| Method | Natural AccuracyBest Final Diff | PGD-10Best Final Diff | PGD-1000Best Final Diff | C&W-1000Best Final Diff | AutoAttackBest Final Diff |
| PGD-ATPGD-AT+TE | 83.75 84.82 -1.0782.35 82.79 -0.44 | [52.64 44.92 7.7255.79 54.83 0.96 | [51.22 42.74 8.4854.65 53.30 1.35 | |50.11 43.63 7.4852.30 51.73 0.57 | |47.74 41.84 5.9050.59 49.62 0.97 |
| TRADESTRADES+TE | [81.19 82.48 -1.29|83.86 83.97 -0.11 | [53.32 50.25 3.0755.15 54.42 0.73 | [52.44 48.67 3.7753.74 53.03 0.71 | |49.88 48.14 1.74|50.77 50.63 0.14 | 49.03 46.80 2.2349.77 49.20 0.57 |
| (a) The evaluation results on CIFAR-10. |
| Method | Natural AccuracyBest FinalDiff | PGD-10Best Final Diff | PGD-1000Best Final Diff | C&W-1000Best Final Diff | AutoAttackBest Final Diff |
| PGD-ATPGD-AT+TE | 57.54 57.510.0356.45 57.12 -0.67 | 29.40 21.75 7.6531.74 30.24 1.50 | 28.54 20.63 7.9131.27 29.80 1.47 | 27.06 21.17 5.8928.27 27.36 0.91 | 24.72 19.34 5.3826.30 25.34 0.96 |
| TRADESTRADES+TE | 57.98 56.321.6659.35 58.72 0.63 | |29.93 27.70 2.23|31.09 30.12 0.97 | 29.51 26.93 2.58|230.54 29.45 1.09 | [25.46 24.42 1.04|26.61 25.94 0.67 | 24.6123.40 1.2125.27 24.55 0.72 |
| (b) The evaluation results on CIFAR-100. |
| Method | Natural AccuracyBest Final Diff | PGD-10Best Final Diff | PGD-1000Best Final Diff | C&W-1000Best Final Diff | AutoAttackBest Final Diff |
| PGD-ATPGD-AT+TE | 89.00 90.55 -1.5590.09 90.91 -0.82 | [54.51 46.97 7.5459.74 59.05 0.69 | [52.22 42.85 9.3757.7156.46 1.25 | 48.66 44.13 4.5354.5553.94 0.61 | 46.61 38.24 8.3751.44 50.61 0.83 |
| TRADESTRADES+TE | 90.88 91.30 -0.4289.01 88.52 0.49 | [59.50 57.04 2.4659.81 58.49 1.32 | [52.78 50.17 2.6158.24 56.66 1.58 | [52.76 50.53 2.2354.00 53.24 0.76 | 40.36 38.88 1.4851.45 50.16 1.29 |
+
+(c) The evaluation results on SVHN.
+
+$$
+\operatorname* { m i n } _ { \pmb { \theta } } \sum _ { i = 1 } ^ { n } \operatorname* { m a x } _ { \mathbf { x } _ { i } ^ { \prime } \in S ( \mathbf { x } _ { i } ) } \left\{ \mathcal { L } ( f _ { \pmb { \theta } } ( \mathbf { x } _ { i } ^ { \prime } ) , y _ { i } ) + w \cdot | | f _ { \pmb { \theta } } ( \mathbf { x } _ { i } ^ { \prime } ) - \hat { \mathbf { p } } _ { i } | | _ { 2 } ^ { 2 } \right\} ,
+$$
+
+where $\hat { { \bf p } } _ { i }$ is the normalization of $\mathbf { p } _ { i }$ as a probability vector and $w$ is a balancing weight. TE can be similarly integrated with TRADES with the same regularization term. The network would learn to fit relatively easy samples with one-hot labels in the initial training stage, as shown in Fig. 7(a). After the learning rate decays, the network can keep assigning low confidence for hard samples with the regularization term in Eq. (7) and avoid fitting one-hot labels. Therefore, the proposed algorithm enables to learn under label noise in AT and alleviates the robust overfitting problem.
+
+# 5 EMPIRICAL EVALUATION ON MITIGATING ROBUST OVERFITTING
+
+In this section, we provide the experimental results on CIFAR-10, CIFAR-100 (Krizhevsky & Hinton, 2009), and SVHN (Netzer et al., 2011) datasets to validate the effectiveness of our proposed method. Code is available at https://github.com/dongyp13/memorization-AT.
+
+Training details. We adopt the common setting that the perturbation budget is $\epsilon = 8 / 2 5 5$ under the $\ell _ { \infty }$ norm in most experiments. We consider PGD-AT and TRADES as two typical AT baselines and integrate the proposed TE approach into them, respectively. We use the ResNet-18 (He et al., 2016) model as the classifier in most experiments. In training, we use the 10-step PGD adversary with $\alpha = 2 / 2 5 5$ . The models are trained via the SGD optimizer with momentum 0.9, weight decay 0.0005, and batch size 128. For CIFAR-10/100, we set the learning rate as 0.1 initially which is decayed by 0.1 at 100 and 150 epochs with totally 200 training epochs. For SVHN, the learning rate starts from 0.01 with a cosine annealing schedule for a total number of 80 training epochs. In our method, We set $\eta = 0 . 9$ and $w = 3 0$ along a Gaussian ramp-up curve (Laine & Aila, 2017).
+
+Evaluation results. We adopt PGD-10, PGD-1000, C&W-1000 (Carlini & Wagner, 2017), and AutoAttack (Croce & Hein, 2020b) for evaluating adversarial robustness rigorously. AutoAttack is a strong attack to evaluate model robustness, which is composed of an ensemble of diverse attacks, including APGD-CE (Croce & Hein, 2020b), APGD-DLR (Croce & Hein, 2020b), FAB (Croce & Hein, 2020a), and Square attack (Andriushchenko et al., 2020). To show the performance of robust overfitting, we report the test accuracy on the best checkpoint that achieves the highest robust test accuracy under PGD-10 and the final checkpoint, as well as the difference between these two checkpoints. The results of PGD-AT, TRADES, and the combinations of them with our proposed approach (denoted as PGD- $\mathbf { A T + T E }$ and TRADES ${ \bf \nabla } + { \bf T } { \bf E }$ ) on the CIFAR-10, CIFAR-100, and SVHN datasets are shown in Table 1.
+
+We can observe that the differences between best and final test accuracies of our method are reduced to around $1 \%$ , while the accuracy gaps of PGDAT and TRADES are much larger. It indicates that our method largely eliminates robust overfitting. Due to being less affected by robust overfitting, our method achieves higher robust accuracies than the baselines. We also show the learning curves of these methods in Fig. 8. We consistently demonstrate the effectiveness of our method on different network architectures (including WRN-34-10
+
+
+Figure 8: The natural and robust test accuracy curves (under PGD10) of PGD-AT, TRADES, and their extensions by integrating the proposed TE approach. The models are trained on CIFAR-10 under the $\ell _ { \infty }$ norm with $\epsilon = 8 / 2 5 5$ based on the ResNet-18 architecture.
+
+and VGG-16) and threat models (including $\ell _ { 2 }$ norm), which will be shown in Appendix C.2.
+
+Discussion and comparison with related works. Our method is kind of similar to the label smoothing (LS) technique, which is studied in AT (Pang et al., 2021). Recent works have also introduced the smoothness in training labels and model weights (Chen et al., 2021; Huang et al., 2020), which can alleviate robust overfitting to some extent. The significant difference between our work and them is that we provide a reasonable explanation for robust overfitting—one-hot labels are noisy for AT, while previous methods did not give such an explanation and could be viewed as solutions to our identified problem. To empirically compare with these methods, we conduct experiments on CIFAR-10 with the ResNet-18 network. Under the PGD-AT framework, we compare with the baseline PGD-AT, PGD-AT+LS, self-adaptive training (SAT) (Huang et al., 2020), and knowledge distillation with stochastic weight averaging (KD-SWA) (Chen et al., 2021). We also adopt the $C o$ - teaching approach (Han et al., 2018) adapted to PGD-AT, which jointly trains two models using the filtered samples given by each other. The results under the adopted attacks are presented in Table 2. Although various techniques can alleviate robust overfitting, our method achieves better robustness than the others, validating its effectiveness. For Co-teaching, though robust overfitting is alleviated, the performance is worse than our proposed method due to the reduction of training data.
+
+Table 2: Test accuracy $( \% )$ of the proposed method and other methods on CIFAR-10 under the $\ell _ { \infty }$ norm with $\epsilon = 8 / 2 5 5$ based on the ResNet-18 architecture.
