009; Mao et al., 2019; Hudson &", "type": "text" } ], "index": 14 }, { "bbox": [ 106, 289, 505, 302 ], "spans": [ { "bbox": [ 106, 289, 505, 302 ], "score": 1.0, "content": "Manning, 2019; d’Avila Garcez et al., 2019; Jiang & Ahn, 2020; d’Avila Garcez & Lamb, 2020) is", "type": "text" } ], "index": 15 }, { "bbox": [ 106, 300, 505, 313 ], "spans": [ { "bbox": [ 106, 300, 505, 313 ], "score": 1.0, "content": "to combine the advantages of symbolic and neural representations into a joint system. This is often", "type": "text" } ], "index": 16 }, { "bbox": [ 106, 311, 506, 324 ], "spans": [ { "bbox": [ 106, 311, 506, 324 ], "score": 1.0, "content": "done in a hybrid approach where a neural network acts as a perception module that interfaces with a", "type": "text" } ], "index": 17 }, { "bbox": [ 105, 322, 506, 334 ], "spans": [ { "bbox": [ 105, 322, 506, 334 ], "score": 1.0, "content": "symbolic reasoning system, e.g. (Mao et al., 2019; Yi et al., 2018). The goal of such an approach is", "type": "text" } ], "index": 18 }, { "bbox": [ 105, 332, 506, 346 ], "spans": [ { "bbox": [ 105, 332, 506, 346 ], "score": 1.0, "content": "to mitigate the issues of one type of representation by the other, e.g. using the power of symbolic", "type": "text" } ], "index": 19 }, { "bbox": [ 105, 344, 506, 356 ], "spans": [ { "bbox": [ 105, 344, 506, 356 ], "score": 1.0, "content": "reasoning systems to handle the generalizability issues of neural networks and on the other hand", "type": "text" } ], "index": 20 }, { "bbox": [ 105, 354, 506, 368 ], "spans": [ { "bbox": [ 105, 354, 506, 368 ], "score": 1.0, "content": "handle the difficulty of noisy data for symbolic systems via neural networks. Recent work has also", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 365, 505, 379 ], "spans": [ { "bbox": [ 105, 365, 505, 379 ], "score": 1.0, "content": "shown the advantage of Neuro-Symbolic approaches for explaining and revising incorrect decisions", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 376, 506, 390 ], "spans": [ { "bbox": [ 105, 376, 506, 390 ], "score": 1.0, "content": "(Ciravegna et al., 2020; Stammer et al., 2021). Many of these previous works, however, train the", "type": "text" } ], "index": 23 }, { "bbox": [ 106, 388, 299, 400 ], "spans": [ { "bbox": [ 106, 388, 299, 400 ], "score": 1.0, "content": "sub-symbolic and symbolic modules separately.", "type": "text" } ], "index": 24 } ], "index": 19, "bbox_fs": [ 105, 278, 506, 400 ] }, { "type": "text", "bbox": [ 107, 405, 505, 493 ], "lines": [ { "bbox": [ 105, 403, 506, 419 ], "spans": [ { "bbox": [ 105, 403, 506, 419 ], "score": 1.0, "content": "Deep Probabilistic Programming Languages (DPPLs) are programming languages that combine deep", "type": "text" } ], "index": 25 }, { "bbox": [ 105, 416, 506, 428 ], "spans": [ { "bbox": [ 105, 416, 506, 428 ], "score": 1.0, "content": "neural networks with probabilistic models and allow a user to express a probabilistic model via a", "type": "text" } ], "index": 26 }, { "bbox": [ 105, 426, 506, 439 ], "spans": [ { "bbox": [ 105, 426, 506, 439 ], "score": 1.0, "content": "logical program. Similar to Neuro-Symbolic architectures, DPPLs thereby unite the advantages of", "type": "text" } ], "index": 27 }, { "bbox": [ 105, 437, 506, 450 ], "spans": [ { "bbox": [ 105, 437, 506, 450 ], "score": 1.0, "content": "different paradigms. DPPLs are related to earlier works such as Markov Logic Networks (MLNs)", "type": "text" } ], "index": 28 }, { "bbox": [ 105, 448, 506, 461 ], "spans": [ { "bbox": [ 105, 448, 506, 461 ], "score": 1.0, "content": "(Richardson & Domingos, 2006). Thereby, the binding link is the Weighted Model Counting (WMC)", "type": "text" } ], "index": 29 }, { "bbox": [ 104, 456, 507, 474 ], "spans": [ { "bbox": [ 104, 456, 160, 474 ], "score": 1.0, "content": "introduced in", "type": "text" }, { "bbox": [ 160, 459, 189, 470 ], "score": 0.41, "content": "\\mathrm { L P ^ { M L N } }", "type": "inline_equation" }, { "bbox": [ 190, 456, 507, 474 ], "score": 1.0, "content": "(Lee & Wang, 2016). Several DPPLs have been proposed by now, among which", "type": "text" } ], "index": 30 }, { "bbox": [ 105, 470, 507, 484 ], "spans": [ { "bbox": [ 105, 470, 507, 484 ], "score": 1.0, "content": "are Pyro (Bingham et al., 2019), Edward (Tran et al., 2017), DeepProbLog (Manhaeve et al., 2018),", "type": "text" } ], "index": 31 }, { "bbox": [ 105, 482, 243, 494 ], "spans": [ { "bbox": [ 105, 482, 243, 494 ], "score": 1.0, "content": "and NeurASP (Yang et al., 2020).", "type": "text" } ], "index": 32 } ], "index": 28.5, "bbox_fs": [ 104, 403, 507, 494 ] }, { "type": "text", "bbox": [ 107, 498, 506, 597 ], "lines": [ { "bbox": [ 106, 498, 506, 511 ], "spans": [ { "bbox": [ 106, 498, 506, 511 ], "score": 1.0, "content": "To resolve the scalability issues of DeepProbLog, which use Sentential Decision Diagrams (SDDs)", "type": "text" } ], "index": 33 }, { "bbox": [ 106, 509, 506, 521 ], "spans": [ { "bbox": [ 106, 509, 506, 521 ], "score": 1.0, "content": "(Darwiche, 2011) as the underlying data structure to evaluate queries, NeurASP (Yang et al., 2020),", "type": "text" } ], "index": 34 }, { "bbox": [ 105, 520, 506, 533 ], "spans": [ { "bbox": [ 105, 520, 506, 533 ], "score": 1.0, "content": "offers a solution by utilizing Answer Set Programming (ASP) (Dimopoulos et al., 1997; Soininen &", "type": "text" } ], "index": 35 }, { "bbox": [ 105, 529, 506, 545 ], "spans": [ { "bbox": [ 105, 529, 506, 545 ], "score": 1.0, "content": "Niemelä, 1999; Marek & Truszczynski, 1999; Calimeri et al., 2020). In this way, NeurASP changes", "type": "text" } ], "index": 36 }, { "bbox": [ 105, 541, 506, 556 ], "spans": [ { "bbox": [ 105, 541, 506, 556 ], "score": 1.0, "content": "the paradigm from query evaluation to model generation, i.e. instead of constructing an SDD or", "type": "text" } ], "index": 37 }, { "bbox": [ 106, 554, 505, 565 ], "spans": [ { "bbox": [ 106, 554, 505, 565 ], "score": 1.0, "content": "similar knowledge representation system, NeurASP generates a set of all possible solutions (one", "type": "text" } ], "index": 38 }, { "bbox": [ 106, 564, 505, 576 ], "spans": [ { "bbox": [ 106, 564, 505, 576 ], "score": 1.0, "content": "model per solution) and estimates the probability for the truth value of each of these solutions. Of", "type": "text" } ], "index": 39 }, { "bbox": [ 106, 575, 506, 588 ], "spans": [ { "bbox": [ 106, 575, 506, 588 ], "score": 1.0, "content": "those DPPLs that handle learning in a relational, probabilistic setting and in an end-to-end fashion,", "type": "text" } ], "index": 40 }, { "bbox": [ 105, 585, 396, 599 ], "spans": [ { "bbox": [ 105, 585, 396, 599 ], "score": 1.0, "content": "all of these are limited to estimating only conditional class probabilities.", "type": "text" } ], "index": 41 } ], "index": 37, "bbox_fs": [ 105, 498, 506, 599 ] }, { "type": "title", "bbox": [ 108, 613, 263, 626 ], "lines": [ { "bbox": [ 105, 612, 265, 628 ], "spans": [ { "bbox": [ 105, 612, 265, 628 ], "score": 1.0, "content": "3 THE SLASH FRAMEWORK", "type": "text" } ], "index": 42 } ], "index": 42 }, { "type": "text", "bbox": [ 108, 637, 504, 671 ], "lines": [ { "bbox": [ 105, 638, 505, 650 ], "spans": [ { "bbox": [ 105, 638, 505, 650 ], "score": 1.0, "content": "In this section, we introduce our novel DPPL, SLASH. Before we dive into the details of this, it is", "type": "text" } ], "index": 43 }, { "bbox": [ 105, 649, 506, 661 ], "spans": [ { "bbox": [ 105, 649, 506, 661 ], "score": 1.0, "content": "necessary to first introduce Neural-Probabilistic Predicates, for which we require an understanding of", "type": "text" } ], "index": 44 }, { "bbox": [ 105, 660, 430, 672 ], "spans": [ { "bbox": [ 105, 660, 430, 672 ], "score": 1.0, "content": "Probabilistic Circuits. Finally, we will present the learning paradigm of SLASH.", "type": "text" } ], "index": 45 } ], "index": 44, "bbox_fs": [ 105, 638, 506, 672 ] }, { "type": "text", "bbox": [ 107, 676, 504, 731 ], "lines": [ { "bbox": [ 106, 677, 506, 689 ], "spans": [ { "bbox": [ 106, 677, 506, 689 ], "score": 1.0, "content": "The term probabilistic circuit (PC) (Choi et al., 2020) represents a unifying framework that en-", "type": "text" } ], "index": 46 }, { "bbox": [ 105, 687, 506, 700 ], "spans": [ { "bbox": [ 105, 687, 506, 700 ], "score": 1.0, "content": "compasses all computational graphs which encode probability distributions and guarantee tractable", "type": "text" } ], "index": 47 }, { "bbox": [ 105, 698, 506, 712 ], "spans": [ { "bbox": [ 105, 698, 506, 712 ], "score": 1.0, "content": "probabilistic modelling. These include Sum-Product Networks (SPNs) (Poon & Domingos, 2011)", "type": "text" } ], "index": 48 }, { "bbox": [ 105, 709, 505, 723 ], "spans": [ { "bbox": [ 105, 709, 505, 723 ], "score": 1.0, "content": "which are deep mixture models represented via a rooted directed acyclic graphs with a recursively", "type": "text" } ], "index": 49 }, { "bbox": [ 105, 720, 178, 733 ], "spans": [ { "bbox": [ 105, 720, 178, 733 ], "score": 1.0, "content": "defined structure.", "type": "text" } ], "index": 50 } ], "index": 48, "bbox_fs": [ 105, 677, 506, 733 ] } ] }, { "preproc_blocks": [ { "type": "image", "bbox": [ 138, 82, 299, 190 ], "blocks": [ { "type": "image_body", "bbox": [ 138, 82, 299, 190 ], "group_id": 1, "lines": [ { "bbox": [ 138, 82, 299, 190 ], "spans": [ { "bbox": [ 138, 82, 299, 190 ], "score": 0.968, "type": "image", "image_path": "fe12c632ab68913a19fb5c0639cac79765cabf16e331224edc0cd2c84bb8eebe.jpg" } ] } ], "index": 3.5, "virtual_lines": [ { "bbox": [ 138, 82, 299, 95.5 ], "spans": [], "index": 0 }, { "bbox": [ 138, 95.5, 299, 109.0 ], "spans": [], "index": 1 }, { "bbox": [ 138, 109.0, 299, 122.5 ], "spans": [], "index": 2 }, { "bbox": [ 138, 122.5, 299, 136.0 ], "spans": [], "index": 3 }, { "bbox": [ 138, 136.0, 299, 149.5 ], "spans": [], "index": 4 }, { "bbox": [ 138, 149.5, 299, 163.0 ], "spans": [], "index": 5 }, { "bbox": [ 138, 163.0, 299, 176.5 ], "spans": [], "index": 6 }, { "bbox": [ 138, 176.5, 299, 190.0 ], "spans": [], "index": 7 } ] } ], "index": 3.5 }, { "type": "text", "bbox": [ 128, 195, 309, 215 ], "lines": [ { "bbox": [ 128, 194, 309, 207 ], "spans": [ { "bbox": [ 128, 194, 309, 207 ], "score": 1.0, "content": "(a) NPPs come in various flavors depending on the", "type": "text" } ], "index": 9 }, { "bbox": [ 128, 205, 221, 215 ], "spans": [ { "bbox": [ 128, 205, 221, 215 ], "score": 1.0, "content": "data and underlying task.", "type": "text" } ], "index": 11 } ], "index": 10.0 }, { "type": "image", "bbox": [ 329, 80, 474, 189 ], "blocks": [ { "type": "image_body", "bbox": [ 329, 80, 474, 189 ], "group_id": 0, "lines": [ { "bbox": [ 329, 80, 474, 189 ], "spans": [ { "bbox": [ 329, 80, 474, 189 ], "score": 0.954, "type": "image", "image_path": "deaa2f06b12c44d64f5ac8f226a3fdb61c0d8bba50b15128cddc5ccafbad3205.jpg" } ] } ], "index": 9.0, "virtual_lines": [ { "bbox": [ 329, 80, 474, 134.5 ], "spans": [], "index": 8 }, { "bbox": [ 329, 134.5, 474, 189.0 ], "spans": [], "index": 10 } ] }, { "type": "image_caption", "bbox": [ 321, 195, 482, 214 ], "group_id": 0, "lines": [ { "bbox": [ 321, 194, 483, 206 ], "spans": [ { "bbox": [ 321, 194, 483, 206 ], "score": 1.0, "content": "(b) Minimal SLASH program and query for", "type": "text" } ], "index": 12 }, { "bbox": [ 321, 205, 375, 216 ], "spans": [ { "bbox": [ 321, 205, 375, 216 ], "score": 1.0, "content": "set prediction.", "type": "text" } ], "index": 13 } ], "index": 12.5 }, { "type": "image_caption", "bbox": [ 106, 225, 506, 280 ], "group_id": 0, "lines": [ { "bbox": [ 105, 224, 507, 237 ], "spans": [ { "bbox": [ 105, 224, 507, 237 ], "score": 1.0, "content": "Figure 2: (a) Depending on the data set and underlying task, SLASH requires a suitable Neural-", "type": "text" } ], "index": 14 }, { "bbox": [ 105, 235, 505, 248 ], "spans": [ { "bbox": [ 105, 235, 505, 248 ], "score": 1.0, "content": "Probabilistic Predicate (NPP) that computes query-dependent probability estimates. An NPP can be", "type": "text" } ], "index": 15 }, { "bbox": [ 105, 246, 507, 260 ], "spans": [ { "bbox": [ 105, 246, 507, 260 ], "score": 1.0, "content": "composed of neural and probabilistic modules, or (depicted via slash symbol) only one of these two.", "type": "text" } ], "index": 16 }, { "bbox": [ 105, 257, 506, 270 ], "spans": [ { "bbox": [ 105, 257, 506, 270 ], "score": 1.0, "content": "(b) A minimal SLASH program and query for the set prediction task, here only showing the NPP that", "type": "text" } ], "index": 17 }, { "bbox": [ 105, 268, 447, 281 ], "spans": [ { "bbox": [ 105, 268, 447, 281 ], "score": 1.0, "content": "models the color category per object. For the full program, we refer to the Appendix.", "type": "text" } ], "index": 18 } ], "index": 16 } ], "index": 12.5 }, { "type": "title", "bbox": [ 108, 302, 294, 313 ], "lines": [ { "bbox": [ 106, 302, 295, 315 ], "spans": [ { "bbox": [ 106, 302, 295, 315 ], "score": 1.0, "content": "3.1 NEURAL-PROBABILISTIC PREDICATES", "type": "text" } ], "index": 19 } ], "index": 19 }, { "type": "text", "bbox": [ 106, 322, 505, 378 ], "lines": [ { "bbox": [ 105, 322, 505, 336 ], "spans": [ { "bbox": [ 105, 322, 505, 336 ], "score": 1.0, "content": "Previous DPPLs, DeepProbLog (Manhaeve et al., 2018) and NeurASP (Yang et al., 2020), introduced", "type": "text" } ], "index": 20 }, { "bbox": [ 105, 333, 505, 347 ], "spans": [ { "bbox": [ 105, 333, 505, 347 ], "score": 1.0, "content": "the Neural Predicate as an annotated-disjunction or as a propositional atom, respectively, to acquire", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 343, 506, 359 ], "spans": [ { "bbox": [ 105, 343, 237, 359 ], "score": 1.0, "content": "conditional class probabilities,", "type": "text" }, { "bbox": [ 237, 345, 273, 357 ], "score": 0.93, "content": "P ( C | X )", "type": "inline_equation" }, { "bbox": [ 273, 343, 506, 359 ], "score": 1.0, "content": ", via the softmax function at the output of an arbitrary", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 355, 505, 369 ], "spans": [ { "bbox": [ 105, 355, 505, 369 ], "score": 1.0, "content": "DNN. As mentioned in the introduction, this approach has certain limitations concerning inference", "type": "text" } ], "index": 23 }, { "bbox": [ 106, 366, 463, 380 ], "spans": [ { "bbox": [ 106, 366, 463, 380 ], "score": 1.0, "content": "capabilities. To resolve this issue, we introduce Neural-Probabilisitic Predicates (NPPs).", "type": "text" } ], "index": 24 } ], "index": 22 }, { "type": "text", "bbox": [ 107, 384, 210, 395 ], "lines": [ { "bbox": [ 106, 383, 210, 396 ], "spans": [ { "bbox": [ 106, 383, 210, 396 ], "score": 1.0, "content": "Formally, we denote with", "type": "text" } ], "index": 25 } ], "index": 25 }, { "type": "interline_equation", "bbox": [ 256, 395, 354, 408 ], "lines": [ { "bbox": [ 256, 395, 354, 408 ], "spans": [ { "bbox": [ 256, 395, 354, 408 ], "score": 0.91, "content": "n p p \\left( h ( x ) , [ v _ { 1 } , . . . , v _ { n } ] \\right)", "type": "interline_equation", "image_path": "187dcc297a855ad44e5c575f0d91e7583f0c038c9c3016485793aabe684f440a.jpg" } ] } ], "index": 26, "virtual_lines": [ { "bbox": [ 256, 395, 354, 408 ], "spans": [], "index": 26 } ] }, { "type": "text", "bbox": [ 107, 411, 505, 466 ], "lines": [ { "bbox": [ 105, 411, 506, 425 ], "spans": [ { "bbox": [ 105, 411, 240, 425 ], "score": 1.0, "content": "a Neural-Probabilistic Predicate", "type": "text" }, { "bbox": [ 241, 412, 248, 422 ], "score": 0.8, "content": "h", "type": "inline_equation" }, { "bbox": [ 248, 411, 489, 425 ], "score": 1.0, "content": ". Thereby, (i) npp is a reserved word to label an NPP, (ii)", "type": "text" }, { "bbox": [ 489, 412, 497, 422 ], "score": 0.74, "content": "h", "type": "inline_equation" }, { "bbox": [ 497, 411, 506, 425 ], "score": 1.0, "content": "a", "type": "text" } ], "index": 27 }, { "bbox": [ 105, 422, 505, 435 ], "spans": [ { "bbox": [ 105, 422, 505, 435 ], "score": 1.0, "content": "symbolic name of either a PC, NN or a joint of a PC and NN (cf. Fig. 2a), e.g., color_attr is the name", "type": "text" } ], "index": 28 }, { "bbox": [ 105, 433, 506, 447 ], "spans": [ { "bbox": [ 105, 433, 267, 447 ], "score": 1.0, "content": "of an NPP of Fig. 2b. Additionally, (iii)", "type": "text" }, { "bbox": [ 267, 435, 275, 444 ], "score": 0.79, "content": "x", "type": "inline_equation" }, { "bbox": [ 275, 433, 379, 447 ], "score": 1.0, "content": "denotes a “term” and (iv)", "type": "text" }, { "bbox": [ 379, 435, 422, 445 ], "score": 0.9, "content": "v _ { 1 } , \\ldots , v _ { n }", "type": "inline_equation" }, { "bbox": [ 422, 433, 506, 447 ], "score": 1.0, "content": "are placeholders for", "type": "text" } ], "index": 29 }, { "bbox": [ 105, 444, 505, 457 ], "spans": [ { "bbox": [ 105, 444, 154, 457 ], "score": 1.0, "content": "each of the", "type": "text" }, { "bbox": [ 155, 446, 162, 455 ], "score": 0.71, "content": "n", "type": "inline_equation" }, { "bbox": [ 163, 444, 253, 457 ], "score": 1.0, "content": "possible outcomes of", "type": "text" }, { "bbox": [ 253, 445, 260, 455 ], "score": 0.82, "content": "h", "type": "inline_equation" }, { "bbox": [ 260, 444, 505, 457 ], "score": 1.0, "content": ". For example, the placeholders for color_attr are the color", "type": "text" } ], "index": 30 }, { "bbox": [ 106, 455, 294, 468 ], "spans": [ { "bbox": [ 106, 455, 294, 468 ], "score": 1.0, "content": "attributes of an object (Red, Blue, Green, etc.).", "type": "text" } ], "index": 31 } ], "index": 29 }, { "type": "text", "bbox": [ 107, 467, 505, 511 ], "lines": [ { "bbox": [ 105, 466, 506, 479 ], "spans": [ { "bbox": [ 105, 466, 321, 479 ], "score": 1.0, "content": "An NPP abbreviates an arithmetic literal of the form", "type": "text" }, { "bbox": [ 321, 468, 345, 477 ], "score": 0.88, "content": "c = v", "type": "inline_equation" }, { "bbox": [ 346, 466, 367, 479 ], "score": 1.0, "content": "with", "type": "text" }, { "bbox": [ 367, 466, 414, 479 ], "score": 0.92, "content": "c \\in \\{ h ( x ) \\}", "type": "inline_equation" }, { "bbox": [ 414, 466, 433, 479 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 433, 466, 503, 479 ], "score": 0.92, "content": "v \\in \\{ v _ { 1 } , \\ldots , v _ { n } \\}", "type": "inline_equation" }, { "bbox": [ 503, 466, 506, 479 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 32 }, { "bbox": [ 105, 477, 506, 489 ], "spans": [ { "bbox": [ 105, 477, 224, 489 ], "score": 1.0, "content": "Furthermore, we denote with", "type": "text" }, { "bbox": [ 224, 478, 246, 488 ], "score": 0.88, "content": "\\Pi ^ { n p p }", "type": "inline_equation" }, { "bbox": [ 246, 477, 433, 489 ], "score": 1.0, "content": "a set of NPPs of the form stated in (Eq. 1) and", "type": "text" }, { "bbox": [ 433, 479, 452, 487 ], "score": 0.87, "content": "r ^ { n p p }", "type": "inline_equation" }, { "bbox": [ 452, 477, 506, 489 ], "score": 1.0, "content": "the set of all", "type": "text" } ], "index": 33 }, { "bbox": [ 104, 486, 507, 502 ], "spans": [ { "bbox": [ 104, 486, 128, 502 ], "score": 1.0, "content": "rules", "type": "text" }, { "bbox": [ 129, 490, 153, 498 ], "score": 0.86, "content": "c = v", "type": "inline_equation" }, { "bbox": [ 153, 486, 463, 502 ], "score": 1.0, "content": "of one NPP, which denotes the possible outcomes, obtained from an NPP in", "type": "text" }, { "bbox": [ 464, 489, 485, 499 ], "score": 0.88, "content": "\\Pi ^ { n p p }", "type": "inline_equation" }, { "bbox": [ 486, 486, 507, 502 ], "score": 1.0, "content": ", e.g.", "type": "text" } ], "index": 34 }, { "bbox": [ 106, 496, 460, 514 ], "spans": [ { "bbox": [ 106, 499, 313, 512 ], "score": 0.88, "content": "r ^ { c o l o r \\_ a t t r } = \\{ c = R e d , c = B l u e , c = G r \\bar { e } e n , . . . \\}", "type": "inline_equation" }, { "bbox": [ 314, 496, 460, 514 ], "score": 1.0, "content": "for the example depicted in Fig. 2b.", "type": "text" } ], "index": 35 } ], "index": 33.5 }, { "type": "text", "bbox": [ 106, 515, 505, 561 ], "lines": [ { "bbox": [ 105, 515, 506, 529 ], "spans": [ { "bbox": [ 105, 515, 181, 529 ], "score": 1.0, "content": "Rules of the form", "type": "text" }, { "bbox": [ 182, 516, 317, 528 ], "score": 0.86, "content": "n p p ( h ( x ) , [ v _ { 1 } , \\ldots , v _ { n } ] ) B o d y", "type": "inline_equation" }, { "bbox": [ 317, 515, 506, 529 ], "score": 1.0, "content": "are used as an abbreviation for application to", "type": "text" } ], "index": 36 }, { "bbox": [ 106, 528, 505, 540 ], "spans": [ { "bbox": [ 106, 528, 505, 540 ], "score": 1.0, "content": "multiple entities, e.g. multiple slots for the task of set prediction (cf. Fig. 2b). Hereby, Body of the", "type": "text" } ], "index": 37 }, { "bbox": [ 105, 538, 505, 551 ], "spans": [ { "bbox": [ 105, 538, 188, 551 ], "score": 1.0, "content": "rule is identified by", "type": "text" }, { "bbox": [ 189, 539, 198, 548 ], "score": 0.73, "content": "\\top", "type": "inline_equation" }, { "bbox": [ 198, 538, 278, 551 ], "score": 1.0, "content": "(tautology, true) or", "type": "text" }, { "bbox": [ 278, 539, 288, 549 ], "score": 0.74, "content": "\\perp", "type": "inline_equation" }, { "bbox": [ 288, 538, 505, 551 ], "score": 1.0, "content": "(contradiction, false) during grounding. Rules of the", "type": "text" } ], "index": 38 }, { "bbox": [ 104, 547, 403, 562 ], "spans": [ { "bbox": [ 104, 547, 128, 562 ], "score": 1.0, "content": "form", "type": "text" }, { "bbox": [ 128, 549, 186, 560 ], "score": 0.81, "content": "H e a d \\gets B o d y", "type": "inline_equation" }, { "bbox": [ 187, 547, 208, 562 ], "score": 1.0, "content": "with", "type": "text" }, { "bbox": [ 208, 550, 227, 559 ], "score": 0.87, "content": "r ^ { n p p }", "type": "inline_equation" }, { "bbox": [ 228, 547, 378, 562 ], "score": 1.0, "content": "appearing in Head are prohibited for", "type": "text" }, { "bbox": [ 378, 549, 399, 559 ], "score": 0.88, "content": "\\Pi ^ { n p p }", "type": "inline_equation" }, { "bbox": [ 400, 547, 403, 562 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 39 } ], "index": 37.5 }, { "type": "text", "bbox": [ 108, 565, 504, 600 ], "lines": [ { "bbox": [ 105, 565, 506, 579 ], "spans": [ { "bbox": [ 105, 565, 506, 579 ], "score": 1.0, "content": "In this work, we largely make use of NPPs that contain probabilistic circuits (specifically SPNs)", "type": "text" } ], "index": 40 }, { "bbox": [ 106, 577, 506, 590 ], "spans": [ { "bbox": [ 106, 577, 506, 590 ], "score": 1.0, "content": "which allow for tractable density estimation and modelling of joint probabilities. In this way, it is", "type": "text" } ], "index": 41 }, { "bbox": [ 106, 588, 502, 601 ], "spans": [ { "bbox": [ 106, 588, 365, 601 ], "score": 1.0, "content": "possible to answer a much richer set of probabilistic queries, i.e.", "type": "text" }, { "bbox": [ 366, 588, 403, 600 ], "score": 0.85, "content": "P ( X , C )", "type": "inline_equation" }, { "bbox": [ 403, 588, 407, 601 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 407, 588, 443, 600 ], "score": 0.87, "content": "P ( X | C )", "type": "inline_equation" }, { "bbox": [ 443, 588, 461, 601 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 461, 588, 497, 600 ], "score": 0.92, "content": "P ( \\bar { C } | X )", "type": "inline_equation" }, { "bbox": [ 498, 588, 502, 601 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 42 } ], "index": 41 }, { "type": "text", "bbox": [ 107, 604, 505, 671 ], "lines": [ { "bbox": [ 105, 605, 505, 618 ], "spans": [ { "bbox": [ 105, 605, 505, 618 ], "score": 1.0, "content": "In addition to this, we introduce the arguably more interesting type of NPP that combines a neural", "type": "text" } ], "index": 43 }, { "bbox": [ 105, 616, 505, 628 ], "spans": [ { "bbox": [ 105, 616, 505, 628 ], "score": 1.0, "content": "module with a PC. Hereby, the neural module learns to map the raw input data into an optimal", "type": "text" } ], "index": 44 }, { "bbox": [ 105, 627, 505, 639 ], "spans": [ { "bbox": [ 105, 627, 505, 639 ], "score": 1.0, "content": "latent representation, e.g. object-based slot representations. The PC, in turn, learns to model the", "type": "text" } ], "index": 45 }, { "bbox": [ 105, 638, 506, 651 ], "spans": [ { "bbox": [ 105, 638, 506, 651 ], "score": 1.0, "content": "joint distribution of these latent variables and produces the final probability estimates. This type of", "type": "text" } ], "index": 46 }, { "bbox": [ 105, 648, 505, 661 ], "spans": [ { "bbox": [ 105, 648, 505, 661 ], "score": 1.0, "content": "NPP nicely combines the representational power of neural networks with the advantages of PCs in", "type": "text" } ], "index": 47 }, { "bbox": [ 105, 660, 282, 673 ], "spans": [ { "bbox": [ 105, 660, 282, 673 ], "score": 1.0, "content": "probability estimation and query flexibility.", "type": "text" } ], "index": 48 } ], "index": 45.5 }, { "type": "text", "bbox": [ 107, 676, 505, 732 ], "lines": [ { "bbox": [ 105, 676, 505, 690 ], "spans": [ { "bbox": [ 105, 676, 505, 690 ], "score": 1.0, "content": "For making the different probabilistic queries distinguishable in a SLASH program, we introduce the", "type": "text" } ], "index": 49 }, { "bbox": [ 105, 687, 505, 700 ], "spans": [ { "bbox": [ 105, 687, 318, 700 ], "score": 1.0, "content": "following notation. We denote a given variable with", "type": "text" }, { "bbox": [ 319, 689, 328, 698 ], "score": 0.69, "content": "^ +", "type": "inline_equation" }, { "bbox": [ 329, 687, 505, 700 ], "score": 1.0, "content": "and the query variable with −. E.g., within", "type": "text" } ], "index": 50 }, { "bbox": [ 105, 697, 506, 713 ], "spans": [ { "bbox": [ 105, 697, 262, 713 ], "score": 1.0, "content": "the running example of set prediction", "type": "text" }, { "bbox": [ 263, 699, 273, 710 ], "score": 0.31, "content": "\\cdot e f .", "type": "inline_equation" }, { "bbox": [ 273, 697, 441, 713 ], "score": 1.0, "content": "Fig. 1 and 2b), with the query color_attr", "type": "text" }, { "bbox": [ 442, 699, 486, 711 ], "score": 0.87, "content": "\\cdot ( + X , - C )", "type": "inline_equation" }, { "bbox": [ 487, 697, 506, 713 ], "score": 1.0, "content": "one", "type": "text" } ], "index": 51 }, { "bbox": [ 105, 708, 507, 723 ], "spans": [ { "bbox": [ 105, 708, 158, 723 ], "score": 1.0, "content": "is asking for", "type": "text" }, { "bbox": [ 158, 710, 194, 721 ], "score": 0.91, "content": "P ( C | X )", "type": "inline_equation" }, { "bbox": [ 195, 708, 305, 723 ], "score": 1.0, "content": ". Similarly, with color_attr", "type": "text" }, { "bbox": [ 305, 711, 349, 722 ], "score": 0.86, "content": "( - X , + C )", "type": "inline_equation" }, { "bbox": [ 349, 708, 420, 723 ], "score": 1.0, "content": "one is asking for", "type": "text" }, { "bbox": [ 420, 710, 456, 722 ], "score": 0.93, "content": "P ( X | C )", "type": "inline_equation" }, { "bbox": [ 456, 708, 507, 723 ], "score": 1.0, "content": "and, finally,", "type": "text" } ], "index": 52 }, { "bbox": [ 105, 719, 272, 734 ], "spans": [ { "bbox": [ 105, 719, 171, 734 ], "score": 1.0, "content": "with color_attr", "type": "text" }, { "bbox": [ 171, 721, 215, 732 ], "score": 0.82, "content": "( - X , - C )", "type": "inline_equation" }, { "bbox": [ 216, 719, 231, 734 ], "score": 1.0, "content": "for", "type": "text" }, { "bbox": [ 232, 722, 268, 732 ], "score": 0.93, "content": "P ( X , C )", "type": "inline_equation" }, { "bbox": [ 268, 719, 272, 734 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 53 } ], "index": 51 } ], "page_idx": 3, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 107, 27, 308, 37 ], "lines": [ { "bbox": [ 106, 25, 309, 39 ], "spans": [ { "bbox": [ 106, 25, 309, 39 ], "score": 1.0, "content": "Under review as a conference paper at ICLR 2022", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 302, 752, 308, 759 ], "lines": [] } ], "para_blocks": [ { "type": "image", "bbox": [ 138, 82, 299, 190 ], "blocks": [ { "type": "image_body", "bbox": [ 138, 82, 299, 190 ], "group_id": 1, "lines": [ { "bbox": [ 138, 82, 299, 190 ], "spans": [ { "bbox": [ 138, 82, 299, 190 ], "score": 0.968, "type": "image", "image_path": "fe12c632ab68913a19fb5c0639cac79765cabf16e331224edc0cd2c84bb8eebe.jpg" } ] } ], "index": 3.5, "virtual_lines": [ { "bbox": [ 138, 82, 299, 95.5 ], "spans": [], "index": 0 }, { "bbox": [ 138, 95.5, 299, 109.0 ], "spans": [], "index": 1 }, { "bbox": [ 138, 109.0, 299, 122.5 ], "spans": [], "index": 2 }, { "bbox": [ 138, 122.5, 299, 136.0 ], "spans": [], "index": 3 }, { "bbox": [ 138, 136.0, 299, 149.5 ], "spans": [], "index": 4 }, { "bbox": [ 138, 149.5, 299, 163.0 ], "spans": [], "index": 5 }, { "bbox": [ 138, 163.0, 299, 176.5 ], "spans": [], "index": 6 }, { "bbox": [ 138, 176.5, 299, 190.0 ], "spans": [], "index": 7 } ] } ], "index": 3.5 }, { "type": "text", "bbox": [ 128, 195, 309, 215 ], "lines": [ { "bbox": [ 128, 194, 309, 207 ], "spans": [ { "bbox": [ 128, 194, 309, 207 ], "score": 1.0, "content": "(a) NPPs come in various flavors depending on the", "type": "text" } ], "index": 9 }, { "bbox": [ 128, 205, 221, 215 ], "spans": [ { "bbox": [ 128, 205, 221, 215 ], "score": 1.0, "content": "data and underlying task.", "type": "text" } ], "index": 11 } ], "index": 10.0, "bbox_fs": [ 128, 194, 309, 215 ] }, { "type": "image", "bbox": [ 329, 80, 474, 189 ], "blocks": [ { "type": "image_body", "bbox": [ 329, 80, 474, 189 ], "group_id": 0, "lines": [ { "bbox": [ 329, 80, 474, 189 ], "spans": [ { "bbox": [ 329, 80, 474, 189 ], "score": 0.954, "type": "image", "image_path": "deaa2f06b12c44d64f5ac8f226a3fdb61c0d8bba50b15128cddc5ccafbad3205.jpg" } ] } ], "index": 9.0, "virtual_lines": [ { "bbox": [ 329, 80, 474, 134.5 ], "spans": [], "index": 8 }, { "bbox": [ 329, 134.5, 474, 189.0 ], "spans": [], "index": 10 } ] }, { "type": "image_caption", "bbox": [ 321, 195, 482, 214 ], "group_id": 0, "lines": [ { "bbox": [ 321, 194, 483, 206 ], "spans": [ { "bbox": [ 321, 194, 483, 206 ], "score": 1.0, "content": "(b) Minimal SLASH program and query for", "type": "text" } ], "index": 12 }, { "bbox": [ 321, 205, 375, 216 ], "spans": [ { "bbox": [ 321, 205, 375, 216 ], "score": 1.0, "content": "set prediction.", "type": "text" } ], "index": 13 } ], "index": 12.5 }, { "type": "image_caption", "bbox": [ 106, 225, 506, 280 ], "group_id": 0, "lines": [ { "bbox": [ 105, 224, 507, 237 ], "spans": [ { "bbox": [ 105, 224, 507, 237 ], "score": 1.0, "content": "Figure 2: (a) Depending on the data set and underlying task, SLASH requires a suitable Neural-", "type": "text" } ], "index": 14 }, { "bbox": [ 105, 235, 505, 248 ], "spans": [ { "bbox": [ 105, 235, 505, 248 ], "score": 1.0, "content": "Probabilistic Predicate (NPP) that computes query-dependent probability estimates. An NPP can be", "type": "text" } ], "index": 15 }, { "bbox": [ 105, 246, 507, 260 ], "spans": [ { "bbox": [ 105, 246, 507, 260 ], "score": 1.0, "content": "composed of neural and probabilistic modules, or (depicted via slash symbol) only one of these two.", "type": "text" } ], "index": 16 }, { "bbox": [ 105, 257, 506, 270 ], "spans": [ { "bbox": [ 105, 257, 506, 270 ], "score": 1.0, "content": "(b) A minimal SLASH program and query for the set prediction task, here only showing the NPP that", "type": "text" } ], "index": 17 }, { "bbox": [ 105, 268, 447, 281 ], "spans": [ { "bbox": [ 105, 268, 447, 281 ], "score": 1.0, "content": "models the color category per object. For the full program, we refer to the Appendix.", "type": "text" } ], "index": 18 } ], "index": 16 } ], "index": 12.5 }, { "type": "title", "bbox": [ 108, 302, 294, 313 ], "lines": [ { "bbox": [ 106, 302, 295, 315 ], "spans": [ { "bbox": [ 106, 302, 295, 315 ], "score": 1.0, "content": "3.1 NEURAL-PROBABILISTIC PREDICATES", "type": "text" } ], "index": 19 } ], "index": 19 }, { "type": "text", "bbox": [ 106, 322, 505, 378 ], "lines": [ { "bbox": [ 105, 322, 505, 336 ], "spans": [ { "bbox": [ 105, 322, 505, 336 ], "score": 1.0, "content": "Previous DPPLs, DeepProbLog (Manhaeve et al., 2018) and NeurASP (Yang et al., 2020), introduced", "type": "text" } ], "index": 20 }, { "bbox": [ 105, 333, 505, 347 ], "spans": [ { "bbox": [ 105, 333, 505, 347 ], "score": 1.0, "content": "the Neural Predicate as an annotated-disjunction or as a propositional atom, respectively, to acquire", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 343, 506, 359 ], "spans": [ { "bbox": [ 105, 343, 237, 359 ], "score": 1.0, "content": "conditional class probabilities,", "type": "text" }, { "bbox": [ 237, 345, 273, 357 ], "score": 0.93, "content": "P ( C | X )", "type": "inline_equation" }, { "bbox": [ 273, 343, 506, 359 ], "score": 1.0, "content": ", via the softmax function at the output of an arbitrary", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 355, 505, 369 ], "spans": [ { "bbox": [ 105, 355, 505, 369 ], "score": 1.0, "content": "DNN. As mentioned in the introduction, this approach has certain limitations concerning inference", "type": "text" } ], "index": 23 }, { "bbox": [ 106, 366, 463, 380 ], "spans": [ { "bbox": [ 106, 366, 463, 380 ], "score": 1.0, "content": "capabilities. To resolve this issue, we introduce Neural-Probabilisitic Predicates (NPPs).", "type": "text" } ], "index": 24 } ], "index": 22, "bbox_fs": [ 105, 322, 506, 380 ] }, { "type": "text", "bbox": [ 107, 384, 210, 395 ], "lines": [ { "bbox": [ 106, 383, 210, 396 ], "spans": [ { "bbox": [ 106, 383, 210, 396 ], "score": 1.0, "content": "Formally, we denote with", "type": "text" } ], "index": 25 } ], "index": 25, "bbox_fs": [ 106, 383, 210, 396 ] }, { "type": "interline_equation", "bbox": [ 256, 395, 354, 408 ], "lines": [ { "bbox": [ 256, 395, 354, 408 ], "spans": [ { "bbox": [ 256, 395, 354, 408 ], "score": 0.91, "content": "n p p \\left( h ( x ) , [ v _ { 1 } , . . . , v _ { n } ] \\right)", "type": "interline_equation", "image_path": "187dcc297a855ad44e5c575f0d91e7583f0c038c9c3016485793aabe684f440a.jpg" } ] } ], "index": 26, "virtual_lines": [ { "bbox": [ 256, 395, 354, 408 ], "spans": [], "index": 26 } ] }, { "type": "text", "bbox": [ 107, 411, 505, 466 ], "lines": [ { "bbox": [ 105, 411, 506, 425 ], "spans": [ { "bbox": [ 105, 411, 240, 425 ], "score": 1.0, "content": "a Neural-Probabilistic Predicate", "type": "text" }, { "bbox": [ 241, 412, 248, 422 ], "score": 0.8, "content": "h", "type": "inline_equation" }, { "bbox": [ 248, 411, 489, 425 ], "score": 1.0, "content": ". Thereby, (i) npp is a reserved word to label an NPP, (ii)", "type": "text" }, { "bbox": [ 489, 412, 497, 422 ], "score": 0.74, "content": "h", "type": "inline_equation" }, { "bbox": [ 497, 411, 506, 425 ], "score": 1.0, "content": "a", "type": "text" } ], "index": 27 }, { "bbox": [ 105, 422, 505, 435 ], "spans": [ { "bbox": [ 105, 422, 505, 435 ], "score": 1.0, "content": "symbolic name of either a PC, NN or a joint of a PC and NN (cf. Fig. 2a), e.g., color_attr is the name", "type": "text" } ], "index": 28 }, { "bbox": [ 105, 433, 506, 447 ], "spans": [ { "bbox": [ 105, 433, 267, 447 ], "score": 1.0, "content": "of an NPP of Fig. 2b. Additionally, (iii)", "type": "text" }, { "bbox": [ 267, 435, 275, 444 ], "score": 0.79, "content": "x", "type": "inline_equation" }, { "bbox": [ 275, 433, 379, 447 ], "score": 1.0, "content": "denotes a “term” and (iv)", "type": "text" }, { "bbox": [ 379, 435, 422, 445 ], "score": 0.9, "content": "v _ { 1 } , \\ldots , v _ { n }", "type": "inline_equation" }, { "bbox": [ 422, 433, 506, 447 ], "score": 1.0, "content": "are placeholders for", "type": "text" } ], "index": 29 }, { "bbox": [ 105, 444, 505, 457 ], "spans": [ { "bbox": [ 105, 444, 154, 457 ], "score": 1.0, "content": "each of the", "type": "text" }, { "bbox": [ 155, 446, 162, 455 ], "score": 0.71, "content": "n", "type": "inline_equation" }, { "bbox": [ 163, 444, 253, 457 ], "score": 1.0, "content": "possible outcomes of", "type": "text" }, { "bbox": [ 253, 445, 260, 455 ], "score": 0.82, "content": "h", "type": "inline_equation" }, { "bbox": [ 260, 444, 505, 457 ], "score": 1.0, "content": ". For example, the placeholders for color_attr are the color", "type": "text" } ], "index": 30 }, { "bbox": [ 106, 455, 294, 468 ], "spans": [ { "bbox": [ 106, 455, 294, 468 ], "score": 1.0, "content": "attributes of an object (Red, Blue, Green, etc.).", "type": "text" } ], "index": 31 } ], "index": 29, "bbox_fs": [ 105, 411, 506, 468 ] }, { "type": "text", "bbox": [ 107, 467, 505, 511 ], "lines": [ { "bbox": [ 105, 466, 506, 479 ], "spans": [ { "bbox": [ 105, 466, 321, 479 ], "score": 1.0, "content": "An NPP abbreviates an arithmetic literal of the form", "type": "text" }, { "bbox": [ 321, 468, 345, 477 ], "score": 0.88, "content": "c = v", "type": "inline_equation" }, { "bbox": [ 346, 466, 367, 479 ], "score": 1.0, "content": "with", "type": "text" }, { "bbox": [ 367, 466, 414, 479 ], "score": 0.92, "content": "c \\in \\{ h ( x ) \\}", "type": "inline_equation" }, { "bbox": [ 414, 466, 433, 479 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 433, 466, 503, 479 ], "score": 0.92, "content": "v \\in \\{ v _ { 1 } , \\ldots , v _ { n } \\}", "type": "inline_equation" }, { "bbox": [ 503, 466, 506, 479 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 32 }, { "bbox": [ 105, 477, 506, 489 ], "spans": [ { "bbox": [ 105, 477, 224, 489 ], "score": 1.0, "content": "Furthermore, we denote with", "type": "text" }, { "bbox": [ 224, 478, 246, 488 ], "score": 0.88, "content": "\\Pi ^ { n p p }", "type": "inline_equation" }, { "bbox": [ 246, 477, 433, 489 ], "score": 1.0, "content": "a set of NPPs of the form stated in (Eq. 1) and", "type": "text" }, { "bbox": [ 433, 479, 452, 487 ], "score": 0.87, "content": "r ^ { n p p }", "type": "inline_equation" }, { "bbox": [ 452, 477, 506, 489 ], "score": 1.0, "content": "the set of all", "type": "text" } ], "index": 33 }, { "bbox": [ 104, 486, 507, 502 ], "spans": [ { "bbox": [ 104, 486, 128, 502 ], "score": 1.0, "content": "rules", "type": "text" }, { "bbox": [ 129, 490, 153, 498 ], "score": 0.86, "content": "c = v", "type": "inline_equation" }, { "bbox": [ 153, 486, 463, 502 ], "score": 1.0, "content": "of one NPP, which denotes the possible outcomes, obtained from an NPP in", "type": "text" }, { "bbox": [ 464, 489, 485, 499 ], "score": 0.88, "content": "\\Pi ^ { n p p }", "type": "inline_equation" }, { "bbox": [ 486, 486, 507, 502 ], "score": 1.0, "content": ", e.g.", "type": "text" } ], "index": 34 }, { "bbox": [ 106, 496, 460, 514 ], "spans": [ { "bbox": [ 106, 499, 313, 512 ], "score": 0.88, "content": "r ^ { c o l o r \\_ a t t r } = \\{ c = R e d , c = B l u e , c = G r \\bar { e } e n , . . . \\}", "type": "inline_equation" }, { "bbox": [ 314, 496, 460, 514 ], "score": 1.0, "content": "for the example depicted in Fig. 2b.", "type": "text" } ], "index": 35 } ], "index": 33.5, "bbox_fs": [ 104, 466, 507, 514 ] }, { "type": "text", "bbox": [ 106, 515, 505, 561 ], "lines": [ { "bbox": [ 105, 515, 506, 529 ], "spans": [ { "bbox": [ 105, 515, 181, 529 ], "score": 1.0, "content": "Rules of the form", "type": "text" }, { "bbox": [ 182, 516, 317, 528 ], "score": 0.86, "content": "n p p ( h ( x ) , [ v _ { 1 } , \\ldots , v _ { n } ] ) B o d y", "type": "inline_equation" }, { "bbox": [ 317, 515, 506, 529 ], "score": 1.0, "content": "are used as an abbreviation for application to", "type": "text" } ], "index": 36 }, { "bbox": [ 106, 528, 505, 540 ], "spans": [ { "bbox": [ 106, 528, 505, 540 ], "score": 1.0, "content": "multiple entities, e.g. multiple slots for the task of set prediction (cf. Fig. 2b). Hereby, Body of the", "type": "text" } ], "index": 37 }, { "bbox": [ 105, 538, 505, 551 ], "spans": [ { "bbox": [ 105, 538, 188, 551 ], "score": 1.0, "content": "rule is identified by", "type": "text" }, { "bbox": [ 189, 539, 198, 548 ], "score": 0.73, "content": "\\top", "type": "inline_equation" }, { "bbox": [ 198, 538, 278, 551 ], "score": 1.0, "content": "(tautology, true) or", "type": "text" }, { "bbox": [ 278, 539, 288, 549 ], "score": 0.74, "content": "\\perp", "type": "inline_equation" }, { "bbox": [ 288, 538, 505, 551 ], "score": 1.0, "content": "(contradiction, false) during grounding. Rules of the", "type": "text" } ], "index": 38 }, { "bbox": [ 104, 547, 403, 562 ], "spans": [ { "bbox": [ 104, 547, 128, 562 ], "score": 1.0, "content": "form", "type": "text" }, { "bbox": [ 128, 549, 186, 560 ], "score": 0.81, "content": "H e a d \\gets B o d y", "type": "inline_equation" }, { "bbox": [ 187, 547, 208, 562 ], "score": 1.0, "content": "with", "type": "text" }, { "bbox": [ 208, 550, 227, 559 ], "score": 0.87, "content": "r ^ { n p p }", "type": "inline_equation" }, { "bbox": [ 228, 547, 378, 562 ], "score": 1.0, "content": "appearing in Head are prohibited for", "type": "text" }, { "bbox": [ 378, 549, 399, 559 ], "score": 0.88, "content": "\\Pi ^ { n p p }", "type": "inline_equation" }, { "bbox": [ 400, 547, 403, 562 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 39 } ], "index": 37.5, "bbox_fs": [ 104, 515, 506, 562 ] }, { "type": "text", "bbox": [ 108, 565, 504, 600 ], "lines": [ { "bbox": [ 105, 565, 506, 579 ], "spans": [ { "bbox": [ 105, 565, 506, 579 ], "score": 1.0, "content": "In this work, we largely make use of NPPs that contain probabilistic circuits (specifically SPNs)", "type": "text" } ], "index": 40 }, { "bbox": [ 106, 577, 506, 590 ], "spans": [ { "bbox": [ 106, 577, 506, 590 ], "score": 1.0, "content": "which allow for tractable density estimation and modelling of joint probabilities. In this way, it is", "type": "text" } ], "index": 41 }, { "bbox": [ 106, 588, 502, 601 ], "spans": [ { "bbox": [ 106, 588, 365, 601 ], "score": 1.0, "content": "possible to answer a much richer set of probabilistic queries, i.e.", "type": "text" }, { "bbox": [ 366, 588, 403, 600 ], "score": 0.85, "content": "P ( X , C )", "type": "inline_equation" }, { "bbox": [ 403, 588, 407, 601 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 407, 588, 443, 600 ], "score": 0.87, "content": "P ( X | C )", "type": "inline_equation" }, { "bbox": [ 443, 588, 461, 601 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 461, 588, 497, 600 ], "score": 0.92, "content": "P ( \\bar { C } | X )", "type": "inline_equation" }, { "bbox": [ 498, 588, 502, 601 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 42 } ], "index": 41, "bbox_fs": [ 105, 565, 506, 601 ] }, { "type": "text", "bbox": [ 107, 604, 505, 671 ], "lines": [ { "bbox": [ 105, 605, 505, 618 ], "spans": [ { "bbox": [ 105, 605, 505, 618 ], "score": 1.0, "content": "In addition to this, we introduce the arguably more interesting type of NPP that combines a neural", "type": "text" } ], "index": 43 }, { "bbox": [ 105, 616, 505, 628 ], "spans": [ { "bbox": [ 105, 616, 505, 628 ], "score": 1.0, "content": "module with a PC. Hereby, the neural module learns to map the raw input data into an optimal", "type": "text" } ], "index": 44 }, { "bbox": [ 105, 627, 505, 639 ], "spans": [ { "bbox": [ 105, 627, 505, 639 ], "score": 1.0, "content": "latent representation, e.g. object-based slot representations. The PC, in turn, learns to model the", "type": "text" } ], "index": 45 }, { "bbox": [ 105, 638, 506, 651 ], "spans": [ { "bbox": [ 105, 638, 506, 651 ], "score": 1.0, "content": "joint distribution of these latent variables and produces the final probability estimates. This type of", "type": "text" } ], "index": 46 }, { "bbox": [ 105, 648, 505, 661 ], "spans": [ { "bbox": [ 105, 648, 505, 661 ], "score": 1.0, "content": "NPP nicely combines the representational power of neural networks with the advantages of PCs in", "type": "text" } ], "index": 47 }, { "bbox": [ 105, 660, 282, 673 ], "spans": [ { "bbox": [ 105, 660, 282, 673 ], "score": 1.0, "content": "probability estimation and query flexibility.", "type": "text" } ], "index": 48 } ], "index": 45.5, "bbox_fs": [ 105, 605, 506, 673 ] }, { "type": "text", "bbox": [ 107, 676, 505, 732 ], "lines": [ { "bbox": [ 105, 676, 505, 690 ], "spans": [ { "bbox": [ 105, 676, 505, 690 ], "score": 1.0, "content": "For making the different probabilistic queries distinguishable in a SLASH program, we introduce the", "type": "text" } ], "index": 49 }, { "bbox": [ 105, 687, 505, 700 ], "spans": [ { "bbox": [ 105, 687, 318, 700 ], "score": 1.0, "content": "following notation. We denote a given variable with", "type": "text" }, { "bbox": [ 319, 689, 328, 698 ], "score": 0.69, "content": "^ +", "type": "inline_equation" }, { "bbox": [ 329, 687, 505, 700 ], "score": 1.0, "content": "and the query variable with −. E.g., within", "type": "text" } ], "index": 50 }, { "bbox": [ 105, 697, 506, 713 ], "spans": [ { "bbox": [ 105, 697, 262, 713 ], "score": 1.0, "content": "the running example of set prediction", "type": "text" }, { "bbox": [ 263, 699, 273, 710 ], "score": 0.31, "content": "\\cdot e f .", "type": "inline_equation" }, { "bbox": [ 273, 697, 441, 713 ], "score": 1.0, "content": "Fig. 1 and 2b), with the query color_attr", "type": "text" }, { "bbox": [ 442, 699, 486, 711 ], "score": 0.87, "content": "\\cdot ( + X , - C )", "type": "inline_equation" }, { "bbox": [ 487, 697, 506, 713 ], "score": 1.0, "content": "one", "type": "text" } ], "index": 51 }, { "bbox": [ 105, 708, 507, 723 ], "spans": [ { "bbox": [ 105, 708, 158, 723 ], "score": 1.0, "content": "is asking for", "type": "text" }, { "bbox": [ 158, 710, 194, 721 ], "score": 0.91, "content": "P ( C | X )", "type": "inline_equation" }, { "bbox": [ 195, 708, 305, 723 ], "score": 1.0, "content": ". Similarly, with color_attr", "type": "text" }, { "bbox": [ 305, 711, 349, 722 ], "score": 0.86, "content": "( - X , + C )", "type": "inline_equation" }, { "bbox": [ 349, 708, 420, 723 ], "score": 1.0, "content": "one is asking for", "type": "text" }, { "bbox": [ 420, 710, 456, 722 ], "score": 0.93, "content": "P ( X | C )", "type": "inline_equation" }, { "bbox": [ 456, 708, 507, 723 ], "score": 1.0, "content": "and, finally,", "type": "text" } ], "index": 52 }, { "bbox": [ 105, 719, 272, 734 ], "spans": [ { "bbox": [ 105, 719, 171, 734 ], "score": 1.0, "content": "with color_attr", "type": "text" }, { "bbox": [ 171, 721, 215, 732 ], "score": 0.82, "content": "( - X , - C )", "type": "inline_equation" }, { "bbox": [ 216, 719, 231, 734 ], "score": 1.0, "content": "for", "type": "text" }, { "bbox": [ 232, 722, 268, 732 ], "score": 0.93, "content": "P ( X , C )", "type": "inline_equation" }, { "bbox": [ 268, 719, 272, 734 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 53 } ], "index": 51, "bbox_fs": [ 105, 676, 507, 734 ] } ] }, { "preproc_blocks": [ { "type": "text", "bbox": [ 106, 81, 505, 149 ], "lines": [ { "bbox": [ 105, 81, 506, 96 ], "spans": [ { "bbox": [ 105, 81, 506, 96 ], "score": 1.0, "content": "To summarize, an NPP can consist of neural and/or probabilistic modules and produces query-", "type": "text" } ], "index": 0 }, { "bbox": [ 105, 93, 505, 106 ], "spans": [ { "bbox": [ 105, 93, 505, 106 ], "score": 1.0, "content": "dependent probability estimates. Due to the flexibility of its definition, the term NPP contains the", "type": "text" } ], "index": 1 }, { "bbox": [ 104, 104, 506, 118 ], "spans": [ { "bbox": [ 104, 104, 506, 118 ], "score": 1.0, "content": "predicates of previous works (Manhaeve et al., 2018; Yang et al., 2020), but also more interesting", "type": "text" } ], "index": 2 }, { "bbox": [ 104, 115, 506, 128 ], "spans": [ { "bbox": [ 104, 115, 506, 128 ], "score": 1.0, "content": "predicates discussed above. The specific “flavor” of an NPP should be chosen depending on what", "type": "text" } ], "index": 3 }, { "bbox": [ 105, 126, 506, 139 ], "spans": [ { "bbox": [ 105, 126, 506, 139 ], "score": 1.0, "content": "type of probability estimation is required (cf. Fig 2a). Lastly, NPPs have the unified loss function of", "type": "text" } ], "index": 4 }, { "bbox": [ 106, 137, 218, 150 ], "spans": [ { "bbox": [ 106, 137, 218, 150 ], "score": 1.0, "content": "the negative log-likelihood:", "type": "text" } ], "index": 5 } ], "index": 2.5 }, { "type": "interline_equation", "bbox": [ 111, 154, 504, 187 ], "lines": [ { "bbox": [ 111, 154, 504, 187 ], "spans": [ { "bbox": [ 111, 154, 504, 187 ], "score": 0.93, "content": "L _ { N P P } : = - \\log L H ( x , \\hat { x } ) = \\sum _ { i = 1 } ^ { n } L H ( x _ { i } , \\hat { x } _ { i } ) = - \\sum _ { i = 1 } ^ { n } x _ { i } \\cdot \\log ( P _ { \\xi } ^ { ( X , C ) } ( x _ { i } ) ) = - \\sum _ { i = 1 } ^ { n } \\log ( P _ { \\xi } ^ { ( X , C ) } )", "type": "interline_equation", "image_path": "2df2b15aa55bfa392ca1eea024da8df01763335460315fe384a335f6d03243b3.jpg" } ] } ], "index": 7, "virtual_lines": [ { "bbox": [ 111, 154, 504, 165.0 ], "spans": [], "index": 6 }, { "bbox": [ 111, 165.0, 504, 176.0 ], "spans": [], "index": 7 }, { "bbox": [ 111, 176.0, 504, 187.0 ], "spans": [], "index": 8 } ] }, { "type": "text", "bbox": [ 107, 195, 506, 222 ], "lines": [ { "bbox": [ 105, 194, 505, 208 ], "spans": [ { "bbox": [ 105, 194, 347, 208 ], "score": 1.0, "content": "whereby we are assuming the data to be i.i.d., ground truth", "type": "text" }, { "bbox": [ 347, 197, 357, 207 ], "score": 0.85, "content": "x _ { i }", "type": "inline_equation" }, { "bbox": [ 357, 194, 460, 208 ], "score": 1.0, "content": "to be the all-ones vector,", "type": "text" }, { "bbox": [ 460, 196, 467, 207 ], "score": 0.85, "content": "\\xi", "type": "inline_equation" }, { "bbox": [ 467, 194, 505, 208 ], "score": 1.0, "content": "to be the", "type": "text" } ], "index": 9 }, { "bbox": [ 101, 205, 506, 228 ], "spans": [ { "bbox": [ 101, 205, 216, 228 ], "score": 1.0, "content": "parameters of the NPP and", "type": "text" }, { "bbox": [ 216, 207, 246, 223 ], "score": 0.94, "content": "P _ { \\xi } ^ { ( X , C ) }", "type": "inline_equation" }, { "bbox": [ 247, 205, 324, 228 ], "score": 1.0, "content": "are the predictions", "type": "text" }, { "bbox": [ 324, 210, 334, 220 ], "score": 0.88, "content": "{ \\hat { x } } _ { i }", "type": "inline_equation" }, { "bbox": [ 335, 205, 506, 228 ], "score": 1.0, "content": "obtained from the PC encoded in the NPP.", "type": "text" } ], "index": 10 } ], "index": 9.5 }, { "type": "title", "bbox": [ 107, 235, 313, 247 ], "lines": [ { "bbox": [ 106, 235, 313, 248 ], "spans": [ { "bbox": [ 106, 235, 313, 248 ], "score": 1.0, "content": "3.2 THE SLASH LANGUAGE AND SEMANTICS", "type": "text" } ], "index": 11 } ], "index": 11 }, { "type": "text", "bbox": [ 106, 255, 505, 301 ], "lines": [ { "bbox": [ 105, 256, 505, 269 ], "spans": [ { "bbox": [ 105, 256, 505, 269 ], "score": 1.0, "content": "Fig. 1 presents an illustration of SLASH, exemplified for the task of set prediction, with all of its key", "type": "text" } ], "index": 12 }, { "bbox": [ 105, 267, 505, 280 ], "spans": [ { "bbox": [ 105, 267, 505, 280 ], "score": 1.0, "content": "components. Having introduced the NPPs previously, which produce probability estimates, we now", "type": "text" } ], "index": 13 }, { "bbox": [ 105, 279, 505, 292 ], "spans": [ { "bbox": [ 105, 279, 505, 292 ], "score": 1.0, "content": "continue in the pipeline on how to use these probability estimates for answering logical queries. We", "type": "text" } ], "index": 14 }, { "bbox": [ 105, 289, 295, 302 ], "spans": [ { "bbox": [ 105, 289, 295, 302 ], "score": 1.0, "content": "begin by formally defining a SLASH program.", "type": "text" } ], "index": 15 } ], "index": 13.5 }, { "type": "text", "bbox": [ 106, 304, 505, 338 ], "lines": [ { "bbox": [ 105, 302, 507, 317 ], "spans": [ { "bbox": [ 105, 302, 324, 317 ], "score": 1.0, "content": "Definition 1. A SLASH program Π is the union of", "type": "text" }, { "bbox": [ 324, 304, 345, 315 ], "score": 0.8, "content": "\\Pi ^ { a s p }", "type": "inline_equation" }, { "bbox": [ 345, 302, 350, 317 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 351, 304, 372, 315 ], "score": 0.59, "content": "\\Pi ^ { n p p }", "type": "inline_equation" }, { "bbox": [ 373, 302, 428, 317 ], "score": 1.0, "content": ". Therewith,", "type": "text" }, { "bbox": [ 429, 304, 450, 315 ], "score": 0.85, "content": "\\Pi ^ { a s p }", "type": "inline_equation" }, { "bbox": [ 451, 302, 507, 317 ], "score": 1.0, "content": "is the set of", "type": "text" } ], "index": 16 }, { "bbox": [ 104, 313, 506, 328 ], "spans": [ { "bbox": [ 104, 313, 440, 328 ], "score": 1.0, "content": "propositional rules (standard rules from ASP-Core-2 (Calimeri et al., 2020)), and", "type": "text" }, { "bbox": [ 441, 315, 462, 325 ], "score": 0.8, "content": "\\Pi ^ { n p p }", "type": "inline_equation" }, { "bbox": [ 463, 313, 506, 328 ], "score": 1.0, "content": "is a set of", "type": "text" } ], "index": 17 }, { "bbox": [ 104, 326, 344, 339 ], "spans": [ { "bbox": [ 104, 326, 344, 339 ], "score": 1.0, "content": "Neural-Probabilistic Predicates of the form stated in Eq. 1.", "type": "text" } ], "index": 18 } ], "index": 17 }, { "type": "text", "bbox": [ 106, 346, 505, 414 ], "lines": [ { "bbox": [ 105, 347, 505, 360 ], "spans": [ { "bbox": [ 105, 347, 505, 360 ], "score": 1.0, "content": "Fig. 2b depicts a minimal SLASH program for the task of set prediction, exemplifying a set of", "type": "text" } ], "index": 19 }, { "bbox": [ 105, 358, 505, 371 ], "spans": [ { "bbox": [ 105, 358, 505, 371 ], "score": 1.0, "content": "propositional rules and neural predicates. Similar to NeurASP, SLASH requires ASP and as such", "type": "text" } ], "index": 20 }, { "bbox": [ 106, 370, 505, 381 ], "spans": [ { "bbox": [ 106, 370, 505, 381 ], "score": 1.0, "content": "adopts its syntax to most part. We therefore now address integrating our NPPs into an ASP compatible", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 379, 505, 393 ], "spans": [ { "bbox": [ 105, 379, 505, 393 ], "score": 1.0, "content": "form to obtain the success probability for the logical query given all possible solutions. Thus, we", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 391, 505, 404 ], "spans": [ { "bbox": [ 105, 391, 369, 404 ], "score": 1.0, "content": "define SLASH’s semantics. For SLASH to translate the program", "type": "text" }, { "bbox": [ 369, 391, 378, 401 ], "score": 0.63, "content": "\\Pi", "type": "inline_equation" }, { "bbox": [ 379, 391, 505, 404 ], "score": 1.0, "content": "to the ASP-solver’s compatible", "type": "text" } ], "index": 23 }, { "bbox": [ 106, 403, 339, 414 ], "spans": [ { "bbox": [ 106, 403, 339, 414 ], "score": 1.0, "content": "form, the rules (Eq. 1) will be rewritten to the set of rules:", "type": "text" } ], "index": 24 } ], "index": 21.5 }, { "type": "interline_equation", "bbox": [ 241, 419, 370, 433 ], "lines": [ { "bbox": [ 241, 419, 370, 433 ], "spans": [ { "bbox": [ 241, 419, 370, 433 ], "score": 0.93, "content": "1 \\{ h ( x ) = v _ { 1 } ; \\ldots ; h ( x ) = v _ { n } \\} 1", "type": "interline_equation", "image_path": "e4c76dbbc6e4f0607f36cfe65c5c74b6f382d229457f6b2b9ff4996044984268.jpg" } ] } ], "index": 25, "virtual_lines": [ { "bbox": [ 241, 419, 370, 433 ], "spans": [], "index": 25 } ] }, { "type": "text", "bbox": [ 106, 438, 505, 516 ], "lines": [ { "bbox": [ 105, 438, 505, 450 ], "spans": [ { "bbox": [ 105, 438, 505, 450 ], "score": 1.0, "content": "The ASP-solver should understand this as “Pick exactly one rule from the set”. After the translation", "type": "text" } ], "index": 26 }, { "bbox": [ 106, 450, 505, 461 ], "spans": [ { "bbox": [ 106, 450, 326, 461 ], "score": 1.0, "content": "is done, we can ask an ASP-solver for the solutions for", "type": "text" }, { "bbox": [ 326, 450, 335, 459 ], "score": 0.52, "content": "\\Pi", "type": "inline_equation" }, { "bbox": [ 335, 450, 505, 461 ], "score": 1.0, "content": ". We denote a set of ASP constraints in the", "type": "text" } ], "index": 27 }, { "bbox": [ 107, 460, 505, 473 ], "spans": [ { "bbox": [ 107, 461, 163, 471 ], "score": 0.79, "content": "\\mathrm { f o r m } \\gets B o d y", "type": "inline_equation" }, { "bbox": [ 163, 460, 209, 473 ], "score": 1.0, "content": ", as queries", "type": "text" }, { "bbox": [ 210, 461, 219, 472 ], "score": 0.84, "content": "Q", "type": "inline_equation" }, { "bbox": [ 219, 460, 438, 473 ], "score": 1.0, "content": "(annotation). and each of the solutions with respect to", "type": "text" }, { "bbox": [ 439, 461, 448, 472 ], "score": 0.85, "content": "Q", "type": "inline_equation" }, { "bbox": [ 448, 460, 505, 473 ], "score": 1.0, "content": "as a potential", "type": "text" } ], "index": 28 }, { "bbox": [ 105, 471, 505, 484 ], "spans": [ { "bbox": [ 105, 471, 143, 484 ], "score": 1.0, "content": "solution,", "type": "text" }, { "bbox": [ 144, 472, 150, 481 ], "score": 0.73, "content": "I", "type": "inline_equation" }, { "bbox": [ 150, 471, 324, 484 ], "score": 1.0, "content": ", (referred to as stable model in ASP). With", "type": "text" }, { "bbox": [ 324, 471, 349, 483 ], "score": 0.92, "content": "I | _ { r ^ { n _ { P } p } }", "type": "inline_equation" }, { "bbox": [ 349, 471, 477, 484 ], "score": 1.0, "content": "we denote the projection of the", "type": "text" }, { "bbox": [ 477, 472, 483, 481 ], "score": 0.8, "content": "I", "type": "inline_equation" }, { "bbox": [ 484, 471, 505, 484 ], "score": 1.0, "content": "onto", "type": "text" } ], "index": 29 }, { "bbox": [ 107, 480, 503, 496 ], "spans": [ { "bbox": [ 107, 482, 126, 492 ], "score": 0.72, "content": "r ^ { n p p }", "type": "inline_equation" }, { "bbox": [ 126, 480, 130, 496 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 131, 482, 204, 494 ], "score": 0.77, "content": "N u m ( I | _ { r ^ { n p p } } , \\Pi ) \\ -", "type": "inline_equation" }, { "bbox": [ 204, 480, 412, 496 ], "score": 1.0, "content": "– the number of the possible solutions of the program", "type": "text" }, { "bbox": [ 412, 483, 421, 492 ], "score": 0.67, "content": "\\Pi", "type": "inline_equation" }, { "bbox": [ 421, 480, 478, 496 ], "score": 1.0, "content": "agreeing with", "type": "text" }, { "bbox": [ 478, 483, 503, 494 ], "score": 0.91, "content": "I | _ { r ^ { n p p } }", "type": "inline_equation" } ], "index": 30 }, { "bbox": [ 105, 492, 505, 506 ], "spans": [ { "bbox": [ 105, 492, 119, 506 ], "score": 1.0, "content": "on", "type": "text" }, { "bbox": [ 120, 494, 139, 504 ], "score": 0.88, "content": "r ^ { n p p }", "type": "inline_equation" }, { "bbox": [ 140, 492, 419, 506 ], "score": 1.0, "content": ". Because we aim to calculate the success probability of the query", "type": "text" }, { "bbox": [ 419, 494, 428, 505 ], "score": 0.84, "content": "Q", "type": "inline_equation" }, { "bbox": [ 428, 492, 505, 506 ], "score": 1.0, "content": ", we formalize the", "type": "text" } ], "index": 31 }, { "bbox": [ 105, 504, 300, 516 ], "spans": [ { "bbox": [ 105, 504, 242, 516 ], "score": 1.0, "content": "probability of a potential solution", "type": "text" }, { "bbox": [ 243, 505, 249, 514 ], "score": 0.78, "content": "I", "type": "inline_equation" }, { "bbox": [ 249, 504, 300, 516 ], "score": 1.0, "content": "beforehand.", "type": "text" } ], "index": 32 } ], "index": 29 }, { "type": "text", "bbox": [ 106, 519, 504, 552 ], "lines": [ { "bbox": [ 105, 519, 505, 532 ], "spans": [ { "bbox": [ 105, 519, 363, 532 ], "score": 1.0, "content": "Definition 2. We specify the probability of the potential solution,", "type": "text" }, { "bbox": [ 364, 520, 370, 529 ], "score": 0.4, "content": "I", "type": "inline_equation" }, { "bbox": [ 370, 519, 437, 532 ], "score": 1.0, "content": ", for the program", "type": "text" }, { "bbox": [ 438, 519, 446, 529 ], "score": 0.31, "content": "\\Pi", "type": "inline_equation" }, { "bbox": [ 447, 519, 505, 532 ], "score": 1.0, "content": "as the product", "type": "text" } ], "index": 33 }, { "bbox": [ 105, 530, 504, 543 ], "spans": [ { "bbox": [ 105, 530, 237, 543 ], "score": 1.0, "content": "of the probabilities of all atoms", "type": "text" }, { "bbox": [ 238, 532, 262, 540 ], "score": 0.86, "content": "c = v", "type": "inline_equation" }, { "bbox": [ 263, 530, 274, 543 ], "score": 1.0, "content": "in", "type": "text" }, { "bbox": [ 275, 531, 299, 542 ], "score": 0.92, "content": "I | _ { r ^ { n p p } }", "type": "inline_equation" }, { "bbox": [ 300, 530, 495, 543 ], "score": 1.0, "content": "divided by the number of potential solutions of", "type": "text" }, { "bbox": [ 495, 532, 504, 540 ], "score": 0.27, "content": "\\Pi", "type": "inline_equation" } ], "index": 34 }, { "bbox": [ 105, 540, 228, 554 ], "spans": [ { "bbox": [ 105, 540, 228, 554 ], "score": 1.0, "content": "agreeing with I|rnpp on rnpp:", "type": "text" } ], "index": 35 } ], "index": 34 }, { "type": "interline_equation", "bbox": [ 171, 558, 439, 592 ], "lines": [ { "bbox": [ 171, 558, 439, 592 ], "spans": [ { "bbox": [ 171, 558, 439, 592 ], "score": 0.92, "content": "P _ { \\Pi } ( I ) = \\left\\{ \\begin{array} { l l } { \\frac { \\prod _ { c = v \\in { I \\vert _ { r } n p p } } P _ { \\Pi } ( c = v ) } { N u m ( I \\vert _ { r ^ { n p p } } , \\Pi ) } , } & { i f I i s a p o t e n t i a l s o l u t i o n o f \\Pi , } \\\\ { 0 , } & { o t h e r w i s e . } \\end{array} \\right.", "type": "interline_equation", "image_path": "8412f1d11746ec4046c38cac73f234346a59ec206ee4309a3774af105b0f2f9a.jpg" } ] } ], "index": 37, "virtual_lines": [ { "bbox": [ 171, 558, 439, 569.3333333333334 ], "spans": [], "index": 36 }, { "bbox": [ 171, 569.3333333333334, 439, 580.6666666666667 ], "spans": [], "index": 37 }, { "bbox": [ 171, 580.6666666666667, 439, 592.0000000000001 ], "spans": [], "index": 38 } ] }, { "type": "text", "bbox": [ 106, 611, 361, 624 ], "lines": [ { "bbox": [ 106, 612, 360, 625 ], "spans": [ { "bbox": [ 106, 612, 360, 625 ], "score": 1.0, "content": "Therefore, the probability of a query can be defined as follows.", "type": "text" } ], "index": 39 } ], "index": 39 }, { "type": "text", "bbox": [ 105, 627, 484, 639 ], "lines": [ { "bbox": [ 105, 626, 484, 641 ], "spans": [ { "bbox": [ 105, 626, 277, 641 ], "score": 1.0, "content": "Definition 3. The probability of the query", "type": "text" }, { "bbox": [ 277, 628, 287, 639 ], "score": 0.83, "content": "Q", "type": "inline_equation" }, { "bbox": [ 287, 626, 424, 641 ], "score": 1.0, "content": "given the set of possible solutions", "type": "text" }, { "bbox": [ 424, 628, 430, 637 ], "score": 0.7, "content": "I", "type": "inline_equation" }, { "bbox": [ 431, 626, 484, 641 ], "score": 1.0, "content": "is defined as", "type": "text" } ], "index": 40 } ], "index": 40 }, { "type": "interline_equation", "bbox": [ 258, 644, 352, 673 ], "lines": [ { "bbox": [ 258, 644, 352, 673 ], "spans": [ { "bbox": [ 258, 644, 352, 673 ], "score": 0.94, "content": "P _ { \\Pi } ( Q ) : = \\sum _ { I \\ v { = } Q } P _ { \\Pi } ( I ) .", "type": "interline_equation", "image_path": "85f8048e7bb05e818b56885781d4a00f2c446c1eab1ad62d131afc2c89182051.jpg" } ] } ], "index": 41.5, "virtual_lines": [ { "bbox": [ 258, 644, 352, 658.5 ], "spans": [], "index": 41 }, { "bbox": [ 258, 658.5, 352, 673.0 ], "spans": [], "index": 42 } ] }, { "type": "text", "bbox": [ 106, 678, 506, 702 ], "lines": [ { "bbox": [ 105, 677, 506, 692 ], "spans": [ { "bbox": [ 105, 677, 142, 692 ], "score": 1.0, "content": "Thereby,", "type": "text" }, { "bbox": [ 142, 679, 171, 690 ], "score": 0.91, "content": "I \\models Q", "type": "inline_equation" }, { "bbox": [ 171, 677, 208, 692 ], "score": 1.0, "content": "reads as", "type": "text" }, { "bbox": [ 209, 679, 219, 689 ], "score": 0.44, "content": "^ { * } I", "type": "inline_equation" }, { "bbox": [ 220, 677, 254, 692 ], "score": 1.0, "content": "satisfies", "type": "text" }, { "bbox": [ 254, 679, 264, 690 ], "score": 0.75, "content": "Q", "type": "inline_equation" }, { "bbox": [ 264, 677, 416, 692 ], "score": 1.0, "content": "”. The probability of the set of queries", "type": "text" }, { "bbox": [ 416, 678, 495, 691 ], "score": 0.93, "content": "\\mathbf { Q } = \\{ Q _ { 1 } , \\ldots , Q _ { l } \\}", "type": "inline_equation" }, { "bbox": [ 495, 677, 506, 692 ], "score": 1.0, "content": "is", "type": "text" } ], "index": 43 }, { "bbox": [ 106, 690, 317, 703 ], "spans": [ { "bbox": [ 106, 690, 317, 703 ], "score": 1.0, "content": "defined as the product of the probability of each. I.e.", "type": "text" } ], "index": 44 } ], "index": 43.5 }, { "type": "interline_equation", "bbox": [ 211, 706, 400, 735 ], "lines": [ { "bbox": [ 211, 706, 400, 735 ], "spans": [ { "bbox": [ 211, 706, 400, 735 ], "score": 0.93, "content": "P _ { \\Pi } \\left( \\mathbf { Q } \\right) : = \\prod _ { Q _ { i } \\in \\mathbf { Q } } P _ { \\Pi } ( Q _ { i } ) = \\prod _ { Q _ { i } \\in \\mathbf { Q } } \\sum _ { I \\ v { | } = Q } P _ { \\Pi } ( I ) .", "type": "interline_equation", "image_path": "8fbd6787ba9865c4142b45115e9ee1a16382065a884dc2a918eaf718c77fae02.jpg" } ] } ], "index": 45.5, "virtual_lines": [ { "bbox": [ 211, 706, 400, 720.5 ], "spans": [], "index": 45 }, { "bbox": [ 211, 720.5, 400, 735.0 ], "spans": [], "index": 46 } ] } ], "page_idx": 4, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 106, 27, 308, 37 ], "lines": [ { "bbox": [ 106, 26, 308, 38 ], "spans": [ { "bbox": [ 106, 26, 308, 38 ], "score": 1.0, "content": "Under review as a conference paper at ICLR 2022", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 302, 751, 309, 760 ], "lines": [ { "bbox": [ 302, 750, 309, 762 ], "spans": [ { "bbox": [ 302, 750, 309, 762 ], "score": 1.0, "content": "5", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "text", "bbox": [ 106, 81, 505, 149 ], "lines": [ { "bbox": [ 105, 81, 506, 96 ], "spans": [ { "bbox": [ 105, 81, 506, 96 ], "score": 1.0, "content": "To summarize, an NPP can consist of neural and/or probabilistic modules and produces query-", "type": "text" } ], "index": 0 }, { "bbox": [ 105, 93, 505, 106 ], "spans": [ { "bbox": [ 105, 93, 505, 106 ], "score": 1.0, "content": "dependent probability estimates. Due to the flexibility of its definition, the term NPP contains the", "type": "text" } ], "index": 1 }, { "bbox": [ 104, 104, 506, 118 ], "spans": [ { "bbox": [ 104, 104, 506, 118 ], "score": 1.0, "content": "predicates of previous works (Manhaeve et al., 2018; Yang et al., 2020), but also more interesting", "type": "text" } ], "index": 2 }, { "bbox": [ 104, 115, 506, 128 ], "spans": [ { "bbox": [ 104, 115, 506, 128 ], "score": 1.0, "content": "predicates discussed above. The specific “flavor” of an NPP should be chosen depending on what", "type": "text" } ], "index": 3 }, { "bbox": [ 105, 126, 506, 139 ], "spans": [ { "bbox": [ 105, 126, 506, 139 ], "score": 1.0, "content": "type of probability estimation is required (cf. Fig 2a). Lastly, NPPs have the unified loss function of", "type": "text" } ], "index": 4 }, { "bbox": [ 106, 137, 218, 150 ], "spans": [ { "bbox": [ 106, 137, 218, 150 ], "score": 1.0, "content": "the negative log-likelihood:", "type": "text" } ], "index": 5 } ], "index": 2.5, "bbox_fs": [ 104, 81, 506, 150 ] }, { "type": "interline_equation", "bbox": [ 111, 154, 504, 187 ], "lines": [ { "bbox": [ 111, 154, 504, 187 ], "spans": [ { "bbox": [ 111, 154, 504, 187 ], "score": 0.93, "content": "L _ { N P P } : = - \\log L H ( x , \\hat { x } ) = \\sum _ { i = 1 } ^ { n } L H ( x _ { i } , \\hat { x } _ { i } ) = - \\sum _ { i = 1 } ^ { n } x _ { i } \\cdot \\log ( P _ { \\xi } ^ { ( X , C ) } ( x _ { i } ) ) = - \\sum _ { i = 1 } ^ { n } \\log ( P _ { \\xi } ^ { ( X , C ) } )", "type": "interline_equation", "image_path": "2df2b15aa55bfa392ca1eea024da8df01763335460315fe384a335f6d03243b3.jpg" } ] } ], "index": 7, "virtual_lines": [ { "bbox": [ 111, 154, 504, 165.0 ], "spans": [], "index": 6 }, { "bbox": [ 111, 165.0, 504, 176.0 ], "spans": [], "index": 7 }, { "bbox": [ 111, 176.0, 504, 187.0 ], "spans": [], "index": 8 } ] }, { "type": "text", "bbox": [ 107, 195, 506, 222 ], "lines": [ { "bbox": [ 105, 194, 505, 208 ], "spans": [ { "bbox": [ 105, 194, 347, 208 ], "score": 1.0, "content": "whereby we are assuming the data to be i.i.d., ground truth", "type": "text" }, { "bbox": [ 347, 197, 357, 207 ], "score": 0.85, "content": "x _ { i }", "type": "inline_equation" }, { "bbox": [ 357, 194, 460, 208 ], "score": 1.0, "content": "to be the all-ones vector,", "type": "text" }, { "bbox": [ 460, 196, 467, 207 ], "score": 0.85, "content": "\\xi", "type": "inline_equation" }, { "bbox": [ 467, 194, 505, 208 ], "score": 1.0, "content": "to be the", "type": "text" } ], "index": 9 }, { "bbox": [ 101, 205, 506, 228 ], "spans": [ { "bbox": [ 101, 205, 216, 228 ], "score": 1.0, "content": "parameters of the NPP and", "type": "text" }, { "bbox": [ 216, 207, 246, 223 ], "score": 0.94, "content": "P _ { \\xi } ^ { ( X , C ) }", "type": "inline_equation" }, { "bbox": [ 247, 205, 324, 228 ], "score": 1.0, "content": "are the predictions", "type": "text" }, { "bbox": [ 324, 210, 334, 220 ], "score": 0.88, "content": "{ \\hat { x } } _ { i }", "type": "inline_equation" }, { "bbox": [ 335, 205, 506, 228 ], "score": 1.0, "content": "obtained from the PC encoded in the NPP.", "type": "text" } ], "index": 10 } ], "index": 9.5, "bbox_fs": [ 101, 194, 506, 228 ] }, { "type": "title", "bbox": [ 107, 235, 313, 247 ], "lines": [ { "bbox": [ 106, 235, 313, 248 ], "spans": [ { "bbox": [ 106, 235, 313, 248 ], "score": 1.0, "content": "3.2 THE SLASH LANGUAGE AND SEMANTICS", "type": "text" } ], "index": 11 } ], "index": 11 }, { "type": "text", "bbox": [ 106, 255, 505, 301 ], "lines": [ { "bbox": [ 105, 256, 505, 269 ], "spans": [ { "bbox": [ 105, 256, 505, 269 ], "score": 1.0, "content": "Fig. 1 presents an illustration of SLASH, exemplified for the task of set prediction, with all of its key", "type": "text" } ], "index": 12 }, { "bbox": [ 105, 267, 505, 280 ], "spans": [ { "bbox": [ 105, 267, 505, 280 ], "score": 1.0, "content": "components. Having introduced the NPPs previously, which produce probability estimates, we now", "type": "text" } ], "index": 13 }, { "bbox": [ 105, 279, 505, 292 ], "spans": [ { "bbox": [ 105, 279, 505, 292 ], "score": 1.0, "content": "continue in the pipeline on how to use these probability estimates for answering logical queries. We", "type": "text" } ], "index": 14 }, { "bbox": [ 105, 289, 295, 302 ], "spans": [ { "bbox": [ 105, 289, 295, 302 ], "score": 1.0, "content": "begin by formally defining a SLASH program.", "type": "text" } ], "index": 15 } ], "index": 13.5, "bbox_fs": [ 105, 256, 505, 302 ] }, { "type": "text", "bbox": [ 106, 304, 505, 338 ], "lines": [ { "bbox": [ 105, 302, 507, 317 ], "spans": [ { "bbox": [ 105, 302, 324, 317 ], "score": 1.0, "content": "Definition 1. A SLASH program Π is the union of", "type": "text" }, { "bbox": [ 324, 304, 345, 315 ], "score": 0.8, "content": "\\Pi ^ { a s p }", "type": "inline_equation" }, { "bbox": [ 345, 302, 350, 317 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 351, 304, 372, 315 ], "score": 0.59, "content": "\\Pi ^ { n p p }", "type": "inline_equation" }, { "bbox": [ 373, 302, 428, 317 ], "score": 1.0, "content": ". Therewith,", "type": "text" }, { "bbox": [ 429, 304, 450, 315 ], "score": 0.85, "content": "\\Pi ^ { a s p }", "type": "inline_equation" }, { "bbox": [ 451, 302, 507, 317 ], "score": 1.0, "content": "is the set of", "type": "text" } ], "index": 16 }, { "bbox": [ 104, 313, 506, 328 ], "spans": [ { "bbox": [ 104, 313, 440, 328 ], "score": 1.0, "content": "propositional rules (standard rules from ASP-Core-2 (Calimeri et al., 2020)), and", "type": "text" }, { "bbox": [ 441, 315, 462, 325 ], "score": 0.8, "content": "\\Pi ^ { n p p }", "type": "inline_equation" }, { "bbox": [ 463, 313, 506, 328 ], "score": 1.0, "content": "is a set of", "type": "text" } ], "index": 17 }, { "bbox": [ 104, 326, 344, 339 ], "spans": [ { "bbox": [ 104, 326, 344, 339 ], "score": 1.0, "content": "Neural-Probabilistic Predicates of the form stated in Eq. 1.", "type": "text" } ], "index": 18 } ], "index": 17, "bbox_fs": [ 104, 302, 507, 339 ] }, { "type": "text", "bbox": [ 106, 346, 505, 414 ], "lines": [ { "bbox": [ 105, 347, 505, 360 ], "spans": [ { "bbox": [ 105, 347, 505, 360 ], "score": 1.0, "content": "Fig. 2b depicts a minimal SLASH program for the task of set prediction, exemplifying a set of", "type": "text" } ], "index": 19 }, { "bbox": [ 105, 358, 505, 371 ], "spans": [ { "bbox": [ 105, 358, 505, 371 ], "score": 1.0, "content": "propositional rules and neural predicates. Similar to NeurASP, SLASH requires ASP and as such", "type": "text" } ], "index": 20 }, { "bbox": [ 106, 370, 505, 381 ], "spans": [ { "bbox": [ 106, 370, 505, 381 ], "score": 1.0, "content": "adopts its syntax to most part. We therefore now address integrating our NPPs into an ASP compatible", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 379, 505, 393 ], "spans": [ { "bbox": [ 105, 379, 505, 393 ], "score": 1.0, "content": "form to obtain the success probability for the logical query given all possible solutions. Thus, we", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 391, 505, 404 ], "spans": [ { "bbox": [ 105, 391, 369, 404 ], "score": 1.0, "content": "define SLASH’s semantics. For SLASH to translate the program", "type": "text" }, { "bbox": [ 369, 391, 378, 401 ], "score": 0.63, "content": "\\Pi", "type": "inline_equation" }, { "bbox": [ 379, 391, 505, 404 ], "score": 1.0, "content": "to the ASP-solver’s compatible", "type": "text" } ], "index": 23 }, { "bbox": [ 106, 403, 339, 414 ], "spans": [ { "bbox": [ 106, 403, 339, 414 ], "score": 1.0, "content": "form, the rules (Eq. 1) will be rewritten to the set of rules:", "type": "text" } ], "index": 24 } ], "index": 21.5, "bbox_fs": [ 105, 347, 505, 414 ] }, { "type": "interline_equation", "bbox": [ 241, 419, 370, 433 ], "lines": [ { "bbox": [ 241, 419, 370, 433 ], "spans": [ { "bbox": [ 241, 419, 370, 433 ], "score": 0.93, "content": "1 \\{ h ( x ) = v _ { 1 } ; \\ldots ; h ( x ) = v _ { n } \\} 1", "type": "interline_equation", "image_path": "e4c76dbbc6e4f0607f36cfe65c5c74b6f382d229457f6b2b9ff4996044984268.jpg" } ] } ], "index": 25, "virtual_lines": [ { "bbox": [ 241, 419, 370, 433 ], "spans": [], "index": 25 } ] }, { "type": "text", "bbox": [ 106, 438, 505, 516 ], "lines": [ { "bbox": [ 105, 438, 505, 450 ], "spans": [ { "bbox": [ 105, 438, 505, 450 ], "score": 1.0, "content": "The ASP-solver should understand this as “Pick exactly one rule from the set”. After the translation", "type": "text" } ], "index": 26 }, { "bbox": [ 106, 450, 505, 461 ], "spans": [ { "bbox": [ 106, 450, 326, 461 ], "score": 1.0, "content": "is done, we can ask an ASP-solver for the solutions for", "type": "text" }, { "bbox": [ 326, 450, 335, 459 ], "score": 0.52, "content": "\\Pi", "type": "inline_equation" }, { "bbox": [ 335, 450, 505, 461 ], "score": 1.0, "content": ". We denote a set of ASP constraints in the", "type": "text" } ], "index": 27 }, { "bbox": [ 107, 460, 505, 473 ], "spans": [ { "bbox": [ 107, 461, 163, 471 ], "score": 0.79, "content": "\\mathrm { f o r m } \\gets B o d y", "type": "inline_equation" }, { "bbox": [ 163, 460, 209, 473 ], "score": 1.0, "content": ", as queries", "type": "text" }, { "bbox": [ 210, 461, 219, 472 ], "score": 0.84, "content": "Q", "type": "inline_equation" }, { "bbox": [ 219, 460, 438, 473 ], "score": 1.0, "content": "(annotation). and each of the solutions with respect to", "type": "text" }, { "bbox": [ 439, 461, 448, 472 ], "score": 0.85, "content": "Q", "type": "inline_equation" }, { "bbox": [ 448, 460, 505, 473 ], "score": 1.0, "content": "as a potential", "type": "text" } ], "index": 28 }, { "bbox": [ 105, 471, 505, 484 ], "spans": [ { "bbox": [ 105, 471, 143, 484 ], "score": 1.0, "content": "solution,", "type": "text" }, { "bbox": [ 144, 472, 150, 481 ], "score": 0.73, "content": "I", "type": "inline_equation" }, { "bbox": [ 150, 471, 324, 484 ], "score": 1.0, "content": ", (referred to as stable model in ASP). With", "type": "text" }, { "bbox": [ 324, 471, 349, 483 ], "score": 0.92, "content": "I | _ { r ^ { n _ { P } p } }", "type": "inline_equation" }, { "bbox": [ 349, 471, 477, 484 ], "score": 1.0, "content": "we denote the projection of the", "type": "text" }, { "bbox": [ 477, 472, 483, 481 ], "score": 0.8, "content": "I", "type": "inline_equation" }, { "bbox": [ 484, 471, 505, 484 ], "score": 1.0, "content": "onto", "type": "text" } ], "index": 29 }, { "bbox": [ 107, 480, 503, 496 ], "spans": [ { "bbox": [ 107, 482, 126, 492 ], "score": 0.72, "content": "r ^ { n p p }", "type": "inline_equation" }, { "bbox": [ 126, 480, 130, 496 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 131, 482, 204, 494 ], "score": 0.77, "content": "N u m ( I | _ { r ^ { n p p } } , \\Pi ) \\ -", "type": "inline_equation" }, { "bbox": [ 204, 480, 412, 496 ], "score": 1.0, "content": "– the number of the possible solutions of the program", "type": "text" }, { "bbox": [ 412, 483, 421, 492 ], "score": 0.67, "content": "\\Pi", "type": "inline_equation" }, { "bbox": [ 421, 480, 478, 496 ], "score": 1.0, "content": "agreeing with", "type": "text" }, { "bbox": [ 478, 483, 503, 494 ], "score": 0.91, "content": "I | _ { r ^ { n p p } }", "type": "inline_equation" } ], "index": 30 }, { "bbox": [ 105, 492, 505, 506 ], "spans": [ { "bbox": [ 105, 492, 119, 506 ], "score": 1.0, "content": "on", "type": "text" }, { "bbox": [ 120, 494, 139, 504 ], "score": 0.88, "content": "r ^ { n p p }", "type": "inline_equation" }, { "bbox": [ 140, 492, 419, 506 ], "score": 1.0, "content": ". Because we aim to calculate the success probability of the query", "type": "text" }, { "bbox": [ 419, 494, 428, 505 ], "score": 0.84, "content": "Q", "type": "inline_equation" }, { "bbox": [ 428, 492, 505, 506 ], "score": 1.0, "content": ", we formalize the", "type": "text" } ], "index": 31 }, { "bbox": [ 105, 504, 300, 516 ], "spans": [ { "bbox": [ 105, 504, 242, 516 ], "score": 1.0, "content": "probability of a potential solution", "type": "text" }, { "bbox": [ 243, 505, 249, 514 ], "score": 0.78, "content": "I", "type": "inline_equation" }, { "bbox": [ 249, 504, 300, 516 ], "score": 1.0, "content": "beforehand.", "type": "text" } ], "index": 32 } ], "index": 29, "bbox_fs": [ 105, 438, 505, 516 ] }, { "type": "text", "bbox": [ 106, 519, 504, 552 ], "lines": [ { "bbox": [ 105, 519, 505, 532 ], "spans": [ { "bbox": [ 105, 519, 363, 532 ], "score": 1.0, "content": "Definition 2. We specify the probability of the potential solution,", "type": "text" }, { "bbox": [ 364, 520, 370, 529 ], "score": 0.4, "content": "I", "type": "inline_equation" }, { "bbox": [ 370, 519, 437, 532 ], "score": 1.0, "content": ", for the program", "type": "text" }, { "bbox": [ 438, 519, 446, 529 ], "score": 0.31, "content": "\\Pi", "type": "inline_equation" }, { "bbox": [ 447, 519, 505, 532 ], "score": 1.0, "content": "as the product", "type": "text" } ], "index": 33 }, { "bbox": [ 105, 530, 504, 543 ], "spans": [ { "bbox": [ 105, 530, 237, 543 ], "score": 1.0, "content": "of the probabilities of all atoms", "type": "text" }, { "bbox": [ 238, 532, 262, 540 ], "score": 0.86, "content": "c = v", "type": "inline_equation" }, { "bbox": [ 263, 530, 274, 543 ], "score": 1.0, "content": "in", "type": "text" }, { "bbox": [ 275, 531, 299, 542 ], "score": 0.92, "content": "I | _ { r ^ { n p p } }", "type": "inline_equation" }, { "bbox": [ 300, 530, 495, 543 ], "score": 1.0, "content": "divided by the number of potential solutions of", "type": "text" }, { "bbox": [ 495, 532, 504, 540 ], "score": 0.27, "content": "\\Pi", "type": "inline_equation" } ], "index": 34 }, { "bbox": [ 105, 540, 228, 554 ], "spans": [ { "bbox": [ 105, 540, 228, 554 ], "score": 1.0, "content": "agreeing with I|rnpp on rnpp:", "type": "text" } ], "index": 35 } ], "index": 34, "bbox_fs": [ 105, 519, 505, 554 ] }, { "type": "interline_equation", "bbox": [ 171, 558, 439, 592 ], "lines": [ { "bbox": [ 171, 558, 439, 592 ], "spans": [ { "bbox": [ 171, 558, 439, 592 ], "score": 0.92, "content": "P _ { \\Pi } ( I ) = \\left\\{ \\begin{array} { l l } { \\frac { \\prod _ { c = v \\in { I \\vert _ { r } n p p } } P _ { \\Pi } ( c = v ) } { N u m ( I \\vert _ { r ^ { n p p } } , \\Pi ) } , } & { i f I i s a p o t e n t i a l s o l u t i o n o f \\Pi , } \\\\ { 0 , } & { o t h e r w i s e . } \\end{array} \\right.", "type": "interline_equation", "image_path": "8412f1d11746ec4046c38cac73f234346a59ec206ee4309a3774af105b0f2f9a.jpg" } ] } ], "index": 37, "virtual_lines": [ { "bbox": [ 171, 558, 439, 569.3333333333334 ], "spans": [], "index": 36 }, { "bbox": [ 171, 569.3333333333334, 439, 580.6666666666667 ], "spans": [], "index": 37 }, { "bbox": [ 171, 580.6666666666667, 439, 592.0000000000001 ], "spans": [], "index": 38 } ] }, { "type": "text", "bbox": [ 106, 611, 361, 624 ], "lines": [ { "bbox": [ 106, 612, 360, 625 ], "spans": [ { "bbox": [ 106, 612, 360, 625 ], "score": 1.0, "content": "Therefore, the probability of a query can be defined as follows.", "type": "text" } ], "index": 39 } ], "index": 39, "bbox_fs": [ 106, 612, 360, 625 ] }, { "type": "text", "bbox": [ 105, 627, 484, 639 ], "lines": [ { "bbox": [ 105, 626, 484, 641 ], "spans": [ { "bbox": [ 105, 626, 277, 641 ], "score": 1.0, "content": "Definition 3. The probability of the query", "type": "text" }, { "bbox": [ 277, 628, 287, 639 ], "score": 0.83, "content": "Q", "type": "inline_equation" }, { "bbox": [ 287, 626, 424, 641 ], "score": 1.0, "content": "given the set of possible solutions", "type": "text" }, { "bbox": [ 424, 628, 430, 637 ], "score": 0.7, "content": "I", "type": "inline_equation" }, { "bbox": [ 431, 626, 484, 641 ], "score": 1.0, "content": "is defined as", "type": "text" } ], "index": 40 } ], "index": 40, "bbox_fs": [ 105, 626, 484, 641 ] }, { "type": "interline_equation", "bbox": [ 258, 644, 352, 673 ], "lines": [ { "bbox": [ 258, 644, 352, 673 ], "spans": [ { "bbox": [ 258, 644, 352, 673 ], "score": 0.94, "content": "P _ { \\Pi } ( Q ) : = \\sum _ { I \\ v { = } Q } P _ { \\Pi } ( I ) .", "type": "interline_equation", "image_path": "85f8048e7bb05e818b56885781d4a00f2c446c1eab1ad62d131afc2c89182051.jpg" } ] } ], "index": 41.5, "virtual_lines": [ { "bbox": [ 258, 644, 352, 658.5 ], "spans": [], "index": 