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Additionally, to allow the", "type": "text" } ], "index": 35 }, { "bbox": [ 105, 609, 506, 622 ], "spans": [ { "bbox": [ 105, 609, 506, 622 ], "score": 1.0, "content": "models to learn these long-term patterns, the historical input to the models should also be long. To", "type": "text" } ], "index": 36 }, { "bbox": [ 105, 620, 318, 634 ], "spans": [ { "bbox": [ 105, 620, 318, 634 ], "score": 1.0, "content": "this end, low time and space complexity is a priority.", "type": "text" } ], "index": 37 } ], "index": 33, "bbox_fs": [ 105, 532, 506, 634 ] }, { "type": "text", "bbox": [ 108, 637, 502, 693 ], "lines": [ { "bbox": [ 106, 638, 503, 650 ], "spans": [ { "bbox": [ 106, 638, 503, 650 ], "score": 1.0, "content": "Unfortunately, the present state-of-the-art methods fail to accomplish these two objectives simul-", "type": "text" } ], "index": 38 }, { "bbox": [ 106, 648, 504, 661 ], "spans": [ { "bbox": [ 106, 648, 504, 661 ], "score": 1.0, "content": "taneously. On one end, RNN (Salinas et al., 2020) and CNN (Munir et al., 2018) achieve a low", "type": "text" } ], "index": 39 }, { "bbox": [ 105, 659, 505, 673 ], "spans": [ { "bbox": [ 105, 659, 361, 673 ], "score": 1.0, "content": "time complexity that is linear in terms of the time series length", "type": "text" }, { "bbox": [ 361, 660, 369, 669 ], "score": 0.63, "content": "L", "type": "inline_equation" }, { "bbox": [ 369, 659, 505, 673 ], "score": 1.0, "content": ", yet their maximum length of the", "type": "text" } ], "index": 40 }, { "bbox": [ 106, 670, 505, 683 ], "spans": [ { "bbox": [ 106, 670, 205, 683 ], "score": 1.0, "content": "signal traversing path is", "type": "text" }, { "bbox": [ 205, 670, 228, 682 ], "score": 0.91, "content": "\\mathcal { O } ( L )", "type": "inline_equation" }, { "bbox": [ 229, 670, 505, 683 ], "score": 1.0, "content": ", thus rendering them difficult to learn dependencies between distant", "type": "text" } ], "index": 41 }, { "bbox": [ 106, 681, 504, 694 ], "spans": [ { "bbox": [ 106, 681, 482, 694 ], "score": 1.0, "content": "positions. On the other extreme, Transformer dramatically shortens the maximum path to be", "type": "text" }, { "bbox": [ 482, 681, 504, 693 ], "score": 0.82, "content": "\\mathcal { O } ( 1 )", "type": "inline_equation" } ], "index": 42 } ], "index": 40, "bbox_fs": [ 105, 638, 505, 694 ] } ] }, { "preproc_blocks": [ { "type": "image", "bbox": [ 106, 79, 505, 249 ], "blocks": [ { "type": "image_body", "bbox": [ 106, 79, 505, 249 ], "group_id": 0, "lines": [ { "bbox": [ 106, 79, 505, 249 ], "spans": [ { "bbox": [ 106, 79, 505, 249 ], "score": 0.975, "type": "image", "image_path": "cdf77eb19d8c5bb6faae2ef9f8506ac4c8d47ead1a5b2a18785a17c7a43fae51.jpg" } ] } ], "index": 1, "virtual_lines": [ { "bbox": [ 106, 79, 505, 135.66666666666666 ], "spans": [], "index": 0 }, { "bbox": [ 106, 135.66666666666666, 505, 192.33333333333331 ], "spans": [], "index": 1 }, { "bbox": [ 106, 192.33333333333331, 505, 248.99999999999997 ], "spans": [], "index": 2 } ] }, { "type": "image_caption", "bbox": [ 147, 261, 462, 273 ], "group_id": 0, "lines": [ { "bbox": [ 146, 259, 464, 276 ], "spans": [ { "bbox": [ 146, 259, 464, 276 ], "score": 1.0, "content": "Figure 1: Graphs of commonly used neural network models for sequence data.", "type": "text" } ], "index": 3 } ], "index": 3 } ], "index": 2.0 }, { "type": "text", "bbox": [ 106, 282, 506, 315 ], "lines": [ { "bbox": [ 105, 281, 505, 295 ], "spans": [ { "bbox": [ 105, 281, 505, 295 ], "score": 1.0, "content": "Table 1: Comparison of the complexity and the maximum signal traveling path for different models,", "type": "text" } ], "index": 4 }, { "bbox": [ 106, 293, 505, 306 ], "spans": [ { "bbox": [ 106, 293, 134, 306 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 134, 294, 144, 303 ], "score": 0.79, "content": "G", "type": "inline_equation" }, { "bbox": [ 144, 293, 375, 306 ], "score": 1.0, "content": "is the number of global tokens in ETC. In practice, the", "type": "text" }, { "bbox": [ 375, 293, 384, 303 ], "score": 0.8, "content": "G", "type": "inline_equation" }, { "bbox": [ 385, 293, 447, 306 ], "score": 1.0, "content": "increases with", "type": "text" }, { "bbox": [ 447, 294, 455, 303 ], "score": 0.77, "content": "L", "type": "inline_equation" }, { "bbox": [ 456, 293, 505, 306 ], "score": 1.0, "content": ", and so the", "type": "text" } ], "index": 5 }, { "bbox": [ 106, 304, 247, 317 ], "spans": [ { "bbox": [ 106, 304, 247, 317 ], "score": 1.0, "content": "complexity of ETC is super-linear.", "type": "text" } ], "index": 6 } ], "index": 5 }, { "type": "table", "bbox": [ 146, 320, 461, 427 ], "blocks": [ { "type": "table_body", "bbox": [ 146, 320, 461, 427 ], "group_id": 0, "lines": [ { "bbox": [ 146, 320, 461, 427 ], "spans": [ { "bbox": [ 146, 320, 461, 427 ], "score": 0.98, "html": "
MethodComplexityper layerMaximum path length
CNN (Munir et al., 2018)O(L)O(L)
RNN (Salinas et al.,2020)O(L)O(L)
Full-Attention (Vaswani et al.,2017)O(L2)0(1)
ETC (Ainslie et al., 2020)O(GL)0(1)
Longformer (Beltagy et al., 2020)O(L)O(L)
LogTrans (Li et al., 2019)O(L log L)O(log L)
PyraformerO(L)0(1)
", "type": "table", "image_path": "626481153aa20b388c15fa64bb9c13cbb3dc9d0c3b61a4d2103faa3becc1404b.jpg" } ] } ], "index": 8, "virtual_lines": [ { "bbox": [ 146, 320, 461, 355.6666666666667 ], "spans": [], "index": 7 }, { "bbox": [ 146, 355.6666666666667, 461, 391.33333333333337 ], "spans": [], "index": 8 }, { "bbox": [ 146, 391.33333333333337, 461, 427.00000000000006 ], "spans": [], "index": 9 } ] } ], "index": 8 }, { "type": "text", "bbox": [ 106, 437, 505, 493 ], "lines": [ { "bbox": [ 105, 437, 506, 450 ], "spans": [ { "bbox": [ 105, 437, 323, 450 ], "score": 1.0, "content": "at the sacrifice of increasing the time complexity to", "type": "text" }, { "bbox": [ 323, 437, 351, 450 ], "score": 0.92, "content": "\\mathcal { O } ( L ^ { 2 } )", "type": "inline_equation" }, { "bbox": [ 352, 437, 506, 450 ], "score": 1.0, "content": ". As a consequence, it cannot tackle", "type": "text" } ], "index": 10 }, { "bbox": [ 105, 448, 505, 461 ], "spans": [ { "bbox": [ 105, 448, 505, 461 ], "score": 1.0, "content": "very long sequences. To find a compromise between the model capacity and complexity, variants", "type": "text" } ], "index": 11 }, { "bbox": [ 106, 459, 505, 472 ], "spans": [ { "bbox": [ 106, 459, 505, 472 ], "score": 1.0, "content": "of Transformer are proposed, such as Longformer (Beltagy et al., 2020), Reformer (Kitaev et al.,", "type": "text" } ], "index": 12 }, { "bbox": [ 105, 469, 505, 483 ], "spans": [ { "bbox": [ 105, 469, 505, 483 ], "score": 1.0, "content": "2019), and Informer (Zhou et al., 2021). However, few of them can achieve a maximum path length", "type": "text" } ], "index": 13 }, { "bbox": [ 105, 479, 387, 496 ], "spans": [ { "bbox": [ 105, 479, 144, 496 ], "score": 1.0, "content": "less than", "type": "text" }, { "bbox": [ 144, 481, 167, 493 ], "score": 0.91, "content": "\\mathcal { O } ( L )", "type": "inline_equation" }, { "bbox": [ 168, 479, 387, 496 ], "score": 1.0, "content": "while greatly reducing the time and space complexity.", "type": "text" } ], "index": 14 } ], "index": 12 }, { "type": "text", "bbox": [ 106, 498, 505, 663 ], "lines": [ { "bbox": [ 105, 497, 505, 512 ], "spans": [ { "bbox": [ 105, 497, 505, 512 ], "score": 1.0, "content": "In this paper, we propose a novel pyramidal attention based Transformer (Pyraformer) to bridge", "type": "text" } ], "index": 15 }, { "bbox": [ 105, 508, 505, 523 ], "spans": [ { "bbox": [ 105, 508, 505, 523 ], "score": 1.0, "content": "the gap between capturing the long-range dependencies and achieving a low time and space com-", "type": "text" } ], "index": 16 }, { "bbox": [ 106, 520, 505, 533 ], "spans": [ { "bbox": [ 106, 520, 505, 533 ], "score": 1.0, "content": "plexity. Specifically, we develop the pyramidal attention mechanism by passing messages based on", "type": "text" } ], "index": 17 }, { "bbox": [ 105, 532, 505, 543 ], "spans": [ { "bbox": [ 105, 532, 505, 543 ], "score": 1.0, "content": "attention in the pyramidal graph as shown in Figure 1(d). The edges in this graph can be divided", "type": "text" } ], "index": 18 }, { "bbox": [ 105, 542, 505, 554 ], "spans": [ { "bbox": [ 105, 542, 505, 554 ], "score": 1.0, "content": "into two groups: the inter-scale and the intra-scale connections. The inter-scale connections build", "type": "text" } ], "index": 19 }, { "bbox": [ 105, 553, 505, 565 ], "spans": [ { "bbox": [ 105, 553, 505, 565 ], "score": 1.0, "content": "a multiresolution representation of the original sequence: nodes at the finest scale correspond to", "type": "text" } ], "index": 20 }, { "bbox": [ 105, 564, 506, 577 ], "spans": [ { "bbox": [ 105, 564, 506, 577 ], "score": 1.0, "content": "the time points in the original time series (e.g., hourly observations), while nodes in the coarser", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 575, 505, 588 ], "spans": [ { "bbox": [ 105, 575, 505, 588 ], "score": 1.0, "content": "scales represent features with lower resolutions (e.g., daily, weekly, and monthly patterns). Such", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 585, 505, 599 ], "spans": [ { "bbox": [ 105, 585, 505, 599 ], "score": 1.0, "content": "latent coarser-scale nodes are initially introduced via a coarser-scale construction module. On the", "type": "text" } ], "index": 23 }, { "bbox": [ 105, 597, 505, 610 ], "spans": [ { "bbox": [ 105, 597, 505, 610 ], "score": 1.0, "content": "other hand, the intra-scale edges capture the temporal dependencies at each resolution by connecting", "type": "text" } ], "index": 24 }, { "bbox": [ 105, 606, 505, 622 ], "spans": [ { "bbox": [ 105, 606, 505, 622 ], "score": 1.0, "content": "neighboring nodes together. As a result, this model provides a compact representation for long-range", "type": "text" } ], "index": 25 }, { "bbox": [ 106, 618, 505, 631 ], "spans": [ { "bbox": [ 106, 618, 505, 631 ], "score": 1.0, "content": "temporal dependencies among far-apart positions by capturing such behavior at coarser resolutions,", "type": "text" } ], "index": 26 }, { "bbox": [ 105, 629, 505, 643 ], "spans": [ { "bbox": [ 105, 629, 505, 643 ], "score": 1.0, "content": "leading to a smaller length of the signal traversing path. Moreover, modeling temporal dependencies", "type": "text" } ], "index": 27 }, { "bbox": [ 105, 640, 505, 654 ], "spans": [ { "bbox": [ 105, 640, 505, 654 ], "score": 1.0, "content": "of different ranges at different scales with sparse neighboring intra-scale connections significantly", "type": "text" } ], "index": 28 }, { "bbox": [ 105, 652, 399, 664 ], "spans": [ { "bbox": [ 105, 652, 399, 664 ], "score": 1.0, "content": "reduces the computational cost. In short, our key contributions comprise:", "type": "text" } ], "index": 29 } ], "index": 22 }, { "type": "text", "bbox": [ 133, 672, 505, 732 ], "lines": [ { "bbox": [ 132, 671, 506, 685 ], "spans": [ { "bbox": [ 132, 671, 506, 685 ], "score": 1.0, "content": "• We propose Pyraformer to simultaneously capture temporal dependencies of different", "type": "text" } ], "index": 30 }, { "bbox": [ 141, 684, 504, 696 ], "spans": [ { "bbox": [ 141, 684, 504, 696 ], "score": 1.0, "content": "ranges in a compact multi-resolution fashion. To distinguish Pyraformer from the state-", "type": "text" } ], "index": 31 }, { "bbox": [ 141, 695, 498, 707 ], "spans": [ { "bbox": [ 141, 695, 498, 707 ], "score": 1.0, "content": "of-the-art methods, we summarize all models from the perspective of graphs in Figure 1.", "type": "text" } ], "index": 32 }, { "bbox": [ 132, 708, 506, 723 ], "spans": [ { "bbox": [ 132, 708, 506, 723 ], "score": 1.0, "content": "• Theoretically, we prove that by choosing parameters appropriately, the maximum path", "type": "text" } ], "index": 33 }, { "bbox": [ 141, 721, 505, 732 ], "spans": [ { "bbox": [ 141, 721, 180, 732 ], "score": 1.0, "content": "length of", "type": "text" }, { "bbox": [ 181, 721, 202, 732 ], "score": 0.91, "content": "\\mathcal { O } ( 1 )", "type": "inline_equation" }, { "bbox": [ 202, 721, 352, 732 ], "score": 1.0, "content": "and the time and space complexity of", "type": "text" }, { "bbox": [ 352, 722, 375, 732 ], "score": 0.92, "content": "\\mathcal O ( L )", "type": "inline_equation" }, { "bbox": [ 376, 721, 505, 732 ], "score": 1.0, "content": "can be reached concurrently. To", "type": "text" } ], "index": 34 } ], "index": 32 } ], "page_idx": 1, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 107, 27, 293, 37 ], "lines": [ { "bbox": [ 106, 25, 294, 39 ], "spans": [ { "bbox": [ 106, 25, 294, 39 ], "score": 1.0, "content": "Published 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": [ 106, 79, 505, 249 ], "blocks": [ { "type": "image_body", "bbox": [ 106, 79, 505, 249 ], "group_id": 0, "lines": [ { "bbox": [ 106, 79, 505, 249 ], "spans": [ { "bbox": [ 106, 79, 505, 249 ], "score": 0.975, "type": "image", "image_path": "cdf77eb19d8c5bb6faae2ef9f8506ac4c8d47ead1a5b2a18785a17c7a43fae51.jpg" } ] } ], "index": 1, "virtual_lines": [ { "bbox": [ 106, 79, 505, 135.66666666666666 ], "spans": [], "index": 0 }, { "bbox": [ 106, 135.66666666666666, 505, 192.33333333333331 ], "spans": [], "index": 1 }, { "bbox": [ 106, 192.33333333333331, 505, 248.99999999999997 ], "spans": [], "index": 2 } ] }, { "type": "image_caption", "bbox": [ 147, 261, 462, 273 ], "group_id": 0, "lines": [ { "bbox": [ 146, 259, 464, 276 ], "spans": [ { "bbox": [ 146, 259, 464, 276 ], "score": 1.0, "content": "Figure 1: Graphs of commonly used neural network models for sequence data.", "type": "text" } ], "index": 3 } ], "index": 3 } ], "index": 2.0 }, { "type": "text", "bbox": [ 106, 282, 506, 315 ], "lines": [ { "bbox": [ 105, 281, 505, 295 ], "spans": [ { "bbox": [ 105, 281, 505, 295 ], "score": 1.0, "content": "Table 1: Comparison of the complexity and the maximum signal traveling path for different models,", "type": "text" } ], "index": 4 }, { "bbox": [ 106, 293, 505, 306 ], "spans": [ { "bbox": [ 106, 293, 134, 306 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 134, 294, 144, 303 ], "score": 0.79, "content": "G", "type": "inline_equation" }, { "bbox": [ 144, 293, 375, 306 ], "score": 1.0, "content": "is the number of global tokens in ETC. In practice, the", "type": "text" }, { "bbox": [ 375, 293, 384, 303 ], "score": 0.8, "content": "G", "type": "inline_equation" }, { "bbox": [ 385, 293, 447, 306 ], "score": 1.0, "content": "increases with", "type": "text" }, { "bbox": [ 447, 294, 455, 303 ], "score": 0.77, "content": "L", "type": "inline_equation" }, { "bbox": [ 456, 293, 505, 306 ], "score": 1.0, "content": ", and so the", "type": "text" } ], "index": 5 }, { "bbox": [ 106, 304, 247, 317 ], "spans": [ { "bbox": [ 106, 304, 247, 317 ], "score": 1.0, "content": "complexity of ETC is super-linear.", "type": "text" } ], "index": 6 } ], "index": 5, "bbox_fs": [ 105, 281, 505, 317 ] }, { "type": "table", "bbox": [ 146, 320, 461, 427 ], "blocks": [ { "type": "table_body", "bbox": [ 146, 320, 461, 427 ], "group_id": 0, "lines": [ { "bbox": [ 146, 320, 461, 427 ], "spans": [ { "bbox": [ 146, 320, 461, 427 ], "score": 0.98, "html": "
MethodComplexityper layerMaximum path length
CNN (Munir et al., 2018)O(L)O(L)
RNN (Salinas et al.,2020)O(L)O(L)
Full-Attention (Vaswani et al.,2017)O(L2)0(1)
ETC (Ainslie et al., 2020)O(GL)0(1)
Longformer (Beltagy et al., 2020)O(L)O(L)
LogTrans (Li et al., 2019)O(L log L)O(log L)
PyraformerO(L)0(1)
", "type": "table", "image_path": "626481153aa20b388c15fa64bb9c13cbb3dc9d0c3b61a4d2103faa3becc1404b.jpg" } ] } ], "index": 8, "virtual_lines": [ { "bbox": [ 146, 320, 461, 355.6666666666667 ], "spans": [], "index": 7 }, { "bbox": [ 146, 355.6666666666667, 461, 391.33333333333337 ], "spans": [], "index": 8 }, { "bbox": [ 146, 391.33333333333337, 461, 427.00000000000006 ], "spans": [], "index": 9 } ] } ], "index": 8 }, { "type": "text", "bbox": [ 106, 437, 505, 493 ], "lines": [ { "bbox": [ 105, 437, 506, 450 ], "spans": [ { "bbox": [ 105, 437, 323, 450 ], "score": 1.0, "content": "at the sacrifice of increasing the time complexity to", "type": "text" }, { "bbox": [ 323, 437, 351, 450 ], "score": 0.92, "content": "\\mathcal { O } ( L ^ { 2 } )", "type": "inline_equation" }, { "bbox": [ 352, 437, 506, 450 ], "score": 1.0, "content": ". As a consequence, it cannot tackle", "type": "text" } ], "index": 10 }, { "bbox": [ 105, 448, 505, 461 ], "spans": [ { "bbox": [ 105, 448, 505, 461 ], "score": 1.0, "content": "very long sequences. To find a compromise between the model capacity and complexity, variants", "type": "text" } ], "index": 11 }, { "bbox": [ 106, 459, 505, 472 ], "spans": [ { "bbox": [ 106, 459, 505, 472 ], "score": 1.0, "content": "of Transformer are proposed, such as Longformer (Beltagy et al., 2020), Reformer (Kitaev et al.,", "type": "text" } ], "index": 12 }, { "bbox": [ 105, 469, 505, 483 ], "spans": [ { "bbox": [ 105, 469, 505, 483 ], "score": 1.0, "content": "2019), and Informer (Zhou et al., 2021). However, few of them can achieve a maximum path length", "type": "text" } ], "index": 13 }, { "bbox": [ 105, 479, 387, 496 ], "spans": [ { "bbox": [ 105, 479, 144, 496 ], "score": 1.0, "content": "less than", "type": "text" }, { "bbox": [ 144, 481, 167, 493 ], "score": 0.91, "content": "\\mathcal { O } ( L )", "type": "inline_equation" }, { "bbox": [ 168, 479, 387, 496 ], "score": 1.0, "content": "while greatly reducing the time and space complexity.", "type": "text" } ], "index": 14 } ], "index": 12, "bbox_fs": [ 105, 437, 506, 496 ] }, { "type": "text", "bbox": [ 106, 498, 505, 663 ], "lines": [ { "bbox": [ 105, 497, 505, 512 ], "spans": [ { "bbox": [ 105, 497, 505, 512 ], "score": 1.0, "content": "In this paper, we propose a novel pyramidal attention based Transformer (Pyraformer) to bridge", "type": "text" } ], "index": 15 }, { "bbox": [ 105, 508, 505, 523 ], "spans": [ { "bbox": [ 105, 508, 505, 523 ], "score": 1.0, "content": "the gap between capturing the long-range dependencies and achieving a low time and space com-", "type": "text" } ], "index": 16 }, { "bbox": [ 106, 520, 505, 533 ], "spans": [ { "bbox": [ 106, 520, 505, 533 ], "score": 1.0, "content": "plexity. Specifically, we develop the pyramidal attention mechanism by passing messages based on", "type": "text" } ], "index": 17 }, { "bbox": [ 105, 532, 505, 543 ], "spans": [ { "bbox": [ 105, 532, 505, 543 ], "score": 1.0, "content": "attention in the pyramidal graph as shown in Figure 1(d). The edges in this graph can be divided", "type": "text" } ], "index": 18 }, { "bbox": [ 105, 542, 505, 554 ], "spans": [ { "bbox": [ 105, 542, 505, 554 ], "score": 1.0, "content": "into two groups: the inter-scale and the intra-scale connections. The inter-scale connections build", "type": "text" } ], "index": 19 }, { "bbox": [ 105, 553, 505, 565 ], "spans": [ { "bbox": [ 105, 553, 505, 565 ], "score": 1.0, "content": "a multiresolution representation of the original sequence: nodes at the finest scale correspond to", "type": "text" } ], "index": 20 }, { "bbox": [ 105, 564, 506, 577 ], "spans": [ { "bbox": [ 105, 564, 506, 577 ], "score": 1.0, "content": "the time points in the original time series (e.g., hourly observations), while nodes in the coarser", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 575, 505, 588 ], "spans": [ { "bbox": [ 105, 575, 505, 588 ], "score": 1.0, "content": "scales represent features with lower resolutions (e.g., daily, weekly, and monthly patterns). Such", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 585, 505, 599 ], "spans": [ { "bbox": [ 105, 585, 505, 599 ], "score": 1.0, "content": "latent coarser-scale nodes are initially introduced via a coarser-scale construction module. On the", "type": "text" } ], "index": 23 }, { "bbox": [ 105, 597, 505, 610 ], "spans": [ { "bbox": [ 105, 597, 505, 610 ], "score": 1.0, "content": "other hand, the intra-scale edges capture the temporal dependencies at each resolution by connecting", "type": "text" } ], "index": 24 }, { "bbox": [ 105, 606, 505, 622 ], "spans": [ { "bbox": [ 105, 606, 505, 622 ], "score": 1.0, "content": "neighboring nodes together. As a result, this model provides a compact representation for long-range", "type": "text" } ], "index": 25 }, { "bbox": [ 106, 618, 505, 631 ], "spans": [ { "bbox": [ 106, 618, 505, 631 ], "score": 1.0, "content": "temporal dependencies among far-apart positions by capturing such behavior at coarser resolutions,", "type": "text" } ], "index": 26 }, { "bbox": [ 105, 629, 505, 643 ], "spans": [ { "bbox": [ 105, 629, 505, 643 ], "score": 1.0, "content": "leading to a smaller length of the signal traversing path. Moreover, modeling temporal dependencies", "type": "text" } ], "index": 27 }, { "bbox": [ 105, 640, 505, 654 ], "spans": [ { "bbox": [ 105, 640, 505, 654 ], "score": 1.0, "content": "of different ranges at different scales with sparse neighboring intra-scale connections significantly", "type": "text" } ], "index": 28 }, { "bbox": [ 105, 652, 399, 664 ], "spans": [ { "bbox": [ 105, 652, 399, 664 ], "score": 1.0, "content": "reduces the computational cost. In short, our key contributions comprise:", "type": "text" } ], "index": 29 } ], "index": 22, "bbox_fs": [ 105, 497, 506, 664 ] }, { "type": "text", "bbox": [ 133, 672, 505, 732 ], "lines": [ { "bbox": [ 132, 671, 506, 685 ], "spans": [ { "bbox": [ 132, 671, 506, 685 ], "score": 1.0, "content": "• We propose Pyraformer to simultaneously capture temporal dependencies of different", "type": "text" } ], "index": 30 }, { "bbox": [ 141, 684, 504, 696 ], "spans": [ { "bbox": [ 141, 684, 504, 696 ], "score": 1.0, "content": "ranges in a compact multi-resolution fashion. To distinguish Pyraformer from the state-", "type": "text" } ], "index": 31 }, { "bbox": [ 141, 695, 498, 707 ], "spans": [ { "bbox": [ 141, 695, 498, 707 ], "score": 1.0, "content": "of-the-art methods, we summarize all models from the perspective of graphs in Figure 1.", "type": "text" } ], "index": 32 }, { "bbox": [ 132, 708, 506, 723 ], "spans": [ { "bbox": [ 132, 708, 506, 723 ], "score": 1.0, "content": "• Theoretically, we prove that by choosing parameters appropriately, the maximum path", "type": "text" } ], "index": 33 }, { "bbox": [ 141, 721, 505, 732 ], "spans": [ { "bbox": [ 141, 721, 180, 732 ], "score": 1.0, "content": "length of", "type": "text" }, { "bbox": [ 181, 721, 202, 732 ], "score": 0.91, "content": "\\mathcal { O } ( 1 )", "type": "inline_equation" }, { "bbox": [ 202, 721, 352, 732 ], "score": 1.0, "content": "and the time and space complexity of", "type": "text" }, { "bbox": [ 352, 722, 375, 732 ], "score": 0.92, "content": "\\mathcal O ( L )", "type": "inline_equation" }, { "bbox": [ 376, 721, 505, 732 ], "score": 1.0, "content": "can be reached concurrently. To", "type": "text" } ], "index": 34 }, { "bbox": [ 141, 82, 505, 95 ], "spans": [ { "bbox": [ 141, 82, 505, 95 ], "score": 1.0, "content": "highlight the appeal of the proposed model, we further compare different models in terms", "type": "text", "cross_page": true } ], "index": 0 }, { "bbox": [ 142, 93, 353, 105 ], "spans": [ { "bbox": [ 142, 93, 353, 105 ], "score": 1.0, "content": "of the maximum path and the complexity in Table 1.", "type": "text", "cross_page": true } ], "index": 1 } ], "index": 32, "bbox_fs": [ 132, 671, 506, 732 ] } ] }, { "preproc_blocks": [ { "type": "text", "bbox": [ 138, 82, 503, 105 ], "lines": [ { "bbox": [ 141, 82, 505, 95 ], "spans": [ { "bbox": [ 141, 82, 505, 95 ], "score": 1.0, "content": "highlight the appeal of the proposed model, we further compare different models in terms", "type": "text" } ], "index": 0 }, { "bbox": [ 142, 93, 353, 105 ], "spans": [ { "bbox": [ 142, 93, 353, 105 ], "score": 1.0, "content": "of the maximum path and the complexity in Table 1.", "type": "text" } ], "index": 1 } ], "index": 0.5 }, { "type": "text", "bbox": [ 134, 109, 504, 153 ], "lines": [ { "bbox": [ 133, 109, 505, 122 ], "spans": [ { "bbox": [ 133, 109, 505, 122 ], "score": 1.0, "content": "• Experimentally, we show that the proposed Pyraformer yields more accurate predictions", "type": "text" } ], "index": 2 }, { "bbox": [ 141, 120, 505, 132 ], "spans": [ { "bbox": [ 141, 120, 505, 132 ], "score": 1.0, "content": "than the original Transformer and its variants on various real-world datasets under the sce-", "type": "text" } ], "index": 3 }, { "bbox": [ 141, 131, 505, 143 ], "spans": [ { "bbox": [ 141, 131, 505, 143 ], "score": 1.0, "content": "nario of both single-step and long-range multi-step forecasting, but with lower time and", "type": "text" } ], "index": 4 }, { "bbox": [ 141, 143, 199, 154 ], "spans": [ { "bbox": [ 141, 143, 199, 154 ], "score": 1.0, "content": "memory cost.", "type": "text" } ], "index": 5 } ], "index": 3.5 }, { "type": "title", "bbox": [ 108, 169, 217, 183 ], "lines": [ { "bbox": [ 104, 168, 218, 185 ], "spans": [ { "bbox": [ 104, 168, 218, 185 ], "score": 1.0, "content": "2 RELATED WORKS", "type": "text" } ], "index": 6 } ], "index": 6 }, { "type": "title", "bbox": [ 108, 195, 252, 207 ], "lines": [ { "bbox": [ 106, 195, 253, 208 ], "spans": [ { "bbox": [ 106, 195, 253, 208 ], "score": 1.0, "content": "2.1 TIME SERIES FORECASTING", "type": "text" } ], "index": 7 } ], "index": 7 }, { "type": "text", "bbox": [ 107, 216, 505, 260 ], "lines": [ { "bbox": [ 106, 217, 505, 228 ], "spans": [ { "bbox": [ 106, 217, 505, 228 ], "score": 1.0, "content": "Time series forecasting methods can be roughly divided into statistical methods and neural network", "type": "text" } ], "index": 8 }, { "bbox": [ 105, 227, 505, 240 ], "spans": [ { "bbox": [ 105, 227, 505, 240 ], "score": 1.0, "content": "based methods. The first group involves ARIMA (Box & Jenkins, 1968) and Prophet (Taylor &", "type": "text" } ], "index": 9 }, { "bbox": [ 105, 237, 505, 251 ], "spans": [ { "bbox": [ 105, 237, 505, 251 ], "score": 1.0, "content": "Letham, 2018). However, both of them need to fit each time series separately, and their performance", "type": "text" } ], "index": 10 }, { "bbox": [ 105, 249, 294, 263 ], "spans": [ { "bbox": [ 105, 249, 294, 263 ], "score": 1.0, "content": "pales when it comes to long-range forecasting.", "type": "text" } ], "index": 11 } ], "index": 9.5 }, { "type": "text", "bbox": [ 107, 266, 505, 398 ], "lines": [ { "bbox": [ 105, 266, 505, 280 ], "spans": [ { "bbox": [ 105, 266, 505, 280 ], "score": 1.0, "content": "More recently, the development of deep learning has spawned a tremendous increase in neural net-", "type": "text" } ], "index": 12 }, { "bbox": [ 105, 276, 505, 290 ], "spans": [ { "bbox": [ 105, 276, 505, 290 ], "score": 1.0, "content": "work based time series forecasting methods, including CNN (Munir et al., 2018), RNN (Salinas", "type": "text" } ], "index": 13 }, { "bbox": [ 105, 287, 505, 300 ], "spans": [ { "bbox": [ 105, 287, 505, 300 ], "score": 1.0, "content": "et al., 2020) and Transformer (Li et al., 2019). As mentioned in the previous section, CNN and RNN", "type": "text" } ], "index": 14 }, { "bbox": [ 105, 298, 506, 313 ], "spans": [ { "bbox": [ 105, 298, 282, 313 ], "score": 1.0, "content": "enjoy a low time and space complexity (i.e.,", "type": "text" }, { "bbox": [ 283, 299, 308, 311 ], "score": 0.88, "content": "\\mathcal { O } ( L ) )", "type": "inline_equation" }, { "bbox": [ 308, 298, 389, 313 ], "score": 1.0, "content": "), but entail a path of", "type": "text" }, { "bbox": [ 389, 299, 413, 311 ], "score": 0.92, "content": "\\mathcal O ( L )", "type": "inline_equation" }, { "bbox": [ 413, 298, 506, 313 ], "score": 1.0, "content": "to describe long-range", "type": "text" } ], "index": 15 }, { "bbox": [ 105, 310, 505, 322 ], "spans": [ { "bbox": [ 105, 310, 505, 322 ], "score": 1.0, "content": "dependence. We refer the readers to Appendix A for a more detailed review on related RNN-based", "type": "text" } ], "index": 16 }, { "bbox": [ 105, 321, 505, 333 ], "spans": [ { "bbox": [ 105, 321, 505, 333 ], "score": 1.0, "content": "models. By contrast, Transformer (Vaswani et al., 2017) can effectively capture the long-range de-", "type": "text" } ], "index": 17 }, { "bbox": [ 105, 331, 505, 345 ], "spans": [ { "bbox": [ 105, 331, 204, 345 ], "score": 1.0, "content": "pendence with a path of", "type": "text" }, { "bbox": [ 205, 332, 226, 344 ], "score": 0.89, "content": "\\mathcal { O } ( 1 )", "type": "inline_equation" }, { "bbox": [ 227, 331, 437, 345 ], "score": 1.0, "content": "steps, whereas the complexity increases vastly from", "type": "text" }, { "bbox": [ 438, 332, 461, 344 ], "score": 0.91, "content": "\\mathcal O ( L )", "type": "inline_equation" }, { "bbox": [ 461, 331, 473, 345 ], "score": 1.0, "content": "to", "type": "text" }, { "bbox": [ 473, 331, 501, 344 ], "score": 0.91, "content": "\\check { \\mathcal { O } } ( L ^ { 2 } )", "type": "inline_equation" }, { "bbox": [ 501, 331, 505, 345 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 18 }, { "bbox": [ 106, 343, 505, 355 ], "spans": [ { "bbox": [ 106, 343, 505, 355 ], "score": 1.0, "content": "To alleviate this computational burden, LogTrans (Li et al., 2019) and Informer (Zhou et al., 2021)", "type": "text" } ], "index": 19 }, { "bbox": [ 105, 353, 506, 367 ], "spans": [ { "bbox": [ 105, 353, 506, 367 ], "score": 1.0, "content": "are proposed: the former constrains that each point in the sequence can only attend to the point that", "type": "text" } ], "index": 20 }, { "bbox": [ 105, 363, 506, 378 ], "spans": [ { "bbox": [ 105, 363, 116, 378 ], "score": 1.0, "content": "is", "type": "text" }, { "bbox": [ 116, 365, 128, 375 ], "score": 0.86, "content": "2 ^ { n }", "type": "inline_equation" }, { "bbox": [ 128, 363, 219, 378 ], "score": 1.0, "content": "steps before it, where", "type": "text" }, { "bbox": [ 219, 365, 273, 376 ], "score": 0.92, "content": "n = 1 , 2 , \\cdots", "type": "inline_equation" }, { "bbox": [ 273, 363, 506, 378 ], "score": 1.0, "content": ", and the latter utilizes the sparsity of the attention score,", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 375, 506, 389 ], "spans": [ { "bbox": [ 105, 375, 331, 389 ], "score": 1.0, "content": "resulting in substantial decrease in the complexity (i.e.,", "type": "text" }, { "bbox": [ 331, 375, 377, 388 ], "score": 0.93, "content": "\\mathcal { O } ( L \\log L )", "type": "inline_equation" }, { "bbox": [ 378, 375, 506, 389 ], "score": 1.0, "content": "at the expense of introducing a", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 386, 226, 399 ], "spans": [ { "bbox": [ 105, 386, 226, 399 ], "score": 1.0, "content": "longer maximum path length.", "type": "text" } ], "index": 23 } ], "index": 17.5 }, { "type": "title", "bbox": [ 108, 412, 237, 423 ], "lines": [ { "bbox": [ 106, 412, 239, 424 ], "spans": [ { "bbox": [ 106, 412, 239, 424 ], "score": 1.0, "content": "2.2 SPARSE TRANSFORMERS", "type": "text" } ], "index": 24 } ], "index": 24 }, { "type": "text", "bbox": [ 107, 433, 505, 609 ], "lines": [ { "bbox": [ 105, 433, 505, 446 ], "spans": [ { "bbox": [ 105, 433, 505, 446 ], "score": 1.0, "content": "In addition to the literature on time series forecasting, a plethora of methods have been proposed for", "type": "text" } ], "index": 25 }, { "bbox": [ 105, 444, 505, 456 ], "spans": [ { "bbox": [ 105, 444, 505, 456 ], "score": 1.0, "content": "enhancing the efficiency of Transformer in the field of natural language processing (NLP). Similar", "type": "text" } ], "index": 26 }, { "bbox": [ 105, 454, 506, 469 ], "spans": [ { "bbox": [ 105, 454, 506, 469 ], "score": 1.0, "content": "to CNN, Longformer (Beltagy et al., 2020) computes attention within a local sliding window or a", "type": "text" } ], "index": 27 }, { "bbox": [ 105, 466, 505, 478 ], "spans": [ { "bbox": [ 105, 466, 355, 478 ], "score": 1.0, "content": "dilated sliding window. Although the complexity is reduced to", "type": "text" }, { "bbox": [ 355, 466, 386, 478 ], "score": 0.92, "content": "\\mathcal { O } ( A L )", "type": "inline_equation" }, { "bbox": [ 387, 466, 416, 478 ], "score": 1.0, "content": ", where", "type": "text" }, { "bbox": [ 417, 466, 425, 476 ], "score": 0.73, "content": "A", "type": "inline_equation" }, { "bbox": [ 426, 466, 505, 478 ], "score": 1.0, "content": "is the local window", "type": "text" } ], "index": 28 }, { "bbox": [ 105, 477, 505, 489 ], "spans": [ { "bbox": [ 105, 477, 505, 489 ], "score": 1.0, "content": "size, the limited window size makes it difficult to exchange information globally. The consequent", "type": "text" } ], "index": 29 }, { "bbox": [ 105, 488, 504, 500 ], "spans": [ { "bbox": [ 105, 488, 205, 500 ], "score": 1.0, "content": "maximum path length is", "type": "text" }, { "bbox": [ 205, 488, 240, 500 ], "score": 0.93, "content": "\\mathcal { O } ( L / A )", "type": "inline_equation" }, { "bbox": [ 241, 488, 504, 500 ], "score": 1.0, "content": ". As an alternative, Reformer (Kitaev et al., 2019) exploits locality", "type": "text" } ], "index": 30 }, { "bbox": [ 105, 499, 505, 511 ], "spans": [ { "bbox": [ 105, 499, 505, 511 ], "score": 1.0, "content": "sensitive hashing (LSH) to divide the sequence into several buckets, and then performs attention", "type": "text" } ], "index": 31 }, { "bbox": [ 106, 509, 505, 523 ], "spans": [ { "bbox": [ 106, 509, 505, 523 ], "score": 1.0, "content": "within each bucket. It also employs reversible Transformer to further reduce memory consumption,", "type": "text" } ], "index": 32 }, { "bbox": [ 105, 521, 505, 533 ], "spans": [ { "bbox": [ 105, 521, 505, 533 ], "score": 1.0, "content": "and so an extremely long sequence can be processed. Its maximum path length is proportional", "type": "text" } ], "index": 33 }, { "bbox": [ 105, 532, 505, 544 ], "spans": [ { "bbox": [ 105, 532, 505, 544 ], "score": 1.0, "content": "to the number of buckets though, and worse still, a large bucket number is required to reduce the", "type": "text" } ], "index": 34 }, { "bbox": [ 105, 542, 506, 556 ], "spans": [ { "bbox": [ 105, 542, 506, 556 ], "score": 1.0, "content": "complexity. On the other hand, ETC (Ainslie et al., 2020) introduces an extra set of global tokens", "type": "text" } ], "index": 35 }, { "bbox": [ 105, 553, 505, 567 ], "spans": [ { "bbox": [ 105, 553, 336, 567 ], "score": 1.0, "content": "for the sake of global information exchange, leading to an", "type": "text" }, { "bbox": [ 337, 554, 368, 565 ], "score": 0.92, "content": "\\mathcal { O } ( G L )", "type": "inline_equation" }, { "bbox": [ 368, 553, 505, 567 ], "score": 1.0, "content": "time and space complexity and an", "type": "text" } ], "index": 36 }, { "bbox": [ 107, 564, 506, 578 ], "spans": [ { "bbox": [ 107, 564, 128, 576 ], "score": 0.87, "content": "\\mathcal { O } ( 1 )", "type": "inline_equation" }, { "bbox": [ 129, 564, 246, 578 ], "score": 1.0, "content": "maximum path length, where", "type": "text" }, { "bbox": [ 246, 565, 255, 575 ], "score": 0.83, "content": "G", "type": "inline_equation" }, { "bbox": [ 256, 564, 419, 578 ], "score": 1.0, "content": "is the number of global tokens. However,", "type": "text" }, { "bbox": [ 420, 565, 429, 575 ], "score": 0.79, "content": "G", "type": "inline_equation" }, { "bbox": [ 429, 564, 506, 578 ], "score": 1.0, "content": "typically increases", "type": "text" } ], "index": 37 }, { "bbox": [ 105, 575, 506, 588 ], "spans": [ { "bbox": [ 105, 575, 127, 588 ], "score": 1.0, "content": "with", "type": "text" }, { "bbox": [ 127, 576, 135, 586 ], "score": 0.78, "content": "L", "type": "inline_equation" }, { "bbox": [ 135, 575, 506, 588 ], "score": 1.0, "content": ", and the consequent complexity is still super-linear. Akin to ETC, the proposed Pyraformer", "type": "text" } ], "index": 38 }, { "bbox": [ 105, 586, 505, 599 ], "spans": [ { "bbox": [ 105, 586, 505, 599 ], "score": 1.0, "content": "also introduces global tokens, but in a multiscale manner, successfully reducing the complexity to", "type": "text" } ], "index": 39 }, { "bbox": [ 107, 597, 485, 610 ], "spans": [ { "bbox": [ 107, 597, 130, 609 ], "score": 0.9, "content": "\\mathcal { O } ( L )", "type": "inline_equation" }, { "bbox": [ 131, 597, 485, 610 ], "score": 1.0, "content": "without increasing the order of the maximum path length as in the original Transformer.", "type": "text" } ], "index": 40 } ], "index": 32.5 }, { "type": "title", "bbox": [ 109, 623, 271, 634 ], "lines": [ { "bbox": [ 106, 622, 273, 636 ], "spans": [ { "bbox": [ 106, 622, 273, 636 ], "score": 1.0, "content": "2.3 HIERARCHICAL TRANSFORMERS", "type": "text" } ], "index": 41 } ], "index": 41 }, { "type": "text", "bbox": [ 107, 643, 505, 732 ], "lines": [ { "bbox": [ 105, 643, 505, 656 ], "spans": [ { "bbox": [ 105, 643, 505, 656 ], "score": 1.0, "content": "Finally, we provide a brief review on methods that improve Transformer’s ability to capture the", "type": "text" } ], "index": 42 }, { "bbox": [ 105, 655, 505, 667 ], "spans": [ { "bbox": [ 105, 655, 505, 667 ], "score": 1.0, "content": "hierarchical structure of natural language, although they have never been used for time series fore-", "type": "text" } ], "index": 43 }, { "bbox": [ 105, 666, 506, 677 ], "spans": [ { "bbox": [ 105, 666, 506, 677 ], "score": 1.0, "content": "casting. HIBERT (Miculicich et al., 2018) first uses a Sent Encoder to extract the features of a", "type": "text" } ], "index": 44 }, { "bbox": [ 105, 677, 505, 689 ], "spans": [ { "bbox": [ 105, 677, 505, 689 ], "score": 1.0, "content": "sentence, and then forms the EOS tokens of sentences in the document as a new sequence and input", "type": "text" } ], "index": 45 }, { "bbox": [ 105, 688, 505, 700 ], "spans": [ { "bbox": [ 105, 688, 505, 700 ], "score": 1.0, "content": "it into the Doc Encoder. However, it is specialized for natural language and cannot be generalized", "type": "text" } ], "index": 46 }, { "bbox": [ 105, 699, 505, 711 ], "spans": [ { "bbox": [ 105, 699, 505, 711 ], "score": 1.0, "content": "to other sequence data. Multi-scale Transformer (Subramanian et al., 2020) learns the multi-scale", "type": "text" } ], "index": 47 }, { "bbox": [ 105, 710, 505, 722 ], "spans": [ { "bbox": [ 105, 710, 505, 722 ], "score": 1.0, "content": "representations of sequence data using both the top-down and bottom-up network structures. Such", "type": "text" } ], "index": 48 }, { "bbox": [ 106, 721, 505, 733 ], "spans": [ { "bbox": [ 106, 721, 505, 733 ], "score": 1.0, "content": "multi-scale representations help reduce the time and memory cost of the original Transformer, but", "type": "text" } ], "index": 49 } ], "index": 45.5 } ], "page_idx": 2, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 108, 27, 293, 37 ], "lines": [ { "bbox": [ 106, 26, 294, 38 ], "spans": [ { "bbox": [ 106, 26, 294, 38 ], "score": 1.0, "content": "Published as a conference paper at ICLR 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": [ 138, 82, 503, 105 ], "lines": [], "index": 0.5, "bbox_fs": [ 141, 82, 505, 105 ], "lines_deleted": true }, { "type": "text", "bbox": [ 134, 109, 504, 153 ], "lines": [ { "bbox": [ 133, 109, 505, 122 ], "spans": [ { "bbox": [ 133, 109, 505, 122 ], "score": 1.0, "content": "• Experimentally, we show that the proposed Pyraformer yields more accurate predictions", "type": "text" } ], "index": 2 }, { "bbox": [ 141, 120, 505, 132 ], "spans": [ { "bbox": [ 141, 120, 505, 132 ], "score": 1.0, "content": "than the original Transformer and its variants on various real-world datasets under the sce-", "type": "text" } ], "index": 3 }, { "bbox": [ 141, 131, 505, 143 ], "spans": [ { "bbox": [ 141, 131, 505, 143 ], "score": 1.0, "content": "nario of both single-step and long-range multi-step forecasting, but with lower time and", "type": "text" } ], "index": 4 }, { "bbox": [ 141, 143, 199, 154 ], "spans": [ { "bbox": [ 141, 143, 199, 154 ], "score": 1.0, "content": "memory cost.", "type": "text" } ], "index": 5 } ], "index": 3.5, "bbox_fs": [ 133, 109, 505, 154 ] }, { "type": "title", "bbox": [ 108, 169, 217, 183 ], "lines": [ { "bbox": [ 104, 168, 218, 185 ], "spans": [ { "bbox": [ 104, 168, 218, 185 ], "score": 1.0, "content": "2 RELATED WORKS", "type": "text" } ], "index": 6 } ], "index": 6 }, { "type": "title", "bbox": [ 108, 195, 252, 207 ], "lines": [ { "bbox": [ 106, 195, 253, 208 ], "spans": [ { "bbox": [ 106, 195, 253, 208 ], "score": 1.0, "content": "2.1 TIME SERIES FORECASTING", "type": "text" } ], "index": 7 } ], "index": 7 }, { "type": "text", "bbox": [ 107, 216, 505, 260 ], "lines": [ { "bbox": [ 106, 217, 505, 228 ], "spans": [ { "bbox": [ 106, 217, 505, 228 ], "score": 1.0, "content": "Time series forecasting methods can be roughly divided into statistical methods and neural network", "type": "text" } ], "index": 8 }, { "bbox": [ 105, 227, 505, 240 ], "spans": [ { "bbox": [ 105, 227, 505, 240 ], "score": 1.0, "content": "based methods. The first group involves ARIMA (Box & Jenkins, 1968) and Prophet (Taylor &", "type": "text" } ], "index": 9 }, { "bbox": [ 105, 237, 505, 251 ], "spans": [ { "bbox": [ 105, 237, 505, 251 ], "score": 1.0, "content": "Letham, 2018). However, both of them need to fit each time series separately, and their performance", "type": "text" } ], "index": 10 }, { "bbox": [ 105, 249, 294, 263 ], "spans": [ { "bbox": [ 105, 249, 294, 263 ], "score": 1.0, "content": "pales when it comes to long-range forecasting.", "type": "text" } ], "index": 11 } ], "index": 9.5, "bbox_fs": [ 105, 217, 505, 263 ] }, { "type": "text", "bbox": [ 107, 266, 505, 398 ], "lines": [ { "bbox": [ 105, 266, 505, 280 ], "spans": [ { "bbox": [ 105, 266, 505, 280 ], "score": 1.0, "content": "More recently, the development of deep learning has spawned a tremendous increase in neural net-", "type": "text" } ], "index": 12 }, { "bbox": [ 105, 276, 505, 290 ], "spans": [ { "bbox": [ 105, 276, 505, 290 ], "score": 1.0, "content": "work based time series forecasting methods, including CNN (Munir et al., 2018), RNN (Salinas", "type": "text" } ], "index": 13 }, { "bbox": [ 105, 287, 505, 300 ], "spans": [ { "bbox": [ 105, 287, 505, 300 ], "score": 1.0, "content": "et al., 2020) and Transformer (Li et al., 2019). As mentioned in the previous section, CNN and RNN", "type": "text" } ], "index": 14 }, { "bbox": [ 105, 298, 506, 313 ], "spans": [ { "bbox": [ 105, 298, 282, 313 ], "score": 1.0, "content": "enjoy a low time and space complexity (i.e.,", "type": "text" }, { "bbox": [ 283, 299, 308, 311 ], "score": 0.88, "content": "\\mathcal { O } ( L ) )", "type": "inline_equation" }, { "bbox": [ 308, 298, 389, 313 ], "score": 1.0, "content": "), but entail a path of", "type": "text" }, { "bbox": [ 389, 299, 413, 311 ], "score": 0.92, "content": "\\mathcal O ( L )", "type": "inline_equation" }, { "bbox": [ 413, 298, 506, 313 ], "score": 1.0, "content": "to describe long-range", "type": "text" } ], "index": 15 }, { "bbox": [ 105, 310, 505, 322 ], "spans": [ { "bbox": [ 105, 310, 505, 322 ], "score": 1.0, "content": "dependence. We refer the readers to Appendix A for a more detailed review on related RNN-based", "type": "text" } ], "index": 16 }, { "bbox": [ 105, 321, 505, 333 ], "spans": [ { "bbox": [ 105, 321, 505, 333 ], "score": 1.0, "content": "models. By contrast, Transformer (Vaswani et al., 2017) can effectively capture the long-range de-", "type": "text" } ], "index": 17 }, { "bbox": [ 105, 331, 505, 345 ], "spans": [ { "bbox": [ 105, 331, 204, 345 ], "score": 1.0, "content": "pendence with a path of", "type": "text" }, { "bbox": [ 205, 332, 226, 344 ], "score": 0.89, "content": "\\mathcal { O } ( 1 )", "type": "inline_equation" }, { "bbox": [ 227, 331, 437, 345 ], "score": 1.0, "content": "steps, whereas the complexity increases vastly from", "type": "text" }, { "bbox": [ 438, 332, 461, 344 ], "score": 0.91, "content": "\\mathcal O ( L )", "type": "inline_equation" }, { "bbox": [ 461, 331, 473, 345 ], "score": 1.0, "content": "to", "type": "text" }, { "bbox": [ 473, 331, 501, 344 ], "score": 0.91, "content": "\\check { \\mathcal { O } } ( L ^ { 2 } )", "type": "inline_equation" }, { "bbox": [ 501, 331, 505, 345 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 18 }, { "bbox": [ 106, 343, 505, 355 ], "spans": [ { "bbox": [ 106, 343, 505, 355 ], "score": 1.0, "content": "To alleviate this computational burden, LogTrans (Li et al., 2019) and Informer (Zhou et al., 2021)", "type": "text" } ], "index": 19 }, { "bbox": [ 105, 353, 506, 367 ], "spans": [ { "bbox": [ 105, 353, 506, 367 ], "score": 1.0, "content": "are proposed: the former constrains that each point in the sequence can only attend to the point that", "type": "text" } ], "index": 20 }, { "bbox": [ 105, 363, 506, 378 ], "spans": [ { "bbox": [ 105, 363, 116, 378 ], "score": 1.0, "content": "is", "type": "text" }, { "bbox": [ 116, 365, 128, 375 ], "score": 0.86, "content": "2 ^ { n }", "type": "inline_equation" }, { "bbox": [ 128, 363, 219, 378 ], "score": 1.0, "content": "steps before it, where", "type": "text" }, { "bbox": [ 219, 365, 273, 376 ], "score": 0.92, "content": "n = 1 , 2 , \\cdots", "type": "inline_equation" }, { "bbox": [ 273, 363, 506, 378 ], "score": 1.0, "content": ", and the latter utilizes the sparsity of the attention score,", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 375, 506, 389 ], "spans": [ { "bbox": [ 105, 375, 331, 389 ], "score": 1.0, "content": "resulting in substantial decrease in the complexity (i.e.,", "type": "text" }, { "bbox": [ 331, 375, 377, 388 ], "score": 0.93, "content": "\\mathcal { O } ( L \\log L )", "type": "inline_equation" }, { "bbox": [ 378, 375, 506, 389 ], "score": 1.0, "content": "at the expense of introducing a", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 386, 226, 399 ], "spans": [ { "bbox": [ 105, 386, 226, 399 ], "score": 1.0, "content": "longer maximum path length.", "type": "text" } ], "index": 23 } ], "index": 17.5, "bbox_fs": [ 105, 266, 506, 399 ] }, { "type": "title", "bbox": [ 108, 412, 237, 423 ], "lines": [ { "bbox": [ 106, 412, 239, 424 ], "spans": [ { "bbox": [ 106, 412, 239, 424 ], "score": 1.0, "content": "2.2 SPARSE TRANSFORMERS", "type": "text" } ], "index": 24 } ], "index": 24 }, { "type": "text", "bbox": [ 107, 433, 505, 609 ], "lines": [ { "bbox": [ 105, 433, 505, 446 ], "spans": [ { "bbox": [ 105, 433, 505, 446 ], "score": 1.0, "content": "In addition to the literature on time series forecasting, a plethora of methods have been proposed for", "type": "text" } ], "index": 25 }, { "bbox": [ 105, 444, 505, 456 ], "spans": [ { "bbox": [ 105, 444, 505, 456 ], "score": 1.0, "content": "enhancing the efficiency of Transformer in the field of natural language processing (NLP). Similar", "type": "text" } ], "index": 26 }, { "bbox": [ 105, 454, 506, 469 ], "spans": [ { "bbox": [ 105, 454, 506, 469 ], "score": 1.0, "content": "to CNN, Longformer (Beltagy et al., 2020) computes attention within a local sliding window or a", "type": "text" } ], "index": 27 }, { "bbox": [ 105, 466, 505, 478 ], "spans": [ { "bbox": [ 105, 466, 355, 478 ], "score": 1.0, "content": "dilated sliding window. Although the complexity is reduced to", "type": "text" }, { "bbox": [ 355, 466, 386, 478 ], "score": 0.92, "content": "\\mathcal { O } ( A L )", "type": "inline_equation" }, { "bbox": [ 387, 466, 416, 478 ], "score": 1.0, "content": ", where", "type": "text" }, { "bbox": [ 417, 466, 425, 476 ], "score": 0.73, "content": "A", "type": "inline_equation" }, { "bbox": [ 426, 466, 505, 478 ], "score": 1.0, "content": "is the local window", "type": "text" } ], "index": 28 }, { "bbox": [ 105, 477, 505, 489 ], "spans": [ { "bbox": [ 105, 477, 505, 489 ], "score": 1.0, "content": "size, the limited window size makes it difficult to exchange information globally. The consequent", "type": "text" } ], "index": 29 }, { "bbox": [ 105, 488, 504, 500 ], "spans": [ { "bbox": [ 105, 488, 205, 500 ], "score": 1.0, "content": "maximum path length is", "type": "text" }, { "bbox": [ 205, 488, 240, 500 ], "score": 0.93, "content": "\\mathcal { O } ( L / A )", "type": "inline_equation" }, { "bbox": [ 241, 488, 504, 500 ], "score": 1.0, "content": ". As an alternative, Reformer (Kitaev et al., 2019) exploits locality", "type": "text" } ], "index": 30 }, { "bbox": [ 105, 499, 505, 511 ], "spans": [ { "bbox": [ 105, 499, 505, 511 ], "score": 1.0, "content": "sensitive hashing (LSH) to divide the sequence into several buckets, and then performs attention", "type": "text" } ], "index": 31 }, { "bbox": [ 106, 509, 505, 523 ], "spans": [ { "bbox": [ 106, 509, 505, 523 ], "score": 1.0, "content": "within each bucket. It also employs reversible Transformer to further reduce memory consumption,", "type": "text" } ], "index": 32 }, { "bbox": [ 105, 521, 505, 533 ], "spans": [ { "bbox": [ 105, 521, 505, 533 ], "score": 1.0, "content": "and so an extremely long sequence can be processed. Its maximum path length is proportional", "type": "text" } ], "index": 33 }, { "bbox": [ 105, 532, 505, 544 ], "spans": [ { "bbox": [ 105, 532, 505, 544 ], "score": 1.0, "content": "to the number of buckets though, and worse still, a large bucket number is required to reduce the", "type": "text" } ], "index": 34 }, { "bbox": [ 105, 542, 506, 556 ], "spans": [ { "bbox": [ 105, 542, 506, 556 ], "score": 1.0, "content": "complexity. On the other hand, ETC (Ainslie et al., 2020) introduces an extra set of global tokens", "type": "text" } ], "index": 35 }, { "bbox": [ 105, 553, 505, 567 ], "spans": [ { "bbox": [ 105, 553, 336, 567 ], "score": 1.0, "content": "for the sake of global information exchange, leading to an", "type": "text" }, { "bbox": [ 337, 554, 368, 565 ], "score": 0.92, "content": "\\mathcal { O } ( G L )", "type": "inline_equation" }, { "bbox": [ 368, 553, 505, 567 ], "score": 1.0, "content": "time and space complexity and an", "type": "text" } ], "index": 36 }, { "bbox": [ 107, 564, 506, 578 ], "spans": [ { "bbox": [ 107, 564, 128, 576 ], "score": 0.87, "content": "\\mathcal { O } ( 1 )", "type": "inline_equation" }, { "bbox": [ 129, 564, 246, 578 ], "score": 1.0, "content": "maximum path length, where", "type": "text" }, { "bbox": [ 246, 565, 255, 575 ], "score": 0.83, "content": "G", "type": "inline_equation" }, { "bbox": [ 256, 564, 419, 578 ], "score": 1.0, "content": "is the number of global tokens. However,", "type": "text" }, { "bbox": [ 420, 565, 429, 575 ], "score": 0.79, "content": "G", "type": "inline_equation" }, { "bbox": [ 429, 564, 506, 578 ], "score": 1.0, "content": "typically increases", "type": "text" } ], "index": 37 }, { "bbox": [ 105, 575, 506, 588 ], "spans": [ { "bbox": [ 105, 575, 127, 588 ], "score": 1.0, "content": "with", "type": "text" }, { "bbox": [ 127, 576, 135, 586 ], "score": 0.78, "content": "L", "type": "inline_equation" }, { "bbox": [ 135, 575, 506, 588 ], "score": 1.0, "content": ", and the consequent complexity is still super-linear. Akin to ETC, the proposed Pyraformer", "type": "text" } ], "index": 38 }, { "bbox": [ 105, 586, 505, 599 ], "spans": [ { "bbox": [ 105, 586, 505, 599 ], "score": 1.0, "content": "also introduces global tokens, but in a multiscale manner, successfully reducing the complexity to", "type": "text" } ], "index": 39 }, { "bbox": [ 107, 597, 485, 610 ], "spans": [ { "bbox": [ 107, 597, 130, 609 ], "score": 0.9, "content": "\\mathcal { O } ( L )", "type": "inline_equation" }, { "bbox": [ 131, 597, 485, 610 ], "score": 1.0, "content": "without increasing the order of the maximum path length as in the original Transformer.", "type": "text" } ], "index": 40 } ], "index": 32.5, "bbox_fs": [ 105, 433, 506, 610 ] }, { "type": "title", "bbox": [ 109, 623, 271, 634 ], "lines": [ { "bbox": [ 106, 622, 273, 636 ], "spans": [ { "bbox": [ 106, 622, 273, 636 ], "score": 1.0, "content": "2.3 HIERARCHICAL TRANSFORMERS", "type": "text" } ], "index": 41 } ], "index": 41 }, { "type": "text", "bbox": [ 107, 643, 505, 732 ], "lines": [ { "bbox": [ 105, 643, 505, 656 ], "spans": [ { "bbox": [ 105, 643, 505, 656 ], "score": 1.0, "content": "Finally, we provide a brief review on methods that improve Transformer’s ability to capture the", "type": "text" } ], "index": 42 }, { "bbox": [ 105, 655, 505, 667 ], "spans": [ { "bbox": [ 105, 655, 505, 667 ], "score": 1.0, "content": "hierarchical structure of natural language, although they have never been used for time series fore-", "type": "text" } ], "index": 43 }, { "bbox": [ 105, 666, 506, 677 ], "spans": [ { "bbox": [ 105, 666, 506, 677 ], "score": 1.0, "content": "casting. HIBERT (Miculicich et al., 2018) first uses a Sent Encoder to extract the features of a", "type": "text" } ], "index": 44 }, { "bbox": [ 105, 677, 505, 689 ], "spans": [ { "bbox": [ 105, 677, 505, 689 ], "score": 1.0, "content": "sentence, and then forms the EOS tokens of sentences in the document as a new sequence and input", "type": "text" } ], "index": 45 }, { "bbox": [ 105, 688, 505, 700 ], "spans": [ { "bbox": [ 105, 688, 505, 700 ], "score": 1.0, "content": "it into the Doc Encoder. However, it is specialized for natural language and cannot be generalized", "type": "text" } ], "index": 46 }, { "bbox": [ 105, 699, 505, 711 ], "spans": [ { "bbox": [ 105, 699, 505, 711 ], "score": 1.0, "content": "to other sequence data. Multi-scale Transformer (Subramanian et al., 2020) learns the multi-scale", "type": "text" } ], "index": 47 }, { "bbox": [ 105, 710, 505, 722 ], "spans": [ { "bbox": [ 105, 710, 505, 722 ], "score": 1.0, "content": "representations of sequence data using both the top-down and bottom-up network structures. Such", "type": "text" } ], "index": 48 }, { "bbox": [ 106, 721, 505, 733 ], "spans": [ { "bbox": [ 106, 721, 505, 733 ], "score": 1.0, "content": "multi-scale representations help reduce the time and memory cost of the original Transformer, but", "type": "text" } ], "index": 49 }, { "bbox": [ 105, 250, 505, 262 ], "spans": [ { "bbox": [ 105, 250, 505, 262 ], "score": 1.0, "content": "it still suffers from the pitfall of the quadratic complexity. Alternatively, BP-Transformer (Ye et al.,", "type": "text", "cross_page": true } ], "index": 6 }, { "bbox": [ 105, 260, 505, 273 ], "spans": [ { "bbox": [ 105, 260, 505, 273 ], "score": 1.0, "content": "2019) recursively partitions the entire input sequence into two until a partition only contains a single", "type": "text", "cross_page": true } ], "index": 7 }, { "bbox": [ 105, 272, 505, 284 ], "spans": [ { "bbox": [ 105, 272, 505, 284 ], "score": 1.0, "content": "token. The partitioned sequences then form a binary tree. In the attention layer, each upper-scale", "type": "text", "cross_page": true } ], "index": 8 }, { "bbox": [ 106, 283, 504, 294 ], "spans": [ { "bbox": [ 106, 283, 495, 294 ], "score": 1.0, "content": "node can attend to its own children, while the nodes at the bottom scale can attend to the adjacent", "type": "text", "cross_page": true }, { "bbox": [ 495, 283, 504, 292 ], "score": 0.59, "content": "A", "type": "inline_equation", "cross_page": true } ], "index": 9 }, { "bbox": [ 105, 294, 505, 305 ], "spans": [ { "bbox": [ 105, 294, 505, 305 ], "score": 1.0, "content": "nodes at the same scale and all coarser-scale nodes. Note that BP-Transformer initializes the nodes", "type": "text", "cross_page": true } ], "index": 10 }, { "bbox": [ 105, 305, 504, 316 ], "spans": [ { "bbox": [ 105, 305, 504, 316 ], "score": 1.0, "content": "at coarser scale with zeros, whereas Pyraformer introduces the coarser-scale nodes using a construc-", "type": "text", "cross_page": true } ], "index": 11 }, { "bbox": [ 105, 315, 505, 328 ], "spans": [ { "bbox": [ 105, 315, 505, 328 ], "score": 1.0, "content": "tion module in a more flexible manner. Moreover, BP-Transformer is associated with a denser graph", "type": "text", "cross_page": true } ], "index": 12 }, { "bbox": [ 105, 325, 394, 339 ], "spans": [ { "bbox": [ 105, 325, 343, 339 ], "score": 1.0, "content": "than Pyraformer, thus giving rise to a higher complexity of", "type": "text", "cross_page": true }, { "bbox": [ 343, 326, 389, 338 ], "score": 0.93, "content": "\\mathcal { O } ( L \\log L )", "type": "inline_equation", "cross_page": true }, { "bbox": [ 390, 325, 394, 339 ], "score": 1.0, "content": ".", "type": "text", "cross_page": true } ], "index": 13 } ], "index": 45.5, "bbox_fs": [ 105, 643, 506, 733 ] } ] }, { "preproc_blocks": [ { "type": "image", "bbox": [ 106, 79, 505, 190 ], "blocks": [ { "type": "image_body", "bbox": [ 106, 79, 505, 190 ], "group_id": 0, "lines": [ { "bbox": [ 106, 79, 505, 190 ], "spans": [ { "bbox": [ 106, 79, 505, 190 ], "score": 0.971, "type": "image", "image_path": "56cd8894c208a5e364b1dc0bb7b398b69c875aae166d907b12cd98ff3ac99fe5.jpg" } ] } ], "index": 1, "virtual_lines": [ { "bbox": [ 106, 79, 505, 116.0 ], "spans": [], "index": 0 }, { "bbox": [ 106, 116.0, 505, 153.0 ], "spans": [], "index": 1 }, { "bbox": [ 106, 153.0, 505, 190.0 ], "spans": [], "index": 2 } ] }, { "type": "image_caption", "bbox": [ 107, 200, 503, 234 ], "group_id": 0, "lines": [ { "bbox": [ 106, 200, 505, 213 ], "spans": [ { "bbox": [ 106, 200, 505, 213 ], "score": 1.0, "content": "Figure 2: The architecture of Pyraformer: The CSCM summarizes the embedded sequence at differ-", "type": "text" } ], "index": 3 }, { "bbox": [ 105, 210, 505, 224 ], "spans": [ { "bbox": [ 105, 210, 505, 224 ], "score": 1.0, "content": "ent scales and builds a multi-resolution tree structure. Then the PAM is used to exchange information", "type": "text" } ], "index": 4 }, { "bbox": [ 105, 222, 213, 235 ], "spans": [ { "bbox": [ 105, 222, 213, 235 ], "score": 1.0, "content": "between nodes efficiently.", "type": "text" } ], "index": 5 } ], "index": 4 } ], "index": 2.5 }, { "type": "text", "bbox": [ 107, 249, 505, 338 ], "lines": [ { "bbox": [ 105, 250, 505, 262 ], "spans": [ { "bbox": [ 105, 250, 505, 262 ], "score": 1.0, "content": "it still suffers from the pitfall of the quadratic complexity. Alternatively, BP-Transformer (Ye et al.,", "type": "text" } ], "index": 6 }, { "bbox": [ 105, 260, 505, 273 ], "spans": [ { "bbox": [ 105, 260, 505, 273 ], "score": 1.0, "content": "2019) recursively partitions the entire input sequence into two until a partition only contains a single", "type": "text" } ], "index": 7 }, { "bbox": [ 105, 272, 505, 284 ], "spans": [ { "bbox": [ 105, 272, 505, 284 ], "score": 1.0, "content": "token. The partitioned sequences then form a binary tree. In the attention layer, each upper-scale", "type": "text" } ], "index": 8 }, { "bbox": [ 106, 283, 504, 294 ], "spans": [ { "bbox": [ 106, 283, 495, 294 ], "score": 1.0, "content": "node can attend to its own children, while the nodes at the bottom scale can attend to the adjacent", "type": "text" }, { "bbox": [ 495, 283, 504, 292 ], "score": 0.59, "content": "A", "type": "inline_equation" } ], "index": 9 }, { "bbox": [ 105, 294, 505, 305 ], "spans": [ { "bbox": [ 105, 294, 505, 305 ], "score": 1.0, "content": "nodes at the same scale and all coarser-scale nodes. Note that BP-Transformer initializes the nodes", "type": "text" } ], "index": 10 }, { "bbox": [ 105, 305, 504, 316 ], "spans": [ { "bbox": [ 105, 305, 504, 316 ], "score": 1.0, "content": "at coarser scale with zeros, whereas Pyraformer introduces the coarser-scale nodes using a construc-", "type": "text" } ], "index": 11 }, { "bbox": [ 105, 315, 505, 328 ], "spans": [ { "bbox": [ 105, 315, 505, 328 ], "score": 1.0, "content": "tion module in a more flexible manner. Moreover, BP-Transformer is associated with a denser graph", "type": "text" } ], "index": 12 }, { "bbox": [ 105, 325, 394, 339 ], "spans": [ { "bbox": [ 105, 325, 343, 339 ], "score": 1.0, "content": "than Pyraformer, thus giving rise to a higher complexity of", "type": "text" }, { "bbox": [ 343, 326, 389, 338 ], "score": 0.93, "content": "\\mathcal { O } ( L \\log L )", "type": "inline_equation" }, { "bbox": [ 390, 325, 394, 339 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 13 } ], "index": 9.5 }, { "type": "title", "bbox": [ 108, 353, 172, 366 ], "lines": [ { "bbox": [ 104, 351, 174, 370 ], "spans": [ { "bbox": [ 104, 351, 174, 370 ], "score": 1.0, "content": "3 METHOD", "type": "text" } ], "index": 14 } ], "index": 14 }, { "type": "text", "bbox": [ 106, 379, 505, 510 ], "lines": [ { "bbox": [ 105, 378, 504, 393 ], "spans": [ { "bbox": [ 105, 378, 427, 393 ], "score": 1.0, "content": "The time series forecasting problem can be formulated as predicting the future", "type": "text" }, { "bbox": [ 427, 380, 439, 389 ], "score": 0.77, "content": "M", "type": "inline_equation" }, { "bbox": [ 439, 378, 463, 393 ], "score": 1.0, "content": "steps", "type": "text" }, { "bbox": [ 464, 381, 504, 391 ], "score": 0.87, "content": "z _ { t + 1 : t + M }", "type": "inline_equation" } ], "index": 15 }, { "bbox": [ 104, 389, 504, 405 ], "spans": [ { "bbox": [ 104, 389, 186, 405 ], "score": 1.0, "content": "given the previous", "type": "text" }, { "bbox": [ 186, 390, 195, 400 ], "score": 0.77, "content": "L", "type": "inline_equation" }, { "bbox": [ 195, 389, 287, 405 ], "score": 1.0, "content": "steps of observations", "type": "text" }, { "bbox": [ 287, 392, 325, 402 ], "score": 0.89, "content": "z _ { t - L + 1 : t }", "type": "inline_equation" }, { "bbox": [ 325, 389, 451, 405 ], "score": 1.0, "content": "and the associated covariates", "type": "text" }, { "bbox": [ 451, 393, 504, 402 ], "score": 0.87, "content": "\\pmb { x } _ { t - L + 1 : t + M }", "type": "inline_equation" } ], "index": 16 }, { "bbox": [ 105, 401, 506, 414 ], "spans": [ { "bbox": [ 105, 401, 506, 414 ], "score": 1.0, "content": "(e.g., hour-of-the-day). To move forward to this goal, we propose Pyraformer in this paper, whose", "type": "text" } ], "index": 17 }, { "bbox": [ 105, 412, 505, 424 ], "spans": [ { "bbox": [ 105, 412, 505, 424 ], "score": 1.0, "content": "overall architecture is summarized in Figure 2. As shown in the figure, we first embed the observed", "type": "text" } ], "index": 18 }, { "bbox": [ 106, 423, 505, 435 ], "spans": [ { "bbox": [ 106, 423, 505, 435 ], "score": 1.0, "content": "data, the covariates, and the positions separately and then add them together, in the same vein with", "type": "text" } ], "index": 19 }, { "bbox": [ 105, 433, 506, 446 ], "spans": [ { "bbox": [ 105, 433, 385, 446 ], "score": 1.0, "content": "Informer (Zhou et al., 2021). Next, we construct a multi-resolution", "type": "text" }, { "bbox": [ 385, 434, 394, 444 ], "score": 0.83, "content": "C", "type": "inline_equation" }, { "bbox": [ 394, 433, 506, 446 ], "score": 1.0, "content": "-ary tree using the coarser-", "type": "text" } ], "index": 20 }, { "bbox": [ 106, 445, 506, 457 ], "spans": [ { "bbox": [ 106, 445, 506, 457 ], "score": 1.0, "content": "scale construction module (CSCM), where nodes at a coarser scale summarize the information of", "type": "text" } ], "index": 21 }, { "bbox": [ 107, 456, 505, 468 ], "spans": [ { "bbox": [ 107, 456, 115, 466 ], "score": 0.77, "content": "C", "type": "inline_equation" }, { "bbox": [ 116, 456, 505, 468 ], "score": 1.0, "content": "nodes at the corresponding finer scale. To further capture the temporal dependencies of different", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 466, 506, 480 ], "spans": [ { "bbox": [ 105, 466, 506, 480 ], "score": 1.0, "content": "ranges, we introduce the pyramidal attention module (PAM) by passing messages using the attention", "type": "text" } ], "index": 23 }, { "bbox": [ 105, 477, 506, 491 ], "spans": [ { "bbox": [ 105, 477, 506, 491 ], "score": 1.0, "content": "mechanism in the pyramidal graph. Finally, depending on the downstream task, we employ different", "type": "text" } ], "index": 24 }, { "bbox": [ 105, 489, 505, 501 ], "spans": [ { "bbox": [ 105, 489, 505, 501 ], "score": 1.0, "content": "network structures to output the final predictions. In the sequel, we elaborate on each part of the", "type": "text" } ], "index": 25 }, { "bbox": [ 105, 500, 483, 512 ], "spans": [ { "bbox": [ 105, 500, 483, 512 ], "score": 1.0, "content": "proposed model. 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We", "type": "text" } ], "index": 32 }, { "bbox": [ 106, 600, 504, 612 ], "spans": [ { "bbox": [ 106, 600, 504, 612 ], "score": 1.0, "content": "can decompose the pyramidal graph into two parts: the inter-scale and the intra-scale connections.", "type": "text" } ], "index": 33 }, { "bbox": [ 105, 610, 505, 624 ], "spans": [ { "bbox": [ 105, 610, 249, 624 ], "score": 1.0, "content": "The inter-scale connections form a", "type": "text" }, { "bbox": [ 249, 612, 258, 621 ], "score": 0.82, "content": "C", "type": "inline_equation" }, { "bbox": [ 259, 610, 399, 624 ], "score": 1.0, "content": "-ary tree, in which each parent has", "type": "text" }, { "bbox": [ 400, 611, 409, 621 ], "score": 0.78, "content": "C", "type": "inline_equation" }, { "bbox": [ 409, 610, 505, 624 ], "score": 1.0, "content": "children. 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We", "type": "text" } ], "index": 32 }, { "bbox": [ 106, 600, 504, 612 ], "spans": [ { "bbox": [ 106, 600, 504, 612 ], "score": 1.0, "content": "can decompose the pyramidal graph into two parts: the inter-scale and the intra-scale connections.", "type": "text" } ], "index": 33 }, { "bbox": [ 105, 610, 505, 624 ], "spans": [ { "bbox": [ 105, 610, 249, 624 ], "score": 1.0, "content": "The inter-scale connections form a", "type": "text" }, { "bbox": [ 249, 612, 258, 621 ], "score": 0.82, "content": "C", "type": "inline_equation" }, { "bbox": [ 259, 610, 399, 624 ], "score": 1.0, "content": "-ary tree, in which each parent has", "type": "text" }, { "bbox": [ 400, 611, 409, 621 ], "score": 0.78, "content": "C", "type": "inline_equation" }, { "bbox": [ 409, 610, 505, 624 ], "score": 1.0, "content": "children. 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Let", "type": "text" }, { "bbox": [ 462, 122, 474, 132 ], "score": 0.77, "content": "\\boldsymbol { X }", "type": "inline_equation" }, { "bbox": [ 474, 121, 493, 134 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 493, 121, 504, 132 ], "score": 0.75, "content": "\\mathbf { Y }", "type": "inline_equation" } ], "index": 3 }, { "bbox": [ 106, 132, 505, 145 ], "spans": [ { "bbox": [ 106, 132, 505, 145 ], "score": 1.0, "content": "denote the input and output of a single attention head respectively. Note that multiple heads can", "type": "text" } ], "index": 4 }, { "bbox": [ 105, 143, 505, 156 ], "spans": [ { "bbox": [ 105, 143, 406, 156 ], "score": 1.0, "content": "be introduced to describe the temporal pattern from different perspectives.", "type": "text" }, { "bbox": [ 407, 144, 418, 153 ], "score": 0.78, "content": "\\boldsymbol { X }", "type": "inline_equation" }, { "bbox": [ 418, 143, 505, 156 ], "score": 1.0, "content": "is first linearly trans-", "type": "text" } ], "index": 5 }, { "bbox": [ 105, 154, 505, 168 ], "spans": [ { "bbox": [ 105, 154, 325, 168 ], "score": 1.0, "content": "formed into three distinct matrices, namely, the query", "type": "text" }, { "bbox": [ 326, 154, 377, 167 ], "score": 0.92, "content": "Q = X W _ { Q }", "type": "inline_equation" }, { "bbox": [ 378, 154, 414, 168 ], "score": 1.0, "content": ", the key", "type": "text" }, { "bbox": [ 414, 154, 469, 165 ], "score": 0.92, "content": "\\pmb { K } = \\pmb { X } \\pmb { W } _ { K }", "type": "inline_equation" }, { "bbox": [ 469, 154, 505, 168 ], "score": 1.0, "content": ", and the", "type": "text" } ], "index": 6 }, { "bbox": [ 104, 162, 506, 181 ], "spans": [ { "bbox": [ 104, 162, 131, 181 ], "score": 1.0, "content": "value", "type": "text" }, { "bbox": [ 131, 166, 182, 177 ], "score": 0.91, "content": "\\pmb { V } = \\pmb { X } \\pmb { W } _ { V }", "type": "inline_equation" }, { "bbox": [ 182, 162, 212, 181 ], "score": 1.0, "content": ", where", "type": "text" }, { "bbox": [ 213, 166, 231, 178 ], "score": 0.83, "content": "W _ { Q }", "type": "inline_equation" }, { "bbox": [ 231, 162, 235, 181 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 235, 166, 255, 178 ], "score": 0.76, "content": "W _ { K }", "type": "inline_equation" }, { "bbox": [ 255, 162, 259, 181 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 259, 165, 322, 177 ], "score": 0.91, "content": "W _ { V } \\in \\mathbb { R } ^ { L \\times D _ { K } }", "type": "inline_equation" }, { "bbox": [ 322, 162, 357, 181 ], "score": 1.0, "content": ". For the", "type": "text" }, { "bbox": [ 358, 168, 362, 176 ], "score": 0.78, "content": "i", "type": "inline_equation" }, { "bbox": [ 362, 162, 392, 181 ], "score": 1.0, "content": "-th row", "type": "text" }, { "bbox": [ 392, 168, 402, 178 ], "score": 0.84, "content": "\\pmb q _ { i }", "type": "inline_equation" }, { "bbox": [ 402, 162, 413, 181 ], "score": 1.0, "content": "in", "type": "text" }, { "bbox": [ 414, 167, 423, 178 ], "score": 0.85, "content": "Q", "type": "inline_equation" }, { "bbox": [ 424, 162, 506, 181 ], "score": 1.0, "content": ", it can attend to any", "type": "text" } ], "index": 7 }, { "bbox": [ 105, 177, 459, 190 ], "spans": [ { "bbox": [ 105, 177, 183, 190 ], "score": 1.0, "content": "rows (i.e., keys) in", "type": "text" }, { "bbox": [ 183, 178, 194, 187 ], "score": 0.77, "content": "\\kappa", "type": "inline_equation" }, { "bbox": [ 195, 177, 363, 190 ], "score": 1.0, "content": ". In other words, the corresponding output", "type": "text" }, { "bbox": [ 364, 180, 374, 189 ], "score": 0.86, "content": "\\mathbf { \\nabla } _ { \\mathbf { \\psi } _ { 3 } } \\psi _ { i }", "type": "inline_equation" }, { "bbox": [ 374, 177, 459, 190 ], "score": 1.0, "content": "can be expressed as:", "type": "text" } ], "index": 8 } ], "index": 5.5 }, { "type": "interline_equation", "bbox": [ 232, 190, 378, 224 ], "lines": [ { "bbox": [ 232, 190, 378, 224 ], "spans": [ { "bbox": [ 232, 190, 378, 224 ], "score": 0.95, "content": "{ \\pmb y } _ { i } = \\sum _ { \\ell = 1 } ^ { L } \\frac { \\mathrm { e x p } ( { \\pmb q } _ { i } { \\pmb k } _ { \\ell } ^ { T } / \\sqrt { D _ { K } } ) { \\pmb v } _ { \\ell } } { \\sum _ { \\ell = 1 } ^ { L } \\mathrm { e x p } ( { \\pmb q } _ { i } { \\pmb k } _ { \\ell } ^ { T } / \\sqrt { D _ { K } } ) } ,", "type": "interline_equation", "image_path": "a4ab598b546f6b7ebb82b39ce9ff06398783e40dac4c9cea4945eff5dbeb48ba.jpg" } ] } ], "index": 9.5, "virtual_lines": [ { "bbox": [ 232, 190, 378, 207.0 ], "spans": [], "index": 9 }, { "bbox": [ 232, 207.0, 378, 224.0 ], "spans": [], "index": 10 } ] }, { "type": "text", "bbox": [ 106, 226, 505, 282 ], "lines": [ { "bbox": [ 106, 226, 506, 239 ], "spans": [ { "bbox": [ 106, 226, 133, 239 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 134, 226, 148, 239 ], "score": 0.9, "content": "k _ { \\ell } ^ { T }", "type": "inline_equation" }, { "bbox": [ 148, 226, 269, 239 ], "score": 1.0, "content": "denotes the transpose of row", "type": "text" }, { "bbox": [ 270, 227, 276, 236 ], "score": 0.79, "content": "\\ell", "type": "inline_equation" }, { "bbox": [ 276, 226, 288, 239 ], "score": 1.0, "content": "in", "type": "text" }, { "bbox": [ 288, 227, 299, 236 ], "score": 0.81, "content": "\\kappa", "type": "inline_equation" }, { "bbox": [ 300, 226, 506, 239 ], "score": 1.0, "content": ". We emphasize that the number of query-key dot", "type": "text" } ], "index": 11 }, { "bbox": [ 106, 237, 506, 251 ], "spans": [ { "bbox": [ 106, 237, 506, 251 ], "score": 1.0, "content": "products (Q-K pairs) that need to be calculated and stored dictates the time and space complexity of", "type": "text" } ], "index": 12 }, { "bbox": [ 106, 249, 505, 261 ], "spans": [ { "bbox": [ 106, 249, 505, 261 ], "score": 1.0, "content": "the attention mechanism. Viewed another way, this number is proportional to the number of edges", "type": "text" } ], "index": 13 }, { "bbox": [ 106, 260, 505, 272 ], "spans": [ { "bbox": [ 106, 260, 505, 272 ], "score": 1.0, "content": "in the graph (see Figure 1(a)). Since all Q-K pairs are computed and stored in the full attention", "type": "text" } ], "index": 14 }, { "bbox": [ 105, 270, 373, 283 ], "spans": [ { "bbox": [ 105, 270, 341, 283 ], "score": 1.0, "content": "mechanism (1), the resulting time and space complexity is", "type": "text" }, { "bbox": [ 342, 270, 369, 282 ], "score": 0.94, "content": "\\mathcal { O } ( L ^ { 2 } )", "type": "inline_equation" }, { "bbox": [ 369, 270, 373, 283 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 15 } ], "index": 13 }, { "type": "text", "bbox": [ 106, 286, 506, 380 ], "lines": [ { "bbox": [ 105, 286, 506, 300 ], "spans": [ { "bbox": [ 105, 286, 506, 300 ], "score": 1.0, "content": "As opposed to the above full attention mechanism, every node only pays attention to a limited set of", "type": "text" } ], "index": 16 }, { "bbox": [ 103, 297, 504, 317 ], "spans": [ { "bbox": [ 103, 297, 486, 317 ], "score": 1.0, "content": "keys in the PAM, corresponding to the pyramidal graph in Figure 1d. Concretely, suppose that", "type": "text" }, { "bbox": [ 486, 298, 504, 313 ], "score": 0.9, "content": "n _ { \\ell } ^ { ( s ) }", "type": "inline_equation" } ], "index": 17 }, { "bbox": [ 106, 312, 505, 324 ], "spans": [ { "bbox": [ 106, 312, 155, 324 ], "score": 1.0, "content": "denotes the", "type": "text" }, { "bbox": [ 155, 312, 161, 322 ], "score": 0.53, "content": "\\ell \\cdot", "type": "inline_equation" }, { "bbox": [ 162, 312, 229, 324 ], "score": 1.0, "content": "-th node at scale", "type": "text" }, { "bbox": [ 230, 314, 236, 322 ], "score": 0.7, "content": "s", "type": "inline_equation" }, { "bbox": [ 236, 312, 268, 324 ], "score": 1.0, "content": ", where", "type": "text" }, { "bbox": [ 268, 312, 325, 324 ], "score": 0.92, "content": "s = 1 , \\cdots , S", "type": "inline_equation" }, { "bbox": [ 326, 312, 505, 324 ], "score": 1.0, "content": "represents the bottom scale to the top scale", "type": "text" } ], "index": 18 }, { "bbox": [ 103, 323, 504, 341 ], "spans": [ { "bbox": [ 103, 323, 474, 341 ], "score": 1.0, "content": "sequentially. In general, each node in the graph can attend to a set of neighboring nodes", "type": "text" }, { "bbox": [ 474, 323, 493, 338 ], "score": 0.91, "content": "\\mathbb { N } _ { \\ell } ^ { ( s ) }", "type": "inline_equation" }, { "bbox": [ 493, 323, 504, 341 ], "score": 1.0, "content": "at", "type": "text" } ], "index": 19 }, { "bbox": [ 105, 338, 507, 353 ], "spans": [ { "bbox": [ 105, 338, 209, 353 ], "score": 1.0, "content": "three scales: the adjacent", "type": "text" }, { "bbox": [ 210, 340, 218, 350 ], "score": 0.79, "content": "A", "type": "inline_equation" }, { "bbox": [ 219, 338, 464, 353 ], "score": 1.0, "content": "nodes at the same scale including the node itself (denoted as", "type": "text" }, { "bbox": [ 464, 338, 483, 352 ], "score": 0.9, "content": "\\mathbb { A } _ { \\ell } ^ { ( s ) }", "type": "inline_equation" }, { "bbox": [ 483, 338, 507, 353 ], "score": 1.0, "content": "), the", "type": "text" } ], "index": 20 }, { "bbox": [ 107, 352, 506, 368 ], "spans": [ { "bbox": [ 107, 354, 116, 365 ], "score": 0.78, "content": "C", "type": "inline_equation" }, { "bbox": [ 116, 352, 201, 368 ], "score": 1.0, "content": "children it has in the", "type": "text" }, { "bbox": [ 201, 354, 210, 364 ], "score": 0.83, "content": "C", "type": "inline_equation" }, { "bbox": [ 211, 352, 294, 368 ], "score": 1.0, "content": "-ary tree (denoted as", "type": "text" }, { "bbox": [ 294, 352, 313, 367 ], "score": 0.88, "content": "\\mathbb { C } _ { \\ell } ^ { ( s ) } .", "type": "inline_equation" }, { "bbox": [ 313, 352, 423, 368 ], "score": 1.0, "content": "), and the parent of it in the", "type": "text" }, { "bbox": [ 424, 354, 433, 364 ], "score": 0.82, "content": "C", "type": "inline_equation" }, { "bbox": [ 433, 352, 506, 368 ], "score": 1.0, "content": "-ary tree (denoted", "type": "text" } ], "index": 21 }, { "bbox": [ 106, 363, 163, 384 ], "spans": [ { "bbox": [ 106, 366, 127, 381 ], "score": 0.87, "content": "\\mathbb { P } _ { \\ell } ^ { ( s ) } )", "type": "inline_equation" }, { "bbox": [ 127, 363, 163, 384 ], "score": 1.0, "content": ", that is,", "type": "text" } ], "index": 22 } ], "index": 19 }, { "type": "interline_equation", "bbox": [ 174, 381, 437, 447 ], "lines": [ { "bbox": [ 174, 381, 437, 447 ], "spans": [ { "bbox": [ 174, 381, 437, 447 ], "score": 0.95, "content": "\\left\\{ \\begin{array} { l l l l l l l l l l l l l } { \\mathbb { N } _ { \\ell } ^ { ( s ) } } & { = } & { \\mathbb { A } _ { \\ell } ^ { ( s ) } \\cup \\mathbb { C } _ { \\ell } ^ { ( s ) } \\cup \\mathbb { P } _ { l } ^ { ( s ) } } & & & & & \\\\ { \\mathbb { A } _ { \\ell } ^ { ( s ) } } & { = } & { \\{ n _ { j } ^ { ( s ) } : | j - \\ell | \\leq \\frac { A - 1 } { 2 } , 1 \\leq j \\leq \\frac { L } { C ^ { s - 1 } } \\} } & & & & & \\\\ { \\mathbb { C } _ { \\ell } ^ { ( s ) } } & { = } & { \\{ n _ { j } ^ { ( s - 1 ) } : ( \\ell - 1 ) C < j \\leq \\ell C \\} } & { \\mathrm { i f } s \\geq 2 \\mathrm { e l s e } \\emptyset } & & & \\\\ { \\mathbb { P } _ { \\ell } ^ { ( s ) } } & { = } & { \\{ n _ { j } ^ { ( s + 1 ) } : j = \\lceil \\frac { \\ell } { C } \\rceil \\} } & { \\mathrm { i f } s \\leq S - 1 \\mathrm { e l s e } \\emptyset } & & & & \\end{array} \\right. .", "type": "interline_equation", "image_path": "c1c6b42e71ce1a071839b917e297e956e538cb8315d07f3cdb9c25ccecde1a58.jpg" } ] } ], "index": 24, "virtual_lines": [ { "bbox": [ 174, 381, 437, 403.0 ], "spans": [], "index": 23 }, { "bbox": [ 174, 403.0, 437, 425.0 ], "spans": [], "index": 24 }, { "bbox": [ 174, 425.0, 437, 447.0 ], "spans": [], "index": 25 } ] }, { "type": "text", "bbox": [ 106, 449, 351, 463 ], "lines": [ { "bbox": [ 100, 446, 352, 470 ], "spans": [ { "bbox": [ 100, 446, 248, 470 ], "score": 1.0, "content": "It follows that the attention at node", "type": "text" }, { "bbox": [ 248, 448, 266, 464 ], "score": 0.92, "content": "n _ { \\ell } ^ { ( s ) }", "type": "inline_equation" }, { "bbox": [ 266, 446, 352, 470 ], "score": 1.0, "content": "can be simplified as:√", "type": "text" } ], "index": 26 } ], "index": 26 }, { "type": "interline_equation", "bbox": [ 225, 464, 385, 500 ], "lines": [ { "bbox": [ 225, 464, 385, 500 ], "spans": [ { "bbox": [ 225, 464, 385, 500 ], "score": 0.94, "content": "\\pmb { y } _ { i } = \\sum _ { \\ell \\in \\mathbb { N } _ { \\ell } ^ { ( s ) } } \\frac { \\exp ( \\pmb { q } _ { i } \\pmb { k } _ { \\ell } ^ { T } / \\sqrt { d _ { K } } ) \\pmb { v } _ { \\ell } } { \\sum _ { \\ell \\in \\mathbb { N } _ { l } ^ { ( s ) } } \\exp ( \\pmb { q } _ { i } \\pmb { k } _ { \\ell } ^ { T } / \\sqrt { d _ { K } } ) } ,", "type": "interline_equation", "image_path": "65219bb1ee3e17b2e49fcc877e25c0f00a110a80ddb131298165d032413727b3.jpg" } ] } ], "index": 27.5, "virtual_lines": [ { "bbox": [ 225, 464, 385, 482.0 ], "spans": [], "index": 27 }, { "bbox": [ 225, 482.0, 385, 500.0 ], "spans": [], "index": 28 } ] }, { "type": "text", "bbox": [ 107, 507, 505, 541 ], "lines": [ { "bbox": [ 106, 507, 505, 519 ], "spans": [ { "bbox": [ 106, 507, 316, 519 ], "score": 1.0, "content": "We further denote the number of attention layers as", "type": "text" }, { "bbox": [ 316, 508, 326, 517 ], "score": 0.83, "content": "N", "type": "inline_equation" }, { "bbox": [ 327, 507, 505, 519 ], "score": 1.0, "content": ". Without loss of generality, we assume that", "type": "text" } ], "index": 29 }, { "bbox": [ 107, 516, 506, 532 ], "spans": [ { "bbox": [ 107, 519, 115, 528 ], "score": 0.79, "content": "L", "type": "inline_equation" }, { "bbox": [ 115, 516, 175, 532 ], "score": 1.0, "content": "is divisible by", "type": "text" }, { "bbox": [ 175, 518, 200, 529 ], "score": 0.91, "content": "C ^ { S - 1 }", "type": "inline_equation" }, { "bbox": [ 200, 516, 424, 532 ], "score": 1.0, "content": ". We can then have the following lemma (cf. Appendix", "type": "text" }, { "bbox": [ 425, 519, 433, 528 ], "score": 0.44, "content": "\\mathbf { B }", "type": "inline_equation" }, { "bbox": [ 433, 516, 506, 532 ], "score": 1.0, "content": "for the proof and", "type": "text" } ], "index": 30 }, { "bbox": [ 105, 528, 278, 542 ], "spans": [ { "bbox": [ 105, 528, 278, 542 ], "score": 1.0, "content": "Table 4 for the meanings of the notations).", "type": "text" } ], "index": 31 } ], "index": 30 }, { "type": "text", "bbox": [ 106, 542, 502, 565 ], "lines": [ { "bbox": [ 105, 541, 505, 556 ], "spans": [ { "bbox": [ 105, 541, 182, 556 ], "score": 1.0, "content": "Lemma 1. Given", "type": "text" }, { "bbox": [ 182, 542, 232, 554 ], "score": 0.68, "content": "A , C , L , N", "type": "inline_equation" }, { "bbox": [ 233, 541, 255, 556 ], "score": 1.0, "content": ", and", "type": "text" }, { "bbox": [ 255, 543, 263, 552 ], "score": 0.77, "content": "S", "type": "inline_equation" }, { "bbox": [ 264, 541, 392, 556 ], "score": 1.0, "content": "that satisfy Equation (4), after", "type": "text" }, { "bbox": [ 392, 543, 402, 552 ], "score": 0.78, "content": "N", "type": "inline_equation" }, { "bbox": [ 403, 541, 505, 556 ], "score": 1.0, "content": "stacked attention layers,", "type": "text" } ], "index": 32 }, { "bbox": [ 106, 553, 356, 566 ], "spans": [ { "bbox": [ 106, 553, 356, 566 ], "score": 1.0, "content": "nodes at the coarsest scale can obtain a global receptive field.", "type": "text" } ], "index": 33 } ], "index": 32.5 }, { "type": "interline_equation", "bbox": [ 253, 566, 358, 591 ], "lines": [ { "bbox": [ 253, 566, 358, 591 ], "spans": [ { "bbox": [ 253, 566, 358, 591 ], "score": 0.94, "content": "\\frac { L } { C ^ { S - 1 } } - 1 \\leq \\frac { ( A - 1 ) N } { 2 } .", "type": "interline_equation", "image_path": "eafe1980cccd7d86fbe5781b52170d343501370e89e2c447787bffb17b333aa3.jpg" } ] } ], "index": 34, "virtual_lines": [ { "bbox": [ 253, 566, 358, 591 ], "spans": [], "index": 34 } ] }, { "type": "text", "bbox": [ 108, 596, 505, 630 ], "lines": [ { "bbox": [ 106, 597, 505, 609 ], "spans": [ { "bbox": [ 106, 597, 269, 609 ], "score": 1.0, "content": "In addition, when the number of scales", "type": "text" }, { "bbox": [ 270, 597, 278, 607 ], "score": 0.83, "content": "S", "type": "inline_equation" }, { "bbox": [ 278, 597, 505, 609 ], "score": 1.0, "content": "is fixed, the following two propositions summarize the", "type": "text" } ], "index": 35 }, { "bbox": [ 106, 608, 505, 620 ], "spans": [ { "bbox": [ 106, 608, 505, 620 ], "score": 1.0, "content": "time and space complexity and the order of the maximum path length for the proposed pyramidal", "type": "text" } ], "index": 36 }, { "bbox": [ 105, 619, 408, 631 ], "spans": [ { "bbox": [ 105, 619, 357, 631 ], "score": 1.0, "content": "attention mechanism. We refer the readers to Appendix C and", "type": "text" }, { "bbox": [ 357, 619, 367, 629 ], "score": 0.55, "content": "\\mathrm { D }", "type": "inline_equation" }, { "bbox": [ 367, 619, 408, 631 ], "score": 1.0, "content": "for proof.", "type": "text" } ], "index": 37 } ], "index": 36 }, { "type": "text", "bbox": [ 105, 632, 505, 654 ], "lines": [ { "bbox": [ 105, 630, 505, 645 ], "spans": [ { "bbox": [ 105, 630, 459, 645 ], "score": 1.0, "content": "Proposition 1. The time and space complexity for the pyramidal attention mechanism is", "type": "text" }, { "bbox": [ 459, 632, 489, 644 ], "score": 0.84, "content": "\\mathcal { O } ( A L )", "type": "inline_equation" }, { "bbox": [ 490, 630, 505, 645 ], "score": 1.0, "content": "for", "type": "text" } ], "index": 38 }, { "bbox": [ 105, 643, 376, 655 ], "spans": [ { "bbox": [ 105, 643, 131, 655 ], "score": 1.0, "content": "given", "type": "text" }, { "bbox": [ 131, 644, 140, 653 ], "score": 0.47, "content": "A", "type": "inline_equation" }, { "bbox": [ 140, 643, 158, 655 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 158, 644, 166, 653 ], "score": 0.69, "content": "L", "type": "inline_equation" }, { "bbox": [ 166, 643, 231, 655 ], "score": 1.0, "content": "and amounts to", "type": "text" }, { "bbox": [ 232, 644, 255, 655 ], "score": 0.91, "content": "\\mathcal O ( L )", "type": "inline_equation" }, { "bbox": [ 255, 643, 280, 655 ], "score": 1.0, "content": "when", "type": "text" }, { "bbox": [ 280, 644, 289, 653 ], "score": 0.73, "content": "A", "type": "inline_equation" }, { "bbox": [ 289, 643, 365, 655 ], "score": 1.0, "content": "is a constant w.r.t.", "type": "text" }, { "bbox": [ 365, 644, 373, 653 ], "score": 0.44, "content": "L", "type": "inline_equation" }, { "bbox": [ 373, 643, 376, 655 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 39 } ], "index": 38.5 }, { "type": "text", "bbox": [ 106, 656, 505, 701 ], "lines": [ { "bbox": [ 105, 655, 506, 669 ], "spans": [ { "bbox": [ 105, 655, 506, 669 ], "score": 1.0, "content": "Proposition 2. Let the signal traversing path between two nodes in a graph denote the shortest path", "type": "text" } ], "index": 40 }, { "bbox": [ 106, 668, 505, 679 ], "spans": [ { "bbox": [ 106, 668, 505, 679 ], "score": 1.0, "content": "connecting them. Then the maximum length of signal traversing path between two arbitrary nodes", "type": "text" } ], "index": 41 }, { "bbox": [ 104, 676, 506, 691 ], "spans": [ { "bbox": [ 104, 676, 210, 691 ], "score": 1.0, "content": "in the pyramidal graph is", "type": "text" }, { "bbox": [ 210, 679, 293, 690 ], "score": 0.91, "content": "\\mathcal { O } ( S + L / C ^ { S - 1 } / A )", "type": "inline_equation" }, { "bbox": [ 294, 676, 333, 691 ], "score": 1.0, "content": "for given", "type": "text" }, { "bbox": [ 333, 680, 341, 687 ], "score": 0.7, "content": "A", "type": "inline_equation" }, { "bbox": [ 341, 676, 345, 691 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 345, 680, 353, 687 ], "score": 0.73, "content": "C", "type": "inline_equation" }, { "bbox": [ 354, 676, 358, 691 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 358, 680, 365, 688 ], "score": 0.65, "content": "L", "type": "inline_equation" }, { "bbox": [ 366, 676, 387, 691 ], "score": 1.0, "content": ", and", "type": "text" }, { "bbox": [ 388, 680, 394, 688 ], "score": 0.77, "content": "S", "type": "inline_equation" }, { "bbox": [ 395, 676, 453, 691 ], "score": 1.0, "content": ". Suppose that", "type": "text" }, { "bbox": [ 453, 678, 462, 688 ], "score": 0.72, "content": "A", "type": "inline_equation" }, { "bbox": [ 462, 676, 480, 691 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 481, 678, 488, 688 ], "score": 0.79, "content": "S", "type": "inline_equation" }, { "bbox": [ 489, 676, 506, 691 ], "score": 1.0, "content": "are", "type": "text" } ], "index": 42 }, { "bbox": [ 104, 689, 500, 702 ], "spans": [ { "bbox": [ 104, 689, 145, 702 ], "score": 1.0, "content": "fixed and", "type": "text" }, { "bbox": [ 146, 691, 154, 699 ], "score": 0.78, "content": "C", "type": "inline_equation" }, { "bbox": [ 154, 689, 358, 702 ], "score": 1.0, "content": "satisfies Equation (5), the maximum path length is", "type": "text" }, { "bbox": [ 358, 690, 380, 701 ], "score": 0.9, "content": "\\mathcal { O } ( 1 )", "type": "inline_equation" }, { "bbox": [ 380, 689, 488, 702 ], "score": 1.0, "content": "for time series with length", "type": "text" }, { "bbox": [ 488, 690, 496, 699 ], "score": 0.73, "content": "L", "type": "inline_equation" }, { "bbox": [ 496, 689, 500, 702 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 43 } ], "index": 41.5 }, { "type": "interline_equation", "bbox": [ 231, 702, 384, 735 ], "lines": [ { "bbox": [ 231, 702, 384, 735 ], "spans": [ { "bbox": [ 231, 702, 384, 735 ], "score": 0.96, "content": "\\sqrt [ s - 1 ] { L } \\geq C \\geq \\sqrt [ s - 1 ] { \\frac { L } { ( A - 1 ) N / 2 + 1 } } .", "type": "interline_equation", "image_path": "ba62517cd5c850e97d37eb87878ea0cbbdcaa9bcd04097cae7bcc4d114e94a4b.jpg" } ] } ], "index": 44.5, "virtual_lines": [ { "bbox": [ 231, 702, 384, 718.5 ], "spans": [], "index": 44 }, { "bbox": [ 231, 718.5, 384, 735.0 ], "spans": [], "index": 45 } ] } ], "page_idx": 4, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 108, 27, 293, 37 ], "lines": [ { "bbox": [ 106, 26, 294, 38 ], "spans": [ { "bbox": [ 106, 26, 294, 38 ], "score": 1.0, "content": "Published as a conference paper at ICLR 2022", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 302, 751, 309, 760 ], "lines": [ { "bbox": [ 301, 750, 310, 762 ], "spans": [ { "bbox": [ 301, 750, 310, 762 ], "score": 1.0, "content": "5", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "text", "bbox": [ 107, 82, 504, 116 ], "lines": [], "index": 1, "bbox_fs": [ 105, 81, 506, 117 ], "lines_deleted": true }, { "type": "text", "bbox": [ 106, 120, 505, 189 ], "lines": [ { "bbox": [ 106, 121, 504, 134 ], "spans": [ { "bbox": [ 106, 121, 462, 134 ], "score": 1.0, "content": "Before delving into the PAM, we first introduce the original attention mechanism. Let", "type": "text" }, { "bbox": [ 462, 122, 474, 132 ], "score": 0.77, "content": "\\boldsymbol { X }", "type": "inline_equation" }, { "bbox": [ 474, 121, 493, 134 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 493, 121, 504, 132 ], "score": 0.75, "content": "\\mathbf { Y }", "type": "inline_equation" } ], "index": 3 }, { "bbox": [ 106, 132, 505, 145 ], "spans": [ { "bbox": [ 106, 132, 505, 145 ], "score": 1.0, "content": "denote the input and output of a single attention head respectively. Note that multiple heads can", "type": "text" } ], "index": 4 }, { "bbox": [ 105, 143, 505, 156 ], "spans": [ { "bbox": [ 105, 143, 406, 156 ], "score": 1.0, "content": "be introduced to describe the temporal pattern from different perspectives.", "type": "text" }, { "bbox": [ 407, 144, 418, 153 ], "score": 0.78, "content": "\\boldsymbol { X }", "type": "inline_equation" }, { "bbox": [ 418, 143, 505, 156 ], "score": 1.0, "content": "is first linearly trans-", "type": "text" } ], "index": 5 }, { "bbox": [ 105, 154, 505, 168 ], "spans": [ { "bbox": [ 105, 154, 325, 168 ], "score": 1.0, "content": "formed into three distinct matrices, namely, the query", "type": "text" }, { "bbox": [ 326, 154, 377, 167 ], "score": 0.92, "content": "Q = X W _ { Q }", "type": "inline_equation" }, { "bbox": [ 378, 154, 414, 168 ], "score": 1.0, "content": ", the key", "type": "text" }, { "bbox": [ 414, 154, 469, 165 ], "score": 0.92, "content": "\\pmb { K } = \\pmb { X } \\pmb { W } _ { K }", "type": "inline_equation" }, { "bbox": [ 469, 154, 505, 168 ], "score": 1.0, "content": ", and the", "type": "text" } ], "index": 6 }, { "bbox": [ 104, 162, 506, 181 ], "spans": [ { "bbox": [ 104, 162, 131, 181 ], "score": 1.0, "content": "value", "type": "text" }, { "bbox": [ 131, 166, 182, 177 ], "score": 0.91, "content": "\\pmb { V } = \\pmb { X } \\pmb { W } _ { V }", "type": "inline_equation" }, { "bbox": [ 182, 162, 212, 181 ], "score": 1.0, "content": ", where", "type": "text" }, { "bbox": [ 213, 166, 231, 178 ], "score": 0.83, "content": "W _ { Q }", "type": "inline_equation" }, { "bbox": [ 231, 162, 235, 181 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 235, 166, 255, 178 ], "score": 0.76, "content": "W _ { K }", "type": "inline_equation" }, { "bbox": [ 255, 162, 259, 181 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 259, 165, 322, 177 ], "score": 0.91, "content": "W _ { V } \\in \\mathbb { R } ^ { L \\times D _ { K } }", "type": "inline_equation" }, { "bbox": [ 322, 162, 357, 181 ], "score": 1.0, "content": ". For the", "type": "text" }, { "bbox": [ 358, 168, 362, 176 ], "score": 0.78, "content": "i", "type": "inline_equation" }, { "bbox": [ 362, 162, 392, 181 ], "score": 1.0, "content": "-th row", "type": "text" }, { "bbox": [ 392, 168, 402, 178 ], "score": 0.84, "content": "\\pmb q _ { i }", "type": "inline_equation" }, { "bbox": [ 402, 162, 413, 181 ], "score": 1.0, "content": "in", "type": "text" }, { "bbox": [ 414, 167, 423, 178 ], "score": 0.85, "content": "Q", "type": "inline_equation" }, { "bbox": [ 424, 162, 506, 181 ], "score": 1.0, "content": ", it can attend to any", "type": "text" } ], "index": 7 }, { "bbox": [ 105, 177, 459, 190 ], "spans": [ { "bbox": [ 105, 177, 183, 190 ], "score": 1.0, "content": "rows (i.e., keys) in", "type": "text" }, { "bbox": [ 183, 178, 194, 187 ], "score": 0.77, "content": "\\kappa", "type": "inline_equation" }, { "bbox": [ 195, 177, 363, 190 ], "score": 1.0, "content": ". In other words, the corresponding output", "type": "text" }, { "bbox": [ 364, 180, 374, 189 ], "score": 0.86, "content": "\\mathbf { \\nabla } _ { \\mathbf { \\psi } _ { 3 } } \\psi _ { i }", "type": "inline_equation" }, { "bbox": [ 374, 177, 459, 190 ], "score": 1.0, "content": "can be expressed as:", "type": "text" } ], "index": 8 } ], "index": 5.5, "bbox_fs": [ 104, 121, 506, 190 ] }, { "type": "interline_equation", "bbox": [ 232, 190, 378, 224 ], "lines": [ { "bbox": [ 232, 190, 378, 224 ], "spans": [ { "bbox": [ 232, 190, 378, 224 ], "score": 0.95, "content": "{ \\pmb y } _ { i } = \\sum _ { \\ell = 1 } ^ { L } \\frac { \\mathrm { e x p } ( { \\pmb q } _ { i } { \\pmb k } _ { \\ell } ^ { T } / \\sqrt { D _ { K } } ) { \\pmb v } _ { \\ell } } { \\sum _ { \\ell = 1 } ^ { L } \\mathrm { e x p } ( { \\pmb q } _ { i } { \\pmb k } _ { \\ell } ^ { T } / \\sqrt { D _ { K } } ) } ,", "type": "interline_equation", "image_path": "a4ab598b546f6b7ebb82b39ce9ff06398783e40dac4c9cea4945eff5dbeb48ba.jpg" } ] } ], "index": 9.5, "virtual_lines": [ { "bbox": [ 232, 190, 378, 207.0 ], "spans": [], "index": 9 }, { "bbox": [ 232, 207.0, 378, 224.0 ], "spans": [], "index": 10 } ] }, { "type": "text", "bbox": [ 106, 226, 505, 282 ], "lines": [ { "bbox": [ 106, 226, 506, 239 ], "spans": [ { "bbox": [ 106, 226, 133, 239 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 134, 226, 148, 239 ], "score": 0.9, "content": "k _ { \\ell } ^ { T }", "type": "inline_equation" }, { "bbox": [ 148, 226, 269, 239 ], "score": 1.0, "content": "denotes the transpose of row", "type": "text" }, { "bbox": [ 270, 227, 276, 236 ], "score": 0.79, "content": "\\ell", "type": "inline_equation" }, { "bbox": [ 276, 226, 288, 239 ], "score": 1.0, "content": "in", "type": "text" }, { "bbox": [ 288, 227, 299, 236 ], "score": 0.81, "content": "\\kappa", "type": "inline_equation" }, { "bbox": [ 300, 226, 506, 239 ], "score": 1.0, "content": ". We emphasize that the number of query-key dot", "type": "text" } ], "index": 11 }, { "bbox": [ 106, 237, 506, 251 ], "spans": [ { "bbox": [ 106, 237, 506, 251 ], "score": 1.0, "content": "products (Q-K pairs) that need to be calculated and stored dictates the time and space complexity of", "type": "text" } ], "index": 12 }, { "bbox": [ 106, 249, 505, 261 ], "spans": [ { "bbox": [ 106, 249, 505, 261 ], "score": 1.0, "content": "the attention mechanism. Viewed another way, this number is proportional to the number of edges", "type": "text" } ], "index": 13 }, { "bbox": [ 106, 260, 505, 272 ], "spans": [ { "bbox": [ 106, 260, 505, 272 ], "score": 1.0, "content": "in the graph (see Figure 1(a)). Since all Q-K pairs are computed and stored in the full attention", "type": "text" } ], "index": 14 }, { "bbox": [ 105, 270, 373, 283 ], "spans": [ { "bbox": [ 105, 270, 341, 283 ], "score": 1.0, "content": "mechanism (1), the resulting time and space complexity is", "type": "text" }, { "bbox": [ 342, 270, 369, 282 ], "score": 0.94, "content": "\\mathcal { O } ( L ^ { 2 } )", "type": "inline_equation" }, { "bbox": [ 369, 270, 373, 283 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 15 } ], "index": 13, "bbox_fs": [ 105, 226, 506, 283 ] }, { "type": "text", "bbox": [ 106, 286, 506, 380 ], "lines": [ { "bbox": [ 105, 286, 506, 300 ], "spans": [ { "bbox": [ 105, 286, 506, 300 ], "score": 1.0, "content": "As opposed to the above full attention mechanism, every node only pays attention to a limited set of", "type": "text" } ], "index": 16 }, { "bbox": [ 103, 297, 504, 317 ], "spans": [ { "bbox": [ 103, 297, 486, 317 ], "score": 1.0, "content": "keys in the PAM, corresponding to the pyramidal graph in Figure 1d. Concretely, suppose that", "type": "text" }, { "bbox": [ 486, 298, 504, 313 ], "score": 0.9, "content": "n _ { \\ell } ^ { ( s ) }", "type": "inline_equation" } ], "index": 17 }, { "bbox": [ 106, 312, 505, 324 ], "spans": [ { "bbox": [ 106, 312, 155, 324 ], "score": 1.0, "content": "denotes the", "type": "text" }, { "bbox": [ 155, 312, 161, 322 ], "score": 0.53, "content": "\\ell \\cdot", "type": "inline_equation" }, { "bbox": [ 162, 312, 229, 324 ], "score": 1.0, "content": "-th node at scale", "type": "text" }, { "bbox": [ 230, 314, 236, 322 ], "score": 0.7, "content": "s", "type": "inline_equation" }, { "bbox": [ 236, 312, 268, 324 ], "score": 1.0, "content": ", where", "type": "text" }, { "bbox": [ 268, 312, 325, 324 ], "score": 0.92, "content": "s = 1 , \\cdots , S", "type": "inline_equation" }, { "bbox": [ 326, 312, 505, 324 ], "score": 1.0, "content": "represents the bottom scale to the top scale", "type": "text" } ], "index": 18 }, { "bbox": [ 103, 323, 504, 341 ], "spans": [ { "bbox": [ 103, 323, 474, 341 ], "score": 1.0, "content": "sequentially. 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\\ell | \\leq \\frac { A - 1 } { 2 } , 1 \\leq j \\leq \\frac { L } { C ^ { s - 1 } } \\} } & & & & & \\\\ { \\mathbb { C } _ { \\ell } ^ { ( s ) } } & { = } & { \\{ n _ { j } ^ { ( s - 1 ) } : ( \\ell - 1 ) C < j \\leq \\ell C \\} } & { \\mathrm { i f } s \\geq 2 \\mathrm { e l s e } \\emptyset } & & & \\\\ { \\mathbb { P } _ { \\ell } ^ { ( s ) } } & { = } & { \\{ n _ { j } ^ { ( s + 1 ) } : j = \\lceil \\frac { \\ell } { C } \\rceil \\} } & { \\mathrm { i f } s \\leq S - 1 \\mathrm { e l s e } \\emptyset } & & & & \\end{array} \\right. .", "type": "interline_equation", "image_path": "c1c6b42e71ce1a071839b917e297e956e538cb8315d07f3cdb9c25ccecde1a58.jpg" } ] } ], "index": 24, "virtual_lines": [ { "bbox": [ 174, 381, 437, 403.0 ], "spans": [], "index": 23 }, { "bbox": [ 174, 403.0, 437, 425.0 ], "spans": [], "index": 24 }, { "bbox": [ 174, 425.0, 437, 447.0 ], "spans": [], "index": 25 } ] }, { "type": "text", "bbox": [ 106, 449, 351, 463 ], "lines": [ { "bbox": [ 100, 446, 352, 470 ], "spans": [ { "bbox": [ 100, 446, 248, 470 ], "score": 1.0, "content": "It follows that the attention at node", "type": "text" }, { "bbox": [ 248, 448, 266, 464 ], "score": 0.92, "content": "n _ { \\ell } ^ { ( s ) }", "type": "inline_equation" }, { "bbox": [ 266, 446, 352, 470 ], "score": 1.0, "content": "can be simplified as:√", "type": "text" } ], "index": 26 } ], "index": 26, "bbox_fs": [ 100, 446, 352, 470 ] }, { "type": "interline_equation", "bbox": [ 225, 464, 385, 500 ], "lines": [ { "bbox": [ 225, 464, 385, 500 ], "spans": [ { "bbox": [ 225, 464, 385, 500 ], "score": 0.94, "content": "\\pmb { y } _ { i } = \\sum _ { \\ell \\in \\mathbb { N } _ { \\ell } ^ { ( s ) } } \\frac { \\exp ( \\pmb { q } _ { i } \\pmb { k } _ { \\ell } ^ { T } / \\sqrt { d _ { K } } ) \\pmb { v } _ { \\ell } } { \\sum _ { \\ell \\in \\mathbb { N } _ { l } ^ { ( s ) } } \\exp ( \\pmb { q } _ { i } \\pmb { k } _ { \\ell } ^ { T } / \\sqrt { d _ { K } } ) } ,", "type": "interline_equation", "image_path": "65219bb1ee3e17b2e49fcc877e25c0f00a110a80ddb131298165d032413727b3.jpg" } ] } ], "index": 27.5, "virtual_lines": [ { "bbox": [ 225, 464, 385, 482.0 ], "spans": [], "index": 27 }, { "bbox": [ 225, 482.0, 385, 500.0 ], "spans": [], "index": 28 } ] }, { "type": "text", "bbox": [ 107, 507, 505, 541 ], "lines": [ { "bbox": [ 106, 507, 505, 519 ], "spans": [ { "bbox": [ 106, 507, 316, 519 ], "score": 1.0, "content": "We further denote the number of attention layers as", "type": "text" }, { "bbox": [ 316, 508, 326, 517 ], "score": 0.83, "content": "N", "type": "inline_equation" }, { "bbox": [ 327, 507, 505, 519 ], "score": 1.0, "content": ". Without loss of generality, we assume that", "type": "text" } ], "index": 29 }, { "bbox": [ 107, 516, 506, 532 ], "spans": [ { "bbox": [ 107, 519, 115, 528 ], "score": 0.79, "content": "L", "type": "inline_equation" }, { "bbox": [ 115, 516, 175, 532 ], "score": 1.0, "content": "is divisible by", "type": "text" }, { "bbox": [ 175, 518, 200, 529 ], "score": 0.91, "content": "C ^ { S - 1 }", "type": "inline_equation" }, { "bbox": [ 200, 516, 424, 532 ], "score": 1.0, "content": ". We can then have the following lemma (cf. 