+
+| Method | Natural AccuracyBest Final Diff | PGD-10Best Final Diff | PGD-1000Best Final Diff | C&W-1000Best Final Diff | AutoAttackBest Final Diff |
| PGD-ATPGD-AT+TE | 83.75 84.82 -1.0782.35 82.79 -0.44 | [52.64 44.92 7.7255.79 54.83 0.96 | [51.22 42.74 8.4854.65 53.30 1.35 | |50.11 43.63 7.4852.30 51.73 0.57 | |47.74 41.84 5.9050.59 49.62 0.97 |
| TRADESTRADES+TE | [81.19 82.48 -1.29|83.86 83.97 -0.11 | [53.32 50.25 3.0755.15 54.42 0.73 | [52.44 48.67 3.7753.74 53.03 0.71 | |49.88 48.14 1.74|50.77 50.63 0.14 | 49.03 46.80 2.2349.77 49.20 0.57 |
| (a) The evaluation results on CIFAR-10. |
| Method | Natural AccuracyBest FinalDiff | PGD-10Best Final Diff | PGD-1000Best Final Diff | C&W-1000Best Final Diff | AutoAttackBest Final Diff |
| PGD-ATPGD-AT+TE | 57.54 57.510.0356.45 57.12 -0.67 | 29.40 21.75 7.6531.74 30.24 1.50 | 28.54 20.63 7.9131.27 29.80 1.47 | 27.06 21.17 5.8928.27 27.36 0.91 | 24.72 19.34 5.3826.30 25.34 0.96 |
| TRADESTRADES+TE | 57.98 56.321.6659.35 58.72 0.63 | |29.93 27.70 2.23|31.09 30.12 0.97 | 29.51 26.93 2.58|230.54 29.45 1.09 | [25.46 24.42 1.04|26.61 25.94 0.67 | 24.6123.40 1.2125.27 24.55 0.72 |
| (b) The evaluation results on CIFAR-100. |
| Method | Natural AccuracyBest Final Diff | PGD-10Best Final Diff | PGD-1000Best Final Diff | C&W-1000Best Final Diff | AutoAttackBest Final Diff |
| PGD-ATPGD-AT+TE | 89.00 90.55 -1.5590.09 90.91 -0.82 | [54.51 46.97 7.5459.74 59.05 0.69 | [52.22 42.85 9.3757.7156.46 1.25 | 48.66 44.13 4.5354.5553.94 0.61 | 46.61 38.24 8.3751.44 50.61 0.83 |
| TRADESTRADES+TE | 90.88 91.30 -0.4289.01 88.52 0.49 | [59.50 57.04 2.4659.81 58.49 1.32 | [52.78 50.17 2.6158.24 56.66 1.58 | [52.76 50.53 2.2354.00 53.24 0.76 | 40.36 38.88 1.4851.45 50.16 1.29 |
",
+ "bbox": [
+ 173,
+ 128,
+ 823,
+ 457
+ ],
+ "page_idx": 7
+ },
+ {
+ "type": "text",
+ "text": "",
+ "bbox": [
+ 173,
+ 479,
+ 823,
+ 508
+ ],
+ "page_idx": 7
+ },
+ {
+ "type": "equation",
+ "img_path": "images/fed72a46edc93d5572f2299efdef066518df16ea3a3e6956a9ea36faa88b9d55.jpg",
+ "text": "$$\n\\operatorname* { m i n } _ { \\pmb { \\theta } } \\sum _ { i = 1 } ^ { n } \\operatorname* { m a x } _ { \\mathbf { x } _ { i } ^ { \\prime } \\in S ( \\mathbf { x } _ { i } ) } \\left\\{ \\mathcal { L } ( f _ { \\pmb { \\theta } } ( \\mathbf { x } _ { i } ^ { \\prime } ) , y _ { i } ) + w \\cdot | | f _ { \\pmb { \\theta } } ( \\mathbf { x } _ { i } ^ { \\prime } ) - \\hat { \\mathbf { p } } _ { i } | | _ { 2 } ^ { 2 } \\right\\} ,\n$$",
+ "text_format": "latex",
+ "bbox": [
+ 308,
+ 513,
+ 689,
+ 555
+ ],
+ "page_idx": 7
+ },
+ {
+ "type": "text",
+ "text": "where $\\hat { { \\bf p } } _ { i }$ is the normalization of $\\mathbf { p } _ { i }$ as a probability vector and $w$ is a balancing weight. TE can be similarly integrated with TRADES with the same regularization term. The network would learn to fit relatively easy samples with one-hot labels in the initial training stage, as shown in Fig. 7(a). After the learning rate decays, the network can keep assigning low confidence for hard samples with the regularization term in Eq. (7) and avoid fitting one-hot labels. Therefore, the proposed algorithm enables to learn under label noise in AT and alleviates the robust overfitting problem. ",
+ "bbox": [
+ 174,
+ 559,
+ 825,
+ 643
+ ],
+ "page_idx": 7
+ },
+ {
+ "type": "text",
+ "text": "5 EMPIRICAL EVALUATION ON MITIGATING ROBUST OVERFITTING ",
+ "text_level": 1,
+ "bbox": [
+ 176,
+ 660,
+ 736,
+ 674
+ ],
+ "page_idx": 7
+ },
+ {
+ "type": "text",
+ "text": "In this section, we provide the experimental results on CIFAR-10, CIFAR-100 (Krizhevsky & Hinton, 2009), and SVHN (Netzer et al., 2011) datasets to validate the effectiveness of our proposed method. Code is available at https://github.com/dongyp13/memorization-AT. ",
+ "bbox": [
+ 174,
+ 686,
+ 825,
+ 729
+ ],
+ "page_idx": 7
+ },
+ {
+ "type": "text",
+ "text": "Training details. We adopt the common setting that the perturbation budget is $\\epsilon = 8 / 2 5 5$ under the $\\ell _ { \\infty }$ norm in most experiments. We consider PGD-AT and TRADES as two typical AT baselines and integrate the proposed TE approach into them, respectively. We use the ResNet-18 (He et al., 2016) model as the classifier in most experiments. In training, we use the 10-step PGD adversary with $\\alpha = 2 / 2 5 5$ . The models are trained via the SGD optimizer with momentum 0.9, weight decay 0.0005, and batch size 128. For CIFAR-10/100, we set the learning rate as 0.1 initially which is decayed by 0.1 at 100 and 150 epochs with totally 200 training epochs. For SVHN, the learning rate starts from 0.01 with a cosine annealing schedule for a total number of 80 training epochs. In our method, We set $\\eta = 0 . 9$ and $w = 3 0$ along a Gaussian ramp-up curve (Laine & Aila, 2017). ",
+ "bbox": [
+ 174,
+ 734,
+ 825,
+ 861
+ ],
+ "page_idx": 7
+ },
+ {
+ "type": "text",
+ "text": "Evaluation results. We adopt PGD-10, PGD-1000, C&W-1000 (Carlini & Wagner, 2017), and AutoAttack (Croce & Hein, 2020b) for evaluating adversarial robustness rigorously. AutoAttack is a strong attack to evaluate model robustness, which is composed of an ensemble of diverse attacks, including APGD-CE (Croce & Hein, 2020b), APGD-DLR (Croce & Hein, 2020b), FAB (Croce & Hein, 2020a), and Square attack (Andriushchenko et al., 2020). To show the performance of robust overfitting, we report the test accuracy on the best checkpoint that achieves the highest robust test accuracy under PGD-10 and the final checkpoint, as well as the difference between these two checkpoints. The results of PGD-AT, TRADES, and the combinations of them with our proposed approach (denoted as PGD- $\\mathbf { A T + T E }$ and TRADES ${ \\bf \\nabla } + { \\bf T } { \\bf E }$ ) on the CIFAR-10, CIFAR-100, and SVHN datasets are shown in Table 1. ",
+ "bbox": [
+ 174,
+ 867,
+ 823,
+ 924
+ ],
+ "page_idx": 7
+ },
+ {
+ "type": "text",
+ "text": "",
+ "bbox": [
+ 174,
+ 103,
+ 825,
+ 186
+ ],
+ "page_idx": 8
+ },
+ {
+ "type": "text",
+ "text": "We can observe that the differences between best and final test accuracies of our method are reduced to around $1 \\%$ , while the accuracy gaps of PGDAT and TRADES are much larger. It indicates that our method largely eliminates robust overfitting. Due to being less affected by robust overfitting, our method achieves higher robust accuracies than the baselines. We also show the learning curves of these methods in Fig. 8. We consistently demonstrate the effectiveness of our method on different network architectures (including WRN-34-10 ",
+ "bbox": [
+ 174,
+ 194,
+ 419,
+ 400
+ ],
+ "page_idx": 8
+ },
+ {
+ "type": "image",
+ "img_path": "images/60741a322624e7a344d48dd39e648d3bdbec90b00204725405a64e566413b5ab.jpg",
+ "image_caption": [
+ "Figure 8: The natural and robust test accuracy curves (under PGD10) of PGD-AT, TRADES, and their extensions by integrating the proposed TE approach. The models are trained on CIFAR-10 under the $\\ell _ { \\infty }$ norm with $\\epsilon = 8 / 2 5 5$ based on the ResNet-18 architecture. "
+ ],
+ "image_footnote": [],
+ "bbox": [
+ 436,
+ 200,
+ 820,
+ 319
+ ],
+ "page_idx": 8
+ },
+ {
+ "type": "text",
+ "text": "and VGG-16) and threat models (including $\\ell _ { 2 }$ norm), which will be shown in Appendix C.2. ",
+ "bbox": [
+ 169,
+ 401,
+ 776,
+ 416
+ ],
+ "page_idx": 8
+ },
+ {
+ "type": "text",
+ "text": "Discussion and comparison with related works. Our method is kind of similar to the label smoothing (LS) technique, which is studied in AT (Pang et al., 2021). Recent works have also introduced the smoothness in training labels and model weights (Chen et al., 2021; Huang et al., 2020), which can alleviate robust overfitting to some extent. The significant difference between our work and them is that we provide a reasonable explanation for robust overfitting—one-hot labels are noisy for AT, while previous methods did not give such an explanation and could be viewed as solutions to our identified problem. To empirically compare with these methods, we conduct experiments on CIFAR-10 with the ResNet-18 network. Under the PGD-AT framework, we compare with the baseline PGD-AT, PGD-AT+LS, self-adaptive training (SAT) (Huang et al., 2020), and knowledge distillation with stochastic weight averaging (KD-SWA) (Chen et al., 2021). We also adopt the $C o$ - teaching approach (Han et al., 2018) adapted to PGD-AT, which jointly trains two models using the filtered samples given by each other. The results under the adopted attacks are presented in Table 2. Although various techniques can alleviate robust overfitting, our method achieves better robustness than the others, validating its effectiveness. For Co-teaching, though robust overfitting is alleviated, the performance is worse than our proposed method due to the reduction of training data. ",
+ "bbox": [
+ 173,
+ 422,
+ 825,
+ 631
+ ],
+ "page_idx": 8
+ },
+ {
+ "type": "table",
+ "img_path": "images/361dd795551945a1d42eb0e90f9f90e6ae9b6708b7daa3d7a81d92dc95fd5ad3.jpg",
+ "table_caption": [
+ "Table 2: Test accuracy $( \\% )$ of the proposed method and other methods on CIFAR-10 under the $\\ell _ { \\infty }$ norm with $\\epsilon = 8 / 2 5 5$ based on the ResNet-18 architecture. "
+ ],
+ "table_footnote": [],
+ "table_body": "| Method | Natural AccuracyBest Final Diff | PGD-10Best Final Diff | PGD-1000Best Final Diff | C&W-1000Best Final Diff | AutoAttackBest Final Diff |
| PGD-ATPGD-AT+TE | 83.75 84.82 -1.0782.35 82.79 -0.44 | [52.64 44.92 7.7255.79 54.83 0.96 | [51.22 42.74 8.4854.65 53.30 1.35 | |50.11 43.63 7.4852.30 51.73 0.57 | |47.74 41.84 5.9050.59 49.62 0.97 |