41 }, { "bbox": [ 258, 658.5, 352, 673.0 ], "spans": [], "index": 42 } ] }, { "type": "text", "bbox": [ 106, 678, 506, 702 ], "lines": [ { "bbox": [ 105, 677, 506, 692 ], "spans": [ { "bbox": [ 105, 677, 142, 692 ], "score": 1.0, "content": "Thereby,", "type": "text" }, { "bbox": [ 142, 679, 171, 690 ], "score": 0.91, "content": "I \\models Q", "type": "inline_equation" }, { "bbox": [ 171, 677, 208, 692 ], "score": 1.0, "content": "reads as", "type": "text" }, { "bbox": [ 209, 679, 219, 689 ], "score": 0.44, "content": "^ { * } I", "type": "inline_equation" }, { "bbox": [ 220, 677, 254, 692 ], "score": 1.0, "content": "satisfies", "type": "text" }, { "bbox": [ 254, 679, 264, 690 ], "score": 0.75, "content": "Q", "type": "inline_equation" }, { "bbox": [ 264, 677, 416, 692 ], "score": 1.0, "content": "”. The probability of the set of queries", "type": "text" }, { "bbox": [ 416, 678, 495, 691 ], "score": 0.93, "content": "\\mathbf { Q } = \\{ Q _ { 1 } , \\ldots , Q _ { l } \\}", "type": "inline_equation" }, { "bbox": [ 495, 677, 506, 692 ], "score": 1.0, "content": "is", "type": "text" } ], "index": 43 }, { "bbox": [ 106, 690, 317, 703 ], "spans": [ { "bbox": [ 106, 690, 317, 703 ], "score": 1.0, "content": "defined as the product of the probability of each. I.e.", "type": "text" } ], "index": 44 } ], "index": 43.5, "bbox_fs": [ 105, 677, 506, 703 ] }, { "type": "interline_equation", "bbox": [ 211, 706, 400, 735 ], "lines": [ { "bbox": [ 211, 706, 400, 735 ], "spans": [ { "bbox": [ 211, 706, 400, 735 ], "score": 0.93, "content": "P _ { \\Pi } \\left( \\mathbf { Q } \\right) : = \\prod _ { Q _ { i } \\in \\mathbf { Q } } P _ { \\Pi } ( Q _ { i } ) = \\prod _ { Q _ { i } \\in \\mathbf { Q } } \\sum _ { I \\ v { | } = Q } P _ { \\Pi } ( I ) .", "type": "interline_equation", "image_path": "8fbd6787ba9865c4142b45115e9ee1a16382065a884dc2a918eaf718c77fae02.jpg" } ] } ], "index": 45.5, "virtual_lines": [ { "bbox": [ 211, 706, 400, 720.5 ], "spans": [], "index": 45 }, { "bbox": [ 211, 720.5, 400, 735.0 ], "spans": [], "index": 46 } ] } ] }, { "preproc_blocks": [ { "type": "title", "bbox": [ 108, 82, 282, 94 ], "lines": [ { "bbox": [ 106, 82, 283, 95 ], "spans": [ { "bbox": [ 106, 82, 283, 95 ], "score": 1.0, "content": "3.3 PARAMETER LEARNING IN SLASH", "type": "text" } ], "index": 0 } ], "index": 0 }, { "type": "text", "bbox": [ 106, 102, 505, 170 ], "lines": [ { "bbox": [ 105, 102, 505, 116 ], "spans": [ { "bbox": [ 105, 102, 169, 116 ], "score": 1.0, "content": "We denote with", "type": "text" }, { "bbox": [ 169, 103, 191, 115 ], "score": 0.86, "content": "\\Pi ( \\pmb \\theta )", "type": "inline_equation" }, { "bbox": [ 192, 102, 390, 116 ], "score": 1.0, "content": "the SLASH program under consideration, thereby", "type": "text" }, { "bbox": [ 390, 104, 397, 113 ], "score": 0.77, "content": "\\pmb \\theta", "type": "inline_equation" }, { "bbox": [ 398, 102, 505, 116 ], "score": 1.0, "content": "is the set of the parameters", "type": "text" } ], "index": 1 }, { "bbox": [ 105, 114, 505, 127 ], "spans": [ { "bbox": [ 105, 114, 403, 127 ], "score": 1.0, "content": "associated with Π. Further, making the i.i.d. assumption of the query set", "type": "text" }, { "bbox": [ 403, 114, 414, 126 ], "score": 0.49, "content": "\\mathbf { Q }", "type": "inline_equation" }, { "bbox": [ 414, 114, 505, 127 ], "score": 1.0, "content": ", we follow Manhaeve", "type": "text" } ], "index": 2 }, { "bbox": [ 106, 125, 505, 138 ], "spans": [ { "bbox": [ 106, 125, 505, 138 ], "score": 1.0, "content": "et al. (2018) and Skryagin et al. (2020), and use the learning from entailment setting. That is, the", "type": "text" } ], "index": 3 }, { "bbox": [ 105, 135, 505, 150 ], "spans": [ { "bbox": [ 105, 135, 460, 150 ], "score": 1.0, "content": "training examples are logical queries that are known to be true in the SLASH program", "type": "text" }, { "bbox": [ 460, 136, 482, 148 ], "score": 0.74, "content": "\\Pi ( \\theta )", "type": "inline_equation" }, { "bbox": [ 482, 135, 505, 150 ], "score": 1.0, "content": ". The", "type": "text" } ], "index": 4 }, { "bbox": [ 105, 146, 505, 161 ], "spans": [ { "bbox": [ 105, 146, 249, 161 ], "score": 1.0, "content": "goal is now to learn the parameters", "type": "text" }, { "bbox": [ 249, 147, 256, 157 ], "score": 0.78, "content": "\\pmb \\theta", "type": "inline_equation" }, { "bbox": [ 257, 146, 353, 161 ], "score": 1.0, "content": "of the SLASH program", "type": "text" }, { "bbox": [ 353, 147, 375, 159 ], "score": 0.85, "content": "\\Pi ( \\pmb \\theta )", "type": "inline_equation" }, { "bbox": [ 376, 146, 505, 161 ], "score": 1.0, "content": "so that the observed queries are", "type": "text" } ], "index": 5 }, { "bbox": [ 105, 156, 155, 172 ], "spans": [ { "bbox": [ 105, 156, 155, 172 ], "score": 1.0, "content": "most likely.", "type": "text" } ], "index": 6 } ], "index": 3.5 }, { "type": "text", "bbox": [ 107, 174, 505, 210 ], "lines": [ { "bbox": [ 105, 173, 505, 188 ], "spans": [ { "bbox": [ 105, 173, 505, 188 ], "score": 1.0, "content": "To this end, we employ the negative log-likelihood and the cross-entropy of the observed queries", "type": "text" } ], "index": 7 }, { "bbox": [ 106, 185, 507, 200 ], "spans": [ { "bbox": [ 106, 186, 150, 199 ], "score": 0.88, "content": "P _ { \\mathrm { { I I } } ( \\theta ) } ( Q _ { i } )", "type": "inline_equation" }, { "bbox": [ 150, 185, 312, 200 ], "score": 1.0, "content": "and their predicted probability value", "type": "text" }, { "bbox": [ 312, 186, 371, 199 ], "score": 0.93, "content": "P ^ { ( X _ { \\mathbf { Q } } , C ) } ( x _ { Q _ { i } } )", "type": "inline_equation" }, { "bbox": [ 372, 185, 507, 200 ], "score": 1.0, "content": ", assuming the NPPs are fixed:", "type": "text" } ], "index": 8 }, { "bbox": [ 106, 200, 149, 210 ], "spans": [ { "bbox": [ 106, 200, 149, 210 ], "score": 0.78, "content": "L _ { E N T } : =", "type": "inline_equation" } ], "index": 9 } ], "index": 8 }, { "type": "interline_equation", "bbox": [ 109, 214, 490, 248 ], "lines": [ { "bbox": [ 109, 214, 490, 248 ], "spans": [ { "bbox": [ 109, 214, 490, 248 ], "score": 0.94, "content": "- \\log L H \\left( \\log ( P _ { \\Pi ( \\theta ) } ( \\mathbf { Q } ) ) , P ^ { ( X _ { \\mathbf { Q } } , C ) } ( x _ { \\mathbf { Q } } ) \\right) = - \\sum _ { j = 1 } ^ { m } \\log ( P _ { \\Pi ( \\theta ) } ( Q _ { i j } ) ) \\cdot \\log \\left( P ^ { ( X _ { \\mathbf { Q } } , C ) } ( x _ { Q _ { i j } } ) \\right) .", "type": "interline_equation", "image_path": "73189bdbb324a7a5de4d157ea631cf184596aa216a9062f2a7d12c533ba1b87c.jpg" } ] } ], "index": 11, "virtual_lines": [ { "bbox": [ 109, 214, 490, 225.33333333333334 ], "spans": [], "index": 10 }, { "bbox": [ 109, 225.33333333333334, 490, 236.66666666666669 ], "spans": [], "index": 11 }, { "bbox": [ 109, 236.66666666666669, 490, 248.00000000000003 ], "spans": [], "index": 12 } ] }, { "type": "text", "bbox": [ 106, 253, 506, 299 ], "lines": [ { "bbox": [ 106, 253, 505, 265 ], "spans": [ { "bbox": [ 106, 253, 505, 265 ], "score": 1.0, "content": "This loss function aims at maximizing the estimated success probability. We remark that the defined", "type": "text" } ], "index": 13 }, { "bbox": [ 105, 263, 506, 277 ], "spans": [ { "bbox": [ 105, 263, 506, 277 ], "score": 1.0, "content": "loss function is true regardless of the NPP’s form (NN with Softmax, PC or PC jointly with NN). The", "type": "text" } ], "index": 14 }, { "bbox": [ 105, 274, 506, 289 ], "spans": [ { "bbox": [ 105, 274, 282, 289 ], "score": 1.0, "content": "only difference will be the second term, i.e.", "type": "text" }, { "bbox": [ 282, 276, 339, 289 ], "score": 0.92, "content": "P ^ { ( C | X _ { \\mathbf { Q } } ) } ( x _ { \\mathbf { Q } } )", "type": "inline_equation" }, { "bbox": [ 339, 274, 351, 289 ], "score": 1.0, "content": "or", "type": "text" }, { "bbox": [ 352, 275, 411, 289 ], "score": 0.91, "content": "P ^ { ( X _ { \\mathbf { Q } } | C ) } ( x _ { \\mathbf { Q } } ) )", "type": "inline_equation" }, { "bbox": [ 411, 274, 506, 289 ], "score": 1.0, "content": "depending on the NPP", "type": "text" } ], "index": 15 }, { "bbox": [ 106, 288, 383, 299 ], "spans": [ { "bbox": [ 106, 288, 347, 299 ], "score": 1.0, "content": "and task. Furthermore, we assume that for the set of queries", "type": "text" }, { "bbox": [ 348, 288, 357, 299 ], "score": 0.8, "content": "\\mathbf { Q }", "type": "inline_equation" }, { "bbox": [ 358, 288, 383, 299 ], "score": 1.0, "content": "holds", "type": "text" } ], "index": 16 } ], "index": 14.5 }, { "type": "interline_equation", "bbox": [ 251, 304, 360, 319 ], "lines": [ { "bbox": [ 251, 304, 360, 319 ], "spans": [ { "bbox": [ 251, 304, 360, 319 ], "score": 0.92, "content": "P _ { \\Pi ( \\pmb \\theta ) } ( Q ) > 0 \\quad \\forall Q \\in { \\bf Q } .", "type": "interline_equation", "image_path": "38f649600323f2cd8117cf3828720d0ed67716c406e417f4cd2dd9149045732f.jpg" } ] } ], "index": 17, "virtual_lines": [ { "bbox": [ 251, 304, 360, 319 ], "spans": [], "index": 17 } ] }, { "type": "text", "bbox": [ 107, 323, 505, 357 ], "lines": [ { "bbox": [ 105, 321, 506, 338 ], "spans": [ { "bbox": [ 105, 321, 398, 338 ], "score": 1.0, "content": "In accordance with the semantics, we seek to reward the right solutions", "type": "text" }, { "bbox": [ 398, 326, 422, 334 ], "score": 0.86, "content": "v = c", "type": "inline_equation" }, { "bbox": [ 423, 321, 506, 338 ], "score": 1.0, "content": "and penalize wrong", "type": "text" } ], "index": 18 }, { "bbox": [ 105, 334, 506, 348 ], "spans": [ { "bbox": [ 105, 334, 127, 348 ], "score": 1.0, "content": "ones", "type": "text" }, { "bbox": [ 127, 335, 151, 347 ], "score": 0.9, "content": "v \\neq c", "type": "inline_equation" }, { "bbox": [ 151, 334, 282, 348 ], "score": 1.0, "content": ". Referring to the probabilities in", "type": "text" }, { "bbox": [ 282, 335, 302, 345 ], "score": 0.86, "content": "r ^ { n p p }", "type": "inline_equation" }, { "bbox": [ 302, 334, 506, 348 ], "score": 1.0, "content": "(the set of logical rules denoting NPPs, see Def. 2)", "type": "text" } ], "index": 19 }, { "bbox": [ 105, 346, 384, 359 ], "spans": [ { "bbox": [ 105, 346, 117, 359 ], "score": 1.0, "content": "as", "type": "text" }, { "bbox": [ 117, 348, 125, 357 ], "score": 0.69, "content": "\\mathbf { p }", "type": "inline_equation" }, { "bbox": [ 125, 346, 281, 359 ], "score": 1.0, "content": ", one can compute their gradients w.r.t.", "type": "text" }, { "bbox": [ 281, 347, 288, 356 ], "score": 0.8, "content": "\\pmb { \\theta }", "type": "inline_equation" }, { "bbox": [ 288, 346, 384, 359 ], "score": 1.0, "content": "via backpropagation as", "type": "text" } ], "index": 20 } ], "index": 19 }, { "type": "interline_equation", "bbox": [ 189, 362, 424, 397 ], "lines": [ { "bbox": [ 189, 362, 424, 397 ], "spans": [ { "bbox": [ 189, 362, 424, 397 ], "score": 0.94, "content": "\\sum _ { Q \\in { \\bf Q } } \\frac { \\partial \\log \\left( P _ { \\Pi ( \\pmb \\theta ) } ( Q ) \\right) } { \\partial \\pmb \\theta } = \\sum _ { Q \\in { \\bf Q } } \\frac { \\partial \\log \\left( P _ { \\Pi ( \\pmb \\theta ) } ( Q ) \\right) } { \\partial { \\bf p } } \\times \\frac { \\partial { \\bf p } } { \\partial \\pmb \\theta } .", "type": "interline_equation", "image_path": "c71b1b57c842f235633b5397660cc117df3a103d2b93a4c5089f527d193524eb.jpg" } ] } ], "index": 21.5, "virtual_lines": [ { "bbox": [ 189, 362, 424, 379.5 ], "spans": [], "index": 21 }, { "bbox": [ 189, 379.5, 424, 397.0 ], "spans": [], "index": 22 } ] }, { "type": "text", "bbox": [ 108, 403, 504, 445 ], "lines": [ { "bbox": [ 106, 400, 506, 419 ], "spans": [ { "bbox": [ 106, 400, 145, 419 ], "score": 1.0, "content": "The term", "type": "text" }, { "bbox": [ 145, 402, 158, 417 ], "score": 0.9, "content": "\\textstyle { \\frac { \\partial \\mathbf { p } } { \\partial \\theta } }", "type": "inline_equation" }, { "bbox": [ 158, 400, 506, 419 ], "score": 1.0, "content": "can now be computed as usual via backward propagation through the NPPs (see Eq. 13", "type": "text" } ], "index": 23 }, { "bbox": [ 105, 413, 504, 427 ], "spans": [ { "bbox": [ 105, 413, 263, 427 ], "score": 1.0, "content": "in the appendix for details). By letting", "type": "text" }, { "bbox": [ 263, 417, 270, 426 ], "score": 0.79, "content": "p", "type": "inline_equation" }, { "bbox": [ 270, 413, 448, 427 ], "score": 1.0, "content": "to be the label of the probability of an atom", "type": "text" }, { "bbox": [ 449, 416, 473, 425 ], "score": 0.88, "content": "c = v", "type": "inline_equation" }, { "bbox": [ 473, 413, 484, 427 ], "score": 1.0, "content": "in", "type": "text" }, { "bbox": [ 484, 415, 504, 425 ], "score": 0.87, "content": "r ^ { n p p }", "type": "inline_equation" } ], "index": 24 }, { "bbox": [ 106, 426, 499, 446 ], "spans": [ { "bbox": [ 106, 431, 160, 444 ], "score": 1.0, "content": "and denoting", "type": "text" }, { "bbox": [ 161, 430, 216, 444 ], "score": 0.91, "content": "P _ { \\Pi ( \\pmb { \\theta } ) } ( c = v )", "type": "inline_equation" }, { "bbox": [ 216, 431, 255, 444 ], "score": 1.0, "content": ", the term", "type": "text" }, { "bbox": [ 256, 426, 315, 446 ], "score": 0.94, "content": "\\frac { \\partial \\log \\left( P _ { \\mathrm { I I } ( \\pmb { \\theta } ) } ( Q ) \\right) } { \\partial \\mathbf { p } }", "type": "inline_equation" }, { "bbox": [ 313, 429, 499, 444 ], "score": 1.0, "content": "follows from NeurASP (Yang et al., 2020) as", "type": "text" } ], "index": 25 } ], "index": 24 }, { "type": "interline_equation", "bbox": [ 171, 450, 440, 509 ], "lines": [ { "bbox": [ 171, 450, 440, 509 ], "spans": [ { "bbox": [ 171, 450, 440, 509 ], "score": 0.95, "content": "\\frac { \\partial \\log \\left( P _ { \\Pi ( \\theta ) } ( Q ) \\right) } { \\partial \\mathbf { p } } = \\frac { I \\underset { I : I \\mid = Q } { \\sum } \\frac { P _ { \\Pi ( \\theta ) } ( I ) } { P _ { \\Pi ( \\theta ) } ( c = v ) } - \\underset { I : I , v ^ { \\prime } \\mid = Q } { \\sum } \\frac { P _ { \\Pi ( \\theta ) } ( I ) } { P _ { \\Pi ( \\theta ) } ( c = v ^ { \\prime } ) } } { \\underset { I : I \\mid = Q } { \\sum } P _ { \\Pi ( \\theta ) } ( I ) } .", "type": "interline_equation", "image_path": "16ead2545e3a3b0901ab36b2cc5263e0781b7c70ac9da887d966ab918f15d447.jpg" } ] } ], "index": 27, "virtual_lines": [ { "bbox": [ 171, 450, 440, 469.6666666666667 ], "spans": [], "index": 26 }, { "bbox": [ 171, 469.6666666666667, 440, 489.33333333333337 ], "spans": [], "index": 27 }, { "bbox": [ 171, 489.33333333333337, 440, 509.00000000000006 ], "spans": [], "index": 28 } ] }, { "type": "text", "bbox": [ 105, 520, 505, 542 ], "lines": [ { "bbox": [ 105, 519, 507, 533 ], "spans": [ { "bbox": [ 105, 519, 507, 533 ], "score": 1.0, "content": "This is sensible. For instance, if a query to be true is not likely to be entailed, the gradient is positive.", "type": "text" } ], "index": 29 }, { "bbox": [ 106, 530, 318, 543 ], "spans": [ { "bbox": [ 106, 530, 318, 543 ], "score": 1.0, "content": "Putting everything together, the final loss function is", "type": "text" } ], "index": 30 } ], "index": 29.5 }, { "type": "interline_equation", "bbox": [ 246, 549, 364, 561 ], "lines": [ { "bbox": [ 246, 549, 364, 561 ], "spans": [ { "bbox": [ 246, 549, 364, 561 ], "score": 0.9, "content": "{ \\cal L } _ { S L A S H } = { \\cal L } _ { N P P } + { \\cal L } _ { E N T }", "type": "interline_equation", "image_path": "f145bb6a4a4cfb00ec9261d4033d6ebfe1f151e47554c538aff178796d7c3820.jpg" } ] } ], "index": 31, "virtual_lines": [ { "bbox": [ 246, 549, 364, 561 ], "spans": [], "index": 31 } ] }, { "type": "text", "bbox": [ 105, 568, 505, 590 ], "lines": [ { "bbox": [ 106, 568, 505, 580 ], "spans": [ { "bbox": [ 106, 568, 505, 580 ], "score": 1.0, "content": "and we perform training using coordinate descent, i.e., we train the NPPs, the train the program with", "type": "text" } ], "index": 32 }, { "bbox": [ 105, 578, 355, 591 ], "spans": [ { "bbox": [ 105, 578, 355, 591 ], "score": 1.0, "content": "fixed NPPs, train the NPPs with the program fixed, and so on.", "type": "text" } ], "index": 33 } ], "index": 32.5 }, { "type": "text", "bbox": [ 106, 595, 505, 640 ], "lines": [ { "bbox": [ 105, 595, 505, 607 ], "spans": [ { "bbox": [ 105, 595, 505, 607 ], "score": 1.0, "content": "In hindsight, rather than requiring a novel loss function for each individual task and data set, with", "type": "text" } ], "index": 34 }, { "bbox": [ 106, 606, 505, 619 ], "spans": [ { "bbox": [ 106, 606, 505, 619 ], "score": 1.0, "content": "SLASH, it is possible to simply incorporate the specific requirements into the logic program. The", "type": "text" } ], "index": 35 }, { "bbox": [ 105, 617, 505, 631 ], "spans": [ { "bbox": [ 105, 617, 505, 631 ], "score": 1.0, "content": "training loss, however, remains the same. We refer to the Appendix A for further details, including", "type": "text" } ], "index": 36 }, { "bbox": [ 106, 629, 264, 641 ], "spans": [ { "bbox": [ 106, 629, 264, 641 ], "score": 1.0, "content": "the derivation of the total loss gradient.", "type": "text" } ], "index": 37 } ], "index": 35.5 }, { "type": "title", "bbox": [ 108, 656, 234, 669 ], "lines": [ { "bbox": [ 105, 655, 235, 671 ], "spans": [ { "bbox": [ 105, 655, 235, 671 ], "score": 1.0, "content": "4 EMPIRICAL RESULTS", "type": "text" } ], "index": 38 } ], "index": 38 }, { "type": "text", "bbox": [ 106, 681, 503, 704 ], "lines": [ { "bbox": [ 105, 680, 505, 694 ], "spans": [ { "bbox": [ 105, 680, 505, 694 ], "score": 1.0, "content": "The advantage of SLASH lies in the efficient integration of neural, probabilistic and symbolic", "type": "text" } ], "index": 39 }, { "bbox": [ 106, 693, 443, 705 ], "spans": [ { "bbox": [ 106, 693, 443, 705 ], "score": 1.0, "content": "computations. To emphasize this, we conduct a variety of experimental evaluations.", "type": "text" } ], "index": 40 } ], "index": 39.5 }, { "type": "text", "bbox": [ 106, 709, 503, 732 ], "lines": [ { "bbox": [ 106, 709, 505, 722 ], "spans": [ { "bbox": [ 106, 709, 505, 722 ], "score": 1.0, "content": "Experimental Details. We use two benchmark data sets, namely MNIST (LeCun et al., 1998b) for", "type": "text" } ], "index": 41 }, { "bbox": [ 106, 720, 505, 733 ], "spans": [ { "bbox": [ 106, 720, 505, 733 ], "score": 1.0, "content": "the task of MNIST-Addition and a variant of the ShapeWorld data set (Kuhnle & Copestake, 2017) for", "type": "text" } ], "index": 42 } ], "index": 41.5 } ], "page_idx": 5, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 107, 26, 308, 37 ], "lines": [ { "bbox": [ 106, 25, 308, 38 ], "spans": [ { "bbox": [ 106, 25, 308, 38 ], "score": 1.0, "content": "Under review as a conference paper at ICLR 2022", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 302, 751, 309, 760 ], "lines": [ { "bbox": [ 302, 750, 310, 761 ], "spans": [ { "bbox": [ 302, 750, 310, 761 ], "score": 1.0, "content": "6", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "title", "bbox": [ 108, 82, 282, 94 ], "lines": [ { "bbox": [ 106, 82, 283, 95 ], "spans": [ { "bbox": [ 106, 82, 283, 95 ], "score": 1.0, "content": "3.3 PARAMETER LEARNING IN SLASH", "type": "text" } ], "index": 0 } ], "index": 0 }, { "type": "text", "bbox": [ 106, 102, 505, 170 ], "lines": [ { "bbox": [ 105, 102, 505, 116 ], "spans": [ { "bbox": [ 105, 102, 169, 116 ], "score": 1.0, "content": "We denote with", "type": "text" }, { "bbox": [ 169, 103, 191, 115 ], "score": 0.86, "content": "\\Pi ( \\pmb \\theta )", "type": "inline_equation" }, { "bbox": [ 192, 102, 390, 116 ], "score": 1.0, "content": "the SLASH program under consideration, thereby", "type": "text" }, { "bbox": [ 390, 104, 397, 113 ], "score": 0.77, "content": "\\pmb \\theta", "type": "inline_equation" }, { "bbox": [ 398, 102, 505, 116 ], "score": 1.0, "content": "is the set of the parameters", "type": "text" } ], "index": 1 }, { "bbox": [ 105, 114, 505, 127 ], "spans": [ { "bbox": [ 105, 114, 403, 127 ], "score": 1.0, "content": "associated with Π. Further, making the i.i.d. assumption of the query set", "type": "text" }, { "bbox": [ 403, 114, 414, 126 ], "score": 0.49, "content": "\\mathbf { Q }", "type": "inline_equation" }, { "bbox": [ 414, 114, 505, 127 ], "score": 1.0, "content": ", we follow Manhaeve", "type": "text" } ], "index": 2 }, { "bbox": [ 106, 125, 505, 138 ], "spans": [ { "bbox": [ 106, 125, 505, 138 ], "score": 1.0, "content": "et al. (2018) and Skryagin et al. (2020), and use the learning from entailment setting. That is, the", "type": "text" } ], "index": 3 }, { "bbox": [ 105, 135, 505, 150 ], "spans": [ { "bbox": [ 105, 135, 460, 150 ], "score": 1.0, "content": "training examples are logical queries that are known to be true in the SLASH program", "type": "text" }, { "bbox": [ 460, 136, 482, 148 ], "score": 0.74, "content": "\\Pi ( \\theta )", "type": "inline_equation" }, { "bbox": [ 482, 135, 505, 150 ], "score": 1.0, "content": ". The", "type": "text" } ], "index": 4 }, { "bbox": [ 105, 146, 505, 161 ], "spans": [ { "bbox": [ 105, 146, 249, 161 ], "score": 1.0, "content": "goal is now to learn the parameters", "type": "text" }, { "bbox": [ 249, 147, 256, 157 ], "score": 0.78, "content": "\\pmb \\theta", "type": "inline_equation" }, { "bbox": [ 257, 146, 353, 161 ], "score": 1.0, "content": "of the SLASH program", "type": "text" }, { "bbox": [ 353, 147, 375, 159 ], "score": 0.85, "content": "\\Pi ( \\pmb \\theta )", "type": "inline_equation" }, { "bbox": [ 376, 146, 505, 161 ], "score": 1.0, "content": "so that the observed queries are", "type": "text" } ], "index": 5 }, { "bbox": [ 105, 156, 155, 172 ], "spans": [ { "bbox": [ 105, 156, 155, 172 ], "score": 1.0, "content": "most likely.", "type": "text" } ], "index": 6 } ], "index": 3.5, "bbox_fs": [ 105, 102, 505, 172 ] }, { "type": "text", "bbox": [ 107, 174, 505, 210 ], "lines": [ { "bbox": [ 105, 173, 505, 188 ], "spans": [ { "bbox": [ 105, 173, 505, 188 ], "score": 1.0, "content": "To this end, we employ the negative log-likelihood and the cross-entropy of the observed queries", "type": "text" } ], "index": 7 }, { "bbox": [ 106, 185, 507, 200 ], "spans": [ { "bbox": [ 106, 186, 150, 199 ], "score": 0.88, "content": "P _ { \\mathrm { { I I } } ( \\theta ) } ( Q _ { i } )", "type": "inline_equation" }, { "bbox": [ 150, 185, 312, 200 ], "score": 1.0, "content": "and their predicted probability value", "type": "text" }, { "bbox": [ 312, 186, 371, 199 ], "score": 0.93, "content": "P ^ { ( X _ { \\mathbf { Q } } , C ) } ( x _ { Q _ { i } } )", "type": "inline_equation" }, { "bbox": [ 372, 185, 507, 200 ], "score": 1.0, "content": ", assuming the NPPs are fixed:", "type": "text" } ], "index": 8 }, { "bbox": [ 106, 200, 149, 210 ], "spans": [ { "bbox": [ 106, 200, 149, 210 ], "score": 0.78, "content": "L _ { E N T } : =", "type": "inline_equation" } ], "index": 9 } ], "index": 8, "bbox_fs": [ 105, 173, 507, 210 ] }, { "type": "interline_equation", "bbox": [ 109, 214, 490, 248 ], "lines": [ { "bbox": [ 109, 214, 490, 248 ], "spans": [ { "bbox": [ 109, 214, 490, 248 ], "score": 0.94, "content": "- \\log L H \\left( \\log ( P _ { \\Pi ( \\theta ) } ( \\mathbf { Q } ) ) , P ^ { ( X _ { \\mathbf { Q } } , C ) } ( x _ { \\mathbf { Q } } ) \\right) = - \\sum _ { j = 1 } ^ { m } \\log ( P _ { \\Pi ( \\theta ) } ( Q _ { i j } ) ) \\cdot \\log \\left( P ^ { ( X _ { \\mathbf { Q } } , C ) } ( x _ { Q _ { i j } } ) \\right) .", "type": "interline_equation", "image_path": "73189bdbb324a7a5de4d157ea631cf184596aa216a9062f2a7d12c533ba1b87c.jpg" } ] } ], "index": 11, "virtual_lines": [ { "bbox": [ 109, 214, 490, 225.33333333333334 ], "spans": [], "index": 10 }, { "bbox": [ 109, 225.33333333333334, 490, 236.66666666666669 ], "spans": [], "index": 11 }, { "bbox": [ 109, 236.66666666666669, 490, 248.00000000000003 ], "spans": [], "index": 12 } ] }, { "type": "text", "bbox": [ 106, 253, 506, 299 ], "lines": [ { "bbox": [ 106, 253, 505, 265 ], "spans": [ { "bbox": [ 106, 253, 505, 265 ], "score": 1.0, "content": "This loss function aims at maximizing the estimated success probability. We remark that the defined", "type": "text" } ], "index": 13 }, { "bbox": [ 105, 263, 506, 277 ], "spans": [ { "bbox": [ 105, 263, 506, 277 ], "score": 1.0, "content": "loss function is true regardless of the NPP’s form (NN with Softmax, PC or PC jointly with NN). The", "type": "text" } ], "index": 14 }, { "bbox": [ 105, 274, 506, 289 ], "spans": [ { "bbox": [ 105, 274, 282, 289 ], "score": 1.0, "content": "only difference will be the second term, i.e.", "type": "text" }, { "bbox": [ 282, 276, 339, 289 ], "score": 0.92, "content": "P ^ { ( C | X _ { \\mathbf { Q } } ) } ( x _ { \\mathbf { Q } } )", "type": "inline_equation" }, { "bbox": [ 339, 274, 351, 289 ], "score": 1.0, "content": "or", "type": "text" }, { "bbox": [ 352, 275, 411, 289 ], "score": 0.91, "content": "P ^ { ( X _ { \\mathbf { Q } } | C ) } ( x _ { \\mathbf { Q } } ) )", "type": "inline_equation" }, { "bbox": [ 411, 274, 506, 289 ], "score": 1.0, "content": "depending on the NPP", "type": "text" } ], "index": 15 }, { "bbox": [ 106, 288, 383, 299 ], "spans": [ { "bbox": [ 106, 288, 347, 299 ], "score": 1.0, "content": "and task. Furthermore, we assume that for the set of queries", "type": "text" }, { "bbox": [ 348, 288, 357, 299 ], "score": 0.8, "content": "\\mathbf { Q }", "type": "inline_equation" }, { "bbox": [ 358, 288, 383, 299 ], "score": 1.0, "content": "holds", "type": "text" } ], "index": 16 } ], "index": 14.5, "bbox_fs": [ 105, 253, 506, 299 ] }, { "type": "interline_equation", "bbox": [ 251, 304, 360, 319 ], "lines": [ { "bbox": [ 251, 304, 360, 319 ], "spans": [ { "bbox": [ 251, 304, 360, 319 ], "score": 0.92, "content": "P _ { \\Pi ( \\pmb \\theta ) } ( Q ) > 0 \\quad \\forall Q \\in { \\bf Q } .", "type": "interline_equation", "image_path": "38f649600323f2cd8117cf3828720d0ed67716c406e417f4cd2dd9149045732f.jpg" } ] } ], "index": 17, "virtual_lines": [ { "bbox": [ 251, 304, 360, 319 ], "spans": [], "index": 17 } ] }, { "type": "text", "bbox": [ 107, 323, 505, 357 ], "lines": [ { "bbox": [ 105, 321, 506, 338 ], "spans": [ { "bbox": [ 105, 321, 398, 338 ], "score": 1.0, "content": "In accordance with the semantics, we seek to reward the right solutions", "type": "text" }, { "bbox": [ 398, 326, 422, 334 ], "score": 0.86, "content": "v = c", "type": "inline_equation" }, { "bbox": [ 423, 321, 506, 338 ], "score": 1.0, "content": "and penalize wrong", "type": "text" } ], "index": 18 }, { "bbox": [ 105, 334, 506, 348 ], "spans": [ { "bbox": [ 105, 334, 127, 348 ], "score": 1.0, "content": "ones", "type": "text" }, { "bbox": [ 127, 335, 151, 347 ], "score": 0.9, "content": "v \\neq c", "type": "inline_equation" }, { "bbox": [ 151, 334, 282, 348 ], "score": 1.0, "content": ". Referring to the probabilities in", "type": "text" }, { "bbox": [ 282, 335, 302, 345 ], "score": 0.86, "content": "r ^ { n p p }", "type": "inline_equation" }, { "bbox": [ 302, 334, 506, 348 ], "score": 1.0, "content": "(the set of logical rules denoting NPPs, see Def. 2)", "type": "text" } ], "index": 19 }, { "bbox": [ 105, 346, 384, 359 ], "spans": [ { "bbox": [ 105, 346, 117, 359 ], "score": 1.0, "content": "as", "type": "text" }, { "bbox": [ 117, 348, 125, 357 ], "score": 0.69, "content": "\\mathbf { p }", "type": "inline_equation" }, { "bbox": [ 125, 346, 281, 359 ], "score": 1.0, "content": ", one can compute their gradients w.r.t.", "type": "text" }, { "bbox": [ 281, 347, 288, 356 ], "score": 0.8, "content": "\\pmb { \\theta }", "type": "inline_equation" }, { "bbox": [ 288, 346, 384, 359 ], "score": 1.0, "content": "via backpropagation as", "type": "text" } ], "index": 20 } ], "index": 19, "bbox_fs": [ 105, 321, 506, 359 ] }, { "type": "interline_equation", "bbox": [ 189, 362, 424, 397 ], "lines": [ { "bbox": [ 189, 362, 424, 397 ], "spans": [ { "bbox": [ 189, 362, 424, 397 ], "score": 0.94, "content": "\\sum _ { Q \\in { \\bf Q } } \\frac { \\partial \\log \\left( P _ { \\Pi ( \\pmb \\theta ) } ( Q ) \\right) } { \\partial \\pmb \\theta } = \\sum _ { Q \\in { \\bf Q } } \\frac { \\partial \\log \\left( P _ { \\Pi ( \\pmb \\theta ) } ( Q ) \\right) } { \\partial { \\bf p } } \\times \\frac { \\partial { \\bf p } } { \\partial \\pmb \\theta } .", "type": "interline_equation", "image_path": "c71b1b57c842f235633b5397660cc117df3a103d2b93a4c5089f527d193524eb.jpg" } ] } ], "index": 21.5, "virtual_lines": [ { "bbox": [ 189, 362, 424, 379.5 ], "spans": [], "index": 21 }, { "bbox": [ 189, 379.5, 424, 397.0 ], "spans": [], "index": 22 } ] }, { "type": "text", "bbox": [ 108, 403, 504, 445 ], "lines": [ { "bbox": [ 106, 400, 506, 419 ], "spans": [ { "bbox": [ 106, 400, 145, 419 ], "score": 1.0, "content": "The term", "type": "text" }, { "bbox": [ 145, 402, 158, 417 ], "score": 0.9, "content": "\\textstyle { \\frac { \\partial \\mathbf { p } } { \\partial \\theta } }", "type": "inline_equation" }, { "bbox": [ 158, 400, 506, 419 ], "score": 1.0, "content": "can now be computed as usual via backward propagation through the NPPs (see Eq. 13", "type": "text" } ], "index": 23 }, { "bbox": [ 105, 413, 504, 427 ], "spans": [ { "bbox": [ 105, 413, 263, 427 ], "score": 1.0, "content": "in the appendix for details). By letting", "type": "text" }, { "bbox": [ 263, 417, 270, 426 ], "score": 0.79, "content": "p", "type": "inline_equation" }, { "bbox": [ 270, 413, 448, 427 ], "score": 1.0, "content": "to be the label of the probability of an atom", "type": "text" }, { "bbox": [ 449, 416, 473, 425 ], "score": 0.88, "content": "c = v", "type": "inline_equation" }, { "bbox": [ 473, 413, 484, 427 ], "score": 1.0, "content": "in", "type": "text" }, { "bbox": [ 484, 415, 504, 425 ], "score": 0.87, "content": "r ^ { n p p }", "type": "inline_equation" } ], "index": 24 }, { "bbox": [ 106, 426, 499, 446 ], "spans": [ { "bbox": [ 106, 431, 160, 444 ], "score": 1.0, "content": "and denoting", "type": "text" }, { "bbox": [ 161, 430, 216, 444 ], "score": 0.91, "content": "P _ { \\Pi ( \\pmb { \\theta } ) } ( c = v )", "type": "inline_equation" }, { "bbox": [ 216, 431, 255, 444 ], "score": 1.0, "content": ", the term", "type": "text" }, { "bbox": [ 256, 426, 315, 446 ], "score": 0.94, "content": "\\frac { \\partial \\log \\left( P _ { \\mathrm { I I } ( \\pmb { \\theta } ) } ( Q ) \\right) } { \\partial \\mathbf { p } }", "type": "inline_equation" }, { "bbox": [ 313, 429, 499, 444 ], "score": 1.0, "content": "follows from NeurASP (Yang et al., 2020) as", "type": "text" } ], "index": 25 } ], "index": 24, "bbox_fs": [ 105, 400, 506, 446 ] }, { "type": "interline_equation", "bbox": [ 171, 450, 440, 509 ], "lines": [ { "bbox": [ 171, 450, 440, 509 ], "spans": [ { "bbox": [ 171, 450, 440, 509 ], "score": 0.95, "content": "\\frac { \\partial \\log \\left( P _ { \\Pi ( \\theta ) } ( Q ) \\right) } { \\partial \\mathbf { p } } = \\frac { I \\underset { I : I \\mid = Q } { \\sum } \\frac { P _ { \\Pi ( \\theta ) } ( I ) } { P _ { \\Pi ( \\theta ) } ( c = v ) } - \\underset { I : I , v ^ { \\prime } \\mid = Q } { \\sum } \\frac { P _ { \\Pi ( \\theta ) } ( I ) } { P _ { \\Pi ( \\theta ) } ( c = v ^ { \\prime } ) } } { \\underset { I : I \\mid = Q } { \\sum } P _ { \\Pi ( \\theta ) } ( I ) } .", "type": "interline_equation", "image_path": "16ead2545e3a3b0901ab36b2cc5263e0781b7c70ac9da887d966ab918f15d447.jpg" } ] } ], "index": 27, "virtual_lines": [ { "bbox": [ 171, 450, 440, 469.6666666666667 ], "spans": [], "index": 26 }, { "bbox": [ 171, 469.6666666666667, 440, 489.33333333333337 ], "spans": [], "index": 27 }, { "bbox": [ 171, 489.33333333333337, 440, 509.00000000000006 ], "spans": [], "index": 28 } ] }, { "type": "text", "bbox": [ 105, 520, 505, 542 ], "lines": [ { "bbox": [ 105, 519, 507, 533 ], "spans": [ { "bbox": [ 105, 519, 507, 533 ], "score": 1.0, "content": "This is sensible. For instance, if a query to be true is not likely to be entailed, the gradient is positive.", "type": "text" } ], "index": 29 }, { "bbox": [ 106, 530, 318, 543 ], "spans": [ { "bbox": [ 106, 530, 318, 543 ], "score": 1.0, "content": "Putting everything together, the final loss function is", "type": "text" } ], "index": 30 } ], "index": 29.5, "bbox_fs": [ 105, 519, 507, 543 ] }, { "type": "interline_equation", "bbox": [ 246, 549, 364, 561 ], "lines": [ { "bbox": [ 246, 549, 364, 561 ], "spans": [ { "bbox": [ 246, 549, 364, 561 ], "score": 0.9, "content": "{ \\cal L } _ { S L A S H } = { \\cal L } _ { N P P } + { \\cal L } _ { E N T }", "type": "interline_equation", "image_path": "f145bb6a4a4cfb00ec9261d4033d6ebfe1f151e47554c538aff178796d7c3820.jpg" } ] } ], "index": 31, "virtual_lines": [ { "bbox": [ 246, 549, 364, 561 ], "spans": [], "index": 31 } ] }, { "type": "text", "bbox": [ 105, 568, 505, 590 ], "lines": [ { "bbox": [ 106, 568, 505, 580 ], "spans": [ { "bbox": [ 106, 568, 505, 580 ], "score": 1.0, "content": "and we perform training using coordinate descent, i.e., we train the NPPs, the train the program with", "type": "text" } ], "index": 32 }, { "bbox": [ 105, 578, 355, 591 ], "spans": [ { "bbox": [ 105, 578, 355, 591 ], "score": 1.0, "content": "fixed NPPs, train the NPPs with the program fixed, and so on.", "type": "text" } ], "index": 33 } ], "index": 32.5, "bbox_fs": [ 105, 568, 505, 591 ] }, { "type": "text", "bbox": [ 106, 595, 505, 640 ], "lines": [ { "bbox": [ 105, 595, 505, 607 ], "spans": [ { "bbox": [ 105, 595, 505, 607 ], "score": 1.0, "content": "In hindsight, rather than requiring a novel loss function for each individual task and data set, with", "type": "text" } ], "index": 34 }, { "bbox": [ 106, 606, 505, 619 ], "spans": [ { "bbox": [ 106, 606, 505, 619 ], "score": 1.0, "content": "SLASH, it is possible to simply incorporate the specific requirements into the logic program. The", "type": "text" } ], "index": 35 }, { "bbox": [ 105, 617, 505, 631 ], "spans": [ { "bbox": [ 105, 617, 505, 631 ], "score": 1.0, "content": "training loss, however, remains the same. We refer to the Appendix A for further details, including", "type": "text" } ], "index": 36 }, { "bbox": [ 106, 629, 264, 641 ], "spans": [ { "bbox": [ 106, 629, 264, 641 ], "score": 1.0, "content": "the derivation of the total loss gradient.", "type": "text" } ], "index": 37 } ], "index": 35.5, "bbox_fs": [ 105, 595, 505, 641 ] }, { "type": "title", "bbox": [ 108, 656, 234, 669 ], "lines": [ { "bbox": [ 105, 655, 235, 671 ], "spans": [ { "bbox": [ 105, 655, 235, 671 ], "score": 1.0, "content": "4 EMPIRICAL RESULTS", "type": "text" } ], "index": 38 } ], "index": 38 }, { "type": "text", "bbox": [ 106, 681, 503, 704 ], "lines": [ { "bbox": [ 105, 680, 505, 694 ], "spans": [ { "bbox": [ 105, 680, 505, 694 ], "score": 1.0, "content": "The advantage of SLASH lies in the efficient integration of neural, probabilistic and symbolic", "type": "text" } ], "index": 39 }, { "bbox": [ 106, 693, 443, 705 ], "spans": [ { "bbox": [ 106, 693, 443, 705 ], "score": 1.0, "content": "computations. To emphasize this, we conduct a variety of experimental evaluations.", "type": "text" } ], "index": 40 } ], "index": 39.5, "bbox_fs": [ 105, 680, 505, 705 ] }, { "type": "text", "bbox": [ 106, 709, 503, 732 ], "lines": [ { "bbox": [ 106, 709, 505, 722 ], "spans": [ { "bbox": [ 106, 709, 505, 722 ], "score": 1.0, "content": "Experimental Details. We use two benchmark data sets, namely MNIST (LeCun et al., 1998b) for", "type": "text" } ], "index": 41 }, { "bbox": [ 106, 720, 505, 733 ], "spans": [ { "bbox": [ 106, 720, 505, 733 ], "score": 1.0, "content": "the task of MNIST-Addition and a variant of the ShapeWorld data set (Kuhnle & Copestake, 2017) for", "type": "text" } ], "index": 42 } ], "index": 41.5, "bbox_fs": [ 106, 709, 505, 733 ] } ] }, { "preproc_blocks": [ { "type": "text", "bbox": [ 106, 80, 505, 147 ], "lines": [ { "bbox": [ 105, 80, 506, 93 ], "spans": [ { "bbox": [ 105, 80, 506, 93 ], "score": 1.0, "content": "Table 1: MNIST Addition Results. Test accuracy corresponds to the percentage of correctly classified", "type": "text" } ], "index": 0 }, { "bbox": [ 105, 90, 505, 104 ], "spans": [ { "bbox": [ 105, 90, 505, 104 ], "score": 1.0, "content": "test images. (a) Test accuracies in percent for the MNIST Addition task with various DPPLs, including", "type": "text" } ], "index": 1 }, { "bbox": [ 105, 101, 505, 115 ], "spans": [ { "bbox": [ 105, 101, 505, 115 ], "score": 1.0, "content": "SLASH with an NPP that models the joint probabilities (SLASH (PC)) and one that models only", "type": "text" } ], "index": 2 }, { "bbox": [ 106, 113, 505, 126 ], "spans": [ { "bbox": [ 106, 113, 505, 126 ], "score": 1.0, "content": "conditional probabilities (SLASH (DNN)). (b) Test accuracies in percent for the MNIST Addition", "type": "text" } ], "index": 3 }, { "bbox": [ 105, 124, 505, 137 ], "spans": [ { "bbox": [ 105, 124, 505, 137 ], "score": 1.0, "content": "task with missing data, comparing DeepProbLog with SLASH (PC). The amount of missing data was", "type": "text" } ], "index": 4 }, { "bbox": [ 105, 134, 324, 149 ], "spans": [ { "bbox": [ 105, 134, 168, 149 ], "score": 1.0, "content": "varied between", "type": "text" }, { "bbox": [ 169, 135, 189, 145 ], "score": 0.85, "content": "50 \\%", "type": "inline_equation" }, { "bbox": [ 189, 134, 206, 149 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 207, 135, 226, 145 ], "score": 0.86, "content": "9 7 \\%", "type": "inline_equation" }, { "bbox": [ 227, 134, 324, 149 ], "score": 1.0, "content": "of the pixels per image.", "type": "text" } ], "index": 5 } ], "index": 2.5 }, { "type": "table", "bbox": [ 119, 171, 274, 232 ], "blocks": [ { "type": "table_caption", "bbox": [ 140, 156, 250, 166 ], "group_id": 0, "lines": [ { "bbox": [ 139, 155, 251, 167 ], "spans": [ { "bbox": [ 139, 155, 251, 167 ], "score": 1.0, "content": "(a) Baseline MNIST Addition.", "type": "text" } ], "index": 6 } ], "index": 6 }, { "type": "table_body", "bbox": [ 119, 171, 274, 232 ], "group_id": 0, "lines": [ { "bbox": [ 119, 171, 274, 232 ], "spans": [ { "bbox": [ 119, 171, 274, 232 ], "score": 0.972, "html": "
Test Acc. (%)
DeepProbLog98.49±0.18
NeurASP98.21 ± 0.30
SLASH (PC)95.39 ± 0.29
SLASH (DNN)98.74±0.21
", "type": "table", "image_path": "3788e55de7e7cfea5762aea98d8f13b0b9a4135918a954edc0d22b46010214d8.jpg" } ] } ], "index": 11.0, "virtual_lines": [ { "bbox": [ 119, 171, 274, 186.25 ], "spans": [], "index": 8 }, { "bbox": [ 119, 186.25, 274, 201.5 ], "spans": [], "index": 10 }, { "bbox": [ 119, 201.5, 274, 216.75 ], "spans": [], "index": 12 }, { "bbox": [ 119, 216.75, 274, 232.0 ], "spans": [], "index": 14 } ] } ], "index": 8.5 }, { "type": "table", "bbox": [ 309, 171, 491, 232 ], "blocks": [ { "type": "table_caption", "bbox": [ 334, 156, 461, 167 ], "group_id": 1, "lines": [ { "bbox": [ 333, 155, 462, 168 ], "spans": [ { "bbox": [ 333, 155, 462, 168 ], "score": 1.0, "content": "(b) Missing data MNIST Addition.", "type": "text" } ], "index": 7 } ], "index": 7 }, { "type": "table_body", "bbox": [ 309, 171, 491, 232 ], "group_id": 1, "lines": [ { "bbox": [ 309, 171, 491, 232 ], "spans": [ { "bbox": [ 309, 171, 491, 232 ], "score": 0.975, "html": "
DeepProbLogSLASH (PC)
50%97.73 ± 0.1297.67±0.12
80%76.07 ± 18.3896.72士0.05
90%69.15 ± 29.1594.85士0.38
97%32.46 ± 22.4882.57士4.66