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Given", "type": "text" }, { "bbox": [ 182, 542, 232, 554 ], "score": 0.68, "content": "A , C , L , N", "type": "inline_equation" }, { "bbox": [ 233, 541, 255, 556 ], "score": 1.0, "content": ", and", "type": "text" }, { "bbox": [ 255, 543, 263, 552 ], "score": 0.77, "content": "S", "type": "inline_equation" }, { "bbox": [ 264, 541, 392, 556 ], "score": 1.0, "content": "that satisfy Equation (4), after", "type": "text" }, { "bbox": [ 392, 543, 402, 552 ], "score": 0.78, "content": "N", "type": "inline_equation" }, { "bbox": [ 403, 541, 505, 556 ], "score": 1.0, "content": "stacked attention layers,", "type": "text" } ], "index": 32 }, { "bbox": [ 106, 553, 356, 566 ], "spans": [ { "bbox": [ 106, 553, 356, 566 ], "score": 1.0, "content": "nodes at the coarsest scale can obtain a global receptive field.", "type": "text" } ], "index": 33 } ], "index": 32.5, "bbox_fs": [ 105, 541, 505, 566 ] }, { "type": "interline_equation", "bbox": [ 253, 566, 358, 591 ], "lines": [ { "bbox": [ 253, 566, 358, 591 ], "spans": [ { "bbox": [ 253, 566, 358, 591 ], "score": 0.94, "content": "\\frac { L } { C ^ { S - 1 } } - 1 \\leq \\frac { ( A - 1 ) N } { 2 } .", "type": "interline_equation", "image_path": "eafe1980cccd7d86fbe5781b52170d343501370e89e2c447787bffb17b333aa3.jpg" } ] } ], "index": 34, "virtual_lines": [ { "bbox": [ 253, 566, 358, 591 ], "spans": [], "index": 34 } ] }, { "type": "text", "bbox": [ 108, 596, 505, 630 ], "lines": [ { "bbox": [ 106, 597, 505, 609 ], "spans": [ { "bbox": [ 106, 597, 269, 609 ], "score": 1.0, "content": "In addition, when the number of scales", "type": "text" }, { "bbox": [ 270, 597, 278, 607 ], "score": 0.83, "content": "S", "type": "inline_equation" }, { "bbox": [ 278, 597, 505, 609 ], "score": 1.0, "content": "is fixed, the following two propositions summarize the", "type": "text" } ], "index": 35 }, { "bbox": [ 106, 608, 505, 620 ], "spans": [ { "bbox": [ 106, 608, 505, 620 ], "score": 1.0, "content": "time and space complexity and the order of the maximum path length for the proposed pyramidal", "type": "text" } ], "index": 36 }, { "bbox": [ 105, 619, 408, 631 ], "spans": [ { "bbox": [ 105, 619, 357, 631 ], "score": 1.0, "content": "attention mechanism. We refer the readers to Appendix C and", "type": "text" }, { "bbox": [ 357, 619, 367, 629 ], "score": 0.55, "content": "\\mathrm { D }", "type": "inline_equation" }, { "bbox": [ 367, 619, 408, 631 ], "score": 1.0, "content": "for proof.", "type": "text" } ], "index": 37 } ], "index": 36, "bbox_fs": [ 105, 597, 505, 631 ] }, { "type": "text", "bbox": [ 105, 632, 505, 654 ], "lines": [ { "bbox": [ 105, 630, 505, 645 ], "spans": [ { "bbox": [ 105, 630, 459, 645 ], "score": 1.0, "content": "Proposition 1. The time and space complexity for the pyramidal attention mechanism is", "type": "text" }, { "bbox": [ 459, 632, 489, 644 ], "score": 0.84, "content": "\\mathcal { O } ( A L )", "type": "inline_equation" }, { "bbox": [ 490, 630, 505, 645 ], "score": 1.0, "content": "for", "type": "text" } ], "index": 38 }, { "bbox": [ 105, 643, 376, 655 ], "spans": [ { "bbox": [ 105, 643, 131, 655 ], "score": 1.0, "content": "given", "type": "text" }, { "bbox": [ 131, 644, 140, 653 ], "score": 0.47, "content": "A", "type": "inline_equation" }, { "bbox": [ 140, 643, 158, 655 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 158, 644, 166, 653 ], "score": 0.69, "content": "L", "type": "inline_equation" }, { "bbox": [ 166, 643, 231, 655 ], "score": 1.0, "content": "and amounts to", "type": "text" }, { "bbox": [ 232, 644, 255, 655 ], "score": 0.91, "content": "\\mathcal O ( L )", "type": "inline_equation" }, { "bbox": [ 255, 643, 280, 655 ], "score": 1.0, "content": "when", "type": "text" }, { "bbox": [ 280, 644, 289, 653 ], "score": 0.73, "content": "A", "type": "inline_equation" }, { "bbox": [ 289, 643, 365, 655 ], "score": 1.0, "content": "is a constant w.r.t.", "type": "text" }, { "bbox": [ 365, 644, 373, 653 ], "score": 0.44, "content": "L", "type": "inline_equation" }, { "bbox": [ 373, 643, 376, 655 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 39 } ], "index": 38.5, "bbox_fs": [ 105, 630, 505, 655 ] }, { "type": "text", "bbox": [ 106, 656, 505, 701 ], "lines": [ { "bbox": [ 105, 655, 506, 669 ], "spans": [ { "bbox": [ 105, 655, 506, 669 ], "score": 1.0, "content": "Proposition 2. Let the signal traversing path between two nodes in a graph denote the shortest path", "type": "text" } ], "index": 40 }, { "bbox": [ 106, 668, 505, 679 ], "spans": [ { "bbox": [ 106, 668, 505, 679 ], "score": 1.0, "content": "connecting them. Then the maximum length of signal traversing path between two arbitrary nodes", "type": "text" } ], "index": 41 }, { "bbox": [ 104, 676, 506, 691 ], "spans": [ { "bbox": [ 104, 676, 210, 691 ], "score": 1.0, "content": "in the pyramidal graph is", "type": "text" }, { "bbox": [ 210, 679, 293, 690 ], "score": 0.91, "content": "\\mathcal { O } ( S + L / C ^ { S - 1 } / A )", "type": "inline_equation" }, { "bbox": [ 294, 676, 333, 691 ], "score": 1.0, "content": "for given", "type": "text" }, { "bbox": [ 333, 680, 341, 687 ], "score": 0.7, "content": "A", "type": "inline_equation" }, { "bbox": [ 341, 676, 345, 691 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 345, 680, 353, 687 ], "score": 0.73, "content": "C", "type": "inline_equation" }, { "bbox": [ 354, 676, 358, 691 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 358, 680, 365, 688 ], "score": 0.65, "content": "L", "type": "inline_equation" }, { "bbox": [ 366, 676, 387, 691 ], "score": 1.0, "content": ", and", "type": "text" }, { "bbox": [ 388, 680, 394, 688 ], "score": 0.77, "content": "S", "type": "inline_equation" }, { "bbox": [ 395, 676, 453, 691 ], "score": 1.0, "content": ". 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Therefore, the proposed PAM achieves the complexity of", "type": "text" }, { "bbox": [ 353, 223, 377, 235 ], "score": 0.92, "content": "\\mathcal { O } ( L )", "type": "inline_equation" }, { "bbox": [ 377, 222, 505, 236 ], "score": 1.0, "content": "with the maximum path length", "type": "text" } ], "index": 9 }, { "bbox": [ 105, 233, 505, 246 ], "spans": [ { "bbox": [ 105, 233, 118, 246 ], "score": 1.0, "content": "of", "type": "text" }, { "bbox": [ 118, 234, 140, 246 ], "score": 0.89, "content": "\\mathcal { O } ( 1 )", "type": "inline_equation" }, { "bbox": [ 140, 233, 362, 246 ], "score": 1.0, "content": ". Note that in the PAM, a node can attend to at most", "type": "text" }, { "bbox": [ 363, 234, 411, 244 ], "score": 0.92, "content": "A + C + 1", "type": "inline_equation" }, { "bbox": [ 412, 233, 505, 246 ], "score": 1.0, "content": "nodes. Unfortunately,", "type": "text" } ], "index": 10 }, { "bbox": [ 105, 245, 505, 258 ], "spans": [ { "bbox": [ 105, 245, 505, 258 ], "score": 1.0, "content": "such a sparse attention mechanism is not supported in the existing deep learning libraries, such as", "type": "text" } ], "index": 11 }, { "bbox": [ 105, 254, 506, 269 ], "spans": [ { "bbox": [ 105, 254, 506, 269 ], "score": 1.0, "content": "Pytorch and TensorFlow. A naive implementation of the PAM that can fully exploit the tensor", "type": "text" } ], "index": 12 }, { "bbox": [ 105, 266, 505, 279 ], "spans": [ { "bbox": [ 105, 266, 444, 279 ], "score": 1.0, "content": "operation framework is to first compute the product between all Q-K pairs, i.e.,", "type": "text" }, { "bbox": [ 444, 266, 466, 279 ], "score": 0.94, "content": "\\mathbf { \\Delta } q _ { i } \\mathbf { \\Delta } k _ { \\ell } ^ { T }", "type": "inline_equation" }, { "bbox": [ 467, 266, 485, 279 ], "score": 1.0, "content": "for", "type": "text" }, { "bbox": [ 485, 267, 505, 278 ], "score": 0.86, "content": "\\ell =", "type": "inline_equation" } ], "index": 13 }, { "bbox": [ 106, 276, 507, 295 ], "spans": [ { "bbox": [ 106, 281, 143, 293 ], "score": 0.88, "content": "1 , \\cdots , L", "type": "inline_equation" }, { "bbox": [ 144, 276, 225, 295 ], "score": 1.0, "content": ", and then mask out", "type": "text" }, { "bbox": [ 226, 278, 263, 293 ], "score": 0.93, "content": "\\ell \\notin \\mathbb { N } _ { \\ell } ^ { ( s ) }", "type": "inline_equation" }, { "bbox": [ 264, 276, 507, 295 ], "score": 1.0, "content": ". However, the resulting time and space complexity of this", "type": "text" } ], "index": 14 }, { "bbox": [ 105, 292, 506, 306 ], "spans": [ { "bbox": [ 105, 292, 201, 306 ], "score": 1.0, "content": "implementation is still", "type": "text" }, { "bbox": [ 201, 293, 229, 305 ], "score": 0.92, "content": "\\mathcal { O } ( L ^ { 2 } )", "type": "inline_equation" }, { "bbox": [ 230, 292, 506, 306 ], "score": 1.0, "content": ". Instead, we build a customized CUDA kernel specialized for the", "type": "text" } ], "index": 15 }, { "bbox": [ 105, 303, 506, 317 ], "spans": [ { "bbox": [ 105, 303, 506, 317 ], "score": 1.0, "content": "PAM using TVM (Chen et al., 2018), practically reducing the computational time and memory cost", "type": "text" } ], "index": 16 }, { "bbox": [ 105, 313, 505, 328 ], "spans": [ { "bbox": [ 105, 313, 505, 328 ], "score": 1.0, "content": "and making the proposed model amenable to long time series. Longer historical input is typically", "type": "text" } ], "index": 17 }, { "bbox": [ 104, 325, 505, 339 ], "spans": [ { "bbox": [ 104, 325, 505, 339 ], "score": 1.0, "content": "helpful for improving the prediction accuracy, as more information is provided, especially when", "type": "text" } ], "index": 18 }, { "bbox": [ 105, 336, 271, 349 ], "spans": [ { "bbox": [ 105, 336, 271, 349 ], "score": 1.0, "content": "long-range dependencies are considered.", "type": "text" } ], "index": 19 } ], "index": 13.5 }, { "type": "title", "bbox": [ 107, 362, 357, 373 ], "lines": [ { "bbox": [ 105, 360, 358, 376 ], "spans": [ { "bbox": [ 105, 360, 358, 376 ], "score": 1.0, "content": "3.2 COARSER-SCALE CONSTRUCTION MODULE (CSCM)", "type": "text" } ], "index": 20 } ], "index": 20 }, { "type": "text", "bbox": [ 106, 381, 505, 505 ], "lines": [ { "bbox": [ 106, 381, 505, 393 ], "spans": [ { "bbox": [ 106, 381, 505, 393 ], "score": 1.0, "content": "CSCM targets at initializing the nodes at the coarser scales of the pyramidal graph, so as to facilitate", "type": "text" } ], "index": 21 }, { "bbox": [ 106, 393, 505, 405 ], "spans": [ { "bbox": [ 106, 393, 505, 405 ], "score": 1.0, "content": "the subsequent PAM to exchange information between these nodes. Specifically, the coarse-scale", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 403, 505, 417 ], "spans": [ { "bbox": [ 105, 403, 505, 417 ], "score": 1.0, "content": "nodes are introduced scale by scale from bottom to top by performing convolutions on the corre-", "type": "text" } ], "index": 23 }, { "bbox": [ 103, 412, 507, 432 ], "spans": [ { "bbox": [ 103, 412, 207, 432 ], "score": 1.0, "content": "sponding children nodes", "type": "text" }, { "bbox": [ 207, 415, 226, 430 ], "score": 0.92, "content": "\\mathbb { C } _ { \\ell } ^ { ( s ) }", "type": "inline_equation" }, { "bbox": [ 226, 412, 507, 432 ], "score": 1.0, "content": ". As demonstrated in Figure 3, several convolution layers with kernel", "type": "text" } ], "index": 24 }, { "bbox": [ 105, 428, 505, 440 ], "spans": [ { "bbox": [ 105, 428, 125, 440 ], "score": 1.0, "content": "size", "type": "text" }, { "bbox": [ 125, 429, 135, 438 ], "score": 0.8, "content": "C", "type": "inline_equation" }, { "bbox": [ 135, 428, 179, 440 ], "score": 1.0, "content": "and stride", "type": "text" }, { "bbox": [ 179, 429, 189, 438 ], "score": 0.8, "content": "C", "type": "inline_equation" }, { "bbox": [ 189, 428, 505, 440 ], "score": 1.0, "content": "are sequentially applied to the embedded sequence in the dimension of time,", "type": "text" } ], "index": 25 }, { "bbox": [ 106, 439, 506, 452 ], "spans": [ { "bbox": [ 106, 439, 237, 452 ], "score": 1.0, "content": "yielding a sequence with length", "type": "text" }, { "bbox": [ 238, 439, 262, 451 ], "score": 0.91, "content": "L / C ^ { s }", "type": "inline_equation" }, { "bbox": [ 263, 439, 297, 452 ], "score": 1.0, "content": "at scale", "type": "text" }, { "bbox": [ 298, 442, 304, 449 ], "score": 0.59, "content": "s", "type": "inline_equation" }, { "bbox": [ 304, 439, 506, 452 ], "score": 1.0, "content": ". The resulting sequences at different scales form", "type": "text" } ], "index": 26 }, { "bbox": [ 105, 450, 506, 462 ], "spans": [ { "bbox": [ 105, 450, 114, 462 ], "score": 1.0, "content": "a", "type": "text" }, { "bbox": [ 114, 451, 123, 460 ], "score": 0.84, "content": "C", "type": "inline_equation" }, { "bbox": [ 123, 450, 506, 462 ], "score": 1.0, "content": "-ary tree. We concatenate these fine-to-coarse sequences before inputting them to the PAM. In", "type": "text" } ], "index": 27 }, { "bbox": [ 106, 461, 505, 474 ], "spans": [ { "bbox": [ 106, 461, 505, 474 ], "score": 1.0, "content": "order to reduce the amount of parameters and calculations, we reduce the dimension of each node by", "type": "text" } ], "index": 28 }, { "bbox": [ 105, 472, 505, 485 ], "spans": [ { "bbox": [ 105, 472, 505, 485 ], "score": 1.0, "content": "a fully connected layer before inputting the sequence into the stacked convolution layers and restore", "type": "text" } ], "index": 29 }, { "bbox": [ 105, 482, 506, 496 ], "spans": [ { "bbox": [ 105, 482, 506, 496 ], "score": 1.0, "content": "it after all convolutions. Such a bottleneck structure significantly reduces the number of parameters", "type": "text" } ], "index": 30 }, { "bbox": [ 105, 493, 301, 507 ], "spans": [ { "bbox": [ 105, 493, 301, 507 ], "score": 1.0, "content": "in the module and can guard against over-fitting.", "type": "text" } ], "index": 31 } ], "index": 26 }, { "type": "title", "bbox": [ 108, 519, 226, 531 ], "lines": [ { "bbox": [ 105, 518, 228, 532 ], "spans": [ { "bbox": [ 105, 518, 228, 532 ], "score": 1.0, "content": "3.3 PREDICTION MODULE", "type": "text" } ], "index": 32 } ], "index": 32 }, { "type": "text", "bbox": [ 107, 539, 505, 583 ], "lines": [ { "bbox": [ 105, 539, 505, 552 ], "spans": [ { "bbox": [ 105, 539, 351, 552 ], "score": 1.0, "content": "For single-step forecasting, we add an end token (by setting", "type": "text" }, { "bbox": [ 351, 540, 391, 551 ], "score": 0.9, "content": "z _ { t + 1 } = 0", "type": "inline_equation" }, { "bbox": [ 392, 539, 505, 552 ], "score": 1.0, "content": ") to the end of the historical", "type": "text" } ], "index": 33 }, { "bbox": [ 105, 550, 505, 562 ], "spans": [ { "bbox": [ 105, 550, 145, 562 ], "score": 1.0, "content": "sequence", "type": "text" }, { "bbox": [ 145, 551, 181, 562 ], "score": 0.89, "content": "z _ { t - L + 1 : t }", "type": "inline_equation" }, { "bbox": [ 182, 550, 505, 562 ], "score": 1.0, "content": "before inputting it into the embedding layer. After the sequence is encoded by the", "type": "text" } ], "index": 34 }, { "bbox": [ 105, 560, 505, 574 ], "spans": [ { "bbox": [ 105, 560, 505, 574 ], "score": 1.0, "content": "PAM, we gather the features given by the last nodes at all scales in the pyramidal graph, concatenate", "type": "text" } ], "index": 35 }, { "bbox": [ 106, 573, 359, 585 ], "spans": [ { "bbox": [ 106, 573, 359, 585 ], "score": 1.0, "content": "and then input them into a fully connected layer for prediction.", "type": "text" } ], "index": 36 } ], "index": 34.5 }, { "type": "text", "bbox": [ 107, 589, 505, 732 ], "lines": [ { "bbox": [ 105, 588, 506, 601 ], "spans": [ { "bbox": [ 105, 588, 506, 601 ], "score": 1.0, "content": "For multi-step forecasting, we propose two prediction modules. The first one is the same with the", "type": "text" } ], "index": 37 }, { "bbox": [ 106, 600, 505, 612 ], "spans": [ { "bbox": [ 106, 600, 410, 612 ], "score": 1.0, "content": "single-step forecasting module, but maps the last nodes at all scales to all", "type": "text" }, { "bbox": [ 410, 600, 422, 610 ], "score": 0.77, "content": "M", "type": "inline_equation" }, { "bbox": [ 422, 600, 505, 612 ], "score": 1.0, "content": "future time steps in", "type": "text" } ], "index": 38 }, { "bbox": [ 105, 612, 504, 623 ], "spans": [ { "bbox": [ 105, 612, 504, 623 ], "score": 1.0, "content": "a batch. The second one, on the other hand, resorts to a decoder with two full attention layers.", "type": "text" } ], "index": 39 }, { "bbox": [ 105, 622, 505, 634 ], "spans": [ { "bbox": [ 105, 622, 505, 634 ], "score": 1.0, "content": "Specifically, similar to the original Transformer (Vaswani et al., 2017), we replace the observations", "type": "text" } ], "index": 40 }, { "bbox": [ 105, 632, 505, 645 ], "spans": [ { "bbox": [ 105, 632, 157, 645 ], "score": 1.0, "content": "at the future", "type": "text" }, { "bbox": [ 157, 633, 169, 643 ], "score": 0.7, "content": "M", "type": "inline_equation" }, { "bbox": [ 170, 632, 505, 645 ], "score": 1.0, "content": "time steps with 0, embed them in the same manner with the historical observations,", "type": "text" } ], "index": 41 }, { "bbox": [ 105, 643, 505, 657 ], "spans": [ { "bbox": [ 105, 643, 505, 657 ], "score": 1.0, "content": "and refer to the summation of the observation, covariate, and positional embedding as the “prediction", "type": "text" } ], "index": 42 }, { "bbox": [ 105, 654, 506, 667 ], "spans": [ { "bbox": [ 105, 654, 135, 667 ], "score": 1.0, "content": "token”", "type": "text" }, { "bbox": [ 136, 655, 148, 667 ], "score": 0.88, "content": "F _ { p }", "type": "inline_equation" }, { "bbox": [ 149, 654, 379, 667 ], "score": 1.0, "content": ". The first attention layer then takes the prediction tokens", "type": "text" }, { "bbox": [ 380, 655, 392, 667 ], "score": 0.9, "content": "F _ { p }", "type": "inline_equation" }, { "bbox": [ 393, 654, 506, 667 ], "score": 1.0, "content": "as the query and the output", "type": "text" } ], "index": 43 }, { "bbox": [ 105, 664, 506, 678 ], "spans": [ { "bbox": [ 105, 664, 166, 678 ], "score": 1.0, "content": "of the encoder", "type": "text" }, { "bbox": [ 167, 666, 179, 677 ], "score": 0.87, "content": "\\pmb { F _ { e } }", "type": "inline_equation" }, { "bbox": [ 179, 664, 435, 678 ], "score": 1.0, "content": "(i.e., all nodes in the PAM) as the key and the value, and yields", "type": "text" }, { "bbox": [ 435, 666, 452, 677 ], "score": 0.92, "content": "{ \\bf { { F } } } _ { d 1 }", "type": "inline_equation" }, { "bbox": [ 452, 664, 506, 678 ], "score": 1.0, "content": ". The second", "type": "text" } ], "index": 44 }, { "bbox": [ 105, 677, 506, 690 ], "spans": [ { "bbox": [ 105, 677, 152, 690 ], "score": 1.0, "content": "layer takes", "type": "text" }, { "bbox": [ 153, 677, 169, 687 ], "score": 0.89, "content": "{ \\mathbf { } } F _ { d 1 }", "type": "inline_equation" }, { "bbox": [ 169, 677, 333, 690 ], "score": 1.0, "content": "as the query, but takes the concatenated", "type": "text" }, { "bbox": [ 334, 677, 350, 687 ], "score": 0.9, "content": "{ \\bf { { F } } } _ { d 1 }", "type": "inline_equation" }, { "bbox": [ 351, 677, 369, 690 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 369, 677, 381, 687 ], "score": 0.89, "content": "\\pmb { F _ { e } }", "type": "inline_equation" }, { "bbox": [ 382, 677, 506, 690 ], "score": 1.0, "content": "as the key and the value. The", "type": "text" } ], "index": 45 }, { "bbox": [ 105, 687, 505, 700 ], "spans": [ { "bbox": [ 105, 687, 197, 700 ], "score": 1.0, "content": "historical information", "type": "text" }, { "bbox": [ 198, 688, 209, 699 ], "score": 0.89, "content": "\\pmb { F _ { e } }", "type": "inline_equation" }, { "bbox": [ 210, 687, 505, 700 ], "score": 1.0, "content": "is fed directly into both attention layers, since such information is vital", "type": "text" } ], "index": 46 }, { "bbox": [ 105, 699, 505, 711 ], "spans": [ { "bbox": [ 105, 699, 505, 711 ], "score": 1.0, "content": "for accurate long-range forecasting. The final prediction is then obtained through a fully connected", "type": "text" } ], "index": 47 }, { "bbox": [ 105, 709, 506, 722 ], "spans": [ { "bbox": [ 105, 709, 506, 722 ], "score": 1.0, "content": "layer across the dimension of channels. Again, we output all future predictions together to avoid the", "type": "text" } ], "index": 48 }, { "bbox": [ 105, 720, 414, 733 ], "spans": [ { "bbox": [ 105, 720, 414, 733 ], "score": 1.0, "content": "problem of error accumulation in the autoregressive decoder of Transformer.", "type": "text" } ], "index": 49 } ], "index": 43 } ], "page_idx": 5, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 107, 27, 293, 37 ], "lines": [ { "bbox": [ 106, 25, 294, 38 ], "spans": [ { "bbox": [ 106, 25, 294, 38 ], "score": 1.0, "content": "Published as a conference paper at ICLR 2022", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 302, 752, 308, 760 ], "lines": [ { "bbox": [ 302, 751, 309, 762 ], "spans": [ { "bbox": [ 302, 751, 309, 762 ], "score": 1.0, "content": "6", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "image", "bbox": [ 195, 79, 416, 181 ], "blocks": [ { "type": "image_body", "bbox": [ 195, 79, 416, 181 ], "group_id": 0, "lines": [ { "bbox": [ 195, 79, 416, 181 ], "spans": [ { "bbox": [ 195, 79, 416, 181 ], "score": 0.966, "type": "image", "image_path": "91aa8cb400219056af9686acde2534aae765cbb65f9bf4e842354362211e49b1.jpg" } ] } ], "index": 3, "virtual_lines": [ { "bbox": [ 195, 79, 416, 93.57142857142857 ], "spans": [], "index": 0 }, { "bbox": [ 195, 93.57142857142857, 416, 108.14285714285714 ], "spans": [], "index": 1 }, { "bbox": [ 195, 108.14285714285714, 416, 122.71428571428571 ], "spans": [], "index": 2 }, { "bbox": [ 195, 122.71428571428571, 416, 137.28571428571428 ], "spans": [], "index": 3 }, { "bbox": [ 195, 137.28571428571428, 416, 151.85714285714286 ], "spans": [], "index": 4 }, { "bbox": [ 195, 151.85714285714286, 416, 166.42857142857144 ], "spans": [], "index": 5 }, { "bbox": [ 195, 166.42857142857144, 416, 181.00000000000003 ], "spans": [], "index": 6 } ] }, { "type": "image_caption", "bbox": [ 105, 187, 503, 199 ], "group_id": 0, "lines": [ { "bbox": [ 106, 186, 504, 200 ], "spans": [ { "bbox": [ 106, 186, 290, 200 ], "score": 1.0, "content": "Figure 3: Coarser-scale construction module:", "type": "text" }, { "bbox": [ 291, 188, 300, 198 ], "score": 0.82, "content": "B", "type": "inline_equation" }, { "bbox": [ 300, 186, 383, 200 ], "score": 1.0, "content": "is the batch size and", "type": "text" }, { "bbox": [ 384, 188, 393, 198 ], "score": 0.83, "content": "D", "type": "inline_equation" }, { "bbox": [ 394, 186, 504, 200 ], "score": 1.0, "content": "is the dimension of a node.", "type": "text" } ], "index": 7 } ], "index": 7 } ], "index": 5.0 }, { "type": "text", "bbox": [ 106, 211, 505, 348 ], "lines": [ { "bbox": [ 105, 211, 505, 225 ], "spans": [ { "bbox": [ 105, 211, 215, 225 ], "score": 1.0, "content": "In our experiments, we fix", "type": "text" }, { "bbox": [ 215, 212, 223, 222 ], "score": 0.8, "content": "S", "type": "inline_equation" }, { "bbox": [ 224, 211, 242, 225 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 242, 212, 252, 222 ], "score": 0.81, "content": "N", "type": "inline_equation" }, { "bbox": [ 253, 211, 273, 225 ], "score": 1.0, "content": ", and", "type": "text" }, { "bbox": [ 274, 212, 282, 222 ], "score": 0.74, "content": "A", "type": "inline_equation" }, { "bbox": [ 283, 211, 505, 225 ], "score": 1.0, "content": "can only take 3 or 5, regardless of the sequence length", "type": "text" } ], "index": 8 }, { "bbox": [ 107, 222, 505, 236 ], "spans": [ { "bbox": [ 107, 223, 115, 233 ], "score": 0.67, "content": "L", "type": "inline_equation" }, { "bbox": [ 115, 222, 353, 236 ], "score": 1.0, "content": ". Therefore, the proposed PAM achieves the complexity of", "type": "text" }, { "bbox": [ 353, 223, 377, 235 ], "score": 0.92, "content": "\\mathcal { O } ( L )", "type": "inline_equation" }, { "bbox": [ 377, 222, 505, 236 ], "score": 1.0, "content": "with the maximum path length", "type": "text" } ], "index": 9 }, { "bbox": [ 105, 233, 505, 246 ], "spans": [ { "bbox": [ 105, 233, 118, 246 ], "score": 1.0, "content": "of", "type": "text" }, { "bbox": [ 118, 234, 140, 246 ], "score": 0.89, "content": "\\mathcal { O } ( 1 )", "type": "inline_equation" }, { "bbox": [ 140, 233, 362, 246 ], "score": 1.0, "content": ". Note that in the PAM, a node can attend to at most", "type": "text" }, { "bbox": [ 363, 234, 411, 244 ], "score": 0.92, "content": "A + C + 1", "type": "inline_equation" }, { "bbox": [ 412, 233, 505, 246 ], "score": 1.0, "content": "nodes. Unfortunately,", "type": "text" } ], "index": 10 }, { "bbox": [ 105, 245, 505, 258 ], "spans": [ { "bbox": [ 105, 245, 505, 258 ], "score": 1.0, "content": "such a sparse attention mechanism is not supported in the existing deep learning libraries, such as", "type": "text" } ], "index": 11 }, { "bbox": [ 105, 254, 506, 269 ], "spans": [ { "bbox": [ 105, 254, 506, 269 ], "score": 1.0, "content": "Pytorch and TensorFlow. A naive implementation of the PAM that can fully exploit the tensor", "type": "text" } ], "index": 12 }, { "bbox": [ 105, 266, 505, 279 ], "spans": [ { "bbox": [ 105, 266, 444, 279 ], "score": 1.0, "content": "operation framework is to first compute the product between all Q-K pairs, i.e.,", "type": "text" }, { "bbox": [ 444, 266, 466, 279 ], "score": 0.94, "content": "\\mathbf { \\Delta } q _ { i } \\mathbf { \\Delta } k _ { \\ell } ^ { T }", "type": "inline_equation" }, { "bbox": [ 467, 266, 485, 279 ], "score": 1.0, "content": "for", "type": "text" }, { "bbox": [ 485, 267, 505, 278 ], "score": 0.86, "content": "\\ell =", "type": "inline_equation" } ], "index": 13 }, { "bbox": [ 106, 276, 507, 295 ], "spans": [ { "bbox": [ 106, 281, 143, 293 ], "score": 0.88, "content": "1 , \\cdots , L", "type": "inline_equation" }, { "bbox": [ 144, 276, 225, 295 ], "score": 1.0, "content": ", and then mask out", "type": "text" }, { "bbox": [ 226, 278, 263, 293 ], "score": 0.93, "content": "\\ell \\notin \\mathbb { N } _ { \\ell } ^ { ( s ) }", "type": "inline_equation" }, { "bbox": [ 264, 276, 507, 295 ], "score": 1.0, "content": ". However, the resulting time and space complexity of this", "type": "text" } ], "index": 14 }, { "bbox": [ 105, 292, 506, 306 ], "spans": [ { "bbox": [ 105, 292, 201, 306 ], "score": 1.0, "content": "implementation is still", "type": "text" }, { "bbox": [ 201, 293, 229, 305 ], "score": 0.92, "content": "\\mathcal { O } ( L ^ { 2 } )", "type": "inline_equation" }, { "bbox": [ 230, 292, 506, 306 ], "score": 1.0, "content": ". Instead, we build a customized CUDA kernel specialized for the", "type": "text" } ], "index": 15 }, { "bbox": [ 105, 303, 506, 317 ], "spans": [ { "bbox": [ 105, 303, 506, 317 ], "score": 1.0, "content": "PAM using TVM (Chen et al., 2018), practically reducing the computational time and memory cost", "type": "text" } ], "index": 16 }, { "bbox": [ 105, 313, 505, 328 ], "spans": [ { "bbox": [ 105, 313, 505, 328 ], "score": 1.0, "content": "and making the proposed model amenable to long time series. Longer historical input is typically", "type": "text" } ], "index": 17 }, { "bbox": [ 104, 325, 505, 339 ], "spans": [ { "bbox": [ 104, 325, 505, 339 ], "score": 1.0, "content": "helpful for improving the prediction accuracy, as more information is provided, especially when", "type": "text" } ], "index": 18 }, { "bbox": [ 105, 336, 271, 349 ], "spans": [ { "bbox": [ 105, 336, 271, 349 ], "score": 1.0, "content": "long-range dependencies are considered.", "type": "text" } ], "index": 19 } ], "index": 13.5, "bbox_fs": [ 104, 211, 507, 349 ] }, { "type": "title", "bbox": [ 107, 362, 357, 373 ], "lines": [ { "bbox": [ 105, 360, 358, 376 ], "spans": [ { "bbox": [ 105, 360, 358, 376 ], "score": 1.0, "content": "3.2 COARSER-SCALE CONSTRUCTION MODULE (CSCM)", "type": "text" } ], "index": 20 } ], "index": 20 }, { "type": "text", "bbox": [ 106, 381, 505, 505 ], "lines": [ { "bbox": [ 106, 381, 505, 393 ], "spans": [ { "bbox": [ 106, 381, 505, 393 ], "score": 1.0, "content": "CSCM targets at initializing the nodes at the coarser scales of the pyramidal graph, so as to facilitate", "type": "text" } ], "index": 21 }, { "bbox": [ 106, 393, 505, 405 ], "spans": [ { "bbox": [ 106, 393, 505, 405 ], "score": 1.0, "content": "the subsequent PAM to exchange information between these nodes. Specifically, the coarse-scale", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 403, 505, 417 ], "spans": [ { "bbox": [ 105, 403, 505, 417 ], "score": 1.0, "content": "nodes are introduced scale by scale from bottom to top by performing convolutions on the corre-", "type": "text" } ], "index": 23 }, { "bbox": [ 103, 412, 507, 432 ], "spans": [ { "bbox": [ 103, 412, 207, 432 ], "score": 1.0, "content": "sponding children nodes", "type": "text" }, { "bbox": [ 207, 415, 226, 430 ], "score": 0.92, "content": "\\mathbb { C } _ { \\ell } ^ { ( s ) }", "type": "inline_equation" }, { "bbox": [ 226, 412, 507, 432 ], "score": 1.0, "content": ". As demonstrated in Figure 3, several convolution layers with kernel", "type": "text" } ], "index": 24 }, { "bbox": [ 105, 428, 505, 440 ], "spans": [ { "bbox": [ 105, 428, 125, 440 ], "score": 1.0, "content": "size", "type": "text" }, { "bbox": [ 125, 429, 135, 438 ], "score": 0.8, "content": "C", "type": "inline_equation" }, { "bbox": [ 135, 428, 179, 440 ], "score": 1.0, "content": "and stride", "type": "text" }, { "bbox": [ 179, 429, 189, 438 ], "score": 0.8, "content": "C", "type": "inline_equation" }, { "bbox": [ 189, 428, 505, 440 ], "score": 1.0, "content": "are sequentially applied to the embedded sequence in the dimension of time,", "type": "text" } ], "index": 25 }, { "bbox": [ 106, 439, 506, 452 ], "spans": [ { "bbox": [ 106, 439, 237, 452 ], "score": 1.0, "content": "yielding a sequence with length", "type": "text" }, { "bbox": [ 238, 439, 262, 451 ], "score": 0.91, "content": "L / C ^ { s }", "type": "inline_equation" }, { "bbox": [ 263, 439, 297, 452 ], "score": 1.0, "content": "at scale", "type": "text" }, { "bbox": [ 298, 442, 304, 449 ], "score": 0.59, "content": "s", "type": "inline_equation" }, { "bbox": [ 304, 439, 506, 452 ], "score": 1.0, "content": ". The resulting sequences at different scales form", "type": "text" } ], "index": 26 }, { "bbox": [ 105, 450, 506, 462 ], "spans": [ { "bbox": [ 105, 450, 114, 462 ], "score": 1.0, "content": "a", "type": "text" }, { "bbox": [ 114, 451, 123, 460 ], "score": 0.84, "content": "C", "type": "inline_equation" }, { "bbox": [ 123, 450, 506, 462 ], "score": 1.0, "content": "-ary tree. We concatenate these fine-to-coarse sequences before inputting them to the PAM. In", "type": "text" } ], "index": 27 }, { "bbox": [ 106, 461, 505, 474 ], "spans": [ { "bbox": [ 106, 461, 505, 474 ], "score": 1.0, "content": "order to reduce the amount of parameters and calculations, we reduce the dimension of each node by", "type": "text" } ], "index": 28 }, { "bbox": [ 105, 472, 505, 485 ], "spans": [ { "bbox": [ 105, 472, 505, 485 ], "score": 1.0, "content": "a fully connected layer before inputting the sequence into the stacked convolution layers and restore", "type": "text" } ], "index": 29 }, { "bbox": [ 105, 482, 506, 496 ], "spans": [ { "bbox": [ 105, 482, 506, 496 ], "score": 1.0, "content": "it after all convolutions. Such a bottleneck structure significantly reduces the number of parameters", "type": "text" } ], "index": 30 }, { "bbox": [ 105, 493, 301, 507 ], "spans": [ { "bbox": [ 105, 493, 301, 507 ], "score": 1.0, "content": "in the module and can guard against over-fitting.", "type": "text" } ], "index": 31 } ], "index": 26, "bbox_fs": [ 103, 381, 507, 507 ] }, { "type": "title", "bbox": [ 108, 519, 226, 531 ], "lines": [ { "bbox": [ 105, 518, 228, 532 ], "spans": [ { "bbox": [ 105, 518, 228, 532 ], "score": 1.0, "content": "3.3 PREDICTION MODULE", "type": "text" } ], "index": 32 } ], "index": 32 }, { "type": "text", "bbox": [ 107, 539, 505, 583 ], "lines": [ { "bbox": [ 105, 539, 505, 552 ], "spans": [ { "bbox": [ 105, 539, 351, 552 ], "score": 1.0, "content": "For single-step forecasting, we add an end token (by setting", "type": "text" }, { "bbox": [ 351, 540, 391, 551 ], "score": 0.9, "content": "z _ { t + 1 } = 0", "type": "inline_equation" }, { "bbox": [ 392, 539, 505, 552 ], "score": 1.0, "content": ") to the end of the historical", "type": "text" } ], "index": 33 }, { "bbox": [ 105, 550, 505, 562 ], "spans": [ { "bbox": [ 105, 550, 145, 562 ], "score": 1.0, "content": "sequence", "type": "text" }, { "bbox": [ 145, 551, 181, 562 ], "score": 0.89, "content": "z _ { t - L + 1 : t }", "type": "inline_equation" }, { "bbox": [ 182, 550, 505, 562 ], "score": 1.0, "content": "before inputting it into the embedding layer. After the sequence is encoded by the", "type": "text" } ], "index": 34 }, { "bbox": [ 105, 560, 505, 574 ], "spans": [ { "bbox": [ 105, 560, 505, 574 ], "score": 1.0, "content": "PAM, we gather the features given by the last nodes at all scales in the pyramidal graph, concatenate", "type": "text" } ], "index": 35 }, { "bbox": [ 106, 573, 359, 585 ], "spans": [ { "bbox": [ 106, 573, 359, 585 ], "score": 1.0, "content": "and then input them into a fully connected layer for prediction.", "type": "text" } ], "index": 36 } ], "index": 34.5, "bbox_fs": [ 105, 539, 505, 585 ] }, { "type": "text", "bbox": [ 107, 589, 505, 732 ], "lines": [ { "bbox": [ 105, 588, 506, 601 ], "spans": [ { "bbox": [ 105, 588, 506, 601 ], "score": 1.0, "content": "For multi-step forecasting, we propose two prediction modules. The first one is the same with the", "type": "text" } ], "index": 37 }, { "bbox": [ 106, 600, 505, 612 ], "spans": [ { "bbox": [ 106, 600, 410, 612 ], "score": 1.0, "content": "single-step forecasting module, but maps the last nodes at all scales to all", "type": "text" }, { "bbox": [ 410, 600, 422, 610 ], "score": 0.77, "content": "M", "type": "inline_equation" }, { "bbox": [ 422, 600, 505, 612 ], "score": 1.0, "content": "future time steps in", "type": "text" } ], "index": 38 }, { "bbox": [ 105, 612, 504, 623 ], "spans": [ { "bbox": [ 105, 612, 504, 623 ], "score": 1.0, "content": "a batch. The second one, on the other hand, resorts to a decoder with two full attention layers.", "type": "text" } ], "index": 39 }, { "bbox": [ 105, 622, 505, 634 ], "spans": [ { "bbox": [ 105, 622, 505, 634 ], "score": 1.0, "content": "Specifically, similar to the original Transformer (Vaswani et al., 2017), we replace the observations", "type": "text" } ], "index": 40 }, { "bbox": [ 105, 632, 505, 645 ], "spans": [ { "bbox": [ 105, 632, 157, 645 ], "score": 1.0, "content": "at the future", "type": "text" }, { "bbox": [ 157, 633, 169, 643 ], "score": 0.7, "content": "M", "type": "inline_equation" }, { "bbox": [ 170, 632, 505, 645 ], "score": 1.0, "content": "time steps with 0, embed them in the same manner with the historical observations,", "type": "text" } ], "index": 41 }, { "bbox": [ 105, 643, 505, 657 ], "spans": [ { "bbox": [ 105, 643, 505, 657 ], "score": 1.0, "content": "and refer to the summation of the observation, covariate, and positional embedding as the “prediction", "type": "text" } ], "index": 42 }, { "bbox": [ 105, 654, 506, 667 ], "spans": [ { "bbox": [ 105, 654, 135, 667 ], "score": 1.0, "content": "token”", "type": "text" }, { "bbox": [ 136, 655, 148, 667 ], "score": 0.88, "content": "F _ { p }", "type": "inline_equation" }, { "bbox": [ 149, 654, 379, 667 ], "score": 1.0, "content": ". The first attention layer then takes the prediction tokens", "type": "text" }, { "bbox": [ 380, 655, 392, 667 ], "score": 0.9, "content": "F _ { p }", "type": "inline_equation" }, { "bbox": [ 393, 654, 506, 667 ], "score": 1.0, "content": "as the query and the output", "type": "text" } ], "index": 43 }, { "bbox": [ 105, 664, 506, 678 ], "spans": [ { "bbox": [ 105, 664, 166, 678 ], "score": 1.0, "content": "of the encoder", "type": "text" }, { "bbox": [ 167, 666, 179, 677 ], "score": 0.87, "content": "\\pmb { F _ { e } }", "type": "inline_equation" }, { "bbox": [ 179, 664, 435, 678 ], "score": 1.0, "content": "(i.e., all nodes in the PAM) as the key and the value, and yields", "type": "text" }, { "bbox": [ 435, 666, 452, 677 ], "score": 0.92, "content": "{ \\bf { { F } } } _ { d 1 }", "type": "inline_equation" }, { "bbox": [ 452, 664, 506, 678 ], "score": 1.0, "content": ". 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The", "type": "text" } ], "index": 45 }, { "bbox": [ 105, 687, 505, 700 ], "spans": [ { "bbox": [ 105, 687, 197, 700 ], "score": 1.0, "content": "historical information", "type": "text" }, { "bbox": [ 198, 688, 209, 699 ], "score": 0.89, "content": "\\pmb { F _ { e } }", "type": "inline_equation" }, { "bbox": [ 210, 687, 505, 700 ], "score": 1.0, "content": "is fed directly into both attention layers, since such information is vital", "type": "text" } ], "index": 46 }, { "bbox": [ 105, 699, 505, 711 ], "spans": [ { "bbox": [ 105, 699, 505, 711 ], "score": 1.0, "content": "for accurate long-range forecasting. The final prediction is then obtained through a fully connected", "type": "text" } ], "index": 47 }, { "bbox": [ 105, 709, 506, 722 ], "spans": [ { "bbox": [ 105, 709, 506, 722 ], "score": 1.0, "content": "layer across the dimension of channels. Again, we output all future predictions together to avoid the", "type": "text" } ], "index": 48 }, { "bbox": [ 105, 720, 414, 733 ], "spans": [ { "bbox": [ 105, 720, 414, 733 ], "score": 1.0, "content": "problem of error accumulation in the autoregressive decoder of Transformer.", "type": "text" } ], "index": 49 } ], "index": 43, "bbox_fs": [ 105, 588, 506, 733 ] } ] }, { "preproc_blocks": [ { "type": "table", "bbox": [ 136, 153, 474, 394 ], "blocks": [ { "type": "table_caption", "bbox": [ 107, 89, 504, 145 ], "group_id": 0, "lines": [ { "bbox": [ 105, 88, 505, 102 ], "spans": [ { "bbox": [ 105, 88, 505, 102 ], "score": 1.0, "content": "Table 2: Single-step forecasting results on three datasets. “Q-K pairs” refer to the number of query-", "type": "text" } ], "index": 0 }, { "bbox": [ 105, 100, 506, 114 ], "spans": [ { "bbox": [ 105, 100, 506, 114 ], "score": 1.0, "content": "key dot products performed by all attention layers in the network, which encodes the time and space", "type": "text" } ], "index": 1 }, { "bbox": [ 105, 111, 506, 124 ], "spans": [ { "bbox": [ 105, 111, 329, 124 ], "score": 1.0, "content": "complexity. 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MethodsParametersDatasetsNRMSENDQ-K pairs
Full-attentionO(N(HDDK+DDF))Electricity0.3280.041456976
Wind0.1750.082589824
App Flow0.4070.080589824
LogTransO(N(HDDK+DDF))Electricity0.3330.04150138
Wind0.1730.08158272
App Flow0.3870.07358272
ReformerO(N(HDDK+DDF))Electricity0.3590.047677376
Wind0.1830.086884736
AppFlow0.4630.095884736
ETCO(N(HDDK+DDF))Electricity0.3240.04179536
Wind0.1670.074102144
App Flow0.3970.069102144
LongformerO(N(HDDK+DDF))Electricity0.3300.04141360
Wind0.1660.07552608
AppFlow0.3770.0752608
PyraformerO(N(HDDK+DDF) +(S-1)CD²)Electricity0.3240.04117648
Wind0.1610.07220176
App Flow0.3660.06720176
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The first three datasets were used for single-step", "type": "text" } ], "index": 11 }, { "bbox": [ 105, 468, 505, 481 ], "spans": [ { "bbox": [ 105, 468, 505, 481 ], "score": 1.0, "content": "forecasting, while the last two for long-range multi-step forecasting. We refer the readers to Ap-", "type": "text" } ], "index": 12 }, { "bbox": [ 105, 479, 461, 493 ], "spans": [ { "bbox": [ 105, 479, 461, 493 ], "score": 1.0, "content": "pendix E and F for more details regarding the data description and the experiment setup.", "type": "text" } ], "index": 13 } ], "index": 11.5 }, { "type": "title", "bbox": [ 107, 503, 238, 514 ], "lines": [ { "bbox": [ 105, 501, 239, 516 ], "spans": [ { "bbox": [ 105, 501, 239, 516 ], "score": 1.0, "content": "4.2 RESULTS AND ANALYSIS", "type": "text" } ], "index": 14 } ], "index": 14 }, { "type": "title", "bbox": [ 108, 522, 261, 533 ], "lines": [ { "bbox": [ 106, 522, 262, 533 ], "spans": [ { "bbox": [ 106, 522, 262, 533 ], "score": 1.0, "content": "4.2.1 SINGLE-STEP FORECASTING", "type": "text" } ], "index": 15 } ], "index": 15 }, { "type": "text", "bbox": [ 106, 540, 505, 704 ], "lines": [ { "bbox": [ 106, 539, 505, 552 ], "spans": [ { "bbox": [ 106, 539, 505, 552 ], "score": 1.0, "content": "We conducted single-step prediction experiments on three datasets: Electricity, Wind and App Flow.", "type": "text" } ], "index": 16 }, { "bbox": [ 106, 550, 505, 563 ], "spans": [ { "bbox": [ 106, 550, 505, 563 ], "score": 1.0, "content": "The historical length is 169, 192 and 192, respectively, including the end token. We benchmarked", "type": "text" } ], "index": 17 }, { "bbox": [ 105, 561, 505, 574 ], "spans": [ { "bbox": [ 105, 561, 505, 574 ], "score": 1.0, "content": "Pyraformer against 5 other attention mechanisms, including the original full-attention (Vaswani", "type": "text" } ], "index": 18 }, { "bbox": [ 105, 572, 505, 585 ], "spans": [ { "bbox": [ 105, 572, 505, 585 ], "score": 1.0, "content": "et al., 2017), the log-sparse attention (i.e., LogTrans) (Li et al., 2019), the LSH attention (i.e., Re-", "type": "text" } ], "index": 19 }, { "bbox": [ 105, 583, 505, 595 ], "spans": [ { "bbox": [ 105, 583, 505, 595 ], "score": 1.0, "content": "former) (Kitaev et al., 2019), the sliding window attention with global nodes (i.e., ETC) (Ainslie", "type": "text" } ], "index": 20 }, { "bbox": [ 105, 594, 505, 606 ], "spans": [ { "bbox": [ 105, 594, 505, 606 ], "score": 1.0, "content": "et al., 2020), and the dilated sliding window attention (i.e., Longformer) (Beltagy et al., 2020). In", "type": "text" } ], "index": 21 }, { "bbox": [ 104, 604, 506, 617 ], "spans": [ { "bbox": [ 104, 604, 506, 617 ], "score": 1.0, "content": "particular for ETC, some nodes with equal intervals at the finest scale were selected as the global", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 616, 506, 628 ], "spans": [ { "bbox": [ 105, 616, 506, 628 ], "score": 1.0, "content": "nodes. A global node can attend to all nodes across the sequence and all nodes can attend to it", "type": "text" } ], "index": 23 }, { "bbox": [ 105, 627, 505, 639 ], "spans": [ { "bbox": [ 105, 627, 505, 639 ], "score": 1.0, "content": "in turn(see Figure 1(e)). The training and testing schemes were the same for all models. We fur-", "type": "text" } ], "index": 24 }, { "bbox": [ 106, 638, 505, 651 ], "spans": [ { "bbox": [ 106, 638, 505, 651 ], "score": 1.0, "content": "ther investigated the usefulness of the pretraining strategy (see Appendix G), the weighted sampler,", "type": "text" } ], "index": 25 }, { "bbox": [ 105, 649, 505, 662 ], "spans": [ { "bbox": [ 105, 649, 505, 662 ], "score": 1.0, "content": "and the hard sample mining on all methods, and the best results were presented. We adopted the", "type": "text" } ], "index": 26 }, { "bbox": [ 105, 659, 505, 673 ], "spans": [ { "bbox": [ 105, 659, 505, 673 ], "score": 1.0, "content": "NRMSE (Normalized RMSE) and the ND (Normalized Deviation) as the evaluation indicators (see", "type": "text" } ], "index": 27 }, { "bbox": [ 106, 671, 504, 683 ], "spans": [ { "bbox": [ 106, 671, 504, 683 ], "score": 1.0, "content": "Appendix H for the definitions). The results are summarized in Table 2. For a fair comparison,", "type": "text" } ], "index": 28 }, { "bbox": [ 105, 682, 505, 694 ], "spans": [ { "bbox": [ 105, 682, 505, 694 ], "score": 1.0, "content": "except for full-attention, the overall dot product number of all attention mechanisms was controlled", "type": "text" } ], "index": 29 }, { "bbox": [ 105, 693, 236, 705 ], "spans": [ { "bbox": [ 105, 693, 236, 705 ], "score": 1.0, "content": "to the same order of magnitude.", "type": "text" } ], "index": 30 } ], "index": 23 }, { "type": "text", "bbox": [ 106, 709, 503, 732 ], "lines": [ { "bbox": [ 106, 709, 505, 722 ], "spans": [ { "bbox": [ 106, 709, 505, 722 ], "score": 1.0, "content": "Our experimental results show that Pyraformer outperforms Transformer and its variants in terms", "type": "text" } ], "index": 31 }, { "bbox": [ 105, 720, 505, 733 ], "spans": [ { "bbox": [ 105, 720, 505, 733 ], "score": 1.0, "content": "of NRMSE and ND, with the least number of query-key dot products (a.k.a. Q-K pairs). Con-", "type": "text" } ], "index": 32 } ], "index": 31.5 } ], "page_idx": 6, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 108, 27, 293, 37 ], "lines": [ { "bbox": [ 106, 26, 294, 38 ], "spans": [ { "bbox": [ 106, 26, 294, 38 ], "score": 1.0, "content": "Published as a conference paper at ICLR 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": "7", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "table", "bbox": [ 136, 153, 474, 394 ], "blocks": [ { "type": "table_caption", "bbox": [ 107, 89, 504, 145 ], "group_id": 0, "lines": [ { "bbox": [ 105, 88, 505, 102 ], "spans": [ { "bbox": [ 105, 88, 505, 102 ], "score": 1.0, "content": "Table 2: Single-step forecasting results on three datasets. “Q-K pairs” refer to the number of query-", "type": "text" } ], "index": 0 }, { "bbox": [ 105, 100, 506, 114 ], "spans": [ { "bbox": [ 105, 100, 506, 114 ], "score": 1.0, "content": "key dot products performed by all attention layers in the network, which encodes the time and space", "type": "text" } ], "index": 1 }, { "bbox": [ 105, 111, 506, 124 ], "spans": [ { "bbox": [ 105, 111, 329, 124 ], "score": 1.0, "content": "complexity. We write the number of attention layers by", "type": "text" }, { "bbox": [ 329, 111, 339, 121 ], "score": 0.77, "content": "N", "type": "inline_equation" }, { "bbox": [ 340, 111, 476, 124 ], "score": 1.0, "content": ", the number of attention heads by", "type": "text" }, { "bbox": [ 477, 111, 487, 121 ], "score": 0.77, "content": "H", "type": "inline_equation" }, { "bbox": [ 487, 111, 506, 124 ], "score": 1.0, "content": ", the", "type": "text" } ], "index": 2 }, { "bbox": [ 105, 121, 506, 135 ], "spans": [ { "bbox": [ 105, 121, 188, 135 ], "score": 1.0, "content": "number of scales by", "type": "text" }, { "bbox": [ 188, 123, 196, 132 ], "score": 0.75, "content": "S", "type": "inline_equation" }, { "bbox": [ 196, 121, 309, 135 ], "score": 1.0, "content": ", the dimension of a node by", "type": "text" }, { "bbox": [ 309, 123, 319, 132 ], "score": 0.77, "content": "D", "type": "inline_equation" }, { "bbox": [ 319, 121, 426, 135 ], "score": 1.0, "content": ", the dimension of a key by", "type": "text" }, { "bbox": [ 427, 122, 444, 133 ], "score": 0.9, "content": "D _ { K }", "type": "inline_equation" }, { "bbox": [ 444, 121, 506, 135 ], "score": 1.0, "content": ", the maximum", "type": "text" } ], "index": 3 }, { "bbox": [ 106, 133, 401, 146 ], "spans": [ { "bbox": [ 106, 134, 250, 146 ], "score": 1.0, "content": "dimension of feed-forward layer by", "type": "text" }, { "bbox": [ 251, 133, 266, 144 ], "score": 0.89, "content": "D _ { F }", "type": "inline_equation" }, { "bbox": [ 267, 134, 388, 146 ], "score": 1.0, "content": ", and the convolution stride by", "type": "text" }, { "bbox": [ 389, 134, 398, 144 ], "score": 0.75, "content": "C", "type": "inline_equation" }, { "bbox": [ 398, 134, 401, 146 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 4 } ], "index": 2 }, { "type": "table_body", "bbox": [ 136, 153, 474, 394 ], "group_id": 0, "lines": [ { "bbox": [ 136, 153, 474, 394 ], "spans": [ { "bbox": [ 136, 153, 474, 394 ], "score": 0.986, "html": "
MethodsParametersDatasetsNRMSENDQ-K pairs
Full-attentionO(N(HDDK+DDF))Electricity0.3280.041456976
Wind0.1750.082589824
App Flow0.4070.080589824
LogTransO(N(HDDK+DDF))Electricity0.3330.04150138
Wind0.1730.08158272
App Flow0.3870.07358272
ReformerO(N(HDDK+DDF))Electricity0.3590.047677376
Wind0.1830.086884736
AppFlow0.4630.095884736
ETCO(N(HDDK+DDF))Electricity0.3240.04179536
Wind0.1670.074102144
App Flow0.3970.069102144
LongformerO(N(HDDK+DDF))Electricity0.3300.04141360
Wind0.1660.07552608
AppFlow0.3770.0752608
PyraformerO(N(HDDK+DDF) +(S-1)CD²)Electricity0.3240.04117648
Wind0.1610.07220176
App Flow0.3660.06720176