| TRADESTRADES+TE | [81.19 82.48 -1.29|83.86 83.97 -0.11 | [53.32 50.25 3.0755.15 54.42 0.73 | [52.44 48.67 3.7753.74 53.03 0.71 | |49.88 48.14 1.74|50.77 50.63 0.14 | 49.03 46.80 2.2349.77 49.20 0.57 |
| (a) The evaluation results on CIFAR-10. |
| Method | Natural AccuracyBest FinalDiff | PGD-10Best Final Diff | PGD-1000Best Final Diff | C&W-1000Best Final Diff | AutoAttackBest Final Diff |
| PGD-ATPGD-AT+TE | 57.54 57.510.0356.45 57.12 -0.67 | 29.40 21.75 7.6531.74 30.24 1.50 | 28.54 20.63 7.9131.27 29.80 1.47 | 27.06 21.17 5.8928.27 27.36 0.91 | 24.72 19.34 5.3826.30 25.34 0.96 |
| TRADESTRADES+TE | 57.98 56.321.6659.35 58.72 0.63 | |29.93 27.70 2.23|31.09 30.12 0.97 | 29.51 26.93 2.58|230.54 29.45 1.09 | [25.46 24.42 1.04|26.61 25.94 0.67 | 24.6123.40 1.2125.27 24.55 0.72 |
| (b) The evaluation results on CIFAR-100. |
| Method | Natural AccuracyBest Final Diff | PGD-10Best Final Diff | PGD-1000Best Final Diff | C&W-1000Best Final Diff | AutoAttackBest Final Diff |
| PGD-ATPGD-AT+TE | 89.00 90.55 -1.5590.09 90.91 -0.82 | [54.51 46.97 7.5459.74 59.05 0.69 | [52.22 42.85 9.3757.7156.46 1.25 | 48.66 44.13 4.5354.5553.94 0.61 | 46.61 38.24 8.3751.44 50.61 0.83 |
| TRADESTRADES+TE | 90.88 91.30 -0.4289.01 88.52 0.49 | [59.50 57.04 2.4659.81 58.49 1.32 | [52.78 50.17 2.6158.24 56.66 1.58 | [52.76 50.53 2.2354.00 53.24 0.76 | 40.36 38.88 1.4851.45 50.16 1.29 |
",
+ "type": "table",
+ "image_path": "eed08a7a5d061280010122522fec3399a1b93d22e4d87b986b59c25987162ecf.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 4,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 106,
+ 102,
+ 504,
+ 188.66666666666669
+ ],
+ "spans": [],
+ "index": 3
+ },
+ {
+ "bbox": [
+ 106,
+ 188.66666666666669,
+ 504,
+ 275.33333333333337
+ ],
+ "spans": [],
+ "index": 4
+ },
+ {
+ "bbox": [
+ 106,
+ 275.33333333333337,
+ 504,
+ 362.00000000000006
+ ],
+ "spans": [],
+ "index": 5
+ }
+ ]
+ },
+ {
+ "type": "table_footnote",
+ "bbox": [
+ 239,
+ 363,
+ 372,
+ 373
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 238,
+ 362,
+ 372,
+ 374
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 238,
+ 362,
+ 372,
+ 374
+ ],
+ "score": 1.0,
+ "content": "(c) The evaluation results on SVHN.",
+ "type": "text"
+ }
+ ],
+ "index": 6
+ }
+ ],
+ "index": 6
+ }
+ ],
+ "index": 4
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 380,
+ 504,
+ 403
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 380,
+ 505,
+ 393
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 380,
+ 176,
+ 393
+ ],
+ "score": 1.0,
+ "content": "a training sample",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 177,
+ 383,
+ 187,
+ 392
+ ],
+ "score": 0.85,
+ "content": "\\mathbf { x } _ { i }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 188,
+ 380,
+ 198,
+ 393
+ ],
+ "score": 1.0,
+ "content": "as",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 199,
+ 383,
+ 210,
+ 393
+ ],
+ "score": 0.87,
+ "content": "\\mathbf { p } _ { i }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 210,
+ 380,
+ 383,
+ 393
+ ],
+ "score": 1.0,
+ "content": ", which is updated in each training epoch as",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 383,
+ 380,
+ 502,
+ 393
+ ],
+ "score": 0.92,
+ "content": "\\mathbf { p } _ { i } \\eta \\cdot \\mathbf { p } _ { i } + ( 1 - \\eta ) \\cdot f _ { \\pmb { \\theta } } ( \\mathbf { x } _ { i } )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 502,
+ 380,
+ 505,
+ 393
+ ],
+ "score": 1.0,
+ "content": ",",
+ "type": "text"
+ }
+ ],
+ "index": 7
+ },
+ {
+ "bbox": [
+ 105,
+ 390,
+ 492,
+ 405
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 390,
+ 133,
+ 405
+ ],
+ "score": 1.0,
+ "content": "where",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 133,
+ 394,
+ 140,
+ 403
+ ],
+ "score": 0.81,
+ "content": "\\eta",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 140,
+ 390,
+ 492,
+ 405
+ ],
+ "score": 1.0,
+ "content": "is the momentum term. The training objective of PGD-AT with TE can be expressed as",
+ "type": "text"
+ }
+ ],
+ "index": 8
+ }
+ ],
+ "index": 7.5
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 189,
+ 407,
+ 422,
+ 440
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 189,
+ 407,
+ 422,
+ 440
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 189,
+ 407,
+ 422,
+ 440
+ ],
+ "score": 0.94,
+ "content": "\\operatorname* { m i n } _ { \\pmb { \\theta } } \\sum _ { i = 1 } ^ { n } \\operatorname* { m a x } _ { \\mathbf { x } _ { i } ^ { \\prime } \\in S ( \\mathbf { x } _ { i } ) } \\left\\{ \\mathcal { L } ( f _ { \\pmb { \\theta } } ( \\mathbf { x } _ { i } ^ { \\prime } ) , y _ { i } ) + w \\cdot | | f _ { \\pmb { \\theta } } ( \\mathbf { x } _ { i } ^ { \\prime } ) - \\hat { \\mathbf { p } } _ { i } | | _ { 2 } ^ { 2 } \\right\\} ,",
+ "type": "interline_equation",
+ "image_path": "fed72a46edc93d5572f2299efdef066518df16ea3a3e6956a9ea36faa88b9d55.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 9.5,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 189,
+ 407,
+ 422,
+ 423.5
+ ],
+ "spans": [],
+ "index": 9
+ },
+ {
+ "bbox": [
+ 189,
+ 423.5,
+ 422,
+ 440.0
+ ],
+ "spans": [],
+ "index": 10
+ }
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 443,
+ 505,
+ 510
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 444,
+ 505,
+ 457
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 444,
+ 133,
+ 457
+ ],
+ "score": 1.0,
+ "content": "where",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 134,
+ 444,
+ 145,
+ 456
+ ],
+ "score": 0.89,
+ "content": "\\hat { { \\bf p } } _ { i }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 145,
+ 444,
+ 242,
+ 457
+ ],
+ "score": 1.0,
+ "content": "is the normalization of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 243,
+ 446,
+ 254,
+ 456
+ ],
+ "score": 0.87,
+ "content": "\\mathbf { p } _ { i }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 254,
+ 444,
+ 367,
+ 457
+ ],
+ "score": 1.0,
+ "content": "as a probability vector and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 368,
+ 446,
+ 376,
+ 454
+ ],
+ "score": 0.78,
+ "content": "w",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 377,
+ 444,
+ 505,
+ 457
+ ],
+ "score": 1.0,
+ "content": "is a balancing weight. TE can",
+ "type": "text"
+ }
+ ],
+ "index": 11
+ },
+ {
+ "bbox": [
+ 105,
+ 455,
+ 505,
+ 467
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 455,
+ 505,
+ 467
+ ],
+ "score": 1.0,
+ "content": "be similarly integrated with TRADES with the same regularization term. The network would learn",
+ "type": "text"
+ }
+ ],
+ "index": 12
+ },
+ {
+ "bbox": [
+ 105,
+ 466,
+ 505,
+ 479
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 466,
+ 505,
+ 479
+ ],
+ "score": 1.0,
+ "content": "to fit relatively easy samples with one-hot labels in the initial training stage, as shown in Fig. 7(a).",
+ "type": "text"
+ }
+ ],
+ "index": 13
+ },
+ {
+ "bbox": [
+ 105,
+ 476,
+ 505,
+ 489
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 476,
+ 505,
+ 489
+ ],
+ "score": 1.0,
+ "content": "After the learning rate decays, the network can keep assigning low confidence for hard samples with",
+ "type": "text"
+ }
+ ],
+ "index": 14
+ },
+ {
+ "bbox": [
+ 105,
+ 488,
+ 505,
+ 501
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 488,
+ 505,
+ 501
+ ],
+ "score": 1.0,
+ "content": "the regularization term in Eq. (7) and avoid fitting one-hot labels. Therefore, the proposed algorithm",
+ "type": "text"
+ }
+ ],
+ "index": 15
+ },
+ {
+ "bbox": [
+ 106,
+ 498,
+ 447,
+ 511
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 498,
+ 447,
+ 511
+ ],
+ "score": 1.0,
+ "content": "enables to learn under label noise in AT and alleviates the robust overfitting problem.",
+ "type": "text"
+ }
+ ],
+ "index": 16
+ }
+ ],
+ "index": 13.5
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 108,
+ 523,
+ 451,
+ 534
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 522,
+ 453,
+ 536
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 522,
+ 453,
+ 536
+ ],
+ "score": 1.0,
+ "content": "5 EMPIRICAL EVALUATION ON MITIGATING ROBUST OVERFITTING",
+ "type": "text"
+ }
+ ],
+ "index": 17
+ }
+ ],
+ "index": 17
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 544,
+ 505,
+ 578
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 544,
+ 505,
+ 556
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 544,
+ 505,
+ 556
+ ],
+ "score": 1.0,
+ "content": "In this section, we provide the experimental results on CIFAR-10, CIFAR-100 (Krizhevsky & Hin-",
+ "type": "text"
+ }
+ ],
+ "index": 18
+ },
+ {
+ "bbox": [
+ 105,
+ 554,
+ 505,
+ 567
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 554,
+ 505,
+ 567
+ ],
+ "score": 1.0,
+ "content": "ton, 2009), and SVHN (Netzer et al., 2011) datasets to validate the effectiveness of our proposed",
+ "type": "text"
+ }
+ ],
+ "index": 19
+ },
+ {
+ "bbox": [
+ 105,
+ 565,
+ 485,
+ 579
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 565,
+ 485,
+ 579
+ ],
+ "score": 1.0,
+ "content": "method. Code is available at https://github.com/dongyp13/memorization-AT.",
+ "type": "text"
+ }
+ ],
+ "index": 20
+ }
+ ],
+ "index": 19
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 582,
+ 505,
+ 682
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 582,
+ 506,
+ 596
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 582,
+ 432,
+ 596
+ ],
+ "score": 1.0,
+ "content": "Training details. We adopt the common setting that the perturbation budget is",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 432,
+ 583,
+ 478,
+ 595
+ ],
+ "score": 0.9,
+ "content": "\\epsilon = 8 / 2 5 5",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 479,
+ 582,
+ 506,
+ 596
+ ],
+ "score": 1.0,
+ "content": "under",
+ "type": "text"
+ }
+ ],
+ "index": 21
+ },
+ {
+ "bbox": [
+ 105,
+ 594,