", "type": "table", "image_path": "3c1f935051a7d747c53cdcf34ea4560b6fea51387ad5d49e7ef95ee58f377bb2.jpg" } ] } ], "index": 12.0, "virtual_lines": [ { "bbox": [ 309, 171, 491, 186.25 ], "spans": [], "index": 9 }, { "bbox": [ 309, 186.25, 491, 201.5 ], "spans": [], "index": 11 }, { "bbox": [ 309, 201.5, 491, 216.75 ], "spans": [], "index": 13 }, { "bbox": [ 309, 216.75, 491, 232.0 ], "spans": [], "index": 15 } ] } ], "index": 9.5 }, { "type": "text", "bbox": [ 108, 257, 504, 279 ], "lines": [ { "bbox": [ 106, 258, 505, 269 ], "spans": [ { "bbox": [ 106, 258, 505, 269 ], "score": 1.0, "content": "object-centric set prediction. For all experiments we present the average and the standard deviation", "type": "text" } ], "index": 16 }, { "bbox": [ 105, 267, 388, 281 ], "spans": [ { "bbox": [ 105, 267, 388, 281 ], "score": 1.0, "content": "over five runs with different random seeds for parameter initialization.", "type": "text" } ], "index": 17 } ], "index": 16.5 }, { "type": "text", "bbox": [ 107, 285, 505, 341 ], "lines": [ { "bbox": [ 105, 285, 506, 298 ], "spans": [ { "bbox": [ 105, 285, 506, 298 ], "score": 1.0, "content": "For ShapeWorld experiments, we generate a data set we refer to as ShapeWorld4. Images of", "type": "text" } ], "index": 18 }, { "bbox": [ 105, 295, 506, 309 ], "spans": [ { "bbox": [ 105, 295, 506, 309 ], "score": 1.0, "content": "ShapeWorld4 contain between one and four objects, with each object consisting of four attributes:", "type": "text" } ], "index": 19 }, { "bbox": [ 104, 306, 505, 321 ], "spans": [ { "bbox": [ 104, 306, 505, 321 ], "score": 1.0, "content": "a color (red, blue, green, gray, brown, magenta, cyan or yellow), a shade (bright, or dark), a shape", "type": "text" } ], "index": 20 }, { "bbox": [ 105, 318, 505, 330 ], "spans": [ { "bbox": [ 105, 318, 505, 330 ], "score": 1.0, "content": "(circle, triangle or square) and a size (small or big). Thus, each object can be created from 84 different", "type": "text" } ], "index": 21 }, { "bbox": [ 106, 329, 349, 342 ], "spans": [ { "bbox": [ 106, 329, 349, 342 ], "score": 1.0, "content": "combinations of attributes. Fig. 1 depicts an example image.", "type": "text" } ], "index": 22 } ], "index": 20 }, { "type": "text", "bbox": [ 107, 345, 505, 390 ], "lines": [ { "bbox": [ 105, 345, 507, 358 ], "spans": [ { "bbox": [ 105, 345, 507, 358 ], "score": 1.0, "content": "We measure performance via classification accuracies in the MNIST-Addition task. In our Shape-", "type": "text" } ], "index": 23 }, { "bbox": [ 105, 356, 506, 369 ], "spans": [ { "bbox": [ 105, 356, 506, 369 ], "score": 1.0, "content": "World4 experiments, we present the average precision. We refer to appendix B for the SLASH", "type": "text" } ], "index": 24 }, { "bbox": [ 105, 368, 507, 380 ], "spans": [ { "bbox": [ 105, 368, 507, 380 ], "score": 1.0, "content": "programs and queries of each experiment, and appendix C for a detailed description of hyperparame-", "type": "text" } ], "index": 25 }, { "bbox": [ 106, 379, 200, 390 ], "spans": [ { "bbox": [ 106, 379, 200, 390 ], "score": 1.0, "content": "ters and further details.", "type": "text" } ], "index": 26 } ], "index": 24.5 }, { "type": "text", "bbox": [ 106, 395, 505, 451 ], "lines": [ { "bbox": [ 105, 395, 507, 408 ], "spans": [ { "bbox": [ 105, 395, 507, 408 ], "score": 1.0, "content": "Evaluation 1: SLASH outperforms SOTA DPPLs in MNIST-Addition. The task of MNIST-", "type": "text" } ], "index": 27 }, { "bbox": [ 105, 406, 506, 420 ], "spans": [ { "bbox": [ 105, 406, 506, 420 ], "score": 1.0, "content": "Addition (Manhaeve et al., 2018) is to predict the sum of two MNIST digits, presented only as raw", "type": "text" } ], "index": 28 }, { "bbox": [ 105, 417, 506, 430 ], "spans": [ { "bbox": [ 105, 417, 506, 430 ], "score": 1.0, "content": "images. During test time, however, a model should classify the images directly. Thus, although a", "type": "text" } ], "index": 29 }, { "bbox": [ 105, 429, 505, 442 ], "spans": [ { "bbox": [ 105, 429, 505, 442 ], "score": 1.0, "content": "model does not receive explicit information about the depicted digits, it must learn to identify digits", "type": "text" } ], "index": 30 }, { "bbox": [ 105, 439, 283, 453 ], "spans": [ { "bbox": [ 105, 439, 283, 453 ], "score": 1.0, "content": "via indirect feedback on the sum prediction.", "type": "text" } ], "index": 31 } ], "index": 29 }, { "type": "text", "bbox": [ 106, 456, 505, 523 ], "lines": [ { "bbox": [ 105, 455, 505, 470 ], "spans": [ { "bbox": [ 105, 455, 505, 470 ], "score": 1.0, "content": "We compare the test accuracy after convergence between the three DPPLs: DeepProbLog (Manhaeve", "type": "text" } ], "index": 32 }, { "bbox": [ 104, 466, 506, 481 ], "spans": [ { "bbox": [ 104, 466, 506, 481 ], "score": 1.0, "content": "et al., 2018), NeurASP (Yang et al., 2020) and SLASH, using a probabilistic circuit (PC) or a deep", "type": "text" } ], "index": 33 }, { "bbox": [ 105, 478, 505, 491 ], "spans": [ { "bbox": [ 105, 478, 505, 491 ], "score": 1.0, "content": "neural network (DNN) as NPP. Notably, the DNN used in SLASH (DNN) is the LeNet5 model", "type": "text" } ], "index": 34 }, { "bbox": [ 105, 489, 505, 502 ], "spans": [ { "bbox": [ 105, 489, 505, 502 ], "score": 1.0, "content": "(LeCun et al., 1998a) of DeepProbLog and NeurASP. We note that when using the PC as NPP, we", "type": "text" } ], "index": 35 }, { "bbox": [ 105, 500, 506, 514 ], "spans": [ { "bbox": [ 105, 500, 302, 514 ], "score": 1.0, "content": "have also extracted conditional class probabilities", "type": "text" }, { "bbox": [ 303, 500, 339, 512 ], "score": 0.92, "content": "P ( C | X )", "type": "inline_equation" }, { "bbox": [ 339, 500, 484, 514 ], "score": 1.0, "content": ", by marginalizing the class variables", "type": "text" }, { "bbox": [ 485, 501, 494, 510 ], "score": 0.84, "content": "C", "type": "inline_equation" }, { "bbox": [ 494, 500, 506, 514 ], "score": 1.0, "content": "to", "type": "text" } ], "index": 36 }, { "bbox": [ 106, 511, 475, 524 ], "spans": [ { "bbox": [ 106, 511, 246, 524 ], "score": 1.0, "content": "acquire the normalization constant", "type": "text" }, { "bbox": [ 247, 511, 272, 523 ], "score": 0.92, "content": "P ( X )", "type": "inline_equation" }, { "bbox": [ 272, 511, 331, 524 ], "score": 1.0, "content": "from the joint", "type": "text" }, { "bbox": [ 331, 512, 368, 523 ], "score": 0.93, "content": "P ( X , C )", "type": "inline_equation" }, { "bbox": [ 368, 511, 435, 524 ], "score": 1.0, "content": ", and calculating", "type": "text" }, { "bbox": [ 435, 511, 471, 523 ], "score": 0.93, "content": "P ( X | C )", "type": "inline_equation" }, { "bbox": [ 471, 511, 475, 524 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 37 } ], "index": 34.5 }, { "type": "text", "bbox": [ 106, 528, 505, 628 ], "lines": [ { "bbox": [ 106, 528, 505, 540 ], "spans": [ { "bbox": [ 106, 528, 505, 540 ], "score": 1.0, "content": "The results can be seen in Tab. 1a. We observe that training SLASH with a DNN NPP produces", "type": "text" } ], "index": 38 }, { "bbox": [ 106, 540, 506, 551 ], "spans": [ { "bbox": [ 106, 540, 506, 551 ], "score": 1.0, "content": "SOTA accuracies compared to DeepProbLog and NeurASP, confirming that SLASH’s batch-wise", "type": "text" } ], "index": 39 }, { "bbox": [ 106, 551, 506, 563 ], "spans": [ { "bbox": [ 106, 551, 506, 563 ], "score": 1.0, "content": "loss computation leads to improved performances. We further observe that the test accuracy of", "type": "text" } ], "index": 40 }, { "bbox": [ 105, 560, 506, 574 ], "spans": [ { "bbox": [ 105, 560, 506, 574 ], "score": 1.0, "content": "SLASH with a PC NPP is slightly below the other DPPLs, however we argue that this may be", "type": "text" } ], "index": 41 }, { "bbox": [ 105, 572, 506, 585 ], "spans": [ { "bbox": [ 105, 572, 506, 585 ], "score": 1.0, "content": "since a PC, in comparison to a DNN, is learning a true mixture density rather than just conditional", "type": "text" } ], "index": 42 }, { "bbox": [ 106, 583, 506, 596 ], "spans": [ { "bbox": [ 106, 583, 506, 596 ], "score": 1.0, "content": "probabilities. The advantages of doing so will be investigated in the next experiments. Note that,", "type": "text" } ], "index": 43 }, { "bbox": [ 105, 594, 505, 607 ], "spans": [ { "bbox": [ 105, 594, 505, 607 ], "score": 1.0, "content": "optimal architecture search for PCs, e.g. for computer vision, is an open research question.These", "type": "text" } ], "index": 44 }, { "bbox": [ 106, 605, 505, 617 ], "spans": [ { "bbox": [ 106, 605, 505, 617 ], "score": 1.0, "content": "evaluations show SLASH’s advantages on the benchmark MNIST-Addition task. Additional benefits", "type": "text" } ], "index": 45 }, { "bbox": [ 105, 616, 301, 629 ], "spans": [ { "bbox": [ 105, 616, 301, 629 ], "score": 1.0, "content": "will be made clear in the following experiments.", "type": "text" } ], "index": 46 } ], "index": 42 }, { "type": "text", "bbox": [ 106, 632, 505, 732 ], "lines": [ { "bbox": [ 105, 632, 505, 646 ], "spans": [ { "bbox": [ 105, 632, 505, 646 ], "score": 1.0, "content": "Evaluation 2: Handling Missing Data with SLASH. SLASH offers the advantage of its flexibility", "type": "text" } ], "index": 47 }, { "bbox": [ 105, 643, 505, 657 ], "spans": [ { "bbox": [ 105, 643, 505, 657 ], "score": 1.0, "content": "to use various kinds of NPPs. Thus, in comparison to previous DPPLs, one can easily integrate", "type": "text" } ], "index": 48 }, { "bbox": [ 105, 655, 505, 668 ], "spans": [ { "bbox": [ 105, 655, 505, 668 ], "score": 1.0, "content": "NPPs into SLASH that perform joint probability estimation. For this evaluation, we consider the", "type": "text" } ], "index": 49 }, { "bbox": [ 106, 666, 505, 678 ], "spans": [ { "bbox": [ 106, 666, 505, 678 ], "score": 1.0, "content": "task of MNIST-Addition with missing data. We trained SLASH (PC) and DeepProbLog with the", "type": "text" } ], "index": 50 }, { "bbox": [ 105, 676, 506, 690 ], "spans": [ { "bbox": [ 105, 676, 506, 690 ], "score": 1.0, "content": "MNIST-Addition task with images in which a percentage of pixels per image has been removed. It is", "type": "text" } ], "index": 51 }, { "bbox": [ 105, 688, 505, 701 ], "spans": [ { "bbox": [ 105, 688, 505, 701 ], "score": 1.0, "content": "important to mention here that whereas DeepProbLog handles the missing data simply as background", "type": "text" } ], "index": 52 }, { "bbox": [ 106, 699, 505, 712 ], "spans": [ { "bbox": [ 106, 699, 505, 712 ], "score": 1.0, "content": "pixels, SLASH (PC) specifically models the missing data as uncertain data by marginalizing the", "type": "text" } ], "index": 53 }, { "bbox": [ 105, 709, 505, 723 ], "spans": [ { "bbox": [ 105, 709, 505, 723 ], "score": 1.0, "content": "denoted pixels at inference time. We use DeepProbLog here representative of DPPLs without true", "type": "text" } ], "index": 54 }, { "bbox": [ 106, 721, 183, 733 ], "spans": [ { "bbox": [ 106, 721, 183, 733 ], "score": 1.0, "content": "density estimation.", "type": "text" } ], "index": 55 } ], "index": 51 } ], "page_idx": 6, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 107, 27, 308, 37 ], "lines": [ { "bbox": [ 107, 26, 308, 38 ], "spans": [ { "bbox": [ 107, 26, 308, 38 ], "score": 1.0, "content": "Under review as a conference paper at ICLR 2022", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 302, 751, 308, 759 ], "lines": [ { "bbox": [ 302, 750, 309, 762 ], "spans": [ { "bbox": [ 302, 750, 309, 762 ], "score": 1.0, "content": "7", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "text", "bbox": [ 106, 80, 505, 147 ], "lines": [ { "bbox": [ 105, 80, 506, 93 ], "spans": [ { "bbox": [ 105, 80, 506, 93 ], "score": 1.0, "content": "Table 1: MNIST Addition Results. Test accuracy corresponds to the percentage of correctly classified", "type": "text" } ], "index": 0 }, { "bbox": [ 105, 90, 505, 104 ], "spans": [ { "bbox": [ 105, 90, 505, 104 ], "score": 1.0, "content": "test images. (a) Test accuracies in percent for the MNIST Addition task with various DPPLs, including", "type": "text" } ], "index": 1 }, { "bbox": [ 105, 101, 505, 115 ], "spans": [ { "bbox": [ 105, 101, 505, 115 ], "score": 1.0, "content": "SLASH with an NPP that models the joint probabilities (SLASH (PC)) and one that models only", "type": "text" } ], "index": 2 }, { "bbox": [ 106, 113, 505, 126 ], "spans": [ { "bbox": [ 106, 113, 505, 126 ], "score": 1.0, "content": "conditional probabilities (SLASH (DNN)). (b) Test accuracies in percent for the MNIST Addition", "type": "text" } ], "index": 3 }, { "bbox": [ 105, 124, 505, 137 ], "spans": [ { "bbox": [ 105, 124, 505, 137 ], "score": 1.0, "content": "task with missing data, comparing DeepProbLog with SLASH (PC). The amount of missing data was", "type": "text" } ], "index": 4 }, { "bbox": [ 105, 134, 324, 149 ], "spans": [ { "bbox": [ 105, 134, 168, 149 ], "score": 1.0, "content": "varied between", "type": "text" }, { "bbox": [ 169, 135, 189, 145 ], "score": 0.85, "content": "50 \\%", "type": "inline_equation" }, { "bbox": [ 189, 134, 206, 149 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 207, 135, 226, 145 ], "score": 0.86, "content": "9 7 \\%", "type": "inline_equation" }, { "bbox": [ 227, 134, 324, 149 ], "score": 1.0, "content": "of the pixels per image.", "type": "text" } ], "index": 5 } ], "index": 2.5, "bbox_fs": [ 105, 80, 506, 149 ] }, { "type": "table", "bbox": [ 119, 171, 274, 232 ], "blocks": [ { "type": "table_caption", "bbox": [ 140, 156, 250, 166 ], "group_id": 0, "lines": [ { "bbox": [ 139, 155, 251, 167 ], "spans": [ { "bbox": [ 139, 155, 251, 167 ], "score": 1.0, "content": "(a) Baseline MNIST Addition.", "type": "text" } ], "index": 6 } ], "index": 6 }, { "type": "table_body", "bbox": [ 119, 171, 274, 232 ], "group_id": 0, "lines": [ { "bbox": [ 119, 171, 274, 232 ], "spans": [ { "bbox": [ 119, 171, 274, 232 ], "score": 0.972, "html": "
Test Acc. (%)
DeepProbLog98.49±0.18
NeurASP98.21 ± 0.30
SLASH (PC)95.39 ± 0.29
SLASH (DNN)98.74±0.21
", "type": "table", "image_path": "3788e55de7e7cfea5762aea98d8f13b0b9a4135918a954edc0d22b46010214d8.jpg" } ] } ], "index": 11.0, "virtual_lines": [ { "bbox": [ 119, 171, 274, 186.25 ], "spans": [], "index": 8 }, { "bbox": [ 119, 186.25, 274, 201.5 ], "spans": [], "index": 10 }, { "bbox": [ 119, 201.5, 274, 216.75 ], "spans": [], "index": 12 }, { "bbox": [ 119, 216.75, 274, 232.0 ], "spans": [], "index": 14 } ] } ], "index": 8.5 }, { "type": "table", "bbox": [ 309, 171, 491, 232 ], "blocks": [ { "type": "table_caption", "bbox": [ 334, 156, 461, 167 ], "group_id": 1, "lines": [ { "bbox": [ 333, 155, 462, 168 ], "spans": [ { "bbox": [ 333, 155, 462, 168 ], "score": 1.0, "content": "(b) Missing data MNIST Addition.", "type": "text" } ], "index": 7 } ], "index": 7 }, { "type": "table_body", "bbox": [ 309, 171, 491, 232 ], "group_id": 1, "lines": [ { "bbox": [ 309, 171, 491, 232 ], "spans": [ { "bbox": [ 309, 171, 491, 232 ], "score": 0.975, "html": "
DeepProbLogSLASH (PC)
50%97.73 ± 0.1297.67±0.12
80%76.07 ± 18.3896.72士0.05
90%69.15 ± 29.1594.85士0.38
97%32.46 ± 22.4882.57士4.66
", "type": "table", "image_path": "3c1f935051a7d747c53cdcf34ea4560b6fea51387ad5d49e7ef95ee58f377bb2.jpg" } ] } ], "index": 12.0, "virtual_lines": [ { "bbox": [ 309, 171, 491, 186.25 ], "spans": [], "index": 9 }, { "bbox": [ 309, 186.25, 491, 201.5 ], "spans": [], "index": 11 }, { "bbox": [ 309, 201.5, 491, 216.75 ], "spans": [], "index": 13 }, { "bbox": [ 309, 216.75, 491, 232.0 ], "spans": [], "index": 15 } ] } ], "index": 9.5 }, { "type": "text", "bbox": [ 108, 257, 504, 279 ], "lines": [ { "bbox": [ 106, 258, 505, 269 ], "spans": [ { "bbox": [ 106, 258, 505, 269 ], "score": 1.0, "content": "object-centric set prediction. For all experiments we present the average and the standard deviation", "type": "text" } ], "index": 16 }, { "bbox": [ 105, 267, 388, 281 ], "spans": [ { "bbox": [ 105, 267, 388, 281 ], "score": 1.0, "content": "over five runs with different random seeds for parameter initialization.", "type": "text" } ], "index": 17 } ], "index": 16.5, "bbox_fs": [ 105, 258, 505, 281 ] }, { "type": "text", "bbox": [ 107, 285, 505, 341 ], "lines": [ { "bbox": [ 105, 285, 506, 298 ], "spans": [ { "bbox": [ 105, 285, 506, 298 ], "score": 1.0, "content": "For ShapeWorld experiments, we generate a data set we refer to as ShapeWorld4. Images of", "type": "text" } ], "index": 18 }, { "bbox": [ 105, 295, 506, 309 ], "spans": [ { "bbox": [ 105, 295, 506, 309 ], "score": 1.0, "content": "ShapeWorld4 contain between one and four objects, with each object consisting of four attributes:", "type": "text" } ], "index": 19 }, { "bbox": [ 104, 306, 505, 321 ], "spans": [ { "bbox": [ 104, 306, 505, 321 ], "score": 1.0, "content": "a color (red, blue, green, gray, brown, magenta, cyan or yellow), a shade (bright, or dark), a shape", "type": "text" } ], "index": 20 }, { "bbox": [ 105, 318, 505, 330 ], "spans": [ { "bbox": [ 105, 318, 505, 330 ], "score": 1.0, "content": "(circle, triangle or square) and a size (small or big). Thus, each object can be created from 84 different", "type": "text" } ], "index": 21 }, { "bbox": [ 106, 329, 349, 342 ], "spans": [ { "bbox": [ 106, 329, 349, 342 ], "score": 1.0, "content": "combinations of attributes. Fig. 1 depicts an example image.", "type": "text" } ], "index": 22 } ], "index": 20, "bbox_fs": [ 104, 285, 506, 342 ] }, { "type": "text", "bbox": [ 107, 345, 505, 390 ], "lines": [ { "bbox": [ 105, 345, 507, 358 ], "spans": [ { "bbox": [ 105, 345, 507, 358 ], "score": 1.0, "content": "We measure performance via classification accuracies in the MNIST-Addition task. In our Shape-", "type": "text" } ], "index": 23 }, { "bbox": [ 105, 356, 506, 369 ], "spans": [ { "bbox": [ 105, 356, 506, 369 ], "score": 1.0, "content": "World4 experiments, we present the average precision. We refer to appendix B for the SLASH", "type": "text" } ], "index": 24 }, { "bbox": [ 105, 368, 507, 380 ], "spans": [ { "bbox": [ 105, 368, 507, 380 ], "score": 1.0, "content": "programs and queries of each experiment, and appendix C for a detailed description of hyperparame-", "type": "text" } ], "index": 25 }, { "bbox": [ 106, 379, 200, 390 ], "spans": [ { "bbox": [ 106, 379, 200, 390 ], "score": 1.0, "content": "ters and further details.", "type": "text" } ], "index": 26 } ], "index": 24.5, "bbox_fs": [ 105, 345, 507, 390 ] }, { "type": "text", "bbox": [ 106, 395, 505, 451 ], "lines": [ { "bbox": [ 105, 395, 507, 408 ], "spans": [ { "bbox": [ 105, 395, 507, 408 ], "score": 1.0, "content": "Evaluation 1: SLASH outperforms SOTA DPPLs in MNIST-Addition. The task of MNIST-", "type": "text" } ], "index": 27 }, { "bbox": [ 105, 406, 506, 420 ], "spans": [ { "bbox": [ 105, 406, 506, 420 ], "score": 1.0, "content": "Addition (Manhaeve et al., 2018) is to predict the sum of two MNIST digits, presented only as raw", "type": "text" } ], "index": 28 }, { "bbox": [ 105, 417, 506, 430 ], "spans": [ { "bbox": [ 105, 417, 506, 430 ], "score": 1.0, "content": "images. During test time, however, a model should classify the images directly. Thus, although a", "type": "text" } ], "index": 29 }, { "bbox": [ 105, 429, 505, 442 ], "spans": [ { "bbox": [ 105, 429, 505, 442 ], "score": 1.0, "content": "model does not receive explicit information about the depicted digits, it must learn to identify digits", "type": "text" } ], "index": 30 }, { "bbox": [ 105, 439, 283, 453 ], "spans": [ { "bbox": [ 105, 439, 283, 453 ], "score": 1.0, "content": "via indirect feedback on the sum prediction.", "type": "text" } ], "index": 31 } ], "index": 29, "bbox_fs": [ 105, 395, 507, 453 ] }, { "type": "text", "bbox": [ 106, 456, 505, 523 ], "lines": [ { "bbox": [ 105, 455, 505, 470 ], "spans": [ { "bbox": [ 105, 455, 505, 470 ], "score": 1.0, "content": "We compare the test accuracy after convergence between the three DPPLs: DeepProbLog (Manhaeve", "type": "text" } ], "index": 32 }, { "bbox": [ 104, 466, 506, 481 ], "spans": [ { "bbox": [ 104, 466, 506, 481 ], "score": 1.0, "content": "et al., 2018), NeurASP (Yang et al., 2020) and SLASH, using a probabilistic circuit (PC) or a deep", "type": "text" } ], "index": 33 }, { "bbox": [ 105, 478, 505, 491 ], "spans": [ { "bbox": [ 105, 478, 505, 491 ], "score": 1.0, "content": "neural network (DNN) as NPP. Notably, the DNN used in SLASH (DNN) is the LeNet5 model", "type": "text" } ], "index": 34 }, { "bbox": [ 105, 489, 505, 502 ], "spans": [ { "bbox": [ 105, 489, 505, 502 ], "score": 1.0, "content": "(LeCun et al., 1998a) of DeepProbLog and NeurASP. We note that when using the PC as NPP, we", "type": "text" } ], "index": 35 }, { "bbox": [ 105, 500, 506, 514 ], "spans": [ { "bbox": [ 105, 500, 302, 514 ], "score": 1.0, "content": "have also extracted conditional class probabilities", "type": "text" }, { "bbox": [ 303, 500, 339, 512 ], "score": 0.92, "content": "P ( C | X )", "type": "inline_equation" }, { "bbox": [ 339, 500, 484, 514 ], "score": 1.0, "content": ", by marginalizing the class variables", "type": "text" }, { "bbox": [ 485, 501, 494, 510 ], "score": 0.84, "content": "C", "type": "inline_equation" }, { "bbox": [ 494, 500, 506, 514 ], "score": 1.0, "content": "to", "type": "text" } ], "index": 36 }, { "bbox": [ 106, 511, 475, 524 ], "spans": [ { "bbox": [ 106, 511, 246, 524 ], "score": 1.0, "content": "acquire the normalization constant", "type": "text" }, { "bbox": [ 247, 511, 272, 523 ], "score": 0.92, "content": "P ( X )", "type": "inline_equation" }, { "bbox": [ 272, 511, 331, 524 ], "score": 1.0, "content": "from the joint", "type": "text" }, { "bbox": [ 331, 512, 368, 523 ], "score": 0.93, "content": "P ( X , C )", "type": "inline_equation" }, { "bbox": [ 368, 511, 435, 524 ], "score": 1.0, "content": ", and calculating", "type": "text" }, { "bbox": [ 435, 511, 471, 523 ], "score": 0.93, "content": "P ( X | C )", "type": "inline_equation" }, { "bbox": [ 471, 511, 475, 524 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 37 } ], "index": 34.5, "bbox_fs": [ 104, 455, 506, 524 ] }, { "type": "text", "bbox": [ 106, 528, 505, 628 ], "lines": [ { "bbox": [ 106, 528, 505, 540 ], "spans": [ { "bbox": [ 106, 528, 505, 540 ], "score": 1.0, "content": "The results can be seen in Tab. 1a. We observe that training SLASH with a DNN NPP produces", "type": "text" } ], "index": 38 }, { "bbox": [ 106, 540, 506, 551 ], "spans": [ { "bbox": [ 106, 540, 506, 551 ], "score": 1.0, "content": "SOTA accuracies compared to DeepProbLog and NeurASP, confirming that SLASH’s batch-wise", "type": "text" } ], "index": 39 }, { "bbox": [ 106, 551, 506, 563 ], "spans": [ { "bbox": [ 106, 551, 506, 563 ], "score": 1.0, "content": "loss computation leads to improved performances. We further observe that the test accuracy of", "type": "text" } ], "index": 40 }, { "bbox": [ 105, 560, 506, 574 ], "spans": [ { "bbox": [ 105, 560, 506, 574 ], "score": 1.0, "content": "SLASH with a PC NPP is slightly below the other DPPLs, however we argue that this may be", "type": "text" } ], "index": 41 }, { "bbox": [ 105, 572, 506, 585 ], "spans": [ { "bbox": [ 105, 572, 506, 585 ], "score": 1.0, "content": "since a PC, in comparison to a DNN, is learning a true mixture density rather than just conditional", "type": "text" } ], "index": 42 }, { "bbox": [ 106, 583, 506, 596 ], "spans": [ { "bbox": [ 106, 583, 506, 596 ], "score": 1.0, "content": "probabilities. The advantages of doing so will be investigated in the next experiments. Note that,", "type": "text" } ], "index": 43 }, { "bbox": [ 105, 594, 505, 607 ], "spans": [ { "bbox": [ 105, 594, 505, 607 ], "score": 1.0, "content": "optimal architecture search for PCs, e.g. for computer vision, is an open research question.These", "type": "text" } ], "index": 44 }, { "bbox": [ 106, 605, 505, 617 ], "spans": [ { "bbox": [ 106, 605, 505, 617 ], "score": 1.0, "content": "evaluations show SLASH’s advantages on the benchmark MNIST-Addition task. Additional benefits", "type": "text" } ], "index": 45 }, { "bbox": [ 105, 616, 301, 629 ], "spans": [ { "bbox": [ 105, 616, 301, 629 ], "score": 1.0, "content": "will be made clear in the following experiments.", "type": "text" } ], "index": 46 } ], "index": 42, "bbox_fs": [ 105, 528, 506, 629 ] }, { "type": "text", "bbox": [ 106, 632, 505, 732 ], "lines": [ { "bbox": [ 105, 632, 505, 646 ], "spans": [ { "bbox": [ 105, 632, 505, 646 ], "score": 1.0, "content": "Evaluation 2: Handling Missing Data with SLASH. SLASH offers the advantage of its flexibility", "type": "text" } ], "index": 47 }, { "bbox": [ 105, 643, 505, 657 ], "spans": [ { "bbox": [ 105, 643, 505, 657 ], "score": 1.0, "content": "to use various kinds of NPPs. Thus, in comparison to previous DPPLs, one can easily integrate", "type": "text" } ], "index": 48 }, { "bbox": [ 105, 655, 505, 668 ], "spans": [ { "bbox": [ 105, 655, 505, 668 ], "score": 1.0, "content": "NPPs into SLASH that perform joint probability estimation. For this evaluation, we consider the", "type": "text" } ], "index": 49 }, { "bbox": [ 106, 666, 505, 678 ], "spans": [ { "bbox": [ 106, 666, 505, 678 ], "score": 1.0, "content": "task of MNIST-Addition with missing data. We trained SLASH (PC) and DeepProbLog with the", "type": "text" } ], "index": 50 }, { "bbox": [ 105, 676, 506, 690 ], "spans": [ { "bbox": [ 105, 676, 506, 690 ], "score": 1.0, "content": "MNIST-Addition task with images in which a percentage of pixels per image has been removed. It is", "type": "text" } ], "index": 51 }, { "bbox": [ 105, 688, 505, 701 ], "spans": [ { "bbox": [ 105, 688, 505, 701 ], "score": 1.0, "content": "important to mention here that whereas DeepProbLog handles the missing data simply as background", "type": "text" } ], "index": 52 }, { "bbox": [ 106, 699, 505, 712 ], "spans": [ { "bbox": [ 106, 699, 505, 712 ], "score": 1.0, "content": "pixels, SLASH (PC) specifically models the missing data as uncertain data by marginalizing the", "type": "text" } ], "index": 53 }, { "bbox": [ 105, 709, 505, 723 ], "spans": [ { "bbox": [ 105, 709, 505, 723 ], "score": 1.0, "content": "denoted pixels at inference time. We use DeepProbLog here representative of DPPLs without true", "type": "text" } ], "index": 54 }, { "bbox": [ 106, 721, 183, 733 ], "spans": [ { "bbox": [ 106, 721, 183, 733 ], "score": 1.0, "content": "density estimation.", "type": "text" } ], "index": 55 } ], "index": 51, "bbox_fs": [ 105, 632, 506, 733 ] } ] }, { "preproc_blocks": [ { "type": "table", "bbox": [ 119, 106, 317, 183 ], "blocks": [ { "type": "table_caption", "bbox": [ 119, 83, 288, 104 ], "group_id": 0, "lines": [ { "bbox": [ 118, 83, 288, 93 ], "spans": [ { "bbox": [ 118, 83, 288, 93 ], "score": 0.982, "content": "(a) ShapeWorld4 and ShapeWorld4 CoGenT Test", "type": "text" } ], "index": 0 }, { "bbox": [ 118, 93, 171, 105 ], "spans": [ { "bbox": [ 118, 93, 171, 105 ], "score": 1.0, "content": "Avg.Precision", "type": "text" } ], "index": 1 } ], "index": 0.5 }, { "type": "table_body", "bbox": [ 119, 106, 317, 183 ], "group_id": 0, "lines": [ { "bbox": [ 119, 106, 317, 183 ], "spans": [ { "bbox": [ 119, 106, 317, 183 ], "score": 0.976, "html": "
Slot Att.SLASH Att.
Test SetShapeWorld4 90.24 ± 0.9395.58 ± 0.61
CoGenT
Test Cond. A90.37 ± 2.1996.85 ± 0.43
Test Cond.B27.15 ± 2.3640.58 ± 1.99
", "type": "table", "image_path": "0b06c8954d45055a4eff3ce528d6a37b02d5a2f22cb9bd72bf07133146fdad99.jpg" } ] } ], "index": 4, "virtual_lines": [ { "bbox": [ 119, 106, 317, 121.4 ], "spans": [], "index": 2 }, { "bbox": [ 119, 121.4, 317, 136.8 ], "spans": [], "index": 3 }, { "bbox": [ 119, 136.8, 317, 152.20000000000002 ], "spans": [], "index": 4 }, { "bbox": [ 119, 152.20000000000002, 317, 167.60000000000002 ], "spans": [], "index": 5 }, { "bbox": [ 119, 167.60000000000002, 317, 183.00000000000003 ], "spans": [], "index": 6 } ] }, { "type": "table_caption", "bbox": [ 106, 193, 505, 282 ], "group_id": 0, "lines": [ { "bbox": [ 105, 194, 506, 207 ], "spans": [ { "bbox": [ 105, 194, 506, 207 ], "score": 1.0, "content": "Figure 3: ShapeWorld4 Experiments. (a) Converged test average precision scores for the set prediction", "type": "text" } ], "index": 14 }, { "bbox": [ 105, 204, 505, 218 ], "spans": [ { "bbox": [ 105, 204, 505, 218 ], "score": 1.0, "content": "task with ShapeWorld4 (top) and ShapeWorld4 CoGenT (bottom). (b) Test average precision scores", "type": "text" } ], "index": 15 }, { "bbox": [ 105, 215, 505, 228 ], "spans": [ { "bbox": [ 105, 215, 505, 228 ], "score": 1.0, "content": "for set prediction with ShapeWorld4 over the training epochs. In these experiments we compared", "type": "text" } ], "index": 16 }, { "bbox": [ 106, 227, 505, 238 ], "spans": [ { "bbox": [ 106, 227, 505, 238 ], "score": 1.0, "content": "a baseline slot encoder versus SLASH Attention with slot attention and PC-based NPPs. For the", "type": "text" } ], "index": 17 }, { "bbox": [ 105, 238, 506, 250 ], "spans": [ { "bbox": [ 105, 238, 506, 250 ], "score": 1.0, "content": "CoGenT experiments, a model is trained on one training set and tested on two separate test conditions.", "type": "text" } ], "index": 18 }, { "bbox": [ 105, 248, 505, 261 ], "spans": [ { "bbox": [ 105, 248, 505, 261 ], "score": 1.0, "content": "The Condition A test set contains attribute compositions which were also seen during training. The", "type": "text" } ], "index": 19 }, { "bbox": [ 105, 259, 506, 273 ], "spans": [ { "bbox": [ 105, 259, 506, 273 ], "score": 1.0, "content": "Condition B test set contains attribute compositions which were not seen during training, e.g. yellow", "type": "text" } ], "index": 20 }, { "bbox": [ 106, 271, 418, 283 ], "spans": [ { "bbox": [ 106, 271, 418, 283 ], "score": 1.0, "content": "circles were not present in the training set, but present in Condition B test set.", "type": "text" } ], "index": 21 } ], "index": 17.5 } ], "index": 4 }, { "type": "image", "bbox": [ 323, 84, 493, 171 ], "blocks": [ { "type": "image_body", "bbox": [ 323, 84, 493, 171 ], "group_id": 0, "lines": [ { "bbox": [ 323, 84, 493, 171 ], "spans": [ { "bbox": [ 323, 84, 493, 171 ], "score": 0.959, "type": "image", "image_path": "0eaf19403876aa7fbb58219357ca9c041a53e7685ffb819aff2c0d5f28605742.jpg" } ] } ], "index": 9.5, "virtual_lines": [ { "bbox": [ 323, 84, 493, 98.5 ], "spans": [], "index": 7 }, { "bbox": [ 323, 98.5, 493, 113.0 ], "spans": [], "index": 8 }, { "bbox": [ 323, 113.0, 493, 127.5 ], "spans": [], "index": 9 }, { "bbox": [ 323, 127.5, 493, 142.0 ], "spans": [], "index": 10 }, { "bbox": [ 323, 142.0, 493, 156.5 ], "spans": [], "index": 11 }, { "bbox": [ 323, 156.5, 493, 171.0 ], "spans": [], "index": 12 } ] }, { "type": "image_caption", "bbox": [ 333, 173, 485, 183 ], "group_id": 0, "lines": [ { "bbox": [ 333, 173, 486, 184 ], "spans": [ { "bbox": [ 333, 173, 486, 184 ], "score": 0.98, "content": "(b) Test Avg.Precision over Training Epochs", "type": "text" } ], "index": 13 } ], "index": 13 } ], "index": 11.25 }, { "type": "text", "bbox": [ 107, 295, 505, 384 ], "lines": [ { "bbox": [ 105, 294, 506, 309 ], "spans": [ { "bbox": [ 105, 294, 252, 309 ], "score": 1.0, "content": "The results can be seen in Tab. 1b for", "type": "text" }, { "bbox": [ 252, 296, 272, 307 ], "score": 0.88, "content": "5 0 \\%", "type": "inline_equation" }, { "bbox": [ 272, 294, 275, 309 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 275, 295, 294, 307 ], "score": 0.86, "content": "8 0 \\%", "type": "inline_equation" }, { "bbox": [ 295, 294, 298, 309 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 298, 296, 318, 307 ], "score": 0.86, "content": "9 0 \\%", "type": "inline_equation" }, { "bbox": [ 318, 294, 334, 309 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 335, 296, 354, 307 ], "score": 0.88, "content": "9 7 \\%", "type": "inline_equation" }, { "bbox": [ 355, 294, 506, 309 ], "score": 1.0, "content": "missing pixels per image. We observe", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 306, 506, 321 ], "spans": [ { "bbox": [ 105, 306, 133, 321 ], "score": 1.0, "content": "that at", "type": "text" }, { "bbox": [ 133, 307, 153, 318 ], "score": 0.87, "content": "5 0 \\%", "type": "inline_equation" }, { "bbox": [ 153, 306, 419, 321 ], "score": 1.0, "content": ", DeepProbLog and SLASH produce almost equal accuracies. With", "type": "text" }, { "bbox": [ 420, 307, 439, 318 ], "score": 0.88, "content": "8 0 \\%", "type": "inline_equation" }, { "bbox": [ 440, 306, 506, 321 ], "score": 1.0, "content": "percent missing", "type": "text" } ], "index": 23 }, { "bbox": [ 105, 317, 506, 331 ], "spans": [ { "bbox": [ 105, 317, 506, 331 ], "score": 1.0, "content": "pixels, there is a substantial difference in the ability of the two DPPLs to correctly classify images,", "type": "text" } ], "index": 24 }, { "bbox": [ 105, 329, 505, 341 ], "spans": [ { "bbox": [ 105, 329, 505, 341 ], "score": 1.0, "content": "with SLASH being very stable. By further increasing the percentage of missing pixels, this difference", "type": "text" } ], "index": 25 }, { "bbox": [ 105, 339, 506, 352 ], "spans": [ { "bbox": [ 105, 339, 352, 352 ], "score": 1.0, "content": "becomes even more substantial with SLASH still reaching a", "type": "text" }, { "bbox": [ 352, 340, 372, 351 ], "score": 0.89, "content": "8 2 \\%", "type": "inline_equation" }, { "bbox": [ 372, 339, 473, 352 ], "score": 1.0, "content": "test accuracy even when", "type": "text" }, { "bbox": [ 473, 340, 493, 351 ], "score": 0.88, "content": "9 7 \\%", "type": "inline_equation" }, { "bbox": [ 493, 339, 506, 352 ], "score": 1.0, "content": "of", "type": "text" } ], "index": 26 }, { "bbox": [ 105, 349, 507, 365 ], "spans": [ { "bbox": [ 105, 349, 431, 365 ], "score": 1.0, "content": "the pixels per image are missing, whereas DeepProbLog degrades to an average of", "type": "text" }, { "bbox": [ 432, 351, 451, 362 ], "score": 0.89, "content": "3 2 \\%", "type": "inline_equation" }, { "bbox": [ 451, 349, 507, 365 ], "score": 1.0, "content": "test accuracy.", "type": "text" } ], "index": 27 }, { "bbox": [ 105, 361, 506, 375 ], "spans": [ { "bbox": [ 105, 361, 506, 375 ], "score": 1.0, "content": "We further note that SLASH, in comparison to DeepProbLog, produces largely reduced standard", "type": "text" } ], "index": 28 }, { "bbox": [ 106, 372, 192, 385 ], "spans": [ { "bbox": [ 106, 372, 192, 385 ], "score": 1.0, "content": "deviations over runs.", "type": "text" } ], "index": 29 } ], "index": 25.5 }, { "type": "text", "bbox": [ 107, 389, 505, 434 ], "lines": [ { "bbox": [ 106, 390, 505, 402 ], "spans": [ { "bbox": [ 106, 390, 505, 402 ], "score": 1.0, "content": "Thus, by utilizing the power of true density estimation SLASH, with an appropriate NPP, can produce", "type": "text" } ], "index": 30 }, { "bbox": [ 105, 401, 505, 413 ], "spans": [ { "bbox": [ 105, 401, 505, 413 ], "score": 1.0, "content": "more robust results in comparison to other DPPLs. Further, we refer to Appendix D, which contains", "type": "text" } ], "index": 31 }, { "bbox": [ 105, 412, 506, 425 ], "spans": [ { "bbox": [ 105, 412, 506, 425 ], "score": 1.0, "content": "results of additional experiments where training is performed with the full MNIST data set whereas", "type": "text" } ], "index": 32 }, { "bbox": [ 105, 423, 330, 435 ], "spans": [ { "bbox": [ 105, 423, 330, 435 ], "score": 1.0, "content": "only the test set entails different rates of missing pixels.", "type": "text" } ], "index": 33 } ], "index": 31.5 }, { "type": "text", "bbox": [ 107, 439, 505, 506 ], "lines": [ { "bbox": [ 106, 440, 506, 451 ], "spans": [ { "bbox": [ 106, 440, 506, 451 ], "score": 1.0, "content": "Evaluation 3: Improved Concept Learning via SLASH. We show that SLASH can be very ef-", "type": "text" } ], "index": 34 }, { "bbox": [ 105, 451, 505, 462 ], "spans": [ { "bbox": [ 105, 451, 505, 462 ], "score": 1.0, "content": "fective for the complex task of set prediction, which previous DPPLs have not tackled. We revert", "type": "text" } ], "index": 35 }, { "bbox": [ 105, 461, 505, 474 ], "spans": [ { "bbox": [ 105, 461, 505, 474 ], "score": 1.0, "content": "to the ShapeWorld4 data set for this setting. For set prediction, a model is trained to predict the", "type": "text" } ], "index": 36 }, { "bbox": [ 105, 471, 505, 485 ], "spans": [ { "bbox": [ 105, 471, 505, 485 ], "score": 1.0, "content": "discrete attributes of a set of objects in an image (cf. Fig. 1 for an example ShapeWorld4 image). The", "type": "text" } ], "index": 37 }, { "bbox": [ 105, 482, 505, 496 ], "spans": [ { "bbox": [ 105, 482, 505, 496 ], "score": 1.0, "content": "difficulty for the model lies therein that it must match an unordered set of corresponding attributes", "type": "text" } ], "index": 38 }, { "bbox": [ 105, 493, 478, 508 ], "spans": [ { "bbox": [ 105, 493, 478, 508 ], "score": 1.0, "content": "(with varying number of entities over samples) with its internal representations of the image.", "type": "text" } ], "index": 39 } ], "index": 36.5 }, { "type": "text", "bbox": [ 107, 511, 505, 566 ], "lines": [ { "bbox": [ 106, 511, 505, 523 ], "spans": [ { "bbox": [ 106, 511, 505, 523 ], "score": 1.0, "content": "The slot attention module introduced by Locatello et al. (2020) allows for an attractive object-centric", "type": "text" } ], "index": 40 }, { "bbox": [ 105, 522, 505, 535 ], "spans": [ { "bbox": [ 105, 522, 505, 535 ], "score": 1.0, "content": "approach to this task. Specifically, this module represents a pluggable, differentiable module that can", "type": "text" } ], "index": 41 }, { "bbox": [ 106, 533, 507, 546 ], "spans": [ { "bbox": [ 106, 533, 507, 546 ], "score": 1.0, "content": "be easily added to any architecture and, through a competitive softmax-based attention mechanism,", "type": "text" } ], "index": 42 }, { "bbox": [ 106, 545, 506, 557 ], "spans": [ { "bbox": [ 106, 545, 506, 557 ], "score": 1.0, "content": "can enforce the binding of specific parts of a latent representation into permutation-invariant, task-", "type": "text" } ], "index": 43 }, { "bbox": [ 106, 556, 222, 567 ], "spans": [ { "bbox": [ 106, 556, 222, 567 ], "score": 1.0, "content": "specific vectors, called slots.", "type": "text" } ], "index": 44 } ], "index": 42 }, { "type": "text", "bbox": [ 107, 572, 505, 638 ], "lines": [ { "bbox": [ 105, 572, 505, 585 ], "spans": [ { "bbox": [ 105, 572, 505, 585 ], "score": 1.0, "content": "In our experiments, we wish to show that by adding logical constraints to the training setting, one", "type": "text" } ], "index": 45 }, { "bbox": [ 105, 582, 506, 596 ], "spans": [ { "bbox": [ 105, 582, 506, 596 ], "score": 1.0, "content": "can improve the overall performances and generalization properties of such a model. For this, we", "type": "text" } ], "index": 46 }, { "bbox": [ 105, 594, 506, 607 ], "spans": [ { "bbox": [ 105, 594, 506, 607 ], "score": 1.0, "content": "train SLASH with NPPs as depicted in Fig. 1 consisting of a shared slot encoder and separate PCs,", "type": "text" } ], "index": 47 }, { "bbox": [ 105, 604, 506, 618 ], "spans": [ { "bbox": [ 105, 604, 506, 618 ], "score": 1.0, "content": "each modelling the mixture of latent slot variables and the attributes of one category, e.g. color.", "type": "text" } ], "index": 48 }, { "bbox": [ 106, 616, 506, 628 ], "spans": [ { "bbox": [ 106, 616, 506, 628 ], "score": 1.0, "content": "For ShapeWorld4, we thereby have altogether four NPPs. SLASH is trained via queries of the kind", "type": "text" } ], "index": 49 }, { "bbox": [ 105, 627, 465, 639 ], "spans": [ { "bbox": [ 105, 627, 465, 639 ], "score": 1.0, "content": "exemplified in Fig. 7 in the Appendix. We refer to this configuration as SLASH Attention.", "type": "text" } ], "index": 50 } ], "index": 47.5 }, { "type": "text", "bbox": [ 107, 643, 505, 732 ], "lines": [ { "bbox": [ 106, 644, 505, 656 ], "spans": [ { "bbox": [ 106, 644, 505, 656 ], "score": 1.0, "content": "We compare SLASH Attention to a baseline slot attention encoder using an MLP and Hungarian", "type": "text" } ], "index": 51 }, { "bbox": [ 105, 655, 505, 667 ], "spans": [ { "bbox": [ 105, 655, 505, 667 ], "score": 1.0, "content": "loss for predicting the object properties from the slot encodings as in Locatello et al. (2020). The", "type": "text" } ], "index": 52 }, { "bbox": [ 105, 666, 506, 678 ], "spans": [ { "bbox": [ 105, 666, 506, 678 ], "score": 1.0, "content": "results of these experiments can be found in Fig. 3a (top). We observe that the average precision after", "type": "text" } ], "index": 53 }, { "bbox": [ 105, 677, 505, 689 ], "spans": [ { "bbox": [ 105, 677, 505, 689 ], "score": 1.0, "content": "convergence on the held-out test set with SLASH Attention is greatly improved to that of the baseline", "type": "text" } ], "index": 54 }, { "bbox": [ 105, 687, 505, 700 ], "spans": [ { "bbox": [ 105, 687, 505, 700 ], "score": 1.0, "content": "model. Additionally, in Fig. 3b we observe that SLASH Attention reaches the average precision", "type": "text" } ], "index": 55 }, { "bbox": [ 105, 698, 505, 712 ], "spans": [ { "bbox": [ 105, 698, 505, 712 ], "score": 1.0, "content": "value of the baseline model in much fewer number of epochs. Thus, we can summarize that adding", "type": "text" } ], "index": 56 }, { "bbox": [ 105, 710, 506, 722 ], "spans": [ { "bbox": [ 105, 710, 506, 722 ], "score": 1.0, "content": "logical knowledge in the training procedure via SLASH can greatly improve the capabilities of a", "type": "text" } ], "index": 57 }, { "bbox": [ 105, 721, 239, 733 ], "spans": [ { "bbox": [ 105, 721, 239, 733 ], "score": 1.0, "content": "neural module for set prediction.", "type": "text" } ], "index": 58 } ], "index": 54.5 } ], "page_idx": 7, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 302, 751, 308, 760 ], "lines": [ { "bbox": [ 300, 750, 309, 761 ], "spans": [ { "bbox": [ 300, 750, 309, 761 ], "score": 1.0, "content": "8", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 107, 27, 308, 37 ], "lines": [ { "bbox": [ 107, 26, 308, 38 ], "spans": [ { "bbox": [ 107, 26, 308, 38 ], "score": 1.0, "content": "Under review as a conference paper at ICLR 2022", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "table", "bbox": [ 119, 106, 317, 183 ], "blocks": [ { "type": "table_caption", "bbox": [ 119, 83, 288, 104 ], "group_id": 0, "lines": [ { "bbox": [ 118, 83, 288, 93 ], "spans": [ { "bbox": [ 118, 83, 288, 93 ], "score": 0.982, "content": "(a) ShapeWorld4 and ShapeWorld4 CoGenT Test", "type": "text" } ], "index": 0 }, { "bbox": [ 118, 93, 171, 105 ], "spans": [ { "bbox": [ 118, 93, 171, 105 ], "score": 1.0, "content": "Avg.Precision", "type": "text" } ], "index": 1 } ], "index": 0.5 }, { "type": "table_body", "bbox": [ 119, 106, 317, 183 ], "group_id": 0, "lines": [ { "bbox": [ 119, 106, 317, 183 ], "spans": [ { "bbox": [ 119, 106, 317, 183 ], "score": 0.976, "html": "
Slot Att.SLASH Att.