", "type": "table", "image_path": "c4e321b0e63f0158f9b19bfa056ecdd39d38f0131f3e85e90ddfeae455ab70aa.jpg" } ] } ], "index": 6, "virtual_lines": [ { "bbox": [ 136, 153, 474, 233.33333333333331 ], "spans": [], "index": 5 }, { "bbox": [ 136, 233.33333333333331, 474, 313.66666666666663 ], "spans": [], "index": 6 }, { "bbox": [ 136, 313.66666666666663, 474, 393.99999999999994 ], "spans": [], "index": 7 } ] } ], "index": 4.0 }, { "type": "title", "bbox": [ 107, 403, 201, 416 ], "lines": [ { "bbox": [ 105, 403, 202, 418 ], "spans": [ { "bbox": [ 105, 403, 202, 418 ], "score": 1.0, "content": "4 EXPERIMENTS", "type": "text" } ], "index": 8 } ], "index": 8 }, { "type": "title", "bbox": [ 108, 427, 287, 439 ], "lines": [ { "bbox": [ 106, 427, 288, 439 ], "spans": [ { "bbox": [ 106, 427, 288, 439 ], "score": 1.0, "content": "4.1 DATASETS AND EXPERIMENT SETUP", "type": "text" } ], "index": 9 } ], "index": 9 }, { "type": "text", "bbox": [ 107, 446, 505, 491 ], "lines": [ { "bbox": [ 105, 445, 505, 459 ], "spans": [ { "bbox": [ 105, 445, 505, 459 ], "score": 1.0, "content": "We demonstrated the advantages of the proposed Pyraformer on the four real-world datasets, in-", "type": "text" } ], "index": 10 }, { "bbox": [ 105, 457, 505, 470 ], "spans": [ { "bbox": [ 105, 457, 505, 470 ], "score": 1.0, "content": "cluding Wind, App Flow, Electricity, and ETT. The first three datasets were used for single-step", "type": "text" } ], "index": 11 }, { "bbox": [ 105, 468, 505, 481 ], "spans": [ { "bbox": [ 105, 468, 505, 481 ], "score": 1.0, "content": "forecasting, while the last two for long-range multi-step forecasting. We refer the readers to Ap-", "type": "text" } ], "index": 12 }, { "bbox": [ 105, 479, 461, 493 ], "spans": [ { "bbox": [ 105, 479, 461, 493 ], "score": 1.0, "content": "pendix E and F for more details regarding the data description and the experiment setup.", "type": "text" } ], "index": 13 } ], "index": 11.5, "bbox_fs": [ 105, 445, 505, 493 ] }, { "type": "title", "bbox": [ 107, 503, 238, 514 ], "lines": [ { "bbox": [ 105, 501, 239, 516 ], "spans": [ { "bbox": [ 105, 501, 239, 516 ], "score": 1.0, "content": "4.2 RESULTS AND ANALYSIS", "type": "text" } ], "index": 14 } ], "index": 14 }, { "type": "title", "bbox": [ 108, 522, 261, 533 ], "lines": [ { "bbox": [ 106, 522, 262, 533 ], "spans": [ { "bbox": [ 106, 522, 262, 533 ], "score": 1.0, "content": "4.2.1 SINGLE-STEP FORECASTING", "type": "text" } ], "index": 15 } ], "index": 15 }, { "type": "text", "bbox": [ 106, 540, 505, 704 ], "lines": [ { "bbox": [ 106, 539, 505, 552 ], "spans": [ { "bbox": [ 106, 539, 505, 552 ], "score": 1.0, "content": "We conducted single-step prediction experiments on three datasets: Electricity, Wind and App Flow.", "type": "text" } ], "index": 16 }, { "bbox": [ 106, 550, 505, 563 ], "spans": [ { "bbox": [ 106, 550, 505, 563 ], "score": 1.0, "content": "The historical length is 169, 192 and 192, respectively, including the end token. We benchmarked", "type": "text" } ], "index": 17 }, { "bbox": [ 105, 561, 505, 574 ], "spans": [ { "bbox": [ 105, 561, 505, 574 ], "score": 1.0, "content": "Pyraformer against 5 other attention mechanisms, including the original full-attention (Vaswani", "type": "text" } ], "index": 18 }, { "bbox": [ 105, 572, 505, 585 ], "spans": [ { "bbox": [ 105, 572, 505, 585 ], "score": 1.0, "content": "et al., 2017), the log-sparse attention (i.e., LogTrans) (Li et al., 2019), the LSH attention (i.e., Re-", "type": "text" } ], "index": 19 }, { "bbox": [ 105, 583, 505, 595 ], "spans": [ { "bbox": [ 105, 583, 505, 595 ], "score": 1.0, "content": "former) (Kitaev et al., 2019), the sliding window attention with global nodes (i.e., ETC) (Ainslie", "type": "text" } ], "index": 20 }, { "bbox": [ 105, 594, 505, 606 ], "spans": [ { "bbox": [ 105, 594, 505, 606 ], "score": 1.0, "content": "et al., 2020), and the dilated sliding window attention (i.e., Longformer) (Beltagy et al., 2020). In", "type": "text" } ], "index": 21 }, { "bbox": [ 104, 604, 506, 617 ], "spans": [ { "bbox": [ 104, 604, 506, 617 ], "score": 1.0, "content": "particular for ETC, some nodes with equal intervals at the finest scale were selected as the global", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 616, 506, 628 ], "spans": [ { "bbox": [ 105, 616, 506, 628 ], "score": 1.0, "content": "nodes. A global node can attend to all nodes across the sequence and all nodes can attend to it", "type": "text" } ], "index": 23 }, { "bbox": [ 105, 627, 505, 639 ], "spans": [ { "bbox": [ 105, 627, 505, 639 ], "score": 1.0, "content": "in turn(see Figure 1(e)). The training and testing schemes were the same for all models. We fur-", "type": "text" } ], "index": 24 }, { "bbox": [ 106, 638, 505, 651 ], "spans": [ { "bbox": [ 106, 638, 505, 651 ], "score": 1.0, "content": "ther investigated the usefulness of the pretraining strategy (see Appendix G), the weighted sampler,", "type": "text" } ], "index": 25 }, { "bbox": [ 105, 649, 505, 662 ], "spans": [ { "bbox": [ 105, 649, 505, 662 ], "score": 1.0, "content": "and the hard sample mining on all methods, and the best results were presented. We adopted the", "type": "text" } ], "index": 26 }, { "bbox": [ 105, 659, 505, 673 ], "spans": [ { "bbox": [ 105, 659, 505, 673 ], "score": 1.0, "content": "NRMSE (Normalized RMSE) and the ND (Normalized Deviation) as the evaluation indicators (see", "type": "text" } ], "index": 27 }, { "bbox": [ 106, 671, 504, 683 ], "spans": [ { "bbox": [ 106, 671, 504, 683 ], "score": 1.0, "content": "Appendix H for the definitions). The results are summarized in Table 2. For a fair comparison,", "type": "text" } ], "index": 28 }, { "bbox": [ 105, 682, 505, 694 ], "spans": [ { "bbox": [ 105, 682, 505, 694 ], "score": 1.0, "content": "except for full-attention, the overall dot product number of all attention mechanisms was controlled", "type": "text" } ], "index": 29 }, { "bbox": [ 105, 693, 236, 705 ], "spans": [ { "bbox": [ 105, 693, 236, 705 ], "score": 1.0, "content": "to the same order of magnitude.", "type": "text" } ], "index": 30 } ], "index": 23, "bbox_fs": [ 104, 539, 506, 705 ] }, { "type": "text", "bbox": [ 106, 709, 503, 732 ], "lines": [ { "bbox": [ 106, 709, 505, 722 ], "spans": [ { "bbox": [ 106, 709, 505, 722 ], "score": 1.0, "content": "Our experimental results show that Pyraformer outperforms Transformer and its variants in terms", "type": "text" } ], "index": 31 }, { "bbox": [ 105, 720, 505, 733 ], "spans": [ { "bbox": [ 105, 720, 505, 733 ], "score": 1.0, "content": "of NRMSE and ND, with the least number of query-key dot products (a.k.a. Q-K pairs). Con-", "type": "text" } ], "index": 32 }, { "bbox": [ 106, 348, 505, 361 ], "spans": [ { "bbox": [ 106, 348, 505, 361 ], "score": 1.0, "content": "cretely, there are three major trends that can be gleaned from Table 2: (1) The proposed Pyraformer", "type": "text", "cross_page": true } ], "index": 4 }, { "bbox": [ 105, 360, 505, 372 ], "spans": [ { "bbox": [ 105, 360, 505, 372 ], "score": 1.0, "content": "yields the most accurate prediction results, suggesting that the pyramidal graph can better explain", "type": "text", "cross_page": true } ], "index": 5 }, { "bbox": [ 106, 370, 505, 383 ], "spans": [ { "bbox": [ 106, 370, 505, 383 ], "score": 1.0, "content": "the temporal interactions in the time series by considering dependencies of different ranges. Inter-", "type": "text", "cross_page": true } ], "index": 6 }, { "bbox": [ 106, 382, 504, 393 ], "spans": [ { "bbox": [ 106, 382, 504, 393 ], "score": 1.0, "content": "estingly, for the Wind dataset, sparse attention mechanisms, namely, LogTrans, ETC, Longformer", "type": "text", "cross_page": true } ], "index": 7 }, { "bbox": [ 106, 392, 505, 405 ], "spans": [ { "bbox": [ 106, 392, 505, 405 ], "score": 1.0, "content": "and Pyraformer, outperform the original full attention Transformer, probably because the data con-", "type": "text", "cross_page": true } ], "index": 8 }, { "bbox": [ 105, 403, 505, 416 ], "spans": [ { "bbox": [ 105, 403, 505, 416 ], "score": 1.0, "content": "tains a large number of zeros and the promotion of adequate sparsity can help avoid over-fitting.", "type": "text", "cross_page": true } ], "index": 9 }, { "bbox": [ 105, 414, 505, 427 ], "spans": [ { "bbox": [ 105, 414, 505, 427 ], "score": 1.0, "content": "(2) The number of Q-K pairs in Pyraformer is the smallest. Recall that this number character-", "type": "text", "cross_page": true } ], "index": 10 }, { "bbox": [ 105, 425, 505, 438 ], "spans": [ { "bbox": [ 105, 425, 360, 438 ], "score": 1.0, "content": "izes the time and space complexity. Remarkably enough, it is", "type": "text", "cross_page": true }, { "bbox": [ 360, 425, 388, 436 ], "score": 0.88, "content": "6 5 . 4 \\%", "type": "inline_equation", "cross_page": true }, { "bbox": [ 388, 425, 505, 438 ], "score": 1.0, "content": "fewer than that of LogTrans", "type": "text", "cross_page": true } ], "index": 11 }, { "bbox": [ 105, 435, 505, 449 ], "spans": [ { "bbox": [ 105, 435, 124, 449 ], "score": 1.0, "content": "and", "type": "text", "cross_page": true }, { "bbox": [ 124, 436, 153, 447 ], "score": 0.88, "content": "9 6 . 6 \\%", "type": "inline_equation", "cross_page": true }, { "bbox": [ 153, 435, 505, 449 ], "score": 1.0, "content": "than that of the full attention. It is worth emphasizing that this computational gain", "type": "text", "cross_page": true } ], "index": 12 }, { "bbox": [ 105, 446, 506, 460 ], "spans": [ { "bbox": [ 105, 446, 506, 460 ], "score": 1.0, "content": "will continue to increase for longer time series. (3) The number of parameters for Pyraformer is", "type": "text", "cross_page": true } ], "index": 13 }, { "bbox": [ 105, 456, 505, 472 ], "spans": [ { "bbox": [ 105, 456, 505, 472 ], "score": 1.0, "content": "slightly larger than that of the other models, resulting from the CSCM. However, this module is very", "type": "text", "cross_page": true } ], "index": 14 }, { "bbox": [ 106, 469, 504, 480 ], "spans": [ { "bbox": [ 106, 469, 243, 480 ], "score": 1.0, "content": "lightweight, which incurs merely", "type": "text", "cross_page": true }, { "bbox": [ 244, 469, 259, 480 ], "score": 0.86, "content": "5 \\%", "type": "inline_equation", "cross_page": true }, { "bbox": [ 259, 469, 504, 480 ], "score": 1.0, "content": "overhead in terms of model size compared to other models.", "type": "text", "cross_page": true } ], "index": 15 }, { "bbox": [ 105, 479, 505, 492 ], "spans": [ { "bbox": [ 105, 479, 336, 492 ], "score": 1.0, "content": "Moreover, in practice, we can fix the hyper-parameters", "type": "text", "cross_page": true }, { "bbox": [ 336, 480, 345, 490 ], "score": 0.54, "content": "A", "type": "inline_equation", "cross_page": true }, { "bbox": [ 345, 479, 349, 492 ], "score": 1.0, "content": ",", "type": "text", "cross_page": true }, { "bbox": [ 350, 480, 358, 490 ], "score": 0.67, "content": "S", "type": "inline_equation", "cross_page": true }, { "bbox": [ 358, 479, 378, 492 ], "score": 1.0, "content": "and", "type": "text", "cross_page": true }, { "bbox": [ 378, 480, 388, 490 ], "score": 0.8, "content": "N", "type": "inline_equation", "cross_page": true }, { "bbox": [ 389, 479, 459, 492 ], "score": 1.0, "content": ", and ensure that", "type": "text", "cross_page": true }, { "bbox": [ 460, 480, 469, 490 ], "score": 0.8, "content": "C", "type": "inline_equation", "cross_page": true }, { "bbox": [ 469, 479, 505, 492 ], "score": 1.0, "content": "satisfies", "type": "text", "cross_page": true } ], "index": 16 }, { "bbox": [ 106, 490, 505, 506 ], "spans": [ { "bbox": [ 106, 491, 238, 505 ], "score": 0.91, "content": "C > \\sqrt [ s - 1 ] { L / ( ( A - 1 ) N / 2 + 1 ) }", "type": "inline_equation", "cross_page": true }, { "bbox": [ 239, 490, 505, 506 ], "score": 1.0, "content": ". Consequently, the extra number of parameters introduced by the", "type": "text", "cross_page": true } ], "index": 17 }, { "bbox": [ 105, 504, 297, 519 ], "spans": [ { "bbox": [ 105, 504, 166, 519 ], "score": 1.0, "content": "CSCM is only", "type": "text", "cross_page": true }, { "bbox": [ 167, 504, 293, 518 ], "score": 0.89, "content": "\\mathcal { O } ( ( S - 1 ) C D _ { K } ^ { 2 } ) \\approx \\mathcal { O } ( \\sqrt [ s ] { L } )", "type": "inline_equation", "cross_page": true }, { "bbox": [ 294, 504, 297, 519 ], "score": 1.0, "content": ".", "type": "text", "cross_page": true } ], "index": 18 } ], "index": 31.5, "bbox_fs": [ 105, 709, 505, 733 ] } ] }, { "preproc_blocks": [ { "type": "table", "bbox": [ 107, 103, 505, 329 ], "blocks": [ { "type": "table_caption", "bbox": [ 203, 90, 407, 101 ], "group_id": 0, "lines": [ { "bbox": [ 202, 87, 407, 104 ], "spans": [ { "bbox": [ 202, 87, 407, 104 ], "score": 1.0, "content": "Table 3: Long-range multi-step forecasting results.", "type": "text" } ], "index": 0 } ], "index": 0 }, { "type": "table_body", "bbox": [ 107, 103, 505, 329 ], "group_id": 0, "lines": [ { "bbox": [ 107, 103, 505, 329 ], "spans": [ { "bbox": [ 107, 103, 505, 329 ], "score": 0.986, "html": "
MethodsMetricsETTh1ETTm1Electricity
16833672096288672168336720
InformerMSE1.0751.3291.3840.5560.8410.9210.7451.5794.365
MAE0.8010.9110.9500.5370.7050.7530.2660.3230.371
Q-K pairs188040188040423360276480560640560640188040188040423360
LogTransMSE0.9831.1001.4110.5540.7861.1690.7911.5844.362
MAE0.7660.8390.9910.4990.6760.8680.3400.3360.366
Q-K pairs74664746642167442547606487686487687466474664216744
LongformerMSE0.8600.9751.0910.5260.7671.0210.7661.5914.361
MAE0.7100.7690.8320.5070.6630.7880.3110.3430.368
Q-K pairs6364863648249120329760100713610071366364863648249120
ReformerMSE0.9581.0441.4580.5430.9240.9810.7831.5844.374
MAE0.7410.7870.9870.5280.7220.7780.3320.3340.374
Q-K pairs10160641016064270950453084161445068814450688101606410160642709504
ETCMSE1.0251.0841.1370.7621.2271.2720.7771.5864.361
MAE0.7710.8110.8660.6530.8800.9080.3260.3400.368
Q-K pairs125280125280288720331344836952836952125280125280288720
PyraformerMSE0.8080.9451.0220.4800.7540.8570.7191.5334.312
MAE0.6830.7660.8060.4860.6590.7070.2560.2910.346
Q-K pairs264722647274280572649638496384264722647274280
", "type": "table", "image_path": "fb65d3346450b25f3f272dc1e4074300781d3c61b8a121948273860cdd52fac9.jpg" } ] } ], "index": 2, "virtual_lines": [ { "bbox": [ 107, 103, 505, 178.33333333333331 ], "spans": [], "index": 1 }, { "bbox": [ 107, 178.33333333333331, 505, 253.66666666666663 ], "spans": [], "index": 2 }, { "bbox": [ 107, 253.66666666666663, 505, 328.99999999999994 ], "spans": [], "index": 3 } ] } ], "index": 1.0 }, { "type": "text", "bbox": [ 106, 348, 505, 517 ], "lines": [ { "bbox": [ 106, 348, 505, 361 ], "spans": [ { "bbox": [ 106, 348, 505, 361 ], "score": 1.0, "content": "cretely, there are three major trends that can be gleaned from Table 2: (1) The proposed Pyraformer", "type": "text" } ], "index": 4 }, { "bbox": [ 105, 360, 505, 372 ], "spans": [ { "bbox": [ 105, 360, 505, 372 ], "score": 1.0, "content": "yields the most accurate prediction results, suggesting that the pyramidal graph can better explain", "type": "text" } ], "index": 5 }, { "bbox": [ 106, 370, 505, 383 ], "spans": [ { "bbox": [ 106, 370, 505, 383 ], "score": 1.0, "content": "the temporal interactions in the time series by considering dependencies of different ranges. Inter-", "type": "text" } ], "index": 6 }, { "bbox": [ 106, 382, 504, 393 ], "spans": [ { "bbox": [ 106, 382, 504, 393 ], "score": 1.0, "content": "estingly, for the Wind dataset, sparse attention mechanisms, namely, LogTrans, ETC, Longformer", "type": "text" } ], "index": 7 }, { "bbox": [ 106, 392, 505, 405 ], "spans": [ { "bbox": [ 106, 392, 505, 405 ], "score": 1.0, "content": "and Pyraformer, outperform the original full attention Transformer, probably because the data con-", "type": "text" } ], "index": 8 }, { "bbox": [ 105, 403, 505, 416 ], "spans": [ { "bbox": [ 105, 403, 505, 416 ], "score": 1.0, "content": "tains a large number of zeros and the promotion of adequate sparsity can help avoid over-fitting.", "type": "text" } ], "index": 9 }, { "bbox": [ 105, 414, 505, 427 ], "spans": [ { "bbox": [ 105, 414, 505, 427 ], "score": 1.0, "content": "(2) The number of Q-K pairs in Pyraformer is the smallest. Recall that this number character-", "type": "text" } ], "index": 10 }, { "bbox": [ 105, 425, 505, 438 ], "spans": [ { "bbox": [ 105, 425, 360, 438 ], "score": 1.0, "content": "izes the time and space complexity. Remarkably enough, it is", "type": "text" }, { "bbox": [ 360, 425, 388, 436 ], "score": 0.88, "content": "6 5 . 4 \\%", "type": "inline_equation" }, { "bbox": [ 388, 425, 505, 438 ], "score": 1.0, "content": "fewer than that of LogTrans", "type": "text" } ], "index": 11 }, { "bbox": [ 105, 435, 505, 449 ], "spans": [ { "bbox": [ 105, 435, 124, 449 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 124, 436, 153, 447 ], "score": 0.88, "content": "9 6 . 6 \\%", "type": "inline_equation" }, { "bbox": [ 153, 435, 505, 449 ], "score": 1.0, "content": "than that of the full attention. It is worth emphasizing that this computational gain", "type": "text" } ], "index": 12 }, { "bbox": [ 105, 446, 506, 460 ], "spans": [ { "bbox": [ 105, 446, 506, 460 ], "score": 1.0, "content": "will continue to increase for longer time series. (3) The number of parameters for Pyraformer is", "type": "text" } ], "index": 13 }, { "bbox": [ 105, 456, 505, 472 ], "spans": [ { "bbox": [ 105, 456, 505, 472 ], "score": 1.0, "content": "slightly larger than that of the other models, resulting from the CSCM. However, this module is very", "type": "text" } ], "index": 14 }, { "bbox": [ 106, 469, 504, 480 ], "spans": [ { "bbox": [ 106, 469, 243, 480 ], "score": 1.0, "content": "lightweight, which incurs merely", "type": "text" }, { "bbox": [ 244, 469, 259, 480 ], "score": 0.86, "content": "5 \\%", "type": "inline_equation" }, { "bbox": [ 259, 469, 504, 480 ], "score": 1.0, "content": "overhead in terms of model size compared to other models.", "type": "text" } ], "index": 15 }, { "bbox": [ 105, 479, 505, 492 ], "spans": [ { "bbox": [ 105, 479, 336, 492 ], "score": 1.0, "content": "Moreover, in practice, we can fix the hyper-parameters", "type": "text" }, { "bbox": [ 336, 480, 345, 490 ], "score": 0.54, "content": "A", "type": "inline_equation" }, { "bbox": [ 345, 479, 349, 492 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 350, 480, 358, 490 ], "score": 0.67, "content": "S", "type": "inline_equation" }, { "bbox": [ 358, 479, 378, 492 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 378, 480, 388, 490 ], "score": 0.8, "content": "N", "type": "inline_equation" }, { "bbox": [ 389, 479, 459, 492 ], "score": 1.0, "content": ", and ensure that", "type": "text" }, { "bbox": [ 460, 480, 469, 490 ], "score": 0.8, "content": "C", "type": "inline_equation" }, { "bbox": [ 469, 479, 505, 492 ], "score": 1.0, "content": "satisfies", "type": "text" } ], "index": 16 }, { "bbox": [ 106, 490, 505, 506 ], "spans": [ { "bbox": [ 106, 491, 238, 505 ], "score": 0.91, "content": "C > \\sqrt [ s - 1 ] { L / ( ( A - 1 ) N / 2 + 1 ) }", "type": "inline_equation" }, { "bbox": [ 239, 490, 505, 506 ], "score": 1.0, "content": ". Consequently, the extra number of parameters introduced by the", "type": "text" } ], "index": 17 }, { "bbox": [ 105, 504, 297, 519 ], "spans": [ { "bbox": [ 105, 504, 166, 519 ], "score": 1.0, "content": "CSCM is only", "type": "text" }, { "bbox": [ 167, 504, 293, 518 ], "score": 0.89, "content": "\\mathcal { O } ( ( S - 1 ) C D _ { K } ^ { 2 } ) \\approx \\mathcal { O } ( \\sqrt [ s ] { L } )", "type": "inline_equation" }, { "bbox": [ 294, 504, 297, 519 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 18 } ], "index": 11 }, { "type": "title", "bbox": [ 107, 531, 319, 542 ], "lines": [ { "bbox": [ 105, 531, 320, 544 ], "spans": [ { "bbox": [ 105, 531, 320, 544 ], "score": 1.0, "content": "4.2.2 LONG-RANGE MULTI-STEP FORECASTING", "type": "text" } ], "index": 19 } ], "index": 19 }, { "type": "text", "bbox": [ 107, 550, 505, 605 ], "lines": [ { "bbox": [ 106, 549, 505, 563 ], "spans": [ { "bbox": [ 106, 549, 505, 563 ], "score": 1.0, "content": "We evaluated the performance of Pyraformer for long-range forecasting on three datasets, that is,", "type": "text" } ], "index": 20 }, { "bbox": [ 106, 561, 505, 573 ], "spans": [ { "bbox": [ 106, 561, 505, 573 ], "score": 1.0, "content": "Electricity, ETTh1, and ETTm1. In particular for ETTh1 and ETTm1, we predicted the future oil", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 572, 505, 584 ], "spans": [ { "bbox": [ 105, 572, 505, 584 ], "score": 1.0, "content": "temperature and the 6 power load features at the same time, which is a multivariate time series", "type": "text" } ], "index": 22 }, { "bbox": [ 106, 583, 505, 595 ], "spans": [ { "bbox": [ 106, 583, 505, 595 ], "score": 1.0, "content": "forecasting problem. Both prediction modules introduced in Section 3.3 were tested for all models", "type": "text" } ], "index": 23 }, { "bbox": [ 106, 594, 275, 606 ], "spans": [ { "bbox": [ 106, 594, 275, 606 ], "score": 1.0, "content": "and the better results are listed in Table 3.", "type": "text" } ], "index": 24 } ], "index": 22 }, { "type": "text", "bbox": [ 107, 610, 505, 732 ], "lines": [ { "bbox": [ 105, 611, 505, 623 ], "spans": [ { "bbox": [ 105, 611, 505, 623 ], "score": 1.0, "content": "It is evident that Pyraformer still achieves the best performance with the least number of Q-K", "type": "text" } ], "index": 25 }, { "bbox": [ 105, 622, 505, 635 ], "spans": [ { "bbox": [ 105, 622, 505, 635 ], "score": 1.0, "content": "pairs for all datasets regardless of the prediction length. More precisely, in comparison with In-", "type": "text" } ], "index": 26 }, { "bbox": [ 105, 632, 505, 645 ], "spans": [ { "bbox": [ 105, 632, 443, 645 ], "score": 1.0, "content": "former (Zhou et al., 2021), the MSE given by Pyraformer for ETTh1 is decreased by", "type": "text" }, { "bbox": [ 443, 632, 470, 644 ], "score": 0.83, "content": "2 4 . 8 \\%", "type": "inline_equation" }, { "bbox": [ 471, 632, 474, 645 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 474, 632, 501, 644 ], "score": 0.84, "content": "2 8 . 9 \\%", "type": "inline_equation" }, { "bbox": [ 501, 632, 505, 645 ], "score": 1.0, "content": ",", "type": "text" } ], "index": 27 }, { "bbox": [ 106, 643, 505, 657 ], "spans": [ { "bbox": [ 106, 643, 134, 655 ], "score": 0.88, "content": "2 6 . 2 \\%", "type": "inline_equation" }, { "bbox": [ 135, 643, 505, 657 ], "score": 1.0, "content": "respectively when the prediction length is 168, 336, and 720. Once again, this bolsters our", "type": "text" } ], "index": 28 }, { "bbox": [ 105, 653, 505, 668 ], "spans": [ { "bbox": [ 105, 653, 505, 668 ], "score": 1.0, "content": "belief that it is more beneficial to employ the pyramidal graph when describing the temporal depen-", "type": "text" } ], "index": 29 }, { "bbox": [ 105, 664, 505, 679 ], "spans": [ { "bbox": [ 105, 664, 505, 679 ], "score": 1.0, "content": "dencies. Interestingly, we notice that for Pyraformer, the results given by the first prediction module", "type": "text" } ], "index": 30 }, { "bbox": [ 106, 677, 504, 688 ], "spans": [ { "bbox": [ 106, 677, 504, 688 ], "score": 1.0, "content": "are better than those by the second one. One possible explanation is that the second prediction mod-", "type": "text" } ], "index": 31 }, { "bbox": [ 106, 688, 505, 699 ], "spans": [ { "bbox": [ 106, 688, 505, 699 ], "score": 1.0, "content": "ule based on the full attention layers cannot differentiate features with different resolutions, while", "type": "text" } ], "index": 32 }, { "bbox": [ 105, 698, 505, 711 ], "spans": [ { "bbox": [ 105, 698, 505, 711 ], "score": 1.0, "content": "the first module based on a single fully connected layer can take full advantages of such features", "type": "text" } ], "index": 33 }, { "bbox": [ 104, 708, 505, 723 ], "spans": [ { "bbox": [ 104, 708, 505, 723 ], "score": 1.0, "content": "in an automated fashion. To better elucidate the modeling capacity of Pyraformer for long-range", "type": "text" } ], "index": 34 }, { "bbox": [ 106, 721, 459, 733 ], "spans": [ { "bbox": [ 106, 721, 459, 733 ], "score": 1.0, "content": "forecasting, we refer the readers to Appendix I for a detailed example on synthetic data.", "type": "text" } ], "index": 35 } ], "index": 30 } ], "page_idx": 7, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 108, 27, 293, 37 ], "lines": [ { "bbox": [ 106, 26, 294, 38 ], "spans": [ { "bbox": [ 106, 26, 294, 38 ], "score": 1.0, "content": "Published as a conference paper at ICLR 2022", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 302, 751, 308, 760 ], "lines": [ { "bbox": [ 300, 750, 309, 761 ], "spans": [ { "bbox": [ 300, 750, 309, 761 ], "score": 1.0, "content": "8", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "table", "bbox": [ 107, 103, 505, 329 ], "blocks": [ { "type": "table_caption", "bbox": [ 203, 90, 407, 101 ], "group_id": 0, "lines": [ { "bbox": [ 202, 87, 407, 104 ], "spans": [ { "bbox": [ 202, 87, 407, 104 ], "score": 1.0, "content": "Table 3: Long-range multi-step forecasting results.", "type": "text" } ], "index": 0 } ], "index": 0 }, { "type": "table_body", "bbox": [ 107, 103, 505, 329 ], "group_id": 0, "lines": [ { "bbox": [ 107, 103, 505, 329 ], "spans": [ { "bbox": [ 107, 103, 505, 329 ], "score": 0.986, "html": "
MethodsMetricsETTh1ETTm1Electricity
16833672096288672168336720
InformerMSE1.0751.3291.3840.5560.8410.9210.7451.5794.365
MAE0.8010.9110.9500.5370.7050.7530.2660.3230.371
Q-K pairs188040188040423360276480560640560640188040188040423360
LogTransMSE0.9831.1001.4110.5540.7861.1690.7911.5844.362
MAE0.7660.8390.9910.4990.6760.8680.3400.3360.366
Q-K pairs74664746642167442547606487686487687466474664216744
LongformerMSE0.8600.9751.0910.5260.7671.0210.7661.5914.361
MAE0.7100.7690.8320.5070.6630.7880.3110.3430.368
Q-K pairs6364863648249120329760100713610071366364863648249120
ReformerMSE0.9581.0441.4580.5430.9240.9810.7831.5844.374
MAE0.7410.7870.9870.5280.7220.7780.3320.3340.374
Q-K pairs10160641016064270950453084161445068814450688101606410160642709504
ETCMSE1.0251.0841.1370.7621.2271.2720.7771.5864.361
MAE0.7710.8110.8660.6530.8800.9080.3260.3400.368
Q-K pairs125280125280288720331344836952836952125280125280288720
PyraformerMSE0.8080.9451.0220.4800.7540.8570.7191.5334.312
MAE0.6830.7660.8060.4860.6590.7070.2560.2910.346
Q-K pairs264722647274280572649638496384264722647274280
", "type": "table", "image_path": "fb65d3346450b25f3f272dc1e4074300781d3c61b8a121948273860cdd52fac9.jpg" } ] } ], "index": 2, "virtual_lines": [ { "bbox": [ 107, 103, 505, 178.33333333333331 ], "spans": [], "index": 1 }, { "bbox": [ 107, 178.33333333333331, 505, 253.66666666666663 ], "spans": [], "index": 2 }, { "bbox": [ 107, 253.66666666666663, 505, 328.99999999999994 ], "spans": [], "index": 3 } ] } ], "index": 1.0 }, { "type": "text", "bbox": [ 106, 348, 505, 517 ], "lines": [], "index": 11, "bbox_fs": [ 105, 348, 506, 519 ], "lines_deleted": true }, { "type": "title", "bbox": [ 107, 531, 319, 542 ], "lines": [ { "bbox": [ 105, 531, 320, 544 ], "spans": [ { "bbox": [ 105, 531, 320, 544 ], "score": 1.0, "content": "4.2.2 LONG-RANGE MULTI-STEP FORECASTING", "type": "text" } ], "index": 19 } ], "index": 19 }, { "type": "text", "bbox": [ 107, 550, 505, 605 ], "lines": [ { "bbox": [ 106, 549, 505, 563 ], "spans": [ { "bbox": [ 106, 549, 505, 563 ], "score": 1.0, "content": "We evaluated the performance of Pyraformer for long-range forecasting on three datasets, that is,", "type": "text" } ], "index": 20 }, { "bbox": [ 106, 561, 505, 573 ], "spans": [ { "bbox": [ 106, 561, 505, 573 ], "score": 1.0, "content": "Electricity, ETTh1, and ETTm1. In particular for ETTh1 and ETTm1, we predicted the future oil", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 572, 505, 584 ], "spans": [ { "bbox": [ 105, 572, 505, 584 ], "score": 1.0, "content": "temperature and the 6 power load features at the same time, which is a multivariate time series", "type": "text" } ], "index": 22 }, { "bbox": [ 106, 583, 505, 595 ], "spans": [ { "bbox": [ 106, 583, 505, 595 ], "score": 1.0, "content": "forecasting problem. Both prediction modules introduced in Section 3.3 were tested for all models", "type": "text" } ], "index": 23 }, { "bbox": [ 106, 594, 275, 606 ], "spans": [ { "bbox": [ 106, 594, 275, 606 ], "score": 1.0, "content": "and the better results are listed in Table 3.", "type": "text" } ], "index": 24 } ], "index": 22, "bbox_fs": [ 105, 549, 505, 606 ] }, { "type": "text", "bbox": [ 107, 610, 505, 732 ], "lines": [ { "bbox": [ 105, 611, 505, 623 ], "spans": [ { "bbox": [ 105, 611, 505, 623 ], "score": 1.0, "content": "It is evident that Pyraformer still achieves the best performance with the least number of Q-K", "type": "text" } ], "index": 25 }, { "bbox": [ 105, 622, 505, 635 ], "spans": [ { "bbox": [ 105, 622, 505, 635 ], "score": 1.0, "content": "pairs for all datasets regardless of the prediction length. More precisely, in comparison with In-", "type": "text" } ], "index": 26 }, { "bbox": [ 105, 632, 505, 645 ], "spans": [ { "bbox": [ 105, 632, 443, 645 ], "score": 1.0, "content": "former (Zhou et al., 2021), the MSE given by Pyraformer for ETTh1 is decreased by", "type": "text" }, { "bbox": [ 443, 632, 470, 644 ], "score": 0.83, "content": "2 4 . 8 \\%", "type": "inline_equation" }, { "bbox": [ 471, 632, 474, 645 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 474, 632, 501, 644 ], "score": 0.84, "content": "2 8 . 9 \\%", "type": "inline_equation" }, { "bbox": [ 501, 632, 505, 645 ], "score": 1.0, "content": ",", "type": "text" } ], "index": 27 }, { "bbox": [ 106, 643, 505, 657 ], "spans": [ { "bbox": [ 106, 643, 134, 655 ], "score": 0.88, "content": "2 6 . 2 \\%", "type": "inline_equation" }, { "bbox": [ 135, 643, 505, 657 ], "score": 1.0, "content": "respectively when the prediction length is 168, 336, and 720. Once again, this bolsters our", "type": "text" } ], "index": 28 }, { "bbox": [ 105, 653, 505, 668 ], "spans": [ { "bbox": [ 105, 653, 505, 668 ], "score": 1.0, "content": "belief that it is more beneficial to employ the pyramidal graph when describing the temporal depen-", "type": "text" } ], "index": 29 }, { "bbox": [ 105, 664, 505, 679 ], "spans": [ { "bbox": [ 105, 664, 505, 679 ], "score": 1.0, "content": "dencies. Interestingly, we notice that for Pyraformer, the results given by the first prediction module", "type": "text" } ], "index": 30 }, { "bbox": [ 106, 677, 504, 688 ], "spans": [ { "bbox": [ 106, 677, 504, 688 ], "score": 1.0, "content": "are better than those by the second one. One possible explanation is that the second prediction mod-", "type": "text" } ], "index": 31 }, { "bbox": [ 106, 688, 505, 699 ], "spans": [ { "bbox": [ 106, 688, 505, 699 ], "score": 1.0, "content": "ule based on the full attention layers cannot differentiate features with different resolutions, while", "type": "text" } ], "index": 32 }, { "bbox": [ 105, 698, 505, 711 ], "spans": [ { "bbox": [ 105, 698, 505, 711 ], "score": 1.0, "content": "the first module based on a single fully connected layer can take full advantages of such features", "type": "text" } ], "index": 33 }, { "bbox": [ 104, 708, 505, 723 ], "spans": [ { "bbox": [ 104, 708, 505, 723 ], "score": 1.0, "content": "in an automated fashion. To better elucidate the modeling capacity of Pyraformer for long-range", "type": "text" } ], "index": 34 }, { "bbox": [ 106, 721, 459, 733 ], "spans": [ { "bbox": [ 106, 721, 459, 733 ], "score": 1.0, "content": "forecasting, we refer the readers to Appendix I for a detailed example on synthetic data.", "type": "text" } ], "index": 35 } ], "index": 30, "bbox_fs": [ 104, 611, 505, 733 ] } ] }, { "preproc_blocks": [ { "type": "image", "bbox": [ 119, 83, 492, 239 ], "blocks": [ { "type": "image_body", "bbox": [ 119, 83, 492, 239 ], "group_id": 0, "lines": [ { "bbox": [ 119, 83, 492, 239 ], "spans": [ { "bbox": [ 119, 83, 492, 239 ], "score": 0.974, "type": "image", "image_path": "c76b7f33141c4ef44a7a1594f69b85c5196a20fecbd341cd7e0a1ecd285eb186.jpg" } ] } ], "index": 1, "virtual_lines": [ { "bbox": [ 119, 83, 492, 135.0 ], "spans": [], "index": 0 }, { "bbox": [ 119, 135.0, 492, 187.0 ], "spans": [], "index": 1 }, { "bbox": [ 119, 187.0, 492, 239.0 ], "spans": [], "index": 2 } ] }, { "type": "image_caption", "bbox": [ 107, 245, 504, 268 ], "group_id": 0, "lines": [ { "bbox": [ 106, 245, 505, 258 ], "spans": [ { "bbox": [ 106, 245, 505, 258 ], "score": 1.0, "content": "Figure 4: Comparison of the time and memory consumption between the full, the prob-sparse, and", "type": "text" } ], "index": 3 }, { "bbox": [ 105, 256, 505, 270 ], "spans": [ { "bbox": [ 105, 256, 505, 270 ], "score": 1.0, "content": "the TVM implementation of the pyramidal attention: (a) computation time; (b) memory occupation.", "type": "text" } ], "index": 4 } ], "index": 3.5 } ], "index": 2.25 }, { "type": "title", "bbox": [ 107, 280, 298, 290 ], "lines": [ { "bbox": [ 105, 279, 300, 293 ], "spans": [ { "bbox": [ 105, 279, 300, 293 ], "score": 1.0, "content": "4.2.3 SPEED AND MEMORY CONSUMPTION", "type": "text" } ], "index": 5 } ], "index": 5 }, { "type": "text", "bbox": [ 107, 297, 505, 440 ], "lines": [ { "bbox": [ 106, 297, 505, 310 ], "spans": [ { "bbox": [ 106, 297, 505, 310 ], "score": 1.0, "content": "To check the efficiency of the customized CUDA kernel implemented based on TVM, we depicted", "type": "text" } ], "index": 6 }, { "bbox": [ 105, 307, 505, 322 ], "spans": [ { "bbox": [ 105, 307, 464, 322 ], "score": 1.0, "content": "the empirical computation time and memory cost as a function of the sequence length", "type": "text" }, { "bbox": [ 464, 309, 473, 318 ], "score": 0.74, "content": "L", "type": "inline_equation" }, { "bbox": [ 473, 307, 505, 322 ], "score": 1.0, "content": "in Fig-", "type": "text" } ], "index": 7 }, { "bbox": [ 106, 320, 505, 331 ], "spans": [ { "bbox": [ 106, 320, 505, 331 ], "score": 1.0, "content": "ure 4. Here we only compared Pyraformer with the full attention and the prob-sparse attention in", "type": "text" } ], "index": 8 }, { "bbox": [ 105, 330, 505, 343 ], "spans": [ { "bbox": [ 105, 330, 411, 343 ], "score": 1.0, "content": "Informer (Zhou et al., 2021). All the computations were performed on a", "type": "text" }, { "bbox": [ 412, 331, 441, 341 ], "score": 0.42, "content": "1 2 \\mathrm { \\ G B }", "type": "inline_equation" }, { "bbox": [ 441, 330, 466, 343 ], "score": 1.0, "content": "Titan", "type": "text" }, { "bbox": [ 466, 331, 481, 342 ], "score": 0.36, "content": "\\mathrm { X p }", "type": "inline_equation" }, { "bbox": [ 481, 330, 505, 343 ], "score": 1.0, "content": "GPU", "type": "text" } ], "index": 9 }, { "bbox": [ 105, 340, 506, 354 ], "spans": [ { "bbox": [ 105, 340, 506, 354 ], "score": 1.0, "content": "with Ubuntu 16.04, CUDA 11.0, and TVM 0.8.0. Figure 4 shows that the time and memory cost", "type": "text" } ], "index": 10 }, { "bbox": [ 105, 352, 505, 365 ], "spans": [ { "bbox": [ 105, 352, 440, 365 ], "score": 1.0, "content": "of the proposed Pyraformer based on TVM is approximately a linear function of", "type": "text" }, { "bbox": [ 440, 353, 448, 362 ], "score": 0.71, "content": "L", "type": "inline_equation" }, { "bbox": [ 448, 352, 505, 365 ], "score": 1.0, "content": ", as expected.", "type": "text" } ], "index": 11 }, { "bbox": [ 105, 363, 505, 376 ], "spans": [ { "bbox": [ 105, 363, 505, 376 ], "score": 1.0, "content": "Furthermore, the time and memory consumption of the TVM implementation can be several or-", "type": "text" } ], "index": 12 }, { "bbox": [ 105, 374, 505, 387 ], "spans": [ { "bbox": [ 105, 374, 505, 387 ], "score": 1.0, "content": "ders of magnitude smaller than that of the full attention and the prob-sparse attention, especially for", "type": "text" } ], "index": 13 }, { "bbox": [ 105, 385, 505, 398 ], "spans": [ { "bbox": [ 105, 385, 505, 398 ], "score": 1.0, "content": "relatively long time series. Indeed, for a 12GB Titan Xp GPU, when the sequence length reaches", "type": "text" } ], "index": 14 }, { "bbox": [ 105, 396, 505, 409 ], "spans": [ { "bbox": [ 105, 396, 505, 409 ], "score": 1.0, "content": "5800, full attention encounters the out-of-memory (OOM) problem, yet the TVM implementation", "type": "text" } ], "index": 15 }, { "bbox": [ 105, 407, 505, 420 ], "spans": [ { "bbox": [ 105, 407, 505, 420 ], "score": 1.0, "content": "of Pyraformer only occupies 1GB of memory. When it comes to a sequence with 20000 time points,", "type": "text" } ], "index": 16 }, { "bbox": [ 105, 417, 505, 432 ], "spans": [ { "bbox": [ 105, 417, 505, 432 ], "score": 1.0, "content": "even Informer incurs the OOM problem, whereas the memory cost of Pyraformer is only 1.91GB", "type": "text" } ], "index": 17 }, { "bbox": [ 105, 429, 310, 441 ], "spans": [ { "bbox": [ 105, 429, 310, 441 ], "score": 1.0, "content": "and the computation time per batch is only 0.082s.", "type": "text" } ], "index": 18 } ], "index": 12 }, { "type": "title", "bbox": [ 108, 453, 209, 464 ], "lines": [ { "bbox": [ 106, 452, 210, 465 ], "spans": [ { "bbox": [ 106, 452, 210, 465 ], "score": 1.0, "content": "4.3 ABLATION STUDY", "type": "text" } ], "index": 19 } ], "index": 19 }, { "type": "text", "bbox": [ 107, 472, 505, 550 ], "lines": [ { "bbox": [ 106, 472, 505, 484 ], "spans": [ { "bbox": [ 106, 472, 352, 484 ], "score": 1.0, "content": "We also performed ablation studies to measure the impact of", "type": "text" }, { "bbox": [ 353, 473, 361, 482 ], "score": 0.75, "content": "A", "type": "inline_equation" }, { "bbox": [ 362, 472, 379, 484 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 380, 473, 389, 483 ], "score": 0.8, "content": "C", "type": "inline_equation" }, { "bbox": [ 389, 472, 505, 484 ], "score": 1.0, "content": ", the CSCM architecture, the", "type": "text" } ], "index": 20 }, { "bbox": [ 105, 484, 505, 496 ], "spans": [ { "bbox": [ 105, 484, 505, 496 ], "score": 1.0, "content": "history length, and the PAM on the prediction accuracy of Pyraformer. The results are displayed in", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 493, 505, 507 ], "spans": [ { "bbox": [ 105, 493, 505, 507 ], "score": 1.0, "content": "Tables 7-10. Detailed Discussions on the results can be found in Appendix J. Here, we only provide", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 505, 505, 518 ], "spans": [ { "bbox": [ 105, 505, 342, 518 ], "score": 1.0, "content": "an overview of the major findings: (1) it is better to increase", "type": "text" }, { "bbox": [ 342, 506, 351, 515 ], "score": 0.82, "content": "C", "type": "inline_equation" }, { "bbox": [ 352, 505, 371, 518 ], "score": 1.0, "content": "with", "type": "text" }, { "bbox": [ 371, 506, 379, 515 ], "score": 0.73, "content": "L", "type": "inline_equation" }, { "bbox": [ 380, 505, 407, 518 ], "score": 1.0, "content": "but fix", "type": "text" }, { "bbox": [ 407, 506, 416, 515 ], "score": 0.74, "content": "A", "type": "inline_equation" }, { "bbox": [ 416, 505, 505, 518 ], "score": 1.0, "content": "to a small constant for", "type": "text" } ], "index": 23 }, { "bbox": [ 105, 516, 505, 529 ], "spans": [ { "bbox": [ 105, 516, 505, 529 ], "score": 1.0, "content": "the sake of reducing the prediction error; (2) convolution with bottleneck strikes a balance between", "type": "text" } ], "index": 24 }, { "bbox": [ 105, 528, 505, 540 ], "spans": [ { "bbox": [ 105, 528, 505, 540 ], "score": 1.0, "content": "the prediction accuracy and the number of parameters, and hence, we use it as the CSCM; (3) more", "type": "text" } ], "index": 25 }, { "bbox": [ 105, 537, 502, 552 ], "spans": [ { "bbox": [ 105, 537, 502, 552 ], "score": 1.0, "content": "history helps increase the accuracy of forecasting; (4) the PAM is essential for accurate prediction.", "type": "text" } ], "index": 26 } ], "index": 23 }, { "type": "title", "bbox": [ 108, 565, 275, 578 ], "lines": [ { "bbox": [ 104, 563, 277, 581 ], "spans": [ { "bbox": [ 104, 563, 277, 581 ], "score": 1.0, "content": "5 CONCLUSION AND OUTLOOK", "type": "text" } ], "index": 27 } ], "index": 27 }, { "type": "text", "bbox": [ 107, 589, 505, 732 ], "lines": [ { "bbox": [ 105, 588, 505, 601 ], "spans": [ { "bbox": [ 105, 588, 505, 601 ], "score": 1.0, "content": "In this paper, we propose Pyraformer, a novel model based on pyramidal attention that can effec-", "type": "text" } ], "index": 28 }, { "bbox": [ 105, 600, 505, 613 ], "spans": [ { "bbox": [ 105, 600, 505, 613 ], "score": 1.0, "content": "tively describe both short and long temporal dependencies with low time and space complexity.", "type": "text" } ], "index": 29 }, { "bbox": [ 105, 609, 506, 625 ], "spans": [ { "bbox": [ 105, 609, 322, 625 ], "score": 1.0, "content": "Concretely, we first exploit the CSCM to construct a", "type": "text" }, { "bbox": [ 322, 612, 331, 621 ], "score": 0.83, "content": "C", "type": "inline_equation" }, { "bbox": [ 331, 609, 506, 625 ], "score": 1.0, "content": "-ary tree, and then design the PAM to pass", "type": "text" } ], "index": 30 }, { "bbox": [ 105, 622, 505, 635 ], "spans": [ { "bbox": [ 105, 622, 395, 635 ], "score": 1.0, "content": "messages in both the inter-scale and the intra-scale fashion. 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Experimental results show that the proposed", "type": "text" } ], "index": 33 }, { "bbox": [ 105, 654, 505, 668 ], "spans": [ { "bbox": [ 105, 654, 505, 668 ], "score": 1.0, "content": "model outperforms the state-of-the-art models for both single-step and long-range multi-step pre-", "type": "text" } ], "index": 34 }, { "bbox": [ 105, 666, 505, 678 ], "spans": [ { "bbox": [ 105, 666, 505, 678 ], "score": 1.0, "content": "diction tasks, but with less computational time and memory cost. 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Here we only compared Pyraformer with the full attention and the prob-sparse attention in", "type": "text" } ], "index": 8 }, { "bbox": [ 105, 330, 505, 343 ], "spans": [ { "bbox": [ 105, 330, 411, 343 ], "score": 1.0, "content": "Informer (Zhou et al., 2021). All the computations were performed on a", "type": "text" }, { "bbox": [ 412, 331, 441, 341 ], "score": 0.42, "content": "1 2 \\mathrm { \\ G B }", "type": "inline_equation" }, { "bbox": [ 441, 330, 466, 343 ], "score": 1.0, "content": "Titan", "type": "text" }, { "bbox": [ 466, 331, 481, 342 ], "score": 0.36, "content": "\\mathrm { X p }", "type": "inline_equation" }, { "bbox": [ 481, 330, 505, 343 ], "score": 1.0, "content": "GPU", "type": "text" } ], "index": 9 }, { "bbox": [ 105, 340, 506, 354 ], "spans": [ { "bbox": [ 105, 340, 506, 354 ], "score": 1.0, "content": "with Ubuntu 16.04, CUDA 11.0, and TVM 0.8.0. Figure 4 shows that the time and memory cost", "type": "text" } ], "index": 10 }, { "bbox": [ 105, 352, 505, 365 ], "spans": [ { "bbox": [ 105, 352, 440, 365 ], "score": 1.0, "content": "of the proposed Pyraformer based on TVM is approximately a linear function of", "type": "text" }, { "bbox": [ 440, 353, 448, 362 ], "score": 0.71, "content": "L", "type": "inline_equation" }, { "bbox": [ 448, 352, 505, 365 ], "score": 1.0, "content": ", as expected.", "type": "text" } ], "index": 11 }, { "bbox": [ 105, 363, 505, 376 ], "spans": [ { "bbox": [ 105, 363, 505, 376 ], "score": 1.0, "content": "Furthermore, the time and memory consumption of the TVM implementation can be several or-", "type": "text" } ], "index": 12 }, { "bbox": [ 105, 374, 505, 387 ], "spans": [ { "bbox": [ 105, 374, 505, 387 ], "score": 1.0, "content": "ders of magnitude smaller than that of the full attention and the prob-sparse attention, especially for", "type": "text" } ], "index": 13 }, { "bbox": [ 105, 385, 505, 398 ], "spans": [ { "bbox": [ 105, 385, 505, 398 ], "score": 1.0, "content": "relatively long time series. Indeed, for a 12GB Titan Xp GPU, when the sequence length reaches", "type": "text" } ], "index": 14 }, { "bbox": [ 105, 396, 505, 409 ], "spans": [ { "bbox": [ 105, 396, 505, 409 ], "score": 1.0, "content": "5800, full attention encounters the out-of-memory (OOM) problem, yet the TVM implementation", "type": "text" } ], "index": 15 }, { "bbox": [ 105, 407, 505, 420 ], "spans": [ { "bbox": [ 105, 407, 505, 420 ], "score": 1.0, "content": "of Pyraformer only occupies 1GB of memory. When it comes to a sequence with 20000 time points,", "type": "text" } ], "index": 16 }, { "bbox": [ 105, 417, 505, 432 ], "spans": [ { "bbox": [ 105, 417, 505, 432 ], "score": 1.0, "content": "even Informer incurs the OOM problem, whereas the memory cost of Pyraformer is only 1.91GB", "type": "text" } ], "index": 17 }, { "bbox": [ 105, 429, 310, 441 ], "spans": [ { "bbox": [ 105, 429, 310, 441 ], "score": 1.0, "content": "and the computation time per batch is only 0.082s.", "type": "text" } ], "index": 18 } ], "index": 12, "bbox_fs": [ 105, 297, 506, 441 ] }, { "type": "title", "bbox": [ 108, 453, 209, 464 ], "lines": [ { "bbox": [ 106, 452, 210, 465 ], "spans": [ { "bbox": [ 106, 452, 210, 465 ], "score": 1.0, "content": "4.3 ABLATION STUDY", "type": "text" } ], "index": 19 } ], "index": 19 }, { "type": "text", "bbox": [ 107, 472, 505, 550 ], "lines": [ { "bbox": [ 106, 472, 505, 484 ], "spans": [ { "bbox": [ 106, 472, 352, 484 ], "score": 1.0, "content": "We also performed ablation studies to measure the impact of", "type": "text" }, { "bbox": [ 353, 473, 361, 482 ], "score": 0.75, "content": "A", "type": "inline_equation" }, { "bbox": [ 362, 472, 379, 484 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 380, 473, 389, 483 ], "score": 0.8, "content": "C", "type": "inline_equation" }, { "bbox": [ 389, 472, 505, 484 ], "score": 1.0, "content": ", the CSCM architecture, the", "type": "text" } ], "index": 20 }, { "bbox": [ 105, 484, 505, 496 ], "spans": [ { "bbox": [ 105, 484, 505, 496 ], "score": 1.0, "content": "history length, and the PAM on the prediction accuracy of Pyraformer. The results are displayed in", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 493, 505, 507 ], "spans": [ { "bbox": [ 105, 493, 505, 507 ], "score": 1.0, "content": "Tables 7-10. Detailed Discussions on the results can be found in Appendix J. Here, we only provide", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 505, 505, 518 ], "spans": [ { "bbox": [ 105, 505, 342, 518 ], "score": 1.0, "content": "an overview of the major findings: (1) it is better to increase", "type": "text" }, { "bbox": [ 342, 506, 351, 515 ], "score": 0.82, "content": "C", "type": "inline_equation" }, { "bbox": [ 352, 505, 371, 518 ], "score": 1.0, "content": "with", "type": "text" }, { "bbox": [ 371, 506, 379, 515 ], "score": 0.73, "content": "L", "type": "inline_equation" }, { "bbox": [ 380, 505, 407, 518 ], "score": 1.0, "content": "but fix", "type": "text" }, { "bbox": [ 407, 506, 416, 515 ], "score": 0.74, "content": "A", "type": "inline_equation" }, { "bbox": [ 416, 505, 505, 518 ], "score": 1.0, "content": "to a small constant for", "type": "text" } ], "index": 23 }, { "bbox": [ 105, 516, 505, 529 ], "spans": [ { "bbox": [ 105, 516, 505, 529 ], "score": 1.0, "content": "the sake of reducing the prediction error; (2) convolution with bottleneck strikes a balance between", "type": "text" } ], "index": 24 }, { "bbox": [ 105, 528, 505, 540 ], "spans": [ { "bbox": [ 105, 528, 505, 540 ], "score": 1.0, "content": "the prediction accuracy and the number of parameters, and hence, we use it as the CSCM; (3) more", "type": "text" } ], "index": 25 }, { "bbox": [ 105, 537, 502, 552 ], "spans": [ { "bbox": [ 105, 537, 502, 552 ], "score": 1.0, "content": "history helps increase the accuracy of forecasting; (4) the PAM is essential for accurate prediction.", "type": "text" } ], "index": 26 } ], "index": 23, "bbox_fs": [ 105, 472, 505, 552 ] }, { "type": "title", "bbox": [ 108, 565, 275, 578 ], "lines": [ { "bbox": [ 104, 563, 277, 581 ], "spans": [ { "bbox": [ 104, 563, 277, 581 ], "score": 1.0, "content": "5 CONCLUSION AND OUTLOOK", "type": "text" } ], "index": 27 } ], "index": 27 }, { "type": "text", "bbox": [ 107, 589, 505, 732 ], "lines": [ { "bbox": [ 105, 588, 505, 601 ], "spans": [ { "bbox": [ 105, 588, 505, 601 ], "score": 1.0, "content": "In this paper, we propose Pyraformer, a novel model based on pyramidal attention that can effec-", "type": "text" } ], "index": 28 }, { "bbox": [ 105, 600, 505, 613 ], "spans": [ { "bbox": [ 105, 600, 505, 613 ], "score": 1.0, "content": "tively describe both short and long temporal dependencies with low time and space complexity.", "type": "text" } ], "index": 29 }, { "bbox": [ 105, 609, 506, 625 ], "spans": [ { "bbox": [ 105, 609, 322, 625 ], "score": 1.0, "content": "Concretely, we first exploit the CSCM to construct a", "type": "text" }, { "bbox": [ 322, 612, 331, 621 ], "score": 0.83, "content": "C", "type": "inline_equation" }, { "bbox": [ 331, 609, 506, 625 ], "score": 1.0, "content": "-ary tree, and then design the PAM to pass", "type": "text" } ], "index": 30 }, { "bbox": [ 105, 622, 505, 635 ], "spans": [ { "bbox": [ 105, 622, 395, 635 ], "score": 1.0, "content": "messages in both the inter-scale and the intra-scale fashion. 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On", "type": "text" } ], "index": 36 }, { "bbox": [ 105, 686, 505, 702 ], "spans": [ { "bbox": [ 105, 686, 505, 702 ], "score": 1.0, "content": "the other hand, we have shown in Appendix I that other configurations of the hyper-parameters may", "type": "text" } ], "index": 37 }, { "bbox": [ 105, 698, 505, 712 ], "spans": [ { "bbox": [ 105, 698, 505, 712 ], "score": 1.0, "content": "further improve the performance of Pyraformer. In the future work, we would like to explore how", "type": "text" } ], "index": 38 }, { "bbox": [ 105, 709, 505, 723 ], "spans": [ { "bbox": [ 105, 709, 505, 723 ], "score": 1.0, "content": "to adaptively learn the hyper-parameters from the data. Also, it is interesting to extend Pyraformer", "type": "text" } ], "index": 39 }, { "bbox": [ 105, 720, 406, 734 ], "spans": [ { "bbox": [ 105, 720, 406, 734 ], "score": 1.0, "content": "to other fields, including natural language processing and computer vision.", "type": "text" } ], "index": 40 } ], "index": 34, "bbox_fs": [ 105, 588, 506, 734 ] } ] }, { "preproc_blocks": [ { "type": "title", "bbox": [ 108, 82, 219, 93 ], "lines": [ { "bbox": [ 106, 82, 220, 96 ], "spans": [ { "bbox": [ 106, 82, 220, 96 ], "score": 1.0, "content": "ACKNOWLEDGEMENT", "type": "text" } ], "index": 0 } ], "index": 0 }, { "type": "text", "bbox": [ 106, 105, 503, 128 ], "lines": [ { "bbox": [ 105, 105, 505, 118 ], "spans": [ { "bbox": [ 105, 105, 505, 118 ], "score": 1.0, "content": "In this work, Prof. Weiyao Lin was supported by Ant Group through Ant Research Program and in", "type": "text" } ], "index": 1 }, { "bbox": [ 105, 117, 422, 130 ], "spans": [ { "bbox": [ 105, 117, 422, 130 ], "score": 1.0, "content": "part by National Natural Science Foundation of China under grant U21B2013.", "type": "text" } ], "index": 2 } ], "index": 1.5 }, { "type": "text", "bbox": [ 104, 132, 506, 741 ], "lines": [ { "bbox": [ 106, 144, 176, 159 ], "spans": [ { "bbox": [ 106, 144, 176, 159 ], "score": 1.0, "content": "REFERENCES", "type": "text" } ], "index": 3 }, { "bbox": [ 104, 162, 506, 177 ], "spans": [ { "bbox": [ 104, 162, 506, 177 ], "score": 1.0, "content": "Joshua Ainslie, Santiago Ontanon, Chris Alberti, Vaclav Cvicek, Zachary Fisher, Philip Pham,", "type": "text" } ], "index": 4 }, { "bbox": [ 114, 173, 507, 188 ], "spans": [ { "bbox": [ 114, 173, 507, 188 ], "score": 1.0, "content": "Anirudh Ravula, Sumit Sanghai, Qifan Wang, and Li Yang. 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NotationSizeMeaning
LConstantThe length of historical sequence.