+ 505,
+ 607
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 594,
+ 121,
+ 607
+ ],
+ "score": 1.0,
+ "content": "the",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 121,
+ 594,
+ 135,
+ 605
+ ],
+ "score": 0.89,
+ "content": "\\ell _ { \\infty }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 135,
+ 594,
+ 505,
+ 607
+ ],
+ "score": 1.0,
+ "content": "norm in most experiments. We consider PGD-AT and TRADES as two typical AT baselines",
+ "type": "text"
+ }
+ ],
+ "index": 22
+ },
+ {
+ "bbox": [
+ 105,
+ 604,
+ 505,
+ 617
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 604,
+ 505,
+ 617
+ ],
+ "score": 1.0,
+ "content": "and integrate the proposed TE approach into them, respectively. We use the ResNet-18 (He et al.,",
+ "type": "text"
+ }
+ ],
+ "index": 23
+ },
+ {
+ "bbox": [
+ 105,
+ 614,
+ 505,
+ 630
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 614,
+ 505,
+ 630
+ ],
+ "score": 1.0,
+ "content": "2016) model as the classifier in most experiments. In training, we use the 10-step PGD adversary",
+ "type": "text"
+ }
+ ],
+ "index": 24
+ },
+ {
+ "bbox": [
+ 105,
+ 626,
+ 505,
+ 640
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 626,
+ 126,
+ 640
+ ],
+ "score": 1.0,
+ "content": "with",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 127,
+ 627,
+ 173,
+ 639
+ ],
+ "score": 0.93,
+ "content": "\\alpha = 2 / 2 5 5",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 173,
+ 626,
+ 505,
+ 640
+ ],
+ "score": 1.0,
+ "content": ". The models are trained via the SGD optimizer with momentum 0.9, weight decay",
+ "type": "text"
+ }
+ ],
+ "index": 25
+ },
+ {
+ "bbox": [
+ 105,
+ 637,
+ 506,
+ 651
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 637,
+ 506,
+ 651
+ ],
+ "score": 1.0,
+ "content": "0.0005, and batch size 128. For CIFAR-10/100, we set the learning rate as 0.1 initially which is",
+ "type": "text"
+ }
+ ],
+ "index": 26
+ },
+ {
+ "bbox": [
+ 105,
+ 648,
+ 505,
+ 662
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 648,
+ 505,
+ 662
+ ],
+ "score": 1.0,
+ "content": "decayed by 0.1 at 100 and 150 epochs with totally 200 training epochs. For SVHN, the learning rate",
+ "type": "text"
+ }
+ ],
+ "index": 27
+ },
+ {
+ "bbox": [
+ 105,
+ 659,
+ 506,
+ 673
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 659,
+ 506,
+ 673
+ ],
+ "score": 1.0,
+ "content": "starts from 0.01 with a cosine annealing schedule for a total number of 80 training epochs. In our",
+ "type": "text"
+ }
+ ],
+ "index": 28
+ },
+ {
+ "bbox": [
+ 105,
+ 670,
+ 474,
+ 683
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 670,
+ 170,
+ 683
+ ],
+ "score": 1.0,
+ "content": "method, We set",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 170,
+ 671,
+ 203,
+ 682
+ ],
+ "score": 0.91,
+ "content": "\\eta = 0 . 9",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 204,
+ 670,
+ 221,
+ 683
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 221,
+ 671,
+ 253,
+ 681
+ ],
+ "score": 0.89,
+ "content": "w = 3 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 254,
+ 670,
+ 474,
+ 683
+ ],
+ "score": 1.0,
+ "content": "along a Gaussian ramp-up curve (Laine & Aila, 2017).",
+ "type": "text"
+ }
+ ],
+ "index": 29
+ }
+ ],
+ "index": 25
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 687,
+ 504,
+ 732
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 686,
+ 506,
+ 700
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 686,
+ 506,
+ 700
+ ],
+ "score": 1.0,
+ "content": "Evaluation results. We adopt PGD-10, PGD-1000, C&W-1000 (Carlini & Wagner, 2017), and",
+ "type": "text"
+ }
+ ],
+ "index": 30
+ },
+ {
+ "bbox": [
+ 106,
+ 699,
+ 506,
+ 711
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 699,
+ 506,
+ 711
+ ],
+ "score": 1.0,
+ "content": "AutoAttack (Croce & Hein, 2020b) for evaluating adversarial robustness rigorously. AutoAttack is",
+ "type": "text"
+ }
+ ],
+ "index": 31
+ },
+ {
+ "bbox": [
+ 105,
+ 710,
+ 505,
+ 722
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 710,
+ 505,
+ 722
+ ],
+ "score": 1.0,
+ "content": "a strong attack to evaluate model robustness, which is composed of an ensemble of diverse attacks,",
+ "type": "text"
+ }
+ ],
+ "index": 32
+ },
+ {
+ "bbox": [
+ 105,
+ 720,
+ 506,
+ 733
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 720,
+ 506,
+ 733
+ ],
+ "score": 1.0,
+ "content": "including APGD-CE (Croce & Hein, 2020b), APGD-DLR (Croce & Hein, 2020b), FAB (Croce",
+ "type": "text"
+ }
+ ],
+ "index": 33
+ }
+ ],
+ "index": 31.5
+ }
+ ],
+ "page_idx": 7,
+ "page_size": [
+ 612,
+ 792
+ ],
+ "discarded_blocks": [
+ {
+ "type": "discarded",
+ "bbox": [
+ 107,
+ 27,
+ 293,
+ 37
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 26,
+ 294,
+ 38
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 26,
+ 294,
+ 38
+ ],
+ "score": 1.0,
+ "content": "Published as a conference paper at ICLR 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": [
+ 106,
+ 102,
+ 504,
+ 362
+ ],
+ "blocks": [
+ {
+ "type": "table_caption",
+ "bbox": [
+ 106,
+ 69,
+ 505,
+ 99
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 69,
+ 504,
+ 80
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 69,
+ 189,
+ 80
+ ],
+ "score": 1.0,
+ "content": "Table 1: Test accuracy",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 189,
+ 70,
+ 203,
+ 79
+ ],
+ "score": 0.72,
+ "content": "( \\% )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 204,
+ 69,
+ 451,
+ 80
+ ],
+ "score": 1.0,
+ "content": "of several methods on CIFAR-10, CIFAR-100, and SVHN under the",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 452,
+ 70,
+ 464,
+ 79
+ ],
+ "score": 0.86,
+ "content": "\\ell _ { \\infty }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 464,
+ 69,
+ 504,
+ 80
+ ],
+ "score": 1.0,
+ "content": "norm with",
+ "type": "text"
+ }
+ ],
+ "index": 0
+ },
+ {
+ "bbox": [
+ 107,
+ 79,
+ 505,
+ 90
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 79,
+ 147,
+ 90
+ ],
+ "score": 0.91,
+ "content": "\\epsilon = 8 / 2 5 5",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 147,
+ 79,
+ 505,
+ 90
+ ],
+ "score": 1.0,
+ "content": "based on the ResNet-18 architecture. We choose the best checkpoint according to the highest robust",
+ "type": "text"
+ }
+ ],
+ "index": 1
+ },
+ {
+ "bbox": [
+ 106,
+ 89,
+ 248,
+ 100
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 89,
+ 248,
+ 100
+ ],
+ "score": 1.0,
+ "content": "accuracy on the test set under PGD-10.",
+ "type": "text"
+ }
+ ],
+ "index": 2
+ }
+ ],
+ "index": 1
+ },
+ {
+ "type": "table_body",
+ "bbox": [
+ 106,
+ 102,
+ 504,
+ 362
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 102,
+ 504,
+ 362
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 102,
+ 504,
+ 362
+ ],
+ "score": 0.979,
+ "html": "| Method | Natural AccuracyBest Final Diff | PGD-10Best Final Diff | PGD-1000Best Final Diff | C&W-1000Best Final Diff | AutoAttackBest Final Diff |
| PGD-ATPGD-AT+TE | 83.75 84.82 -1.0782.35 82.79 -0.44 | [52.64 44.92 7.7255.79 54.83 0.96 | [51.22 42.74 8.4854.65 53.30 1.35 | |50.11 43.63 7.4852.30 51.73 0.57 | |47.74 41.84 5.9050.59 49.62 0.97 |
| TRADESTRADES+TE | [81.19 82.48 -1.29|83.86 83.97 -0.11 | [53.32 50.25 3.0755.15 54.42 0.73 | [52.44 48.67 3.7753.74 53.03 0.71 | |49.88 48.14 1.74|50.77 50.63 0.14 | 49.03 46.80 2.2349.77 49.20 0.57 |
| (a) The evaluation results on CIFAR-10. |
| Method | Natural AccuracyBest FinalDiff | PGD-10Best Final Diff | PGD-1000Best Final Diff | C&W-1000Best Final Diff | AutoAttackBest Final Diff |
| PGD-ATPGD-AT+TE | 57.54 57.510.0356.45 57.12 -0.67 | 29.40 21.75 7.6531.74 30.24 1.50 | 28.54 20.63 7.9131.27 29.80 1.47 | 27.06 21.17 5.8928.27 27.36 0.91 | 24.72 19.34 5.3826.30 25.34 0.96 |
| TRADESTRADES+TE | 57.98 56.321.6659.35 58.72 0.63 | |29.93 27.70 2.23|31.09 30.12 0.97 | 29.51 26.93 2.58|230.54 29.45 1.09 | [25.46 24.42 1.04|26.61 25.94 0.67 | 24.6123.40 1.2125.27 24.55 0.72 |
| (b) The evaluation results on CIFAR-100. |
| Method | Natural AccuracyBest Final Diff | PGD-10Best Final Diff | PGD-1000Best Final Diff | C&W-1000Best Final Diff | AutoAttackBest Final Diff |
| PGD-ATPGD-AT+TE | 89.00 90.55 -1.5590.09 90.91 -0.82 | [54.51 46.97 7.5459.74 59.05 0.69 | [52.22 42.85 9.3757.7156.46 1.25 | 48.66 44.13 4.5354.5553.94 0.61 | 46.61 38.24 8.3751.44 50.61 0.83 |
| TRADESTRADES+TE | 90.88 91.30 -0.4289.01 88.52 0.49 | [59.50 57.04 2.4659.81 58.49 1.32 | [52.78 50.17 2.6158.24 56.66 1.58 | [52.76 50.53 2.2354.00 53.24 0.76 | 40.36 38.88 1.4851.45 50.16 1.29 |
",
+ "type": "table",
+ "image_path": "eed08a7a5d061280010122522fec3399a1b93d22e4d87b986b59c25987162ecf.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 4,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 106,
+ 102,
+ 504,
+ 188.66666666666669
+ ],
+ "spans": [],
+ "index": 3
+ },
+ {
+ "bbox": [
+ 106,
+ 188.66666666666669,
+ 504,
+ 275.33333333333337
+ ],
+ "spans": [],
+ "index": 4
+ },
+ {
+ "bbox": [
+ 106,
+ 275.33333333333337,
+ 504,
+ 362.00000000000006
+ ],
+ "spans": [],
+ "index": 5
+ }
+ ]
+ },
+ {
+ "type": "table_footnote",
+ "bbox": [
+ 239,
+ 363,
+ 372,
+ 373
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 238,
+ 362,
+ 372,
+ 374
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 238,
+ 362,
+ 372,
+ 374
+ ],
+ "score": 1.0,
+ "content": "(c) The evaluation results on SVHN.",
+ "type": "text"
+ }
+ ],
+ "index": 6
+ }
+ ],
+ "index": 6
+ }
+ ],
+ "index": 4
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 380,
+ 504,
+ 403
+ ],
+ "lines": [],
+ "index": 7.5,
+ "bbox_fs": [
+ 105,
+ 380,
+ 505,
+ 405
+ ],
+ "lines_deleted": true
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 189,
+ 407,
+ 422,
+ 440
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 189,