Test SetShapeWorld4 90.24 ± 0.9395.58 ± 0.61
CoGenT
Test Cond. A90.37 ± 2.1996.85 ± 0.43
Test Cond.B27.15 ± 2.3640.58 ± 1.99
", "type": "table", "image_path": "0b06c8954d45055a4eff3ce528d6a37b02d5a2f22cb9bd72bf07133146fdad99.jpg" } ] } ], "index": 4, "virtual_lines": [ { "bbox": [ 119, 106, 317, 121.4 ], "spans": [], "index": 2 }, { "bbox": [ 119, 121.4, 317, 136.8 ], "spans": [], "index": 3 }, { "bbox": [ 119, 136.8, 317, 152.20000000000002 ], "spans": [], "index": 4 }, { "bbox": [ 119, 152.20000000000002, 317, 167.60000000000002 ], "spans": [], "index": 5 }, { "bbox": [ 119, 167.60000000000002, 317, 183.00000000000003 ], "spans": [], "index": 6 } ] }, { "type": "table_caption", "bbox": [ 106, 193, 505, 282 ], "group_id": 0, "lines": [ { "bbox": [ 105, 194, 506, 207 ], "spans": [ { "bbox": [ 105, 194, 506, 207 ], "score": 1.0, "content": "Figure 3: ShapeWorld4 Experiments. (a) Converged test average precision scores for the set prediction", "type": "text" } ], "index": 14 }, { "bbox": [ 105, 204, 505, 218 ], "spans": [ { "bbox": [ 105, 204, 505, 218 ], "score": 1.0, "content": "task with ShapeWorld4 (top) and ShapeWorld4 CoGenT (bottom). (b) Test average precision scores", "type": "text" } ], "index": 15 }, { "bbox": [ 105, 215, 505, 228 ], "spans": [ { "bbox": [ 105, 215, 505, 228 ], "score": 1.0, "content": "for set prediction with ShapeWorld4 over the training epochs. In these experiments we compared", "type": "text" } ], "index": 16 }, { "bbox": [ 106, 227, 505, 238 ], "spans": [ { "bbox": [ 106, 227, 505, 238 ], "score": 1.0, "content": "a baseline slot encoder versus SLASH Attention with slot attention and PC-based NPPs. For the", "type": "text" } ], "index": 17 }, { "bbox": [ 105, 238, 506, 250 ], "spans": [ { "bbox": [ 105, 238, 506, 250 ], "score": 1.0, "content": "CoGenT experiments, a model is trained on one training set and tested on two separate test conditions.", "type": "text" } ], "index": 18 }, { "bbox": [ 105, 248, 505, 261 ], "spans": [ { "bbox": [ 105, 248, 505, 261 ], "score": 1.0, "content": "The Condition A test set contains attribute compositions which were also seen during training. The", "type": "text" } ], "index": 19 }, { "bbox": [ 105, 259, 506, 273 ], "spans": [ { "bbox": [ 105, 259, 506, 273 ], "score": 1.0, "content": "Condition B test set contains attribute compositions which were not seen during training, e.g. yellow", "type": "text" } ], "index": 20 }, { "bbox": [ 106, 271, 418, 283 ], "spans": [ { "bbox": [ 106, 271, 418, 283 ], "score": 1.0, "content": "circles were not present in the training set, but present in Condition B test set.", "type": "text" } ], "index": 21 } ], "index": 17.5 } ], "index": 4 }, { "type": "image", "bbox": [ 323, 84, 493, 171 ], "blocks": [ { "type": "image_body", "bbox": [ 323, 84, 493, 171 ], "group_id": 0, "lines": [ { "bbox": [ 323, 84, 493, 171 ], "spans": [ { "bbox": [ 323, 84, 493, 171 ], "score": 0.959, "type": "image", "image_path": "0eaf19403876aa7fbb58219357ca9c041a53e7685ffb819aff2c0d5f28605742.jpg" } ] } ], "index": 9.5, "virtual_lines": [ { "bbox": [ 323, 84, 493, 98.5 ], "spans": [], "index": 7 }, { "bbox": [ 323, 98.5, 493, 113.0 ], "spans": [], "index": 8 }, { "bbox": [ 323, 113.0, 493, 127.5 ], "spans": [], "index": 9 }, { "bbox": [ 323, 127.5, 493, 142.0 ], "spans": [], "index": 10 }, { "bbox": [ 323, 142.0, 493, 156.5 ], "spans": [], "index": 11 }, { "bbox": [ 323, 156.5, 493, 171.0 ], "spans": [], "index": 12 } ] }, { "type": "image_caption", "bbox": [ 333, 173, 485, 183 ], "group_id": 0, "lines": [ { "bbox": [ 333, 173, 486, 184 ], "spans": [ { "bbox": [ 333, 173, 486, 184 ], "score": 0.98, "content": "(b) Test Avg.Precision over Training Epochs", "type": "text" } ], "index": 13 } ], "index": 13 } ], "index": 11.25 }, { "type": "text", "bbox": [ 107, 295, 505, 384 ], "lines": [ { "bbox": [ 105, 294, 506, 309 ], "spans": [ { "bbox": [ 105, 294, 252, 309 ], "score": 1.0, "content": "The results can be seen in Tab. 1b for", "type": "text" }, { "bbox": [ 252, 296, 272, 307 ], "score": 0.88, "content": "5 0 \\%", "type": "inline_equation" }, { "bbox": [ 272, 294, 275, 309 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 275, 295, 294, 307 ], "score": 0.86, "content": "8 0 \\%", "type": "inline_equation" }, { "bbox": [ 295, 294, 298, 309 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 298, 296, 318, 307 ], "score": 0.86, "content": "9 0 \\%", "type": "inline_equation" }, { "bbox": [ 318, 294, 334, 309 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 335, 296, 354, 307 ], "score": 0.88, "content": "9 7 \\%", "type": "inline_equation" }, { "bbox": [ 355, 294, 506, 309 ], "score": 1.0, "content": "missing pixels per image. We observe", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 306, 506, 321 ], "spans": [ { "bbox": [ 105, 306, 133, 321 ], "score": 1.0, "content": "that at", "type": "text" }, { "bbox": [ 133, 307, 153, 318 ], "score": 0.87, "content": "5 0 \\%", "type": "inline_equation" }, { "bbox": [ 153, 306, 419, 321 ], "score": 1.0, "content": ", DeepProbLog and SLASH produce almost equal accuracies. With", "type": "text" }, { "bbox": [ 420, 307, 439, 318 ], "score": 0.88, "content": "8 0 \\%", "type": "inline_equation" }, { "bbox": [ 440, 306, 506, 321 ], "score": 1.0, "content": "percent missing", "type": "text" } ], "index": 23 }, { "bbox": [ 105, 317, 506, 331 ], "spans": [ { "bbox": [ 105, 317, 506, 331 ], "score": 1.0, "content": "pixels, there is a substantial difference in the ability of the two DPPLs to correctly classify images,", "type": "text" } ], "index": 24 }, { "bbox": [ 105, 329, 505, 341 ], "spans": [ { "bbox": [ 105, 329, 505, 341 ], "score": 1.0, "content": "with SLASH being very stable. By further increasing the percentage of missing pixels, this difference", "type": "text" } ], "index": 25 }, { "bbox": [ 105, 339, 506, 352 ], "spans": [ { "bbox": [ 105, 339, 352, 352 ], "score": 1.0, "content": "becomes even more substantial with SLASH still reaching a", "type": "text" }, { "bbox": [ 352, 340, 372, 351 ], "score": 0.89, "content": "8 2 \\%", "type": "inline_equation" }, { "bbox": [ 372, 339, 473, 352 ], "score": 1.0, "content": "test accuracy even when", "type": "text" }, { "bbox": [ 473, 340, 493, 351 ], "score": 0.88, "content": "9 7 \\%", "type": "inline_equation" }, { "bbox": [ 493, 339, 506, 352 ], "score": 1.0, "content": "of", "type": "text" } ], "index": 26 }, { "bbox": [ 105, 349, 507, 365 ], "spans": [ { "bbox": [ 105, 349, 431, 365 ], "score": 1.0, "content": "the pixels per image are missing, whereas DeepProbLog degrades to an average of", "type": "text" }, { "bbox": [ 432, 351, 451, 362 ], "score": 0.89, "content": "3 2 \\%", "type": "inline_equation" }, { "bbox": [ 451, 349, 507, 365 ], "score": 1.0, "content": "test accuracy.", "type": "text" } ], "index": 27 }, { "bbox": [ 105, 361, 506, 375 ], "spans": [ { "bbox": [ 105, 361, 506, 375 ], "score": 1.0, "content": "We further note that SLASH, in comparison to DeepProbLog, produces largely reduced standard", "type": "text" } ], "index": 28 }, { "bbox": [ 106, 372, 192, 385 ], "spans": [ { "bbox": [ 106, 372, 192, 385 ], "score": 1.0, "content": "deviations over runs.", "type": "text" } ], "index": 29 } ], "index": 25.5, "bbox_fs": [ 105, 294, 507, 385 ] }, { "type": "text", "bbox": [ 107, 389, 505, 434 ], "lines": [ { "bbox": [ 106, 390, 505, 402 ], "spans": [ { "bbox": [ 106, 390, 505, 402 ], "score": 1.0, "content": "Thus, by utilizing the power of true density estimation SLASH, with an appropriate NPP, can produce", "type": "text" } ], "index": 30 }, { "bbox": [ 105, 401, 505, 413 ], "spans": [ { "bbox": [ 105, 401, 505, 413 ], "score": 1.0, "content": "more robust results in comparison to other DPPLs. Further, we refer to Appendix D, which contains", "type": "text" } ], "index": 31 }, { "bbox": [ 105, 412, 506, 425 ], "spans": [ { "bbox": [ 105, 412, 506, 425 ], "score": 1.0, "content": "results of additional experiments where training is performed with the full MNIST data set whereas", "type": "text" } ], "index": 32 }, { "bbox": [ 105, 423, 330, 435 ], "spans": [ { "bbox": [ 105, 423, 330, 435 ], "score": 1.0, "content": "only the test set entails different rates of missing pixels.", "type": "text" } ], "index": 33 } ], "index": 31.5, "bbox_fs": [ 105, 390, 506, 435 ] }, { "type": "text", "bbox": [ 107, 439, 505, 506 ], "lines": [ { "bbox": [ 106, 440, 506, 451 ], "spans": [ { "bbox": [ 106, 440, 506, 451 ], "score": 1.0, "content": "Evaluation 3: Improved Concept Learning via SLASH. We show that SLASH can be very ef-", "type": "text" } ], "index": 34 }, { "bbox": [ 105, 451, 505, 462 ], "spans": [ { "bbox": [ 105, 451, 505, 462 ], "score": 1.0, "content": "fective for the complex task of set prediction, which previous DPPLs have not tackled. We revert", "type": "text" } ], "index": 35 }, { "bbox": [ 105, 461, 505, 474 ], "spans": [ { "bbox": [ 105, 461, 505, 474 ], "score": 1.0, "content": "to the ShapeWorld4 data set for this setting. For set prediction, a model is trained to predict the", "type": "text" } ], "index": 36 }, { "bbox": [ 105, 471, 505, 485 ], "spans": [ { "bbox": [ 105, 471, 505, 485 ], "score": 1.0, "content": "discrete attributes of a set of objects in an image (cf. Fig. 1 for an example ShapeWorld4 image). The", "type": "text" } ], "index": 37 }, { "bbox": [ 105, 482, 505, 496 ], "spans": [ { "bbox": [ 105, 482, 505, 496 ], "score": 1.0, "content": "difficulty for the model lies therein that it must match an unordered set of corresponding attributes", "type": "text" } ], "index": 38 }, { "bbox": [ 105, 493, 478, 508 ], "spans": [ { "bbox": [ 105, 493, 478, 508 ], "score": 1.0, "content": "(with varying number of entities over samples) with its internal representations of the image.", "type": "text" } ], "index": 39 } ], "index": 36.5, "bbox_fs": [ 105, 440, 506, 508 ] }, { "type": "text", "bbox": [ 107, 511, 505, 566 ], "lines": [ { "bbox": [ 106, 511, 505, 523 ], "spans": [ { "bbox": [ 106, 511, 505, 523 ], "score": 1.0, "content": "The slot attention module introduced by Locatello et al. (2020) allows for an attractive object-centric", "type": "text" } ], "index": 40 }, { "bbox": [ 105, 522, 505, 535 ], "spans": [ { "bbox": [ 105, 522, 505, 535 ], "score": 1.0, "content": "approach to this task. Specifically, this module represents a pluggable, differentiable module that can", "type": "text" } ], "index": 41 }, { "bbox": [ 106, 533, 507, 546 ], "spans": [ { "bbox": [ 106, 533, 507, 546 ], "score": 1.0, "content": "be easily added to any architecture and, through a competitive softmax-based attention mechanism,", "type": "text" } ], "index": 42 }, { "bbox": [ 106, 545, 506, 557 ], "spans": [ { "bbox": [ 106, 545, 506, 557 ], "score": 1.0, "content": "can enforce the binding of specific parts of a latent representation into permutation-invariant, task-", "type": "text" } ], "index": 43 }, { "bbox": [ 106, 556, 222, 567 ], "spans": [ { "bbox": [ 106, 556, 222, 567 ], "score": 1.0, "content": "specific vectors, called slots.", "type": "text" } ], "index": 44 } ], "index": 42, "bbox_fs": [ 105, 511, 507, 567 ] }, { "type": "text", "bbox": [ 107, 572, 505, 638 ], "lines": [ { "bbox": [ 105, 572, 505, 585 ], "spans": [ { "bbox": [ 105, 572, 505, 585 ], "score": 1.0, "content": "In our experiments, we wish to show that by adding logical constraints to the training setting, one", "type": "text" } ], "index": 45 }, { "bbox": [ 105, 582, 506, 596 ], "spans": [ { "bbox": [ 105, 582, 506, 596 ], "score": 1.0, "content": "can improve the overall performances and generalization properties of such a model. For this, we", "type": "text" } ], "index": 46 }, { "bbox": [ 105, 594, 506, 607 ], "spans": [ { "bbox": [ 105, 594, 506, 607 ], "score": 1.0, "content": "train SLASH with NPPs as depicted in Fig. 1 consisting of a shared slot encoder and separate PCs,", "type": "text" } ], "index": 47 }, { "bbox": [ 105, 604, 506, 618 ], "spans": [ { "bbox": [ 105, 604, 506, 618 ], "score": 1.0, "content": "each modelling the mixture of latent slot variables and the attributes of one category, e.g. color.", "type": "text" } ], "index": 48 }, { "bbox": [ 106, 616, 506, 628 ], "spans": [ { "bbox": [ 106, 616, 506, 628 ], "score": 1.0, "content": "For ShapeWorld4, we thereby have altogether four NPPs. SLASH is trained via queries of the kind", "type": "text" } ], "index": 49 }, { "bbox": [ 105, 627, 465, 639 ], "spans": [ { "bbox": [ 105, 627, 465, 639 ], "score": 1.0, "content": "exemplified in Fig. 7 in the Appendix. We refer to this configuration as SLASH Attention.", "type": "text" } ], "index": 50 } ], "index": 47.5, "bbox_fs": [ 105, 572, 506, 639 ] }, { "type": "text", "bbox": [ 107, 643, 505, 732 ], "lines": [ { "bbox": [ 106, 644, 505, 656 ], "spans": [ { "bbox": [ 106, 644, 505, 656 ], "score": 1.0, "content": "We compare SLASH Attention to a baseline slot attention encoder using an MLP and Hungarian", "type": "text" } ], "index": 51 }, { "bbox": [ 105, 655, 505, 667 ], "spans": [ { "bbox": [ 105, 655, 505, 667 ], "score": 1.0, "content": "loss for predicting the object properties from the slot encodings as in Locatello et al. (2020). The", "type": "text" } ], "index": 52 }, { "bbox": [ 105, 666, 506, 678 ], "spans": [ { "bbox": [ 105, 666, 506, 678 ], "score": 1.0, "content": "results of these experiments can be found in Fig. 3a (top). We observe that the average precision after", "type": "text" } ], "index": 53 }, { "bbox": [ 105, 677, 505, 689 ], "spans": [ { "bbox": [ 105, 677, 505, 689 ], "score": 1.0, "content": "convergence on the held-out test set with SLASH Attention is greatly improved to that of the baseline", "type": "text" } ], "index": 54 }, { "bbox": [ 105, 687, 505, 700 ], "spans": [ { "bbox": [ 105, 687, 505, 700 ], "score": 1.0, "content": "model. Additionally, in Fig. 3b we observe that SLASH Attention reaches the average precision", "type": "text" } ], "index": 55 }, { "bbox": [ 105, 698, 505, 712 ], "spans": [ { "bbox": [ 105, 698, 505, 712 ], "score": 1.0, "content": "value of the baseline model in much fewer number of epochs. Thus, we can summarize that adding", "type": "text" } ], "index": 56 }, { "bbox": [ 105, 710, 506, 722 ], "spans": [ { "bbox": [ 105, 710, 506, 722 ], "score": 1.0, "content": "logical knowledge in the training procedure via SLASH can greatly improve the capabilities of a", "type": "text" } ], "index": 57 }, { "bbox": [ 105, 721, 239, 733 ], "spans": [ { "bbox": [ 105, 721, 239, 733 ], "score": 1.0, "content": "neural module for set prediction.", "type": "text" } ], "index": 58 } ], "index": 54.5, "bbox_fs": [ 105, 644, 506, 733 ] } ] }, { "preproc_blocks": [ { "type": "text", "bbox": [ 107, 82, 505, 137 ], "lines": [ { "bbox": [ 106, 83, 505, 95 ], "spans": [ { "bbox": [ 106, 83, 505, 95 ], "score": 1.0, "content": "Evaluation 4: Improved Compositional Generalization with SLASH. To test the hypothesis that", "type": "text" } ], "index": 0 }, { "bbox": [ 106, 94, 506, 106 ], "spans": [ { "bbox": [ 106, 94, 506, 106 ], "score": 1.0, "content": "SLASH Attention possesses improved generalization properties in comparison to the baseline model,", "type": "text" } ], "index": 1 }, { "bbox": [ 105, 104, 506, 117 ], "spans": [ { "bbox": [ 105, 104, 506, 117 ], "score": 1.0, "content": "we ran experiments on a variant of ShapeWorld4 similar to the CLEVR Compositional Generalization", "type": "text" } ], "index": 2 }, { "bbox": [ 105, 114, 505, 129 ], "spans": [ { "bbox": [ 105, 114, 505, 129 ], "score": 1.0, "content": "Test (CoGenT) (Johnson et al., 2017). The goal of CoGenT is to investigate a model’s ability to", "type": "text" } ], "index": 3 }, { "bbox": [ 105, 125, 405, 141 ], "spans": [ { "bbox": [ 105, 125, 405, 141 ], "score": 1.0, "content": "handle novel combinations of attributes that were not seen during training.", "type": "text" } ], "index": 4 } ], "index": 2 }, { "type": "text", "bbox": [ 106, 143, 505, 221 ], "lines": [ { "bbox": [ 106, 143, 505, 155 ], "spans": [ { "bbox": [ 106, 143, 505, 155 ], "score": 1.0, "content": "For this purpose, we established two conditions within a ShapeWorld4 CoGenT data set: Condition", "type": "text" } ], "index": 5 }, { "bbox": [ 105, 154, 506, 167 ], "spans": [ { "bbox": [ 105, 154, 506, 167 ], "score": 1.0, "content": "(A) – the training and test data set contains squares with the colors gray, blue, brown, or yellow,", "type": "text" } ], "index": 6 }, { "bbox": [ 105, 165, 505, 178 ], "spans": [ { "bbox": [ 105, 165, 505, 178 ], "score": 1.0, "content": "triangles with the colors red, green, magenta, or cyan and circles of all colors. Condition (B) – the", "type": "text" } ], "index": 7 }, { "bbox": [ 105, 176, 507, 190 ], "spans": [ { "bbox": [ 105, 176, 507, 190 ], "score": 1.0, "content": "training set is as in Condition (A). However, the test set contains squares with the colors red, green,", "type": "text" } ], "index": 8 }, { "bbox": [ 105, 187, 505, 199 ], "spans": [ { "bbox": [ 105, 187, 505, 199 ], "score": 1.0, "content": "magenta, or cyan, triangles with the colors gray, blue, brown, or yellow and circles of all colors. The", "type": "text" } ], "index": 9 }, { "bbox": [ 105, 198, 507, 210 ], "spans": [ { "bbox": [ 105, 198, 507, 210 ], "score": 1.0, "content": "goal is to investigate how well a model can generalize that, e.g., also squares can have the color red,", "type": "text" } ], "index": 10 }, { "bbox": [ 105, 209, 350, 223 ], "spans": [ { "bbox": [ 105, 209, 350, 223 ], "score": 1.0, "content": "although never having seen evidence for this during training.", "type": "text" } ], "index": 11 } ], "index": 8 }, { "type": "text", "bbox": [ 107, 226, 505, 292 ], "lines": [ { "bbox": [ 105, 225, 506, 239 ], "spans": [ { "bbox": [ 105, 225, 506, 239 ], "score": 1.0, "content": "The resulting average precision test scores are presented in Fig. 3a (bottom). We observe that, even", "type": "text" } ], "index": 12 }, { "bbox": [ 105, 236, 505, 250 ], "spans": [ { "bbox": [ 105, 236, 505, 250 ], "score": 1.0, "content": "though the SLASH Program used for this experiment was not explicitly written to handle composition", "type": "text" } ], "index": 13 }, { "bbox": [ 104, 246, 506, 261 ], "spans": [ { "bbox": [ 104, 246, 506, 261 ], "score": 1.0, "content": "generalization, SLASH Attention shows greatly improved generalization capabilities. This can be", "type": "text" } ], "index": 14 }, { "bbox": [ 105, 259, 505, 272 ], "spans": [ { "bbox": [ 105, 259, 184, 272 ], "score": 1.0, "content": "seen in the approx.", "type": "text" }, { "bbox": [ 184, 259, 204, 270 ], "score": 0.88, "content": "1 3 \\%", "type": "inline_equation" }, { "bbox": [ 204, 259, 505, 272 ], "score": 1.0, "content": "higher average precision scores on the Condition (B) test set in comparison", "type": "text" } ], "index": 15 }, { "bbox": [ 105, 269, 505, 282 ], "spans": [ { "bbox": [ 105, 269, 505, 282 ], "score": 1.0, "content": "to the baseline model. Importantly, this trend still holds even when subtracting the higher precision", "type": "text" } ], "index": 16 }, { "bbox": [ 105, 281, 244, 293 ], "spans": [ { "bbox": [ 105, 281, 244, 293 ], "score": 1.0, "content": "scores observed in Condition (A).", "type": "text" } ], "index": 17 } ], "index": 14.5 }, { "type": "text", "bbox": [ 107, 298, 505, 375 ], "lines": [ { "bbox": [ 105, 297, 506, 311 ], "spans": [ { "bbox": [ 105, 297, 506, 311 ], "score": 1.0, "content": "To summarize our findings from the experiments on set prediction: we observe that adding prior", "type": "text" } ], "index": 18 }, { "bbox": [ 106, 309, 505, 321 ], "spans": [ { "bbox": [ 106, 309, 505, 321 ], "score": 1.0, "content": "knowledge in the form of logical constraints via SLASH can greatly improve a neural module in", "type": "text" } ], "index": 19 }, { "bbox": [ 106, 320, 506, 332 ], "spans": [ { "bbox": [ 106, 320, 506, 332 ], "score": 1.0, "content": "terms of performance and generalizability. On a side note: training neural networks for novel tasks,", "type": "text" } ], "index": 20 }, { "bbox": [ 106, 331, 506, 343 ], "spans": [ { "bbox": [ 106, 331, 506, 343 ], "score": 1.0, "content": "often involves defining explicit loss functions, e.g. Hungarian loss for set prediction. In contrast", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 342, 506, 354 ], "spans": [ { "bbox": [ 105, 342, 506, 354 ], "score": 1.0, "content": "with SLASH, no matter the choice of NPP and underlying task, the training loss remains the same.", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 352, 505, 365 ], "spans": [ { "bbox": [ 105, 352, 505, 365 ], "score": 1.0, "content": "Task-related requirements simply need to be added as lines of code to the SLASH program. This", "type": "text" } ], "index": 23 }, { "bbox": [ 105, 363, 342, 376 ], "spans": [ { "bbox": [ 105, 363, 342, 376 ], "score": 1.0, "content": "additionally highlights SLASH’s versatility and flexibility.", "type": "text" } ], "index": 24 } ], "index": 21 }, { "type": "text", "bbox": [ 107, 380, 505, 425 ], "lines": [ { "bbox": [ 105, 380, 506, 393 ], "spans": [ { "bbox": [ 105, 380, 506, 393 ], "score": 1.0, "content": "Summary of all Empirical Results. All empirical results together demonstrate that the flexibility of", "type": "text" } ], "index": 25 }, { "bbox": [ 105, 391, 506, 404 ], "spans": [ { "bbox": [ 105, 391, 506, 404 ], "score": 1.0, "content": "SLASH is highly beneficial and can easily outperform state-of-the-art: one can freely combine what", "type": "text" } ], "index": 26 }, { "bbox": [ 105, 403, 505, 416 ], "spans": [ { "bbox": [ 105, 403, 505, 416 ], "score": 1.0, "content": "is required to solve the underlying task — (deep) neural networks, PCs, and logic. Particularly, the", "type": "text" } ], "index": 27 }, { "bbox": [ 106, 414, 395, 425 ], "spans": [ { "bbox": [ 106, 414, 395, 425 ], "score": 1.0, "content": "results indicate the potential of integrating PCs via SLASH into DPPLs.", "type": "text" } ], "index": 28 } ], "index": 26.5 }, { "type": "title", "bbox": [ 107, 465, 302, 479 ], "lines": [ { "bbox": [ 105, 465, 304, 480 ], "spans": [ { "bbox": [ 105, 465, 304, 480 ], "score": 1.0, "content": "5 CONCLUSION AND FUTURE WORK", "type": "text" } ], "index": 29 } ], "index": 29 }, { "type": "text", "bbox": [ 107, 506, 505, 660 ], "lines": [ { "bbox": [ 105, 505, 507, 519 ], "spans": [ { "bbox": [ 105, 505, 507, 519 ], "score": 1.0, "content": "We introduce SLASH, a novel DPPL that integrates neural computations with tractable probability es-", "type": "text" } ], "index": 30 }, { "bbox": [ 106, 518, 505, 529 ], "spans": [ { "bbox": [ 106, 518, 505, 529 ], "score": 1.0, "content": "timates and logical statements. The key ingredient of SLASH to achieve this are Neural-Probabilistic", "type": "text" } ], "index": 31 }, { "bbox": [ 105, 528, 505, 540 ], "spans": [ { "bbox": [ 105, 528, 505, 540 ], "score": 1.0, "content": "Predicates (NPPs) that can be flexibly constructed out of neural and/or probabilistic circuit modules", "type": "text" } ], "index": 32 }, { "bbox": [ 105, 538, 506, 553 ], "spans": [ { "bbox": [ 105, 538, 506, 553 ], "score": 1.0, "content": "based on the data and underlying task. With these NPPs, one can produce task-specific probability", "type": "text" } ], "index": 33 }, { "bbox": [ 106, 550, 506, 563 ], "spans": [ { "bbox": [ 106, 550, 506, 563 ], "score": 1.0, "content": "estimates. The details and additional prior knowledge of a task are neatly encompassed within a", "type": "text" } ], "index": 34 }, { "bbox": [ 105, 561, 506, 574 ], "spans": [ { "bbox": [ 105, 561, 506, 574 ], "score": 1.0, "content": "SLASH program with only few lines of code. Finally, via Answer Set Programming and Weighted", "type": "text" } ], "index": 35 }, { "bbox": [ 105, 571, 506, 585 ], "spans": [ { "bbox": [ 105, 571, 506, 585 ], "score": 1.0, "content": "Model Counting, the logical SLASH program and probability estimates from the NPPs are combined", "type": "text" } ], "index": 36 }, { "bbox": [ 105, 582, 506, 597 ], "spans": [ { "bbox": [ 105, 582, 506, 597 ], "score": 1.0, "content": "to estimate the truth value of a task-specific query. Our experiments show the power and efficiency", "type": "text" } ], "index": 37 }, { "bbox": [ 105, 594, 506, 607 ], "spans": [ { "bbox": [ 105, 594, 506, 607 ], "score": 1.0, "content": "of SLASH, improving upon previous DPPLs in the benchmark MNIST-Addition task in terms of", "type": "text" } ], "index": 38 }, { "bbox": [ 105, 605, 506, 618 ], "spans": [ { "bbox": [ 105, 605, 506, 618 ], "score": 1.0, "content": "performance, efficiency and robustness. Importantly, by integrating a SOTA slot attention encoder", "type": "text" } ], "index": 39 }, { "bbox": [ 106, 616, 506, 629 ], "spans": [ { "bbox": [ 106, 616, 506, 629 ], "score": 1.0, "content": "into NPPs and adding few logical constraints, SLASH demonstrates improved performances and", "type": "text" } ], "index": 40 }, { "bbox": [ 105, 627, 506, 640 ], "spans": [ { "bbox": [ 105, 627, 506, 640 ], "score": 1.0, "content": "generalizability in comparison to the pure slot encoder for the task of object-centric set prediction;", "type": "text" } ], "index": 41 }, { "bbox": [ 105, 637, 505, 650 ], "spans": [ { "bbox": [ 105, 637, 505, 650 ], "score": 1.0, "content": "a setting no DPPL has tackled yet. This shows the great potential of DPPLs to elegantly combine", "type": "text" } ], "index": 42 }, { "bbox": [ 106, 649, 387, 662 ], "spans": [ { "bbox": [ 106, 649, 387, 662 ], "score": 1.0, "content": "logical reasoning with neural computations and uncertainty estimates.", "type": "text" } ], "index": 43 } ], "index": 36.5 }, { "type": "text", "bbox": [ 107, 666, 505, 732 ], "lines": [ { "bbox": [ 105, 665, 505, 678 ], "spans": [ { "bbox": [ 105, 665, 505, 678 ], "score": 1.0, "content": "Interesting avenues for future work include benchmarking SLASH on additional data types and", "type": "text" } ], "index": 44 }, { "bbox": [ 105, 677, 505, 690 ], "spans": [ { "bbox": [ 105, 677, 505, 690 ], "score": 1.0, "content": "tasks. One should explore unsupervised and weakly supervised learning using logic with SLASH and", "type": "text" } ], "index": 45 }, { "bbox": [ 106, 688, 505, 700 ], "spans": [ { "bbox": [ 106, 688, 505, 700 ], "score": 1.0, "content": "investigate how far logical constraints can help unsupervised object discovery. In direct alignment", "type": "text" } ], "index": 46 }, { "bbox": [ 105, 699, 505, 711 ], "spans": [ { "bbox": [ 105, 699, 505, 711 ], "score": 1.0, "content": "with our work, one should also investigate image generation via the beneficial feature of PCs to", "type": "text" } ], "index": 47 }, { "bbox": [ 105, 709, 506, 722 ], "spans": [ { "bbox": [ 105, 709, 506, 722 ], "score": 1.0, "content": "generate random samples. Actually, it should be possible to generate images that encapsulate logical", "type": "text" } ], "index": 48 }, { "bbox": [ 106, 721, 435, 732 ], "spans": [ { "bbox": [ 106, 721, 435, 732 ], "score": 1.0, "content": "knowledge bases. This is important to move from data-rich to knowledge-rich AI.", "type": "text" } ], "index": 49 } ], "index": 46.5 } ], "page_idx": 8, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 107, 27, 308, 37 ], "lines": [ { "bbox": [ 106, 25, 308, 38 ], "spans": [ { "bbox": [ 106, 25, 308, 38 ], "score": 1.0, "content": "Under review as a conference paper at ICLR 2022", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 302, 751, 308, 759 ], "lines": [ { "bbox": [ 302, 751, 309, 762 ], "spans": [ { "bbox": [ 302, 751, 309, 762 ], "score": 1.0, "content": "9", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "text", "bbox": [ 107, 82, 505, 137 ], "lines": [ { "bbox": [ 106, 83, 505, 95 ], "spans": [ { "bbox": [ 106, 83, 505, 95 ], "score": 1.0, "content": "Evaluation 4: Improved Compositional Generalization with SLASH. To test the hypothesis that", "type": "text" } ], "index": 0 }, { "bbox": [ 106, 94, 506, 106 ], "spans": [ { "bbox": [ 106, 94, 506, 106 ], "score": 1.0, "content": "SLASH Attention possesses improved generalization properties in comparison to the baseline model,", "type": "text" } ], "index": 1 }, { "bbox": [ 105, 104, 506, 117 ], "spans": [ { "bbox": [ 105, 104, 506, 117 ], "score": 1.0, "content": "we ran experiments on a variant of ShapeWorld4 similar to the CLEVR Compositional Generalization", "type": "text" } ], "index": 2 }, { "bbox": [ 105, 114, 505, 129 ], "spans": [ { "bbox": [ 105, 114, 505, 129 ], "score": 1.0, "content": "Test (CoGenT) (Johnson et al., 2017). The goal of CoGenT is to investigate a model’s ability to", "type": "text" } ], "index": 3 }, { "bbox": [ 105, 125, 405, 141 ], "spans": [ { "bbox": [ 105, 125, 405, 141 ], "score": 1.0, "content": "handle novel combinations of attributes that were not seen during training.", "type": "text" } ], "index": 4 } ], "index": 2, "bbox_fs": [ 105, 83, 506, 141 ] }, { "type": "text", "bbox": [ 106, 143, 505, 221 ], "lines": [ { "bbox": [ 106, 143, 505, 155 ], "spans": [ { "bbox": [ 106, 143, 505, 155 ], "score": 1.0, "content": "For this purpose, we established two conditions within a ShapeWorld4 CoGenT data set: Condition", "type": "text" } ], "index": 5 }, { "bbox": [ 105, 154, 506, 167 ], "spans": [ { "bbox": [ 105, 154, 506, 167 ], "score": 1.0, "content": "(A) – the training and test data set contains squares with the colors gray, blue, brown, or yellow,", "type": "text" } ], "index": 6 }, { "bbox": [ 105, 165, 505, 178 ], "spans": [ { "bbox": [ 105, 165, 505, 178 ], "score": 1.0, "content": "triangles with the colors red, green, magenta, or cyan and circles of all colors. Condition (B) – the", "type": "text" } ], "index": 7 }, { "bbox": [ 105, 176, 507, 190 ], "spans": [ { "bbox": [ 105, 176, 507, 190 ], "score": 1.0, "content": "training set is as in Condition (A). However, the test set contains squares with the colors red, green,", "type": "text" } ], "index": 8 }, { "bbox": [ 105, 187, 505, 199 ], "spans": [ { "bbox": [ 105, 187, 505, 199 ], "score": 1.0, "content": "magenta, or cyan, triangles with the colors gray, blue, brown, or yellow and circles of all colors. The", "type": "text" } ], "index": 9 }, { "bbox": [ 105, 198, 507, 210 ], "spans": [ { "bbox": [ 105, 198, 507, 210 ], "score": 1.0, "content": "goal is to investigate how well a model can generalize that, e.g., also squares can have the color red,", "type": "text" } ], "index": 10 }, { "bbox": [ 105, 209, 350, 223 ], "spans": [ { "bbox": [ 105, 209, 350, 223 ], "score": 1.0, "content": "although never having seen evidence for this during training.", "type": "text" } ], "index": 11 } ], "index": 8, "bbox_fs": [ 105, 143, 507, 223 ] }, { "type": "text", "bbox": [ 107, 226, 505, 292 ], "lines": [ { "bbox": [ 105, 225, 506, 239 ], "spans": [ { "bbox": [ 105, 225, 506, 239 ], "score": 1.0, "content": "The resulting average precision test scores are presented in Fig. 3a (bottom). We observe that, even", "type": "text" } ], "index": 12 }, { "bbox": [ 105, 236, 505, 250 ], "spans": [ { "bbox": [ 105, 236, 505, 250 ], "score": 1.0, "content": "though the SLASH Program used for this experiment was not explicitly written to handle composition", "type": "text" } ], "index": 13 }, { "bbox": [ 104, 246, 506, 261 ], "spans": [ { "bbox": [ 104, 246, 506, 261 ], "score": 1.0, "content": "generalization, SLASH Attention shows greatly improved generalization capabilities. This can be", "type": "text" } ], "index": 14 }, { "bbox": [ 105, 259, 505, 272 ], "spans": [ { "bbox": [ 105, 259, 184, 272 ], "score": 1.0, "content": "seen in the approx.", "type": "text" }, { "bbox": [ 184, 259, 204, 270 ], "score": 0.88, "content": "1 3 \\%", "type": "inline_equation" }, { "bbox": [ 204, 259, 505, 272 ], "score": 1.0, "content": "higher average precision scores on the Condition (B) test set in comparison", "type": "text" } ], "index": 15 }, { "bbox": [ 105, 269, 505, 282 ], "spans": [ { "bbox": [ 105, 269, 505, 282 ], "score": 1.0, "content": "to the baseline model. Importantly, this trend still holds even when subtracting the higher precision", "type": "text" } ], "index": 16 }, { "bbox": [ 105, 281, 244, 293 ], "spans": [ { "bbox": [ 105, 281, 244, 293 ], "score": 1.0, "content": "scores observed in Condition (A).", "type": "text" } ], "index": 17 } ], "index": 14.5, "bbox_fs": [ 104, 225, 506, 293 ] }, { "type": "text", "bbox": [ 107, 298, 505, 375 ], "lines": [ { "bbox": [ 105, 297, 506, 311 ], "spans": [ { "bbox": [ 105, 297, 506, 311 ], "score": 1.0, "content": "To summarize our findings from the experiments on set prediction: we observe that adding prior", "type": "text" } ], "index": 18 }, { "bbox": [ 106, 309, 505, 321 ], "spans": [ { "bbox": [ 106, 309, 505, 321 ], "score": 1.0, "content": "knowledge in the form of logical constraints via SLASH can greatly improve a neural module in", "type": "text" } ], "index": 19 }, { "bbox": [ 106, 320, 506, 332 ], "spans": [ { "bbox": [ 106, 320, 506, 332 ], "score": 1.0, "content": "terms of performance and generalizability. On a side note: training neural networks for novel tasks,", "type": "text" } ], "index": 20 }, { "bbox": [ 106, 331, 506, 343 ], "spans": [ { "bbox": [ 106, 331, 506, 343 ], "score": 1.0, "content": "often involves defining explicit loss functions, e.g. Hungarian loss for set prediction. In contrast", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 342, 506, 354 ], "spans": [ { "bbox": [ 105, 342, 506, 354 ], "score": 1.0, "content": "with SLASH, no matter the choice of NPP and underlying task, the training loss remains the same.", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 352, 505, 365 ], "spans": [ { "bbox": [ 105, 352, 505, 365 ], "score": 1.0, "content": "Task-related requirements simply need to be added as lines of code to the SLASH program. This", "type": "text" } ], "index": 23 }, { "bbox": [ 105, 363, 342, 376 ], "spans": [ { "bbox": [ 105, 363, 342, 376 ], "score": 1.0, "content": "additionally highlights SLASH’s versatility and flexibility.", "type": "text" } ], "index": 24 } ], "index": 21, "bbox_fs": [ 105, 297, 506, 376 ] }, { "type": "text", "bbox": [ 107, 380, 505, 425 ], "lines": [ { "bbox": [ 105, 380, 506, 393 ], "spans": [ { "bbox": [ 105, 380, 506, 393 ], "score": 1.0, "content": "Summary of all Empirical Results. All empirical results together demonstrate that the flexibility of", "type": "text" } ], "index": 25 }, { "bbox": [ 105, 391, 506, 404 ], "spans": [ { "bbox": [ 105, 391, 506, 404 ], "score": 1.0, "content": "SLASH is highly beneficial and can easily outperform state-of-the-art: one can freely combine what", "type": "text" } ], "index": 26 }, { "bbox": [ 105, 403, 505, 416 ], "spans": [ { "bbox": [ 105, 403, 505, 416 ], "score": 1.0, "content": "is required to solve the underlying task — (deep) neural networks, PCs, and logic. Particularly, the", "type": "text" } ], "index": 27 }, { "bbox": [ 106, 414, 395, 425 ], "spans": [ { "bbox": [ 106, 414, 395, 425 ], "score": 1.0, "content": "results indicate the potential of integrating PCs via SLASH into DPPLs.", "type": "text" } ], "index": 28 } ], "index": 26.5, "bbox_fs": [ 105, 380, 506, 425 ] }, { "type": "title", "bbox": [ 107, 465, 302, 479 ], "lines": [ { "bbox": [ 105, 465, 304, 480 ], "spans": [ { "bbox": [ 105, 465, 304, 480 ], "score": 1.0, "content": "5 CONCLUSION AND FUTURE WORK", "type": "text" } ], "index": 29 } ], "index": 29 }, { "type": "text", "bbox": [ 107, 506, 505, 660 ], "lines": [ { "bbox": [ 105, 505, 507, 519 ], "spans": [ { "bbox": [ 105, 505, 507, 519 ], "score": 1.0, "content": "We introduce SLASH, a novel DPPL that integrates neural computations with tractable probability es-", "type": "text" } ], "index": 30 }, { "bbox": [ 106, 518, 505, 529 ], "spans": [ { "bbox": [ 106, 518, 505, 529 ], "score": 1.0, "content": "timates and logical statements. The key ingredient of SLASH to achieve this are Neural-Probabilistic", "type": "text" } ], "index": 31 }, { "bbox": [ 105, 528, 505, 540 ], "spans": [ { "bbox": [ 105, 528, 505, 540 ], "score": 1.0, "content": "Predicates (NPPs) that can be flexibly constructed out of neural and/or probabilistic circuit modules", "type": "text" } ], "index": 32 }, { "bbox": [ 105, 538, 506, 553 ], "spans": [ { "bbox": [ 105, 538, 506, 553 ], "score": 1.0, "content": "based on the data and underlying task. With these NPPs, one can produce task-specific probability", "type": "text" } ], "index": 33 }, { "bbox": [ 106, 550, 506, 563 ], "spans": [ { "bbox": [ 106, 550, 506, 563 ], "score": 1.0, "content": "estimates. The details and additional prior knowledge of a task are neatly encompassed within a", "type": "text" } ], "index": 34 }, { "bbox": [ 105, 561, 506, 574 ], "spans": [ { "bbox": [ 105, 561, 506, 574 ], "score": 1.0, "content": "SLASH program with only few lines of code. Finally, via Answer Set Programming and Weighted", "type": "text" } ], "index": 35 }, { "bbox": [ 105, 571, 506, 585 ], "spans": [ { "bbox": [ 105, 571, 506, 585 ], "score": 1.0, "content": "Model Counting, the logical SLASH program and probability estimates from the NPPs are combined", "type": "text" } ], "index": 36 }, { "bbox": [ 105, 582, 506, 597 ], "spans": [ { "bbox": [ 105, 582, 506, 597 ], "score": 1.0, "content": "to estimate the truth value of a task-specific query. Our experiments show the power and efficiency", "type": "text" } ], "index": 37 }, { "bbox": [ 105, 594, 506, 607 ], "spans": [ { "bbox": [ 105, 594, 506, 607 ], "score": 1.0, "content": "of SLASH, improving upon previous DPPLs in the benchmark MNIST-Addition task in terms of", "type": "text" } ], "index": 38 }, { "bbox": [ 105, 605, 506, 618 ], "spans": [ { "bbox": [ 105, 605, 506, 618 ], "score": 1.0, "content": "performance, efficiency and robustness. Importantly, by integrating a SOTA slot attention encoder", "type": "text" } ], "index": 39 }, { "bbox": [ 106, 616, 506, 629 ], "spans": [ { "bbox": [ 106, 616, 506, 629 ], "score": 1.0, "content": "into NPPs and adding few logical constraints, SLASH demonstrates improved performances and", "type": "text" } ], "index": 40 }, { "bbox": [ 105, 627, 506, 640 ], "spans": [ { "bbox": [ 105, 627, 506, 640 ], "score": 1.0, "content": "generalizability in comparison to the pure slot encoder for the task of object-centric set prediction;", "type": "text" } ], "index": 41 }, { "bbox": [ 105, 637, 505, 650 ], "spans": [ { "bbox": [ 105, 637, 505, 650 ], "score": 1.0, "content": "a setting no DPPL has tackled yet. This shows the great potential of DPPLs to elegantly combine", "type": "text" } ], "index": 42 }, { "bbox": [ 106, 649, 387, 662 ], "spans": [ { "bbox": [ 106, 649, 387, 662 ], "score": 1.0, "content": "logical reasoning with neural computations and uncertainty estimates.", "type": "text" } ], "index": 43 } ], "index": 36.5, "bbox_fs": [ 105, 505, 507, 662 ] }, { "type": "text", "bbox": [ 107, 666, 505, 732 ], "lines": [ { "bbox": [ 105, 665, 505, 678 ], "spans": [ { "bbox": [ 105, 665, 505, 678 ], "score": 1.0, "content": "Interesting avenues for future work include benchmarking SLASH on additional data types and", "type": "text" } ], "index": 44 }, { "bbox": [ 105, 677, 505, 690 ], "spans": [ { "bbox": [ 105, 677, 505, 690 ], "score": 1.0, "content": "tasks. One should explore unsupervised and weakly supervised learning using logic with SLASH and", "type": "text" } ], "index": 45 }, { "bbox": [ 106, 688, 505, 700 ], "spans": [ { "bbox": [ 106, 688, 505, 700 ], "score": 1.0, "content": "investigate how far logical constraints can help unsupervised object discovery. In direct alignment", "type": "text" } ], "index": 46 }, { "bbox": [ 105, 699, 505, 711 ], "spans": [ { "bbox": [ 105, 699, 505, 711 ], "score": 1.0, "content": "with our work, one should also investigate image generation via the beneficial feature of PCs to", "type": "text" } ], "index": 47 }, { "bbox": [ 105, 709, 506, 722 ], "spans": [ { "bbox": [ 105, 709, 506, 722 ], "score": 1.0, "content": "generate random samples. Actually, it should be possible to generate images that encapsulate logical", "type": "text" } ], "index": 48 }, { "bbox": [ 106, 721, 435, 732 ], "spans": [ { "bbox": [ 106, 721, 435, 732 ], "score": 1.0, "content": "knowledge bases. This is important to move from data-rich to knowledge-rich AI.", "type": "text" } ], "index": 49 } ], "index": 46.5, "bbox_fs": [ 105, 665, 506, 732 ] } ] }, { "preproc_blocks": [ { "type": "title", "bbox": [ 108, 81, 211, 94 ], "lines": [ { "bbox": [ 106, 81, 212, 95 ], "spans": [ { "bbox": [ 106, 81, 212, 95 ], "score": 1.0, "content": "ETHICS STATEMENT", "type": "text" } ], "index": 0 } ], "index": 0 }, { "type": "text", "bbox": [ 107, 105, 505, 172 ], "lines": [ { "bbox": [ 106, 106, 505, 118 ], "spans": [ { "bbox": [ 106, 106, 505, 118 ], "score": 1.0, "content": "With our work, we have shown that one can add prior knowledge and logical constraints to the", "type": "text" } ], "index": 1 }, { "bbox": [ 106, 117, 505, 129 ], "spans": [ { "bbox": [ 106, 117, 505, 129 ], "score": 1.0, "content": "training of learning systems. We postulate that SLASH can therefore additionally be used to identify", "type": "text" } ], "index": 2 }, { "bbox": [ 105, 127, 506, 141 ], "spans": [ { "bbox": [ 105, 127, 506, 141 ], "score": 1.0, "content": "and remove biases or undesirable behavior, by adding constraints within the SLASH program. We", "type": "text" } ], "index": 3 }, { "bbox": [ 105, 137, 507, 153 ], "spans": [ { "bbox": [ 105, 137, 507, 153 ], "score": 1.0, "content": "observe that this feature, however, also has the potential danger to be used in the opposite way, e.g.", "type": "text" } ], "index": 4 }, { "bbox": [ 106, 150, 505, 163 ], "spans": [ { "bbox": [ 106, 150, 505, 163 ], "score": 1.0, "content": "explicitly adding bias and discriminatory factors to a system. To the best of our knowledge, our study", "type": "text" } ], "index": 5 }, { "bbox": [ 106, 162, 370, 173 ], "spans": [ { "bbox": [ 106, 162, 370, 173 ], "score": 1.0, "content": "does not raise any ethical, privacy or conflict of interest concerns.", "type": "text" } ], "index": 6 } ], "index": 3.5 }, { "type": "title", "bbox": [ 108, 188, 267, 200 ], "lines": [ { "bbox": [ 106, 187, 270, 202 ], "spans": [ { "bbox": [ 106, 187, 270, 202 ], "score": 1.0, "content": "REPRODUCIBILITY STATEMENT", "type": "text" } ], "index": 7 } ], "index": 7 }, { "type": "text", "bbox": [ 107, 213, 505, 268 ], "lines": [ { "bbox": [ 106, 213, 504, 225 ], "spans": [ { "bbox": [ 106, 213, 504, 225 ], "score": 1.0, "content": "An official, curated GitHub repository will be made public with the final version, containing the code", "type": "text" } ], "index": 8 }, { "bbox": [ 105, 223, 506, 236 ], "spans": [ { "bbox": [ 105, 223, 506, 236 ], "score": 1.0, "content": "of SLASH, as well as scripts to reproduce the experiments and generate data sets. In addition to this,", "type": "text" } ], "index": 9 }, { "bbox": [ 106, 235, 505, 247 ], "spans": [ { "bbox": [ 106, 235, 505, 247 ], "score": 1.0, "content": "architectural details and hyperparameters are included in the appendix. Preliminary code will be", "type": "text" } ], "index": 10 }, { "bbox": [ 106, 246, 505, 259 ], "spans": [ { "bbox": [ 106, 246, 505, 259 ], "score": 1.0, "content": "uploaded upon submission. Lastly, details on the evaluation metrics and relevant data sets are given", "type": "text" } ], "index": 11 }, { "bbox": [ 106, 257, 253, 269 ], "spans": [ { "bbox": [ 106, 257, 253, 269 ], "score": 1.0, "content": "in the main text as well as appendix.", "type": "text" } ], "index": 12 } ], "index": 10 }, { "type": "title", "bbox": [ 108, 285, 175, 297 ], "lines": [ { "bbox": [ 106, 284, 176, 298 ], "spans": [ { "bbox": [ 106, 284, 176, 298 ], "score": 1.0, "content": "REFERENCES", "type": "text" } ], "index": 13 } ], "index": 13 }, { "type": "text", "bbox": [ 107, 303, 504, 326 ], "lines": [ { "bbox": [ 106, 302, 506, 316 ], "spans": [ { "bbox": [ 106, 302, 506, 316 ], "score": 1.0, "content": "Yoshua Bengio. From System 1 Deep Learning to System 2 Deep Learning. Invited talk NeurIPS,", "type": "text" } ], "index": 14 }, { "bbox": [ 116, 313, 426, 327 ], "spans": [ { "bbox": [ 116, 313, 350, 327 ], "score": 1.0, "content": "2019. URL https://www.youtube.com/watch?v", "type": "text" }, { "bbox": [ 351, 316, 357, 324 ], "score": 0.29, "content": "=", "type": "inline_equation" }, { "bbox": [ 358, 313, 426, 327 ], "score": 1.0, "content": "T3sxeTgT4qc.", "type": "text" } ], "index": 15 } ], "index": 14.5 }, { "type": "text", "bbox": [ 107, 332, 506, 366 ], "lines": [ { "bbox": [ 106, 333, 505, 345 ], "spans": [ { "bbox": [ 106, 333, 505, 345 ], "score": 1.0, "content": "Eli Bingham, Jonathan P. Chen, Martin Jankowiak, Fritz Obermeyer, Neeraj Pradhan, Theofanis", "type": "text" } ], "index": 16 }, { "bbox": [ 115, 342, 506, 358 ], "spans": [ { "bbox": [ 115, 342, 506, 358 ], "score": 1.0, "content": "Karaletsos, Rohit Singh, Paul A. Szerlip, Paul Horsfall, and Noah D. Goodman. Pyro: Deep", "type": "text" } ], "index": 17 }, { "bbox": [ 116, 354, 507, 367 ], "spans": [ { "bbox": [ 116, 354, 507, 367 ], "score": 1.0, "content": "universal probabilistic programming. Journal of Machine Learning Research, pp. 28:1–28:6, 2019.", "type": "text" } ], "index": 18 } ], "index": 17 }, { "type": "text", "bbox": [ 107, 372, 506, 406 ], "lines": [ { "bbox": [ 105, 372, 505, 386 ], "spans": [ { "bbox": [ 105, 372, 505, 386 ], "score": 1.0, "content": "Christopher P. Burgess, Loïc Matthey, Nicholas Watters, Rishabh Kabra, Irina Higgins, Matthew", "type": "text" } ], "index": 19 }, { "bbox": [ 115, 383, 506, 397 ], "spans": [ { "bbox": [ 115, 383, 506, 397 ], "score": 1.0, "content": "Botvinick, and Alexander Lerchner. Monet: Unsupervised scene decomposition and representation.", "type": "text" } ], "index": 20 }, { "bbox": [ 116, 395, 171, 406 ], "spans": [ { "bbox": [ 116, 395, 171, 406 ], "score": 1.0, "content": "CoRR, 2019.", "type": "text" } ], "index": 21 } ], "index": 20 }, { "type": "text", "bbox": [ 106, 413, 506, 457 ], "lines": [ { "bbox": [ 106, 414, 506, 425 ], "spans": [ { "bbox": [ 106, 414, 506, 425 ], "score": 1.0, "content": "Francesco Calimeri, Wolfgang Faber, Martin Gebser, Giovambattista Ianni, Roland Kaminski,", "type": "text" } ], "index": 22 }, { "bbox": [ 115, 424, 506, 437 ], "spans": [ { "bbox": [ 115, 424, 506, 437 ], "score": 1.0, "content": "Thomas Krennwallner, Nicola Leone, Marco Maratea, Francesco Ricca, and Torsten Schaub.", "type": "text" } ], "index": 23 }, { "bbox": [ 115, 435, 507, 448 ], "spans": [ { "bbox": [ 115, 435, 507, 448 ], "score": 1.0, "content": "Asp-core-2 input language format. Theory and Practice of Logic Programming, pp. 294–309,", "type": "text" } ], "index": 24 }, { "bbox": [ 114, 445, 142, 458 ], "spans": [ { "bbox": [ 114, 445, 142, 458 ], "score": 1.0, "content": "2020.", "type": "text" } ], "index": 25 } ], "index": 23.5 }, { "type": "text", "bbox": [ 108, 465, 504, 488 ], "lines": [ { "bbox": [ 106, 464, 506, 479 ], "spans": [ { "bbox": [ 106, 464, 506, 479 ], "score": 1.0, "content": "YooJung Choi, Antonio Vergari, and Guy Van den Broeck. Probabilistic circuits: A unifying", "type": "text" } ], "index": 26 }, { "bbox": [ 116, 476, 465, 488 ], "spans": [ { "bbox": [ 116, 476, 465, 488 ], "score": 1.0, "content": "framework for tractable probabilistic models. Technical report, Technical report, 2020.", "type": "text" } ], "index": 27 } ], "index": 26.5 }, { "type": "text", "bbox": [ 106, 495, 505, 517 ], "lines": [ { "bbox": [ 106, 494, 505, 507 ], "spans": [ { "bbox": [ 106, 494, 505, 507 ], "score": 1.0, "content": "Gabriele Ciravegna, Francesco Giannini, Marco Gori, Marco Maggini, and Stefano Melacci. Human-", "type": "text" } ], "index": 28 }, { "bbox": [ 115, 505, 417, 518 ], "spans": [ { "bbox": [ 115, 505, 417, 518 ], "score": 1.0, "content": "driven FOL explanations of deep learning. In IJCAI, pp. 2234–2240, 2020.", "type": "text" } ], "index": 29 } ], "index": 28.5 }, { "type": "text", "bbox": [ 106, 524, 370, 536 ], "lines": [ { "bbox": [ 106, 524, 371, 537 ], "spans": [ { "bbox": [ 106, 524, 371, 537 ], "score": 1.0, "content": "W. F. Clocksin and Chris Mellish. Programming in Prolog. 