GConstantThe number of global tokens in ETC.
MConstantThe length of future sequence to be predicted.
BConstantBatch size.
DConstantThe dimension of each node.
DKConstantThe dimension of a key.
XB×L×DInput of a single attention head.
YB×L×DOutput of a single attention head.
QB×L×DkThe query.
KB×L×DkThe key.
VB×L×DkThe value.
WQD×DkThe weight matrix of the query.
WKD ×DkThe weight matrix of the key.
WvD×DkThe weight matrix of the value.
SConstantNumber of scales.
AConstantNumber of adjacent nodes at the same scale that a node can attend to.
CConstantNumber of finer scale nodes that a coarser scale node can summarize.
NConstantNumber of attention layers.
n(DThe l-th node at scale s.
Nlen(N)) × D
Alen(A()） × D The adjacent A nodes at the same scale with n(s).
Clen(C()) × D
Plen(P(s) × D The parent node of n(s).
FpB×M×DThe prediction tokens.
FeB ×Ltot ×D The output of the encoder. Ltot represents the output length of the encod
Fd1B×M×DThe output of the first attention-based decoder layer.
HConstantThe number of attention heads.
DF
ConstantThe maximum dimension of the feed-forward layer.
", "type": "table", "image_path": "558772b46478afb75a19e4e430ee43cce92001ed79e2a8acfd3f505ea01429a0.jpg" } ] } ], "index": 2, "virtual_lines": [ { "bbox": [ 110, 108, 488, 258.66666666666663 ], "spans": [], "index": 1 }, { "bbox": [ 110, 258.66666666666663, 488, 409.33333333333326 ], "spans": [], "index": 2 }, { "bbox": [ 110, 409.33333333333326, 488, 559.9999999999999 ], "spans": [], "index": 3 } ] } ], "index": 1.0 }, { "type": "title", "bbox": [ 108, 584, 404, 597 ], "lines": [ { "bbox": [ 106, 584, 405, 599 ], "spans": [ { "bbox": [ 106, 584, 405, 599 ], "score": 1.0, "content": "A A BRIEF REVIEW ON RELATED RNN-BASED MODELS", "type": "text" } ], "index": 4 } ], "index": 4 }, { "type": "text", "bbox": [ 106, 610, 505, 732 ], "lines": [ { "bbox": [ 105, 611, 504, 623 ], "spans": [ { "bbox": [ 105, 611, 504, 623 ], "score": 1.0, "content": "In this section, we provide a brief review on the related RNN-based models. Multiscale tempo-", "type": "text" } ], "index": 5 }, { "bbox": [ 105, 622, 505, 634 ], "spans": [ { "bbox": [ 105, 622, 505, 634 ], "score": 1.0, "content": "ral dependencies are successfully captured in HRNN (Costa-jussa & Fonollosa, 2016) and HM- `", "type": "text" } ], "index": 6 }, { "bbox": [ 105, 632, 506, 645 ], "spans": [ { "bbox": [ 105, 632, 506, 645 ], "score": 1.0, "content": "RNN (Chung et al., 2019). The former requires expert knowledge to partition the sequence into", "type": "text" } ], "index": 7 }, { "bbox": [ 105, 643, 505, 656 ], "spans": [ { "bbox": [ 105, 643, 505, 656 ], "score": 1.0, "content": "different resolutions, while the latter learns the partition automatically from the data. Note that the", "type": "text" } ], "index": 8 }, { "bbox": [ 106, 655, 505, 667 ], "spans": [ { "bbox": [ 106, 655, 424, 667 ], "score": 1.0, "content": "theoretical maximum length of the signal traversing path in both models is still", "type": "text" }, { "bbox": [ 424, 655, 447, 667 ], "score": 0.91, "content": "\\mathcal O ( L )", "type": "inline_equation" }, { "bbox": [ 448, 655, 505, 667 ], "score": 1.0, "content": ". Another line", "type": "text" } ], "index": 9 }, { "bbox": [ 106, 666, 505, 678 ], "spans": [ { "bbox": [ 106, 666, 505, 678 ], "score": 1.0, "content": "of works aim to shorten the signal traversing path by adding residual connections (Kim et al., 2017)", "type": "text" } ], "index": 10 }, { "bbox": [ 106, 677, 505, 689 ], "spans": [ { "bbox": [ 106, 677, 505, 689 ], "score": 1.0, "content": "or dilated connections to LSTMs (Chang et al., 2017). However, they do not consider the multires-", "type": "text" } ], "index": 11 }, { "bbox": [ 105, 687, 505, 702 ], "spans": [ { "bbox": [ 105, 687, 505, 702 ], "score": 1.0, "content": "olution temporal dependencies explicitly. Furthermore, all aforementioned RNNs only propagate", "type": "text" } ], "index": 12 }, { "bbox": [ 105, 698, 505, 712 ], "spans": [ { "bbox": [ 105, 698, 505, 712 ], "score": 1.0, "content": "information in one direction from the past to the future. An appealing approach that allows bidirec-", "type": "text" } ], "index": 13 }, { "bbox": [ 106, 710, 505, 722 ], "spans": [ { "bbox": [ 106, 710, 505, 722 ], "score": 1.0, "content": "tional information exchange is Bi-LSTM (Schuster, 1996). The forward and backward propagation", "type": "text" } ], "index": 14 }, { "bbox": [ 105, 721, 492, 733 ], "spans": [ { "bbox": [ 105, 721, 492, 733 ], "score": 1.0, "content": "is realized through two different LSTMs though, and so still incurs a long signal traversing path.", "type": "text" } ], "index": 15 } ], "index": 10 } ], "page_idx": 11, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 108, 27, 293, 37 ], "lines": [ { "bbox": [ 106, 26, 294, 38 ], "spans": [ { "bbox": [ 106, 26, 294, 38 ], "score": 1.0, "content": "Published as a conference paper at ICLR 2022", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 300, 751, 311, 760 ], "lines": [ { "bbox": [ 299, 750, 313, 764 ], "spans": [ { "bbox": [ 299, 750, 313, 764 ], "score": 1.0, "content": "12", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "table", "bbox": [ 110, 108, 488, 560 ], "blocks": [ { "type": "table_caption", "bbox": [ 241, 89, 370, 101 ], "group_id": 0, "lines": [ { "bbox": [ 240, 88, 370, 102 ], "spans": [ { "bbox": [ 240, 88, 370, 102 ], "score": 1.0, "content": "Table 4: Meanings of notations.", "type": "text" } ], "index": 0 } ], "index": 0 }, { "type": "table_body", "bbox": [ 110, 108, 488, 560 ], "group_id": 0, "lines": [ { "bbox": [ 110, 108, 488, 560 ], "spans": [ { "bbox": [ 110, 108, 488, 560 ], "score": 0.974, "html": "
NotationSizeMeaning
LConstantThe length of historical sequence.
GConstantThe number of global tokens in ETC.
MConstantThe length of future sequence to be predicted.
BConstantBatch size.
DConstantThe dimension of each node.
DKConstantThe dimension of a key.
XB×L×DInput of a single attention head.
YB×L×DOutput of a single attention head.
QB×L×DkThe query.
KB×L×DkThe key.
VB×L×DkThe value.
WQD×DkThe weight matrix of the query.
WKD ×DkThe weight matrix of the key.
WvD×DkThe weight matrix of the value.
SConstantNumber of scales.
AConstantNumber of adjacent nodes at the same scale that a node can attend to.
CConstantNumber of finer scale nodes that a coarser scale node can summarize.
NConstantNumber of attention layers.
n(DThe l-th node at scale s.
Nlen(N)) × D
Alen(A()） × D The adjacent A nodes at the same scale with n(s).
Clen(C()) × D
Plen(P(s) × D The parent node of n(s).
FpB×M×DThe prediction tokens.
FeB ×Ltot ×D The output of the encoder. Ltot represents the output length of the encod
Fd1B×M×DThe output of the first attention-based decoder layer.
HConstantThe number of attention heads.
DF
ConstantThe maximum dimension of the feed-forward layer.
", "type": "table", "image_path": "558772b46478afb75a19e4e430ee43cce92001ed79e2a8acfd3f505ea01429a0.jpg" } ] } ], "index": 2, "virtual_lines": [ { "bbox": [ 110, 108, 488, 258.66666666666663 ], "spans": [], "index": 1 }, { "bbox": [ 110, 258.66666666666663, 488, 409.33333333333326 ], "spans": [], "index": 2 }, { "bbox": [ 110, 409.33333333333326, 488, 559.9999999999999 ], "spans": [], "index": 3 } ] } ], "index": 1.0 }, { "type": "title", "bbox": [ 108, 584, 404, 597 ], "lines": [ { "bbox": [ 106, 584, 405, 599 ], "spans": [ { "bbox": [ 106, 584, 405, 599 ], "score": 1.0, "content": "A A BRIEF REVIEW ON RELATED RNN-BASED MODELS", "type": "text" } ], "index": 4 } ], "index": 4 }, { "type": "text", "bbox": [ 106, 610, 505, 732 ], "lines": [ { "bbox": [ 105, 611, 504, 623 ], "spans": [ { "bbox": [ 105, 611, 504, 623 ], "score": 1.0, "content": "In this section, we provide a brief review on the related RNN-based models. Multiscale tempo-", "type": "text" } ], "index": 5 }, { "bbox": [ 105, 622, 505, 634 ], "spans": [ { "bbox": [ 105, 622, 505, 634 ], "score": 1.0, "content": "ral dependencies are successfully captured in HRNN (Costa-jussa & Fonollosa, 2016) and HM- `", "type": "text" } ], "index": 6 }, { "bbox": [ 105, 632, 506, 645 ], "spans": [ { "bbox": [ 105, 632, 506, 645 ], "score": 1.0, "content": "RNN (Chung et al., 2019). The former requires expert knowledge to partition the sequence into", "type": "text" } ], "index": 7 }, { "bbox": [ 105, 643, 505, 656 ], "spans": [ { "bbox": [ 105, 643, 505, 656 ], "score": 1.0, "content": "different resolutions, while the latter learns the partition automatically from the data. Note that the", "type": "text" } ], "index": 8 }, { "bbox": [ 106, 655, 505, 667 ], "spans": [ { "bbox": [ 106, 655, 424, 667 ], "score": 1.0, "content": "theoretical maximum length of the signal traversing path in both models is still", "type": "text" }, { "bbox": [ 424, 655, 447, 667 ], "score": 0.91, "content": "\\mathcal O ( L )", "type": "inline_equation" }, { "bbox": [ 448, 655, 505, 667 ], "score": 1.0, "content": ". Another line", "type": "text" } ], "index": 9 }, { "bbox": [ 106, 666, 505, 678 ], "spans": [ { "bbox": [ 106, 666, 505, 678 ], "score": 1.0, "content": "of works aim to shorten the signal traversing path by adding residual connections (Kim et al., 2017)", "type": "text" } ], "index": 10 }, { "bbox": [ 106, 677, 505, 689 ], "spans": [ { "bbox": [ 106, 677, 505, 689 ], "score": 1.0, "content": "or dilated connections to LSTMs (Chang et al., 2017). However, they do not consider the multires-", "type": "text" } ], "index": 11 }, { "bbox": [ 105, 687, 505, 702 ], "spans": [ { "bbox": [ 105, 687, 505, 702 ], "score": 1.0, "content": "olution temporal dependencies explicitly. Furthermore, all aforementioned RNNs only propagate", "type": "text" } ], "index": 12 }, { "bbox": [ 105, 698, 505, 712 ], "spans": [ { "bbox": [ 105, 698, 505, 712 ], "score": 1.0, "content": "information in one direction from the past to the future. An appealing approach that allows bidirec-", "type": "text" } ], "index": 13 }, { "bbox": [ 106, 710, 505, 722 ], "spans": [ { "bbox": [ 106, 710, 505, 722 ], "score": 1.0, "content": "tional information exchange is Bi-LSTM (Schuster, 1996). The forward and backward propagation", "type": "text" } ], "index": 14 }, { "bbox": [ 105, 721, 492, 733 ], "spans": [ { "bbox": [ 105, 721, 492, 733 ], "score": 1.0, "content": "is realized through two different LSTMs though, and so still incurs a long signal traversing path.", "type": "text" } ], "index": 15 } ], "index": 10, "bbox_fs": [ 105, 611, 506, 733 ] } ] }, { "preproc_blocks": [ { "type": "text", "bbox": [ 107, 82, 505, 137 ], "lines": [ { "bbox": [ 106, 83, 504, 95 ], "spans": [ { "bbox": [ 106, 83, 504, 95 ], "score": 1.0, "content": "As opposed to the abovementioned RNN-based models, the proposed Pyraformer enables bidirec-", "type": "text" } ], "index": 0 }, { "bbox": [ 105, 93, 506, 107 ], "spans": [ { "bbox": [ 105, 93, 506, 107 ], "score": 1.0, "content": "tional information exchange that can better describe the temporal dependencies, while providing a", "type": "text" } ], "index": 1 }, { "bbox": [ 106, 105, 505, 117 ], "spans": [ { "bbox": [ 106, 105, 505, 117 ], "score": 1.0, "content": "multiresolution representation of the observed sequence at the same time. We also notice that due", "type": "text" } ], "index": 2 }, { "bbox": [ 105, 115, 506, 128 ], "spans": [ { "bbox": [ 105, 115, 506, 128 ], "score": 1.0, "content": "to the unidirectional property of RNNs, it is difficult the realize the pyramidal graph in Figure 1d", "type": "text" } ], "index": 3 }, { "bbox": [ 105, 126, 174, 138 ], "spans": [ { "bbox": [ 105, 126, 174, 138 ], "score": 1.0, "content": "based on RNNs.", "type": "text" } ], "index": 4 } ], "index": 2 }, { "type": "title", "bbox": [ 108, 153, 231, 166 ], "lines": [ { "bbox": [ 105, 152, 233, 168 ], "spans": [ { "bbox": [ 105, 152, 233, 168 ], "score": 1.0, "content": "B PROOF OF LEMMA 1", "type": "text" } ], "index": 5 } ], "index": 5 }, { "type": "text", "bbox": [ 106, 177, 505, 266 ], "lines": [ { "bbox": [ 106, 178, 505, 190 ], "spans": [ { "bbox": [ 106, 178, 152, 190 ], "score": 1.0, "content": "Proof. Let", "type": "text" }, { "bbox": [ 153, 178, 160, 188 ], "score": 0.78, "content": "S", "type": "inline_equation" }, { "bbox": [ 161, 178, 374, 190 ], "score": 1.0, "content": "denote the number of scales in the pyramidal graph,", "type": "text" }, { "bbox": [ 374, 178, 383, 188 ], "score": 0.81, "content": "C", "type": "inline_equation" }, { "bbox": [ 383, 178, 505, 190 ], "score": 1.0, "content": "the number of children nodes", "type": "text" } ], "index": 6 }, { "bbox": [ 105, 189, 504, 201 ], "spans": [ { "bbox": [ 105, 189, 177, 201 ], "score": 1.0, "content": "in the finer scale", "type": "text" }, { "bbox": [ 177, 189, 201, 199 ], "score": 0.87, "content": "s - 1", "type": "inline_equation" }, { "bbox": [ 201, 189, 347, 201 ], "score": 1.0, "content": "that a node in the the coarser scale", "type": "text" }, { "bbox": [ 347, 191, 353, 199 ], "score": 0.49, "content": "s", "type": "inline_equation" }, { "bbox": [ 354, 189, 434, 201 ], "score": 1.0, "content": "can summarize for", "type": "text" }, { "bbox": [ 434, 189, 504, 200 ], "score": 0.87, "content": "s = 2 , \\cdots , S , A", "type": "inline_equation" } ], "index": 7 }, { "bbox": [ 106, 200, 505, 211 ], "spans": [ { "bbox": [ 106, 200, 398, 211 ], "score": 1.0, "content": "the number of adjacent nodes that a node can attend to within each scale,", "type": "text" }, { "bbox": [ 398, 200, 408, 210 ], "score": 0.78, "content": "N", "type": "inline_equation" }, { "bbox": [ 409, 200, 505, 211 ], "score": 1.0, "content": "the number of attention", "type": "text" } ], "index": 8 }, { "bbox": [ 104, 210, 505, 224 ], "spans": [ { "bbox": [ 104, 210, 152, 224 ], "score": 1.0, "content": "layers, and", "type": "text" }, { "bbox": [ 152, 211, 160, 221 ], "score": 0.76, "content": "L", "type": "inline_equation" }, { "bbox": [ 161, 210, 505, 224 ], "score": 1.0, "content": "the length of the input time series. We define the term “receptive field” of an arbitrary", "type": "text" } ], "index": 9 }, { "bbox": [ 105, 222, 505, 234 ], "spans": [ { "bbox": [ 105, 222, 128, 234 ], "score": 1.0, "content": "node", "type": "text" }, { "bbox": [ 129, 223, 141, 233 ], "score": 0.86, "content": "n _ { a }", "type": "inline_equation" }, { "bbox": [ 141, 222, 285, 234 ], "score": 1.0, "content": "in a graph as the set of nodes that", "type": "text" }, { "bbox": [ 285, 223, 297, 233 ], "score": 0.86, "content": "n _ { a }", "type": "inline_equation" }, { "bbox": [ 298, 222, 505, 234 ], "score": 1.0, "content": "can receive messages from. We further define the", "type": "text" } ], "index": 10 }, { "bbox": [ 105, 232, 506, 245 ], "spans": [ { "bbox": [ 105, 232, 506, 245 ], "score": 1.0, "content": "distance between two arbitrary nodes in a graph as the length of the shortest path between them", "type": "text" } ], "index": 11 }, { "bbox": [ 106, 244, 505, 255 ], "spans": [ { "bbox": [ 106, 244, 505, 255 ], "score": 1.0, "content": "(i.e., the number of steps to travel from one node to another). Note that in each attention layer, the", "type": "text" } ], "index": 12 }, { "bbox": [ 105, 255, 308, 267 ], "spans": [ { "bbox": [ 105, 255, 308, 267 ], "score": 1.0, "content": "messages can only travel by one step in the graph.", "type": "text" } ], "index": 13 } ], "index": 9.5 }, { "type": "text", "bbox": [ 106, 271, 505, 360 ], "lines": [ { "bbox": [ 104, 270, 507, 284 ], "spans": [ { "bbox": [ 104, 270, 301, 284 ], "score": 1.0, "content": "Without sacrificing generality, we assume that", "type": "text" }, { "bbox": [ 301, 272, 309, 281 ], "score": 0.78, "content": "L", "type": "inline_equation" }, { "bbox": [ 310, 270, 374, 284 ], "score": 1.0, "content": "is divisible by", "type": "text" }, { "bbox": [ 374, 271, 399, 282 ], "score": 0.9, "content": "C ^ { S - 1 }", "type": "inline_equation" }, { "bbox": [ 399, 270, 507, 284 ], "score": 1.0, "content": ", and then the number of", "type": "text" } ], "index": 14 }, { "bbox": [ 104, 280, 507, 295 ], "spans": [ { "bbox": [ 104, 280, 219, 295 ], "score": 1.0, "content": "nodes at the coarsest scale", "type": "text" }, { "bbox": [ 220, 283, 228, 292 ], "score": 0.81, "content": "S", "type": "inline_equation" }, { "bbox": [ 228, 280, 240, 295 ], "score": 1.0, "content": "is", "type": "text" }, { "bbox": [ 241, 282, 277, 294 ], "score": 0.93, "content": "L / C ^ { S - 1 }", "type": "inline_equation" }, { "bbox": [ 277, 280, 426, 295 ], "score": 1.0, "content": ". Since every node is connected to", "type": "text" }, { "bbox": [ 426, 283, 435, 292 ], "score": 0.69, "content": "A", "type": "inline_equation" }, { "bbox": [ 435, 280, 507, 295 ], "score": 1.0, "content": "closest nodes at", "type": "text" } ], "index": 15 }, { "bbox": [ 106, 294, 505, 305 ], "spans": [ { "bbox": [ 106, 294, 505, 305 ], "score": 1.0, "content": "the same scale, the distance between the leftmost and the rightmost node at the coarsest scale is", "type": "text" } ], "index": 16 }, { "bbox": [ 106, 302, 506, 318 ], "spans": [ { "bbox": [ 106, 304, 212, 316 ], "score": 0.91, "content": "2 ( L / C ^ { S - 1 } - 1 ) / ( A - 1 )", "type": "inline_equation" }, { "bbox": [ 212, 302, 506, 318 ], "score": 1.0, "content": ". Hence, the leftmost and the rightmost node in the coarsest scale are in", "type": "text" } ], "index": 17 }, { "bbox": [ 104, 312, 506, 329 ], "spans": [ { "bbox": [ 104, 312, 315, 329 ], "score": 1.0, "content": "the receptive field of each other after the stack of", "type": "text" }, { "bbox": [ 315, 315, 448, 327 ], "score": 0.92, "content": "N \\geq 2 ( L / C ^ { S - 1 } - 1 ) / ( A - 1 )", "type": "inline_equation" }, { "bbox": [ 449, 312, 506, 329 ], "score": 1.0, "content": "layers of the", "type": "text" } ], "index": 18 }, { "bbox": [ 105, 326, 506, 339 ], "spans": [ { "bbox": [ 105, 326, 506, 339 ], "score": 1.0, "content": "pyramidal attention. In addition, owing to the CSCM, nodes at the coarsest scale can be regarded as", "type": "text" } ], "index": 19 }, { "bbox": [ 106, 338, 506, 349 ], "spans": [ { "bbox": [ 106, 338, 506, 349 ], "score": 1.0, "content": "the summary of the nodes in the finer scales. As a result, when Equation (4) is satisfied, all nodes at", "type": "text" } ], "index": 20 }, { "bbox": [ 106, 348, 506, 361 ], "spans": [ { "bbox": [ 106, 348, 389, 361 ], "score": 1.0, "content": "the coarsest scale have a global receptive field, which closes the proof.", "type": "text" }, { "bbox": [ 494, 348, 506, 359 ], "score": 1.0, "content": "□", "type": "text" } ], "index": 21 } ], "index": 17.5 }, { "type": "title", "bbox": [ 108, 375, 260, 388 ], "lines": [ { "bbox": [ 106, 374, 263, 390 ], "spans": [ { "bbox": [ 106, 374, 263, 390 ], "score": 1.0, "content": "C PROOF OF PROPOSITION 1", "type": "text" } ], "index": 22 } ], "index": 22 }, { "type": "text", "bbox": [ 107, 399, 397, 411 ], "lines": [ { "bbox": [ 104, 397, 399, 414 ], "spans": [ { "bbox": [ 104, 397, 190, 414 ], "score": 1.0, "content": "Proof. Suppose that", "type": "text" }, { "bbox": [ 190, 398, 208, 410 ], "score": 0.9, "content": "L ^ { ( s ) }", "type": "inline_equation" }, { "bbox": [ 208, 397, 358, 414 ], "score": 1.0, "content": "denotes the number of nodes at scale", "type": "text" }, { "bbox": [ 359, 402, 365, 410 ], "score": 0.77, "content": "s", "type": "inline_equation" }, { "bbox": [ 365, 397, 399, 414 ], "score": 1.0, "content": ", that is,", "type": "text" } ], "index": 23 } ], "index": 23 }, { "type": "interline_equation", "bbox": [ 252, 413, 359, 437 ], "lines": [ { "bbox": [ 252, 413, 359, 437 ], "spans": [ { "bbox": [ 252, 413, 359, 437 ], "score": 0.95, "content": "L ^ { ( s ) } = \\frac { L } { C ^ { s - 1 } } , 1 \\leq s \\leq S .", "type": "interline_equation", "image_path": "b5f1b491f95c5fece313706172541d0f149f4f2138e1d758130be0c11d50b1f2.jpg" } ] } ], "index": 24, "virtual_lines": [ { "bbox": [ 252, 413, 359, 437 ], "spans": [], "index": 24 } ] }, { "type": "text", "bbox": [ 107, 440, 504, 465 ], "lines": [ { "bbox": [ 105, 435, 506, 463 ], "spans": [ { "bbox": [ 105, 435, 151, 463 ], "score": 1.0, "content": "For a node", "type": "text" }, { "bbox": [ 152, 439, 169, 455 ], "score": 0.91, "content": "n _ { \\ell } ^ { ( s ) }", "type": "inline_equation" }, { "bbox": [ 170, 435, 378, 463 ], "score": 1.0, "content": "in the pyramidal graph, the number of dot products", "type": "text" }, { "bbox": [ 379, 439, 397, 455 ], "score": 0.92, "content": "P _ { \\ell } ^ { ( s ) }", "type": "inline_equation" }, { "bbox": [ 398, 435, 506, 463 ], "score": 1.0, "content": "it acts as the query can be", "type": "text" } ], "index": 25 }, { "bbox": [ 106, 453, 219, 466 ], "spans": [ { "bbox": [ 106, 453, 219, 466 ], "score": 1.0, "content": "decomposed into two parts:", "type": "text" } ], "index": 26 } ], "index": 25.5 }, { "type": "interline_equation", "bbox": [ 249, 466, 361, 485 ], "lines": [ { "bbox": [ 249, 466, 361, 485 ], "spans": [ { "bbox": [ 249, 466, 361, 485 ], "score": 0.93, "content": "P _ { \\ell } ^ { ( s ) } = P _ { \\ell } ^ { ( s ) } { } _ { \\mathrm { i n t e r } } + P _ { \\ell } ^ { ( s ) } { } _ { \\mathrm { i n t r a } } ,", "type": "interline_equation", "image_path": "2975e1a0f9a1eb1b5bffae2f82beb2acc57c16f054675de0e3c15f7727419fe7.jpg" } ] } ], "index": 27, "virtual_lines": [ { "bbox": [ 249, 466, 361, 485 ], "spans": [], "index": 27 } ] }, { "type": "text", "bbox": [ 105, 489, 504, 513 ], "lines": [ { "bbox": [ 104, 484, 505, 509 ], "spans": [ { "bbox": [ 104, 486, 156, 502 ], "score": 1.0, "content": "where P (s)", "type": "text" }, { "bbox": [ 134, 487, 167, 503 ], "score": 0.92, "content": "P _ { \\ell } ^ { ( s ) } { _ { \\mathrm { i n t r a } } }", "type": "inline_equation" }, { "bbox": [ 162, 484, 212, 509 ], "score": 1.0, "content": "a and P (s)ℓ int", "type": "text" }, { "bbox": [ 218, 488, 505, 505 ], "score": 1.0, "content": "denotes the intra-scale and the inter-scale part respectively. According", "type": "text" } ], "index": 28 }, { "bbox": [ 104, 500, 424, 514 ], "spans": [ { "bbox": [ 104, 500, 424, 514 ], "score": 1.0, "content": "to the structure of the pyramidal graph, we can have the following inequalities:", "type": "text" } ], "index": 29 } ], "index": 28.5 }, { "type": "interline_equation", "bbox": [ 267, 514, 343, 550 ], "lines": [ { "bbox": [ 267, 514, 343, 550 ], "spans": [ { "bbox": [ 267, 514, 343, 550 ], "score": 0.88, "content": "\\begin{array} { r l } & { P _ { \\ell \\mathrm { \\tiny ~ \\min i n t r a } } ^ { ( s ) } \\le A , } \\\\ & { P _ { \\ell \\mathrm { \\tiny ~ \\min t e r } } ^ { ( s ) } \\le C + 1 . } \\end{array}", "type": "interline_equation", "image_path": "cea9cf030e4643444de64d9af5181bbcc2c41561080ae816e36a4dfd39cd8b1c.jpg" } ] } ], "index": 30.5, "virtual_lines": [ { "bbox": [ 267, 514, 343, 532.0 ], "spans": [], "index": 30 }, { "bbox": [ 267, 532.0, 343, 550.0 ], "spans": [], "index": 31 } ] }, { "type": "text", "bbox": [ 107, 550, 505, 606 ], "lines": [ { "bbox": [ 106, 551, 505, 563 ], "spans": [ { "bbox": [ 106, 551, 360, 563 ], "score": 1.0, "content": "The first inequality (8) holds since a node typically attends to", "type": "text" }, { "bbox": [ 361, 551, 369, 561 ], "score": 0.77, "content": "A", "type": "inline_equation" }, { "bbox": [ 370, 551, 505, 563 ], "score": 1.0, "content": "most adjacent nodes at the same", "type": "text" } ], "index": 32 }, { "bbox": [ 106, 561, 505, 573 ], "spans": [ { "bbox": [ 106, 561, 505, 573 ], "score": 1.0, "content": "scale but for the leftmost and the rightmost node, the number of in-scale nodes it can attend to is", "type": "text" } ], "index": 33 }, { "bbox": [ 105, 573, 505, 585 ], "spans": [ { "bbox": [ 105, 573, 160, 585 ], "score": 1.0, "content": "smaller than", "type": "text" }, { "bbox": [ 160, 573, 168, 583 ], "score": 0.75, "content": "A", "type": "inline_equation" }, { "bbox": [ 169, 573, 505, 585 ], "score": 1.0, "content": ". On the other hand, the second inequality (9) holds because a node typically has", "type": "text" } ], "index": 34 }, { "bbox": [ 107, 583, 505, 597 ], "spans": [ { "bbox": [ 107, 584, 115, 594 ], "score": 0.79, "content": "C", "type": "inline_equation" }, { "bbox": [ 116, 583, 505, 597 ], "score": 1.0, "content": "children and 1 parent in the pyramidal graph but nodes at the top and the bottom scale can only", "type": "text" } ], "index": 35 }, { "bbox": [ 105, 595, 315, 608 ], "spans": [ { "bbox": [ 105, 595, 188, 608 ], "score": 1.0, "content": "attend to fewer than", "type": "text" }, { "bbox": [ 188, 595, 214, 605 ], "score": 0.9, "content": "C + 1", "type": "inline_equation" }, { "bbox": [ 215, 595, 315, 608 ], "score": 1.0, "content": "nodes at adjacent scales.", "type": "text" } ], "index": 36 } ], "index": 34 }, { "type": "text", "bbox": [ 106, 611, 427, 623 ], "lines": [ { "bbox": [ 105, 610, 429, 625 ], "spans": [ { "bbox": [ 105, 610, 409, 625 ], "score": 1.0, "content": "In summary, the number of dot products that need to be calculated for scale", "type": "text" }, { "bbox": [ 410, 614, 415, 622 ], "score": 0.74, "content": "s", "type": "inline_equation" }, { "bbox": [ 416, 610, 429, 625 ], "score": 1.0, "content": "is:", "type": "text" } ], "index": 37 } ], "index": 37 }, { "type": "interline_equation", "bbox": [ 195, 625, 416, 662 ], "lines": [ { "bbox": [ 195, 625, 416, 662 ], "spans": [ { "bbox": [ 195, 625, 416, 662 ], "score": 0.94, "content": "P ^ { ( s ) } = \\sum _ { \\ell = 1 } ^ { L ^ { ( s ) } } \\big ( P _ { \\ell \\mathrm { \\tiny ~ \\mathrm { ~ i n t r a } } } ^ { ( s ) } + P _ { \\ell \\mathrm { \\tiny ~ \\mathrm { ~ i n t e r } } } ^ { ( s ) } \\big ) \\le L ^ { ( s ) } ( A + C + 1 ) .", "type": "interline_equation", "image_path": "193fc7d128d8a5850b6d4ec754f889a089ea0c9aa1cf90ffd011664f378c8bcd.jpg" } ] } ], "index": 38.5, "virtual_lines": [ { "bbox": [ 195, 625, 416, 643.5 ], "spans": [], "index": 38 }, { "bbox": [ 195, 643.5, 416, 662.0 ], "spans": [], "index": 39 } ] }, { "type": "text", "bbox": [ 108, 665, 505, 699 ], "lines": [ { "bbox": [ 105, 663, 505, 679 ], "spans": [ { "bbox": [ 105, 663, 147, 679 ], "score": 1.0, "content": "Note that", "type": "text" }, { "bbox": [ 147, 664, 222, 678 ], "score": 0.92, "content": "P ^ { ( 1 ) } \\leq L ( A + 1 )", "type": "inline_equation" }, { "bbox": [ 222, 663, 322, 679 ], "score": 1.0, "content": "for the finest scale (i.e.,", "type": "text" }, { "bbox": [ 323, 667, 349, 676 ], "score": 0.87, "content": "s = 1", "type": "inline_equation" }, { "bbox": [ 350, 663, 505, 679 ], "score": 1.0, "content": ") since nodes at this scale do not have", "type": "text" } ], "index": 40 }, { "bbox": [ 105, 676, 505, 689 ], "spans": [ { "bbox": [ 105, 676, 505, 689 ], "score": 1.0, "content": "any children. It follows that the number of dot products that need to be calculated for the entire", "type": "text" } ], "index": 41 }, { "bbox": [ 105, 687, 222, 700 ], "spans": [ { "bbox": [ 105, 687, 222, 700 ], "score": 1.0, "content": "pyramidal attention layer is:", "type": "text" } ], "index": 42 } ], "index": 41 }, { "type": "interline_equation", "bbox": [ 181, 701, 239, 735 ], "lines": [ { "bbox": [ 181, 701, 239, 735 ], "spans": [ { "bbox": [ 181, 701, 239, 735 ], "score": 0.92, "content": "P = \\sum _ { s = 1 } ^ { S } P ^ { ( s ) }", "type": "interline_equation", "image_path": "27e0bdb332828aee29138af544adda18db9e17d5d6dd7c4f4e6fa80c9ef86533.jpg" } ] } ], "index": 43.5, "virtual_lines": [ { "bbox": [ 181, 701, 239, 718.0 ], "spans": [], "index": 43 }, { "bbox": [ 181, 718.0, 239, 735.0 ], "spans": [], "index": 44 } ] } ], "page_idx": 12, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 108, 27, 293, 37 ], "lines": [ { "bbox": [ 106, 25, 293, 38 ], "spans": [ { "bbox": [ 106, 25, 293, 38 ], "score": 1.0, "content": "Published as a conference paper at ICLR 2022", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 300, 751, 311, 760 ], "lines": [ { "bbox": [ 299, 750, 313, 764 ], "spans": [ { "bbox": [ 299, 750, 313, 764 ], "score": 1.0, "content": "13", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "text", "bbox": [ 107, 82, 505, 137 ], "lines": [ { "bbox": [ 106, 83, 504, 95 ], "spans": [ { "bbox": [ 106, 83, 504, 95 ], "score": 1.0, "content": "As opposed to the abovementioned RNN-based models, the proposed Pyraformer enables bidirec-", "type": "text" } ], "index": 0 }, { "bbox": [ 105, 93, 506, 107 ], "spans": [ { "bbox": [ 105, 93, 506, 107 ], "score": 1.0, "content": "tional information exchange that can better describe the temporal dependencies, while providing a", "type": "text" } ], "index": 1 }, { "bbox": [ 106, 105, 505, 117 ], "spans": [ { "bbox": [ 106, 105, 505, 117 ], "score": 1.0, "content": "multiresolution representation of the observed sequence at the same time. We also notice that due", "type": "text" } ], "index": 2 }, { "bbox": [ 105, 115, 506, 128 ], "spans": [ { "bbox": [ 105, 115, 506, 128 ], "score": 1.0, "content": "to the unidirectional property of RNNs, it is difficult the realize the pyramidal graph in Figure 1d", "type": "text" } ], "index": 3 }, { "bbox": [ 105, 126, 174, 138 ], "spans": [ { "bbox": [ 105, 126, 174, 138 ], "score": 1.0, "content": "based on RNNs.", "type": "text" } ], "index": 4 } ], "index": 2, "bbox_fs": [ 105, 83, 506, 138 ] }, { "type": "title", "bbox": [ 108, 153, 231, 166 ], "lines": [ { "bbox": [ 105, 152, 233, 168 ], "spans": [ { "bbox": [ 105, 152, 233, 168 ], "score": 1.0, "content": "B PROOF OF LEMMA 1", "type": "text" } ], "index": 5 } ], "index": 5 }, { "type": "text", "bbox": [ 106, 177, 505, 266 ], "lines": [ { "bbox": [ 106, 178, 505, 190 ], "spans": [ { "bbox": [ 106, 178, 152, 190 ], "score": 1.0, "content": "Proof. Let", "type": "text" }, { "bbox": [ 153, 178, 160, 188 ], "score": 0.78, "content": "S", "type": "inline_equation" }, { "bbox": [ 161, 178, 374, 190 ], "score": 1.0, "content": "denote the number of scales in the pyramidal graph,", "type": "text" }, { "bbox": [ 374, 178, 383, 188 ], "score": 0.81, "content": "C", "type": "inline_equation" }, { "bbox": [ 383, 178, 505, 190 ], "score": 1.0, "content": "the number of children nodes", "type": "text" } ], "index": 6 }, { "bbox": [ 105, 189, 504, 201 ], "spans": [ { "bbox": [ 105, 189, 177, 201 ], "score": 1.0, "content": "in the finer scale", "type": "text" }, { "bbox": [ 177, 189, 201, 199 ], "score": 0.87, "content": "s - 1", "type": "inline_equation" }, { "bbox": [ 201, 189, 347, 201 ], "score": 1.0, "content": "that a node in the the coarser scale", "type": "text" }, { "bbox": [ 347, 191, 353, 199 ], "score": 0.49, "content": "s", "type": "inline_equation" }, { "bbox": [ 354, 189, 434, 201 ], "score": 1.0, "content": "can summarize for", "type": "text" }, { "bbox": [ 434, 189, 504, 200 ], "score": 0.87, "content": "s = 2 , \\cdots , S , A", "type": "inline_equation" } ], "index": 7 }, { "bbox": [ 106, 200, 505, 211 ], "spans": [ { "bbox": [ 106, 200, 398, 211 ], "score": 1.0, "content": "the number of adjacent nodes that a node can attend to within each scale,", "type": "text" }, { "bbox": [ 398, 200, 408, 210 ], "score": 0.78, "content": "N", "type": "inline_equation" }, { "bbox": [ 409, 200, 505, 211 ], "score": 1.0, "content": "the number of attention", "type": "text" } ], "index": 8 }, { "bbox": [ 104, 210, 505, 224 ], "spans": [ { "bbox": [ 104, 210, 152, 224 ], "score": 1.0, "content": "layers, and", "type": "text" }, { "bbox": [ 152, 211, 160, 221 ], "score": 0.76, "content": "L", "type": "inline_equation" }, { "bbox": [ 161, 210, 505, 224 ], "score": 1.0, "content": "the length of the input time series. We define the term “receptive field” of an arbitrary", "type": "text" } ], "index": 9 }, { "bbox": [ 105, 222, 505, 234 ], "spans": [ { "bbox": [ 105, 222, 128, 234 ], "score": 1.0, "content": "node", "type": "text" }, { "bbox": [ 129, 223, 141, 233 ], "score": 0.86, "content": "n _ { a }", "type": "inline_equation" }, { "bbox": [ 141, 222, 285, 234 ], "score": 1.0, "content": "in a graph as the set of nodes that", "type": "text" }, { "bbox": [ 285, 223, 297, 233 ], "score": 0.86, "content": "n _ { a }", "type": "inline_equation" }, { "bbox": [ 298, 222, 505, 234 ], "score": 1.0, "content": "can receive messages from. We further define the", "type": "text" } ], "index": 10 }, { "bbox": [ 105, 232, 506, 245 ], "spans": [ { "bbox": [ 105, 232, 506, 245 ], "score": 1.0, "content": "distance between two arbitrary nodes in a graph as the length of the shortest path between them", "type": "text" } ], "index": 11 }, { "bbox": [ 106, 244, 505, 255 ], "spans": [ { "bbox": [ 106, 244, 505, 255 ], "score": 1.0, "content": "(i.e., the number of steps to travel from one node to another). Note that in each attention layer, the", "type": "text" } ], "index": 12 }, { "bbox": [ 105, 255, 308, 267 ], "spans": [ { "bbox": [ 105, 255, 308, 267 ], "score": 1.0, "content": "messages can only travel by one step in the graph.", "type": "text" } ], "index": 13 } ], "index": 9.5, "bbox_fs": [ 104, 178, 506, 267 ] }, { "type": "text", "bbox": [ 106, 271, 505, 360 ], "lines": [ { "bbox": [ 104, 270, 507, 284 ], "spans": [ { "bbox": [ 104, 270, 301, 284 ], "score": 1.0, "content": "Without sacrificing generality, we assume that", "type": "text" }, { "bbox": [ 301, 272, 309, 281 ], "score": 0.78, "content": "L", "type": "inline_equation" }, { "bbox": [ 310, 270, 374, 284 ], "score": 1.0, "content": "is divisible by", "type": "text" }, { "bbox": [ 374, 271, 399, 282 ], "score": 0.9, "content": "C ^ { S - 1 }", "type": "inline_equation" }, { "bbox": [ 399, 270, 507, 284 ], "score": 1.0, "content": ", and then the number of", "type": "text" } ], "index": 14 }, { "bbox": [ 104, 280, 507, 295 ], "spans": [ { "bbox": [ 104, 280, 219, 295 ], "score": 1.0, "content": "nodes at the coarsest scale", "type": "text" }, { "bbox": [ 220, 283, 228, 292 ], "score": 0.81, "content": "S", "type": "inline_equation" }, { "bbox": [ 228, 280, 240, 295 ], "score": 1.0, "content": "is", "type": "text" }, { "bbox": [ 241, 282, 277, 294 ], "score": 0.93, "content": "L / C ^ { S - 1 }", "type": "inline_equation" }, { "bbox": [ 277, 280, 426, 295 ], "score": 1.0, "content": ". Since every node is connected to", "type": "text" }, { "bbox": [ 426, 283, 435, 292 ], "score": 0.69, "content": "A", "type": "inline_equation" }, { "bbox": [ 435, 280, 507, 295 ], "score": 1.0, "content": "closest nodes at", "type": "text" } ], "index": 15 }, { "bbox": [ 106, 294, 505, 305 ], "spans": [ { "bbox": [ 106, 294, 505, 305 ], "score": 1.0, "content": "the same scale, the distance between the leftmost and the rightmost node at the coarsest scale is", "type": "text" } ], "index": 16 }, { "bbox": [ 106, 302, 506, 318 ], "spans": [ { "bbox": [ 106, 304, 212, 316 ], "score": 0.91, "content": "2 ( L / C ^ { S - 1 } - 1 ) / ( A - 1 )", "type": "inline_equation" }, { "bbox": [ 212, 302, 506, 318 ], "score": 1.0, "content": ". Hence, the leftmost and the rightmost node in the coarsest scale are in", "type": "text" } ], "index": 17 }, { "bbox": [ 104, 312, 506, 329 ], "spans": [ { "bbox": [ 104, 312, 315, 329 ], "score": 1.0, "content": "the receptive field of each other after the stack of", "type": "text" }, { "bbox": [ 315, 315, 448, 327 ], "score": 0.92, "content": "N \\geq 2 ( L / C ^ { S - 1 } - 1 ) / ( A - 1 )", "type": "inline_equation" }, { "bbox": [ 449, 312, 506, 329 ], "score": 1.0, "content": "layers of the", "type": "text" } ], "index": 18 }, { "bbox": [ 105, 326, 506, 339 ], "spans": [ { "bbox": [ 105, 326, 506, 339 ], "score": 1.0, "content": "pyramidal attention. In addition, owing to the CSCM, nodes at the coarsest scale can be regarded as", "type": "text" } ], "index": 19 }, { "bbox": [ 106, 338, 506, 349 ], "spans": [ { "bbox": [ 106, 338, 506, 349 ], "score": 1.0, "content": "the summary of the nodes in the finer scales. As a result, when Equation (4) is satisfied, all nodes at", "type": "text" } ], "index": 20 }, { "bbox": [ 106, 348, 506, 361 ], "spans": [ { "bbox": [ 106, 348, 389, 361 ], "score": 1.0, "content": "the coarsest scale have a global receptive field, which closes the proof.", "type": "text" }, { "bbox": [ 494, 348, 506, 359 ], "score": 1.0, "content": "□", "type": "text" } ], "index": 21 } ], "index": 17.5, "bbox_fs": [ 104, 270, 507, 361 ] }, { "type": "title", "bbox": [ 108, 375, 260, 388 ], "lines": [ { "bbox": [ 106, 374, 263, 390 ], "spans": [ { "bbox": [ 106, 374, 263, 390 ], "score": 1.0, "content": "C PROOF OF PROPOSITION 1", "type": "text" } ], "index": 22 } ], "index": 22 }, { "type": "text", "bbox": [ 107, 399, 397, 411 ], "lines": [ { "bbox": [ 104, 397, 399, 414 ], "spans": [ { "bbox": [ 104, 397, 190, 414 ], "score": 1.0, "content": "Proof. Suppose that", "type": "text" }, { "bbox": [ 190, 398, 208, 410 ], "score": 0.9, "content": "L ^ { ( s ) }", "type": "inline_equation" }, { "bbox": [ 208, 397, 358, 414 ], "score": 1.0, "content": "denotes the number of nodes at scale", "type": "text" }, { "bbox": [ 359, 402, 365, 410 ], "score": 0.77, "content": "s", "type": "inline_equation" }, { "bbox": [ 365, 397, 399, 414 ], "score": 1.0, "content": ", that is,", "type": "text" } ], "index": 23 } ], "index": 23, "bbox_fs": [ 104, 397, 399, 414 ] }, { "type": "interline_equation", "bbox": [ 252, 413, 359, 437 ], "lines": [ { "bbox": [ 252, 413, 359, 437 ], "spans": [ { "bbox": [ 252, 413, 359, 437 ], "score": 0.95, "content": "L ^ { ( s ) } = \\frac { L } { C ^ { s - 1 } } , 1 \\leq s \\leq S .", "type": "interline_equation", "image_path": "b5f1b491f95c5fece313706172541d0f149f4f2138e1d758130be0c11d50b1f2.jpg" } ] } ], "index": 24, "virtual_lines": [ { "bbox": [ 252, 413, 359, 437 ], "spans": [], "index": 24 } ] }, { "type": "text", "bbox": [ 107, 440, 504, 465 ], "lines": [ { "bbox": [ 105, 435, 506, 463 ], "spans": [ { "bbox": [ 105, 435, 151, 463 ], "score": 1.0, "content": "For a node", "type": "text" }, { "bbox": [ 152, 439, 169, 455 ], "score": 0.91, "content": "n _ { \\ell } ^ { ( s ) }", "type": "inline_equation" }, { "bbox": [ 170, 435, 378, 463 ], "score": 1.0, "content": "in the pyramidal graph, the number of dot products", "type": "text" }, { "bbox": [ 379, 439, 397, 455 ], "score": 0.92, "content": "P _ { \\ell } ^ { ( s ) }", "type": "inline_equation" }, { "bbox": [ 398, 435, 506, 463 ], "score": 1.0, "content": "it acts as the query can be", "type": "text" } ], "index": 25 }, { "bbox": [ 106, 453, 219, 466 ], "spans": [ { "bbox": [ 106, 453, 219, 466 ], "score": 1.0, "content": "decomposed into two parts:", "type": "text" } ], "index": 26 } ], "index": 25.5, "bbox_fs": [ 105, 435, 506, 466 ] }, { "type": "interline_equation", "bbox": [ 249, 466, 361, 485 ], "lines": [ { "bbox": [ 249, 466, 361, 485 ], "spans": [ { "bbox": [ 249, 466, 361, 485 ], "score": 0.93, "content": "P _ { \\ell } ^ { ( s ) } = P _ { \\ell } ^ { ( s ) } { } _ { \\mathrm { i n t e r } } + P _ { \\ell } ^ { ( s ) } { } _ { \\mathrm { i n t r a } } ,", "type": "interline_equation", "image_path": "2975e1a0f9a1eb1b5bffae2f82beb2acc57c16f054675de0e3c15f7727419fe7.jpg" } ] } ], "index": 27, "virtual_lines": [ { "bbox": [ 249, 466, 361, 485 ], "spans": [], "index": 27 } ] }, { "type": "text", "bbox": [ 105, 489, 504, 513 ], "lines": [ { "bbox": [ 104, 484, 505, 509 ], "spans": [ { "bbox": [ 104, 486, 156, 502 ], "score": 1.0, "content": "where P (s)", "type": "text" }, { "bbox": [ 134, 487, 167, 503 ], "score": 0.92, "content": "P _ { \\ell } ^ { ( s ) } { _ { \\mathrm { i n t r a } } }", "type": "inline_equation" }, { "bbox": [ 162, 484, 212, 509 ], "score": 1.0, "content": "a and P (s)ℓ int", "type": "text" }, { "bbox": [ 218, 488, 505, 505 ], "score": 1.0, "content": "denotes the intra-scale and the inter-scale part respectively. According", "type": "text" } ], "index": 28 }, { "bbox": [ 104, 500, 424, 514 ], "spans": [ { "bbox": [ 104, 500, 424, 514 ], "score": 1.0, "content": "to the structure of the pyramidal graph, we can have the following inequalities:", "type": "text" } ], "index": 29 } ], "index": 28.5, "bbox_fs": [ 104, 484, 505, 514 ] }, { "type": "interline_equation", "bbox": [ 267, 514, 343, 550 ], "lines": [ { "bbox": [ 267, 514, 343, 550 ], "spans": [ { "bbox": [ 267, 514, 343, 550 ], "score": 0.88, "content": "\\begin{array} { r l } & { P _ { \\ell \\mathrm { \\tiny ~ \\min i n t r a } } ^ { ( s ) } \\le A , } \\\\ & { P _ { \\ell \\mathrm { \\tiny ~ \\min t e r } } ^ { ( s ) } \\le C + 1 . } \\end{array}", "type": "interline_equation", "image_path": "cea9cf030e4643444de64d9af5181bbcc2c41561080ae816e36a4dfd39cd8b1c.jpg" } ] } ], "index": 30.5, "virtual_lines": [ { "bbox": [ 267, 514, 343, 532.0 ], "spans": [], "index": 30 }, { "bbox": [ 267, 532.0, 343, 550.0 ], "spans": [], "index": 31 } ] }, { "type": "text", "bbox": [ 107, 550, 505, 606 ], "lines": [ { "bbox": [ 106, 551, 505, 563 ], "spans": [ { "bbox": [ 106, 551, 360, 563 ], "score": 1.0, "content": "The first inequality (8) holds since a node typically attends to", "type": "text" }, { "bbox": [ 361, 551, 369, 561 ], "score": 0.77, "content": "A", "type": "inline_equation" }, { "bbox": [ 370, 551, 505, 563 ], "score": 1.0, "content": "most adjacent nodes at the same", "type": "text" } ], "index": 32 }, { "bbox": [ 106, 561, 505, 573 ], "spans": [ { "bbox": [ 106, 561, 505, 573 ], "score": 1.0, "content": "scale but for the leftmost and the rightmost node, the number of in-scale nodes it can attend to is", "type": "text" } ], "index": 33 }, { "bbox": [ 105, 573, 505, 585 ], "spans": [ { "bbox": [ 105, 573, 160, 585 ], "score": 1.0, "content": "smaller than", "type": "text" }, { "bbox": [ 160, 573, 168, 583 ], "score": 0.75, "content": "A", "type": "inline_equation" }, { "bbox": [ 169, 573, 505, 585 ], "score": 1.0, "content": ". On the other hand, the second inequality (9) holds because a node typically has", "type": "text" } ], "index": 34 }, { "bbox": [ 107, 583, 505, 597 ], "spans": [ { "bbox": [ 107, 584, 115, 594 ], "score": 0.79, "content": "C", "type": "inline_equation" }, { "bbox": [ 116, 583, 505, 597 ], "score": 1.0, "content": "children and 1 parent in the pyramidal graph but nodes at the top and the bottom scale can only", "type": "text" } ], "index": 35 }, { "bbox": [ 105, 595, 315, 608 ], "spans": [ { "bbox": [ 105, 595, 188, 608 ], "score": 1.0, "content": "attend to fewer than", "type": "text" }, { "bbox": [ 188, 595, 214, 605 ], "score": 0.9, "content": "C + 1", "type": "inline_equation" }, { "bbox": [ 215, 595, 315, 608 ], "score": 1.0, "content": "nodes at adjacent scales.", "type": "text" } ], "index": 36 } ], "index": 34, "bbox_fs": [ 105, 551, 505, 608 ] }, { "type": "text", "bbox": [ 106, 611, 427, 