+ 407,
+ 422,
+ 440
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 189,
+ 407,
+ 422,
+ 440
+ ],
+ "score": 0.94,
+ "content": "\\operatorname* { m i n } _ { \\pmb { \\theta } } \\sum _ { i = 1 } ^ { n } \\operatorname* { m a x } _ { \\mathbf { x } _ { i } ^ { \\prime } \\in S ( \\mathbf { x } _ { i } ) } \\left\\{ \\mathcal { L } ( f _ { \\pmb { \\theta } } ( \\mathbf { x } _ { i } ^ { \\prime } ) , y _ { i } ) + w \\cdot | | f _ { \\pmb { \\theta } } ( \\mathbf { x } _ { i } ^ { \\prime } ) - \\hat { \\mathbf { p } } _ { i } | | _ { 2 } ^ { 2 } \\right\\} ,",
+ "type": "interline_equation",
+ "image_path": "fed72a46edc93d5572f2299efdef066518df16ea3a3e6956a9ea36faa88b9d55.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 9.5,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 189,
+ 407,
+ 422,
+ 423.5
+ ],
+ "spans": [],
+ "index": 9
+ },
+ {
+ "bbox": [
+ 189,
+ 423.5,
+ 422,
+ 440.0
+ ],
+ "spans": [],
+ "index": 10
+ }
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 443,
+ 505,
+ 510
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 444,
+ 505,
+ 457
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 444,
+ 133,
+ 457
+ ],
+ "score": 1.0,
+ "content": "where",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 134,
+ 444,
+ 145,
+ 456
+ ],
+ "score": 0.89,
+ "content": "\\hat { { \\bf p } } _ { i }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 145,
+ 444,
+ 242,
+ 457
+ ],
+ "score": 1.0,
+ "content": "is the normalization of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 243,
+ 446,
+ 254,
+ 456
+ ],
+ "score": 0.87,
+ "content": "\\mathbf { p } _ { i }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 254,
+ 444,
+ 367,
+ 457
+ ],
+ "score": 1.0,
+ "content": "as a probability vector and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 368,
+ 446,
+ 376,
+ 454
+ ],
+ "score": 0.78,
+ "content": "w",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 377,
+ 444,
+ 505,
+ 457
+ ],
+ "score": 1.0,
+ "content": "is a balancing weight. TE can",
+ "type": "text"
+ }
+ ],
+ "index": 11
+ },
+ {
+ "bbox": [
+ 105,
+ 455,
+ 505,
+ 467
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 455,
+ 505,
+ 467
+ ],
+ "score": 1.0,
+ "content": "be similarly integrated with TRADES with the same regularization term. The network would learn",
+ "type": "text"
+ }
+ ],
+ "index": 12
+ },
+ {
+ "bbox": [
+ 105,
+ 466,
+ 505,
+ 479
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 466,
+ 505,
+ 479
+ ],
+ "score": 1.0,
+ "content": "to fit relatively easy samples with one-hot labels in the initial training stage, as shown in Fig. 7(a).",
+ "type": "text"
+ }
+ ],
+ "index": 13
+ },
+ {
+ "bbox": [
+ 105,
+ 476,
+ 505,
+ 489
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 476,
+ 505,
+ 489
+ ],
+ "score": 1.0,
+ "content": "After the learning rate decays, the network can keep assigning low confidence for hard samples with",
+ "type": "text"
+ }
+ ],
+ "index": 14
+ },
+ {
+ "bbox": [
+ 105,
+ 488,
+ 505,
+ 501
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 488,
+ 505,
+ 501
+ ],
+ "score": 1.0,
+ "content": "the regularization term in Eq. (7) and avoid fitting one-hot labels. Therefore, the proposed algorithm",
+ "type": "text"
+ }
+ ],
+ "index": 15
+ },
+ {
+ "bbox": [
+ 106,
+ 498,
+ 447,
+ 511
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 498,
+ 447,
+ 511
+ ],
+ "score": 1.0,
+ "content": "enables to learn under label noise in AT and alleviates the robust overfitting problem.",
+ "type": "text"
+ }
+ ],
+ "index": 16
+ }
+ ],
+ "index": 13.5,
+ "bbox_fs": [
+ 105,
+ 444,
+ 505,
+ 511
+ ]
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 108,
+ 523,
+ 451,
+ 534
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 522,
+ 453,
+ 536
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 522,
+ 453,
+ 536
+ ],
+ "score": 1.0,
+ "content": "5 EMPIRICAL EVALUATION ON MITIGATING ROBUST OVERFITTING",
+ "type": "text"
+ }
+ ],
+ "index": 17
+ }
+ ],
+ "index": 17
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 544,
+ 505,
+ 578
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 544,
+ 505,
+ 556
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 544,
+ 505,
+ 556
+ ],
+ "score": 1.0,
+ "content": "In this section, we provide the experimental results on CIFAR-10, CIFAR-100 (Krizhevsky & Hin-",
+ "type": "text"
+ }
+ ],
+ "index": 18
+ },
+ {
+ "bbox": [
+ 105,
+ 554,
+ 505,
+ 567
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 554,
+ 505,
+ 567
+ ],
+ "score": 1.0,
+ "content": "ton, 2009), and SVHN (Netzer et al., 2011) datasets to validate the effectiveness of our proposed",
+ "type": "text"
+ }
+ ],
+ "index": 19
+ },
+ {
+ "bbox": [
+ 105,
+ 565,
+ 485,
+ 579
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 565,
+ 485,
+ 579
+ ],
+ "score": 1.0,
+ "content": "method. Code is available at https://github.com/dongyp13/memorization-AT.",
+ "type": "text"
+ }
+ ],
+ "index": 20
+ }
+ ],
+ "index": 19,
+ "bbox_fs": [
+ 105,
+ 544,
+ 505,
+ 579
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 582,
+ 505,
+ 682
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 582,
+ 506,
+ 596
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 582,
+ 432,
+ 596
+ ],
+ "score": 1.0,
+ "content": "Training details. We adopt the common setting that the perturbation budget is",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 432,
+ 583,
+ 478,
+ 595
+ ],
+ "score": 0.9,
+ "content": "\\epsilon = 8 / 2 5 5",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 479,
+ 582,
+ 506,
+ 596
+ ],
+ "score": 1.0,
+ "content": "under",
+ "type": "text"
+ }
+ ],
+ "index": 21
+ },
+ {
+ "bbox": [
+ 105,
+ 594,
+ 505,
+ 607
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 594,
+ 121,
+ 607
+ ],
+ "score": 1.0,
+ "content": "the",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 121,
+ 594,
+ 135,
+ 605
+ ],
+ "score": 0.89,
+ "content": "\\ell _ { \\infty }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 135,
+ 594,
+ 505,
+ 607
+ ],
+ "score": 1.0,
+ "content": "norm in most experiments. We consider PGD-AT and TRADES as two typical AT baselines",
+ "type": "text"
+ }
+ ],
+ "index": 22
+ },
+ {
+ "bbox": [
+ 105,
+ 604,
+ 505,
+ 617
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 604,
+ 505,
+ 617
+ ],
+ "score": 1.0,
+ "content": "and integrate the proposed TE approach into them, respectively. We use the ResNet-18 (He et al.,",
+ "type": "text"
+ }
+ ],
+ "index": 23
+ },
+ {
+ "bbox": [
+ 105,
+ 614,
+ 505,
+ 630
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 614,
+ 505,
+ 630
+ ],
+ "score": 1.0,
+ "content": "2016) model as the classifier in most experiments. In training, we use the 10-step PGD adversary",
+ "type": "text"
+ }
+ ],
+ "index": 24
+ },
+ {
+ "bbox": [
+ 105,
+ 626,
+ 505,
+ 640
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 626,
+ 126,
+ 640
+ ],
+ "score": 1.0,
+ "content": "with",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 127,
+ 627,
+ 173,
+ 639
+ ],
+ "score": 0.93,
+ "content": "\\alpha = 2 / 2 5 5",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 173,
+ 626,
+ 505,
+ 640
+ ],
+ "score": 1.0,
+ "content": ". The models are trained via the SGD optimizer with momentum 0.9, weight decay",
+ "type": "text"
+ }
+ ],
+ "index": 25
+ },
+ {
+ "bbox": [
+ 105,
+ 637,
+ 506,
+ 651
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 637,
+ 506,
+ 651
+ ],
+ "score": 1.0,
+ "content": "0.0005, and batch size 128. For CIFAR-10/100, we set the learning rate as 0.1 initially which is",
+ "type": "text"
+ }
+ ],
+ "index": 26
+ },
+ {
+ "bbox": [
+ 105,
+ 648,
+ 505,
+ 662
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 648,
+ 505,
+ 662
+ ],
+ "score": 1.0,
+ "content": "decayed by 0.1 at 100 and 150 epochs with totally 200 training epochs. For SVHN, the learning rate",
+ "type": "text"
+ }
+ ],
+ "index": 27
+ },
+ {
+ "bbox": [
+ 105,
+ 659,
+ 506,
+ 673
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 659,
+ 506,
+ 673
+ ],
+ "score": 1.0,
+ "content": "starts from 0.01 with a cosine annealing schedule for a total number of 80 training epochs. In our",
+ "type": "text"
+ }
+ ],
+ "index": 28
+ },
+ {
+ "bbox": [
+ 105,
+ 670,
+ 474,
+ 683
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 670,
+ 170,
+ 683
+ ],
+ "score": 1.0,
+ "content": "method, We set",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 170,
+ 671,
+ 203,
+ 682
+ ],
+ "score": 0.91,
+ "content": "\\eta = 0 . 9",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 204,
+ 670,
+ 221,
+ 683
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 221,
+ 671,
+ 253,
+ 681
+ ],
+ "score": 0.89,
+ "content": "w = 3 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 254,
+ 670,
+ 474,
+ 683
+ ],
+ "score": 1.0,
+ "content": "along a Gaussian ramp-up curve (Laine & Aila, 2017).",
+ "type": "text"
+ }
+ ],
+ "index": 29
+ }
+ ],
+ "index": 25,
+ "bbox_fs": [
+ 105,
+ 582,
+ 506,
+ 683
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 687,
+ 504,
+ 732
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 686,
+ 506,
+ 700
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 686,
+ 506,
+ 700
+ ],
+ "score": 1.0,
+ "content": "Evaluation results. We adopt PGD-10, PGD-1000, C&W-1000 (Carlini & Wagner, 2017), and",
+ "type": "text"
+ }
+ ],
+ "index": 30
+ },
+ {
+ "bbox": [
+ 106,
+ 699,
+ 506,
+ 711
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 699,
+ 506,
+ 711
+ ],
+ "score": 1.0,
+ "content": "AutoAttack (Croce & Hein, 2020b) for evaluating adversarial robustness rigorously. AutoAttack is",
+ "type": "text"
+ }
+ ],
+ "index": 31
+ },
+ {
+ "bbox": [
+ 105,
+ 710,
+ 505,
+ 722