1981.", "type": "text" } ], "index": 30 } ], "index": 30 }, { "type": "text", "bbox": [ 109, 543, 475, 555 ], "lines": [ { "bbox": [ 106, 542, 475, 556 ], "spans": [ { "bbox": [ 106, 542, 475, 556 ], "score": 1.0, "content": "Alain Colmerauer and Philippe Roussel. The birth of prolog. In HOPL-II, pp. 37–52, 1993.", "type": "text" } ], "index": 31 } ], "index": 31 }, { "type": "text", "bbox": [ 108, 561, 506, 584 ], "lines": [ { "bbox": [ 105, 560, 506, 575 ], "spans": [ { "bbox": [ 105, 560, 506, 575 ], "score": 1.0, "content": "Adnan Darwiche. SDD: A new canonical representation of propositional knowledge bases. In IJCAI,", "type": "text" } ], "index": 32 }, { "bbox": [ 115, 573, 197, 584 ], "spans": [ { "bbox": [ 115, 573, 197, 584 ], "score": 1.0, "content": "pp. 819–826, 2011.", "type": "text" } ], "index": 33 } ], "index": 32.5 }, { "type": "text", "bbox": [ 106, 591, 462, 603 ], "lines": [ { "bbox": [ 106, 590, 462, 604 ], "spans": [ { "bbox": [ 106, 590, 462, 604 ], "score": 1.0, "content": "Artur d’Avila Garcez and Luís C. Lamb. Neurosymbolic AI: the 3rd wave. CoRR, 2020.", "type": "text" } ], "index": 34 } ], "index": 34 }, { "type": "text", "bbox": [ 108, 609, 504, 632 ], "lines": [ { "bbox": [ 105, 607, 507, 624 ], "spans": [ { "bbox": [ 105, 607, 507, 624 ], "score": 1.0, "content": "Artur S. d’Avila Garcez, Luís C. Lamb, and Dov M. Gabbay. 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URL https://www.youtube.com/watch?v", "type": "text" }, { "bbox": [ 351, 316, 357, 324 ], "score": 0.29, "content": "=", "type": "inline_equation" }, { "bbox": [ 358, 313, 426, 327 ], "score": 1.0, "content": "T3sxeTgT4qc.", "type": "text" } ], "index": 15 } ], "index": 14.5, "bbox_fs": [ 106, 302, 506, 327 ] }, { "type": "text", "bbox": [ 107, 332, 506, 366 ], "lines": [ { "bbox": [ 106, 333, 505, 345 ], "spans": [ { "bbox": [ 106, 333, 505, 345 ], "score": 1.0, "content": "Eli Bingham, Jonathan P. Chen, Martin Jankowiak, Fritz Obermeyer, Neeraj Pradhan, Theofanis", "type": "text" } ], "index": 16 }, { "bbox": [ 115, 342, 506, 358 ], "spans": [ { "bbox": [ 115, 342, 506, 358 ], "score": 1.0, "content": "Karaletsos, Rohit Singh, Paul A. Szerlip, Paul Horsfall, and Noah D. Goodman. Pyro: Deep", "type": "text" } ], "index": 17 }, { "bbox": [ 116, 354, 507, 367 ], "spans": [ { "bbox": [ 116, 354, 507, 367 ], "score": 1.0, "content": "universal probabilistic programming. Journal of Machine Learning Research, pp. 28:1–28:6, 2019.", "type": "text" } ], "index": 18 } ], "index": 17, "bbox_fs": [ 106, 333, 507, 367 ] }, { "type": "text", "bbox": [ 107, 372, 506, 406 ], "lines": [ { "bbox": [ 105, 372, 505, 386 ], "spans": [ { "bbox": [ 105, 372, 505, 386 ], "score": 1.0, "content": "Christopher P. Burgess, Loïc Matthey, Nicholas Watters, Rishabh Kabra, Irina Higgins, Matthew", "type": "text" } ], "index": 19 }, { "bbox": [ 115, 383, 506, 397 ], "spans": [ { "bbox": [ 115, 383, 506, 397 ], "score": 1.0, "content": "Botvinick, and Alexander Lerchner. Monet: Unsupervised scene decomposition and representation.", "type": "text" } ], "index": 20 }, { "bbox": [ 116, 395, 171, 406 ], "spans": [ { "bbox": [ 116, 395, 171, 406 ], "score": 1.0, "content": "CoRR, 2019.", "type": "text" } ], "index": 21 } ], "index": 20, "bbox_fs": [ 105, 372, 506, 406 ] }, { "type": "text", "bbox": [ 106, 413, 506, 457 ], "lines": [ { "bbox": [ 106, 414, 506, 425 ], "spans": [ { "bbox": [ 106, 414, 506, 425 ], "score": 1.0, "content": "Francesco Calimeri, Wolfgang Faber, Martin Gebser, Giovambattista Ianni, Roland Kaminski,", "type": "text" } ], "index": 22 }, { "bbox": [ 115, 424, 506, 437 ], "spans": [ { "bbox": [ 115, 424, 506, 437 ], "score": 1.0, "content": "Thomas Krennwallner, Nicola Leone, Marco Maratea, Francesco Ricca, and Torsten Schaub.", "type": "text" } ], "index": 23 }, { "bbox": [ 115, 435, 507, 448 ], "spans": [ { "bbox": [ 115, 435, 507, 448 ], "score": 1.0, "content": "Asp-core-2 input language format. Theory and Practice of Logic Programming, pp. 294–309,", "type": "text" } ], "index": 24 }, { "bbox": [ 114, 445, 142, 458 ], "spans": [ { "bbox": [ 114, 445, 142, 458 ], "score": 1.0, "content": "2020.", "type": "text" } ], "index": 25 } ], "index": 23.5, "bbox_fs": [ 106, 414, 507, 458 ] }, { "type": "text", "bbox": [ 108, 465, 504, 488 ], "lines": [ { "bbox": [ 106, 464, 506, 479 ], "spans": [ { "bbox": [ 106, 464, 506, 479 ], "score": 1.0, "content": "YooJung Choi, Antonio Vergari, and Guy Van den Broeck. Probabilistic circuits: A unifying", "type": "text" } ], "index": 26 }, { "bbox": [ 116, 476, 465, 488 ], "spans": [ { "bbox": [ 116, 476, 465, 488 ], "score": 1.0, "content": "framework for tractable probabilistic models. Technical report, Technical report, 2020.", "type": "text" } ], "index": 27 } ], "index": 26.5, "bbox_fs": [ 106, 464, 506, 488 ] }, { "type": "text", "bbox": [ 106, 495, 505, 517 ], "lines": [ { "bbox": [ 106, 494, 505, 507 ], "spans": [ { "bbox": [ 106, 494, 505, 507 ], "score": 1.0, "content": "Gabriele Ciravegna, Francesco Giannini, Marco Gori, Marco Maggini, and Stefano Melacci. Human-", "type": "text" } ], "index": 28 }, { "bbox": [ 115, 505, 417, 518 ], "spans": [ { "bbox": [ 115, 505, 417, 518 ], "score": 1.0, "content": "driven FOL explanations of deep learning. In IJCAI, pp. 2234–2240, 2020.", "type": "text" } ], "index": 29 } ], "index": 28.5, "bbox_fs": [ 106, 494, 505, 518 ] }, { "type": "text", "bbox": [ 106, 524, 370, 536 ], "lines": [ { "bbox": [ 106, 524, 371, 537 ], "spans": [ { "bbox": [ 106, 524, 371, 537 ], "score": 1.0, "content": "W. F. Clocksin and Chris Mellish. Programming in Prolog. 1981.", "type": "text" } ], "index": 30 } ], "index": 30, "bbox_fs": [ 106, 524, 371, 537 ] }, { "type": "text", "bbox": [ 109, 543, 475, 555 ], "lines": [ { "bbox": [ 106, 542, 475, 556 ], "spans": [ { "bbox": [ 106, 542, 475, 556 ], "score": 1.0, "content": "Alain Colmerauer and Philippe Roussel. The birth of prolog. In HOPL-II, pp. 37–52, 1993.", "type": "text" } ], "index": 31 } ], "index": 31, "bbox_fs": [ 106, 542, 475, 556 ] }, { "type": "text", "bbox": [ 108, 561, 506, 584 ], "lines": [ { "bbox": [ 105, 560, 506, 575 ], "spans": [ { "bbox": [ 105, 560, 506, 575 ], "score": 1.0, "content": "Adnan Darwiche. SDD: A new canonical representation of propositional knowledge bases. In IJCAI,", "type": "text" } ], "index": 32 }, { "bbox": [ 115, 573, 197, 584 ], "spans": [ { "bbox": [ 115, 573, 197, 584 ], "score": 1.0, "content": "pp. 819–826, 2011.", "type": "text" } ], "index": 33 } ], "index": 32.5, "bbox_fs": [ 105, 560, 506, 584 ] }, { "type": "text", "bbox": [ 106, 591, 462, 603 ], "lines": [ { "bbox": [ 106, 590, 462, 604 ], "spans": [ { "bbox": [ 106, 590, 462, 604 ], "score": 1.0, "content": "Artur d’Avila Garcez and Luís C. Lamb. Neurosymbolic AI: the 3rd wave. CoRR, 2020.", "type": "text" } ], "index": 34 } ], "index": 34, "bbox_fs": [ 106, 590, 462, 604 ] }, { "type": "text", "bbox": [ 108, 609, 504, 632 ], "lines": [ { "bbox": [ 105, 607, 507, 624 ], "spans": [ { "bbox": [ 105, 607, 507, 624 ], "score": 1.0, "content": "Artur S. d’Avila Garcez, Luís C. Lamb, and Dov M. Gabbay. Neural-Symbolic Cognitive Reasoning.", "type": "text" } ], "index": 35 }, { "bbox": [ 116, 621, 241, 633 ], "spans": [ { "bbox": [ 116, 621, 241, 633 ], "score": 1.0, "content": "Cognitive Technologies. 2009.", "type": "text" } ], "index": 36 } ], "index": 35.5, "bbox_fs": [ 105, 607, 507, 633 ] }, { "type": "text", "bbox": [ 107, 639, 505, 673 ], "lines": [ { "bbox": [ 106, 639, 506, 651 ], "spans": [ { "bbox": [ 106, 639, 506, 651 ], "score": 1.0, "content": "Artur S. d’Avila Garcez, Marco Gori, Luís C. Lamb, Luciano Serafini, Michael Spranger, and Son N.", "type": "text" } ], "index": 37 }, { "bbox": [ 115, 650, 505, 663 ], "spans": [ { "bbox": [ 115, 650, 505, 663 ], "score": 1.0, "content": "Tran. Neural-symbolic computing: An effective methodology for principled integration of machine", "type": "text" } ], "index": 38 }, { "bbox": [ 116, 662, 322, 674 ], "spans": [ { "bbox": [ 116, 662, 322, 674 ], "score": 1.0, "content": "learning and reasoning. FLAP, pp. 611–632, 2019.", "type": "text" } ], "index": 39 } ], "index": 38, "bbox_fs": [ 106, 639, 506, 674 ] }, { "type": "text", "bbox": [ 108, 680, 504, 703 ], "lines": [ { "bbox": [ 106, 679, 505, 693 ], "spans": [ { "bbox": [ 106, 679, 505, 693 ], "score": 1.0, "content": "Yannis Dimopoulos, Bernhard Nebel, and Jana Koehler. Encoding planning problems in nonmono-", "type": "text" } ], "index": 40 }, { "bbox": [ 115, 691, 466, 704 ], "spans": [ { "bbox": [ 115, 691, 466, 704 ], "score": 1.0, "content": "tonic logic programs. In ECP, Lecture Notes in Computer Science, pp. 169–181, 1997.", "type": "text" } ], "index": 41 } ], "index": 40.5, "bbox_fs": [ 106, 679, 505, 704 ] }, { "type": "text", "bbox": [ 108, 709, 503, 732 ], "lines": [ { "bbox": [ 106, 709, 505, 722 ], "spans": [ { "bbox": [ 106, 709, 505, 722 ], "score": 1.0, "content": "Martin Engelcke, Adam R. Kosiorek, Oiwi Parker Jones, and Ingmar Posner. GENESIS: generative", "type": "text" } ], "index": 42 }, { "bbox": [ 115, 721, 468, 732 ], "spans": [ { "bbox": [ 115, 721, 468, 732 ], "score": 1.0, "content": "scene inference and sampling with object-centric latent representations. In ICLR, 2020.", "type": "text" } ], "index": 43 } ], "index": 42.5, "bbox_fs": [ 106, 709, 505, 732 ] } ] }, { "preproc_blocks": [ { "type": "text", "bbox": [ 105, 82, 505, 105 ], "lines": [ { "bbox": [ 105, 81, 507, 96 ], "spans": [ { "bbox": [ 105, 81, 507, 96 ], "score": 1.0, "content": "Jerry A Fodor and Zenon W Pylyshyn. Connectionism and cognitive architecture: A critical analysis.", "type": "text" } ], "index": 0 }, { "bbox": [ 116, 93, 226, 106 ], "spans": [ { "bbox": [ 116, 93, 226, 106 ], "score": 1.0, "content": "Cognition, pp. 3–71, 1988.", "type": "text" } ], "index": 1 } ], "index": 0.5 }, { "type": "text", "bbox": [ 108, 112, 503, 146 ], "lines": [ { "bbox": [ 105, 111, 505, 126 ], "spans": [ { "bbox": [ 105, 111, 505, 126 ], "score": 1.0, "content": "Klaus Greff, Raphaël Lopez Kaufman, Rishabh Kabra, Nick Watters, Christopher Burgess, Daniel", "type": "text" } ], "index": 2 }, { "bbox": [ 115, 123, 505, 136 ], "spans": [ { "bbox": [ 115, 123, 505, 136 ], "score": 1.0, "content": "Zoran, Loic Matthey, Matthew Botvinick, and Alexander Lerchner. Multi-object representation", "type": "text" } ], "index": 3 }, { "bbox": [ 115, 134, 424, 147 ], "spans": [ { "bbox": [ 115, 134, 424, 147 ], "score": 1.0, "content": "learning with iterative variational inference. In ICML, pp. 2424–2433, 2019.", "type": "text" } ], "index": 4 } ], "index": 3 }, { "type": "text", "bbox": [ 104, 153, 504, 176 ], "lines": [ { "bbox": [ 105, 153, 506, 167 ], "spans": [ { "bbox": [ 105, 153, 506, 167 ], "score": 1.0, "content": "Klaus Greff, Sjoerd van Steenkiste, and Jürgen Schmidhuber. On the binding problem in artificial", "type": "text" } ], "index": 5 }, { "bbox": [ 116, 165, 239, 176 ], "spans": [ { "bbox": [ 116, 165, 239, 176 ], "score": 1.0, "content": "neural networks. CoRR, 2020.", "type": "text" } ], "index": 6 } ], "index": 5.5 }, { "type": "text", "bbox": [ 105, 183, 505, 207 ], "lines": [ { "bbox": [ 105, 183, 507, 197 ], "spans": [ { "bbox": [ 105, 183, 507, 197 ], "score": 1.0, "content": "Drew A. Hudson and Christopher D. Manning. Learning by abstraction: The neural state machine.", "type": "text" } ], "index": 7 }, { "bbox": [ 116, 195, 256, 207 ], "spans": [ { "bbox": [ 116, 195, 256, 207 ], "score": 1.0, "content": "In NeurIPS, pp. 5901–5914, 2019.", "type": "text" } ], "index": 8 } ], "index": 7.5 }, { "type": "text", "bbox": [ 104, 213, 464, 226 ], "lines": [ { "bbox": [ 105, 213, 464, 226 ], "spans": [ { "bbox": [ 105, 213, 464, 226 ], "score": 1.0, "content": "Jindong Jiang and Sungjin Ahn. Generative neurosymbolic machines. In NeurIPS, 2020.", "type": "text" } ], "index": 9 } ], "index": 9 }, { "type": "text", "bbox": [ 108, 232, 505, 267 ], "lines": [ { "bbox": [ 106, 233, 505, 245 ], "spans": [ { "bbox": [ 106, 233, 505, 245 ], "score": 1.0, "content": "Justin Johnson, Bharath Hariharan, Laurens van der Maaten, Li Fei-Fei, C. Lawrence Zitnick, and", "type": "text" } ], "index": 10 }, { "bbox": [ 115, 244, 505, 257 ], "spans": [ { "bbox": [ 115, 244, 505, 257 ], "score": 1.0, "content": "Ross B. Girshick. CLEVR: A diagnostic dataset for compositional language and elementary visual", "type": "text" } ], "index": 11 }, { "bbox": [ 115, 255, 292, 267 ], "spans": [ { "bbox": [ 115, 255, 292, 267 ], "score": 1.0, "content": "reasoning. In CVPR, pp. 1988–1997, 2017.", "type": "text" } ], "index": 12 } ], "index": 11 }, { "type": "text", "bbox": [ 105, 273, 505, 297 ], "lines": [ { "bbox": [ 105, 273, 506, 287 ], "spans": [ { "bbox": [ 105, 273, 506, 287 ], "score": 1.0, "content": "Diederik P. Kingma and Jimmy Ba. Adam: A method for stochastic optimization. In ICLR 2015,", "type": "text" } ], "index": 13 }, { "bbox": [ 115, 284, 143, 297 ], "spans": [ { "bbox": [ 115, 284, 143, 297 ], "score": 1.0, "content": "2015.", "type": "text" } ], "index": 14 } ], "index": 13.5 }, { "type": "text", "bbox": [ 106, 304, 504, 327 ], "lines": [ { "bbox": [ 106, 303, 505, 317 ], "spans": [ { "bbox": [ 106, 303, 505, 317 ], "score": 1.0, "content": "Alexander Kuhnle and Ann A. Copestake. Shapeworld - A new test methodology for multimodal", "type": "text" } ], "index": 15 }, { "bbox": [ 115, 316, 272, 327 ], "spans": [ { "bbox": [ 115, 316, 272, 327 ], "score": 1.0, "content": "language understanding. CoRR, 2017.", "type": "text" } ], "index": 16 } ], "index": 15.5 }, { "type": "text", "bbox": [ 106, 334, 504, 357 ], "lines": [ { "bbox": [ 106, 334, 505, 347 ], "spans": [ { "bbox": [ 106, 334, 505, 347 ], "score": 1.0, "content": "Yann LeCun, Léon Bottou, Yoshua Bengio, and Patrick Haffner. Gradient-based learning applied to", "type": "text" } ], "index": 17 }, { "bbox": [ 116, 345, 304, 357 ], "spans": [ { "bbox": [ 116, 345, 304, 357 ], "score": 1.0, "content": "document recognition. pp. 2278–2324, 1998a.", "type": "text" } ], "index": 18 } ], "index": 17.5 }, { "type": "text", "bbox": [ 107, 364, 504, 388 ], "lines": [ { "bbox": [ 107, 364, 505, 377 ], "spans": [ { "bbox": [ 107, 364, 505, 377 ], "score": 1.0, "content": "Yann LeCun, Corinna Cortes, and Christopher J.C. Burges. THE MNIST DATABASE of handwritten", "type": "text" } ], "index": 19 }, { "bbox": [ 116, 375, 400, 388 ], "spans": [ { "bbox": [ 116, 375, 400, 388 ], "score": 1.0, "content": "digits, 1998b. URL http://yann.lecun.com/exdb/mnist/.", "type": "text" } ], "index": 20 } ], "index": 19.5 }, { "type": "text", "bbox": [ 106, 394, 504, 417 ], "lines": [ { "bbox": [ 105, 393, 506, 407 ], "spans": [ { "bbox": [ 105, 393, 506, 407 ], "score": 1.0, "content": "Joohyung Lee and Yi Wang. Weighted rules under the stable model semantics. In KR, pp. 145–154,", "type": "text" } ], "index": 21 }, { "bbox": [ 114, 404, 143, 418 ], "spans": [ { "bbox": [ 114, 404, 143, 418 ], "score": 1.0, "content": "2016.", "type": "text" } ], "index": 22 } ], "index": 21.5 }, { "type": "text", "bbox": [ 106, 424, 505, 458 ], "lines": [ { "bbox": [ 105, 423, 505, 438 ], "spans": [ { "bbox": [ 105, 423, 505, 438 ], "score": 1.0, "content": "Zhixuan Lin, Yi-Fu Wu, Skand Vishwanath Peri, Weihao Sun, Gautam Singh, Fei Deng, Jindong", "type": "text" } ], "index": 23 }, { "bbox": [ 115, 436, 505, 448 ], "spans": [ { "bbox": [ 115, 436, 505, 448 ], "score": 1.0, "content": "Jiang, and Sungjin Ahn. SPACE: unsupervised object-oriented scene representation via spatial", "type": "text" } ], "index": 24 }, { "bbox": [ 116, 447, 297, 458 ], "spans": [ { "bbox": [ 116, 447, 297, 458 ], "score": 1.0, "content": "attention and decomposition. In ICLR, 2020.", "type": "text" } ], "index": 25 } ], "index": 24 }, { "type": "text", "bbox": [ 106, 465, 505, 499 ], "lines": [ { "bbox": [ 105, 464, 506, 479 ], "spans": [ { "bbox": [ 105, 464, 506, 479 ], "score": 1.0, "content": "Francesco Locatello, Dirk Weissenborn, Thomas Unterthiner, Aravindh Mahendran, Georg Heigold,", "type": "text" } ], "index": 26 }, { "bbox": [ 115, 477, 507, 489 ], "spans": [ { "bbox": [ 115, 477, 507, 489 ], "score": 1.0, "content": "Jakob Uszkoreit, Alexey Dosovitskiy, and Thomas Kipf. Object-centric learning with slot attention.", "type": "text" } ], "index": 27 }, { "bbox": [ 116, 488, 191, 499 ], "spans": [ { "bbox": [ 116, 488, 191, 499 ], "score": 1.0, "content": "In NeurIPS, 2020.", "type": "text" } ], "index": 28 } ], "index": 27 }, { "type": "text", "bbox": [ 105, 506, 505, 531 ], "lines": [ { "bbox": [ 106, 507, 506, 519 ], "spans": [ { "bbox": [ 106, 507, 506, 519 ], "score": 1.0, "content": "Robin Manhaeve, Sebastijan Dumancic, Angelika Kimmig, Thomas Demeester, and Luc De Raedt.", "type": "text" } ], "index": 29 }, { "bbox": [ 115, 517, 429, 531 ], "spans": [ { "bbox": [ 115, 517, 429, 531 ], "score": 1.0, "content": "Deepproblog: Neural probabilistic logic programming. pp. 3753–3763, 2018.", "type": "text" } ], "index": 30 } ], "index": 29.5 }, { "type": "text", "bbox": [ 106, 536, 505, 570 ], "lines": [ { "bbox": [ 105, 536, 506, 550 ], "spans": [ { "bbox": [ 105, 536, 506, 550 ], "score": 1.0, "content": "Jiayuan Mao, Chuang Gan, Pushmeet Kohli, Joshua B. Tenenbaum, and Jiajun Wu. The neuro-", "type": "text" } ], "index": 31 }, { "bbox": [ 115, 547, 505, 561 ], "spans": [ { "bbox": [ 115, 547, 505, 561 ], "score": 1.0, "content": "symbolic concept learner: Interpreting scenes, words, and sentences from natural supervision. In", "type": "text" } ], "index": 32 }, { "bbox": [ 115, 559, 168, 571 ], "spans": [ { "bbox": [ 115, 559, 168, 571 ], "score": 1.0, "content": "ICLR, 2019.", "type": "text" } ], "index": 33 } ], "index": 32 }, { "type": "text", "bbox": [ 105, 577, 482, 591 ], "lines": [ { "bbox": [ 105, 577, 481, 591 ], "spans": [ { "bbox": [ 105, 577, 481, 591 ], "score": 1.0, "content": "Gary F Marcus. The algebraic mind: Integrating connectionism and cognitive science. 2019.", "type": "text" } ], "index": 34 } ], "index": 34 }, { "type": "text", "bbox": [ 107, 596, 505, 631 ], "lines": [ { "bbox": [ 105, 595, 505, 611 ], "spans": [ { "bbox": [ 105, 595, 505, 611 ], "score": 1.0, "content": "Victor W. Marek and Miroslaw Truszczynski. Stable models and an alternative logic programming", "type": "text" } ], "index": 35 }, { "bbox": [ 115, 608, 507, 622 ], "spans": [ { "bbox": [ 115, 608, 507, 622 ], "score": 1.0, "content": "paradigm. In The Logic Programming Paradigm - A 25-Year Perspective, Artificial Intelligence,", "type": "text" } ], "index": 36 }, { "bbox": [ 114, 619, 196, 631 ], "spans": [ { "bbox": [ 114, 619, 196, 631 ], "score": 1.0, "content": "pp. 375–398. 1999.", "type": "text" } ], "index": 37 } ], "index": 36 }, { "type": "text", "bbox": [ 107, 638, 505, 672 ], "lines": [ { "bbox": [ 104, 636, 506, 653 ], "spans": [ { "bbox": [ 104, 636, 506, 653 ], "score": 1.0, "content": "Robert Peharz, Steven Lang, Antonio Vergari, Karl Stelzner, Alejandro Molina, Martin Trapp,", "type": "text" } ], "index": 38 }, { "bbox": [ 116, 649, 505, 662 ], "spans": [ { "bbox": [ 116, 649, 505, 662 ], "score": 1.0, "content": "Guy Van den Broeck, Kristian Kersting, and Zoubin Ghahramani. Einsum networks: Fast and", "type": "text" } ], "index": 39 }, { "bbox": [ 116, 660, 451, 673 ], "spans": [ { "bbox": [ 116, 660, 451, 673 ], "score": 1.0, "content": "scalable learning of tractable probabilistic circuits. In ICML, pp. 7563–7574, 2020.", "type": "text" } ], "index": 40 } ], "index": 39 }, { "type": "text", "bbox": [ 105, 679, 505, 702 ], "lines": [ { "bbox": [ 105, 679, 506, 692 ], "spans": [ { "bbox": [ 105, 679, 506, 692 ], "score": 1.0, "content": "Hoifung Poon and Pedro M. Domingos. Sum-product networks: A new deep architecture. In UAI,", "type": "text" } ], "index": 41 }, { "bbox": [ 115, 690, 196, 702 ], "spans": [ { "bbox": [ 115, 690, 196, 702 ], "score": 1.0, "content": "pp. 337–346, 2011.", "type": "text" } ], "index": 42 } ], "index": 41.5 }, { "type": "text", "bbox": [ 106, 709, 504, 732 ], "lines": [ { "bbox": [ 104, 706, 506, 724 ], "spans": [ { "bbox": [ 104, 706, 506, 724 ], "score": 1.0, "content": "Matthew Richardson and Pedro M. Domingos. Markov logic networks. Machine Learning, pp.", "type": "text" } ], "index": 43 }, { "bbox": [ 116, 720, 180, 732 ], "spans": [ { "bbox": [ 116, 720, 180, 732 ], "score": 1.0, "content": "107–136, 2006.", "type": "text" } ], "index": 44 } ], "index": 43.5 } ], "page_idx": 10, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 107, 27, 308, 37 ], "lines": [ { "bbox": [ 107, 26, 308, 38 ], "spans": [ { "bbox": [ 107, 26, 308, 38 ], "score": 1.0, "content": "Under review as a conference paper at ICLR 2022", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 300, 751, 310, 760 ], "lines": [ { "bbox": [ 299, 750, 312, 765 ], "spans": [ { "bbox": [ 299, 750, 312, 765 ], "score": 1.0, "content": "", "type": "text", "height": 15, "width": 13 } ] } ] } ], "para_blocks": [ { "type": "text", "bbox": [ 105, 82, 505, 105 ], "lines": [ { "bbox": [ 105, 81, 507, 96 ], "spans": [ { "bbox": [ 105, 81, 507, 96 ], "score": 1.0, "content": "Jerry A Fodor and Zenon W Pylyshyn. Connectionism and cognitive architecture: A critical analysis.", "type": "text" } ], "index": 0 }, { "bbox": [ 116, 93, 226, 106 ], "spans": [ { "bbox": [ 116, 93, 226, 106 ], "score": 1.0, "content": "Cognition, pp. 3–71, 1988.", "type": "text" } ], "index": 1 } ], "index": 0.5, "bbox_fs": [ 105, 81, 507, 106 ] }, { "type": "text", "bbox": [ 108, 112, 503, 146 ], "lines": [ { "bbox": [ 105, 111, 505, 126 ], "spans": [ { "bbox": [ 105, 111, 505, 126 ], "score": 1.0, "content": "Klaus Greff, Raphaël Lopez Kaufman, Rishabh Kabra, Nick Watters, Christopher Burgess, Daniel", "type": "text" } ], "index": 2 }, { "bbox": [ 115, 123, 505, 136 ], "spans": [ { "bbox": [ 115, 123, 505, 136 ], "score": 1.0, "content": "Zoran, Loic Matthey, Matthew Botvinick, and Alexander Lerchner. Multi-object representation", "type": "text" } ], "index": 3 }, { "bbox": [ 115, 134, 424, 147 ], "spans": [ { "bbox": [ 115, 134, 424, 147 ], "score": 1.0, "content": "learning with iterative variational inference. In ICML, pp. 2424–2433, 2019.", "type": "text" } ], "index": 4 } ], "index": 3, "bbox_fs": [ 105, 111, 505, 147 ] }, { "type": "text", "bbox": [ 104, 153, 504, 176 ], "lines": [ { "bbox": [ 105, 153, 506, 167 ], "spans": [ { "bbox": [ 105, 153, 506, 167 ], "score": 1.0, "content": "Klaus Greff, Sjoerd van Steenkiste, and Jürgen Schmidhuber. On the binding problem in artificial", "type": "text" } ], "index": 5 }, { "bbox": [ 116, 165, 239, 176 ], "spans": [ { "bbox": [ 116, 165, 239, 176 ], "score": 1.0, "content": "neural networks. CoRR, 2020.", "type": "text" } ], "index": 6 } ], "index": 5.5, "bbox_fs": [ 105, 153, 506, 176 ] }, { "type": "text", "bbox": [ 105, 183, 505, 207 ], "lines": [ { "bbox": [ 105, 183, 507, 197 ], "spans": [ { "bbox": [ 105, 183, 507, 197 ], "score": 1.0, "content": "Drew A. Hudson and Christopher D. Manning. Learning by abstraction: The neural state machine.", "type": "text" } ], "index": 7 }, { "bbox": [ 116, 195, 256, 207 ], "spans": [ { "bbox": [ 116, 195, 256, 207 ], "score": 1.0, "content": "In NeurIPS, pp. 5901–5914, 2019.", "type": "text" } ], "index": 8 } ], "index": 7.5, "bbox_fs": [ 105, 183, 507, 207 ] }, { "type": "text", "bbox": [ 104, 213, 464, 226 ], "lines": [ { "bbox": [ 105, 213, 464, 226 ], "spans": [ { "bbox": [ 105, 213, 464, 226 ], "score": 1.0, "content": "Jindong Jiang and Sungjin Ahn. Generative neurosymbolic machines. In NeurIPS, 2020.", "type": "text" } ], "index": 9 } ], "index": 9, "bbox_fs": [ 105, 213, 464, 226 ] }, { "type": "text", "bbox": [ 108, 232, 505, 267 ], "lines": [ { "bbox": [ 106, 233, 505, 245 ], "spans": [ { "bbox": [ 106, 233, 505, 245 ], "score": 1.0, "content": "Justin Johnson, Bharath Hariharan, Laurens van der Maaten, Li Fei-Fei, C. Lawrence Zitnick, and", "type": "text" } ], "index": 10 }, { "bbox": [ 115, 244, 505, 257 ], "spans": [ { "bbox": [ 115, 244, 505, 257 ], "score": 1.0, "content": "Ross B. Girshick. CLEVR: A diagnostic dataset for compositional language and elementary visual", "type": "text" } ], "index": 11 }, { "bbox": [ 115, 255, 292, 267 ], "spans": [ { "bbox": [ 115, 255, 292, 267 ], "score": 1.0, "content": "reasoning. In CVPR, pp. 1988–1997, 2017.", "type": "text" } ], "index": 12 } ], "index": 11, "bbox_fs": [ 106, 233, 505, 267 ] }, { "type": "text", "bbox": [ 105, 273, 505, 297 ], "lines": [ { "bbox": [ 105, 273, 506, 287 ], "spans": [ { "bbox": [ 105, 273, 506, 287 ], "score": 1.0, "content": "Diederik P. Kingma and Jimmy Ba. Adam: A method for stochastic optimization. In ICLR 2015,", "type": "text" } ], "index": 13 }, { "bbox": [ 115, 284, 143, 297 ], "spans": [ { "bbox": [ 115, 284, 143, 297 ], "score": 1.0, "content": "2015.", "type": "text" } ], "index": 14 } ], "index": 13.5, "bbox_fs": [ 105, 273, 506, 297 ] }, { "type": "text", "bbox": [ 106, 304, 504, 327 ], "lines": [ { "bbox": [ 106, 303, 505, 317 ], "spans": [ { "bbox": [ 106, 303, 505, 317 ], "score": 1.0, "content": "Alexander Kuhnle and Ann A. Copestake. Shapeworld - A new test methodology for multimodal", "type": "text" } ], "index": 15 }, { "bbox": [ 115, 316, 272, 327 ], "spans": [ { "bbox": [ 115, 316, 272, 327 ], "score": 1.0, "content": "language understanding. CoRR, 2017.", "type": "text" } ], "index": 16 } ], "index": 15.5, "bbox_fs": [ 106, 303, 505, 327 ] }, { "type": "text", "bbox": [ 106, 334, 504, 357 ], "lines": [ { "bbox": [ 106, 334, 505, 347 ], "spans": [ { "bbox": [ 106, 334, 505, 347 ], "score": 1.0, "content": "Yann LeCun, Léon Bottou, Yoshua Bengio, and Patrick Haffner. Gradient-based learning applied to", "type": "text" } ], "index": 17 }, { "bbox": [ 116, 345, 304, 357 ], "spans": [ { "bbox": [ 116, 345, 304, 357 ], "score": 1.0, "content": "document recognition. pp. 2278–2324, 1998a.", "type": "text" } ], "index": 18 } ], "index": 17.5, "bbox_fs": [ 106, 334, 505, 357 ] }, { "type": "text", "bbox": [ 107, 364, 504, 388 ], "lines": [ { "bbox": [ 107, 364, 505, 377 ], "spans": [ { "bbox": [ 107, 364, 505, 377 ], "score": 1.0, "content": "Yann LeCun, Corinna Cortes, and Christopher J.C. Burges. THE MNIST DATABASE of handwritten", "type": "text" } ], "index": 19 }, { "bbox": [ 116, 375, 400, 388 ], "spans": [ { "bbox": [ 116, 375, 400, 388 ], "score": 1.0, "content": "digits, 1998b. URL http://yann.lecun.com/exdb/mnist/.", "type": "text" } ], "index": 20 } ], "index": 19.5, "bbox_fs": [ 107, 364, 505, 388 ] }, { "type": "text", "bbox": [ 106, 394, 504, 417 ], "lines": [ { "bbox": [ 105, 393, 506, 407 ], "spans": [ { "bbox": [ 105, 393, 506, 407 ], "score": 1.0, "content": "Joohyung Lee and Yi Wang. Weighted rules under the stable model semantics. In KR, pp. 145–154,", "type": "text" } ], "index": 21 }, { "bbox": [ 114, 404, 143, 418 ], "spans": [ { "bbox": [ 114, 404, 143, 418 ], "score": 1.0, "content": "2016.", "type": "text" } ], "index": 22 } ], "index": 21.5, "bbox_fs": [ 105, 393, 506, 418 ] }, { "type": "text", "bbox": [ 106, 424, 505, 458 ], "lines": [ { "bbox": [ 105, 423, 505, 438 ], "spans": [ { "bbox": [ 105, 423, 505, 438 ], "score": 1.0, "content": "Zhixuan Lin, Yi-Fu Wu, Skand Vishwanath Peri, Weihao Sun, Gautam Singh, Fei Deng, Jindong", "type": "text" } ], "index": 23 }, { "bbox": [ 115, 436, 505, 448 ], "spans": [ { "bbox": [ 115, 436, 505, 448 ], "score": 1.0, "content": "Jiang, and Sungjin Ahn. SPACE: unsupervised object-oriented scene representation via spatial", "type": "text" } ], "index": 24 }, { "bbox": [ 116, 447, 297, 458 ], "spans": [ { "bbox": [ 116, 447, 297, 458 ], "score": 1.0, "content": "attention and decomposition. In ICLR, 2020.", "type": "text" } ], "index": 25 } ], "index": 24, "bbox_fs": [ 105, 423, 505, 458 ] }, { "type": "text", "bbox": [ 106, 465, 505, 499 ], "lines": [ { "bbox": [ 105, 464, 506, 479 ], "spans": [ { "bbox": [ 105, 464, 506, 479 ], "score": 1.0, "content": "Francesco Locatello, Dirk Weissenborn, Thomas Unterthiner, Aravindh Mahendran, Georg Heigold,", "type": "text" } ], "index": 26 }, { "bbox": [ 115, 477, 507, 489 ], "spans": [ { "bbox": [ 115, 477, 507, 489 ], "score": 1.0, "content": "Jakob Uszkoreit, Alexey Dosovitskiy, and Thomas Kipf. Object-centric learning with slot attention.", "type": "text" } ], "index": 27 }, { "bbox": [ 116, 488, 191, 499 ], "spans": [ { "bbox": [ 116, 488, 191, 499 ], "score": 1.0, "content": "In NeurIPS, 2020.", "type": "text" } ], "index": 28 } ], "index": 27, "bbox_fs": [ 105, 464, 507, 499 ] }, { "type": "text", "bbox": [ 105, 506, 505, 531 ], "lines": [ { "bbox": [ 106, 507, 506, 519 ], "spans": [ { "bbox": [ 106, 507, 506, 519 ], "score": 1.0, "content": "Robin Manhaeve, Sebastijan Dumancic, Angelika Kimmig, Thomas Demeester, and Luc De Raedt.", "type": "text" } ], "index": 29 }, { "bbox": [ 115, 517, 429, 531 ], "spans": [ { "bbox": [ 115, 517, 429, 531 ], "score": 1.0, "content": "Deepproblog: Neural probabilistic logic programming. pp. 3753–3763, 2018.", "type": "text" } ], "index": 30 } ], "index": 29.5, "bbox_fs": [ 106, 507, 506, 531 ] }, { "type": "text", "bbox": [ 106, 536, 505, 570 ], "lines": [ { "bbox": [ 105, 536, 506, 550 ], "spans": [ { "bbox": [ 105, 536, 506, 550 ], "score": 1.0, "content": "Jiayuan Mao, Chuang Gan, Pushmeet Kohli, Joshua B. Tenenbaum, and Jiajun Wu. The neuro-", "type": "text" } ], "index": 31 }, { "bbox": [ 115, 547, 505, 561 ], "spans": [ { "bbox": [ 115, 547, 505, 561 ], "score": 1.0, "content": "symbolic concept learner: Interpreting scenes, words, and sentences from natural supervision. In", "type": "text" } ], "index": 32 }, { "bbox": [ 115, 559, 168, 571 ], "spans": [ { "bbox": [ 115, 559, 168, 571 ], "score": 1.0, "content": "ICLR, 2019.", "type": "text" } ], "index": 33 } ], "index": 32, "bbox_fs": [ 105, 536, 506, 571 ] }, { "type": "text", "bbox": [ 105, 577, 482, 591 ], "lines": [ { "bbox": [ 105, 577, 481, 591 ], "spans": [ { "bbox": [ 105, 577, 481, 591 ], "score": 1.0, "content": "Gary F Marcus. The algebraic mind: Integrating connectionism and cognitive science. 2019.", "type": "text" } ], "index": 34 } ], "index": 34, "bbox_fs": [ 105, 577, 481, 591 ] }, { "type": "text", "bbox": [ 107, 596, 505, 631 ], "lines": [ { "bbox": [ 105, 595, 505, 611 ], "spans": [ { "bbox": [ 105, 595, 505, 611 ], "score": 1.0, "content": "Victor W. Marek and Miroslaw Truszczynski. Stable models and an alternative logic programming", "type": "text" } ], "index": 35 }, { "bbox": [ 115, 608, 507, 622 ], "spans": [ { "bbox": [ 115, 608, 507, 622 ], "score": 1.0, "content": "paradigm. In The Logic Programming Paradigm - A 25-Year Perspective, Artificial Intelligence,", "type": "text" } ], "index": 36 }, { "bbox": [ 114, 619, 196, 631 ], "spans": [ { "bbox": [ 114, 619, 196, 631 ], "score": 1.0, "content": "pp. 375–398. 1999.", "type": "text" } ], "index": 37 } ], "index": 36, "bbox_fs": [ 105, 595, 507, 631 ] }, { "type": "text", "bbox": [ 107, 638, 505, 672 ], "lines": [ { "bbox": [ 104, 636, 506, 653 ], "spans": [ { "bbox": [ 104, 636, 506, 653 ], "score": 1.0, "content": "Robert Peharz, Steven Lang, Antonio Vergari, Karl Stelzner, Alejandro Molina, Martin Trapp,", "type": "text" } ], "index": 38 }, { "bbox": [ 116, 649, 505, 662 ], "spans": [ { "bbox": [ 116, 649, 505, 662 ], "score": 1.0, "content": "Guy Van den Broeck, Kristian Kersting, and Zoubin Ghahramani. Einsum networks: Fast and", "type": "text" } ], "index": 39 }, { "bbox": [ 116, 660, 451, 673 ], "spans": [ { "bbox": [ 116, 660, 451, 673 ], "score": 1.0, "content": "scalable learning of tractable probabilistic circuits. In ICML, pp. 7563–7574, 2020.", "type": "text" } ], "index": 40 } ], "index": 39, "bbox_fs": [ 104, 636, 506, 673 ] }, { "type": "text", "bbox": [ 105, 679, 505, 702 ], "lines": [ { "bbox": [ 105, 679, 506, 692 ], "spans": [ { "bbox": [ 105, 679, 506, 692 ], "score": 1.0, "content": "Hoifung Poon and Pedro M. Domingos. Sum-product networks: A new deep architecture. In UAI,", "type": "text" } ], "index": 41 }, { "bbox": [ 115, 690, 196, 702 ], "spans": [ { "bbox": [ 115, 690, 196, 702 ], "score": 1.0, "content": "pp. 337–346, 2011.", "type": "text" } ], "index": 42 } ], "index": 41.5, "bbox_fs": [ 105, 679, 506, 702 ] }, { "type": "text", "bbox": [ 106, 709, 504, 732 ], "lines": [ { "bbox": [ 104, 706, 506, 724 ], "spans": [ { "bbox": [ 104, 706, 506, 724 ], "score": 1.0, "content": "Matthew Richardson and Pedro M. Domingos. Markov logic networks. Machine Learning, pp.", "type": "text" } ], "index": 43 }, { "bbox": [ 116, 720, 180, 732 ], "spans": [ { "bbox": [ 116, 720, 180, 732 ], "score": 1.0, "content": "107–136, 2006.", "type": "text" } ], "index": 44 } ], "index": 43.5, "bbox_fs": [ 104, 706, 506, 732 ] } ] }, { "preproc_blocks": [ { "type": "text", "bbox": [ 104, 82, 505, 105 ], "lines": [ { "bbox": [ 105, 81, 506, 96 ], "spans": [ { "bbox": [ 105, 81, 506, 96 ], "score": 1.0, "content": "Arseny Skryagin, Karl Stelzner, Alejandro Molina, Fabrizio Ventola, and Kristian Kersting. Splog:", "type": "text" } ], "index": 0 }, { "bbox": [ 116, 93, 276, 106 ], "spans": [ { "bbox": [ 116, 93, 276, 106 ], "score": 1.0, "content": "Sum-product logic. In ProbProg, 2020.", "type": "text" } ], "index": 1 } ], "index": 0.5 }, { "type": "text", "bbox": [ 106, 112, 504, 135 ], "lines": [ { "bbox": [ 106, 111, 505, 125 ], "spans": [ { "bbox": [ 106, 111, 505, 125 ], "score": 1.0, "content": "Timo Soininen and Ilkka Niemelä. Developing a declarative rule language for applications in product", "type": "text" } ], "index": 2 }, { "bbox": [ 116, 123, 442, 136 ], "spans": [ { "bbox": [ 116, 123, 442, 136 ], "score": 1.0, "content": "configuration. In PADL, Lecture Notes in Computer Science, pp. 305–319, 1999.", "type": "text" } ], "index": 3 } ], "index": 2.5 }, { "type": "text", "bbox": [ 106, 142, 504, 165 ], "lines": [ { "bbox": [ 106, 140, 506, 156 ], "spans": [ { "bbox": [ 106, 140, 506, 156 ], "score": 1.0, "content": "Wolfgang Stammer, Patrick Schramowski, and Kristian Kersting. Right for the right concept: Revising", "type": "text" } ], "index": 4 }, { "bbox": [ 115, 153, 505, 166 ], "spans": [ { "bbox": [ 115, 153, 505, 166 ], "score": 1.0, "content": "neuro-symbolic concepts by interacting with their explanations. 