623 ], "lines": [ { "bbox": [ 105, 610, 429, 625 ], "spans": [ { "bbox": [ 105, 610, 409, 625 ], "score": 1.0, "content": "In summary, the number of dot products that need to be calculated for scale", "type": "text" }, { "bbox": [ 410, 614, 415, 622 ], "score": 0.74, "content": "s", "type": "inline_equation" }, { "bbox": [ 416, 610, 429, 625 ], "score": 1.0, "content": "is:", "type": "text" } ], "index": 37 } ], "index": 37, "bbox_fs": [ 105, 610, 429, 625 ] }, { "type": "interline_equation", "bbox": [ 195, 625, 416, 662 ], "lines": [ { "bbox": [ 195, 625, 416, 662 ], "spans": [ { "bbox": [ 195, 625, 416, 662 ], "score": 0.94, "content": "P ^ { ( s ) } = \\sum _ { \\ell = 1 } ^ { L ^ { ( s ) } } \\big ( P _ { \\ell \\mathrm { \\tiny ~ \\mathrm { ~ i n t r a } } } ^ { ( s ) } + P _ { \\ell \\mathrm { \\tiny ~ \\mathrm { ~ i n t e r } } } ^ { ( s ) } \\big ) \\le L ^ { ( s ) } ( A + C + 1 ) .", "type": "interline_equation", "image_path": "193fc7d128d8a5850b6d4ec754f889a089ea0c9aa1cf90ffd011664f378c8bcd.jpg" } ] } ], "index": 38.5, "virtual_lines": [ { "bbox": [ 195, 625, 416, 643.5 ], "spans": [], "index": 38 }, { "bbox": [ 195, 643.5, 416, 662.0 ], "spans": [], "index": 39 } ] }, { "type": "text", "bbox": [ 108, 665, 505, 699 ], "lines": [ { "bbox": [ 105, 663, 505, 679 ], "spans": [ { "bbox": [ 105, 663, 147, 679 ], "score": 1.0, "content": "Note that", "type": "text" }, { "bbox": [ 147, 664, 222, 678 ], "score": 0.92, "content": "P ^ { ( 1 ) } \\leq L ( A + 1 )", "type": "inline_equation" }, { "bbox": [ 222, 663, 322, 679 ], "score": 1.0, "content": "for the finest scale (i.e.,", "type": "text" }, { "bbox": [ 323, 667, 349, 676 ], "score": 0.87, "content": "s = 1", "type": "inline_equation" }, { "bbox": [ 350, 663, 505, 679 ], "score": 1.0, "content": ") since nodes at this scale do not have", "type": "text" } ], "index": 40 }, { "bbox": [ 105, 676, 505, 689 ], "spans": [ { "bbox": [ 105, 676, 505, 689 ], "score": 1.0, "content": "any children. It follows that the number of dot products that need to be calculated for the entire", "type": "text" } ], "index": 41 }, { "bbox": [ 105, 687, 222, 700 ], "spans": [ { "bbox": [ 105, 687, 222, 700 ], "score": 1.0, "content": "pyramidal attention layer is:", "type": "text" } ], "index": 42 } ], "index": 41, "bbox_fs": [ 105, 663, 505, 700 ] }, { "type": "interline_equation", "bbox": [ 181, 701, 239, 735 ], "lines": [ { "bbox": [ 181, 701, 239, 735 ], "spans": [ { "bbox": [ 181, 701, 239, 735 ], "score": 0.92, "content": "P = \\sum _ { s = 1 } ^ { S } P ^ { ( s ) }", "type": "interline_equation", "image_path": "27e0bdb332828aee29138af544adda18db9e17d5d6dd7c4f4e6fa80c9ef86533.jpg" } ] } ], "index": 43.5, "virtual_lines": [ { "bbox": [ 181, 701, 239, 718.0 ], "spans": [], "index": 43 }, { "bbox": [ 181, 718.0, 239, 735.0 ], "spans": [], "index": 44 } ] } ] }, { "preproc_blocks": [ { "type": "interline_equation", "bbox": [ 190, 79, 429, 167 ], "lines": [ { "bbox": [ 190, 79, 429, 167 ], "spans": [ { "bbox": [ 190, 79, 429, 167 ], "score": 0.94, "content": "\\begin{array} { l } { { \\le L ( A + 1 ) + L ^ { ( 2 ) } ( A + C + 1 ) + . . . + L ^ { ( S ) } ( A + C + 1 ) } } \\\\ { { \\displaystyle = L ( \\sum _ { s = 1 } ^ { S } C ^ { - ( s - 1 ) } A + \\sum _ { s = 2 } ^ { S } C ^ { - ( s - 1 ) } + \\sum _ { s = 1 } ^ { S - 1 } C ^ { - ( s - 1 ) } + 1 ) } } \\\\ { { \\displaystyle < L ( ( A + 2 ) \\sum _ { s = 1 } ^ { S } C ^ { - ( s - 1 ) } + 1 ) . } } \\end{array}", "type": "interline_equation", "image_path": "71c6102d4102b1bdd8a922b56682274b1dfbd963fa124950e8fd7ec44c3aa8e4.jpg" } ] } ], "index": 2, "virtual_lines": [ { "bbox": [ 190, 79, 429, 96.6 ], "spans": [], "index": 0 }, { "bbox": [ 190, 96.6, 429, 114.19999999999999 ], "spans": [], "index": 1 }, { "bbox": [ 190, 114.19999999999999, 429, 131.79999999999998 ], "spans": [], "index": 2 }, { "bbox": [ 190, 131.79999999999998, 429, 149.39999999999998 ], "spans": [], "index": 3 }, { "bbox": [ 190, 149.39999999999998, 429, 166.99999999999997 ], "spans": [], "index": 4 } ] }, { "type": "text", "bbox": [ 105, 171, 504, 195 ], "lines": [ { "bbox": [ 106, 171, 504, 183 ], "spans": [ { "bbox": [ 106, 171, 495, 183 ], "score": 1.0, "content": "In order to guarantee that the nodes at the coarsest scale have a global receptive field, we choose", "type": "text" }, { "bbox": [ 495, 172, 504, 181 ], "score": 0.79, "content": "C", "type": "inline_equation" } ], "index": 5 }, { "bbox": [ 104, 180, 475, 197 ], "spans": [ { "bbox": [ 104, 180, 145, 197 ], "score": 1.0, "content": "such that", "type": "text" }, { "bbox": [ 145, 182, 193, 194 ], "score": 0.92, "content": "\\bar { C } \\propto \\ ^ { s - 1 } \\bar { \\sqrt { L } }", "type": "inline_equation" }, { "bbox": [ 193, 180, 475, 197 ], "score": 1.0, "content": ". Consequently, the complexity of the proposed pyramidal attention is:", "type": "text" } ], "index": 6 } ], "index": 5.5 }, { "type": "interline_equation", "bbox": [ 218, 201, 391, 334 ], "lines": [ { "bbox": [ 218, 201, 391, 334 ], "spans": [ { "bbox": [ 218, 201, 391, 334 ], "score": 0.94, "content": "\\begin{array} { r l } { { \\mathcal { O } ( P ) \\leq \\mathcal { O } ( L ( ( A + 2 ) \\sum _ { s = 1 } ^ { s } C ^ { - ( s - 1 ) } + 1 ) ) } } \\\\ & { = \\mathcal { O } ( L ( A + 2 ) \\sum _ { s = 1 } ^ { S } C ^ { - ( s - 1 ) } ) } \\\\ & { = \\mathcal { O } ( \\frac { ( A + 2 ) L ^ { \\frac { S } { s - 1 } } - 1 } { L ^ { \\frac { S - 1 } { s - 1 } } - 1 } ) } \\\\ & { = \\mathcal { O } ( \\frac { A L ^ { \\frac { S - 1 } { s - 1 } } - 1 } { L ^ { \\frac { S - 1 } { s - 1 } } - 1 } ) . } \\end{array}", "type": "interline_equation", "image_path": "0f285521537123eece2567dd5f22e611f7a3ca1ca4616456490cf2911ffc1f86.jpg" } ] } ], "index": 10.5, "virtual_lines": [ { "bbox": [ 218, 201, 391, 217.625 ], "spans": [], "index": 7 }, { "bbox": [ 218, 217.625, 391, 234.25 ], "spans": [], "index": 8 }, { "bbox": [ 218, 234.25, 391, 250.875 ], "spans": [], "index": 9 }, { "bbox": [ 218, 250.875, 391, 267.5 ], "spans": [], "index": 10 }, { "bbox": [ 218, 267.5, 391, 284.125 ], "spans": [], "index": 11 }, { "bbox": [ 218, 284.125, 391, 300.75 ], "spans": [], "index": 12 }, { "bbox": [ 218, 300.75, 391, 317.375 ], "spans": [], "index": 13 }, { "bbox": [ 218, 317.375, 391, 334.0 ], "spans": [], "index": 14 } ] }, { "type": "text", "bbox": [ 106, 337, 504, 361 ], "lines": [ { "bbox": [ 106, 338, 506, 351 ], "spans": [ { "bbox": [ 106, 338, 133, 351 ], "score": 1.0, "content": "When", "type": "text" }, { "bbox": [ 133, 339, 141, 348 ], "score": 0.79, "content": "L", "type": "inline_equation" }, { "bbox": [ 142, 338, 357, 351 ], "score": 1.0, "content": "approaches infinity, the above expression amounts to", "type": "text" }, { "bbox": [ 358, 338, 389, 350 ], "score": 0.93, "content": "\\mathcal { O } ( A L )", "type": "inline_equation" }, { "bbox": [ 389, 338, 419, 351 ], "score": 1.0, "content": ". Since", "type": "text" }, { "bbox": [ 419, 339, 428, 348 ], "score": 0.8, "content": "A", "type": "inline_equation" }, { "bbox": [ 428, 338, 506, 351 ], "score": 1.0, "content": "can be fixed when", "type": "text" } ], "index": 15 }, { "bbox": [ 107, 349, 505, 362 ], "spans": [ { "bbox": [ 107, 349, 115, 359 ], "score": 0.8, "content": "L", "type": "inline_equation" }, { "bbox": [ 115, 349, 317, 362 ], "score": 1.0, "content": "changes, the complexity can be further reduced to", "type": "text" }, { "bbox": [ 317, 349, 340, 361 ], "score": 0.92, "content": "\\mathcal { O } ( L )", "type": "inline_equation" }, { "bbox": [ 341, 349, 345, 362 ], "score": 1.0, "content": ".", "type": "text" }, { "bbox": [ 495, 350, 505, 360 ], "score": 0.999, "content": "□", "type": "text" } ], "index": 16 } ], "index": 15.5 }, { "type": "title", "bbox": [ 108, 377, 262, 390 ], "lines": [ { "bbox": [ 106, 377, 264, 392 ], "spans": [ { "bbox": [ 106, 377, 264, 392 ], "score": 1.0, "content": "D PROOF OF PROPOSITION 2", "type": "text" } ], "index": 17 } ], "index": 17 }, { "type": "text", "bbox": [ 107, 401, 505, 444 ], "lines": [ { "bbox": [ 101, 397, 510, 438 ], "spans": [ { "bbox": [ 101, 408, 123, 438 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 104, 397, 152, 420 ], "score": 1.0, "content": "Proof. Let", "type": "text" }, { "bbox": [ 124, 415, 141, 429 ], "score": 0.91, "content": "n _ { L } ^ { ( 1 ) }", "type": "inline_equation" }, { "bbox": [ 142, 408, 510, 438 ], "score": 1.0, "content": "ℓ is the largest among all pairs of nodes in the pyramidal graph. The shortest path to travel", "type": "text" }, { "bbox": [ 152, 401, 169, 415 ], "score": 0.92, "content": "n _ { \\ell } ^ { ( s ) }", "type": "inline_equation" }, { "bbox": [ 170, 397, 224, 420 ], "score": 1.0, "content": "represent the", "type": "text" }, { "bbox": [ 225, 403, 230, 413 ], "score": 0.67, "content": "\\ell", "type": "inline_equation" }, { "bbox": [ 231, 397, 290, 420 ], "score": 1.0, "content": "-th node of the", "type": "text" }, { "bbox": [ 290, 405, 296, 413 ], "score": 0.77, "content": "s", "type": "inline_equation" }, { "bbox": [ 296, 397, 486, 420 ], "score": 1.0, "content": "-th scale. 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1 ) } } ^ { ( S - 1 ) } \\to \\cdots \\to n _ { L } ^ { ( 1 ) } .", "type": "interline_equation", "image_path": "0efba398a60f73f984c2641162cc48e4b0876cd8500ce0e758c65f3affc9fa1e.jpg" } ] } ], "index": 20, "virtual_lines": [ { "bbox": [ 162, 449, 448, 468 ], "spans": [], "index": 20 } ] }, { "type": "text", "bbox": [ 109, 472, 483, 484 ], "lines": [ { "bbox": [ 106, 471, 486, 486 ], "spans": [ { "bbox": [ 106, 471, 486, 486 ], "score": 1.0, "content": "Correspondingly, the length of the maximum path between two arbitrary nodes in the graph is:", "type": "text" } ], "index": 21 } ], "index": 21, "bbox_fs": [ 106, 471, 486, 486 ] }, { "type": "interline_equation", "bbox": [ 236, 489, 375, 516 ], "lines": [ { "bbox": [ 236, 489, 375, 516 ], "spans": [ { "bbox": [ 236, 489, 375, 516 ], "score": 0.94, "content": "L _ { \\mathrm { m a x } } = 2 ( S - 1 ) + \\frac { 2 ( L ^ { ( S ) } - 1 ) } { A - 1 } .", "type": "interline_equation", "image_path": "4920b1996ec98796735325345250236300724d44c333468078950027461a853b.jpg" } ] } ], "index": 22, "virtual_lines": [ { "bbox": [ 236, 489, 375, 516 ], "spans": [], "index": 22 } ] }, { "type": "text", "bbox": [ 106, 522, 426, 536 ], "lines": [ { "bbox": [ 105, 521, 427, 537 ], "spans": [ { "bbox": [ 105, 521, 133, 537 ], "score": 1.0, "content": "When", "type": "text" }, { "bbox": [ 133, 525, 142, 534 ], "score": 0.85, "content": "C", "type": "inline_equation" }, { "bbox": [ 142, 521, 261, 537 ], "score": 1.0, "content": "satisfies Equation (5), that is,", "type": "text" }, { "bbox": [ 261, 522, 362, 536 ], "score": 0.91, "content": "L ^ { ( S ) } - 1 \\le ( A - 1 ) N / 2", "type": "inline_equation" }, { "bbox": [ 363, 521, 427, 537 ], "score": 1.0, "content": ", we can obtain:", "type": "text" } ], "index": 23 } ], "index": 23, "bbox_fs": [ 105, 521, 427, 537 ] }, { "type": "interline_equation", "bbox": [ 216, 541, 394, 629 ], "lines": [ { "bbox": [ 216, 541, 394, 629 ], "spans": [ { "bbox": [ 216, 541, 394, 629 ], "score": 0.95, "content": "\\begin{array} { l } { \\displaystyle \\mathcal { O } \\big ( L _ { \\mathrm { m a x } } \\big ) = \\mathcal { O } \\bigg ( 2 ( S - 1 ) + \\frac { 2 \\big ( L ^ { ( S ) } - 1 \\big ) } { A - 1 } \\bigg ) } \\\\ { \\displaystyle \\qquad = \\mathcal { O } \\bigg ( 2 ( S - 1 ) + \\frac { 2 \\big ( \\frac { L } { C ^ { S - 1 } } - 1 \\big ) } { A - 1 } \\bigg ) } \\\\ { \\displaystyle \\qquad = \\mathcal { O } \\big ( 2 ( S - 1 ) + N \\big ) } \\\\ { \\displaystyle \\qquad = \\mathcal { O } ( S + N ) . } \\end{array}", "type": "interline_equation", "image_path": "93812212f74da44a00aba80fcb897f2afcc112b35df8b4ea14bd0a7b264abbb7.jpg" } ] } ], "index": 26, "virtual_lines": [ { "bbox": [ 216, 541, 394, 558.6 ], "spans": [], "index": 24 }, { "bbox": [ 216, 558.6, 394, 576.2 ], "spans": [], "index": 25 }, { "bbox": [ 216, 576.2, 394, 593.8000000000001 ], "spans": [], "index": 26 }, { "bbox": [ 216, 593.8000000000001, 394, 611.4000000000001 ], "spans": [], "index": 27 }, { "bbox": [ 216, 611.4000000000001, 394, 629.0000000000001 ], "spans": [], "index": 28 } ] }, { "type": "text", "bbox": [ 106, 633, 505, 657 ], "lines": [ { "bbox": [ 105, 632, 506, 647 ], "spans": [ { "bbox": [ 105, 632, 131, 647 ], "score": 1.0, "content": "Since", "type": "text" }, { "bbox": [ 131, 634, 153, 644 ], "score": 0.26, "content": "A , S", "type": "inline_equation" }, { "bbox": [ 153, 632, 171, 647 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 171, 635, 181, 644 ], "score": 0.85, "content": "N", "type": "inline_equation" }, { "bbox": [ 182, 632, 256, 647 ], "score": 1.0, "content": "are invariant with", "type": "text" }, { "bbox": [ 256, 635, 264, 644 ], "score": 0.82, "content": "L", "type": "inline_equation" }, { "bbox": [ 264, 632, 424, 647 ], "score": 1.0, "content": ", the order of the maximum path length", "type": "text" }, { "bbox": [ 424, 635, 445, 645 ], "score": 0.9, "content": "L _ { \\mathrm { m a x } }", "type": "inline_equation" }, { "bbox": [ 446, 632, 506, 647 ], "score": 1.0, "content": "can be further", "type": "text" } ], "index": 29 }, { "bbox": [ 105, 644, 186, 658 ], "spans": [ { "bbox": [ 105, 644, 160, 658 ], "score": 1.0, "content": "simplified as", "type": "text" }, { "bbox": [ 160, 645, 181, 657 ], "score": 0.91, "content": "\\mathcal { O } ( 1 )", "type": "inline_equation" }, { "bbox": [ 182, 644, 186, 658 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 30 } ], "index": 29.5, "bbox_fs": [ 105, 632, 506, 658 ] }, { "type": "title", "bbox": [ 107, 673, 181, 686 ], "lines": [ { "bbox": [ 105, 671, 182, 688 ], "spans": [ { "bbox": [ 105, 671, 182, 688 ], "score": 1.0, "content": "E DATASETS", "type": "text" } ], "index": 31 } ], "index": 31 }, { "type": "text", "bbox": [ 107, 698, 505, 732 ], "lines": [ { "bbox": [ 106, 698, 505, 712 ], "spans": [ { "bbox": [ 106, 698, 505, 712 ], "score": 1.0, "content": "We demonstrated the advantages of the proposed Pyraformer on the following four datasets. The", "type": "text" } ], "index": 32 }, { "bbox": [ 105, 708, 505, 723 ], "spans": [ { "bbox": [ 105, 708, 505, 723 ], "score": 1.0, "content": "first three datasets were used for single-step forecasting, while the last two for long-range multi-step", "type": "text" } ], "index": 33 }, { "bbox": [ 105, 719, 156, 734 ], "spans": [ { "bbox": [ 105, 719, 156, 734 ], "score": 1.0, "content": "forecasting.", "type": "text" } ], "index": 34 } ], "index": 33, "bbox_fs": [ 105, 698, 505, 734 ] } ] }, { "preproc_blocks": [ { "type": "text", "bbox": [ 107, 82, 504, 127 ], "lines": [ { "bbox": [ 105, 81, 506, 96 ], "spans": [ { "bbox": [ 105, 81, 506, 96 ], "score": 1.0, "content": "Wind2: This dataset contains hourly estimation of the energy potential in 28 countries between", "type": "text" } ], "index": 0 }, { "bbox": [ 105, 92, 506, 107 ], "spans": [ { "bbox": [ 105, 92, 506, 107 ], "score": 1.0, "content": "1986 and 2015 as a percentage of a power plant’s maximum output. Compared with the remaining", "type": "text" } ], "index": 1 }, { "bbox": [ 105, 104, 506, 117 ], "spans": [ { "bbox": [ 105, 104, 506, 117 ], "score": 1.0, "content": "datasets, it is more sparse and periodically exhibits a large number of zeros. Due to the large size of", "type": "text" } ], "index": 2 }, { "bbox": [ 105, 115, 394, 127 ], "spans": [ { "bbox": [ 105, 115, 394, 127 ], "score": 1.0, "content": "this dataset, the ratio between training and testing set was roughly 32:1.", "type": "text" } ], "index": 3 } ], "index": 1.5 }, { "type": "text", "bbox": [ 107, 132, 505, 176 ], "lines": [ { "bbox": [ 105, 132, 506, 144 ], "spans": [ { "bbox": [ 105, 132, 506, 144 ], "score": 1.0, "content": "App Flow: This dataset was collected at Ant Group3. It consists of hourly maximum traffic flow for", "type": "text" } ], "index": 4 }, { "bbox": [ 105, 142, 506, 156 ], "spans": [ { "bbox": [ 105, 142, 506, 156 ], "score": 1.0, "content": "128 systems deployed on 16 logic data centers, resulting in 1083 different time series in total. The", "type": "text" } ], "index": 5 }, { "bbox": [ 105, 154, 505, 166 ], "spans": [ { "bbox": [ 105, 154, 505, 166 ], "score": 1.0, "content": "length of each series is more than 4 months. Each time series was divided into two segments for", "type": "text" } ], "index": 6 }, { "bbox": [ 105, 166, 319, 177 ], "spans": [ { "bbox": [ 105, 166, 319, 177 ], "score": 1.0, "content": "training and testing respectively, with a ratio of 32:1.", "type": "text" } ], "index": 7 } ], "index": 5.5 }, { "type": "text", "bbox": [ 107, 182, 505, 237 ], "lines": [ { "bbox": [ 105, 181, 505, 195 ], "spans": [ { "bbox": [ 105, 181, 505, 195 ], "score": 1.0, "content": "Electricity4 (Yu et al., 2016): This dataset contains time series of electricity consumption recorded", "type": "text" } ], "index": 8 }, { "bbox": [ 105, 192, 505, 206 ], "spans": [ { "bbox": [ 105, 192, 505, 206 ], "score": 1.0, "content": "every 15 minutes from 370 users. Following DeepAR (Salinas et al., 2020), we aggregated every", "type": "text" } ], "index": 9 }, { "bbox": [ 104, 202, 505, 219 ], "spans": [ { "bbox": [ 104, 202, 505, 219 ], "score": 1.0, "content": "4 records to get the hourly observations. This dataset was employed for both single-step and long-", "type": "text" } ], "index": 10 }, { "bbox": [ 104, 212, 506, 230 ], "spans": [ { "bbox": [ 104, 212, 506, 230 ], "score": 1.0, "content": "range forecasting. We trained with data from 2011-01-01 to 2014-09-01 for single-step forecasting,", "type": "text" } ], "index": 11 }, { "bbox": [ 105, 225, 362, 240 ], "spans": [ { "bbox": [ 105, 225, 362, 240 ], "score": 1.0, "content": "and from 2011-04-01 to 2014-04-01 for long-range forecasting.", "type": "text" } ], "index": 12 } ], "index": 10 }, { "type": "text", "bbox": [ 107, 243, 505, 298 ], "lines": [ { "bbox": [ 106, 241, 505, 256 ], "spans": [ { "bbox": [ 106, 242, 132, 254 ], "score": 0.63, "content": "\\mathbf { \\mathbf { E } } \\mathbf { T } \\mathbf { T } ^ { 5 }", "type": "inline_equation" }, { "bbox": [ 132, 241, 505, 256 ], "score": 1.0, "content": "(Zhou et al., 2021): This dataset comprises 2 years of 2 electricity transformers collected", "type": "text" } ], "index": 13 }, { "bbox": [ 105, 253, 505, 267 ], "spans": [ { "bbox": [ 105, 253, 505, 267 ], "score": 1.0, "content": "from 2 stations, including the oil temperature and 6 power load features. Observations every hour", "type": "text" } ], "index": 14 }, { "bbox": [ 105, 264, 506, 277 ], "spans": [ { "bbox": [ 105, 264, 506, 277 ], "score": 1.0, "content": "(i.e., ETTh1) and every 15 minutes (i.e., ETTm1) are provided. This dataset is typically exploited", "type": "text" } ], "index": 15 }, { "bbox": [ 106, 275, 505, 289 ], "spans": [ { "bbox": [ 106, 275, 505, 289 ], "score": 1.0, "content": "for model assessment on long-range forecasting. Here, we followed Informer (Zhou et al., 2021)", "type": "text" } ], "index": 16 }, { "bbox": [ 105, 286, 434, 300 ], "spans": [ { "bbox": [ 105, 286, 434, 300 ], "score": 1.0, "content": "and partitioned the data into 12 and 4 months for training and testing respectively.", "type": "text" } ], "index": 17 } ], "index": 15 }, { "type": "title", "bbox": [ 108, 316, 231, 328 ], "lines": [ { "bbox": [ 105, 315, 232, 330 ], "spans": [ { "bbox": [ 105, 315, 232, 330 ], "score": 1.0, "content": "F EXPERIMENT SETUP", "type": "text" } ], "index": 18 } ], "index": 18 }, { "type": "text", "bbox": [ 106, 342, 505, 528 ], "lines": [ { "bbox": [ 106, 342, 505, 354 ], "spans": [ { "bbox": [ 106, 342, 136, 354 ], "score": 1.0, "content": "We set", "type": "text" }, { "bbox": [ 137, 342, 165, 352 ], "score": 0.9, "content": "S = 4", "type": "inline_equation" }, { "bbox": [ 166, 342, 184, 354 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 185, 342, 216, 352 ], "score": 0.9, "content": "N = 4", "type": "inline_equation" }, { "bbox": [ 216, 342, 470, 354 ], "score": 1.0, "content": "for Pyraformer in all experiments. When the historical length", "type": "text" }, { "bbox": [ 470, 343, 478, 352 ], "score": 0.79, "content": "L", "type": "inline_equation" }, { "bbox": [ 479, 342, 505, 354 ], "score": 1.0, "content": "is not", "type": "text" } ], "index": 19 }, { "bbox": [ 106, 352, 506, 365 ], "spans": [ { "bbox": [ 106, 352, 156, 365 ], "score": 1.0, "content": "divisible by", "type": "text" }, { "bbox": [ 157, 353, 166, 363 ], "score": 0.79, "content": "C", "type": "inline_equation" }, { "bbox": [ 166, 352, 252, 365 ], "score": 1.0, "content": ", we only introduced", "type": "text" }, { "bbox": [ 253, 353, 281, 364 ], "score": 0.89, "content": "\\lfloor L / C \\rfloor", "type": "inline_equation" }, { "bbox": [ 281, 352, 416, 365 ], "score": 1.0, "content": "nodes in the upper scale, where", "type": "text" }, { "bbox": [ 416, 353, 428, 365 ], "score": 0.82, "content": "\\lfloor \\cdot \\rfloor", "type": "inline_equation" }, { "bbox": [ 428, 352, 506, 365 ], "score": 1.0, "content": "denotes the round", "type": "text" } ], "index": 20 }, { "bbox": [ 106, 363, 506, 376 ], "spans": [ { "bbox": [ 106, 363, 211, 376 ], "score": 1.0, "content": "down operation. The last", "type": "text" }, { "bbox": [ 211, 364, 294, 376 ], "score": 0.92, "content": "L - ( \\lfloor L / \\bar { C } \\rfloor - \\bar { 1 } ) C", "type": "inline_equation" }, { "bbox": [ 294, 363, 506, 376 ], "score": 1.0, "content": "nodes at the bottom scale were all connected to the", "type": "text" } ], "index": 21 }, { "bbox": [ 104, 374, 506, 388 ], "spans": [ { "bbox": [ 104, 374, 371, 388 ], "score": 1.0, "content": "last node at the upper scale. For single-step forecasting, we set", "type": "text" }, { "bbox": [ 371, 375, 402, 385 ], "score": 0.83, "content": "C = 4", "type": "inline_equation" }, { "bbox": [ 403, 374, 407, 388 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 407, 375, 438, 385 ], "score": 0.85, "content": "A = 3", "type": "inline_equation" }, { "bbox": [ 438, 374, 460, 388 ], "score": 1.0, "content": ", and", "type": "text" }, { "bbox": [ 460, 375, 493, 385 ], "score": 0.9, "content": "H = 4", "type": "inline_equation" }, { "bbox": [ 493, 374, 506, 388 ], "score": 1.0, "content": "in", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 386, 506, 399 ], "spans": [ { "bbox": [ 105, 386, 506, 399 ], "score": 1.0, "content": "all experiments. Both training and testing used a fixed-size historical sequence to predict the mean", "type": "text" } ], "index": 23 }, { "bbox": [ 106, 397, 505, 408 ], "spans": [ { "bbox": [ 106, 397, 505, 408 ], "score": 1.0, "content": "and variance of the Gaussian distribution of a single future value. We chose the MSE loss and the", "type": "text" } ], "index": 24 }, { "bbox": [ 106, 407, 506, 420 ], "spans": [ { "bbox": [ 106, 407, 506, 420 ], "score": 1.0, "content": "log-likelihood (Zuo et al., 2020) as our loss functions. The ratio between them was set to 100. For", "type": "text" } ], "index": 25 }, { "bbox": [ 104, 417, 505, 432 ], "spans": [ { "bbox": [ 104, 417, 368, 432 ], "score": 1.0, "content": "optimization, we used Adam with the learning rate starting from", "type": "text" }, { "bbox": [ 368, 418, 390, 429 ], "score": 0.89, "content": "1 0 ^ { - 5 }", "type": "inline_equation" }, { "bbox": [ 390, 417, 505, 432 ], "score": 1.0, "content": "and halving in every epoch.", "type": "text" } ], "index": 26 }, { "bbox": [ 106, 429, 506, 442 ], "spans": [ { "bbox": [ 106, 429, 506, 442 ], "score": 1.0, "content": "We trained Pyraformer with 10 epochs. Weighted sampler based on each window’s average value", "type": "text" } ], "index": 27 }, { "bbox": [ 106, 440, 505, 453 ], "spans": [ { "bbox": [ 106, 440, 505, 453 ], "score": 1.0, "content": "and hard sample mining were used to improve the generalization ability of the network. On the", "type": "text" } ], "index": 28 }, { "bbox": [ 105, 450, 506, 465 ], "spans": [ { "bbox": [ 105, 450, 388, 465 ], "score": 1.0, "content": "other hand, for long-range forecasting, we tested four combinations of", "type": "text" }, { "bbox": [ 388, 452, 397, 461 ], "score": 0.74, "content": "A", "type": "inline_equation" }, { "bbox": [ 397, 450, 414, 465 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 415, 452, 423, 462 ], "score": 0.79, "content": "C", "type": "inline_equation" }, { "bbox": [ 424, 450, 506, 465 ], "score": 1.0, "content": "in each experiment,", "type": "text" } ], "index": 29 }, { "bbox": [ 106, 462, 506, 475 ], "spans": [ { "bbox": [ 106, 462, 506, 475 ], "score": 1.0, "content": "and the best results were presented. Specifically, when the prediction length is smaller than 600, we", "type": "text" } ], "index": 30 }, { "bbox": [ 105, 473, 504, 486 ], "spans": [ { "bbox": [ 105, 473, 133, 486 ], "score": 1.0, "content": "tested", "type": "text" }, { "bbox": [ 133, 474, 171, 484 ], "score": 0.89, "content": "A = 3 , 5", "type": "inline_equation" }, { "bbox": [ 172, 473, 190, 486 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 190, 474, 229, 485 ], "score": 0.9, "content": "C = 4 , 5", "type": "inline_equation" }, { "bbox": [ 229, 473, 465, 486 ], "score": 1.0, "content": ". When the prediction length is larger than 600, we tested", "type": "text" }, { "bbox": [ 466, 474, 504, 485 ], "score": 0.89, "content": "A = 3 , 5", "type": "inline_equation" } ], "index": 31 }, { "bbox": [ 106, 484, 505, 497 ], "spans": [ { "bbox": [ 106, 484, 124, 497 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 124, 485, 163, 496 ], "score": 0.9, "content": "C = 5 , 6", "type": "inline_equation" }, { "bbox": [ 164, 484, 505, 497 ], "score": 1.0, "content": ". The resulting choice of hyper-parameters for each experiment is listed in Table 5.", "type": "text" } ], "index": 32 }, { "bbox": [ 105, 495, 505, 507 ], "spans": [ { "bbox": [ 105, 495, 505, 507 ], "score": 1.0, "content": "In addition, the loss function was the MSE loss only. We still used Adam as our optimizer, but the", "type": "text" } ], "index": 33 }, { "bbox": [ 105, 505, 506, 519 ], "spans": [ { "bbox": [ 105, 505, 212, 519 ], "score": 1.0, "content": "learning rate started from", "type": "text" }, { "bbox": [ 213, 506, 234, 516 ], "score": 0.9, "content": "1 0 ^ { - 4 }", "type": "inline_equation" }, { "bbox": [ 235, 505, 506, 519 ], "score": 1.0, "content": "and was reduced to one-tenth every epoch. We set the number of", "type": "text" } ], "index": 34 }, { "bbox": [ 105, 517, 170, 529 ], "spans": [ { "bbox": [ 105, 517, 170, 529 ], "score": 1.0, "content": "epochs to be 5.", "type": "text" } ], "index": 35 } ], "index": 27 }, { "type": "title", "bbox": [ 108, 546, 200, 559 ], "lines": [ { "bbox": [ 105, 545, 201, 561 ], "spans": [ { "bbox": [ 105, 545, 201, 561 ], "score": 1.0, "content": "G PRETRAINING", "type": "text" } ], "index": 36 } ], "index": 36 }, { "type": "text", "bbox": [ 107, 572, 504, 628 ], "lines": [ { "bbox": [ 105, 572, 505, 586 ], "spans": [ { "bbox": [ 105, 572, 505, 586 ], "score": 1.0, "content": "For single-step forecasting, the value to be predicted is usually close to the last value of history.", "type": "text" } ], "index": 37 }, { "bbox": [ 106, 583, 505, 595 ], "spans": [ { "bbox": [ 106, 583, 505, 595 ], "score": 1.0, "content": "Since we only use the last nodes of all scales to predict, the network tends to focus only on short-", "type": "text" } ], "index": 38 }, { "bbox": [ 106, 595, 505, 607 ], "spans": [ { "bbox": [ 106, 595, 505, 607 ], "score": 1.0, "content": "term dependencies. To force the network to capture long-range dependencies, we add additional", "type": "text" } ], "index": 39 }, { "bbox": [ 105, 605, 506, 618 ], "spans": [ { "bbox": [ 105, 605, 506, 618 ], "score": 1.0, "content": "supervision in the first few epochs of training. Specifically, in the first epoch, we form our network", "type": "text" } ], "index": 40 }, { "bbox": [ 105, 617, 505, 628 ], "spans": [ { "bbox": [ 105, 617, 505, 628 ], "score": 1.0, "content": "as an auto-encoder, as shown in Figure 5. Apart from predicting future values, the PAM is also", "type": "text" } ], "index": 41 } ], "index": 39 } ], "page_idx": 14, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 106, 638, 506, 732 ], "lines": [ { "bbox": [ 117, 636, 506, 652 ], "spans": [ { "bbox": [ 117, 636, 506, 652 ], "score": 1.0, "content": "2Wind dataset can be downloaded at https://www.kaggle.com/sohier/30-years-of-european-wind-generation", "type": "text" } ] }, { "bbox": [ 117, 647, 506, 663 ], "spans": [ { "bbox": [ 117, 647, 506, 663 ], "score": 1.0, "content": "3The App Flow dataset does not contain any Personal Identifiable Information and is desensi-", "type": "text" } ] }, { "bbox": [ 105, 659, 506, 672 ], "spans": [ { "bbox": [ 105, 659, 506, 672 ], "score": 1.0, "content": "tized and encrypted. Adequate data protection was carried out during the experiment to prevent the", "type": "text" } ] }, { "bbox": [ 105, 669, 506, 681 ], "spans": [ { "bbox": [ 105, 669, 506, 681 ], "score": 1.0, "content": "risk of data copy leakage, and the dataset was destroyed after the experiment. It is only used", "type": "text" } ] }, { "bbox": [ 105, 679, 506, 691 ], "spans": [ { "bbox": [ 105, 679, 506, 691 ], "score": 1.0, "content": "for academic research, it does not represent any real business situation. The download link is", "type": "text" } ] }, { "bbox": [ 105, 689, 424, 701 ], "spans": [ { "bbox": [ 105, 689, 424, 701 ], "score": 1.0, "content": "https://github.com/alipay/Pyraformer/tree/master/data/app zone rpc hour encrypted.csv", "type": "text" } ] }, { "bbox": [ 117, 698, 506, 713 ], "spans": [ { "bbox": [ 117, 698, 506, 713 ], "score": 1.0, "content": "4Electricity dataset can be downloaded at https://archive.ics.uci.edu/ml/datasets/ElectricityLoadDiagrams20", "type": "text" } ] }, { "bbox": [ 106, 709, 137, 721 ], "spans": [ { "bbox": [ 106, 709, 137, 721 ], "score": 1.0, "content": "112014", "type": "text" } ] }, { "bbox": [ 118, 720, 399, 733 ], "spans": [ { "bbox": [ 118, 720, 399, 733 ], "score": 1.0, "content": "5ETT dataset can be downloaded at https:// github.com/zhouhaoyi/ETDataset", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 108, 27, 293, 37 ], "lines": [ { "bbox": [ 106, 26, 294, 38 ], "spans": [ { "bbox": [ 106, 26, 294, 38 ], "score": 1.0, "content": "Published as a conference paper at ICLR 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": "15", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "text", "bbox": [ 107, 82, 504, 127 ], "lines": [ { "bbox": [ 105, 81, 506, 96 ], "spans": [ { "bbox": [ 105, 81, 506, 96 ], "score": 1.0, "content": "Wind2: This dataset contains hourly estimation of the energy potential in 28 countries between", "type": "text" } ], "index": 0 }, { "bbox": [ 105, 92, 506, 107 ], "spans": [ { "bbox": [ 105, 92, 506, 107 ], "score": 1.0, "content": "1986 and 2015 as a percentage of a power plant’s maximum output. Compared with the remaining", "type": "text" } ], "index": 1 }, { "bbox": [ 105, 104, 506, 117 ], "spans": [ { "bbox": [ 105, 104, 506, 117 ], "score": 1.0, "content": "datasets, it is more sparse and periodically exhibits a large number of zeros. Due to the large size of", "type": "text" } ], "index": 2 }, { "bbox": [ 105, 115, 394, 127 ], "spans": [ { "bbox": [ 105, 115, 394, 127 ], "score": 1.0, "content": "this dataset, the ratio between training and testing set was roughly 32:1.", "type": "text" } ], "index": 3 } ], "index": 1.5, "bbox_fs": [ 105, 81, 506, 127 ] }, { "type": "text", "bbox": [ 107, 132, 505, 176 ], "lines": [ { "bbox": [ 105, 132, 506, 144 ], "spans": [ { "bbox": [ 105, 132, 506, 144 ], "score": 1.0, "content": "App Flow: This dataset was collected at Ant Group3. It consists of hourly maximum traffic flow for", "type": "text" } ], "index": 4 }, { "bbox": [ 105, 142, 506, 156 ], "spans": [ { "bbox": [ 105, 142, 506, 156 ], "score": 1.0, "content": "128 systems deployed on 16 logic data centers, resulting in 1083 different time series in total. The", "type": "text" } ], "index": 5 }, { "bbox": [ 105, 154, 505, 166 ], "spans": [ { "bbox": [ 105, 154, 505, 166 ], "score": 1.0, "content": "length of each series is more than 4 months. Each time series was divided into two segments for", "type": "text" } ], "index": 6 }, { "bbox": [ 105, 166, 319, 177 ], "spans": [ { "bbox": [ 105, 166, 319, 177 ], "score": 1.0, "content": "training and testing respectively, with a ratio of 32:1.", "type": "text" } ], "index": 7 } ], "index": 5.5, "bbox_fs": [ 105, 132, 506, 177 ] }, { "type": "text", "bbox": [ 107, 182, 505, 237 ], "lines": [ { "bbox": [ 105, 181, 505, 195 ], "spans": [ { "bbox": [ 105, 181, 505, 195 ], "score": 1.0, "content": "Electricity4 (Yu et al., 2016): This dataset contains time series of electricity consumption recorded", "type": "text" } ], "index": 8 }, { "bbox": [ 105, 192, 505, 206 ], "spans": [ { "bbox": [ 105, 192, 505, 206 ], "score": 1.0, "content": "every 15 minutes from 370 users. Following DeepAR (Salinas et al., 2020), we aggregated every", "type": "text" } ], "index": 9 }, { "bbox": [ 104, 202, 505, 219 ], "spans": [ { "bbox": [ 104, 202, 505, 219 ], "score": 1.0, "content": "4 records to get the hourly observations. This dataset was employed for both single-step and long-", "type": "text" } ], "index": 10 }, { "bbox": [ 104, 212, 506, 230 ], "spans": [ { "bbox": [ 104, 212, 506, 230 ], "score": 1.0, "content": "range forecasting. We trained with data from 2011-01-01 to 2014-09-01 for single-step forecasting,", "type": "text" } ], "index": 11 }, { "bbox": [ 105, 225, 362, 240 ], "spans": [ { "bbox": [ 105, 225, 362, 240 ], "score": 1.0, "content": "and from 2011-04-01 to 2014-04-01 for long-range forecasting.", "type": "text" } ], "index": 12 } ], "index": 10, "bbox_fs": [ 104, 181, 506, 240 ] }, { "type": "text", "bbox": [ 107, 243, 505, 298 ], "lines": [ { "bbox": [ 106, 241, 505, 256 ], "spans": [ { "bbox": [ 106, 242, 132, 254 ], "score": 0.63, "content": "\\mathbf { \\mathbf { E } } \\mathbf { T } \\mathbf { T } ^ { 5 }", "type": "inline_equation" }, { "bbox": [ 132, 241, 505, 256 ], "score": 1.0, "content": "(Zhou et al., 2021): This dataset comprises 2 years of 2 electricity transformers collected", "type": "text" } ], "index": 13 }, { "bbox": [ 105, 253, 505, 267 ], "spans": [ { "bbox": [ 105, 253, 505, 267 ], "score": 1.0, "content": "from 2 stations, including the oil temperature and 6 power load features. Observations every hour", "type": "text" } ], "index": 14 }, { "bbox": [ 105, 264, 506, 277 ], "spans": [ { "bbox": [ 105, 264, 506, 277 ], "score": 1.0, "content": "(i.e., ETTh1) and every 15 minutes (i.e., ETTm1) are provided. This dataset is typically exploited", "type": "text" } ], "index": 15 }, { "bbox": [ 106, 275, 505, 289 ], "spans": [ { "bbox": [ 106, 275, 505, 289 ], "score": 1.0, "content": "for model assessment on long-range forecasting. Here, we followed Informer (Zhou et al., 2021)", "type": "text" } ], "index": 16 }, { "bbox": [ 105, 286, 434, 300 ], "spans": [ { "bbox": [ 105, 286, 434, 300 ], "score": 1.0, "content": "and partitioned the data into 12 and 4 months for training and testing respectively.", "type": "text" } ], "index": 17 } ], "index": 15, "bbox_fs": [ 105, 241, 506, 300 ] }, { "type": "title", "bbox": [ 108, 316, 231, 328 ], "lines": [ { "bbox": [ 105, 315, 232, 330 ], "spans": [ { "bbox": [ 105, 315, 232, 330 ], "score": 1.0, "content": "F EXPERIMENT SETUP", "type": "text" } ], "index": 18 } ], "index": 18 }, { "type": "text", "bbox": [ 106, 342, 505, 528 ], "lines": [ { "bbox": [ 106, 342, 505, 354 ], "spans": [ { "bbox": [ 106, 342, 136, 354 ], "score": 1.0, "content": "We set", "type": "text" }, { "bbox": [ 137, 342, 165, 352 ], "score": 0.9, "content": "S = 4", "type": "inline_equation" }, { "bbox": [ 166, 342, 184, 354 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 185, 342, 216, 352 ], "score": 0.9, "content": "N = 4", "type": "inline_equation" }, { "bbox": [ 216, 342, 470, 354 ], "score": 1.0, "content": "for Pyraformer in all experiments. When the historical length", "type": "text" }, { "bbox": [ 470, 343, 478, 352 ], "score": 0.79, "content": "L", "type": "inline_equation" }, { "bbox": [ 479, 342, 505, 354 ], "score": 1.0, "content": "is not", "type": "text" } ], "index": 19 }, { "bbox": [ 106, 352, 506, 365 ], "spans": [ { "bbox": [ 106, 352, 156, 365 ], "score": 1.0, "content": "divisible by", "type": "text" }, { "bbox": [ 157, 353, 166, 363 ], "score": 0.79, "content": "C", "type": "inline_equation" }, { "bbox": [ 166, 352, 252, 365 ], "score": 1.0, "content": ", we only introduced", "type": "text" }, { "bbox": [ 253, 353, 281, 364 ], "score": 0.89, "content": "\\lfloor L / C \\rfloor", "type": "inline_equation" }, { "bbox": [ 281, 352, 416, 365 ], "score": 1.0, "content": "nodes in the upper scale, where", "type": "text" }, { "bbox": [ 416, 353, 428, 365 ], "score": 0.82, "content": "\\lfloor \\cdot \\rfloor", "type": "inline_equation" }, { "bbox": [ 428, 352, 506, 365 ], "score": 1.0, "content": "denotes the round", "type": "text" } ], "index": 20 }, { "bbox": [ 106, 363, 506, 376 ], "spans": [ { "bbox": [ 106, 363, 211, 376 ], "score": 1.0, "content": "down operation. The last", "type": "text" }, { "bbox": [ 211, 364, 294, 376 ], "score": 0.92, "content": "L - ( \\lfloor L / \\bar { C } \\rfloor - \\bar { 1 } ) C", "type": "inline_equation" }, { "bbox": [ 294, 363, 506, 376 ], "score": 1.0, "content": "nodes at the bottom scale were all connected to the", "type": "text" } ], "index": 21 }, { "bbox": [ 104, 374, 506, 388 ], "spans": [ { "bbox": [ 104, 374, 371, 388 ], "score": 1.0, "content": "last node at the upper scale. For single-step forecasting, we set", "type": "text" }, { "bbox": [ 371, 375, 402, 385 ], "score": 0.83, "content": "C = 4", "type": "inline_equation" }, { "bbox": [ 403, 374, 407, 388 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 407, 375, 438, 385 ], "score": 0.85, "content": "A = 3", "type": "inline_equation" }, { "bbox": [ 438, 374, 460, 388 ], "score": 1.0, "content": ", and", "type": "text" }, { "bbox": [ 460, 375, 493, 385 ], "score": 0.9, "content": "H = 4", "type": "inline_equation" }, { "bbox": [ 493, 374, 506, 388 ], "score": 1.0, "content": "in", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 386, 506, 399 ], "spans": [ { "bbox": [ 105, 386, 506, 399 ], "score": 1.0, "content": "all experiments. Both training and testing used a fixed-size historical sequence to predict the mean", "type": "text" } ], "index": 23 }, { "bbox": [ 106, 397, 505, 408 ], "spans": [ { "bbox": [ 106, 397, 505, 408 ], "score": 1.0, "content": "and variance of the Gaussian distribution of a single future value. We chose the MSE loss and the", "type": "text" } ], "index": 24 }, { "bbox": [ 106, 407, 506, 420 ], "spans": [ { "bbox": [ 106, 407, 506, 420 ], "score": 1.0, "content": "log-likelihood (Zuo et al., 2020) as our loss functions. The ratio between them was set to 100. For", "type": "text" } ], "index": 25 }, { "bbox": [ 104, 417, 505, 432 ], "spans": [ { "bbox": [ 104, 417, 368, 432 ], "score": 1.0, "content": "optimization, we used Adam with the learning rate starting from", "type": "text" }, { "bbox": [ 368, 418, 390, 429 ], "score": 0.89, "content": "1 0 ^ { - 5 }", "type": "inline_equation" }, { "bbox": [ 390, 417, 505, 432 ], "score": 1.0, "content": "and halving in every epoch.", "type": "text" } ], "index": 26 }, { "bbox": [ 106, 429, 506, 442 ], "spans": [ { "bbox": [ 106, 429, 506, 442 ], "score": 1.0, "content": "We trained Pyraformer with 10 epochs. Weighted sampler based on each window’s average value", "type": "text" } ], "index": 27 }, { "bbox": [ 106, 440, 505, 453 ], "spans": [ { "bbox": [ 106, 440, 505, 453 ], "score": 1.0, "content": "and hard sample mining were used to improve the generalization ability of the network. On the", "type": "text" } ], "index": 28 }, { "bbox": [ 105, 450, 506, 465 ], "spans": [ { "bbox": [ 105, 450, 388, 465 ], "score": 1.0, "content": "other hand, for long-range forecasting, we tested four combinations of", "type": "text" }, { "bbox": [ 388, 452, 397, 461 ], "score": 0.74, "content": "A", "type": "inline_equation" }, { "bbox": [ 397, 450, 414, 465 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 415, 452, 423, 462 ], "score": 0.79, "content": "C", "type": "inline_equation" }, { "bbox": [ 424, 450, 506, 465 ], "score": 1.0, "content": "in each experiment,", "type": "text" } ], "index": 29 }, { "bbox": [ 106, 462, 506, 475 ], "spans": [ { "bbox": [ 106, 462, 506, 475 ], "score": 1.0, "content": "and the best results were presented. Specifically, when the prediction length is smaller than 600, we", "type": "text" } ], "index": 30 }, { "bbox": [ 105, 473, 504, 486 ], "spans": [ { "bbox": [ 105, 473, 133, 486 ], "score": 1.0, "content": "tested", "type": "text" }, { "bbox": [ 133, 474, 171, 484 ], "score": 0.89, "content": "A = 3 , 5", "type": "inline_equation" }, { "bbox": [ 172, 473, 190, 486 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 190, 474, 229, 485 ], "score": 0.9, "content": "C = 4 , 5", "type": "inline_equation" }, { "bbox": [ 229, 473, 465, 486 ], "score": 1.0, "content": ". When the prediction length is larger than 600, we tested", "type": "text" }, { "bbox": [ 466, 474, 504, 485 ], "score": 0.89, "content": "A = 3 , 5", "type": "inline_equation" } ], "index": 31 }, { "bbox": [ 106, 484, 505, 497 ], "spans": [ { "bbox": [ 106, 484, 124, 497 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 124, 485, 163, 496 ], "score": 0.9, "content": "C = 5 , 6", "type": "inline_equation" }, { "bbox": [ 164, 484, 505, 497 ], "score": 1.0, "content": ". The resulting choice of hyper-parameters for each experiment is listed in Table 5.", "type": "text" } ], "index": 32 }, { "bbox": [ 105, 495, 505, 507 ], "spans": [ { "bbox": [ 105, 495, 505, 507 ], "score": 1.0, "content": "In addition, the loss function was the MSE loss only. We still used Adam as our optimizer, but the", "type": "text" } ], "index": 33 }, { "bbox": [ 105, 505, 506, 519 ], "spans": [ { "bbox": [ 105, 505, 212, 519 ], "score": 1.0, "content": "learning rate started from", "type": "text" }, { "bbox": [ 213, 506, 234, 516 ], "score": 0.9, "content": "1 0 ^ { - 4 }", "type": "inline_equation" }, { "bbox": [ 235, 505, 506, 519 ], "score": 1.0, "content": "and was reduced to one-tenth every epoch. We set the number of", "type": "text" } ], "index": 34 }, { "bbox": [ 105, 517, 170, 529 ], "spans": [ { "bbox": [ 105, 517, 170, 529 ], "score": 1.0, "content": "epochs to be 5.", "type": "text" } ], "index": 35 } ], "index": 27, "bbox_fs": [ 104, 342, 506, 529 ] }, { "type": "title", "bbox": [ 108, 546, 200, 559 ], "lines": [ { "bbox": [ 105, 545, 201, 561 ], "spans": [ { "bbox": [ 105, 545, 201, 561 ], "score": 1.0, "content": "G PRETRAINING", "type": "text" } ], "index": 36 } ], "index": 36 }, { "type": "text", "bbox": [ 107, 572, 504, 628 ], "lines": [ { "bbox": [ 105, 572, 505, 586 ], "spans": [ { "bbox": [ 105, 572, 505, 586 ], "score": 1.0, "content": "For single-step forecasting, the value to be predicted is usually close to the last value of history.", "type": "text" } ], "index": 37 }, { "bbox": [ 106, 583, 505, 595 ], "spans": [ { "bbox": [ 106, 583, 505, 595 ], "score": 1.0, "content": "Since we only use the last nodes of all scales to predict, the network tends to focus only on short-", "type": "text" } ], "index": 38 }, { "bbox": [ 106, 595, 505, 607 ], "spans": [ { "bbox": [ 106, 595, 505, 607 ], "score": 1.0, "content": "term dependencies. To force the network to capture long-range dependencies, we add additional", "type": "text" } ], "index": 39 }, { "bbox": [ 105, 605, 506, 618 ], "spans": [ { "bbox": [ 105, 605, 506, 618 ], "score": 1.0, "content": "supervision in the first few epochs of training. Specifically, in the first epoch, we form our network", "type": "text" } ], "index": 40 }, { "bbox": [ 105, 617, 505, 628 ], "spans": [ { "bbox": [ 105, 617, 505, 628 ], "score": 1.0, "content": "as an auto-encoder, as shown in Figure 5. Apart from predicting future values, the PAM is also", "type": "text" } ], "index": 41 }, { "bbox": [ 105, 505, 506, 521 ], "spans": [ { "bbox": [ 105, 505, 506, 521 ], "score": 1.0, "content": "trained to recover the input values. Note that we test all methods with and without this pretraining", "type": "text", "cross_page": true } ], "index": 19 }, { "bbox": [ 106, 518, 325, 530 ], "spans": [ { "bbox": [ 106, 518, 325, 530 ], "score": 1.0, "content": "strategy and the better results are displayed in Table 2.", "type": "text", "cross_page": true } ], "index": 20 } ], "index": 39, "bbox_fs": [ 105, 572, 506, 628 ] } ] }, { "preproc_blocks": [ { "type": "table", "bbox": [ 151, 101, 459, 258 ], "blocks": [ { "type": "table_caption", "bbox": [ 180, 90, 429, 101 ], "group_id": 0, "lines": [ { "bbox": [ 181, 88, 430, 104 ], "spans": [ { "bbox": [ 181, 88, 430, 104 ], "score": 1.0, "content": "Table 5: Hyper-parameter settings of long-range experiments.", "type": "text" } ], "index": 0 } ], "index": 0 }, { "type": "table_body", "bbox": [ 151, 101, 459, 258 ], "group_id": 0, "lines": [ { "bbox": [ 151, 101, 459, 258 ], "spans": [ { "bbox": [ 151, 101, 459, 258 ], "score": 0.983, "html": "