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 710,
+ 505,
+ 722
+ ],
+ "score": 1.0,
+ "content": "a strong attack to evaluate model robustness, which is composed of an ensemble of diverse attacks,",
+ "type": "text"
+ }
+ ],
+ "index": 32
+ },
+ {
+ "bbox": [
+ 105,
+ 720,
+ 506,
+ 733
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 720,
+ 506,
+ 733
+ ],
+ "score": 1.0,
+ "content": "including APGD-CE (Croce & Hein, 2020b), APGD-DLR (Croce & Hein, 2020b), FAB (Croce",
+ "type": "text"
+ }
+ ],
+ "index": 33
+ },
+ {
+ "bbox": [
+ 106,
+ 82,
+ 506,
+ 95
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 82,
+ 506,
+ 95
+ ],
+ "score": 1.0,
+ "content": "& Hein, 2020a), and Square attack (Andriushchenko et al., 2020). To show the performance of",
+ "type": "text",
+ "cross_page": true
+ }
+ ],
+ "index": 0
+ },
+ {
+ "bbox": [
+ 106,
+ 93,
+ 505,
+ 105
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 93,
+ 505,
+ 105
+ ],
+ "score": 1.0,
+ "content": "robust overfitting, we report the test accuracy on the best checkpoint that achieves the highest robust",
+ "type": "text",
+ "cross_page": true
+ }
+ ],
+ "index": 1
+ },
+ {
+ "bbox": [
+ 105,
+ 104,
+ 506,
+ 117
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 104,
+ 506,
+ 117
+ ],
+ "score": 1.0,
+ "content": "test accuracy under PGD-10 and the final checkpoint, as well as the difference between these two",
+ "type": "text",
+ "cross_page": true
+ }
+ ],
+ "index": 2
+ },
+ {
+ "bbox": [
+ 106,
+ 115,
+ 505,
+ 127
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 115,
+ 505,
+ 127
+ ],
+ "score": 1.0,
+ "content": "checkpoints. The results of PGD-AT, TRADES, and the combinations of them with our proposed",
+ "type": "text",
+ "cross_page": true
+ }
+ ],
+ "index": 3
+ },
+ {
+ "bbox": [
+ 105,
+ 126,
+ 506,
+ 139
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 126,
+ 218,
+ 139
+ ],
+ "score": 1.0,
+ "content": "approach (denoted as PGD-",
+ "type": "text",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 218,
+ 126,
+ 250,
+ 137
+ ],
+ "score": 0.58,
+ "content": "\\mathbf { A T + T E }",
+ "type": "inline_equation",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 250,
+ 126,
+ 307,
+ 139
+ ],
+ "score": 1.0,
+ "content": "and TRADES",
+ "type": "text",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 307,
+ 127,
+ 327,
+ 137
+ ],
+ "score": 0.48,
+ "content": "{ \\bf \\nabla } + { \\bf T } { \\bf E }",
+ "type": "inline_equation",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 327,
+ 126,
+ 506,
+ 139
+ ],
+ "score": 1.0,
+ "content": ") on the CIFAR-10, CIFAR-100, and SVHN",
+ "type": "text",
+ "cross_page": true
+ }
+ ],
+ "index": 4
+ },
+ {
+ "bbox": [
+ 106,
+ 138,
+ 228,
+ 148
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 138,
+ 228,
+ 148
+ ],
+ "score": 1.0,
+ "content": "datasets are shown in Table 1.",
+ "type": "text",
+ "cross_page": true
+ }
+ ],
+ "index": 5
+ }
+ ],
+ "index": 31.5,
+ "bbox_fs": [
+ 105,
+ 686,
+ 506,
+ 733
+ ]
+ }
+ ]
+ },
+ {
+ "preproc_blocks": [
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 82,
+ 505,
+ 148
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 82,
+ 506,
+ 95
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 82,
+ 506,
+ 95
+ ],
+ "score": 1.0,
+ "content": "& Hein, 2020a), and Square attack (Andriushchenko et al., 2020). To show the performance of",
+ "type": "text"
+ }
+ ],
+ "index": 0
+ },
+ {
+ "bbox": [
+ 106,
+ 93,
+ 505,
+ 105
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 93,
+ 505,
+ 105
+ ],
+ "score": 1.0,
+ "content": "robust overfitting, we report the test accuracy on the best checkpoint that achieves the highest robust",
+ "type": "text"
+ }
+ ],
+ "index": 1
+ },
+ {
+ "bbox": [
+ 105,
+ 104,
+ 506,
+ 117
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 104,
+ 506,
+ 117
+ ],
+ "score": 1.0,
+ "content": "test accuracy under PGD-10 and the final checkpoint, as well as the difference between these two",
+ "type": "text"
+ }
+ ],
+ "index": 2
+ },
+ {
+ "bbox": [
+ 106,
+ 115,
+ 505,
+ 127
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 115,
+ 505,
+ 127
+ ],
+ "score": 1.0,
+ "content": "checkpoints. The results of PGD-AT, TRADES, and the combinations of them with our proposed",
+ "type": "text"
+ }
+ ],
+ "index": 3
+ },
+ {
+ "bbox": [
+ 105,
+ 126,
+ 506,
+ 139
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 126,
+ 218,
+ 139
+ ],
+ "score": 1.0,
+ "content": "approach (denoted as PGD-",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 218,
+ 126,
+ 250,
+ 137
+ ],
+ "score": 0.58,
+ "content": "\\mathbf { A T + T E }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 250,
+ 126,
+ 307,
+ 139
+ ],
+ "score": 1.0,
+ "content": "and TRADES",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 307,
+ 127,
+ 327,
+ 137
+ ],
+ "score": 0.48,
+ "content": "{ \\bf \\nabla } + { \\bf T } { \\bf E }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 327,
+ 126,
+ 506,
+ 139
+ ],
+ "score": 1.0,
+ "content": ") on the CIFAR-10, CIFAR-100, and SVHN",
+ "type": "text"
+ }
+ ],
+ "index": 4
+ },
+ {
+ "bbox": [
+ 106,
+ 138,
+ 228,
+ 148
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 138,
+ 228,
+ 148
+ ],
+ "score": 1.0,
+ "content": "datasets are shown in Table 1.",
+ "type": "text"
+ }
+ ],
+ "index": 5
+ }
+ ],
+ "index": 2.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 154,
+ 257,
+ 317
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 153,
+ 258,
+ 165
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 153,
+ 258,
+ 165
+ ],
+ "score": 1.0,
+ "content": "We can observe that the differences",
+ "type": "text"
+ }
+ ],
+ "index": 6
+ },
+ {
+ "bbox": [
+ 105,
+ 164,
+ 257,
+ 177
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 164,
+ 257,
+ 177
+ ],
+ "score": 1.0,
+ "content": "between best and final test accuracies",
+ "type": "text"
+ }
+ ],
+ "index": 7
+ },
+ {
+ "bbox": [
+ 106,
+ 176,
+ 258,
+ 187
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 176,
+ 258,
+ 187
+ ],
+ "score": 1.0,
+ "content": "of our method are reduced to around",
+ "type": "text"
+ }
+ ],
+ "index": 8
+ },
+ {
+ "bbox": [
+ 106,
+ 186,
+ 258,
+ 200
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 187,
+ 121,
+ 198
+ ],
+ "score": 0.84,
+ "content": "1 \\%",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 122,
+ 186,
+ 258,
+ 200
+ ],
+ "score": 1.0,
+ "content": ", while the accuracy gaps of PGD-",
+ "type": "text"
+ }
+ ],
+ "index": 9
+ },
+ {
+ "bbox": [
+ 106,
+ 197,
+ 257,
+ 210
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 197,
+ 257,
+ 210
+ ],
+ "score": 1.0,
+ "content": "AT and TRADES are much larger.",
+ "type": "text"
+ }
+ ],
+ "index": 10
+ },
+ {
+ "bbox": [
+ 105,
+ 208,
+ 258,
+ 222
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 208,
+ 258,
+ 222
+ ],
+ "score": 1.0,
+ "content": "It indicates that our method largely",
+ "type": "text"
+ }
+ ],
+ "index": 11
+ },
+ {
+ "bbox": [
+ 105,
+ 219,
+ 258,
+ 232
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 219,
+ 258,
+ 232
+ ],
+ "score": 1.0,
+ "content": "eliminates robust overfitting. Due to",
+ "type": "text"
+ }
+ ],
+ "index": 12
+ },
+ {
+ "bbox": [
+ 105,
+ 230,
+ 258,
+ 243
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 230,
+ 258,
+ 243
+ ],
+ "score": 1.0,
+ "content": "being less affected by robust over-",
+ "type": "text"
+ }
+ ],
+ "index": 13
+ },
+ {
+ "bbox": [
+ 105,
+ 242,
+ 258,
+ 253
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 242,
+ 258,
+ 253
+ ],
+ "score": 1.0,
+ "content": "fitting, our method achieves higher",
+ "type": "text"
+ }
+ ],
+ "index": 14
+ },
+ {
+ "bbox": [
+ 106,
+ 253,
+ 257,
+ 264
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 253,
+ 257,
+ 264
+ ],
+ "score": 1.0,
+ "content": "robust accuracies than the baselines.",
+ "type": "text"
+ }
+ ],
+ "index": 15
+ },
+ {
+ "bbox": [
+ 105,
+ 263,
+ 258,
+ 275
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 263,
+ 258,
+ 275
+ ],
+ "score": 1.0,
+ "content": "We also show the learning curves of",
+ "type": "text"
+ }
+ ],
+ "index": 16
+ },
+ {
+ "bbox": [
+ 106,
+ 275,
+ 258,
+ 286
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 275,
+ 258,
+ 286
+ ],
+ "score": 1.0,
+ "content": "these methods in Fig. 8. We consis-",
+ "type": "text"
+ }
+ ],
+ "index": 17
+ },
+ {
+ "bbox": [
+ 106,
+ 286,
+ 258,
+ 297
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 286,
+ 258,
+ 297
+ ],
+ "score": 1.0,
+ "content": "tently demonstrate the effectiveness",
+ "type": "text"
+ }
+ ],
+ "index": 18
+ },
+ {
+ "bbox": [
+ 106,
+ 297,
+ 258,
+ 308
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 297,
+ 258,
+ 308
+ ],
+ "score": 1.0,
+ "content": "of our method on different network",
+ "type": "text"
+ }
+ ],
+ "index": 19
+ },
+ {
+ "bbox": [
+ 105,
+ 308,
+ 258,
+ 318
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 308,
+ 258,
+ 318
+ ],
+ "score": 1.0,
+ "content": "architectures (including WRN-34-10",
+ "type": "text"
+ }
+ ],
+ "index": 20
+ }
+ ],
+ "index": 13
+ },
+ {
+ "type": "image",
+ "bbox": [
+ 267,
+ 159,
+ 502,
+ 253
+ ],
+ "blocks": [
+ {
+ "type": "image_body",
+ "bbox": [
+ 267,
+ 159,
+ 502,
+ 253
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 267,
+ 159,
+ 502,
+ 253
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 267,
+ 159,
+ 502,
+ 253
+ ],
+ "score": 0.969,
+ "type": "image",
+ "image_path": "60741a322624e7a344d48dd39e648d3bdbec90b00204725405a64e566413b5ab.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 24,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 267,
+ 159,
+ 502,
+ 172.42857142857142
+ ],
+ "spans": [],
+ "index": 21
+ },
+ {
+ "bbox": [
+ 267,