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^ { n } } { \\frac { 1 } { \\left( P _ { \\xi } ^ { ( X _ { \\mathbf { Q } } , C ) } ( x _ { Q _ { i } } ) \\right) } } \\frac { \\partial } { \\partial x _ { Q _ { i } } } \\left( P _ { \\xi } ^ { ( X _ { \\mathbf { Q } } , C ) } ( x _ { Q _ { i } } ) \\right) \\right) } } \\end{array}", "type": "interline_equation", "image_path": "585d7e4126426f14bee1247b8486a079b7133b5af781821d1e81c73d2b32d03d.jpg" } ] } ], "index": 7, "virtual_lines": [ { "bbox": [ 141, 169, 470, 194.33333333333334 ], "spans": [], "index": 6 }, { "bbox": [ 141, 194.33333333333334, 470, 219.66666666666669 ], "spans": [], "index": 7 }, { "bbox": [ 141, 219.66666666666669, 470, 245.00000000000003 ], "spans": [], "index": 8 } ] }, { "type": "text", "bbox": [ 108, 246, 503, 273 ], "lines": [ { "bbox": [ 106, 244, 507, 266 ], "spans": [ { "bbox": [ 106, 248, 193, 262 ], "score": 1.0, "content": "Here, we remark tha", "type": "text" }, { "bbox": [ 190, 244, 507, 266 ], "score": 1.0, "content": "t ∂xQi∂ξ will be carried out by back-propagation and the expression after it is the", "type": "text" } ], "index": 9 }, { "bbox": [ 105, 260, 171, 276 ], "spans": [ { "bbox": [ 105, 260, 171, 276 ], "score": 1.0, "content": "initial gradient.", "type": "text" } ], "index": 10 } ], "index": 9.5 }, { "type": "text", "bbox": [ 107, 279, 505, 301 ], "lines": [ { "bbox": [ 106, 279, 507, 291 ], "spans": [ { "bbox": [ 106, 279, 375, 291 ], "score": 1.0, "content": "Next, we derive the gradient of the logical entailment loss function", "type": "text" }, { "bbox": [ 376, 280, 403, 290 ], "score": 0.9, "content": "\\mathit { L } _ { \\mathit { E N T } }", "type": "inline_equation" }, { "bbox": [ 403, 279, 507, 291 ], "score": 1.0, "content": ", as defined in equation 7.", "type": "text" } ], "index": 11 }, { "bbox": [ 106, 290, 258, 303 ], "spans": [ { "bbox": [ 106, 290, 258, 303 ], "score": 1.0, "content": "One estimates the gradient as follows", "type": "text" } ], "index": 12 } ], "index": 11.5 }, { "type": 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"bbox": [ 115, 489, 495, 529 ], "spans": [ { "bbox": [ 115, 489, 495, 529 ], "score": 0.96, "content": "H ( y _ { i } , \\hat { y } _ { i } ) : = \\sum _ { j = 1 } ^ { m } y _ { i j } \\cdot \\log \\left( \\frac { 1 } { \\hat { y } _ { i j } } \\right) = \\sum _ { j = 1 } ^ { m } \\left( y _ { i j } \\cdot \\underbrace { \\log ( 1 ) } _ { = 0 } - y _ { i j } \\cdot \\log ( \\hat { y } _ { i j } ) \\right) = - \\sum _ { j = 1 } ^ { m } y _ { i j } \\cdot \\log ( \\hat { y } _ { i j } ) .", "type": "interline_equation", "image_path": "14efd2278c7a27ae3caf3aa86dbfe8b6cd10cb590327c2af094e018d78f8a58a.jpg" } ] } ], "index": 27, "virtual_lines": [ { "bbox": [ 115, 489, 495, 502.3333333333333 ], "spans": [], "index": 26 }, { "bbox": [ 115, 502.3333333333333, 495, 515.6666666666666 ], "spans": [], "index": 27 }, { "bbox": [ 115, 515.6666666666666, 495, 529.0 ], "spans": [], "index": 28 } ] }, { "type": "text", "bbox": [ 106, 531, 201, 542 ], "lines": [ { "bbox": [ 106, 530, 201, 543 ], 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"bbox": [ 111, 572, 510, 606 ], "lines": [ { "bbox": [ 111, 572, 510, 606 ], "spans": [ { "bbox": [ 111, 572, 510, 606 ], "score": 0.92, "content": "\\bar { \\cal I } ( y _ { i } , \\hat { y } _ { i } ) = \\cal H \\left( \\log ( P _ { \\Pi ( \\theta ) } ( Q _ { i } ) ) , P ^ { ( X _ { \\mathbf { Q } } , { \\cal C } ) } ( x _ { Q _ { i } } ) \\right) = - \\sum _ { j = 1 } ^ { m } \\log ( P _ { \\Pi ( \\theta ) } ( Q _ { i j } ) ) \\cdot \\log \\Big ( P ^ { ( X _ { \\mathbf { Q } } , { \\cal C } ) } ( x _ { Q _ { i j } } ) \\Big ) .", "type": "interline_equation", "image_path": "2b9e3b0da6436f352dbb7d2b7302116d37bd724246a285e83381b2e506b7e554.jpg" } ] } ], "index": 33, "virtual_lines": [ { "bbox": [ 111, 572, 510, 583.3333333333334 ], "spans": [], "index": 32 }, { "bbox": [ 111, 583.3333333333334, 510, 594.6666666666667 ], "spans": [], "index": 33 }, { "bbox": [ 111, 594.6666666666667, 510, 606.0000000000001 ], "spans": [], "index": 34 } ] }, { "type": "text", "bbox": [ 107, 614, 505, 660 ], "lines": [ { "bbox": [ 105, 614, 506, 627 ], "spans": [ { "bbox": [ 105, 614, 174, 627 ], "score": 1.0, "content": "We remark that", "type": "text" }, { "bbox": [ 174, 617, 184, 625 ], "score": 0.73, "content": "m", "type": "inline_equation" }, { "bbox": [ 185, 614, 506, 627 ], "score": 1.0, "content": "represent the number of classes defined in the domain of an NPP. Now, we", "type": "text" } ], "index": 35 }, { "bbox": [ 105, 626, 506, 639 ], "spans": [ { "bbox": [ 105, 626, 317, 639 ], "score": 1.0, "content": "differentiate the equation (11) with the respect to", "type": "text" }, { "bbox": [ 317, 628, 324, 637 ], "score": 0.78, "content": "p", "type": "inline_equation" }, { "bbox": [ 324, 626, 506, 639 ], "score": 1.0, "content": "depicted as in Eq. 9 to be the label of the", "type": "text" } ], "index": 36 }, { "bbox": [ 105, 636, 506, 650 ], "spans": [ { "bbox": [ 105, 636, 201, 650 ], "score": 1.0, "content": "probability of an atom", "type": "text" }, { "bbox": [ 201, 639, 227, 647 ], "score": 0.88, "content": "c = v", "type": "inline_equation" }, { "bbox": [ 228, 636, 240, 650 ], "score": 1.0, "content": "in", "type": "text" }, { "bbox": [ 240, 637, 259, 647 ], "score": 0.88, "content": "r ^ { n p p }", "type": "inline_equation" }, { "bbox": [ 259, 636, 302, 650 ], "score": 1.0, "content": ", denoting", "type": "text" }, { "bbox": [ 303, 637, 359, 650 ], "score": 0.91, "content": "P _ { \\Pi ( \\pmb { \\theta } ) } ( c = v )", "type": "inline_equation" }, { "bbox": [ 360, 636, 506, 650 ], "score": 1.0, "content": ". Since differentiation is linear, the", "type": "text" } ], "index": 37 }, { "bbox": [ 105, 648, 246, 661 ], "spans": [ { "bbox": [ 105, 648, 246, 661 ], "score": 1.0, "content": "product rule is applicable directly:", "type": "text" } ], "index": 38 } ], "index": 36.5 }, { "type": "interline_equation", "bbox": [ 160, 661, 452, 731 ], "lines": [ { "bbox": [ 160, 661, 452, 731 ], "spans": [ { "bbox": [ 160, 661, 452, 731 ], "score": 0.93, "content": "\\begin{array} { c } { \\displaystyle \\frac { \\partial } { \\partial p } H \\left( y _ { i } , \\hat { y } _ { i } \\right) = - \\sum _ { j = 1 } ^ { m } \\left[ \\frac { \\partial \\log \\left( P _ { \\Pi ( \\theta ) } ( Q _ { i j } ) \\right) } { \\partial p } \\cdot \\log \\left( P ^ { ( X _ { \\mathbf { Q } } , C ) } ( x _ { Q _ { i j } } ) \\right) \\right. } \\\\ { \\displaystyle \\left. + \\log ( P _ { \\Pi ( \\theta ) } ( Q _ { i j } ) ) \\cdot \\frac { \\partial \\log \\left( P ^ { ( X _ { \\mathbf { Q } } , C ) } ( x _ { Q _ { i j } } ) \\right) } { \\partial p } \\right] . } \\end{array}", "type": "interline_equation", "image_path": "3c0bbe5adcca47ae64eee1a8772ec5065bb21a604d9ef8d82b7d7725f3268047.jpg" } ] } ], "index": 40, "virtual_lines": [ { "bbox": [ 160, 661, 452, 684.3333333333334 ], "spans": [], "index": 39 }, { "bbox": [ 160, 684.3333333333334, 452, 707.6666666666667 ], "spans": [], "index": 40 }, { "bbox": [ 160, 707.6666666666667, 452, 731.0000000000001 ], "spans": [], "index": 41 } ] } ], "page_idx": 12, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 106, 26, 308, 38 ], "lines": [ { "bbox": [ 106, 25, 308, 39 ], "spans": [ { "bbox": [ 106, 25, 308, 39 ], "score": 1.0, "content": "Under review as a conference paper at ICLR 2022", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 300, 751, 311, 760 ], "lines": [ { "bbox": [ 299, 750, 312, 764 ], "spans": [ { "bbox": [ 299, 750, 312, 764 ], "score": 1.0, "content": "13", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "title", "bbox": [ 107, 81, 396, 94 ], "lines": [ { "bbox": [ 106, 81, 396, 96 ], "spans": [ { "bbox": [ 106, 81, 396, 96 ], "score": 1.0, "content": "A APPENDIX A – DETAILS ON PARAMETER LEARNING", "type": "text" } ], "index": 0 } ], "index": 0 }, { "type": "text", "bbox": [ 106, 105, 506, 139 ], "lines": [ { "bbox": [ 106, 106, 506, 118 ], "spans": [ { "bbox": [ 106, 106, 506, 118 ], "score": 1.0, "content": "In the Appendix, we want to discuss details on parameter learning in SLASH. Since we use co-", "type": "text" } ], "index": 1 }, { "bbox": [ 105, 116, 505, 129 ], "spans": [ { "bbox": [ 105, 116, 505, 129 ], "score": 1.0, "content": "ordinate descent for training SLASH we present the derivative of each component of the loss function", "type": "text" } ], "index": 2 }, { "bbox": [ 106, 128, 447, 140 ], "spans": [ { "bbox": [ 106, 128, 447, 140 ], "score": 1.0, "content": "defined in equation 10 since while optimization, one component has to be kept fixed.", "type": "text" } ], "index": 3 } ], "index": 2, "bbox_fs": [ 105, 106, 506, 140 ] }, { "type": "text", "bbox": [ 106, 144, 505, 167 ], "lines": [ { "bbox": [ 105, 143, 506, 158 ], "spans": [ { "bbox": [ 105, 143, 309, 158 ], "score": 1.0, "content": "We start with the gradient of the NPP loss function", "type": "text" }, { "bbox": [ 309, 145, 336, 156 ], "score": 0.91, "content": "L _ { N P P }", "type": "inline_equation" }, { "bbox": [ 337, 143, 506, 158 ], "score": 1.0, "content": "i.e. the negative log-likelihood, defined in", "type": "text" } ], "index": 4 }, { "bbox": [ 104, 155, 152, 168 ], "spans": [ { "bbox": [ 104, 155, 152, 168 ], "score": 1.0, "content": "equation 2", "type": "text" } ], "index": 5 } ], "index": 4.5, "bbox_fs": [ 104, 143, 506, 168 ] }, { "type": "interline_equation", "bbox": [ 141, 169, 470, 245 ], "lines": [ { "bbox": [ 141, 169, 470, 245 ], "spans": [ { "bbox": [ 141, 169, 470, 245 ], "score": 0.94, "content": "\\begin{array} { c } { { \\displaystyle { \\frac { \\partial } { \\partial \\xi } } L _ { N P P } = { \\frac { \\partial } { \\partial x _ { Q _ { i } } } \\cdot \\frac { \\partial x _ { Q _ { i } } } { \\partial \\xi } } L _ { N P P } = { \\frac { \\partial x _ { Q _ { i } } } { \\partial \\xi } } \\left( - { \\sum _ { i = 1 } ^ { n } } { \\frac { \\partial } { \\partial x _ { Q _ { i } } } \\log \\left( P _ { \\xi } ^ { ( X _ { \\mathbf { Q } } , C ) } ( x _ { Q _ { i } } ) \\right) } \\right) } } \\\\ { { \\displaystyle { = \\frac { \\partial x _ { Q _ { i } } } { \\partial \\xi } } \\left( - { \\sum _ { i = 1 } ^ { n } } { \\frac { 1 } { \\left( P _ { \\xi } ^ { ( X _ { \\mathbf { Q } } , C ) } ( x _ { Q _ { i } } ) \\right) } } \\frac { \\partial } { \\partial x _ { Q _ { i } } } \\left( P _ { \\xi } ^ { ( X _ { \\mathbf { Q } } , C ) } ( x _ { Q _ { i } } ) \\right) \\right) } } \\end{array}", "type": "interline_equation", "image_path": "585d7e4126426f14bee1247b8486a079b7133b5af781821d1e81c73d2b32d03d.jpg" } ] } ], "index": 7, "virtual_lines": [ { "bbox": [ 141, 169, 470, 194.33333333333334 ], "spans": [], "index": 6 }, { "bbox": [ 141, 194.33333333333334, 470, 219.66666666666669 ], "spans": [], "index": 7 }, { "bbox": [ 141, 219.66666666666669, 470, 245.00000000000003 ], "spans": [], "index": 8 } ] }, { "type": "text", "bbox": [ 108, 246, 503, 273 ], "lines": [ { "bbox": [ 106, 244, 507, 266 ], "spans": [ { "bbox": [ 106, 248, 193, 262 ], "score": 1.0, "content": "Here, we remark tha", "type": "text" }, { "bbox": [ 190, 244, 507, 266 ], "score": 1.0, "content": "t ∂xQi∂ξ will be carried out by back-propagation and the expression after it is the", "type": "text" } ], "index": 9 }, { "bbox": [ 105, 260, 171, 276 ], "spans": [ { "bbox": [ 105, 260, 171, 276 ], "score": 1.0, "content": "initial gradient.", "type": "text" } ], "index": 10 } ], "index": 9.5, "bbox_fs": [ 105, 244, 507, 276 ] }, { "type": "text", "bbox": [ 107, 279, 505, 301 ], "lines": [ { "bbox": [ 106, 279, 507, 291 ], "spans": [ { "bbox": [ 106, 279, 375, 291 ], "score": 1.0, "content": "Next, we derive the gradient of the logical entailment loss function", "type": "text" }, { "bbox": [ 376, 280, 403, 290 ], "score": 0.9, "content": "\\mathit { L } _ { \\mathit { E N T } }", "type": "inline_equation" }, { "bbox": [ 403, 279, 507, 291 ], "score": 1.0, "content": ", as defined in equation 7.", "type": "text" } ], "index": 11 }, { "bbox": [ 106, 290, 258, 303 ], "spans": [ { "bbox": [ 106, 290, 258, 303 ], "score": 1.0, "content": "One estimates the gradient as follows", "type": "text" } ], "index": 12 } ], "index": 11.5, "bbox_fs": [ 106, 279, 507, 303 ] }, { "type": "interline_equation", "bbox": [ 175, 304, 435, 336 ], "lines": [ { "bbox": [ 175, 304, 435, 336 ], "spans": [ { "bbox": [ 175, 304, 435, 336 ], "score": 0.93, "content": "\\frac { 1 } { n } \\frac { \\partial } { \\partial p } L _ { E N T } \\ge - \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\frac { \\partial \\log ( P _ { \\Pi ( \\theta ) } ( Q _ { i } ) ) } { \\partial p } \\cdot \\log \\bigl ( P ^ { ( X _ { \\mathbf { Q } } , C ) } ( x _ { Q _ { i } } ) \\bigr ) ,", "type": "interline_equation", "image_path": "25d52485cccf285e31823cec21b9741e5713aae7acecaaba5018b450f79cbcf7.jpg" } ] } ], "index": 14, "virtual_lines": [ { "bbox": [ 175, 304, 435, 314.6666666666667 ], "spans": [], "index": 13 }, { "bbox": [ 175, 314.6666666666667, 435, 325.33333333333337 ], "spans": [], "index": 14 }, { "bbox": [ 175, 325.33333333333337, 435, 336.00000000000006 ], 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370, 477, 379, 487 ], "score": 0.84, "content": "y _ { i }", "type": "inline_equation" }, { "bbox": [ 379, 473, 397, 489 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 397, 476, 407, 487 ], "score": 0.87, "content": "\\hat { y } _ { i }", "type": "inline_equation" }, { "bbox": [ 407, 473, 411, 489 ], "score": 1.0, "content": ":", "type": "text" } ], "index": 25 } ], "index": 25, "bbox_fs": [ 105, 473, 411, 489 ] }, { "type": "interline_equation", "bbox": [ 115, 489, 495, 529 ], "lines": [ { "bbox": [ 115, 489, 495, 529 ], "spans": [ { "bbox": [ 115, 489, 495, 529 ], "score": 0.96, "content": "H ( y _ { i } , \\hat { y } _ { i } ) : = \\sum _ { j = 1 } ^ { m } y _ { i j } \\cdot \\log \\left( \\frac { 1 } { \\hat { y } _ { i j } } \\right) = \\sum _ { j = 1 } ^ { m } \\left( y _ { i j } \\cdot \\underbrace { \\log ( 1 ) } _ { = 0 } - y _ { i j } \\cdot \\log ( \\hat { y } _ { i j } ) \\right) = - \\sum _ { j = 1 } ^ { m } y _ { i j } \\cdot \\log ( \\hat { y } _ { i j } ) .", "type": "interline_equation", "image_path": "14efd2278c7a27ae3caf3aa86dbfe8b6cd10cb590327c2af094e018d78f8a58a.jpg" } ] } ], "index": 27, "virtual_lines": [ { "bbox": [ 115, 489, 495, 502.3333333333333 ], "spans": [], "index": 26 }, { "bbox": [ 115, 502.3333333333333, 495, 515.6666666666666 ], "spans": [], "index": 27 }, { "bbox": [ 115, 515.6666666666666, 495, 529.0 ], "spans": [], "index": 28 } ] }, { "type": "text", "bbox": [ 106, 531, 201, 542 ], "lines": [ { "bbox": [ 106, 530, 201, 543 ], "spans": [ { "bbox": [ 106, 530, 201, 543 ], "score": 1.0, "content": "Hereafter we substitute", "type": "text" } ], "index": 29 } ], "index": 29, "bbox_fs": [ 106, 530, 201, 543 ] }, { "type": "interline_equation", "bbox": [ 192, 543, 419, 559 ], "lines": [ { "bbox": [ 192, 543, 419, 559 ], "spans": [ { "bbox": [ 192, 543, 419, 559 ], "score": 0.88, "content": "y _ { i } = \\log ( P _ { \\Pi ( \\theta ) } ( Q _ { i } ) ) \\qquad \\mathrm { ~ a n d ~ } \\qquad \\hat { y } _ { i } = P ^ { ( X _ { \\mathbf { Q } } , C ) } ( x _ { Q _ { i } } )", "type": "interline_equation", "image_path": "8d2dc436876b6a02f8a97fb08e101dcf909c86ec9964c1fd88d9161441971688.jpg" } ] } ], "index": 30, "virtual_lines": [ { "bbox": [ 192, 543, 419, 559 ], "spans": [], "index": 30 } ] }, { "type": "text", "bbox": [ 106, 560, 151, 572 ], "lines": [ { "bbox": [ 105, 560, 151, 572 ], "spans": [ { "bbox": [ 105, 560, 151, 572 ], "score": 1.0, "content": "and obtain", "type": "text" } ], "index": 31 } ], "index": 31, "bbox_fs": [ 105, 560, 151, 572 ] }, { "type": "interline_equation", "bbox": [ 111, 572, 510, 606 ], "lines": [ { "bbox": [ 111, 572, 510, 606 ], "spans": [ { "bbox": [ 111, 572, 510, 606 ], "score": 0.92, "content": "\\bar { \\cal I } ( y _ { i } , \\hat { y } _ { i } ) = \\cal H \\left( \\log ( P _ { \\Pi ( \\theta ) } ( Q _ { i } ) ) , P ^ { ( X _ { \\mathbf { Q } } , { \\cal C } ) } ( x _ { Q _ { i } } ) \\right) = - \\sum _ { j = 1 } ^ { m } \\log ( P _ { \\Pi ( \\theta ) } ( Q _ { i j } ) ) \\cdot \\log \\Big ( P ^ { ( X _ { \\mathbf { Q } } , { \\cal C } ) } ( x _ { Q _ { i j } } ) \\Big ) .", "type": "interline_equation", "image_path": "2b9e3b0da6436f352dbb7d2b7302116d37bd724246a285e83381b2e506b7e554.jpg" } ] } ], "index": 33, "virtual_lines": [ { "bbox": [ 111, 572, 510, 583.3333333333334 ], "spans": [], "index": 32 }, { "bbox": [ 111, 583.3333333333334, 510, 594.6666666666667 ], "spans": [], "index": 33 }, { "bbox": [ 111, 594.6666666666667, 510, 606.0000000000001 ], "spans": [], "index": 34 } ] }, { "type": "text", "bbox": [ 107, 614, 505, 660 ], "lines": [ { "bbox": [ 105, 614, 506, 627 ], "spans": [ { "bbox": [ 105, 614, 174, 627 ], "score": 1.0, "content": "We remark that", "type": "text" }, { "bbox": [ 174, 617, 184, 625 ], "score": 0.73, "content": "m", "type": "inline_equation" }, { "bbox": [ 185, 614, 506, 627 ], "score": 1.0, "content": "represent the number of classes defined in the domain of an NPP. Now, we", "type": "text" } ], "index": 35 }, { "bbox": [ 105, 626, 506, 639 ], "spans": [ { "bbox": [ 105, 626, 317, 639 ], "score": 1.0, "content": "differentiate the equation (11) with the respect to", "type": "text" }, { "bbox": [ 317, 628, 324, 637 ], "score": 0.78, "content": "p", "type": "inline_equation" }, { "bbox": [ 324, 626, 506, 639 ], "score": 1.0, "content": "depicted as in Eq. 9 to be the label of the", "type": "text" } ], "index": 36 }, { "bbox": [ 105, 636, 506, 650 ], "spans": [ { "bbox": [ 105, 636, 201, 650 ], "score": 1.0, "content": "probability of an atom", "type": "text" }, { "bbox": [ 201, 639, 227, 647 ], "score": 0.88, "content": "c = v", "type": "inline_equation" }, { "bbox": [ 228, 636, 240, 650 ], "score": 1.0, "content": "in", "type": "text" }, { "bbox": [ 240, 637, 259, 647 ], "score": 0.88, "content": "r ^ { n p p }", "type": "inline_equation" }, { "bbox": [ 259, 636, 302, 650 ], "score": 1.0, "content": ", denoting", "type": "text" }, { "bbox": [ 303, 637, 359, 650 ], "score": 0.91, "content": "P _ { \\Pi ( \\pmb { \\theta } ) } ( c = v )", "type": "inline_equation" }, { "bbox": [ 360, 636, 506, 650 ], "score": 1.0, "content": ". Since differentiation is linear, the", "type": "text" } ], "index": 37 }, { "bbox": [ 105, 648, 246, 661 ], "spans": [ { "bbox": [ 105, 648, 246, 661 ], "score": 1.0, "content": "product rule is applicable directly:", "type": "text" } ], "index": 38 } ], "index": 36.5, "bbox_fs": [ 105, 614, 506, 661 ] }, { "type": "interline_equation", "bbox": [ 160, 661, 452, 731 ], "lines": [ { "bbox": [ 160, 661, 452, 731 ], "spans": [ { "bbox": [ 160, 661, 452, 731 ], "score": 0.93, "content": "\\begin{array} { c } { \\displaystyle \\frac { \\partial } { \\partial p } H \\left( y _ { i } , \\hat { y } _ { i } \\right) = - \\sum _ { j = 1 } ^ { m } \\left[ \\frac { \\partial \\log \\left( P _ { \\Pi ( \\theta ) } ( Q _ { i j } ) \\right) } { \\partial p } \\cdot \\log \\left( P ^ { ( X _ { \\mathbf { Q } } , C ) } ( x _ { Q _ { i j } } ) \\right) \\right. } \\\\ { \\displaystyle \\left. + \\log ( P _ { \\Pi ( \\theta ) } ( Q _ { i j } ) ) \\cdot \\frac { \\partial \\log \\left( P ^ { ( X _ { \\mathbf { Q } } , C ) } ( x _ { Q _ { i j } } ) \\right) } { \\partial p } \\right] . } \\end{array}", "type": "interline_equation", "image_path": "3c0bbe5adcca47ae64eee1a8772ec5065bb21a604d9ef8d82b7d7725f3268047.jpg" } ] } ], "index": 40, "virtual_lines": [ { "bbox": [ 160, 661, 452, 684.3333333333334 ], "spans": [], "index": 39 }, { "bbox": [ 160, 684.3333333333334, 452, 707.6666666666667 ], "spans": [], "index": 40 }, { "bbox": [ 160, 707.6666666666667, 452, 731.0000000000001 ], "spans": [], "index": 41 } ] } ] }, { "preproc_blocks": [ { "type": "text", "bbox": [ 106, 80, 506, 127 ], "lines": [ { "bbox": [ 104, 78, 506, 102 ], "spans": [ { "bbox": [ 104, 78, 277, 102 ], "score": 1.0, "content": "We do not wish to consider the latter term of", "type": "text" }, { "bbox": [ 278, 81, 422, 100 ], "score": 0.64, "content": "\\begin{array} { r } { \\log ( P _ { \\mathrm { { I I } } ( \\theta ) } ( Q _ { i } ) ) \\cdot \\frac { \\partial \\log \\left( P ^ { ( X _ { \\mathbf { Q } } , C ) } ( x _ { Q _ { i } } ) \\right) } { \\partial p } } \\end{array}", "type": "inline_equation" }, { "bbox": [ 422, 78, 506, 102 ], "score": 1.0, "content": "because it represents", "type": "text" } ], "index": 0 }, { "bbox": [ 106, 95, 506, 119 ], "spans": [ { "bbox": [ 106, 95, 330, 119 ], "score": 1.0, "content": "the rescaling and to keep the first since SLASH procure", "type": "text" }, { "bbox": [ 331, 100, 390, 117 ], "score": 0.92, "content": "\\frac { \\partial \\log ( P _ { \\Pi ( \\theta ) } ( Q _ { i } ) ) } { \\partial p }", "type": "inline_equation" }, { "bbox": [ 391, 95, 506, 119 ], "score": 1.0, "content": "following Eq. 9. To achieve", "type": "text" } ], "index": 1 }, { "bbox": [ 106, 116, 319, 127 ], "spans": [ { "bbox": [ 106, 116, 319, 127 ], "score": 1.0, "content": "this, we estimate equation from above downwards as", "type": "text" } ], "index": 2 } ], "index": 1 }, { "type": "interline_equation", "bbox": [ 171, 131, 441, 166 ], "lines": [ { "bbox": [ 171, 131, 441, 166 ], "spans": [ { "bbox": [ 171, 131, 441, 166 ], "score": 0.93, "content": "\\frac { \\partial } { \\partial p } H \\left( y _ { i } , \\hat { y } _ { i } \\right) \\geq - \\sum _ { j = 1 } ^ { m } \\frac { \\partial \\log ( P _ { \\Pi ( \\theta ) } ( Q _ { i j } ) ) } { \\partial p } \\cdot \\log \\left( P ^ { ( X _ { \\mathbf { Q } } , C ) } ( x _ { Q _ { i j } } ) \\right) .", "type": "interline_equation", "image_path": "b2e3c45fb5858cfba6c3db1fb458509db228f8ac82adf03458a47e91d2270af1.jpg" } ] } ], "index": 4, "virtual_lines": [ { "bbox": [ 171, 131, 441, 142.66666666666666 ], "spans": [], "index": 3 }, { "bbox": [ 171, 142.66666666666666, 441, 154.33333333333331 ], "spans": [], "index": 4 }, { "bbox": [ 171, 154.33333333333331, 441, 165.99999999999997 ], "spans": [], "index": 5 } ] }, { "type": "text", "bbox": [ 105, 170, 492, 183 ], "lines": [ { "bbox": [ 106, 170, 490, 183 ], "spans": [ { "bbox": [ 106, 170, 490, 183 ], "score": 1.0, "content": "Furthermore, let us recall that under i.i.d assumption we obtain from the definition of likelihood", "type": "text" } ], "index": 6 } ], "index": 6 }, { "type": "interline_equation", "bbox": [ 247, 187, 363, 220 ], "lines": [ { "bbox": [ 247, 187, 363, 220 ], "spans": [ { "bbox": [ 247, 187, 363, 220 ], "score": 0.94, "content": "L H ( y , \\hat { y } ) = \\prod _ { i = 1 } ^ { n } L H ( y _ { i } , \\hat { y } _ { i } ) ,", "type": "interline_equation", "image_path": "b1cef1055651c022e4fabcdf7453f8c1f11569815c366ea8cad9ac5be1bcf777.jpg" } ] } ], "index": 7.5, "virtual_lines": [ { "bbox": [ 247, 187, 363, 203.5 ], "spans": [], "index": 7 }, { "bbox": [ 247, 203.5, 363, 220.0 ], "spans": [], "index": 8 } ] }, { "type": "text", "bbox": [ 106, 224, 505, 248 ], "lines": [ { "bbox": [ 105, 223, 505, 238 ], "spans": [ { "bbox": [ 105, 223, 470, 238 ], "score": 1.0, "content": "and following the negative likelihood coupled with the knowledge that the log-likelihood of", "type": "text" }, { "bbox": [ 471, 227, 480, 236 ], "score": 0.84, "content": "y _ { i }", "type": "inline_equation" }, { "bbox": [ 480, 223, 505, 238 ], "score": 1.0, "content": "is the", "type": "text" } ], "index": 9 }, { "bbox": [ 105, 235, 223, 249 ], "spans": [ { "bbox": [ 105, 235, 214, 249 ], "score": 1.0, "content": "log of a particular entry of", "type": "text" }, { "bbox": [ 214, 237, 223, 248 ], "score": 0.87, "content": "\\hat { y } _ { i }", "type": "inline_equation" } ], "index": 10 } ], "index": 9.5 }, { "type": "interline_equation", "bbox": [ 145, 252, 466, 327 ], "lines": [ { "bbox": [ 145, 252, 466, 327 ], "spans": [ { "bbox": [ 145, 252, 466, 327 ], "score": 0.94, "content": "\\begin{array} { c } { { { \\cal L } _ { E N T } = - \\displaystyle \\log { \\cal L } H ( y , \\hat { y } ) = - \\sum _ { i = 1 } ^ { n } \\log { \\cal L } H ( y _ { i } , \\hat { y } _ { i } ) = - \\sum _ { i = 1 } ^ { n } \\sum _ { j = 1 } ^ { m } y _ { i j } \\cdot \\log ( \\hat { y } _ { i j } ) = } } \\\\ { { \\displaystyle \\sum _ { i = 1 } ^ { n } \\left[ - \\sum _ { j = 1 } ^ { m } y _ { i j } \\cdot \\log ( \\hat { y } _ { i j } ) \\right] = \\sum _ { i = 1 } ^ { n } H ( y _ { i } , \\hat { y } _ { i } ) . } } \\end{array}", "type": "interline_equation", "image_path": "d0e566d68e2f20b4e54fc7c2b09a43dd98b00de71bda7849e2d912e6c8af305d.jpg" } ] } ], "index": 12, "virtual_lines": [ { "bbox": [ 145, 252, 466, 277.0 ], "spans": [], "index": 11 }, { "bbox": [ 145, 277.0, 466, 302.0 ], "spans": [], "index": 12 }, { "bbox": [ 145, 302.0, 466, 327.0 ], "spans": [], "index": 13 } ] }, { "type": "text", "bbox": [ 105, 331, 371, 344 ], "lines": [ { "bbox": [ 105, 330, 371, 345 ], "spans": [ { "bbox": [ 105, 330, 371, 345 ], "score": 1.0, "content": "Finally, we obtain the following estimate applying inequality (12)", "type": "text" } ], "index": 14 } ], "index": 14 }, { "type": "interline_equation", "bbox": [ 129, 348, 481, 381 ], "lines": [ { "bbox": [ 129, 348, 481, 381 ], "spans": [ { "bbox": [ 129, 348, 481, 381 ], "score": 0.94, "content": "\\frac { 1 } { n } \\frac { \\partial } { \\partial p } L _ { E N T } = \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\frac { \\partial } { \\partial p } H ( y _ { i } , \\hat { y } _ { i } ) \\geq - \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\frac { \\partial \\log \\left( P _ { \\mathrm { { I I } } ( \\theta ) } ( Q _ { i } ) \\right) } { \\partial p } \\cdot \\log \\left( P ^ { ( X _ { \\mathbf { Q } } , C ) } ( x _ { Q _ { i } } ) \\right)", "type": "interline_equation", "image_path": "e89568be2c91580ddf980db40d0e2dcf668767f5e62acf9ad3ab41e0da02c656.jpg" } ] } ], "index": 16, "virtual_lines": [ { "bbox": [ 129, 348, 481, 359.0 ], "spans": [], "index": 15 }, { "bbox": [ 129, 359.0, 481, 370.0 ], "spans": [], "index": 16 }, { "bbox": [ 129, 370.0, 481, 381.0 ], "spans": [], "index": 17 } ] }, { "type": "text", "bbox": [ 106, 385, 506, 453 ], "lines": [ { "bbox": [ 105, 385, 506, 398 ], "spans": [ { "bbox": [ 105, 385, 506, 398 ], "score": 1.0, "content": "Also, we note that the mathematical transformations listed above hold for any type of NPP and the", "type": "text" } ], "index": 18 }, { "bbox": [ 105, 397, 505, 410 ], "spans": [ { "bbox": [ 105, 397, 505, 410 ], "score": 1.0, "content": "task dependent queries (NN with Softmax, PC or PC jointly with NN). The only difference will be", "type": "text" } ], "index": 19 }, { "bbox": [ 104, 406, 507, 424 ], "spans": [ { "bbox": [ 104, 406, 193, 424 ], "score": 1.0, "content": "the second term, i.e.,", "type": "text" }, { "bbox": [ 193, 408, 277, 421 ], "score": 0.88, "content": "\\log ( P ^ { ( C | X _ { \\mathbf { Q } } ) } ( x _ { Q _ { i j } } ) )", "type": "inline_equation" }, { "bbox": [ 277, 406, 289, 424 ], "score": 1.0, "content": "or", "type": "text" }, { "bbox": [ 290, 408, 373, 421 ], "score": 0.9, "content": "\\log ( P ^ { ( \\bar { X } _ { \\mathbf { Q } } | C ) } \\bar { ( x _ { Q _ { i j } } ) } )", "type": "inline_equation" }, { "bbox": [ 374, 406, 507, 424 ], "score": 1.0, "content": "depending on the NPP and task.", "type": "text" } ], "index": 20 }, { "bbox": [ 104, 419, 507, 434 ], "spans": [ { "bbox": [ 104, 419, 334, 434 ], "score": 1.0, "content": "The NPP in a form of a single PC modeling the joint over", "type": "text" }, { "bbox": [ 335, 421, 351, 432 ], "score": 0.87, "content": "X _ { \\mathbf { Q } }", "type": "inline_equation" }, { "bbox": [ 351, 419, 369, 434 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 369, 421, 378, 430 ], "score": 0.82, "content": "C", "type": "inline_equation" }, { "bbox": [ 378, 419, 507, 434 ], "score": 1.0, "content": "was depicted to be the example.", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 430, 506, 444 ], "spans": [ { "bbox": [ 105, 430, 506, 444 ], "score": 1.0, "content": "With that, the derivation of gradients for both loss functions 2 and 7 is complete, and the training is", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 442, 249, 454 ], "spans": [ { "bbox": [ 105, 442, 249, 454 ], "score": 1.0, "content": "carried out by coordinated descent.", "type": "text" } ], "index": 23 } ], "index": 20.5 }, { "type": "text", "bbox": [ 107, 469, 505, 504 ], "lines": [ { "bbox": [ 105, 469, 505, 483 ], "spans": [ { "bbox": [ 105, 469, 440, 483 ], "score": 1.0, "content": "Backpropagation for joint NN and PC NPPs: If within the SLASH program,", "type": "text" }, { "bbox": [ 441, 470, 462, 482 ], "score": 0.83, "content": "\\Pi ( \\pmb \\theta )", "type": "inline_equation" }, { "bbox": [ 463, 469, 505, 483 ], "score": 1.0, "content": ", the NPP", "type": "text" } ], "index": 24 }, { "bbox": [ 106, 480, 505, 493 ], "spans": [ { "bbox": [ 106, 480, 505, 493 ], "score": 1.0, "content": "forwards the data tensor through a NN first, i.e., the NPP models a joint over the NN’s output", "type": "text" } ], "index": 25 }, { "bbox": [ 106, 492, 270, 504 ], "spans": [ { "bbox": [ 106, 492, 270, 504 ], "score": 1.0, "content": "variables by a PC, then we rewrite (8) to", "type": "text" } ], "index": 26 } ], "index": 25 }, { "type": "interline_equation", "bbox": [ 176, 508, 435, 541 ], "lines": [ { "bbox": [ 176, 508, 435, 541 ], "spans": [ { "bbox": [ 176, 508, 435, 541 ], "score": 0.95, "content": "\\sum _ { i = 1 } ^ { n } { \\frac { \\partial \\log \\left( P _ { \\Pi ( \\theta ) } ( Q _ { i } ) \\right) } { \\partial \\theta } } = \\sum _ { i = 1 } ^ { n } { \\frac { \\partial \\log \\left( P _ { \\Pi ( \\theta ) } ( Q _ { i } ) \\right) } { \\partial \\mathbf { p } } } \\times { \\frac { \\partial \\mathbf { p } } { \\partial \\theta } } \\times { \\frac { \\partial \\theta } { \\partial \\gamma } } .", "type": "interline_equation", "image_path": "44e2cbe01c8e9cc1b050627c51a956c1b81d85f83dda7a0471ea236f4081c019.jpg" } ] } ], "index": 28, "virtual_lines": [ { "bbox": [ 176, 508, 435, 519.0 ], "spans": [], "index": 27 }, { "bbox": [ 176, 519.0, 435, 530.0 ], "spans": [], "index": 28 }, { "bbox": [ 176, 530.0, 435, 541.0 ], "spans": [], "index": 29 } ] }, { "type": "text", "bbox": [ 106, 546, 505, 572 ], "lines": [ { "bbox": [ 105, 546, 505, 562 ], "spans": [ { "bbox": [ 105, 546, 145, 561 ], "score": 1.0, "content": "Thereby,", "type": "text" }, { "bbox": [ 145, 550, 154, 559 ], "score": 0.82, "content": "\\gamma", "type": "inline_equation" }, { "bbox": [ 154, 546, 313, 561 ], "score": 1.0, "content": "is the set of the NN’s parameters and", "type": "text" }, { "bbox": [ 314, 546, 327, 562 ], "score": 0.9, "content": "\\frac { \\partial \\pmb { \\theta } } { \\partial \\gamma }", "type": "inline_equation" }, { "bbox": [ 327, 546, 505, 561 ], "score": 1.0, "content": "is computed by the backward propagation", "type": "text" } ], "index": 30 }, { "bbox": [ 106, 560, 174, 572 ], "spans": [ { "bbox": [ 106, 560, 174, 572 ], "score": 1.0, "content": "through the NN.", "type": "text" } ], "index": 31 } ], "index": 30.5 } ], "page_idx": 13, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 106, 26, 308, 38 ], "lines": [ { "bbox": [ 106, 25, 308, 38 ], "spans": [ { "bbox": [ 106, 25, 308, 38 ], "score": 1.0, "content": "Under review as a conference paper at ICLR 2022", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 300, 751, 311, 760 ], "lines": [ { "bbox": [ 299, 750, 312, 764 ], "spans": [ { "bbox": [ 299, 750, 312, 764 ], "score": 1.0, "content": "14", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "text", "bbox": [ 106, 80, 506, 127 ], "lines": [ { "bbox": [ 104, 78, 506, 102 ], "spans": [ { "bbox": [ 104, 78, 277, 102 ], "score": 1.0, "content": "We do not wish to consider the latter term of", "type": "text" }, { "bbox": [ 278, 81, 422, 100 ], "score": 0.64, "content": "\\begin{array} { r } { \\log ( P _ { \\mathrm { { I I } } ( \\theta ) } ( Q _ { i } ) ) \\cdot \\frac { \\partial \\log \\left( P ^ { ( X _ { \\mathbf { Q } } , C ) } ( x _ { Q _ { i } } ) \\right) } { \\partial p } } \\end{array}", "type": "inline_equation" }, { "bbox": [ 422, 78, 506, 102 ], "score": 1.0, "content": "because it represents", "type": "text" } ], "index": 0 }, { "bbox": [ 106, 95, 506, 119 ], "spans": [ { "bbox": [ 106, 95, 330, 119 ], "score": 1.0, "content": "the rescaling and to keep the first since SLASH procure", "type": "text" }, { "bbox": [ 331, 100, 390, 117 ], "score": 0.92, "content": "\\frac { \\partial \\log ( P _ { \\Pi ( \\theta ) } ( Q _ { i } ) ) } { \\partial p }", "type": "inline_equation" }, { "bbox": [ 391, 95, 506, 119 ], "score": 1.0, "content": "following Eq. 9. To achieve", "type": "text" } ], "index": 1 }, { "bbox": [ 106, 116, 319, 127 ], "spans": [ { "bbox": [ 106, 116, 319, 127 ], "score": 1.0, "content": "this, we estimate equation from above downwards as", "type": "text" } ], "index": 2 } ], "index": 1, "bbox_fs": [ 104, 78, 506, 127 ] }, { "type": "interline_equation", "bbox": [ 171, 131, 441, 166 ], "lines": [ { "bbox": [ 171, 131, 441, 166 ], "spans": [ { "bbox": [ 171, 131, 441, 166 ], "score": 0.93, "content": "\\frac { \\partial } { \\partial p } H \\left( y _ { i } , \\hat { y } _ { i } \\right) \\geq - \\sum _ { j = 1 } ^ { m } \\frac { \\partial \\log ( P _ { \\Pi ( \\theta ) } ( Q _ { i j } ) ) } { \\partial p } \\cdot \\log \\left( P ^ { ( X _ { \\mathbf { Q } } , C ) } ( x _ { Q _ { i j } } ) \\right) .", "type": "interline_equation", "image_path": "b2e3c45fb5858cfba6c3db1fb458509db228f8ac82adf03458a47e91d2270af1.jpg" } ] } ], "index": 4, "virtual_lines": [ { "bbox": [ 171, 131, 441, 142.66666666666666 ], "spans": [], "index": 3 }, { "bbox": [ 171, 142.66666666666666, 441, 154.33333333333331 ], "spans": [], "index": 4 }, { "bbox": [ 171, 154.33333333333331, 441, 165.99999999999997 ], "spans": [], "index": 5 } ] }, { "type": "text", "bbox": [ 105, 170, 492, 183 ], "lines": [ { "bbox": [ 106, 170, 490, 183 ], "spans": [ { "bbox": [ 106, 170, 490, 183 ], "score": 1.0, "content": "Furthermore, let us recall that under i.i.d assumption we obtain from the definition of likelihood", "type": "text" } ], "index": 6 } ], "index": 6, "bbox_fs": [ 106, 170, 490, 183 ] }, { "type": "interline_equation", "bbox": [ 247, 187, 363, 220 ], "lines": [ { "bbox": [ 247, 187, 363, 220 ], "spans": [ { "bbox": [ 247, 187, 363, 220 ], "score": 0.94, "content": "L H ( y , \\hat { y } ) = \\prod _ { i = 1 } ^ { n } L H ( y _ { i } , \\hat { y } _ { i } ) ,", "type": "interline_equation", "image_path": "b1cef1055651c022e4fabcdf7453f8c1f11569815c366ea8cad9ac5be1bcf777.jpg" } ] } ], "index": 7.5, "virtual_lines": [ { "bbox": [ 247, 187, 363, 203.5 ], "spans": [], "index": 7 }, { "bbox": [ 247, 203.5, 363, 220.0 ], "spans": [], "index": 8 } ] }, { "type": "text", "bbox": [ 106, 224, 505, 248 ], "lines": [ { "bbox": [ 105, 223, 505, 238 ], "spans": [ { "bbox": [ 105, 223, 470, 238 ], "score": 1.0, "content": "and following the negative likelihood coupled with the knowledge that the log-likelihood of", "type": "text" }, { "bbox": [ 471, 227, 480, 236 ], "score": 0.84, "content": "y _ { i }", "type": "inline_equation" }, { "bbox": [ 480, 223, 505, 238 ], "score": 1.0, "content": "is the", "type": "text" } ], "index": 9 }, { "bbox": [ 105, 235, 223, 249 ], "spans": [ { "bbox": [ 105, 235, 214, 249 ], "score": 1.0, "content": "log of a particular entry of", "type": "text" }, { "bbox": [ 214, 237, 223, 248 ], "score": 0.87, "content": "\\hat { y } _ { i }", "type": "inline_equation" } ], "index": 10 } ], "index": 9.5, "bbox_fs": [ 105, 223, 505, 249 ] }, { "type": "interline_equation", "bbox": [ 145, 252, 466, 327 ], "lines": [ { "bbox": [ 145, 252, 466, 327 ], "spans": [ { "bbox": [ 145, 252, 466, 327 ], "score": 0.94, "content": "\\begin{array} { c } { { { \\cal L } _ { E N T } = - \\displaystyle \\log { \\cal L } H ( y , \\hat { y } ) = - \\sum _ { i = 1 } ^ { n } \\log { \\cal L } H ( y _ { i } , \\hat { y } _ { i } ) = - \\sum _ { i = 1 } ^ { n } \\sum _ { j = 1 } ^ { m } y _ { i j } \\cdot \\log ( \\hat { y } _ { i j } ) = } } \\\\ { { \\displaystyle \\sum _ { i = 1 } ^ { n } \\left[ - \\sum _ { j = 1 } ^ { m } y _ { i j } \\cdot \\log ( \\hat { y } _ { i j } ) \\right] = \\sum _ { i = 1 } ^ { n } H ( y _ { i } , \\hat { y } _ { i } ) . } } \\end{array}", "type": "interline_equation", "image_path": "d0e566d68e2f20b4e54fc7c2b09a43dd98b00de71bda7849e2d912e6c8af305d.jpg" } ] } ], "index": 12, "virtual_lines": [ { "bbox": [ 145, 252, 466, 277.0 ], "spans": [], "index": 11 }, { "bbox": [ 145, 277.0, 466, 302.0 ], "spans": [], "index": 12 }, { "bbox": [ 145, 302.0, 466, 327.0 ], "spans": [], "index": 13 } ] }, { "type": "text", "bbox": [ 105, 331, 371, 344 ], "lines": [ { "bbox": [ 105, 330, 371, 345 ], "spans": [ { "bbox": [ 105, 330, 371, 345 ], "score": 1.0, "content": "Finally, we obtain the following estimate applying inequality (12)", "type": "text" } ], "index": 14 } ], "index": 14, "bbox_fs": [ 105, 330, 371, 345 ] }, { "type": "interline_equation", "bbox": [ 129, 348, 481, 381 ], "lines": [ { "bbox": [ 129, 348, 481, 381 ], "spans": [ { "bbox": [ 129, 348, 481, 381 ], "score": 0.94, "content": "\\frac { 1 } { n } \\frac { \\partial } { \\partial p } L _ { E N T } = \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\frac { \\partial } { \\partial p } H ( y _ { i } , \\hat { y } _ { i } ) \\geq - \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\frac { \\partial \\log \\left( P _ { \\mathrm { { I I } } ( \\theta ) } ( Q _ { i } ) \\right) } { \\partial p } \\cdot \\log \\left( P ^ { ( X _ { \\mathbf { Q } } , C ) } ( x _ { Q _ { i } } ) \\right)", "type": "interline_equation", "image_path": "e89568be2c91580ddf980db40d0e2dcf668767f5e62acf9ad3ab41e0da02c656.jpg" } ] } ], "index": 16, "virtual_lines": [ { "bbox": [ 129, 348, 481, 359.0 ], "spans": [], "index": 15 }, { "bbox": [ 129, 359.0, 481, 370.0 ], "spans": [], "index": 16 }, { "bbox": [ 129, 370.0, 481, 381.0 ], "spans": [], "index": 17 } ] }, { "type": "text", "bbox": [ 106, 385, 506, 453 ], "lines": [ { "bbox": [ 105, 385, 506, 398 ], "spans": [ { "bbox": [ 105, 385, 506, 398 ], "score": 1.0, "content": "Also, we note that the mathematical transformations listed above hold for any type of NPP and the", "type": "text" } ], "index": 18 }, { "bbox": [ 105, 397, 505, 410 ], "spans": [ { "bbox": [ 105, 397, 505, 410 ], "score": 1.0, "content": "task dependent queries (NN with Softmax, PC or PC jointly with NN). The only difference will be", "type": "text" } ], "index": 19 }, { "bbox": [ 104, 406, 507, 424 ], "spans": [ { "bbox": [ 104, 406, 193, 424 ], "score": 1.0, "content": "the second term, i.e.,", "type": "text" }, { "bbox": [ 193, 408, 277, 421 ], "score": 0.88, "content": "\\log ( P ^ { ( C | X _ { \\mathbf { Q } } ) } ( x _ { Q _ { i j } } ) )", "type": "inline_equation" }, { "bbox": [ 277, 406, 289, 424 ], "score": 1.0, "content": "or", "type": "text" }, { "bbox": [ 290, 408, 373, 421 ], "score": 0.9, "content": "\\log ( P ^ { ( \\bar { X } _ { \\mathbf { Q } } | C ) } \\bar { ( x _ { Q _ { i j } } ) } )", "type": "inline_equation" }, { "bbox": [ 374, 406, 507, 424 ], "score": 1.0, "content": "depending on the NPP and task.", "type": "text" } ], "index": 20 }, { "bbox": [ 104, 419, 507, 434 ], "spans": [ { "bbox": [ 104, 419, 334, 434 ], "score": 1.0, "content": "The NPP in a form of a single PC modeling the joint over", "type": "text" }, { "bbox": [ 335, 421, 351, 432 ], "score": 0.87, "content": "X _ { \\mathbf { Q } }", "type": "inline_equation" }, { "bbox": [ 351, 419, 369, 434 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 369, 421, 378, 430 ], "score": 0.82, "content": "C", "type": "inline_equation" }, { "bbox": [ 378, 419, 507, 434 ], "score": 1.0, "content": "was depicted to be the example.", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 430, 506, 444 ], "spans": [ { "bbox": [ 105, 430, 506, 444 ], "score": 1.0, "content": "With that, the derivation of gradients for both loss functions 2 and 7 is complete, and the training is", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 442, 249, 454 ], "spans": [ { "bbox": [ 105, 442, 249, 454 ], "score": 1.0, "content": "carried out by coordinated descent.", "type": "text" } ], "index": 23 } ], "index": 20.5, "bbox_fs": [ 104, 385, 507, 454 ] }, { "type": "text", "bbox": [ 107, 469, 505, 504 ], "lines": [ { "bbox": [ 105, 469, 505, 483 ], "spans": [ { "bbox": [ 105, 469, 440, 483 ], "score": 1.0, "content": "Backpropagation for joint NN and PC NPPs: If within the SLASH program,", "type": "text" }, { "bbox": [ 441, 470, 462, 482 ], "score": 0.83, "content": "\\Pi ( \\pmb \\theta )", "type": "inline_equation" }, { "bbox": [ 463, 469, 505, 483 ], "score": 1.0, "content": ", the NPP", "type": "text" } ], "index": 24 }, { "bbox": [ 106, 480, 505, 493 ], "spans": [ { "bbox": [ 106, 480, 505, 493 ], "score": 1.0, "content": "forwards the data tensor through a NN first, i.e., the NPP models a joint over the NN’s output", "type": "text" } ], "index": 25 }, { "bbox": [ 106, 492, 270, 504 ], "spans": [ { "bbox": [ 106, 492, 270, 504 ], "score": 1.0, "content": "variables by a PC, then we rewrite (8) to", "type": "text" } ], "index": 26 } ], "index": 25, "bbox_fs": [ 105, 469, 505, 504 ] }, { "type": "interline_equation", "bbox": [ 176, 508, 435, 541 ], "lines": [ { "bbox": [ 176, 508, 435, 541 ], "spans": [ { "bbox": [ 176, 508, 435, 541 ], "score": 0.95, "content": "\\sum _ { i = 1 } ^ { n } { \\frac { \\partial \\log \\left( P _ { \\Pi ( \\theta ) } ( Q _ { i } ) \\right) } { \\partial \\theta } } = \\sum _ { i = 1 } ^ { n } { \\frac { \\partial \\log \\left( P _ { \\Pi ( \\theta ) } ( Q _ { i } ) \\right) } { \\partial \\mathbf { p } } } \\times { \\frac { \\partial \\mathbf { p } } { \\partial \\theta } } \\times { \\frac { \\partial \\theta } { \\partial \\gamma } } .", "type": "interline_equation", "image_path": "44e2cbe01c8e9cc1b050627c51a956c1b81d85f83dda7a0471ea236f4081c019.jpg" } ] } ], "index": 28, "virtual_lines": [ { "bbox": [ 176, 508, 435, 519.0 ], "spans": [], "index": 27 }, { "bbox": [ 176, 519.0, 435, 530.0 ], "spans": [], "index": 28 }, { "bbox": [ 176, 530.0, 435, 541.0 ], "spans": [], "index": 29 } ] }, { "type": "text", "bbox": [ 106, 546, 505, 572 ], "lines": [ { "bbox": [ 105, 546, 505, 562 ], "spans": [ { "bbox": [ 105, 546, 145, 561 ], "score": 1.0, "content": "Thereby,", "type": "text" }, { "bbox": [ 145, 550, 154, 559 ], "score": 0.82, "content": "\\gamma", "type": "inline_equation" }, { "bbox": [ 154, 546, 313, 561 ], "score": 1.0, "content": "is the set of the NN’s parameters and", "type": "text" }, { "bbox": [ 314, 546, 327, 562 ], "score": 0.9, "content": "\\frac { \\partial \\pmb { \\theta } } { \\partial \\gamma }", "type": "inline_equation" }, { "bbox": [ 327, 546, 505, 561 ], "score": 1.0, "content": "is computed by the backward propagation", "type": "text" } ], "index": 30 }, { "bbox": [ 106, 560, 174, 572 ], "spans": [ { "bbox": [ 106, 560, 174, 572 ], "score": 1.0, "content": "through the NN.", "type": "text" } ], "index": 31 } ], "index": 30.5, "bbox_fs": [ 105, 546, 505, 572 ] } ] }, { "preproc_blocks": [ { "type": "title", "bbox": [ 107, 81, 309, 94 ], "lines": [ { "bbox": [ 104, 79, 309, 96 ], "spans": [ { "bbox": [ 104, 79, 309, 96 ], "score": 1.0, "content": "B APPENDIX B – SLASH PROGRAMS", "type": "text" } ], "index": 0 } ], "index": 0 }, { "type": "text", "bbox": [ 106, 106, 506, 151 ], "lines": [ { "bbox": [ 105, 106, 506, 119 ], "spans": [ { "bbox": [ 105, 106, 506, 119 ], "score": 1.0, "content": "Here, the interested reader will find the SLASH programs which we compiled for our experiments.", "type": "text" } ], "index": 1 }, { "bbox": [ 106, 117, 505, 129 ], "spans": [ { "bbox": [ 106, 117, 505, 129 ], "score": 1.0, "content": "Figure 4 presents the one for the MNIST Addition task, Figure 6 – for the set prediction task with", "type": "text" } ], "index": 2 }, { "bbox": [ 105, 128, 506, 140 ], "spans": [ { "bbox": [ 105, 128, 407, 140 ], "score": 1.0, "content": "slot attention encoder and the subsequent CoGenT test. Note the use of the", "type": "text" }, { "bbox": [ 408, 128, 424, 138 ], "score": 0.71, "content": "\" + \"", "type": "inline_equation" }, { "bbox": [ 424, 128, 506, 140 ], "score": 1.0, "content": "and “-” notation for", "type": "text" } ], "index": 3 }, { "bbox": [ 105, 138, 375, 152 ], "spans": [ { "bbox": [ 105, 138, 375, 152 ], "score": 1.0, "content": "indicating whether a random variable is given or being queried for.", "type": "text" } ], "index": 4 } ], "index": 2.5 }, { "type": "text", "bbox": [ 99, 167, 500, 228 ], "lines": [ { "bbox": [ 105, 166, 191, 178 ], "spans": [ { "bbox": [ 105, 166, 191, 178 ], "score": 1.0, "content": "# Define images", "type": "text" } ], "index": 5 }, { "bbox": [ 106, 176, 200, 189 ], "spans": [ { "bbox": [ 106, 177, 147, 188 ], "score": 1.0, "content": "img(i1).", "type": "text" }, { "bbox": [ 153, 176, 200, 189 ], "score": 1.0, "content": "img(i2).", "type": "text" } ], "index": 6 }, { "bbox": [ 98, 186, 320, 199 ], "spans": [ { "bbox": [ 98, 190, 104, 195 ], "score": 1.0, "content": "3", "type": "text" }, { "bbox": [ 105, 186, 320, 199 ], "score": 1.0, "content": "# Define Neural-Probabilistic Predicate", "type": "text" } ], "index": 7 }, { "bbox": [ 99, 195, 361, 209 ], "spans": [ { "bbox": [ 99, 200, 103, 205 ], "score": 1.0, "content": "4", "type": "text" }, { "bbox": [ 105, 195, 361, 209 ], "score": 1.0, "content": "npp(digit(X), [0,1,2,3,4,5,6,7,8,9]) :- img(X).", "type": "text" } ], "index": 8 }, { "bbox": [ 99, 207, 486, 219 ], "spans": [ { "bbox": [ 99, 210, 104, 216 ], "score": 1.0, "content": "5", "type": "text" }, { "bbox": [ 107, 207, 486, 219 ], "score": 1.0, "content": "# Define the addition of digits given two images and the resulting sum", "type": "text" } ], "index": 9 }, { "bbox": [ 98, 217, 459, 228 ], "spans": [ { "bbox": [ 98, 219, 104, 226 ], "score": 1.0, "content": "6", "type": "text" }, { "bbox": [ 105, 217, 250, 228 ], "score": 1.0, "content": "addition(A, B, N) :- digit", "type": "text" }, { "bbox": [ 251, 218, 264, 227 ], "score": 0.52, "content": "\\mathrm { \\Omega } + \\tt { A }", "type": "inline_equation" }, { "bbox": [ 264, 217, 338, 228 ], "score": 1.0, "content": ", -N1), digit(", "type": "text" }, { "bbox": [ 338, 218, 350, 227 ], "score": 0.57, "content": "+ \\mathtt { B }", "type": "inline_equation" }, { "bbox": [ 351, 217, 391, 228 ], "score": 1.0, "content": ", -N2),", "type": "text" }, { "bbox": [ 391, 217, 453, 227 ], "score": 0.55, "content": "\\mathrm { ~ N ~ } = \\mathrm { ~ N ~ 1 ~ } + \\mathrm { ~ N ~ 2 ~ }", "type": "inline_equation" }, { "bbox": [ 453, 217, 459, 228 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 10 } ], "index": 7.5 }, { "type": "text", "bbox": [ 98, 289, 355, 309 ], "lines": [ { "bbox": [ 105, 288, 356, 299 ], "spans": [ { "bbox": [ 105, 288, 356, 299 ], "score": 1.0, "content": "# Is 7 the sum of the digits in img1 and img2?", "type": "text" } ], "index": 13 }, { "bbox": [ 98, 298, 302, 310 ], "spans": [ { "bbox": [ 98, 301, 104, 308 ], "score": 1.0, "content": "2", "type": "text" }, { "bbox": [ 105, 298, 302, 310 ], "score": 1.0, "content": ":- addition(image_id1, image_id2, 7)", "type": "text" } ], "index": 14 } ], "index": 13.5 }, { "type": "image", "bbox": [ 96, 370, 492, 602 ], "blocks": [ { "type": "image_caption", "bbox": [ 104, 243, 505, 265 ], "group_id": 0, "lines": [ { "bbox": [ 105, 241, 505, 256 ], "spans": [ { "bbox": [ 105, 241, 505, 256 ], "score": 1.0, "content": "Figure 4: SLASH Program for MNIST addition. The same program was used for the training with", "type": "text" } ], "index": 11 }, { "bbox": [ 105, 253, 162, 266 ], "spans": [ { "bbox": [ 105, 253, 162, 266 ], "score": 1.0, "content": "missing data.", "type": "text" } ], "index": 12 } ], "index": 11.5 }, { "type": "image_caption", "bbox": [ 105, 324, 504, 347 ], "group_id": 0, "lines": [ { "bbox": [ 105, 323, 505, 337 ], "spans": [ { "bbox": [ 105, 323, 505, 337 ], "score": 1.0, "content": "Figure 5: Example SLASH Query for MNIST addition. The same type of query was used for the", "type": "text" } ], "index": 15 }, { "bbox": [ 106, 334, 213, 348 ], "spans": [ { "bbox": [ 106, 334, 213, 348 ], "score": 1.0, "content": "training with missing data", "type": "text" } ], "index": 16 } ], "index": 15.5 }, { "type": "image_body", "bbox": [ 96, 370, 492, 602 ], "group_id": 0, "lines": [ { "bbox": [ 96, 370, 492, 602 ], "spans": [ { "bbox": [ 96, 370, 492, 602 ], "score": 0.793, "type": "image", "image_path": "5d942cadea3c1d2f75797d835c60baf51150bd1a4748aee77f10460fe3c0d331.jpg" } ] } ], "index": 18, "virtual_lines": [ { "bbox": [ 96, 370, 492, 447.3333333333333 ], "spans": [], "index": 17 }, { "bbox": [ 96, 447.3333333333333, 492, 524.6666666666666 ], "spans": [], "index": 18 }, { "bbox": [ 96, 524.6666666666666, 492, 602.0 ], "spans": [], "index": 19 } ] }, { "type": "image_caption", "bbox": [ 107, 615, 504, 638 ], "group_id": 0, "lines": [ { "bbox": [ 106, 615, 506, 628 ], "spans": [ { "bbox": [ 106, 615, 506, 628 ], "score": 1.0, "content": "Figure 6: SLASH Program for ShapeWorld4. The same program was used for the CoGenT experi-", "type": "text" } ], "index": 20 }, { "bbox": [ 105, 626, 136, 638 ], "spans": [ { "bbox": [ 105, 626, 136, 638 ], "score": 1.0, "content": "ments.", "type": "text" } ], "index": 21 } ], "index": 20.5 } ], "index": 16.75 }, { "type": "text", "bbox": [ 100, 661, 452, 682 ], "lines": [ { "bbox": [ 105, 661, 452, 671 ], "spans": [ { "bbox": [ 105, 661, 452, 671 ], "score": 1.0, "content": "# Does object o1 have the attributes red, circle, bright, small?", "type": "text" } ], "index": 22 }, { "bbox": [ 107, 670, 371, 682 ], "spans": [ { "bbox": [ 107, 670, 371, 682 ], "score": 1.0, "content": ":- has_attributes(o1, red, circle, bright, small)", "type": "text" } ], "index": 23 } ], "index": 22.5 }, { "type": "text", "bbox": [ 105, 696, 505, 720 ], "lines": [ { "bbox": [ 105, 696, 506, 709 ], "spans": [ { "bbox": [ 105, 696, 506, 709 ], "score": 1.0, "content": "Figure 7: Example SLASH Query for ShapeWorld4 experiments. In other words, this query corre-", "type": "text" } ], "index": 24 }, { "bbox": [ 105, 707, 372, 720 ], "spans": [ { "bbox": [ 105, 707, 372, 720 ], "score": 1.0, "content": "sponds to asking SLASH: “Is object 1 a small, bright red circle?”