Datasetprediction lengthNSHAChistorical length
ETTh116844634168
33644634168
72044654336
ETTm19644635384
28844655672
67244636672
Elect16844634168
33644634168
72044635336
", "type": "table", "image_path": "26ecf047e6583e1cbb074f61df77c1a887082b1dd64d8db53e55094ae880fd9b.jpg" } ] } ], "index": 2, "virtual_lines": [ { "bbox": [ 151, 101, 459, 153.33333333333334 ], "spans": [], "index": 1 }, { "bbox": [ 151, 153.33333333333334, 459, 205.66666666666669 ], "spans": [], "index": 2 }, { "bbox": [ 151, 205.66666666666669, 459, 258.0 ], "spans": [], "index": 3 } ] } ], "index": 1.0 }, { "type": "image", "bbox": [ 213, 267, 396, 454 ], "blocks": [ { "type": "image_body", "bbox": [ 213, 267, 396, 454 ], "group_id": 0, "lines": [ { "bbox": [ 213, 267, 396, 454 ], "spans": [ { "bbox": [ 213, 267, 396, 454 ], "score": 0.964, "type": "image", "image_path": "16bca2db6aa268152dfacebb3a8ccf6bd2e2df85f5fcbe3870acda17a4461454.jpg" } ] } ], "index": 10, "virtual_lines": [ { "bbox": [ 213, 267, 396, 281.38461538461536 ], "spans": [], "index": 4 }, { "bbox": [ 213, 281.38461538461536, 396, 295.7692307692307 ], "spans": [], "index": 5 }, { "bbox": [ 213, 295.7692307692307, 396, 310.1538461538461 ], "spans": [], "index": 6 }, { "bbox": [ 213, 310.1538461538461, 396, 324.53846153846143 ], "spans": [], "index": 7 }, { "bbox": [ 213, 324.53846153846143, 396, 338.9230769230768 ], "spans": [], "index": 8 }, { "bbox": [ 213, 338.9230769230768, 396, 353.30769230769215 ], "spans": [], "index": 9 }, { "bbox": [ 213, 353.30769230769215, 396, 367.6923076923075 ], "spans": [], "index": 10 }, { "bbox": [ 213, 367.6923076923075, 396, 382.07692307692287 ], "spans": [], "index": 11 }, { "bbox": [ 213, 382.07692307692287, 396, 396.4615384615382 ], "spans": [], "index": 12 }, { "bbox": [ 213, 396.4615384615382, 396, 410.8461538461536 ], "spans": [], "index": 13 }, { "bbox": [ 213, 410.8461538461536, 396, 425.23076923076894 ], "spans": [], "index": 14 }, { "bbox": [ 213, 425.23076923076894, 396, 439.6153846153843 ], "spans": [], "index": 15 }, { "bbox": [ 213, 439.6153846153843, 396, 453.99999999999966 ], "spans": [], "index": 16 } ] }, { "type": "image_caption", "bbox": [ 106, 464, 505, 487 ], "group_id": 0, "lines": [ { "bbox": [ 105, 462, 505, 477 ], "spans": [ { "bbox": [ 105, 462, 505, 477 ], "score": 1.0, "content": "Figure 5: The pretraining strategy for one-step prediction. Features of nodes surrounded by the", "type": "text" } ], "index": 17 }, { "bbox": [ 105, 474, 404, 488 ], "spans": [ { "bbox": [ 105, 474, 404, 488 ], "score": 1.0, "content": "dashed ellipses are concatenated to recover the corresponding input value.", "type": "text" } ], "index": 18 } ], "index": 17.5 } ], "index": 13.75 }, { "type": "text", "bbox": [ 106, 507, 504, 530 ], "lines": [ { "bbox": [ 105, 505, 506, 521 ], "spans": [ { "bbox": [ 105, 505, 506, 521 ], "score": 1.0, "content": "trained to recover the input values. Note that we test all methods with and without this pretraining", "type": "text" } ], "index": 19 }, { "bbox": [ 106, 518, 325, 530 ], "spans": [ { "bbox": [ 106, 518, 325, 530 ], "score": 1.0, "content": "strategy and the better results are displayed in Table 2.", "type": "text" } ], "index": 20 } ], "index": 19.5 }, { "type": "title", "bbox": [ 107, 546, 177, 559 ], "lines": [ { "bbox": [ 106, 546, 178, 560 ], "spans": [ { "bbox": [ 106, 546, 178, 560 ], "score": 1.0, "content": "H METRICS", "type": "text" } ], "index": 21 } ], "index": 21 }, { "type": "text", "bbox": [ 107, 571, 504, 594 ], "lines": [ { "bbox": [ 105, 570, 506, 585 ], "spans": [ { "bbox": [ 105, 570, 213, 585 ], "score": 1.0, "content": "Denote the target value as", "type": "text" }, { "bbox": [ 214, 573, 229, 583 ], "score": 0.87, "content": "z _ { j , t }", "type": "inline_equation" }, { "bbox": [ 229, 570, 338, 585 ], "score": 1.0, "content": "and the predicted value as", "type": "text" }, { "bbox": [ 338, 572, 353, 583 ], "score": 0.9, "content": "\\hat { z } _ { j , t }", "type": "inline_equation" }, { "bbox": [ 353, 570, 384, 585 ], "score": 1.0, "content": ", where", "type": "text" }, { "bbox": [ 384, 572, 390, 583 ], "score": 0.82, "content": "j", "type": "inline_equation" }, { "bbox": [ 390, 570, 489, 585 ], "score": 1.0, "content": "is the sample index and", "type": "text" }, { "bbox": [ 489, 573, 494, 581 ], "score": 0.79, "content": "t", "type": "inline_equation" }, { "bbox": [ 495, 570, 506, 585 ], "score": 1.0, "content": "is", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 582, 366, 594 ], "spans": [ { "bbox": [ 105, 582, 366, 594 ], "score": 1.0, "content": "the time index. Then NRMSE and ND are calculated as follows:", "type": "text" } ], "index": 23 } ], "index": 22.5 }, { "type": "interline_equation", "bbox": [ 212, 597, 397, 674 ], "lines": [ { "bbox": [ 212, 597, 397, 674 ], "spans": [ { "bbox": [ 212, 597, 397, 674 ], "score": 0.95, "content": "\\begin{array} { r l } & { \\mathrm { N R M S E } = \\frac { \\sqrt { \\frac { 1 } { N T } \\sum _ { j = 1 } ^ { N } \\sum _ { t = 1 } ^ { T } ( z _ { j , t } - \\hat { z } _ { j , t } ) ^ { 2 } } } { \\frac { 1 } { N T } \\sum _ { j = 1 } ^ { N } \\sum _ { t = 1 } ^ { T } | z _ { j , t } | } , } \\\\ & { \\quad \\quad \\quad \\quad \\mathrm { N D } = \\frac { \\sum _ { j = 1 } ^ { N } \\sum _ { t = 1 } ^ { T } | z _ { j , t } - \\hat { z } _ { j , t } | } { \\sum _ { j = 1 } ^ { N } \\sum _ { t = 1 } ^ { T } | z _ { j , t } | } . } \\end{array}", "type": "interline_equation", "image_path": "2ce9de10c5adaecb9130fa6a5a862d53df598ea735cf99671fb521a8ec0f3504.jpg" } ] } ], "index": 26, "virtual_lines": [ { "bbox": [ 212, 597, 397, 612.4 ], "spans": [], "index": 24 }, { "bbox": [ 212, 612.4, 397, 627.8 ], "spans": [], "index": 25 }, { "bbox": [ 212, 627.8, 397, 643.1999999999999 ], "spans": [], "index": 26 }, { "bbox": [ 212, 643.1999999999999, 397, 658.5999999999999 ], "spans": [], "index": 27 }, { "bbox": [ 212, 658.5999999999999, 397, 673.9999999999999 ], "spans": [], "index": 28 } ] }, { "type": "title", "bbox": [ 106, 684, 310, 697 ], "lines": [ { "bbox": [ 105, 683, 311, 699 ], "spans": [ { "bbox": [ 105, 683, 311, 699 ], "score": 1.0, "content": "I EXPERIMENTS ON SYNTHETIC DATA", "type": "text" } ], "index": 29 } ], "index": 29 }, { "type": "text", "bbox": [ 106, 709, 506, 732 ], "lines": [ { "bbox": [ 106, 709, 505, 723 ], "spans": [ { "bbox": [ 106, 709, 505, 723 ], "score": 1.0, "content": "To further evaluate Pyraformer’s ability to capture different ranges of temporal dependencies, we", "type": "text" } ], "index": 30 }, { "bbox": [ 105, 720, 488, 734 ], "spans": [ { "bbox": [ 105, 720, 488, 734 ], "score": 1.0, "content": "synthesized an hourly dataset with multi-range dependencies and carried out experiments on it.", "type": "text" } ], "index": 31 } ], "index": 30.5 } ], "page_idx": 15, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 107, 27, 293, 37 ], "lines": [ { "bbox": [ 106, 26, 294, 38 ], "spans": [ { "bbox": [ 106, 26, 294, 38 ], "score": 1.0, "content": "Published as a conference paper at ICLR 2022", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 300, 751, 311, 760 ], "lines": [ { "bbox": [ 299, 750, 313, 764 ], "spans": [ { "bbox": [ 299, 750, 313, 764 ], "score": 1.0, "content": "16", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "table", "bbox": [ 151, 101, 459, 258 ], "blocks": [ { "type": "table_caption", "bbox": [ 180, 90, 429, 101 ], "group_id": 0, "lines": [ { "bbox": [ 181, 88, 430, 104 ], "spans": [ { "bbox": [ 181, 88, 430, 104 ], "score": 1.0, "content": "Table 5: Hyper-parameter settings of long-range experiments.", "type": "text" } ], "index": 0 } ], "index": 0 }, { "type": "table_body", "bbox": [ 151, 101, 459, 258 ], "group_id": 0, "lines": [ { "bbox": [ 151, 101, 459, 258 ], "spans": [ { "bbox": [ 151, 101, 459, 258 ], "score": 0.983, "html": "
Datasetprediction lengthNSHAChistorical length
ETTh116844634168
33644634168
72044654336
ETTm19644635384
28844655672
67244636672
Elect16844634168
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", "type": "table", "image_path": "26ecf047e6583e1cbb074f61df77c1a887082b1dd64d8db53e55094ae880fd9b.jpg" } ] } ], "index": 2, "virtual_lines": [ { "bbox": [ 151, 101, 459, 153.33333333333334 ], "spans": [], "index": 1 }, { "bbox": [ 151, 153.33333333333334, 459, 205.66666666666669 ], "spans": [], "index": 2 }, { "bbox": [ 151, 205.66666666666669, 459, 258.0 ], "spans": [], "index": 3 } ] } ], "index": 1.0 }, { "type": "image", "bbox": [ 213, 267, 396, 454 ], "blocks": [ { "type": "image_body", "bbox": [ 213, 267, 396, 454 ], "group_id": 0, "lines": [ { "bbox": [ 213, 267, 396, 454 ], "spans": [ { "bbox": [ 213, 267, 396, 454 ], "score": 0.964, "type": "image", "image_path": "16bca2db6aa268152dfacebb3a8ccf6bd2e2df85f5fcbe3870acda17a4461454.jpg" } ] } ], "index": 10, "virtual_lines": [ { "bbox": [ 213, 267, 396, 281.38461538461536 ], "spans": [], "index": 4 }, { "bbox": [ 213, 281.38461538461536, 396, 295.7692307692307 ], "spans": [], "index": 5 }, { "bbox": [ 213, 295.7692307692307, 396, 310.1538461538461 ], "spans": [], "index": 6 }, { "bbox": [ 213, 310.1538461538461, 396, 324.53846153846143 ], "spans": [], "index": 7 }, { "bbox": [ 213, 324.53846153846143, 396, 338.9230769230768 ], "spans": [], "index": 8 }, { "bbox": [ 213, 338.9230769230768, 396, 353.30769230769215 ], "spans": [], "index": 9 }, { "bbox": [ 213, 353.30769230769215, 396, 367.6923076923075 ], "spans": [], "index": 10 }, { "bbox": [ 213, 367.6923076923075, 396, 382.07692307692287 ], "spans": [], "index": 11 }, { "bbox": [ 213, 382.07692307692287, 396, 396.4615384615382 ], "spans": [], "index": 12 }, { "bbox": [ 213, 396.4615384615382, 396, 410.8461538461536 ], "spans": [], "index": 13 }, { "bbox": [ 213, 410.8461538461536, 396, 425.23076923076894 ], "spans": [], "index": 14 }, { "bbox": [ 213, 425.23076923076894, 396, 439.6153846153843 ], "spans": [], "index": 15 }, { "bbox": [ 213, 439.6153846153843, 396, 453.99999999999966 ], "spans": [], "index": 16 } ] }, { "type": "image_caption", "bbox": [ 106, 464, 505, 487 ], "group_id": 0, "lines": [ { "bbox": [ 105, 462, 505, 477 ], "spans": [ { "bbox": [ 105, 462, 505, 477 ], "score": 1.0, "content": "Figure 5: The pretraining strategy for one-step prediction. Features of nodes surrounded by the", "type": "text" } ], "index": 17 }, { "bbox": [ 105, 474, 404, 488 ], "spans": [ { "bbox": [ 105, 474, 404, 488 ], "score": 1.0, "content": "dashed ellipses are concatenated to recover the corresponding input value.", "type": "text" } ], "index": 18 } ], "index": 17.5 } ], "index": 13.75 }, { "type": "text", "bbox": [ 106, 507, 504, 530 ], "lines": [], "index": 19.5, "bbox_fs": [ 105, 505, 506, 530 ], "lines_deleted": true }, { "type": "title", "bbox": [ 107, 546, 177, 559 ], "lines": [ { "bbox": [ 106, 546, 178, 560 ], "spans": [ { "bbox": [ 106, 546, 178, 560 ], "score": 1.0, "content": "H METRICS", "type": "text" } ], "index": 21 } ], "index": 21 }, { "type": "text", "bbox": [ 107, 571, 504, 594 ], "lines": [ { "bbox": [ 105, 570, 506, 585 ], "spans": [ { "bbox": [ 105, 570, 213, 585 ], "score": 1.0, "content": "Denote the target value as", "type": "text" }, { "bbox": [ 214, 573, 229, 583 ], "score": 0.87, "content": "z _ { j , t }", "type": "inline_equation" }, { "bbox": [ 229, 570, 338, 585 ], "score": 1.0, "content": "and the predicted value as", "type": "text" }, { "bbox": [ 338, 572, 353, 583 ], "score": 0.9, "content": "\\hat { z } _ { j , t }", "type": "inline_equation" }, { "bbox": [ 353, 570, 384, 585 ], "score": 1.0, "content": ", where", "type": "text" }, { "bbox": [ 384, 572, 390, 583 ], "score": 0.82, "content": "j", "type": "inline_equation" }, { "bbox": [ 390, 570, 489, 585 ], "score": 1.0, "content": "is the sample index and", "type": "text" }, { "bbox": [ 489, 573, 494, 581 ], "score": 0.79, "content": "t", "type": "inline_equation" }, { "bbox": [ 495, 570, 506, 585 ], "score": 1.0, "content": "is", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 582, 366, 594 ], "spans": [ { "bbox": [ 105, 582, 366, 594 ], "score": 1.0, "content": "the time index. Then NRMSE and ND are calculated as follows:", "type": "text" } ], "index": 23 } ], "index": 22.5, "bbox_fs": [ 105, 570, 506, 594 ] }, { "type": "interline_equation", "bbox": [ 212, 597, 397, 674 ], "lines": [ { "bbox": [ 212, 597, 397, 674 ], "spans": [ { "bbox": [ 212, 597, 397, 674 ], "score": 0.95, "content": "\\begin{array} { r l } & { \\mathrm { N R M S E } = \\frac { \\sqrt { \\frac { 1 } { N T } \\sum _ { j = 1 } ^ { N } \\sum _ { t = 1 } ^ { T } ( z _ { j , t } - \\hat { z } _ { j , t } ) ^ { 2 } } } { \\frac { 1 } { N T } \\sum _ { j = 1 } ^ { N } \\sum _ { t = 1 } ^ { T } | z _ { j , t } | } , } \\\\ & { \\quad \\quad \\quad \\quad \\mathrm { N D } = \\frac { \\sum _ { j = 1 } ^ { N } \\sum _ { t = 1 } ^ { T } | z _ { j , t } - \\hat { z } _ { j , t } | } { \\sum _ { j = 1 } ^ { N } \\sum _ { t = 1 } ^ { T } | z _ { j , t } | } . } \\end{array}", "type": "interline_equation", "image_path": "2ce9de10c5adaecb9130fa6a5a862d53df598ea735cf99671fb521a8ec0f3504.jpg" } ] } ], "index": 26, "virtual_lines": [ { "bbox": [ 212, 597, 397, 612.4 ], "spans": [], "index": 24 }, { "bbox": [ 212, 612.4, 397, 627.8 ], "spans": [], "index": 25 }, { "bbox": [ 212, 627.8, 397, 643.1999999999999 ], "spans": [], "index": 26 }, { "bbox": [ 212, 643.1999999999999, 397, 658.5999999999999 ], "spans": [], "index": 27 }, { "bbox": [ 212, 658.5999999999999, 397, 673.9999999999999 ], "spans": [], "index": 28 } ] }, { "type": "title", "bbox": [ 106, 684, 310, 697 ], "lines": [ { "bbox": [ 105, 683, 311, 699 ], "spans": [ { "bbox": [ 105, 683, 311, 699 ], "score": 1.0, "content": "I EXPERIMENTS ON SYNTHETIC DATA", "type": "text" } ], "index": 29 } ], "index": 29 }, { "type": "text", "bbox": [ 106, 709, 506, 732 ], "lines": [ { "bbox": [ 106, 709, 505, 723 ], "spans": [ { "bbox": [ 106, 709, 505, 723 ], "score": 1.0, "content": "To further evaluate Pyraformer’s ability to capture different ranges of temporal dependencies, we", "type": "text" } ], "index": 30 }, { "bbox": [ 105, 720, 488, 734 ], "spans": [ { "bbox": [ 105, 720, 488, 734 ], "score": 1.0, "content": "synthesized an hourly dataset with multi-range dependencies and carried out experiments on it.", "type": "text" } ], "index": 31 } ], "index": 30.5, "bbox_fs": [ 105, 709, 505, 734 ] } ] }, { "preproc_blocks": [ { "type": "text", "bbox": [ 106, 82, 504, 105 ], "lines": [ { "bbox": [ 106, 82, 505, 95 ], "spans": [ { "bbox": [ 106, 82, 505, 95 ], "score": 1.0, "content": "Specifically, each time series in the synthetic dataset is a linear combination of three sine functions", "type": "text" } ], "index": 0 }, { "bbox": [ 106, 94, 288, 105 ], "spans": [ { "bbox": [ 106, 94, 288, 105 ], "score": 1.0, "content": "of different periods: 24, 168 and 720, that is,", "type": "text" } ], "index": 1 } ], "index": 0.5 }, { "type": "interline_equation", "bbox": [ 186, 109, 424, 133 ], "lines": [ { "bbox": [ 186, 109, 424, 133 ], "spans": [ { "bbox": [ 186, 109, 424, 133 ], "score": 0.92, "content": "f ( t ) = \\beta _ { 0 } + \\beta _ { 1 } \\sin ( \\frac { 2 \\pi } { 2 4 } t ) + \\beta _ { 2 } \\sin ( \\frac { 2 \\pi } { 1 6 8 } t ) + \\beta _ { 3 } \\sin ( \\frac { 2 \\pi } { 7 2 0 } t ) .", "type": "interline_equation", "image_path": "463937176375f114a98c6d72dd24dd030190e1bdec26ec4688589e1dc21c0ac9.jpg" } ] } ], "index": 2, "virtual_lines": [ { "bbox": [ 186, 109, 424, 133 ], "spans": [], "index": 2 } ] }, { "type": "text", "bbox": [ 106, 136, 505, 279 ], "lines": [ { "bbox": [ 106, 136, 505, 148 ], "spans": [ { "bbox": [ 106, 136, 381, 148 ], "score": 1.0, "content": "In the above equation, the coefficients of the three sine functions", "type": "text" }, { "bbox": [ 381, 136, 409, 148 ], "score": 0.53, "content": "\\beta _ { 1 } , \\beta _ { 2 }", "type": "inline_equation" }, { "bbox": [ 410, 136, 433, 148 ], "score": 1.0, "content": ", and", "type": "text" }, { "bbox": [ 433, 136, 444, 148 ], "score": 0.88, "content": "\\beta _ { 3 }", "type": "inline_equation" }, { "bbox": [ 445, 136, 505, 148 ], "score": 1.0, "content": "for each time", "type": "text" } ], "index": 3 }, { "bbox": [ 105, 146, 505, 159 ], "spans": [ { "bbox": [ 105, 146, 286, 159 ], "score": 1.0, "content": "series are uniformly sampled from [5, 10].", "type": "text" }, { "bbox": [ 286, 147, 298, 159 ], "score": 0.87, "content": "\\beta _ { 0 }", "type": "inline_equation" }, { "bbox": [ 299, 146, 505, 159 ], "score": 1.0, "content": "is a Gaussian process with a covariance function", "type": "text" } ], "index": 4 }, { "bbox": [ 107, 156, 506, 172 ], "spans": [ { "bbox": [ 107, 158, 191, 171 ], "score": 0.92, "content": "\\Sigma _ { t _ { 1 } , t _ { 2 } } = | t _ { 1 } - t _ { 2 } | ^ { - 1 }", "type": "inline_equation" }, { "bbox": [ 191, 156, 209, 172 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 210, 158, 273, 170 ], "score": 0.92, "content": "\\Sigma _ { t _ { 1 } } = \\Sigma _ { t _ { 2 } } = 1", "type": "inline_equation" }, { "bbox": [ 274, 156, 304, 172 ], "score": 1.0, "content": ", where", "type": "text" }, { "bbox": [ 305, 159, 314, 169 ], "score": 0.87, "content": "t _ { 1 }", "type": "inline_equation" }, { "bbox": [ 315, 156, 333, 172 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 333, 159, 342, 169 ], "score": 0.87, "content": "t _ { 2 }", "type": "inline_equation" }, { "bbox": [ 343, 156, 506, 172 ], "score": 1.0, "content": "denote two arbitrary time stamps. Such", "type": "text" } ], "index": 5 }, { "bbox": [ 106, 169, 505, 181 ], "spans": [ { "bbox": [ 106, 169, 505, 181 ], "score": 1.0, "content": "polynomially decaying covariance functions are known to have long-range dependence, as oppose to", "type": "text" } ], "index": 6 }, { "bbox": [ 105, 180, 506, 192 ], "spans": [ { "bbox": [ 105, 180, 304, 192 ], "score": 1.0, "content": "the exponentially decaying covariance functions (", "type": "text" }, { "bbox": [ 304, 181, 317, 190 ], "score": 0.26, "content": "\\mathrm { Y u }", "type": "inline_equation" }, { "bbox": [ 317, 180, 506, 192 ], "score": 1.0, "content": "et al., 2019). The start time of each time series", "type": "text" } ], "index": 7 }, { "bbox": [ 106, 190, 505, 203 ], "spans": [ { "bbox": [ 106, 192, 115, 202 ], "score": 0.86, "content": "t _ { 0 }", "type": "inline_equation" }, { "bbox": [ 116, 190, 505, 203 ], "score": 1.0, "content": "is uniformly sampled from [0, 719]. We first generate 60 time series of length 14400, and then", "type": "text" } ], "index": 8 }, { "bbox": [ 105, 201, 506, 216 ], "spans": [ { "bbox": [ 105, 201, 506, 216 ], "score": 1.0, "content": "split each time series into sliding windows of width 1440 with a stride of 24. In our experiments, we", "type": "text" } ], "index": 9 }, { "bbox": [ 105, 213, 506, 226 ], "spans": [ { "bbox": [ 105, 213, 506, 226 ], "score": 1.0, "content": "use the historical 720 time points to predict the future 720 points. Since both the deterministic and", "type": "text" } ], "index": 10 }, { "bbox": [ 104, 223, 506, 237 ], "spans": [ { "bbox": [ 104, 223, 506, 237 ], "score": 1.0, "content": "stochastic parts of the synthetic time series have long-range correlations, such dependencies should", "type": "text" } ], "index": 11 }, { "bbox": [ 106, 235, 505, 247 ], "spans": [ { "bbox": [ 106, 235, 505, 247 ], "score": 1.0, "content": "be well captured in the model in order to yield accurate predictions of the next 720 points. The", "type": "text" } ], "index": 12 }, { "bbox": [ 105, 245, 505, 258 ], "spans": [ { "bbox": [ 105, 245, 505, 258 ], "score": 1.0, "content": "results are summarized in Table 6. Here, we consider two different configurations of Pyraformer: 1)", "type": "text" } ], "index": 13 }, { "bbox": [ 107, 257, 506, 270 ], "spans": [ { "bbox": [ 107, 257, 136, 267 ], "score": 0.89, "content": "C = 6", "type": "inline_equation" }, { "bbox": [ 136, 257, 417, 270 ], "score": 1.0, "content": "for all scales in the pyramidal graph (denoted as Pyraformer6,6,6); 2)", "type": "text" }, { "bbox": [ 417, 257, 451, 267 ], "score": 0.79, "content": "C = 1 2", "type": "inline_equation" }, { "bbox": [ 451, 257, 506, 270 ], "score": 1.0, "content": ", 7, and 4 for", "type": "text" } ], "index": 14 }, { "bbox": [ 105, 266, 422, 282 ], "spans": [ { "bbox": [ 105, 266, 422, 282 ], "score": 1.0, "content": "the three layers sequentially from bottom to top (denoted as Pyraformer12,7,4).", "type": "text" } ], "index": 15 } ], "index": 9 }, { "type": "table", "bbox": [ 229, 311, 380, 450 ], "blocks": [ { "type": "table_caption", "bbox": [ 176, 299, 433, 310 ], "group_id": 0, "lines": [ { "bbox": [ 176, 298, 434, 312 ], "spans": [ { "bbox": [ 176, 298, 434, 312 ], "score": 1.0, "content": "Table 6: Long-range forecasting results on the synthetic dataset.", "type": "text" } ], "index": 16 } ], "index": 16 }, { "type": "table_body", "bbox": [ 229, 311, 380, 450 ], "group_id": 0, "lines": [ { "bbox": [ 229, 311, 380, 450 ], "spans": [ { "bbox": [ 229, 311, 380, 450 ], "score": 0.979, "html": "
MethodMSEMAE
Full attention3.5501.477
LogTrans3.0071.366
ETC4.7425.509
Informer7.5462.092
Longformer2.0321.116
Reformer1.5383.069
Pyraformer6,6,61.2580.877
Pyraformer12,7,41.1760.849
", "type": "table", "image_path": "91fa68487755d17c2f286bf23144e06467b5bd3a504816ccc92ed58d65d5b1c6.jpg" } ] } ], "index": 17.5, "virtual_lines": [ { "bbox": [ 229, 311, 380, 380.5 ], "spans": [], "index": 17 }, { "bbox": [ 229, 380.5, 380, 450.0 ], "spans": [], "index": 18 } ] } ], "index": 16.75 }, { "type": "text", "bbox": [ 106, 460, 505, 549 ], "lines": [ { "bbox": [ 104, 460, 506, 474 ], "spans": [ { "bbox": [ 104, 460, 335, 474 ], "score": 1.0, "content": "It can be observed that Pyraformer6,6,6 with the same", "type": "text" }, { "bbox": [ 335, 461, 344, 470 ], "score": 0.8, "content": "C", "type": "inline_equation" }, { "bbox": [ 345, 460, 506, 474 ], "score": 1.0, "content": "for all scales already outperforms the", "type": "text" } ], "index": 19 }, { "bbox": [ 105, 471, 505, 484 ], "spans": [ { "bbox": [ 105, 471, 505, 484 ], "score": 1.0, "content": "benchmark methods by a large margin. In particular, the MSE given by Pyraformer is decreased by", "type": "text" } ], "index": 20 }, { "bbox": [ 107, 481, 506, 496 ], "spans": [ { "bbox": [ 107, 482, 134, 493 ], "score": 0.88, "content": "1 8 . 2 \\%", "type": "inline_equation" }, { "bbox": [ 135, 481, 506, 496 ], "score": 1.0, "content": "compared with Reformer, which produces the smallest MSE among the existing variants of", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 492, 506, 508 ], "spans": [ { "bbox": [ 105, 492, 506, 508 ], "score": 1.0, "content": "Transformer. On the other hand, by exploiting the information of the known period, Pyraformer12,7,4", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 504, 506, 519 ], "spans": [ { "bbox": [ 105, 504, 376, 519 ], "score": 1.0, "content": "performs even better than Pyraformer6,6,6. Note that in Pyraformer", "type": "text" }, { "bbox": [ 376, 506, 398, 516 ], "score": 0.4, "content": "^ { \\cdot _ { 1 2 , 7 , 4 } }", "type": "inline_equation" }, { "bbox": [ 398, 504, 506, 519 ], "score": 1.0, "content": ", nodes at scale 2, 3, and 4", "type": "text" } ], "index": 23 }, { "bbox": [ 105, 515, 505, 528 ], "spans": [ { "bbox": [ 105, 515, 505, 528 ], "score": 1.0, "content": "characterizes coarser temporal resolutions respectively corresponding to half a day, half a week, and", "type": "text" } ], "index": 24 }, { "bbox": [ 105, 526, 506, 540 ], "spans": [ { "bbox": [ 105, 526, 339, 540 ], "score": 1.0, "content": "half a month. We also tested Pyraformer24,7,4, but setting", "type": "text" }, { "bbox": [ 340, 527, 373, 537 ], "score": 0.9, "content": "C = 2 4", "type": "inline_equation" }, { "bbox": [ 374, 526, 506, 540 ], "score": 1.0, "content": "in the second scale degrades the", "type": "text" } ], "index": 25 }, { "bbox": [ 105, 537, 492, 549 ], "spans": [ { "bbox": [ 105, 537, 492, 549 ], "score": 1.0, "content": "performance, probably because the convolution layer with a kernel size of 24 is difficult to train.", "type": "text" } ], "index": 26 } ], "index": 22.5 }, { "type": "text", "bbox": [ 107, 554, 505, 599 ], "lines": [ { "bbox": [ 105, 553, 505, 568 ], "spans": [ { "bbox": [ 105, 553, 389, 568 ], "score": 1.0, "content": "We further visualized the forecasting results produced by Pyraformer", "type": "text" }, { "bbox": [ 389, 556, 411, 567 ], "score": 0.76, "content": "^ { 1 2 , 7 , 4 }", "type": "inline_equation" }, { "bbox": [ 411, 553, 505, 568 ], "score": 1.0, "content": "in Figure 6. The blue", "type": "text" } ], "index": 27 }, { "bbox": [ 105, 563, 505, 579 ], "spans": [ { "bbox": [ 105, 563, 505, 579 ], "score": 1.0, "content": "solid curve and red dashed curve denote the true and predicted time series respectively. By capturing", "type": "text" } ], "index": 28 }, { "bbox": [ 105, 576, 505, 589 ], "spans": [ { "bbox": [ 105, 576, 505, 589 ], "score": 1.0, "content": "the temporal dependencies with different ranges, the prediction resulting from Pyraformer closely", "type": "text" } ], "index": 29 }, { "bbox": [ 105, 587, 207, 599 ], "spans": [ { "bbox": [ 105, 587, 207, 599 ], "score": 1.0, "content": "follows the ground truth.", "type": "text" } ], "index": 30 } ], "index": 28.5 }, { "type": "text", "bbox": [ 107, 604, 505, 648 ], "lines": [ { "bbox": [ 105, 604, 505, 616 ], "spans": [ { "bbox": [ 105, 604, 505, 616 ], "score": 1.0, "content": "On the other hand, to check whether Pyraformer can extract features with different temporal resolu-", "type": "text" } ], "index": 31 }, { "bbox": [ 106, 615, 505, 627 ], "spans": [ { "bbox": [ 106, 615, 505, 627 ], "score": 1.0, "content": "tions, we depicted the extracted features in a randomly selected channel across time at each scale in", "type": "text" } ], "index": 32 }, { "bbox": [ 105, 626, 505, 638 ], "spans": [ { "bbox": [ 105, 626, 505, 638 ], "score": 1.0, "content": "the pyramidal graph in Figure 7. It is apparent that the features at the coarser scales can be regarded", "type": "text" } ], "index": 33 }, { "bbox": [ 105, 637, 356, 648 ], "spans": [ { "bbox": [ 105, 637, 356, 648 ], "score": 1.0, "content": "as a lower resolution version of the features at the finer scales.", "type": "text" } ], "index": 34 } ], "index": 32.5 }, { "type": "title", "bbox": [ 108, 664, 217, 677 ], "lines": [ { "bbox": [ 105, 663, 219, 679 ], "spans": [ { "bbox": [ 105, 663, 219, 679 ], "score": 1.0, "content": "J ABLATION STUDY", "type": "text" } ], "index": 35 } ], "index": 35 }, { "type": "title", "bbox": [ 107, 689, 218, 700 ], "lines": [ { "bbox": [ 105, 687, 218, 702 ], "spans": [ { "bbox": [ 105, 687, 178, 702 ], "score": 1.0, "content": "J.1 IMPACT OF", "type": "text" }, { "bbox": [ 178, 690, 187, 699 ], "score": 0.42, "content": "A", "type": "inline_equation" }, { "bbox": [ 187, 687, 209, 702 ], "score": 1.0, "content": "AND", "type": "text" }, { "bbox": [ 209, 690, 218, 699 ], "score": 0.68, "content": "C", "type": "inline_equation" } ], "index": 36 } ], "index": 36 }, { "type": "text", "bbox": [ 107, 709, 504, 732 ], "lines": [ { "bbox": [ 106, 708, 505, 721 ], "spans": [ { "bbox": [ 106, 708, 213, 721 ], "score": 1.0, "content": "We studied the impact of", "type": "text" }, { "bbox": [ 213, 710, 222, 720 ], "score": 0.74, "content": "A", "type": "inline_equation" }, { "bbox": [ 222, 708, 242, 721 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 243, 710, 251, 720 ], "score": 0.82, "content": "C", "type": "inline_equation" }, { "bbox": [ 252, 708, 505, 721 ], "score": 1.0, "content": "on the performance of Pyraformer for long-range time series", "type": "text" } ], "index": 37 }, { "bbox": [ 105, 719, 506, 734 ], "spans": [ { "bbox": [ 105, 719, 506, 734 ], "score": 1.0, "content": "forecasting, and showed the results in Table 7. Here, we focus on the dataset ETTh1. The history", "type": "text" } ], "index": 38 } ], "index": 37.5 } ], "page_idx": 16, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 107, 27, 293, 37 ], "lines": [ { "bbox": [ 106, 26, 294, 38 ], "spans": [ { "bbox": [ 106, 26, 294, 38 ], "score": 1.0, "content": "Published as a conference paper at ICLR 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": "17", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "text", "bbox": [ 106, 82, 504, 105 ], "lines": [ { "bbox": [ 106, 82, 505, 95 ], "spans": [ { "bbox": [ 106, 82, 505, 95 ], "score": 1.0, "content": "Specifically, each time series in the synthetic dataset is a linear combination of three sine functions", "type": "text" } ], "index": 0 }, { "bbox": [ 106, 94, 288, 105 ], "spans": [ { "bbox": [ 106, 94, 288, 105 ], "score": 1.0, "content": "of different periods: 24, 168 and 720, that is,", "type": "text" } ], "index": 1 } ], "index": 0.5, "bbox_fs": [ 106, 82, 505, 105 ] }, { "type": "interline_equation", "bbox": [ 186, 109, 424, 133 ], "lines": [ { "bbox": [ 186, 109, 424, 133 ], "spans": [ { "bbox": [ 186, 109, 424, 133 ], "score": 0.92, "content": "f ( t ) = \\beta _ { 0 } + \\beta _ { 1 } \\sin ( \\frac { 2 \\pi } { 2 4 } t ) + \\beta _ { 2 } \\sin ( \\frac { 2 \\pi } { 1 6 8 } t ) + \\beta _ { 3 } \\sin ( \\frac { 2 \\pi } { 7 2 0 } t ) .", "type": "interline_equation", "image_path": "463937176375f114a98c6d72dd24dd030190e1bdec26ec4688589e1dc21c0ac9.jpg" } ] } ], "index": 2, "virtual_lines": [ { "bbox": [ 186, 109, 424, 133 ], "spans": [], "index": 2 } ] }, { "type": "text", "bbox": [ 106, 136, 505, 279 ], "lines": [ { "bbox": [ 106, 136, 505, 148 ], "spans": [ { "bbox": [ 106, 136, 381, 148 ], "score": 1.0, "content": "In the above equation, the coefficients of the three sine functions", "type": "text" }, { "bbox": [ 381, 136, 409, 148 ], "score": 0.53, "content": "\\beta _ { 1 } , \\beta _ { 2 }", "type": "inline_equation" }, { "bbox": [ 410, 136, 433, 148 ], "score": 1.0, "content": ", and", "type": "text" }, { "bbox": [ 433, 136, 444, 148 ], "score": 0.88, "content": "\\beta _ { 3 }", "type": "inline_equation" }, { "bbox": [ 445, 136, 505, 148 ], "score": 1.0, "content": "for each time", "type": "text" } ], "index": 3 }, { "bbox": [ 105, 146, 505, 159 ], "spans": [ { "bbox": [ 105, 146, 286, 159 ], "score": 1.0, "content": "series are uniformly sampled from [5, 10].", "type": "text" }, { "bbox": [ 286, 147, 298, 159 ], "score": 0.87, "content": "\\beta _ { 0 }", "type": "inline_equation" }, { "bbox": [ 299, 146, 505, 159 ], "score": 1.0, "content": "is a Gaussian process with a covariance function", "type": "text" } ], "index": 4 }, { "bbox": [ 107, 156, 506, 172 ], "spans": [ { "bbox": [ 107, 158, 191, 171 ], "score": 0.92, "content": "\\Sigma _ { t _ { 1 } , t _ { 2 } } = | t _ { 1 } - t _ { 2 } | ^ { - 1 }", "type": "inline_equation" }, { "bbox": [ 191, 156, 209, 172 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 210, 158, 273, 170 ], "score": 0.92, "content": "\\Sigma _ { t _ { 1 } } = \\Sigma _ { t _ { 2 } } = 1", "type": "inline_equation" }, { "bbox": [ 274, 156, 304, 172 ], "score": 1.0, "content": ", where", "type": "text" }, { "bbox": [ 305, 159, 314, 169 ], "score": 0.87, "content": "t _ { 1 }", "type": "inline_equation" }, { "bbox": [ 315, 156, 333, 172 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 333, 159, 342, 169 ], "score": 0.87, "content": "t _ { 2 }", "type": "inline_equation" }, { "bbox": [ 343, 156, 506, 172 ], "score": 1.0, "content": "denote two arbitrary time stamps. Such", "type": "text" } ], "index": 5 }, { "bbox": [ 106, 169, 505, 181 ], "spans": [ { "bbox": [ 106, 169, 505, 181 ], "score": 1.0, "content": "polynomially decaying covariance functions are known to have long-range dependence, as oppose to", "type": "text" } ], "index": 6 }, { "bbox": [ 105, 180, 506, 192 ], "spans": [ { "bbox": [ 105, 180, 304, 192 ], "score": 1.0, "content": "the exponentially decaying covariance functions (", "type": "text" }, { "bbox": [ 304, 181, 317, 190 ], "score": 0.26, "content": "\\mathrm { Y u }", "type": "inline_equation" }, { "bbox": [ 317, 180, 506, 192 ], "score": 1.0, "content": "et al., 2019). The start time of each time series", "type": "text" } ], "index": 7 }, { "bbox": [ 106, 190, 505, 203 ], "spans": [ { "bbox": [ 106, 192, 115, 202 ], "score": 0.86, "content": "t _ { 0 }", "type": "inline_equation" }, { "bbox": [ 116, 190, 505, 203 ], "score": 1.0, "content": "is uniformly sampled from [0, 719]. We first generate 60 time series of length 14400, and then", "type": "text" } ], "index": 8 }, { "bbox": [ 105, 201, 506, 216 ], "spans": [ { "bbox": [ 105, 201, 506, 216 ], "score": 1.0, "content": "split each time series into sliding windows of width 1440 with a stride of 24. In our experiments, we", "type": "text" } ], "index": 9 }, { "bbox": [ 105, 213, 506, 226 ], "spans": [ { "bbox": [ 105, 213, 506, 226 ], "score": 1.0, "content": "use the historical 720 time points to predict the future 720 points. Since both the deterministic and", "type": "text" } ], "index": 10 }, { "bbox": [ 104, 223, 506, 237 ], "spans": [ { "bbox": [ 104, 223, 506, 237 ], "score": 1.0, "content": "stochastic parts of the synthetic time series have long-range correlations, such dependencies should", "type": "text" } ], "index": 11 }, { "bbox": [ 106, 235, 505, 247 ], "spans": [ { "bbox": [ 106, 235, 505, 247 ], "score": 1.0, "content": "be well captured in the model in order to yield accurate predictions of the next 720 points. The", "type": "text" } ], "index": 12 }, { "bbox": [ 105, 245, 505, 258 ], "spans": [ { "bbox": [ 105, 245, 505, 258 ], "score": 1.0, "content": "results are summarized in Table 6. Here, we consider two different configurations of Pyraformer: 1)", "type": "text" } ], "index": 13 }, { "bbox": [ 107, 257, 506, 270 ], "spans": [ { "bbox": [ 107, 257, 136, 267 ], "score": 0.89, "content": "C = 6", "type": "inline_equation" }, { "bbox": [ 136, 257, 417, 270 ], "score": 1.0, "content": "for all scales in the pyramidal graph (denoted as Pyraformer6,6,6); 2)", "type": "text" }, { "bbox": [ 417, 257, 451, 267 ], "score": 0.79, "content": "C = 1 2", "type": "inline_equation" }, { "bbox": [ 451, 257, 506, 270 ], "score": 1.0, "content": ", 7, and 4 for", "type": "text" } ], "index": 14 }, { "bbox": [ 105, 266, 422, 282 ], "spans": [ { "bbox": [ 105, 266, 422, 282 ], "score": 1.0, "content": "the three layers sequentially from bottom to top (denoted as Pyraformer12,7,4).", "type": "text" } ], "index": 15 } ], "index": 9, "bbox_fs": [ 104, 136, 506, 282 ] }, { "type": "table", "bbox": [ 229, 311, 380, 450 ], "blocks": [ { "type": "table_caption", "bbox": [ 176, 299, 433, 310 ], "group_id": 0, "lines": [ { "bbox": [ 176, 298, 434, 312 ], "spans": [ { "bbox": [ 176, 298, 434, 312 ], "score": 1.0, "content": "Table 6: Long-range forecasting results on the synthetic dataset.", "type": "text" } ], "index": 16 } ], "index": 16 }, { "type": "table_body", "bbox": [ 229, 311, 380, 450 ], "group_id": 0, "lines": [ { "bbox": [ 229, 311, 380, 450 ], "spans": [ { "bbox": [ 229, 311, 380, 450 ], "score": 0.979, "html": "
MethodMSEMAE
Full attention3.5501.477
LogTrans3.0071.366
ETC4.7425.509
Informer7.5462.092
Longformer2.0321.116
Reformer1.5383.069
Pyraformer6,6,61.2580.877
Pyraformer12,7,41.1760.849