+ 172.42857142857142,
+ 502,
+ 185.85714285714283
+ ],
+ "spans": [],
+ "index": 22
+ },
+ {
+ "bbox": [
+ 267,
+ 185.85714285714283,
+ 502,
+ 199.28571428571425
+ ],
+ "spans": [],
+ "index": 23
+ },
+ {
+ "bbox": [
+ 267,
+ 199.28571428571425,
+ 502,
+ 212.71428571428567
+ ],
+ "spans": [],
+ "index": 24
+ },
+ {
+ "bbox": [
+ 267,
+ 212.71428571428567,
+ 502,
+ 226.14285714285708
+ ],
+ "spans": [],
+ "index": 25
+ },
+ {
+ "bbox": [
+ 267,
+ 226.14285714285708,
+ 502,
+ 239.5714285714285
+ ],
+ "spans": [],
+ "index": 26
+ },
+ {
+ "bbox": [
+ 267,
+ 239.5714285714285,
+ 502,
+ 252.99999999999991
+ ],
+ "spans": [],
+ "index": 27
+ }
+ ]
+ },
+ {
+ "type": "image_caption",
+ "bbox": [
+ 264,
+ 257,
+ 504,
+ 307
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 264,
+ 257,
+ 504,
+ 268
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 264,
+ 257,
+ 504,
+ 268
+ ],
+ "score": 1.0,
+ "content": "Figure 8: The natural and robust test accuracy curves (under PGD-",
+ "type": "text"
+ }
+ ],
+ "index": 28
+ },
+ {
+ "bbox": [
+ 265,
+ 267,
+ 505,
+ 279
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 265,
+ 267,
+ 505,
+ 279
+ ],
+ "score": 1.0,
+ "content": "10) of PGD-AT, TRADES, and their extensions by integrating the",
+ "type": "text"
+ }
+ ],
+ "index": 29
+ },
+ {
+ "bbox": [
+ 264,
+ 278,
+ 505,
+ 288
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 264,
+ 278,
+ 505,
+ 288
+ ],
+ "score": 1.0,
+ "content": "proposed TE approach. The models are trained on CIFAR-10 un-",
+ "type": "text"
+ }
+ ],
+ "index": 30
+ },
+ {
+ "bbox": [
+ 264,
+ 286,
+ 505,
+ 298
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 264,
+ 286,
+ 292,
+ 297
+ ],
+ "score": 1.0,
+ "content": "der the",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 293,
+ 288,
+ 306,
+ 297
+ ],
+ "score": 0.9,
+ "content": "\\ell _ { \\infty }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 306,
+ 286,
+ 347,
+ 297
+ ],
+ "score": 1.0,
+ "content": "norm with",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 348,
+ 287,
+ 390,
+ 298
+ ],
+ "score": 0.89,
+ "content": "\\epsilon = 8 / 2 5 5",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 391,
+ 286,
+ 505,
+ 297
+ ],
+ "score": 1.0,
+ "content": "based on the ResNet-18 archi-",
+ "type": "text"
+ }
+ ],
+ "index": 31
+ },
+ {
+ "bbox": [
+ 264,
+ 297,
+ 294,
+ 308
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 264,
+ 297,
+ 294,
+ 308
+ ],
+ "score": 1.0,
+ "content": "tecture.",
+ "type": "text"
+ }
+ ],
+ "index": 32
+ }
+ ],
+ "index": 30
+ }
+ ],
+ "index": 27.0
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 104,
+ 318,
+ 475,
+ 330
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 317,
+ 477,
+ 332
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 317,
+ 280,
+ 332
+ ],
+ "score": 1.0,
+ "content": "and VGG-16) and threat models (including",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 281,
+ 319,
+ 290,
+ 330
+ ],
+ "score": 0.84,
+ "content": "\\ell _ { 2 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 291,
+ 317,
+ 477,
+ 332
+ ],
+ "score": 1.0,
+ "content": "norm), which will be shown in Appendix C.2.",
+ "type": "text"
+ }
+ ],
+ "index": 33
+ }
+ ],
+ "index": 33
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 335,
+ 505,
+ 500
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 336,
+ 504,
+ 347
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 336,
+ 504,
+ 347
+ ],
+ "score": 1.0,
+ "content": "Discussion and comparison with related works. Our method is kind of similar to the label smooth-",
+ "type": "text"
+ }
+ ],
+ "index": 34
+ },
+ {
+ "bbox": [
+ 105,
+ 347,
+ 505,
+ 358
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 347,
+ 505,
+ 358
+ ],
+ "score": 1.0,
+ "content": "ing (LS) technique, which is studied in AT (Pang et al., 2021). Recent works have also introduced",
+ "type": "text"
+ }
+ ],
+ "index": 35
+ },
+ {
+ "bbox": [
+ 105,
+ 357,
+ 505,
+ 370
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 357,
+ 505,
+ 370
+ ],
+ "score": 1.0,
+ "content": "the smoothness in training labels and model weights (Chen et al., 2021; Huang et al., 2020), which",
+ "type": "text"
+ }
+ ],
+ "index": 36
+ },
+ {
+ "bbox": [
+ 106,
+ 369,
+ 505,
+ 380
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 369,
+ 505,
+ 380
+ ],
+ "score": 1.0,
+ "content": "can alleviate robust overfitting to some extent. The significant difference between our work and",
+ "type": "text"
+ }
+ ],
+ "index": 37
+ },
+ {
+ "bbox": [
+ 105,
+ 379,
+ 505,
+ 392
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 379,
+ 505,
+ 392
+ ],
+ "score": 1.0,
+ "content": "them is that we provide a reasonable explanation for robust overfitting—one-hot labels are noisy",
+ "type": "text"
+ }
+ ],
+ "index": 38
+ },
+ {
+ "bbox": [
+ 105,
+ 390,
+ 506,
+ 403
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 390,
+ 506,
+ 403
+ ],
+ "score": 1.0,
+ "content": "for AT, while previous methods did not give such an explanation and could be viewed as solutions",
+ "type": "text"
+ }
+ ],
+ "index": 39
+ },
+ {
+ "bbox": [
+ 105,
+ 401,
+ 505,
+ 414
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 401,
+ 505,
+ 414
+ ],
+ "score": 1.0,
+ "content": "to our identified problem. To empirically compare with these methods, we conduct experiments",
+ "type": "text"
+ }
+ ],
+ "index": 40
+ },
+ {
+ "bbox": [
+ 105,
+ 412,
+ 505,
+ 425
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 412,
+ 505,
+ 425
+ ],
+ "score": 1.0,
+ "content": "on CIFAR-10 with the ResNet-18 network. Under the PGD-AT framework, we compare with the",
+ "type": "text"
+ }
+ ],
+ "index": 41
+ },
+ {
+ "bbox": [
+ 105,
+ 422,
+ 506,
+ 437
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 422,
+ 506,
+ 437
+ ],
+ "score": 1.0,
+ "content": "baseline PGD-AT, PGD-AT+LS, self-adaptive training (SAT) (Huang et al., 2020), and knowledge",
+ "type": "text"
+ }
+ ],
+ "index": 42
+ },
+ {
+ "bbox": [
+ 105,
+ 433,
+ 505,
+ 447
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 433,
+ 487,
+ 447
+ ],
+ "score": 1.0,
+ "content": "distillation with stochastic weight averaging (KD-SWA) (Chen et al., 2021). We also adopt the",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 488,
+ 434,
+ 501,
+ 444
+ ],
+ "score": 0.61,
+ "content": "C o",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 501,
+ 433,
+ 505,
+ 447
+ ],
+ "score": 1.0,
+ "content": "-",
+ "type": "text"
+ }
+ ],
+ "index": 43
+ },
+ {
+ "bbox": [
+ 106,
+ 445,
+ 505,
+ 458
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 445,
+ 505,
+ 458
+ ],
+ "score": 1.0,
+ "content": "teaching approach (Han et al., 2018) adapted to PGD-AT, which jointly trains two models using the",
+ "type": "text"
+ }
+ ],
+ "index": 44
+ },
+ {
+ "bbox": [
+ 105,
+ 456,
+ 505,
+ 469
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 456,
+ 505,
+ 469
+ ],
+ "score": 1.0,
+ "content": "filtered samples given by each other. The results under the adopted attacks are presented in Table 2.",
+ "type": "text"
+ }
+ ],
+ "index": 45
+ },
+ {
+ "bbox": [
+ 105,
+ 466,
+ 506,
+ 480
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 466,
+ 506,
+ 480
+ ],
+ "score": 1.0,
+ "content": "Although various techniques can alleviate robust overfitting, our method achieves better robustness",
+ "type": "text"
+ }
+ ],
+ "index": 46
+ },
+ {
+ "bbox": [
+ 106,
+ 479,
+ 505,
+ 490
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 479,
+ 505,
+ 490
+ ],
+ "score": 1.0,
+ "content": "than the others, validating its effectiveness. For Co-teaching, though robust overfitting is alleviated,",
+ "type": "text"
+ }
+ ],
+ "index": 47
+ },
+ {
+ "bbox": [
+ 105,
+ 489,
+ 463,
+ 502
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 489,
+ 463,
+ 502
+ ],
+ "score": 1.0,
+ "content": "the performance is worse than our proposed method due to the reduction of training data.",
+ "type": "text"
+ }
+ ],
+ "index": 48
+ }
+ ],
+ "index": 41
+ },
+ {
+ "type": "table",
+ "bbox": [
+ 108,
+ 530,
+ 502,
+ 623
+ ],
+ "blocks": [
+ {
+ "type": "table_caption",
+ "bbox": [
+ 106,
+ 507,
+ 503,
+ 527
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 506,
+ 505,
+ 518
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 506,
+ 190,
+ 518
+ ],
+ "score": 1.0,
+ "content": "Table 2: Test accuracy",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 191,
+ 507,
+ 205,
+ 517
+ ],
+ "score": 0.78,
+ "content": "( \\% )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 205,
+ 506,
+ 451,
+ 518
+ ],
+ "score": 1.0,
+ "content": "of the proposed method and other methods on CIFAR-10 under the",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 451,
+ 507,
+ 464,
+ 517
+ ],
+ "score": 0.87,
+ "content": "\\ell _ { \\infty }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 464,
+ 506,
+ 505,
+ 518
+ ],
+ "score": 1.0,
+ "content": "norm with",
+ "type": "text"
+ }
+ ],
+ "index": 49
+ },
+ {
+ "bbox": [
+ 107,
+ 516,
+ 283,
+ 528
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 517,
+ 147,
+ 528
+ ],
+ "score": 0.9,
+ "content": "\\epsilon = 8 / 2 5 5",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 147,
+ 516,
+ 283,
+ 528
+ ],
+ "score": 1.0,
+ "content": "based on the ResNet-18 architecture.",
+ "type": "text"
+ }
+ ],
+ "index": 50
+ }
+ ],
+ "index": 49.5
+ },
+ {
+ "type": "table_body",
+ "bbox": [
+ 108,
+ 530,
+ 502,
+ 623
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 108,
+ 530,
+ 502,
+ 623
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 108,
+ 530,
+ 502,
+ 623
+ ],
+ "score": 0.982,
+ "html": "| Method | Natural AccuracyBest Final Diff | PGD-10Best Final Diff | PGD-1000Best Final Diff | C&W-1000Best Final Diff | AutoAttackBest Final Diff |