.", "type": "text" } ], "index": 25 } ], "index": 24.5 } ], "page_idx": 14, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 106, 26, 308, 37 ], "lines": [ { "bbox": [ 106, 25, 309, 38 ], "spans": [ { "bbox": [ 106, 25, 309, 38 ], "score": 1.0, "content": "Under review as a conference paper at ICLR 2022", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 300, 751, 310, 760 ], "lines": [ { "bbox": [ 299, 750, 312, 764 ], "spans": [ { "bbox": [ 299, 750, 312, 764 ], "score": 1.0, "content": "", "type": "text", "height": 14, "width": 13 } ] } ] } ], "para_blocks": [ { "type": "title", "bbox": [ 107, 81, 309, 94 ], "lines": [ { "bbox": [ 104, 79, 309, 96 ], "spans": [ { "bbox": [ 104, 79, 309, 96 ], "score": 1.0, "content": "B APPENDIX B – SLASH PROGRAMS", "type": "text" } ], "index": 0 } ], "index": 0 }, { "type": "text", "bbox": [ 106, 106, 506, 151 ], "lines": [ { "bbox": [ 105, 106, 506, 119 ], "spans": [ { "bbox": [ 105, 106, 506, 119 ], "score": 1.0, "content": "Here, the interested reader will find the SLASH programs which we compiled for our experiments.", "type": "text" } ], "index": 1 }, { "bbox": [ 106, 117, 505, 129 ], "spans": [ { "bbox": [ 106, 117, 505, 129 ], "score": 1.0, "content": "Figure 4 presents the one for the MNIST Addition task, Figure 6 – for the set prediction task with", "type": "text" } ], "index": 2 }, { "bbox": [ 105, 128, 506, 140 ], "spans": [ { "bbox": [ 105, 128, 407, 140 ], "score": 1.0, "content": "slot attention encoder and the subsequent CoGenT test. Note the use of the", "type": "text" }, { "bbox": [ 408, 128, 424, 138 ], "score": 0.71, "content": "\" + \"", "type": "inline_equation" }, { "bbox": [ 424, 128, 506, 140 ], "score": 1.0, "content": "and “-” notation for", "type": "text" } ], "index": 3 }, { "bbox": [ 105, 138, 375, 152 ], "spans": [ { "bbox": [ 105, 138, 375, 152 ], "score": 1.0, "content": "indicating whether a random variable is given or being queried for.", "type": "text" } ], "index": 4 } ], "index": 2.5, "bbox_fs": [ 105, 106, 506, 152 ] }, { "type": "text", "bbox": [ 99, 167, 500, 228 ], "lines": [ { "bbox": [ 105, 166, 191, 178 ], "spans": [ { "bbox": [ 105, 166, 191, 178 ], "score": 1.0, "content": "# Define images", "type": "text" } ], "index": 5 }, { "bbox": [ 106, 176, 200, 189 ], "spans": [ { "bbox": [ 106, 177, 147, 188 ], "score": 1.0, "content": "img(i1).", "type": "text" }, { "bbox": [ 153, 176, 200, 189 ], "score": 1.0, "content": "img(i2).", "type": "text" } ], "index": 6 }, { "bbox": [ 98, 186, 320, 199 ], "spans": [ { "bbox": [ 98, 190, 104, 195 ], "score": 1.0, "content": "3", "type": "text" }, { "bbox": [ 105, 186, 320, 199 ], "score": 1.0, "content": "# Define Neural-Probabilistic Predicate", "type": "text" } ], "index": 7 }, { "bbox": [ 99, 195, 361, 209 ], "spans": [ { "bbox": [ 99, 200, 103, 205 ], "score": 1.0, "content": "4", "type": "text" }, { "bbox": [ 105, 195, 361, 209 ], "score": 1.0, "content": "npp(digit(X), [0,1,2,3,4,5,6,7,8,9]) :- img(X).", "type": "text" } ], "index": 8 }, { "bbox": [ 99, 207, 486, 219 ], "spans": [ { "bbox": [ 99, 210, 104, 216 ], "score": 1.0, "content": "5", "type": "text" }, { "bbox": [ 107, 207, 486, 219 ], "score": 1.0, "content": "# Define the addition of digits given two images and the resulting sum", "type": "text" } ], "index": 9 }, { "bbox": [ 98, 217, 459, 228 ], "spans": [ { "bbox": [ 98, 219, 104, 226 ], "score": 1.0, "content": "6", "type": "text" }, { "bbox": [ 105, 217, 250, 228 ], "score": 1.0, "content": "addition(A, B, N) :- digit", "type": "text" }, { "bbox": [ 251, 218, 264, 227 ], "score": 0.52, "content": "\\mathrm { \\Omega } + \\tt { A }", "type": "inline_equation" }, { "bbox": [ 264, 217, 338, 228 ], "score": 1.0, "content": ", -N1), digit(", "type": "text" }, { "bbox": [ 338, 218, 350, 227 ], "score": 0.57, "content": "+ \\mathtt { B }", "type": "inline_equation" }, { "bbox": [ 351, 217, 391, 228 ], "score": 1.0, "content": ", -N2),", "type": "text" }, { "bbox": [ 391, 217, 453, 227 ], "score": 0.55, "content": "\\mathrm { ~ N ~ } = \\mathrm { ~ N ~ 1 ~ } + \\mathrm { ~ N ~ 2 ~ }", "type": "inline_equation" }, { "bbox": [ 453, 217, 459, 228 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 10 } ], "index": 7.5, "bbox_fs": [ 98, 166, 486, 228 ] }, { "type": "text", "bbox": [ 98, 289, 355, 309 ], "lines": [ { "bbox": [ 105, 288, 356, 299 ], "spans": [ { "bbox": [ 105, 288, 356, 299 ], "score": 1.0, "content": "# Is 7 the sum of the digits in img1 and img2?", "type": "text" } ], "index": 13 }, { "bbox": [ 98, 298, 302, 310 ], "spans": [ { "bbox": [ 98, 301, 104, 308 ], "score": 1.0, "content": "2", "type": "text" }, { "bbox": [ 105, 298, 302, 310 ], "score": 1.0, "content": ":- addition(image_id1, image_id2, 7)", "type": "text" } ], "index": 14 } ], "index": 13.5, "bbox_fs": [ 98, 288, 356, 310 ] }, { "type": "image", "bbox": [ 96, 370, 492, 602 ], "blocks": [ { "type": "image_caption", "bbox": [ 104, 243, 505, 265 ], "group_id": 0, "lines": [ { "bbox": [ 105, 241, 505, 256 ], "spans": [ { "bbox": [ 105, 241, 505, 256 ], "score": 1.0, "content": "Figure 4: SLASH Program for MNIST addition. The same program was used for the training with", "type": "text" } ], "index": 11 }, { "bbox": [ 105, 253, 162, 266 ], "spans": [ { "bbox": [ 105, 253, 162, 266 ], "score": 1.0, "content": "missing data.", "type": "text" } ], "index": 12 } ], "index": 11.5 }, { "type": "image_caption", "bbox": [ 105, 324, 504, 347 ], "group_id": 0, "lines": [ { "bbox": [ 105, 323, 505, 337 ], "spans": [ { "bbox": [ 105, 323, 505, 337 ], "score": 1.0, "content": "Figure 5: Example SLASH Query for MNIST addition. The same type of query was used for the", "type": "text" } ], "index": 15 }, { "bbox": [ 106, 334, 213, 348 ], "spans": [ { "bbox": [ 106, 334, 213, 348 ], "score": 1.0, "content": "training with missing data", "type": "text" } ], "index": 16 } ], "index": 15.5 }, { "type": "image_body", "bbox": [ 96, 370, 492, 602 ], "group_id": 0, "lines": [ { "bbox": [ 96, 370, 492, 602 ], "spans": [ { "bbox": [ 96, 370, 492, 602 ], "score": 0.793, "type": "image", "image_path": "5d942cadea3c1d2f75797d835c60baf51150bd1a4748aee77f10460fe3c0d331.jpg" } ] } ], "index": 18, "virtual_lines": [ { "bbox": [ 96, 370, 492, 447.3333333333333 ], "spans": [], "index": 17 }, { "bbox": [ 96, 447.3333333333333, 492, 524.6666666666666 ], "spans": [], "index": 18 }, { "bbox": [ 96, 524.6666666666666, 492, 602.0 ], "spans": [], "index": 19 } ] }, { "type": "image_caption", "bbox": [ 107, 615, 504, 638 ], "group_id": 0, "lines": [ { "bbox": [ 106, 615, 506, 628 ], "spans": [ { "bbox": [ 106, 615, 506, 628 ], "score": 1.0, "content": "Figure 6: SLASH Program for ShapeWorld4. The same program was used for the CoGenT experi-", "type": "text" } ], "index": 20 }, { "bbox": [ 105, 626, 136, 638 ], "spans": [ { "bbox": [ 105, 626, 136, 638 ], "score": 1.0, "content": "ments.", "type": "text" } ], "index": 21 } ], "index": 20.5 } ], "index": 16.75 }, { "type": "text", "bbox": [ 100, 661, 452, 682 ], "lines": [ { "bbox": [ 105, 661, 452, 671 ], "spans": [ { "bbox": [ 105, 661, 452, 671 ], "score": 1.0, "content": "# Does object o1 have the attributes red, circle, bright, small?", "type": "text" } ], "index": 22 }, { "bbox": [ 107, 670, 371, 682 ], "spans": [ { "bbox": [ 107, 670, 371, 682 ], "score": 1.0, "content": ":- has_attributes(o1, red, circle, bright, small)", "type": "text" } ], "index": 23 } ], "index": 22.5, "bbox_fs": [ 105, 661, 452, 682 ] }, { "type": "text", "bbox": [ 105, 696, 505, 720 ], "lines": [ { "bbox": [ 105, 696, 506, 709 ], "spans": [ { "bbox": [ 105, 696, 506, 709 ], "score": 1.0, "content": "Figure 7: Example SLASH Query for ShapeWorld4 experiments. In other words, this query corre-", "type": "text" } ], "index": 24 }, { "bbox": [ 105, 707, 372, 720 ], "spans": [ { "bbox": [ 105, 707, 372, 720 ], "score": 1.0, "content": "sponds to asking SLASH: “Is object 1 a small, bright red circle?”.", "type": "text" } ], "index": 25 } ], "index": 24.5, "bbox_fs": [ 105, 696, 506, 720 ] } ] }, { "preproc_blocks": [ { "type": "title", "bbox": [ 107, 81, 335, 94 ], "lines": [ { "bbox": [ 106, 79, 337, 96 ], "spans": [ { "bbox": [ 106, 79, 337, 96 ], "score": 1.0, "content": "C APPENDIX C – EXPERIMENTAL DETAILS", "type": "text" } ], "index": 0 } ], "index": 0 }, { "type": "title", "bbox": [ 108, 106, 260, 118 ], "lines": [ { "bbox": [ 106, 105, 262, 119 ], "spans": [ { "bbox": [ 106, 105, 262, 119 ], "score": 1.0, "content": "C.1 SHAPEWORLD4 GENERATION", "type": "text" } ], "index": 1 } ], "index": 1 }, { "type": "text", "bbox": [ 107, 127, 505, 160 ], "lines": [ { "bbox": [ 105, 126, 505, 141 ], "spans": [ { "bbox": [ 105, 126, 505, 141 ], "score": 1.0, "content": "The ShapeWorld4 and ShapeWorld4 CoGenT data sets were generated using the original scripts", "type": "text" } ], "index": 2 }, { "bbox": [ 105, 138, 505, 151 ], "spans": [ { "bbox": [ 105, 138, 505, 151 ], "score": 1.0, "content": "of (Kuhnle & Copestake, 2017) (https://github.com/AlexKuhnle/ShapeWorld). The", "type": "text" } ], "index": 3 }, { "bbox": [ 105, 150, 371, 162 ], "spans": [ { "bbox": [ 105, 150, 371, 162 ], "score": 1.0, "content": "exact scripts will be added together with the SLASH source code.", "type": "text" } ], "index": 4 } ], "index": 3 }, { "type": "title", "bbox": [ 108, 175, 366, 186 ], "lines": [ { "bbox": [ 105, 173, 367, 189 ], "spans": [ { "bbox": [ 105, 173, 367, 189 ], "score": 1.0, "content": "C.2 AVERAGE PRECISION COMPUTATION (SHAPEWORLD4)", "type": "text" } ], "index": 5 } ], "index": 5 }, { "type": "text", "bbox": [ 107, 195, 505, 273 ], "lines": [ { "bbox": [ 105, 195, 505, 209 ], "spans": [ { "bbox": [ 105, 195, 505, 209 ], "score": 1.0, "content": "For the baseline slot encoder experiments on ShapeWorld4 we measured the average precision score", "type": "text" } ], "index": 6 }, { "bbox": [ 105, 207, 505, 220 ], "spans": [ { "bbox": [ 105, 207, 505, 220 ], "score": 1.0, "content": "as in Locatello et al. (2020). In comparison to the baseline slot encoder, when applying SLASH", "type": "text" } ], "index": 7 }, { "bbox": [ 105, 217, 506, 231 ], "spans": [ { "bbox": [ 105, 217, 506, 231 ], "score": 1.0, "content": "Attention, however, we handled the case of a slot not containing an object, e.g. only background", "type": "text" } ], "index": 8 }, { "bbox": [ 106, 229, 506, 241 ], "spans": [ { "bbox": [ 106, 229, 506, 241 ], "score": 1.0, "content": "variables, differently. Whereas Locatello et al. (2020) add an additional binary identifier to the", "type": "text" } ], "index": 9 }, { "bbox": [ 105, 239, 506, 252 ], "spans": [ { "bbox": [ 105, 239, 506, 252 ], "score": 1.0, "content": "multi-label ground truth vectors, we have added a background (bg) attribute to each category (cf.", "type": "text" } ], "index": 10 }, { "bbox": [ 105, 250, 505, 263 ], "spans": [ { "bbox": [ 105, 250, 505, 263 ], "score": 1.0, "content": "Fig. 6). A slot is thus considered to be empty (i.e. not containing an object) if each NPP returns the", "type": "text" } ], "index": 11 }, { "bbox": [ 105, 262, 316, 275 ], "spans": [ { "bbox": [ 105, 262, 266, 275 ], "score": 1.0, "content": "maximal conditional probability for the", "type": "text" }, { "bbox": [ 266, 262, 278, 273 ], "score": 0.81, "content": "b g", "type": "inline_equation" }, { "bbox": [ 278, 262, 316, 275 ], "score": 1.0, "content": "attribute.", "type": "text" } ], "index": 12 } ], "index": 9 }, { "type": "text", "bbox": [ 108, 273, 505, 306 ], "lines": [ { "bbox": [ 106, 273, 505, 285 ], "spans": [ { "bbox": [ 106, 273, 505, 285 ], "score": 1.0, "content": "As the ShapeWorld4 prediction task only included discrete object properties both for Slot Attention", "type": "text" } ], "index": 13 }, { "bbox": [ 105, 283, 505, 296 ], "spans": [ { "bbox": [ 105, 283, 505, 296 ], "score": 1.0, "content": "as well as for SLASH Attention the distance threshold for the average precision computation was", "type": "text" } ], "index": 14 }, { "bbox": [ 106, 295, 287, 307 ], "spans": [ { "bbox": [ 106, 295, 287, 307 ], "score": 1.0, "content": "infinity (thus corresponding to no threshold).", "type": "text" } ], "index": 15 } ], "index": 14 }, { "type": "title", "bbox": [ 108, 320, 206, 331 ], "lines": [ { "bbox": [ 105, 318, 208, 334 ], "spans": [ { "bbox": [ 105, 318, 208, 334 ], "score": 1.0, "content": "C.3 MODEL DETAILS", "type": "text" } ], "index": 16 } ], "index": 16 }, { "type": "text", "bbox": [ 107, 341, 505, 385 ], "lines": [ { "bbox": [ 105, 339, 506, 355 ], "spans": [ { "bbox": [ 105, 339, 506, 355 ], "score": 1.0, "content": "For those experiments using NPPs with PC we have used Einsum Networks (EiNets) for implementing", "type": "text" } ], "index": 17 }, { "bbox": [ 105, 352, 506, 365 ], "spans": [ { "bbox": [ 105, 352, 506, 365 ], "score": 1.0, "content": "the probabilistic circuits. EiNets are a novel implementation design for SPNs introduced by Peharz", "type": "text" } ], "index": 18 }, { "bbox": [ 105, 364, 506, 375 ], "spans": [ { "bbox": [ 105, 364, 506, 375 ], "score": 1.0, "content": "et al. (2020) that minimize the issue of computational costs that initial SPNs had suffered. This is", "type": "text" } ], "index": 19 }, { "bbox": [ 105, 374, 507, 388 ], "spans": [ { "bbox": [ 105, 374, 507, 388 ], "score": 1.0, "content": "accomplished by combining several arithmetic operations via a single monolithic einsum-operation.", "type": "text" } ], "index": 20 } ], "index": 18.5 }, { "type": "text", "bbox": [ 107, 390, 504, 413 ], "lines": [ { "bbox": [ 106, 390, 502, 403 ], "spans": [ { "bbox": [ 106, 390, 394, 403 ], "score": 1.0, "content": "For all experiments, the ADAM optimizer (Kingma & Ba, 2015) with", "type": "text" }, { "bbox": [ 395, 391, 434, 402 ], "score": 0.91, "content": "\\beta 1 = 0 . 9", "type": "inline_equation" }, { "bbox": [ 435, 390, 453, 403 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 453, 391, 502, 403 ], "score": 0.9, "content": "\\beta 2 = 0 . 9 9 9", "type": "inline_equation" } ], "index": 21 }, { "bbox": [ 107, 402, 279, 414 ], "spans": [ { "bbox": [ 107, 402, 152, 412 ], "score": 0.89, "content": "\\epsilon = 1 e - 8", "type": "inline_equation" }, { "bbox": [ 152, 402, 279, 414 ], "score": 1.0, "content": "and no weight decay was used.", "type": "text" } ], "index": 22 } ], "index": 21.5 }, { "type": "text", "bbox": [ 106, 419, 505, 496 ], "lines": [ { "bbox": [ 105, 418, 505, 433 ], "spans": [ { "bbox": [ 105, 418, 505, 433 ], "score": 1.0, "content": "MNIST-Addition Experiments For the MNIST-Addition experiments, we ran the DeepProbLog", "type": "text" } ], "index": 23 }, { "bbox": [ 105, 429, 506, 443 ], "spans": [ { "bbox": [ 105, 429, 506, 443 ], "score": 1.0, "content": "and NeurASP programs with their original configurations, as stated in (Manhaeve et al., 2018) and", "type": "text" } ], "index": 24 }, { "bbox": [ 106, 441, 505, 453 ], "spans": [ { "bbox": [ 106, 441, 505, 453 ], "score": 1.0, "content": "(Yang et al., 2020), respectively. For the SLASH MNIST-Addition experiments, we have used the", "type": "text" } ], "index": 25 }, { "bbox": [ 105, 452, 505, 464 ], "spans": [ { "bbox": [ 105, 452, 505, 464 ], "score": 1.0, "content": "same neural module as in DeepProbLog and NeurASP, when training SLASH with the neural NPP", "type": "text" } ], "index": 26 }, { "bbox": [ 105, 462, 506, 476 ], "spans": [ { "bbox": [ 105, 462, 506, 476 ], "score": 1.0, "content": "(SLASH (DNN)) represented in Tab. 2. When using a PC NPP (SLASH (PC)) we have used an EiNet", "type": "text" } ], "index": 27 }, { "bbox": [ 105, 474, 506, 487 ], "spans": [ { "bbox": [ 105, 474, 506, 487 ], "score": 1.0, "content": "with the Poon-Domingos (PD) structure (Poon & Domingos, 2011) and normal distribution for the", "type": "text" } ], "index": 28 }, { "bbox": [ 105, 485, 392, 497 ], "spans": [ { "bbox": [ 105, 485, 392, 497 ], "score": 1.0, "content": "leafs. The formal hyperparameters for the EiNet are depicted in Tab. 3.", "type": "text" } ], "index": 29 } ], "index": 26 }, { "type": "text", "bbox": [ 107, 501, 504, 524 ], "lines": [ { "bbox": [ 105, 500, 505, 515 ], "spans": [ { "bbox": [ 105, 500, 505, 515 ], "score": 1.0, "content": "The learning rate and batch size for the DNN were 0.005 and 100, for DeepProbLog, NeurASP and", "type": "text" } ], "index": 30 }, { "bbox": [ 106, 513, 334, 524 ], "spans": [ { "bbox": [ 106, 513, 334, 524 ], "score": 1.0, "content": "SLASH (DNN). For the EiNet, these were 0.01 and 100.", "type": "text" } ], "index": 31 } ], "index": 30.5 }, { "type": "table", "bbox": [ 165, 555, 446, 676 ], "blocks": [ { "type": "table_caption", "bbox": [ 165, 534, 441, 546 ], "group_id": 0, "lines": [ { "bbox": [ 165, 533, 442, 548 ], "spans": [ { "bbox": [ 165, 533, 442, 548 ], "score": 1.0, "content": "Table 2: Neural module – LeNet5 for MNIST-Addition experiments.", "type": "text" } ], "index": 32 } ], "index": 32 }, { "type": "table_body", "bbox": [ 165, 555, 446, 676 ], "group_id": 0, "lines": [ { "bbox": [ 165, 555, 446, 676 ], "spans": [ { "bbox": [ 165, 555, 446, 676 ], "score": 0.982, "html": "
TypeSize/ChannelsActivationComment
Encoder--
Conv 5 x 51x28x281stride 1
MaxPool2d6x24x24ReLUkernel size 2, stride 2
Conv 5 x 56x12x121stride 1
MaxPool2d16x8x8ReLUkernel size 2,stride 2
Classifier1--
MLP16x4x4,120ReLU-
MLP120,84ReLU-
MLP84,101Softmax
", "type": "table", "image_path": "f70c0188a3a6753bf5fd03b4241a0ce55a0b8fc7d6e5338c2bb42fd3d7818bc9.jpg" } ] } ], "index": 34, "virtual_lines": [ { "bbox": [ 165, 555, 446, 595.3333333333334 ], "spans": [], "index": 33 }, { "bbox": [ 165, 595.3333333333334, 446, 635.6666666666667 ], "spans": [], "index": 34 }, { "bbox": [ 165, 635.6666666666667, 446, 676.0000000000001 ], "spans": [], "index": 35 } ] } ], "index": 33.0 }, { "type": "text", "bbox": [ 107, 687, 505, 732 ], "lines": [ { "bbox": [ 105, 687, 506, 700 ], "spans": [ { "bbox": [ 105, 687, 506, 700 ], "score": 1.0, "content": "ShapeWorld4 Experiments For the baseline slot attention experiments with the ShapeWorld4 data", "type": "text" } ], "index": 36 }, { "bbox": [ 105, 699, 505, 711 ], "spans": [ { "bbox": [ 105, 699, 505, 711 ], "score": 1.0, "content": "set we have used the architecture presented in Tab. 4. For further details on this, we refer to the", "type": "text" } ], "index": 37 }, { "bbox": [ 105, 710, 505, 721 ], "spans": [ { "bbox": [ 105, 710, 505, 721 ], "score": 1.0, "content": "original work of Locatello et al. (2020). The slot encoder had a number of 4 slots and 3 attention", "type": "text" } ], "index": 38 }, { "bbox": [ 105, 721, 231, 732 ], "spans": [ { "bbox": [ 105, 721, 231, 732 ], "score": 1.0, "content": "iterations over all experiments.", "type": "text" } ], "index": 39 } ], "index": 37.5 } ], "page_idx": 15, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 107, 27, 308, 37 ], "lines": [ { "bbox": [ 107, 26, 308, 38 ], "spans": [ { "bbox": [ 107, 26, 308, 38 ], "score": 1.0, "content": "Under review as a conference paper at ICLR 2022", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 300, 751, 311, 760 ], "lines": [ { "bbox": [ 299, 750, 312, 764 ], "spans": [ { "bbox": [ 299, 750, 312, 764 ], "score": 1.0, "content": "16", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "title", "bbox": [ 107, 81, 335, 94 ], "lines": [ { "bbox": [ 106, 79, 337, 96 ], "spans": [ { "bbox": [ 106, 79, 337, 96 ], "score": 1.0, "content": "C APPENDIX C – EXPERIMENTAL DETAILS", "type": "text" } ], "index": 0 } ], "index": 0 }, { "type": "title", "bbox": [ 108, 106, 260, 118 ], "lines": [ { "bbox": [ 106, 105, 262, 119 ], "spans": [ { "bbox": [ 106, 105, 262, 119 ], "score": 1.0, "content": "C.1 SHAPEWORLD4 GENERATION", "type": "text" } ], "index": 1 } ], "index": 1 }, { "type": "text", "bbox": [ 107, 127, 505, 160 ], "lines": [ { "bbox": [ 105, 126, 505, 141 ], "spans": [ { "bbox": [ 105, 126, 505, 141 ], "score": 1.0, "content": "The ShapeWorld4 and ShapeWorld4 CoGenT data sets were generated using the original scripts", "type": "text" } ], "index": 2 }, { "bbox": [ 105, 138, 505, 151 ], "spans": [ { "bbox": [ 105, 138, 505, 151 ], "score": 1.0, "content": "of (Kuhnle & Copestake, 2017) (https://github.com/AlexKuhnle/ShapeWorld). The", "type": "text" } ], "index": 3 }, { "bbox": [ 105, 150, 371, 162 ], "spans": [ { "bbox": [ 105, 150, 371, 162 ], "score": 1.0, "content": "exact scripts will be added together with the SLASH source code.", "type": "text" } ], "index": 4 } ], "index": 3, "bbox_fs": [ 105, 126, 505, 162 ] }, { "type": "title", "bbox": [ 108, 175, 366, 186 ], "lines": [ { "bbox": [ 105, 173, 367, 189 ], "spans": [ { "bbox": [ 105, 173, 367, 189 ], "score": 1.0, "content": "C.2 AVERAGE PRECISION COMPUTATION (SHAPEWORLD4)", "type": "text" } ], "index": 5 } ], "index": 5 }, { "type": "text", "bbox": [ 107, 195, 505, 273 ], "lines": [ { "bbox": [ 105, 195, 505, 209 ], "spans": [ { "bbox": [ 105, 195, 505, 209 ], "score": 1.0, "content": "For the baseline slot encoder experiments on ShapeWorld4 we measured the average precision score", "type": "text" } ], "index": 6 }, { "bbox": [ 105, 207, 505, 220 ], "spans": [ { "bbox": [ 105, 207, 505, 220 ], "score": 1.0, "content": "as in Locatello et al. (2020). In comparison to the baseline slot encoder, when applying SLASH", "type": "text" } ], "index": 7 }, { "bbox": [ 105, 217, 506, 231 ], "spans": [ { "bbox": [ 105, 217, 506, 231 ], "score": 1.0, "content": "Attention, however, we handled the case of a slot not containing an object, e.g. only background", "type": "text" } ], "index": 8 }, { "bbox": [ 106, 229, 506, 241 ], "spans": [ { "bbox": [ 106, 229, 506, 241 ], "score": 1.0, "content": "variables, differently. Whereas Locatello et al. (2020) add an additional binary identifier to the", "type": "text" } ], "index": 9 }, { "bbox": [ 105, 239, 506, 252 ], "spans": [ { "bbox": [ 105, 239, 506, 252 ], "score": 1.0, "content": "multi-label ground truth vectors, we have added a background (bg) attribute to each category (cf.", "type": "text" } ], "index": 10 }, { "bbox": [ 105, 250, 505, 263 ], "spans": [ { "bbox": [ 105, 250, 505, 263 ], "score": 1.0, "content": "Fig. 6). A slot is thus considered to be empty (i.e. not containing an object) if each NPP returns the", "type": "text" } ], "index": 11 }, { "bbox": [ 105, 262, 316, 275 ], "spans": [ { "bbox": [ 105, 262, 266, 275 ], "score": 1.0, "content": "maximal conditional probability for the", "type": "text" }, { "bbox": [ 266, 262, 278, 273 ], "score": 0.81, "content": "b g", "type": "inline_equation" }, { "bbox": [ 278, 262, 316, 275 ], "score": 1.0, "content": "attribute.", "type": "text" } ], "index": 12 } ], "index": 9, "bbox_fs": [ 105, 195, 506, 275 ] }, { "type": "text", "bbox": [ 108, 273, 505, 306 ], "lines": [ { "bbox": [ 106, 273, 505, 285 ], "spans": [ { "bbox": [ 106, 273, 505, 285 ], "score": 1.0, "content": "As the ShapeWorld4 prediction task only included discrete object properties both for Slot Attention", "type": "text" } ], "index": 13 }, { "bbox": [ 105, 283, 505, 296 ], "spans": [ { "bbox": [ 105, 283, 505, 296 ], "score": 1.0, "content": "as well as for SLASH Attention the distance threshold for the average precision computation was", "type": "text" } ], "index": 14 }, { "bbox": [ 106, 295, 287, 307 ], "spans": [ { "bbox": [ 106, 295, 287, 307 ], "score": 1.0, "content": "infinity (thus corresponding to no threshold).", "type": "text" } ], "index": 15 } ], "index": 14, "bbox_fs": [ 105, 273, 505, 307 ] }, { "type": "title", "bbox": [ 108, 320, 206, 331 ], "lines": [ { "bbox": [ 105, 318, 208, 334 ], "spans": [ { "bbox": [ 105, 318, 208, 334 ], "score": 1.0, "content": "C.3 MODEL DETAILS", "type": "text" } ], "index": 16 } ], "index": 16 }, { "type": "text", "bbox": [ 107, 341, 505, 385 ], "lines": [ { "bbox": [ 105, 339, 506, 355 ], "spans": [ { "bbox": [ 105, 339, 506, 355 ], "score": 1.0, "content": "For those experiments using NPPs with PC we have used Einsum Networks (EiNets) for implementing", "type": "text" } ], "index": 17 }, { "bbox": [ 105, 352, 506, 365 ], "spans": [ { "bbox": [ 105, 352, 506, 365 ], "score": 1.0, "content": "the probabilistic circuits. EiNets are a novel implementation design for SPNs introduced by Peharz", "type": "text" } ], "index": 18 }, { "bbox": [ 105, 364, 506, 375 ], "spans": [ { "bbox": [ 105, 364, 506, 375 ], "score": 1.0, "content": "et al. (2020) that minimize the issue of computational costs that initial SPNs had suffered. This is", "type": "text" } ], "index": 19 }, { "bbox": [ 105, 374, 507, 388 ], "spans": [ { "bbox": [ 105, 374, 507, 388 ], "score": 1.0, "content": "accomplished by combining several arithmetic operations via a single monolithic einsum-operation.", "type": "text" } ], "index": 20 } ], "index": 18.5, "bbox_fs": [ 105, 339, 507, 388 ] }, { "type": "text", "bbox": [ 107, 390, 504, 413 ], "lines": [ { "bbox": [ 106, 390, 502, 403 ], "spans": [ { "bbox": [ 106, 390, 394, 403 ], "score": 1.0, "content": "For all experiments, the ADAM optimizer (Kingma & Ba, 2015) with", "type": "text" }, { "bbox": [ 395, 391, 434, 402 ], "score": 0.91, "content": "\\beta 1 = 0 . 9", "type": "inline_equation" }, { "bbox": [ 435, 390, 453, 403 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 453, 391, 502, 403 ], "score": 0.9, "content": "\\beta 2 = 0 . 9 9 9", "type": "inline_equation" } ], "index": 21 }, { "bbox": [ 107, 402, 279, 414 ], "spans": [ { "bbox": [ 107, 402, 152, 412 ], "score": 0.89, "content": "\\epsilon = 1 e - 8", "type": "inline_equation" }, { "bbox": [ 152, 402, 279, 414 ], "score": 1.0, "content": "and no weight decay was used.", "type": "text" } ], "index": 22 } ], "index": 21.5, "bbox_fs": [ 106, 390, 502, 414 ] }, { "type": "text", "bbox": [ 106, 419, 505, 496 ], "lines": [ { "bbox": [ 105, 418, 505, 433 ], "spans": [ { "bbox": [ 105, 418, 505, 433 ], "score": 1.0, "content": "MNIST-Addition Experiments For the MNIST-Addition experiments, we ran the DeepProbLog", "type": "text" } ], "index": 23 }, { "bbox": [ 105, 429, 506, 443 ], "spans": [ { "bbox": [ 105, 429, 506, 443 ], "score": 1.0, "content": "and NeurASP programs with their original configurations, as stated in (Manhaeve et al., 2018) and", "type": "text" } ], "index": 24 }, { "bbox": [ 106, 441, 505, 453 ], "spans": [ { "bbox": [ 106, 441, 505, 453 ], "score": 1.0, "content": "(Yang et al., 2020), respectively. For the SLASH MNIST-Addition experiments, we have used the", "type": "text" } ], "index": 25 }, { "bbox": [ 105, 452, 505, 464 ], "spans": [ { "bbox": [ 105, 452, 505, 464 ], "score": 1.0, "content": "same neural module as in DeepProbLog and NeurASP, when training SLASH with the neural NPP", "type": "text" } ], "index": 26 }, { "bbox": [ 105, 462, 506, 476 ], "spans": [ { "bbox": [ 105, 462, 506, 476 ], "score": 1.0, "content": "(SLASH (DNN)) represented in Tab. 2. When using a PC NPP (SLASH (PC)) we have used an EiNet", "type": "text" } ], "index": 27 }, { "bbox": [ 105, 474, 506, 487 ], "spans": [ { "bbox": [ 105, 474, 506, 487 ], "score": 1.0, "content": "with the Poon-Domingos (PD) structure (Poon & Domingos, 2011) and normal distribution for the", "type": "text" } ], "index": 28 }, { "bbox": [ 105, 485, 392, 497 ], "spans": [ { "bbox": [ 105, 485, 392, 497 ], "score": 1.0, "content": "leafs. The formal hyperparameters for the EiNet are depicted in Tab. 3.", "type": "text" } ], "index": 29 } ], "index": 26, "bbox_fs": [ 105, 418, 506, 497 ] }, { "type": "text", "bbox": [ 107, 501, 504, 524 ], "lines": [ { "bbox": [ 105, 500, 505, 515 ], "spans": [ { "bbox": [ 105, 500, 505, 515 ], "score": 1.0, "content": "The learning rate and batch size for the DNN were 0.005 and 100, for DeepProbLog, NeurASP and", "type": "text" } ], "index": 30 }, { "bbox": [ 106, 513, 334, 524 ], "spans": [ { "bbox": [ 106, 513, 334, 524 ], "score": 1.0, "content": "SLASH (DNN). For the EiNet, these were 0.01 and 100.", "type": "text" } ], "index": 31 } ], "index": 30.5, "bbox_fs": [ 105, 500, 505, 524 ] }, { "type": "table", "bbox": [ 165, 555, 446, 676 ], "blocks": [ { "type": "table_caption", "bbox": [ 165, 534, 441, 546 ], "group_id": 0, "lines": [ { "bbox": [ 165, 533, 442, 548 ], "spans": [ { "bbox": [ 165, 533, 442, 548 ], "score": 1.0, "content": "Table 2: Neural module – LeNet5 for MNIST-Addition experiments.", "type": "text" } ], "index": 32 } ], "index": 32 }, { "type": "table_body", "bbox": [ 165, 555, 446, 676 ], "group_id": 0, "lines": [ { "bbox": [ 165, 555, 446, 676 ], "spans": [ { "bbox": [ 165, 555, 446, 676 ], "score": 0.982, "html": "
TypeSize/ChannelsActivationComment
Encoder--
Conv 5 x 51x28x281stride 1
MaxPool2d6x24x24ReLUkernel size 2, stride 2
Conv 5 x 56x12x121stride 1
MaxPool2d16x8x8ReLUkernel size 2,stride 2
Classifier1--
MLP16x4x4,120ReLU-
MLP120,84ReLU-
MLP84,101Softmax
", "type": "table", "image_path": "f70c0188a3a6753bf5fd03b4241a0ce55a0b8fc7d6e5338c2bb42fd3d7818bc9.jpg" } ] } ], "index": 34, "virtual_lines": [ { "bbox": [ 165, 555, 446, 595.3333333333334 ], "spans": [], "index": 33 }, { "bbox": [ 165, 595.3333333333334, 446, 635.6666666666667 ], "spans": [], "index": 34 }, { "bbox": [ 165, 635.6666666666667, 446, 676.0000000000001 ], "spans": [], "index": 35 } ] } ], "index": 33.0 }, { "type": "text", "bbox": [ 107, 687, 505, 732 ], "lines": [ { "bbox": [ 105, 687, 506, 700 ], "spans": [ { "bbox": [ 105, 687, 506, 700 ], "score": 1.0, "content": "ShapeWorld4 Experiments For the baseline slot attention experiments with the ShapeWorld4 data", "type": "text" } ], "index": 36 }, { "bbox": [ 105, 699, 505, 711 ], "spans": [ { "bbox": [ 105, 699, 505, 711 ], "score": 1.0, "content": "set we have used the architecture presented in Tab. 4. For further details on this, we refer to the", "type": "text" } ], "index": 37 }, { "bbox": [ 105, 710, 505, 721 ], "spans": [ { "bbox": [ 105, 710, 505, 721 ], "score": 1.0, "content": "original work of Locatello et al. (2020). The slot encoder had a number of 4 slots and 3 attention", "type": "text" } ], "index": 38 }, { "bbox": [ 105, 721, 231, 732 ], "spans": [ { "bbox": [ 105, 721, 231, 732 ], "score": 1.0, "content": "iterations over all experiments.", "type": "text" } ], "index": 39 } ], "index": 37.5, "bbox_fs": [ 105, 687, 506, 732 ] } ] }, { "preproc_blocks": [ { "type": "table", "bbox": [ 171, 101, 440, 130 ], "blocks": [ { "type": "table_caption", "bbox": [ 141, 80, 465, 92 ], "group_id": 0, "lines": [ { "bbox": [ 141, 78, 465, 94 ], "spans": [ { "bbox": [ 141, 78, 465, 94 ], "score": 1.0, "content": "Table 3: Probabilistic Circuit module – EiNet for MNIST-Addition experiments.", "type": "text" } ], "index": 0 } ], "index": 0 }, { "type": "table_body", "bbox": [ 171, 101, 440, 130 ], "group_id": 0, "lines": [ { "bbox": [ 171, 101, 440, 130 ], "spans": [ { "bbox": [ 171, 101, 440, 130 ], "score": 0.953, "html": "
VariablesWidthHeightNumber of PiecesClass count
7842828[4,7,28]10
", "type": "table", "image_path": "055619a651ca94f260eb57626f398d17ef7693cb21b9b36feb10922133cfeb38.jpg" } ] } ], "index": 2, "virtual_lines": [ { "bbox": [ 171, 101, 440, 110.66666666666667 ], "spans": [], "index": 1 }, { "bbox": [ 171, 110.66666666666667, 440, 120.33333333333334 ], "spans": [], "index": 2 }, { "bbox": [ 171, 120.33333333333334, 440, 130.0 ], "spans": [], "index": 3 } ] } ], "index": 1.0 }, { "type": "table", "bbox": [ 150, 162, 461, 315 ], "blocks": [ { "type": "table_caption", "bbox": [ 180, 140, 427, 152 ], "group_id": 1, "lines": [ { "bbox": [ 179, 138, 428, 154 ], "spans": [ { "bbox": [ 179, 138, 428, 154 ], "score": 1.0, "content": "Table 4: Baseline slot encoder for ShapeWorld4 experiments.", "type": "text" } ], "index": 4 } ], "index": 4 }, { "type": "table_body", "bbox": [ 150, 162, 461, 315 ], "group_id": 1, "lines": [ { "bbox": [ 150, 162, 461, 315 ], "spans": [ { "bbox": [ 150, 162, 461, 315 ], "score": 0.985, "html": "
TypeSize/ChannelsActivationComment
Conv 5 x 532ReLUstride 1
Conv 5 x 532ReLUstride 1
Conv 5 x 532ReLUstride 1
Conv 5 x 532ReLUstride 1
Position Embedding1-
Flattenaxis: [0, 1,2 x 3]-flatten x, y pos.
Layer Norm-
MLP (per location)32ReLU
MLP (per location)32-
SlotAttentionModule32ReLU
MLP32ReLU
MLP16Sigmoid二 1
", "type": "table", "image_path": "7a6e8fe39959ba94229a5e2822d9f8343254e45215672d154eb62542dc807e0b.jpg" } ] } ], "index": 6, "virtual_lines": [ { "bbox": [ 150, 162, 461, 213.0 ], "spans": [], "index": 5 }, { "bbox": [ 150, 213.0, 461, 264.0 ], "spans": [], "index": 6 }, { "bbox": [ 150, 264.0, 461, 315.0 ], "spans": [], "index": 7 } ] } ], "index": 5.0 }, { "type": "text", "bbox": [ 107, 335, 505, 369 ], "lines": [ { "bbox": [ 105, 335, 505, 347 ], "spans": [ { "bbox": [ 105, 335, 505, 347 ], "score": 1.0, "content": "For the SLASH Attention experiments with ShapeWorld4 we have used the same slot encoder as in", "type": "text" } ], "index": 8 }, { "bbox": [ 105, 345, 505, 359 ], "spans": [ { "bbox": [ 105, 345, 505, 359 ], "score": 1.0, "content": "Tab. 4, however, we replaced the final MLPs with 4 individual EiNets with Poon-Domingos structure", "type": "text" } ], "index": 9 }, { "bbox": [ 106, 357, 414, 370 ], "spans": [ { "bbox": [ 106, 357, 414, 370 ], "score": 1.0, "content": "(Poon & Domingos, 2011). Their hyperparameters are represented in Tab. 5.", "type": "text" } ], "index": 10 } ], "index": 9 }, { "type": "table", "bbox": [ 152, 400, 458, 463 ], "blocks": [ { "type": "table_caption", "bbox": [ 150, 379, 456, 391 ], "group_id": 2, "lines": [ { "bbox": [ 149, 376, 458, 394 ], "spans": [ { "bbox": [ 149, 376, 458, 394 ], "score": 1.0, "content": "Table 5: Probabilistic Circuit module – EiNet for ShapeWorld4 experiments.", "type": "text" } ], "index": 11 } ], "index": 11 }, { "type": "table_body", "bbox": [ 152, 400, 458, 463 ], "group_id": 2, "lines": [ { "bbox": [ 152, 400, 458, 463 ], "spans": [ { "bbox": [ 152, 400, 458, 463 ], "score": 0.979, "html": "
EiNetVariablesWidthHeightNumber ofPiecesClass count
Color3284[4]9
Shape3284[4]4
Shade3284[4]3
Size3284[4]3
", "type": "table", "image_path": "bf6ad54fdca37e6aada941896a644326dea1a55ea9b6a6022392ceabd19cbc57.jpg" } ] } ], "index": 13, "virtual_lines": [ { "bbox": [ 152, 400, 458, 421.0 ], "spans": [], "index": 12 }, { "bbox": [ 152, 421.0, 458, 442.0 ], "spans": [], "index": 13 }, { "bbox": [ 152, 442.0, 458, 463.0 ], "spans": [], "index": 14 } ] } ], "index": 12.0 }, { "type": "text", "bbox": [ 107, 474, 505, 497 ], "lines": [ { "bbox": [ 106, 474, 505, 486 ], "spans": [ { "bbox": [ 106, 474, 505, 486 ], "score": 1.0, "content": "The learning rate and batch size for SLASH Attention were 0.01 and 512, for ShapeWorld4 and", "type": "text" } ], "index": 15 }, { "bbox": [ 106, 485, 475, 498 ], "spans": [ { "bbox": [ 106, 485, 475, 498 ], "score": 1.0, "content": "ShapeWorld4 CoGenT. The learning rate for the baseline slot encoder were 0.0004 and 512.", "type": "text" } ], "index": 16 } ], "index": 15.5 } ], "page_idx": 16, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 106, 27, 308, 37 ], "lines": [ { "bbox": [ 106, 26, 308, 38 ], "spans": [ { "bbox": [ 106, 26, 308, 38 ], "score": 1.0, "content": "Under review as a conference paper at ICLR 2022", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 300, 751, 310, 760 ], "lines": [ { "bbox": [ 299, 750, 312, 764 ], "spans": [ { "bbox": [ 299, 750, 312, 764 ], "score": 1.0, "content": "", "type": "text", "height": 14, "width": 13 } ] } ] } ], "para_blocks": [ { "type": "table", "bbox": [ 171, 101, 440, 130 ], "blocks": [ { "type": "table_caption", "bbox": [ 141, 80, 465, 92 ], "group_id": 0, "lines": [ { "bbox": [ 141, 78, 465, 94 ], "spans": [ { "bbox": [ 141, 78, 465, 94 ], "score": 1.0, "content": "Table 3: Probabilistic Circuit module – EiNet for MNIST-Addition experiments.", "type": "text" } ], "index": 0 } ], "index": 0 }, { "type": "table_body", "bbox": [ 171, 101, 440, 130 ], "group_id": 0, "lines": [ { "bbox": [ 171, 101, 440, 130 ], "spans": [ { "bbox": [ 171, 101, 440, 130 ], "score": 0.953, "html": "
VariablesWidthHeightNumber of PiecesClass count
7842828[4,7,28]10
", "type": "table", "image_path": "055619a651ca94f260eb57626f398d17ef7693cb21b9b36feb10922133cfeb38.jpg" } ] } ], "index": 2, "virtual_lines": [ { "bbox": [ 171, 101, 440, 110.66666666666667 ], "spans": [], "index": 1 }, { "bbox": [ 171, 110.66666666666667, 440, 120.33333333333334 ], "spans": [], "index": 2 }, { "bbox": [ 171, 120.33333333333334, 440, 130.0 ], "spans": [], "index": 3 } ] } ], "index": 1.0 }, { "type": "table", "bbox": [ 150, 162, 461, 315 ], "blocks": [ { "type": "table_caption", "bbox": [ 180, 140, 427, 152 ], "group_id": 1, "lines": [ { "bbox": [ 179, 138, 428, 154 ], "spans": [ { "bbox": [ 179, 138, 428, 154 ], "score": 1.0, "content": "Table 4: Baseline slot encoder for ShapeWorld4 experiments.", "type": "text" } ], "index": 4 } ], "index": 4 }, { "type": "table_body", "bbox": [ 150, 162, 461, 315 ], "group_id": 1, "lines": [ { "bbox": [ 150, 162, 461, 315 ], "spans": [ { "bbox": [ 150, 162, 461, 315 ], "score": 0.985, "html": "
TypeSize/ChannelsActivationComment
Conv 5 x 532ReLUstride 1
Conv 5 x 532ReLUstride 1
Conv 5 x 532ReLUstride 1
Conv 5 x 532ReLUstride 1
Position Embedding1-
Flattenaxis: [0, 1,2 x 3]-flatten x, y pos.
Layer Norm-
MLP (per location)32ReLU
MLP (per location)32-
SlotAttentionModule32ReLU
MLP32ReLU
MLP16Sigmoid二 1
", "type": "table", "image_path": "7a6e8fe39959ba94229a5e2822d9f8343254e45215672d154eb62542dc807e0b.jpg" } ] } ], "index": 6, "virtual_lines": [ { "bbox": [ 150, 162, 461, 213.0 ], "spans": [], "index": 5 }, { "bbox": [ 150, 213.0, 461, 264.0 ], "spans": [], "index": 6 }, { "bbox": [ 150, 264.0, 461, 315.0 ], "spans": [], "index": 7 } ] } ], "index": 5.0 }, { "type": "text", "bbox": [ 107, 335, 505, 369 ], "lines": [ { "bbox": [ 105, 335, 505, 347 ], "spans": [ { "bbox": [ 105, 335, 505, 347 ], "score": 1.0, "content": "For the SLASH Attention experiments with ShapeWorld4 we have used the same slot encoder as in", "type": "text" } ], "index": 8 }, { "bbox": [ 105, 345, 505, 359 ], "spans": [ { "bbox": [ 105, 345, 505, 359 ], "score": 1.0, "content": "Tab. 4, however, we replaced the final MLPs with 4 individual EiNets with Poon-Domingos structure", "type": "text" } ], "index": 9 }, { "bbox": [ 106, 357, 414, 370 ], "spans": [ { "bbox": [ 106, 357, 414, 370 ], "score": 1.0, "content": "(Poon & Domingos, 2011). Their hyperparameters are represented in Tab. 5.", "type": "text" } ], "index": 10 } ], "index": 9, "bbox_fs": [ 105, 335, 505, 370 ] }, { "type": "table", "bbox": [ 152, 400, 458, 463 ], "blocks": [ { "type": "table_caption", "bbox": [ 150, 379, 456, 391 ], "group_id": 2, "lines": [ { "bbox": [ 149, 376, 458, 394 ], "spans": [ { "bbox": [ 149, 376, 458, 394 ], "score": 1.0, "content": "Table 5: Probabilistic Circuit module – EiNet for ShapeWorld4 experiments.", "type": "text" } ], "index": 11 } ], "index": 11 }, { "type": "table_body", "bbox": [ 152, 400, 458, 463 ], "group_id": 2, "lines": [ { "bbox": [ 152, 400, 458, 463 ], "spans": [ { "bbox": [ 152, 400, 458, 463 ], "score": 0.979, "html": "
EiNetVariablesWidthHeightNumber ofPiecesClass count
Color3284[4]9
Shape3284[4]4
Shade3284[4]3
Size3284[4]3
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DeepProbLogSLASH (PC)
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80%31.6± 6.0844.2± 8.23
90%16.94 ± 1.7629.6± 5.77
97%12.33 ± 0.4717.6 ± 2.97
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DeepProbLogSLASH (PC)
50%79.94 ± 7.272.2 ± 12.15
80%31.6± 6.0844.2± 8.23
90%16.94 ± 1.7629.6± 5.77
97%12.33 ± 0.4717.6 ± 2.97
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