", "type": "table", "image_path": "91fa68487755d17c2f286bf23144e06467b5bd3a504816ccc92ed58d65d5b1c6.jpg" } ] } ], "index": 17.5, "virtual_lines": [ { "bbox": [ 229, 311, 380, 380.5 ], "spans": [], "index": 17 }, { "bbox": [ 229, 380.5, 380, 450.0 ], "spans": [], "index": 18 } ] } ], "index": 16.75 }, { "type": "text", "bbox": [ 106, 460, 505, 549 ], "lines": [ { "bbox": [ 104, 460, 506, 474 ], "spans": [ { "bbox": [ 104, 460, 335, 474 ], "score": 1.0, "content": "It can be observed that Pyraformer6,6,6 with the same", "type": "text" }, { "bbox": [ 335, 461, 344, 470 ], "score": 0.8, "content": "C", "type": "inline_equation" }, { "bbox": [ 345, 460, 506, 474 ], "score": 1.0, "content": "for all scales already outperforms the", "type": "text" } ], "index": 19 }, { "bbox": [ 105, 471, 505, 484 ], "spans": [ { "bbox": [ 105, 471, 505, 484 ], "score": 1.0, "content": "benchmark methods by a large margin. In particular, the MSE given by Pyraformer is decreased by", "type": "text" } ], "index": 20 }, { "bbox": [ 107, 481, 506, 496 ], "spans": [ { "bbox": [ 107, 482, 134, 493 ], "score": 0.88, "content": "1 8 . 2 \\%", "type": "inline_equation" }, { "bbox": [ 135, 481, 506, 496 ], "score": 1.0, "content": "compared with Reformer, which produces the smallest MSE among the existing variants of", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 492, 506, 508 ], "spans": [ { "bbox": [ 105, 492, 506, 508 ], "score": 1.0, "content": "Transformer. On the other hand, by exploiting the information of the known period, Pyraformer12,7,4", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 504, 506, 519 ], "spans": [ { "bbox": [ 105, 504, 376, 519 ], "score": 1.0, "content": "performs even better than Pyraformer6,6,6. Note that in Pyraformer", "type": "text" }, { "bbox": [ 376, 506, 398, 516 ], "score": 0.4, "content": "^ { \\cdot _ { 1 2 , 7 , 4 } }", "type": "inline_equation" }, { "bbox": [ 398, 504, 506, 519 ], "score": 1.0, "content": ", nodes at scale 2, 3, and 4", "type": "text" } ], "index": 23 }, { "bbox": [ 105, 515, 505, 528 ], "spans": [ { "bbox": [ 105, 515, 505, 528 ], "score": 1.0, "content": "characterizes coarser temporal resolutions respectively corresponding to half a day, half a week, and", "type": "text" } ], "index": 24 }, { "bbox": [ 105, 526, 506, 540 ], "spans": [ { "bbox": [ 105, 526, 339, 540 ], "score": 1.0, "content": "half a month. We also tested Pyraformer24,7,4, but setting", "type": "text" }, { "bbox": [ 340, 527, 373, 537 ], "score": 0.9, "content": "C = 2 4", "type": "inline_equation" }, { "bbox": [ 374, 526, 506, 540 ], "score": 1.0, "content": "in the second scale degrades the", "type": "text" } ], "index": 25 }, { "bbox": [ 105, 537, 492, 549 ], "spans": [ { "bbox": [ 105, 537, 492, 549 ], "score": 1.0, "content": "performance, probably because the convolution layer with a kernel size of 24 is difficult to train.", "type": "text" } ], "index": 26 } ], "index": 22.5, "bbox_fs": [ 104, 460, 506, 549 ] }, { "type": "text", "bbox": [ 107, 554, 505, 599 ], "lines": [ { "bbox": [ 105, 553, 505, 568 ], "spans": [ { "bbox": [ 105, 553, 389, 568 ], "score": 1.0, "content": "We further visualized the forecasting results produced by Pyraformer", "type": "text" }, { "bbox": [ 389, 556, 411, 567 ], "score": 0.76, "content": "^ { 1 2 , 7 , 4 }", "type": "inline_equation" }, { "bbox": [ 411, 553, 505, 568 ], "score": 1.0, "content": "in Figure 6. The blue", "type": "text" } ], "index": 27 }, { "bbox": [ 105, 563, 505, 579 ], "spans": [ { "bbox": [ 105, 563, 505, 579 ], "score": 1.0, "content": "solid curve and red dashed curve denote the true and predicted time series respectively. By capturing", "type": "text" } ], "index": 28 }, { "bbox": [ 105, 576, 505, 589 ], "spans": [ { "bbox": [ 105, 576, 505, 589 ], "score": 1.0, "content": "the temporal dependencies with different ranges, the prediction resulting from Pyraformer closely", "type": "text" } ], "index": 29 }, { "bbox": [ 105, 587, 207, 599 ], "spans": [ { "bbox": [ 105, 587, 207, 599 ], "score": 1.0, "content": "follows the ground truth.", "type": "text" } ], "index": 30 } ], "index": 28.5, "bbox_fs": [ 105, 553, 505, 599 ] }, { "type": "text", "bbox": [ 107, 604, 505, 648 ], "lines": [ { "bbox": [ 105, 604, 505, 616 ], "spans": [ { "bbox": [ 105, 604, 505, 616 ], "score": 1.0, "content": "On the other hand, to check whether Pyraformer can extract features with different temporal resolu-", "type": "text" } ], "index": 31 }, { "bbox": [ 106, 615, 505, 627 ], "spans": [ { "bbox": [ 106, 615, 505, 627 ], "score": 1.0, "content": "tions, we depicted the extracted features in a randomly selected channel across time at each scale in", "type": "text" } ], "index": 32 }, { "bbox": [ 105, 626, 505, 638 ], "spans": [ { "bbox": [ 105, 626, 505, 638 ], "score": 1.0, "content": "the pyramidal graph in Figure 7. It is apparent that the features at the coarser scales can be regarded", "type": "text" } ], "index": 33 }, { "bbox": [ 105, 637, 356, 648 ], "spans": [ { "bbox": [ 105, 637, 356, 648 ], "score": 1.0, "content": "as a lower resolution version of the features at the finer scales.", "type": "text" } ], "index": 34 } ], "index": 32.5, "bbox_fs": [ 105, 604, 505, 648 ] }, { "type": "title", "bbox": [ 108, 664, 217, 677 ], "lines": [ { "bbox": [ 105, 663, 219, 679 ], "spans": [ { "bbox": [ 105, 663, 219, 679 ], "score": 1.0, "content": "J ABLATION STUDY", "type": "text" } ], "index": 35 } ], "index": 35 }, { "type": "title", "bbox": [ 107, 689, 218, 700 ], "lines": [ { "bbox": [ 105, 687, 218, 702 ], "spans": [ { "bbox": [ 105, 687, 178, 702 ], "score": 1.0, "content": "J.1 IMPACT OF", "type": "text" }, { "bbox": [ 178, 690, 187, 699 ], "score": 0.42, "content": "A", "type": "inline_equation" }, { "bbox": [ 187, 687, 209, 702 ], "score": 1.0, "content": "AND", "type": "text" }, { "bbox": [ 209, 690, 218, 699 ], "score": 0.68, "content": "C", "type": "inline_equation" } ], "index": 36 } ], "index": 36 }, { "type": "text", "bbox": [ 107, 709, 504, 732 ], "lines": [ { "bbox": [ 106, 708, 505, 721 ], "spans": [ { "bbox": [ 106, 708, 213, 721 ], "score": 1.0, "content": "We studied the impact of", "type": "text" }, { "bbox": [ 213, 710, 222, 720 ], "score": 0.74, "content": "A", "type": "inline_equation" }, { "bbox": [ 222, 708, 242, 721 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 243, 710, 251, 720 ], "score": 0.82, "content": "C", "type": "inline_equation" }, { "bbox": [ 252, 708, 505, 721 ], "score": 1.0, "content": "on the performance of Pyraformer for long-range time series", "type": "text" } ], "index": 37 }, { "bbox": [ 105, 719, 506, 734 ], "spans": [ { "bbox": [ 105, 719, 506, 734 ], "score": 1.0, "content": "forecasting, and showed the results in Table 7. Here, we focus on the dataset ETTh1. The history", "type": "text" } ], "index": 38 }, { "bbox": [ 106, 496, 505, 509 ], "spans": [ { "bbox": [ 106, 496, 505, 509 ], "score": 1.0, "content": "length is 336 and the prediction length is 720. From Table 7, we can conclude that the receptive fields", "type": "text", "cross_page": true } ], "index": 9 }, { "bbox": [ 106, 507, 506, 520 ], "spans": [ { "bbox": [ 106, 507, 506, 520 ], "score": 1.0, "content": "of the nodes at the coarsest scale in the PAM play an indispensable role in reducing the prediction", "type": "text", "cross_page": true } ], "index": 10 }, { "bbox": [ 106, 518, 506, 531 ], "spans": [ { "bbox": [ 106, 518, 424, 531 ], "score": 1.0, "content": "error of Pyraformer. For instance, there are 42 nodes at the coarsest scale when", "type": "text", "cross_page": true }, { "bbox": [ 424, 518, 451, 528 ], "score": 0.89, "content": "C = 2", "type": "inline_equation", "cross_page": true }, { "bbox": [ 452, 518, 506, 531 ], "score": 1.0, "content": ". Without the", "type": "text", "cross_page": true } ], "index": 11 }, { "bbox": [ 106, 529, 505, 541 ], "spans": [ { "bbox": [ 106, 529, 505, 541 ], "score": 1.0, "content": "intra-scale connections, each node can only receive messages from 16 nodes at the finest scale. As", "type": "text", "cross_page": true } ], "index": 12 }, { "bbox": [ 106, 541, 505, 552 ], "spans": [ { "bbox": [ 106, 541, 253, 552 ], "score": 1.0, "content": "the number of adjacent connections", "type": "text", "cross_page": true }, { "bbox": [ 254, 541, 262, 550 ], "score": 0.68, "content": "A", "type": "inline_equation", "cross_page": true }, { "bbox": [ 263, 541, 505, 552 ], "score": 1.0, "content": "in each scale increases, the receptive fields of the coarsest-", "type": "text", "cross_page": true } ], "index": 13 }, { "bbox": [ 105, 550, 506, 565 ], "spans": [ { "bbox": [ 105, 550, 506, 565 ], "score": 1.0, "content": "scale nodes also extend, and therefore, the prediction error decreases accordingly. However, as long", "type": "text", "cross_page": true } ], "index": 14 }, { "bbox": [ 105, 562, 505, 575 ], "spans": [ { "bbox": [ 105, 562, 415, 575 ], "score": 1.0, "content": "as the nodes at the top scale have a global receptive field, further increasing", "type": "text", "cross_page": true }, { "bbox": [ 415, 563, 424, 572 ], "score": 0.59, "content": "A", "type": "inline_equation", "cross_page": true }, { "bbox": [ 424, 562, 505, 575 ], "score": 1.0, "content": "will not bring large", "type": "text", "cross_page": true } ], "index": 15 }, { "bbox": [ 105, 572, 506, 586 ], "spans": [ { "bbox": [ 105, 572, 150, 586 ], "score": 1.0, "content": "gains. For", "type": "text", "cross_page": true }, { "bbox": [ 151, 573, 179, 583 ], "score": 0.9, "content": "C = 5", "type": "inline_equation", "cross_page": true }, { "bbox": [ 180, 572, 376, 586 ], "score": 1.0, "content": ", the performance does not improve even though", "type": "text", "cross_page": true }, { "bbox": [ 376, 573, 385, 583 ], "score": 0.71, "content": "A", "type": "inline_equation", "cross_page": true }, { "bbox": [ 385, 572, 506, 586 ], "score": 1.0, "content": "increases. Such observations", "type": "text", "cross_page": true } ], "index": 16 }, { "bbox": [ 105, 584, 506, 596 ], "spans": [ { "bbox": [ 105, 584, 229, 596 ], "score": 1.0, "content": "indicate that it is better to set", "type": "text", "cross_page": true }, { "bbox": [ 230, 584, 239, 594 ], "score": 0.72, "content": "A", "type": "inline_equation", "cross_page": true }, { "bbox": [ 239, 584, 506, 596 ], "score": 1.0, "content": "to be small once the uppermost nodes in the PAM have a global", "type": "text", "cross_page": true } ], "index": 17 }, { "bbox": [ 104, 594, 460, 608 ], "spans": [ { "bbox": [ 104, 594, 285, 608 ], "score": 1.0, "content": "receptive field. 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From Table 7, we can conclude that the receptive fields", "type": "text" } ], "index": 9 }, { "bbox": [ 106, 507, 506, 520 ], "spans": [ { "bbox": [ 106, 507, 506, 520 ], "score": 1.0, "content": "of the nodes at the coarsest scale in the PAM play an indispensable role in reducing the prediction", "type": "text" } ], "index": 10 }, { "bbox": [ 106, 518, 506, 531 ], "spans": [ { "bbox": [ 106, 518, 424, 531 ], "score": 1.0, "content": "error of Pyraformer. For instance, there are 42 nodes at the coarsest scale when", "type": "text" }, { "bbox": [ 424, 518, 451, 528 ], "score": 0.89, "content": "C = 2", "type": "inline_equation" }, { "bbox": [ 452, 518, 506, 531 ], "score": 1.0, "content": ". Without the", "type": "text" } ], "index": 11 }, { "bbox": [ 106, 529, 505, 541 ], "spans": [ { "bbox": [ 106, 529, 505, 541 ], "score": 1.0, "content": "intra-scale connections, each node can only receive messages from 16 nodes at the finest scale. As", "type": "text" } ], "index": 12 }, { "bbox": [ 106, 541, 505, 552 ], "spans": [ { "bbox": [ 106, 541, 253, 552 ], "score": 1.0, "content": "the number of adjacent connections", "type": "text" }, { "bbox": [ 254, 541, 262, 550 ], "score": 0.68, "content": "A", "type": "inline_equation" }, { "bbox": [ 263, 541, 505, 552 ], "score": 1.0, "content": "in each scale increases, the receptive fields of the coarsest-", "type": "text" } ], "index": 13 }, { "bbox": [ 105, 550, 506, 565 ], "spans": [ { "bbox": [ 105, 550, 506, 565 ], "score": 1.0, "content": "scale nodes also extend, and therefore, the prediction error decreases accordingly. However, as long", "type": "text" } ], "index": 14 }, { "bbox": [ 105, 562, 505, 575 ], "spans": [ { "bbox": [ 105, 562, 415, 575 ], "score": 1.0, "content": "as the nodes at the top scale have a global receptive field, further increasing", "type": "text" }, { "bbox": [ 415, 563, 424, 572 ], "score": 0.59, "content": "A", "type": "inline_equation" }, { "bbox": [ 424, 562, 505, 575 ], "score": 1.0, "content": "will not bring large", "type": "text" } ], "index": 15 }, { "bbox": [ 105, 572, 506, 586 ], "spans": [ { "bbox": [ 105, 572, 150, 586 ], "score": 1.0, "content": "gains. For", "type": "text" }, { "bbox": [ 151, 573, 179, 583 ], "score": 0.9, "content": "C = 5", "type": "inline_equation" }, { "bbox": [ 180, 572, 376, 586 ], "score": 1.0, "content": ", the performance does not improve even though", "type": "text" }, { "bbox": [ 376, 573, 385, 583 ], "score": 0.71, "content": "A", "type": "inline_equation" }, { "bbox": [ 385, 572, 506, 586 ], "score": 1.0, "content": "increases. Such observations", "type": "text" } ], "index": 16 }, { "bbox": [ 105, 584, 506, 596 ], "spans": [ { "bbox": [ 105, 584, 229, 596 ], "score": 1.0, "content": "indicate that it is better to set", "type": "text" }, { "bbox": [ 230, 584, 239, 594 ], "score": 0.72, "content": "A", "type": "inline_equation" }, { "bbox": [ 239, 584, 506, 596 ], "score": 1.0, "content": "to be small once the uppermost nodes in the PAM have a global", "type": "text" } ], "index": 17 }, { "bbox": [ 104, 594, 460, 608 ], "spans": [ { "bbox": [ 104, 594, 285, 608 ], "score": 1.0, "content": "receptive field. In practice, we only increase", "type": "text" }, { "bbox": [ 285, 595, 294, 605 ], "score": 0.82, "content": "C", "type": "inline_equation" }, { "bbox": [ 295, 594, 376, 608 ], "score": 1.0, "content": "with the increase of", "type": "text" }, { "bbox": [ 376, 595, 384, 605 ], "score": 0.79, "content": "L", "type": "inline_equation" }, { "bbox": [ 384, 594, 424, 608 ], "score": 1.0, "content": ", but keep", "type": "text" }, { "bbox": [ 424, 595, 433, 605 ], "score": 0.73, "content": "A", "type": "inline_equation" }, { "bbox": [ 433, 594, 460, 608 ], "score": 1.0, "content": "small.", "type": "text" } ], "index": 18 } ], "index": 13.5 }, { "type": "title", "bbox": [ 108, 622, 297, 633 ], "lines": [ { "bbox": [ 105, 621, 300, 636 ], "spans": [ { "bbox": [ 105, 621, 300, 636 ], "score": 1.0, "content": "J.2 IMPACT OF THE CSCM ARCHITECTURE", "type": "text" } ], "index": 19 } ], "index": 19 }, { "type": "text", "bbox": [ 107, 643, 505, 732 ], "lines": [ { "bbox": [ 105, 644, 505, 656 ], "spans": [ { "bbox": [ 105, 644, 424, 656 ], "score": 1.0, "content": "In addition to convolution, there exist other mechanisms for constructing the", "type": "text" }, { "bbox": [ 424, 644, 433, 654 ], "score": 0.81, "content": "C", "type": "inline_equation" }, { "bbox": [ 433, 644, 505, 656 ], "score": 1.0, "content": "-ary tree, such as", "type": "text" } ], "index": 20 }, { "bbox": [ 105, 655, 505, 667 ], "spans": [ { "bbox": [ 105, 655, 505, 667 ], "score": 1.0, "content": "max pooling and average pooling. We studied the impact of different CSCM architectures on the", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 666, 505, 678 ], "spans": [ { "bbox": [ 105, 666, 505, 678 ], "score": 1.0, "content": "performance for long-range forecasting on dataset ETTh1. The history and the prediction length", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 676, 506, 690 ], "spans": [ { "bbox": [ 105, 676, 178, 690 ], "score": 1.0, "content": "are both 168 and", "type": "text" }, { "bbox": [ 178, 677, 208, 687 ], "score": 0.9, "content": "C = 4", "type": "inline_equation" }, { "bbox": [ 208, 676, 506, 690 ], "score": 1.0, "content": "for all mechanisms. The results are listed in Table 8. From Table 8, we", "type": "text" } ], "index": 23 }, { "bbox": [ 105, 687, 505, 700 ], "spans": [ { "bbox": [ 105, 687, 505, 700 ], "score": 1.0, "content": "can tell that: (1) Using pooling layers instead of convolution typically degrades the performance.", "type": "text" } ], "index": 24 }, { "bbox": [ 105, 698, 505, 712 ], "spans": [ { "bbox": [ 105, 698, 505, 712 ], "score": 1.0, "content": "However, the performance of Pyraformer based on max pooling is still superior to that of Informer,", "type": "text" } ], "index": 25 }, { "bbox": [ 105, 709, 505, 722 ], "spans": [ { "bbox": [ 105, 709, 505, 722 ], "score": 1.0, "content": "demonstrating the advantages of the PAM over the prob-sparse attention in Informer. (2) The MSE", "type": "text" } ], "index": 26 }, { "bbox": [ 105, 720, 505, 732 ], "spans": [ { "bbox": [ 105, 720, 273, 732 ], "score": 1.0, "content": "of convolution with the bottleneck is only", "type": "text" }, { "bbox": [ 273, 720, 301, 731 ], "score": 0.87, "content": "1 . 5 1 \\%", "type": "inline_equation" }, { "bbox": [ 301, 720, 505, 732 ], "score": 1.0, "content": "larger than that without bottleneck, but the number", "type": "text" } ], "index": 27 } ], "index": 23.5 } ], "page_idx": 17, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 107, 27, 293, 37 ], "lines": [ { "bbox": [ 106, 25, 294, 39 ], "spans": [ { "bbox": [ 106, 25, 294, 39 ], "score": 1.0, "content": "Published as a conference paper at ICLR 2022", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 301, 751, 311, 760 ], "lines": [ { "bbox": [ 299, 750, 312, 763 ], "spans": [ { "bbox": [ 299, 750, 312, 763 ], "score": 1.0, "content": "18", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "image", "bbox": [ 114, 79, 496, 282 ], "blocks": [ { "type": "image_body", "bbox": [ 114, 79, 496, 282 ], "group_id": 0, "lines": [ { "bbox": [ 114, 79, 496, 282 ], "spans": [ { "bbox": [ 114, 79, 496, 282 ], "score": 0.97, "type": "image", "image_path": "0e107628896de0c58e44310218af31bad1997b6ef59d90b98d324d9c9992721a.jpg" } ] } ], "index": 1, "virtual_lines": [ { "bbox": [ 114, 79, 496, 146.66666666666669 ], "spans": [], "index": 0 }, { "bbox": [ 114, 146.66666666666669, 496, 214.33333333333337 ], "spans": [], "index": 1 }, { "bbox": [ 114, 214.33333333333337, 496, 282.00000000000006 ], "spans": [], "index": 2 } ] }, { "type": "image_caption", "bbox": [ 168, 299, 442, 312 ], "group_id": 0, "lines": [ { "bbox": [ 167, 297, 443, 315 ], "spans": [ { "bbox": [ 167, 297, 443, 315 ], "score": 1.0, "content": "Figure 6: Visualization of prediction results on the synthetic dataset.", "type": "text" } ], "index": 3 } ], "index": 3 } ], "index": 2.0 }, { "type": "image", "bbox": [ 112, 327, 499, 434 ], "blocks": [ { "type": "image_body", "bbox": [ 112, 327, 499, 434 ], "group_id": 1, "lines": [ { "bbox": [ 112, 327, 499, 434 ], "spans": [ { "bbox": [ 112, 327, 499, 434 ], "score": 0.97, "type": "image", "image_path": "78b6ff45bfe2e9e637686e94056b78164c4a40e9a46ac871d93338b981fb0e83.jpg" } ] } ], "index": 5, "virtual_lines": [ { "bbox": [ 112, 327, 499, 362.6666666666667 ], "spans": [], "index": 4 }, { "bbox": [ 112, 362.6666666666667, 499, 398.33333333333337 ], "spans": [], "index": 5 }, { "bbox": [ 112, 398.33333333333337, 499, 434.00000000000006 ], "spans": [], "index": 6 } ] }, { "type": "image_caption", "bbox": [ 105, 451, 503, 474 ], "group_id": 1, "lines": [ { "bbox": [ 106, 450, 505, 463 ], "spans": [ { "bbox": [ 106, 450, 505, 463 ], "score": 1.0, "content": "Figure 7: Visualization of the extracted features across time in second channel at different scales:", "type": "text" } ], "index": 7 }, { "bbox": [ 106, 461, 245, 474 ], "spans": [ { "bbox": [ 106, 461, 245, 474 ], "score": 1.0, "content": "(a) scale 1; (b) scale 2; (c) scale 3.", "type": "text" } ], "index": 8 } ], "index": 7.5 } ], "index": 6.25 }, { "type": "text", "bbox": [ 106, 495, 505, 606 ], "lines": [], "index": 13.5, "bbox_fs": [ 104, 496, 506, 608 ], "lines_deleted": true }, { "type": "title", "bbox": [ 108, 622, 297, 633 ], "lines": [ { "bbox": [ 105, 621, 300, 636 ], "spans": [ { "bbox": [ 105, 621, 300, 636 ], "score": 1.0, "content": "J.2 IMPACT OF THE CSCM ARCHITECTURE", "type": "text" } ], "index": 19 } ], "index": 19 }, { "type": "text", "bbox": [ 107, 643, 505, 732 ], "lines": [ { "bbox": [ 105, 644, 505, 656 ], "spans": [ { "bbox": [ 105, 644, 424, 656 ], "score": 1.0, "content": "In addition to convolution, there exist other mechanisms for constructing the", "type": "text" }, { "bbox": [ 424, 644, 433, 654 ], "score": 0.81, "content": "C", "type": "inline_equation" }, { "bbox": [ 433, 644, 505, 656 ], "score": 1.0, "content": "-ary tree, such as", "type": "text" } ], "index": 20 }, { "bbox": [ 105, 655, 505, 667 ], "spans": [ { "bbox": [ 105, 655, 505, 667 ], "score": 1.0, "content": "max pooling and average pooling. 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From Table 8, we", "type": "text" } ], "index": 23 }, { "bbox": [ 105, 687, 505, 700 ], "spans": [ { "bbox": [ 105, 687, 505, 700 ], "score": 1.0, "content": "can tell that: (1) Using pooling layers instead of convolution typically degrades the performance.", "type": "text" } ], "index": 24 }, { "bbox": [ 105, 698, 505, 712 ], "spans": [ { "bbox": [ 105, 698, 505, 712 ], "score": 1.0, "content": "However, the performance of Pyraformer based on max pooling is still superior to that of Informer,", "type": "text" } ], "index": 25 }, { "bbox": [ 105, 709, 505, 722 ], "spans": [ { "bbox": [ 105, 709, 505, 722 ], "score": 1.0, "content": "demonstrating the advantages of the PAM over the prob-sparse attention in Informer. (2) The MSE", "type": "text" } ], "index": 26 }, { "bbox": [ 105, 720, 505, 732 ], "spans": [ { "bbox": [ 105, 720, 273, 732 ], "score": 1.0, "content": "of convolution with the bottleneck is only", "type": "text" }, { "bbox": [ 273, 720, 301, 731 ], "score": 0.87, "content": "1 . 5 1 \\%", "type": "inline_equation" }, { "bbox": [ 301, 720, 505, 732 ], "score": 1.0, "content": "larger than that without bottleneck, but the number", "type": "text" } ], "index": 27 }, { "bbox": [ 106, 401, 505, 414 ], "spans": [ { "bbox": [ 106, 401, 249, 414 ], "score": 1.0, "content": "of parameters is reduced by almost", "type": "text", "cross_page": true }, { "bbox": [ 249, 401, 268, 412 ], "score": 0.88, "content": "9 0 \\%", "type": "inline_equation", "cross_page": true }, { "bbox": [ 269, 401, 505, 414 ], "score": 1.0, "content": ". Thus, we adopt the more compact module of convolution", "type": "text", "cross_page": true } ], "index": 21 }, { "bbox": [ 107, 412, 230, 424 ], "spans": [ { "bbox": [ 107, 412, 230, 424 ], "score": 1.0, "content": "with bottleneck as our CSCM.", "type": "text", "cross_page": true } ], "index": 22 } ], "index": 23.5, "bbox_fs": [ 105, 644, 506, 732 ] } ] }, { "preproc_blocks": [ { "type": "table", "bbox": [ 146, 101, 466, 175 ], "blocks": [ { "type": "table_caption", "bbox": [ 142, 89, 467, 101 ], "group_id": 0, "lines": [ { "bbox": [ 141, 87, 469, 103 ], "spans": [ { "bbox": [ 141, 87, 219, 103 ], "score": 1.0, "content": "Table 7: Impact of", "type": "text" }, { "bbox": [ 219, 90, 228, 99 ], "score": 0.67, "content": "A", "type": "inline_equation" }, { "bbox": [ 228, 87, 245, 103 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 246, 90, 255, 99 ], "score": 0.78, "content": "C", "type": "inline_equation" }, { "bbox": [ 255, 87, 469, 103 ], "score": 1.0, "content": "on long-range forecasting. The history length is 336.", "type": "text" } ], "index": 0 } ], "index": 0 }, { "type": "table_body", "bbox": [ 146, 101, 466, 175 ], "group_id": 0, "lines": [ { "bbox": [ 146, 101, 466, 175 ], "spans": [ { "bbox": [ 146, 101, 466, 175 ], "score": 0.981, "html": "
A=3A=9A = 13
MSEMAEQ-K pairsMSEMAEQ-K pairsMSE MAEQ-K pairs
C=21.0350.811735121.0290.8151626481.0030.807221112
C=31.0290.817589921.0090.7981289761.0560.805174672
C=41.0010.802532081.0280.8061158481.0270.804156696
C=50.9990.796499921.0050.7961087441.0170.797147192
", "type": "table", "image_path": "54bcb3f8e84f20c0a7a29ad24e936fa518bed68768ab8ad6f0b028e860c576a2.jpg" } ] } ], "index": 2, "virtual_lines": [ { "bbox": [ 146, 101, 466, 125.66666666666667 ], "spans": [], "index": 1 }, { "bbox": [ 146, 125.66666666666667, 466, 150.33333333333334 ], "spans": [], "index": 2 }, { "bbox": [ 146, 150.33333333333334, 466, 175.0 ], "spans": [], "index": 3 } ] } ], "index": 1.0 }, { "type": "table", "bbox": [ 142, 218, 320, 282 ], "blocks": [ { "type": "table_caption", "bbox": [ 110, 183, 350, 217 ], "group_id": 1, "lines": [ { "bbox": [ 109, 182, 351, 197 ], "spans": [ { "bbox": [ 109, 182, 351, 197 ], "score": 1.0, "content": "Table 8: Impact of the CSCM architecture on long-range", "type": "text" } ], "index": 4 }, { "bbox": [ 110, 194, 351, 207 ], "spans": [ { "bbox": [ 110, 194, 351, 207 ], "score": 1.0, "content": "forecasting. Parameters introduced by the normalization", "type": "text" } ], "index": 6 }, { "bbox": [ 109, 206, 298, 218 ], "spans": [ { "bbox": [ 109, 206, 298, 218 ], "score": 1.0, "content": "layers are relatively few, and thus, are ignored.", "type": "text" } ], "index": 8 } ], "index": 6 }, { "type": "table_body", "bbox": [ 142, 218, 320, 282 ], "group_id": 1, "lines": [ { "bbox": [ 142, 218, 320, 282 ], "spans": [ { "bbox": [ 142, 218, 320, 282 ], "score": 0.964, "html": "
CSCMMSEMAEParameters
Max-pooling0.8420.7000
Average-pooling0.8330.6930
Conv.0.7960.6793147264
Conv.w/bottleneck0.8080.683328704
", "type": "table", "image_path": "e056bfab6233f92ef23ff3bb8565237fff87783e4aeb769f13f48b77d51bc013.jpg" } ] } ], "index": 11.5, "virtual_lines": [ { "bbox": [ 142, 218, 320, 234.0 ], "spans": [], "index": 9 }, { "bbox": [ 142, 234.0, 320, 250.0 ], "spans": [], "index": 11 }, { "bbox": [ 142, 250.0, 320, 266.0 ], "spans": [], "index": 12 }, { "bbox": [ 142, 266.0, 320, 282.0 ], "spans": [], "index": 13 } ] } ], "index": 8.75 }, { "type": "table", "bbox": [ 371, 208, 490, 282 ], "blocks": [ { "type": "table_caption", "bbox": [ 361, 184, 499, 206 ], "group_id": 3, "lines": [ { "bbox": [ 359, 182, 501, 196 ], "spans": [ { "bbox": [ 359, 182, 501, 196 ], "score": 1.0, "content": "Table 9: Impact of history length.", "type": "text" } ], "index": 5 }, { "bbox": [ 359, 194, 482, 206 ], "spans": [ { "bbox": [ 359, 194, 482, 206 ], "score": 1.0, "content": "The prediction length is 1344.", "type": "text" } ], "index": 7 } ], "index": 6.0 }, { "type": "table_body", "bbox": [ 371, 208, 490, 282 ], "group_id": 3, "lines": [ { "bbox": [ 371, 208, 490, 282 ], "spans": [ { "bbox": [ 371, 208, 490, 282 ], "score": 0.97, "html": "
History LengthMSEMAE
841.2340.856
1681.2260.868
3361.1080.835
6721.0570.806
13441.0620.806
", "type": "table", "image_path": "87fa7e2a7d596a8b15c5cfdae6ec9ec9d2af9d9bf2290d28144c6d8149258ce7.jpg" } ] } ], "index": 12.0, "virtual_lines": [ { "bbox": [ 371, 208, 490, 245.0 ], "spans": [], "index": 10 }, { "bbox": [ 371, 245.0, 490, 282.0 ], "spans": [], "index": 14 } ] } ], "index": 9.0 }, { "type": "table", "bbox": [ 207, 306, 406, 381 ], "blocks": [ { "type": "table_caption", "bbox": [ 245, 292, 365, 303 ], "group_id": 2, "lines": [ { "bbox": [ 245, 291, 366, 304 ], "spans": [ { "bbox": [ 245, 291, 366, 304 ], "score": 1.0, "content": "Table 10: Impact of the PAM.", "type": "text" } ], "index": 15 } ], "index": 15 }, { "type": "table_body", "bbox": [ 207, 306, 406, 381 ], "group_id": 2, "lines": [ { "bbox": [ 207, 306, 406, 381 ], "spans": [ { "bbox": [ 207, 306, 406, 381 ], "score": 0.978, "html": "
MethodMetrics96288672
CSCM OnlyMSE0.5760.7820.883
MAE0.5440.6830.752
PyraformerMSE0.4800.7540.857
MAE0.4860.6590.707
", "type": "table", "image_path": "a0c5396a148c8233fb5102e7e18188438e622a1e8a25d1a1db2594e43de3523c.jpg" } ] } ], "index": 18, "virtual_lines": [ { "bbox": [ 207, 306, 406, 321.0 ], "spans": [], "index": 16 }, { "bbox": [ 207, 321.0, 406, 336.0 ], "spans": [], "index": 17 }, { "bbox": [ 207, 336.0, 406, 351.0 ], "spans": [], "index": 18 }, { "bbox": [ 207, 351.0, 406, 366.0 ], "spans": [], "index": 19 }, { "bbox": [ 207, 366.0, 406, 381.0 ], "spans": [], "index": 20 } ] } ], "index": 16.5 }, { "type": "text", "bbox": [ 106, 401, 504, 423 ], "lines": [ { "bbox": [ 106, 401, 505, 414 ], "spans": [ { "bbox": [ 106, 401, 249, 414 ], "score": 1.0, "content": "of parameters is reduced by almost", "type": "text" }, { "bbox": [ 249, 401, 268, 412 ], "score": 0.88, "content": "9 0 \\%", "type": "inline_equation" }, { "bbox": [ 269, 401, 505, 414 ], "score": 1.0, "content": ". Thus, we adopt the more compact module of convolution", "type": "text" } ], "index": 21 }, { "bbox": [ 107, 412, 230, 424 ], "spans": [ { "bbox": [ 107, 412, 230, 424 ], "score": 1.0, "content": "with bottleneck as our CSCM.", "type": "text" } ], "index": 22 } ], "index": 21.5 }, { "type": "title", "bbox": [ 108, 437, 273, 448 ], "lines": [ { "bbox": [ 105, 437, 275, 449 ], "spans": [ { "bbox": [ 105, 437, 275, 449 ], "score": 1.0, "content": "J.3 IMPACT OF THE HISTORY LENGTH", "type": "text" } ], "index": 23 } ], "index": 23 }, { "type": "text", "bbox": [ 107, 457, 505, 534 ], "lines": [ { "bbox": [ 106, 456, 505, 470 ], "spans": [ { "bbox": [ 106, 456, 505, 470 ], "score": 1.0, "content": "We also checked the influence of the history length on the prediction accuracy. The dataset is", "type": "text" } ], "index": 24 }, { "bbox": [ 105, 469, 505, 481 ], "spans": [ { "bbox": [ 105, 469, 505, 481 ], "score": 1.0, "content": "ETTm1, since its granularity is minute and contains more long-range dependencies. We fixed the", "type": "text" } ], "index": 25 }, { "bbox": [ 105, 480, 505, 492 ], "spans": [ { "bbox": [ 105, 480, 505, 492 ], "score": 1.0, "content": "prediction length to 1344 and changed the history length from 84 to 1344 in Table 9. As expected,", "type": "text" } ], "index": 26 }, { "bbox": [ 105, 491, 505, 503 ], "spans": [ { "bbox": [ 105, 491, 505, 503 ], "score": 1.0, "content": "a longer history typically improves prediction accuracy. On the other hand, this performance gain", "type": "text" } ], "index": 27 }, { "bbox": [ 105, 501, 505, 514 ], "spans": [ { "bbox": [ 105, 501, 505, 514 ], "score": 1.0, "content": "starts to level off when introducing more history stops providing new information. As shown in", "type": "text" } ], "index": 28 }, { "bbox": [ 105, 513, 505, 524 ], "spans": [ { "bbox": [ 105, 513, 505, 524 ], "score": 1.0, "content": "Figure 8, the time series with length 672 contains almost all periodicity information that is essential", "type": "text" } ], "index": 29 }, { "bbox": [ 105, 524, 333, 536 ], "spans": [ { "bbox": [ 105, 524, 333, 536 ], "score": 1.0, "content": "for prediction, while length 1344 introduces more noise.", "type": "text" } ], "index": 30 } ], "index": 27 }, { "type": "title", "bbox": [ 107, 548, 221, 560 ], "lines": [ { "bbox": [ 105, 548, 222, 561 ], "spans": [ { "bbox": [ 105, 548, 222, 561 ], "score": 1.0, "content": "J.4 IMPACT OF THE PAM", "type": "text" } ], "index": 31 } ], "index": 31 }, { "type": "text", "bbox": [ 107, 569, 505, 624 ], "lines": [ { "bbox": [ 105, 568, 505, 582 ], "spans": [ { "bbox": [ 105, 568, 505, 582 ], "score": 1.0, "content": "Finally, we investigated the importance of the PAM. We compared the performance of Pyraformer", "type": "text" } ], "index": 32 }, { "bbox": [ 106, 579, 505, 592 ], "spans": [ { "bbox": [ 106, 579, 505, 592 ], "score": 1.0, "content": "with and without the PAM on the dataset ETTm1. For a fair comparison, the number of parameters", "type": "text" } ], "index": 33 }, { "bbox": [ 105, 590, 506, 604 ], "spans": [ { "bbox": [ 105, 590, 506, 604 ], "score": 1.0, "content": "of the two methods were controlled to be within the same order of magnitude. More precisely, we", "type": "text" } ], "index": 34 }, { "bbox": [ 106, 602, 505, 614 ], "spans": [ { "bbox": [ 106, 602, 505, 614 ], "score": 1.0, "content": "increased the bottleneck dimension of ”Conv. w/bottleneck” for the model only with the CSCM.", "type": "text" } ], "index": 35 }, { "bbox": [ 105, 612, 476, 626 ], "spans": [ { "bbox": [ 105, 612, 476, 626 ], "score": 1.0, "content": "The results are shown in Table 10. Obviously, the PAM is vital to yield accurate predictions.", "type": "text" } ], "index": 36 } ], "index": 34 }, { "type": "title", "bbox": [ 108, 641, 421, 653 ], "lines": [ { "bbox": [ 105, 640, 423, 657 ], "spans": [ { "bbox": [ 105, 640, 423, 657 ], "score": 1.0, "content": "K DISCUSSION ON THE SELECTION OF HYPER-PARAMETERS", "type": "text" } ], "index": 37 } ], "index": 37 }, { "type": "text", "bbox": [ 107, 666, 505, 732 ], "lines": [ { "bbox": [ 106, 664, 505, 678 ], "spans": [ { "bbox": [ 106, 664, 360, 678 ], "score": 1.0, "content": "We recommend to first determine the number of attention layers", "type": "text" }, { "bbox": [ 361, 667, 370, 675 ], "score": 0.84, "content": "N", "type": "inline_equation" }, { "bbox": [ 370, 664, 505, 678 ], "score": 1.0, "content": "based on the available computing", "type": "text" } ], "index": 38 }, { "bbox": [ 105, 677, 505, 689 ], "spans": [ { "bbox": [ 105, 677, 467, 689 ], "score": 1.0, "content": "resources, as this number is directly related to the model size. Next, the number of scales", "type": "text" }, { "bbox": [ 467, 677, 475, 687 ], "score": 0.79, "content": "S", "type": "inline_equation" }, { "bbox": [ 476, 677, 505, 689 ], "score": 1.0, "content": "can be", "type": "text" } ], "index": 39 }, { "bbox": [ 105, 687, 505, 700 ], "spans": [ { "bbox": [ 105, 687, 505, 700 ], "score": 1.0, "content": "determined by the granularity of the time series. For example, for hourly observations, we typically", "type": "text" } ], "index": 40 }, { "bbox": [ 105, 699, 505, 712 ], "spans": [ { "bbox": [ 105, 699, 462, 712 ], "score": 1.0, "content": "assume that it may also have daily, weekly and monthly periods. Therefore, we can set", "type": "text" }, { "bbox": [ 462, 699, 470, 709 ], "score": 0.83, "content": "S", "type": "inline_equation" }, { "bbox": [ 471, 699, 505, 712 ], "score": 1.0, "content": "to be 4.", "type": "text" } ], "index": 41 }, { "bbox": [ 106, 709, 506, 721 ], "spans": [ { "bbox": [ 106, 709, 245, 721 ], "score": 1.0, "content": "We then focus on the selection of", "type": "text" }, { "bbox": [ 245, 710, 253, 720 ], "score": 0.81, "content": "A", "type": "inline_equation" }, { "bbox": [ 254, 709, 272, 721 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 273, 712, 281, 720 ], "score": 0.84, "content": "C", "type": "inline_equation" }, { "bbox": [ 281, 709, 506, 721 ], "score": 1.0, "content": ". According to the ablation study, we typically prefer a", "type": "text" } ], "index": 42 }, { "bbox": [ 105, 720, 505, 733 ], "spans": [ { "bbox": [ 105, 720, 131, 733 ], "score": 1.0, "content": "small", "type": "text" }, { "bbox": [ 131, 721, 140, 730 ], "score": 0.75, "content": "A", "type": "inline_equation" }, { "bbox": [ 140, 720, 462, 733 ], "score": 1.0, "content": ", such as 3 and 5. Lastly, in order to ensure the network has a receptive field of", "type": "text" }, { "bbox": [ 462, 721, 470, 730 ], "score": 0.81, "content": "L", "type": "inline_equation" }, { "bbox": [ 470, 720, 505, 733 ], "score": 1.0, "content": ", we can", "type": "text" } ], "index": 43 } ], "index": 40.5 } ], "page_idx": 18, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 107, 27, 293, 37 ], "lines": [ { "bbox": [ 106, 26, 294, 38 ], "spans": [ { "bbox": [ 106, 26, 294, 38 ], "score": 1.0, "content": "Published as a conference paper at ICLR 2022", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 301, 751, 311, 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": [ 146, 101, 466, 175 ], "blocks": [ { "type": "table_caption", "bbox": [ 142, 89, 467, 101 ], "group_id": 0, "lines": [ { "bbox": [ 141, 87, 469, 103 ], "spans": [ { "bbox": [ 141, 87, 219, 103 ], "score": 1.0, "content": "Table 7: Impact of", "type": "text" }, { "bbox": [ 219, 90, 228, 99 ], "score": 0.67, "content": "A", "type": "inline_equation" }, { "bbox": [ 228, 87, 245, 103 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 246, 90, 255, 99 ], "score": 0.78, "content": "C", "type": "inline_equation" }, { "bbox": [ 255, 87, 469, 103 ], "score": 1.0, "content": "on long-range forecasting. The history length is 336.", "type": "text" } ], "index": 0 } ], "index": 0 }, { "type": "table_body", "bbox": [ 146, 101, 466, 175 ], "group_id": 0, "lines": [ { "bbox": [ 146, 101, 466, 175 ], "spans": [ { "bbox": [ 146, 101, 466, 175 ], "score": 0.981, "html": "
A=3A=9A = 13
MSEMAEQ-K pairsMSEMAEQ-K pairsMSE MAEQ-K pairs
C=21.0350.811735121.0290.8151626481.0030.807221112
C=31.0290.817589921.0090.7981289761.0560.805174672
C=41.0010.802532081.0280.8061158481.0270.804156696
C=50.9990.796499921.0050.7961087441.0170.797147192
", "type": "table", "image_path": "54bcb3f8e84f20c0a7a29ad24e936fa518bed68768ab8ad6f0b028e860c576a2.jpg" } ] } ], "index": 2, "virtual_lines": [ { "bbox": [ 146, 101, 466, 125.66666666666667 ], "spans": [], "index": 1 }, { "bbox": [ 146, 125.66666666666667, 466, 150.33333333333334 ], "spans": [], "index": 2 }, { "bbox": [ 146, 150.33333333333334, 466, 175.0 ], "spans": [], "index": 3 } ] } ], "index": 1.0 }, { "type": "table", "bbox": [ 142, 218, 320, 282 ], "blocks": [ { "type": "table_caption", "bbox": [ 110, 183, 350, 217 ], "group_id": 1, "lines": [ { "bbox": [ 109, 182, 351, 197 ], "spans": [ { "bbox": [ 109, 182, 351, 197 ], "score": 1.0, "content": "Table 8: Impact of the CSCM architecture on long-range", "type": "text" } ], "index": 4 }, { "bbox": [ 110, 194, 351, 207 ], "spans": [ { "bbox": [ 110, 194, 351, 207 ], "score": 1.0, "content": "forecasting. Parameters introduced by the normalization", "type": "text" } ], "index": 6 }, { "bbox": [ 109, 206, 298, 218 ], "spans": [ { "bbox": [ 109, 206, 298, 218 ], "score": 1.0, "content": "layers are relatively few, and thus, are ignored.", "type": "text" } ], "index": 8 } ], "index": 6 }, { "type": "table_body", "bbox": [ 142, 218, 320, 282 ], "group_id": 1, "lines": [ { "bbox": [ 142, 218, 320, 282 ], "spans": [ { "bbox": [ 142, 218, 320, 282 ], "score": 0.964, "html": "
CSCMMSEMAEParameters
Max-pooling0.8420.7000
Average-pooling0.8330.6930
Conv.0.7960.6793147264
Conv.w/bottleneck0.8080.683328704
", "type": "table", "image_path": "e056bfab6233f92ef23ff3bb8565237fff87783e4aeb769f13f48b77d51bc013.jpg" } ] } ], "index": 11.5, "virtual_lines": [ { "bbox": [ 142, 218, 320, 234.0 ], "spans": [], "index": 9 }, { "bbox": [ 142, 234.0, 320, 250.0 ], "spans": [], "index": 11 }, { "bbox": [ 142, 250.0, 320, 266.0 ], "spans": [], "index": 12 }, { "bbox": [ 142, 266.0, 320, 282.0 ], "spans": [], "index": 13 } ] } ], "index": 8.75 }, { "type": "table", "bbox": [ 371, 208, 490, 282 ], "blocks": [ { "type": "table_caption", "bbox": [ 361, 184, 499, 206 ], "group_id": 3, "lines": [ { "bbox": [ 359, 182, 501, 196 ], "spans": [ { "bbox": [ 359, 182, 501, 196 ], "score": 1.0, "content": "Table 9: Impact of history length.", "type": "text" } ], "index": 5 }, { "bbox": [ 359, 194, 482, 206 ], "spans": [ { "bbox": [ 359, 194, 482, 206 ], "score": 1.0, "content": "The prediction length is 1344.", "type": "text" } ], "index": 7 } ], "index": 6.0 }, { "type": "table_body", "bbox": [ 371, 208, 490, 282 ], "group_id": 3, "lines": [ { "bbox": [ 371, 208, 490, 282 ], "spans": [ { "bbox": [ 371, 208, 490, 282 ], "score": 0.97, "html": "
History LengthMSEMAE
841.2340.856
1681.2260.868
3361.1080.835
6721.0570.806
13441.0620.806
", "type": "table", "image_path": "87fa7e2a7d596a8b15c5cfdae6ec9ec9d2af9d9bf2290d28144c6d8149258ce7.jpg" } ] } ], "index": 12.0, "virtual_lines": [ { "bbox": [ 371, 208, 490, 245.0 ], "spans": [], "index": 10 }, { "bbox": [ 371, 245.0, 490, 282.0 ], "spans": [], "index": 14 } ] } ], "index": 9.0 }, { "type": "table", "bbox": [ 207, 306, 406, 381 ], "blocks": [ { "type": "table_caption", "bbox": [ 245, 292, 365, 303 ], "group_id": 2, "lines": [ { "bbox": [ 245, 291, 366, 304 ], "spans": [ { "bbox": [ 245, 291, 366, 304 ], "score": 1.0, "content": "Table 10: Impact of the PAM.", "type": "text" } ], "index": 15 } ], "index": 15 }, { "type": "table_body", "bbox": [ 207, 306, 406, 381 ], "group_id": 2, "lines": [ { "bbox": [ 207, 306, 406, 381 ], "spans": [ { "bbox": [ 207, 306, 406, 381 ], "score": 0.978, "html": "
MethodMetrics96288672
CSCM OnlyMSE0.5760.7820.883
MAE0.5440.6830.752
PyraformerMSE0.4800.7540.857
MAE0.4860.6590.707
", "type": "table", "image_path": "a0c5396a148c8233fb5102e7e18188438e622a1e8a25d1a1db2594e43de3523c.jpg" } ] } ], "index": 18, "virtual_lines": [ { "bbox": [ 207, 306, 406, 321.0 ], "spans": [], "index": 16 }, { "bbox": [ 207, 321.0, 406, 336.0 ], "spans": [], "index": 17 }, { "bbox": [ 207, 336.0, 406, 351.0 ], "spans": [], "index": 18 }, { "bbox": [ 207, 351.0, 406, 366.0 ], "spans": [], "index": 19 }, { "bbox": [ 207, 366.0, 406, 381.0 ], "spans": [], "index": 20 } ] } ], "index": 16.5 }, { "type": "text", "bbox": [ 106, 401, 504, 423 ], "lines": [], "index": 21.5, "bbox_fs": [ 106, 401, 505, 424 ], "lines_deleted": true }, { "type": "title", "bbox": [ 108, 437, 273, 448 ], "lines": [ { "bbox": [ 105, 437, 275, 449 ], "spans": [ { "bbox": [ 105, 437, 275, 449 ], "score": 1.0, "content": "J.3 IMPACT OF THE HISTORY LENGTH", "type": "text" } ], "index": 23 } ], "index": 23 }, { "type": "text", "bbox": [ 107, 457, 505, 534 ], "lines": [ { "bbox": [ 106, 456, 505, 470 ], "spans": [ { "bbox": [ 106, 456, 505, 470 ], "score": 1.0, "content": "We also checked the influence of the history length on the prediction accuracy. The dataset is", "type": "text" } ], "index": 24 }, { "bbox": [ 105, 469, 505, 481 ], "spans": [ { "bbox": [ 105, 469, 505, 481 ], "score": 1.0, "content": "ETTm1, since its granularity is minute and contains more long-range dependencies. We fixed the", "type": "text" } ], "index": 25 }, { "bbox": [ 105, 480, 505, 492 ], "spans": [ { "bbox": [ 105, 480, 505, 492 ], "score": 1.0, "content": "prediction length to 1344 and changed the history length from 84 to 1344 in Table 9. As expected,", "type": "text" } ], "index": 26 }, { "bbox": [ 105, 491, 505, 503 ], "spans": [ { "bbox": [ 105, 491, 505, 503 ], "score": 1.0, "content": "a longer history typically improves prediction accuracy. On the other hand, this performance gain", "type": "text" } ], "index": 27 }, { "bbox": [ 105, 501, 505, 514 ], "spans": [ { "bbox": [ 105, 501, 505, 514 ], "score": 1.0, "content": "starts to level off when introducing more history stops providing new information. As shown in", "type": "text" } ], "index": 28 }, { "bbox": [ 105, 513, 505, 524 ], "spans": [ { "bbox": [ 105, 513, 505, 524 ], "score": 1.0, "content": "Figure 8, the time series with length 672 contains almost all periodicity information that is essential", "type": "text" } ], "index": 29 }, { "bbox": [ 105, 524, 333, 536 ], "spans": [ { "bbox": [ 105, 524, 333, 536 ], "score": 1.0, "content": "for prediction, while length 1344 introduces more noise.", "type": "text" } ], "index": 30 } ], "index": 27, "bbox_fs": [ 105, 456, 505, 536 ] }, { "type": "title", "bbox": [ 107, 548, 221, 560 ], "lines": [ { "bbox": [ 105, 548, 222, 561 ], "spans": [ { "bbox": [ 105, 548, 222, 561 ], "score": 1.0, "content": "J.4 IMPACT OF THE PAM", "type": "text" } ], "index": 31 } ], "index": 31 }, { "type": "text", "bbox": [ 107, 569, 505, 624 ], "lines": [ { "bbox": [ 105, 568, 505, 582 ], "spans": [ { "bbox": [ 105, 568, 505, 582 ], "score": 1.0, "content": "Finally, we investigated the importance of the PAM. We compared the performance of Pyraformer", "type": "text" } ], "index": 32 }, { "bbox": [ 106, 579, 505, 592 ], "spans": [ { "bbox": [ 106, 579, 505, 592 ], "score": 1.0, "content": "with and without the PAM on the dataset ETTm1. For a fair comparison, the number of parameters", "type": "text" } ], "index": 33 }, { "bbox": [ 105, 590, 506, 604 ], "spans": [ { "bbox": [ 105, 590, 506, 604 ], "score": 1.0, "content": "of the two methods were controlled to be within the same order of magnitude. More precisely, we", "type": "text" } ], "index": 34 }, { "bbox": [ 106, 602, 505, 614 ], "spans": [ { "bbox": [ 106, 602, 505, 614 ], "score": 1.0, "content": "increased the bottleneck dimension of ”Conv. w/bottleneck” for the model only with the CSCM.", "type": "text" } ], "index": 35 }, { "bbox": [ 105, 612, 476, 626 ], "spans": [ { "bbox": [ 105, 612, 476, 626 ], "score": 1.0, "content": "The results are shown in Table 10. Obviously, the PAM is vital to yield accurate predictions.", "type": "text" } ], "index": 36 } ], "index": 34, "bbox_fs": [ 105, 568, 506, 626 ] }, { "type": "title", "bbox": [ 108, 641, 421, 653 ], "lines": [ { "bbox": [ 105, 640, 423, 657 ], "spans": [ { "bbox": [ 105, 640, 423, 657 ], "score": 1.0, "content": "K DISCUSSION ON THE SELECTION OF HYPER-PARAMETERS", "type": "text" } ], "index": 37 } ], "index": 37 }, { "type": "text", "bbox": [ 107, 666, 505, 732 ], "lines": [ { "bbox": [ 106, 664, 505, 678 ], "spans": [ { "bbox": [ 106, 664, 360, 678 ], "score": 1.0, "content": "We recommend to first determine the number of attention layers", "type": "text" }, { "bbox": [ 361, 667, 370, 675 ], "score": 0.84, "content": "N", "type": "inline_equation" }, { "bbox": [ 370, 664, 505, 678 ], "score": 1.0, "content": "based on the available computing", "type": "text" } ], "index": 38 }, { "bbox": [ 105, 677, 505, 689 ], "spans": [ { "bbox": [ 105, 677, 467, 689 ], "score": 1.0, "content": "resources, as this number is directly related to the model size. Next, the number of scales", "type": "text" }, { "bbox": [ 467, 677, 475, 687 ], "score": 0.79, "content": "S", "type": "inline_equation" }, { "bbox": [ 476, 677, 505, 689 ], "score": 1.0, "content": "can be", "type": "text" } ], "index": 39 }, { "bbox": [ 105, 687, 505, 700 ], "spans": [ { "bbox": [ 105, 687, 505, 700 ], "score": 1.0, "content": "determined by the granularity of the time series. For example, for hourly observations, we typically", "type": "text" } ], "index": 40 }, { "bbox": [ 105, 699, 505, 712 ], "spans": [ { "bbox": [ 105, 699, 462, 712 ], "score": 1.0, "content": "assume that it may also have daily, weekly and monthly periods. Therefore, we can set", "type": "text" }, { "bbox": [ 462, 699, 470, 709 ], "score": 0.83, "content": "S", "type": "inline_equation" }, { "bbox": [ 471, 699, 505, 712 ], "score": 1.0, "content": "to be 4.", "type": "text" } ], "index": 41 }, { "bbox": [ 106, 709, 506, 721 ], "spans": [ { "bbox": [ 106, 709, 245, 721 ], "score": 1.0, "content": "We then focus on the selection of", "type": "text" }, { "bbox": [ 245, 710, 253, 720 ], "score": 0.81, "content": "A", "type": "inline_equation" }, { "bbox": [ 254, 709, 272, 721 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 273, 712, 281, 720 ], "score": 0.84, "content": "C", "type": "inline_equation" }, { "bbox": [ 281, 709, 506, 721 ], "score": 1.0, "content": ". According to the ablation study, we typically prefer a", "type": "text" } ], "index": 42 }, { "bbox": [ 105, 720, 505, 733 ], "spans": [ { "bbox": [ 105, 720, 131, 733 ], "score": 1.0, "content": "small", "type": "text" }, { "bbox": [ 131, 721, 140, 730 ], "score": 0.75, "content": "A", "type": "inline_equation" }, { "bbox": [ 140, 720, 462, 733 ], "score": 1.0, "content": ", such as 3 and 5. 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