| PGD-ATPGD-AT+TE | 83.75 84.82 -1.0782.35 82.79 -0.44 | [52.64 44.92 7.7255.79 54.83 0.96 | [51.22 42.74 8.4854.65 53.30 1.35 | |50.11 43.63 7.4852.30 51.73 0.57 | |47.74 41.84 5.9050.59 49.62 0.97 |
| TRADESTRADES+TE | [81.19 82.48 -1.29|83.86 83.97 -0.11 | [53.32 50.25 3.0755.15 54.42 0.73 | [52.44 48.67 3.7753.74 53.03 0.71 | |49.88 48.14 1.74|50.77 50.63 0.14 | 49.03 46.80 2.2349.77 49.20 0.57 |
| (a) The evaluation results on CIFAR-10. |
| Method | Natural AccuracyBest FinalDiff | PGD-10Best Final Diff | PGD-1000Best Final Diff | C&W-1000Best Final Diff | AutoAttackBest Final Diff |
| PGD-ATPGD-AT+TE | 57.54 57.510.0356.45 57.12 -0.67 | 29.40 21.75 7.6531.74 30.24 1.50 | 28.54 20.63 7.9131.27 29.80 1.47 | 27.06 21.17 5.8928.27 27.36 0.91 | 24.72 19.34 5.3826.30 25.34 0.96 |
| TRADESTRADES+TE | 57.98 56.321.6659.35 58.72 0.63 | |29.93 27.70 2.23|31.09 30.12 0.97 | 29.51 26.93 2.58|230.54 29.45 1.09 | [25.46 24.42 1.04|26.61 25.94 0.67 | 24.6123.40 1.2125.27 24.55 0.72 |
| (b) The evaluation results on CIFAR-100. |
| Method | Natural AccuracyBest Final Diff | PGD-10Best Final Diff | PGD-1000Best Final Diff | C&W-1000Best Final Diff | AutoAttackBest Final Diff |
| PGD-ATPGD-AT+TE | 89.00 90.55 -1.5590.09 90.91 -0.82 | [54.51 46.97 7.5459.74 59.05 0.69 | [52.22 42.85 9.3757.7156.46 1.25 | 48.66 44.13 4.5354.5553.94 0.61 | 46.61 38.24 8.3751.44 50.61 0.83 |
| TRADESTRADES+TE | 90.88 91.30 -0.4289.01 88.52 0.49 | [59.50 57.04 2.4659.81 58.49 1.32 | [52.78 50.17 2.6158.24 56.66 1.58 | [52.76 50.53 2.2354.00 53.24 0.76 | 40.36 38.88 1.4851.45 50.16 1.29 |
"
+ },
+ {
+ "category_id": 1,
+ "poly": [
+ 299,
+ 1910,
+ 1402,
+ 1910,
+ 1402,
+ 2034,
+ 299,
+ 2034
+ ],
+ "score": 0.974
+ },
+ {
+ "category_id": 1,
+ "poly": [
+ 299,
+ 1512,
+ 1404,
+ 1512,
+ 1404,
+ 1606,
+ 299,
+ 1606
+ ],
+ "score": 0.973
+ },
+ {
+ "category_id": 1,
+ "poly": [
+ 296,
+ 1058,
+ 1401,
+ 1058,
+ 1401,
+ 1122,
+ 296,
+ 1122
+ ],
+ "score": 0.951
+ },
+ {
+ "category_id": 8,
+ "poly": [
+ 529,
+ 1135,
+ 1173,
+ 1135,
+ 1173,
+ 1220,
+ 529,
+ 1220
+ ],
+ "score": 0.948
+ },
+ {
+ "category_id": 6,
+ "poly": [
+ 297,
+ 193,
+ 1404,
+ 193,
+ 1404,
+ 276,
+ 297,
+ 276
+ ],
+ "score": 0.92
+ },
+ {
+ "category_id": 0,
+ "poly": [
+ 301,
+ 1453,
+ 1254,
+ 1453,
+ 1254,
+ 1486,
+ 301,
+ 1486
+ ],
+ "score": 0.916
+ },
+ {
+ "category_id": 2,
+ "poly": [
+ 299,
+ 75,
+ 815,
+ 75,
+ 815,
+ 105,
+ 299,
+ 105
+ ],
+ "score": 0.899
+ },
+ {
+ "category_id": 7,
+ "poly": [
+ 664,
+ 1010,
+ 1034,
+ 1010,
+ 1034,
+ 1038,
+ 664,
+ 1038
+ ],
+ "score": 0.886
+ },
+ {
+ "category_id": 9,
+ "poly": [
+ 1366,
+ 1161,
+ 1400,
+ 1161,
+ 1400,
+ 1191,
+ 1366,
+ 1191
+ ],
+ "score": 0.877
+ },
+ {
+ "category_id": 2,
+ "poly": [
+ 840,
+ 2088,
+ 858,
+ 2088,
+ 858,
+ 2112,
+ 840,
+ 2112
+ ],
+ "score": 0.791
+ },
+ {
+ "category_id": 14,
+ "poly": [
+ 525,
+ 1132,
+ 1174,
+ 1132,
+ 1174,
+ 1224,
+ 525,
+ 1224
+ ],
+ "score": 0.94,
+ "latex": "\\operatorname* { m i n } _ { \\pmb { \\theta } } \\sum _ { i = 1 } ^ { n } \\operatorname* { m a x } _ { \\mathbf { x } _ { i } ^ { \\prime } \\in S ( \\mathbf { x } _ { i } ) } \\left\\{ \\mathcal { L } ( f _ { \\pmb { \\theta } } ( \\mathbf { x } _ { i } ^ { \\prime } ) , y _ { i } ) + w \\cdot | | f _ { \\pmb { \\theta } } ( \\mathbf { x } _ { i } ^ { \\prime } ) - \\hat { \\mathbf { p } } _ { i } | | _ { 2 } ^ { 2 } \\right\\} ,"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 353,
+ 1742,
+ 481,
+ 1742,
+ 481,
+ 1775,
+ 353,
+ 1775
+ ],
+ "score": 0.93,
+ "latex": "\\alpha = 2 / 2 5 5"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 1066,
+ 1058,
+ 1395,
+ 1058,
+ 1395,
+ 1092,
+ 1066,
+ 1092
+ ],
+ "score": 0.92,
+ "latex": "\\mathbf { p } _ { i } \\eta \\cdot \\mathbf { p } _ { i } + ( 1 - \\eta ) \\cdot f _ { \\pmb { \\theta } } ( \\mathbf { x } _ { i } )"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 298,
+ 221,
+ 410,
+ 221,
+ 410,
+ 251,
+ 298,
+ 251
+ ],
+ "score": 0.91,
+ "latex": "\\epsilon = 8 / 2 5 5"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 474,
+ 1866,
+ 566,
+ 1866,
+ 566,
+ 1897,
+ 474,
+ 1897
+ ],
+ "score": 0.91,
+ "latex": "\\eta = 0 . 9"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 1201,
+ 1620,
+ 1330,
+ 1620,
+ 1330,
+ 1653,
+ 1201,
+ 1653
+ ],
+ "score": 0.9,
+ "latex": "\\epsilon = 8 / 2 5 5"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 616,
+ 1866,
+ 705,
+ 1866,
+ 705,
+ 1893,
+ 616,
+ 1893
+ ],
+ "score": 0.89,
+ "latex": "w = 3 0"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 338,
+ 1652,
+ 376,
+ 1652,
+ 376,
+ 1682,
+ 338,
+ 1682
+ ],
+ "score": 0.89,
+ "latex": "\\ell _ { \\infty }"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 373,
+ 1236,
+ 404,
+ 1236,
+ 404,
+ 1268,
+ 373,
+ 1268
+ ],
+ "score": 0.89,
+ "latex": "\\hat { { \\bf p } } _ { i }"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 675,
+ 1239,
+ 706,
+ 1239,
+ 706,
+ 1268,
+ 675,
+ 1268
+ ],
+ "score": 0.87,
+ "latex": "\\mathbf { p } _ { i }"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 553,
+ 1064,
+ 584,
+ 1064,
+ 584,
+ 1092,
+ 553,
+ 1092
+ ],
+ "score": 0.87,
+ "latex": "\\mathbf { p } _ { i }"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 1256,
+ 195,
+ 1290,
+ 195,
+ 1290,
+ 220,
+ 1256,
+ 220
+ ],
+ "score": 0.86,
+ "latex": "\\ell _ { \\infty }"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 492,
+ 1064,
+ 522,
+ 1064,
+ 522,
+ 1091,
+ 492,
+ 1091
+ ],
+ "score": 0.85,
+ "latex": "\\mathbf { x } _ { i }"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 372,
+ 1095,
+ 390,
+ 1095,
+ 390,
+ 1122,
+ 372,
+ 1122
+ ],
+ "score": 0.81,
+ "latex": "\\eta"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 1023,
+ 1240,
+ 1047,
+ 1240,
+ 1047,
+ 1262,
+ 1023,
+ 1262
+ ],
+ "score": 0.78,
+ "latex": "w"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 527,
+ 195,
+ 566,
+ 195,
+ 566,
+ 222,
+ 527,
+ 222
+ ],
+ "score": 0.72,
+ "latex": "( \\% )"
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 295.0,
+ 192.0,
+ 526.0,
+ 192.0,
+ 526.0,
+ 224.0,
+ 295.0,
+ 224.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 567.0,
+ 192.0,
+ 1255.0,
+ 192.0,
+ 1255.0,
+ 224.0,
+ 567.0,
+ 224.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1291.0,
+ 192.0,
+ 1402.0,
+ 192.0,
+ 1402.0,
+ 224.0,
+ 1291.0,
+ 224.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 294.0,
+ 220.0,
+ 297.0,
+ 220.0,
+ 297.0,
+ 252.0,
+ 294.0,
+ 252.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 411.0,
+ 220.0,
+ 1405.0,
+ 220.0,
+ 1405.0,
+ 252.0,
+ 411.0,
+ 252.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 295.0,
+ 249.0,
+ 691.0,
+ 249.0,
+ 691.0,
+ 278.0,
+ 295.0,
+ 278.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 292.0,
+ 1450.0,
+ 1259.0,
+ 1450.0,
+ 1259.0,
+ 1491.0,
+ 292.0,
+ 1491.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 295.0,
+ 73.0,
+ 817.0,
+ 73.0,
+ 817.0,
+ 108.0,
+ 295.0,
+ 108.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 662.0,
+ 1007.0,
+ 1035.0,
+ 1007.0,
+ 1035.0,
+ 1039.0,
+ 662.0,
+ 1039.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 836.0,
+ 2085.0,
+ 859.0,
+ 2085.0,
+ 859.0,
+ 2116.0,
+ 836.0,
+ 2116.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 296.0,
+ 1234.0,
+ 372.0,
+ 1234.0,
+ 372.0,
+ 1270.0,
+ 296.0,
+ 1270.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 405.0,
+ 1234.0,
+ 674.0,
+ 1234.0,
+ 674.0,
+ 1270.0,
+ 405.0,
+ 1270.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 707.0,
+ 1234.0,
+ 1022.0,
+ 1234.0,
+ 1022.0,
+ 1270.0,
+ 707.0,
+ 1270.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1048.0,
+ 1234.0,
+ 1404.0,
+ 1234.0,
+ 1404.0,
+ 1270.0,
+ 1048.0,
+ 1270.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 293.0,
+ 1264.0,
+ 1405.0,
+ 1264.0,
+ 1405.0,
+ 1299.0,
+ 293.0,
+ 1299.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 293.0,
+ 1295.0,
+ 1404.0,
+ 1295.0,
+ 1404.0,
+ 1331.0,
+ 293.0,
+ 1331.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 294.0,
+ 1324.0,
+ 1404.0,
+ 1324.0,
+ 1404.0,
+ 1360.0,
+ 294.0,
+ 1360.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 294.0,
+ 1356.0,
+ 1404.0,
+ 1356.0,
+ 1404.0,
+ 1392.0,
+ 294.0,
+ 1392.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 295.0,
+ 1385.0,
+ 1242.0,
+ 1385.0,
+ 1242.0,
+ 1420.0,
+ 295.0,
+ 1420.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 294.0,
+ 1618.0,
+ 1200.0,
+ 1618.0,
+ 1200.0,
+ 1658.0,
+ 294.0,
+ 1658.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1331.0,
+ 1618.0,
+ 1408.0,
+ 1618.0,
+ 1408.0,
+ 1658.0,
+ 1331.0,
+ 1658.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 294.0,
+ 1650.0,
+ 337.0,
+ 1650.0,
+ 337.0,
+ 1687.0,
+ 294.0,
+ 1687.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 377.0,
+ 1650.0,
+ 1405.0,
+ 1650.0,
+ 1405.0,
+ 1687.0,
+ 377.0,
+ 1687.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 293.0,
+ 1679.0,
+ 1404.0,
+ 1679.0,
+ 1404.0,
+ 1716.0,
+ 293.0,
+ 1716.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 292.0,
+ 1707.0,
+ 1405.0,
+ 1707.0,
+ 1405.0,
+ 1751.0,
+ 292.0,
+ 1751.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 292.0,
+ 1740.0,
+ 352.0,
+ 1740.0,
+ 352.0,
+ 1779.0,
+ 292.0,
+ 1779.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 482.0,
+ 1740.0,
+ 1405.0,
+ 1740.0,
+ 1405.0,
+ 1779.0,
+ 482.0,
+ 1779.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 294.0,
+ 1772.0,
+ 1407.0,
+ 1772.0,
+ 1407.0,
+ 1809.0,
+ 294.0,
+ 1809.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 294.0,
+ 1801.0,
+ 1404.0,
+ 1801.0,
+ 1404.0,
+ 1839.0,
+ 294.0,
+ 1839.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 294.0,
+ 1833.0,
+ 1407.0,
+ 1833.0,
+ 1407.0,
+ 1870.0,
+ 294.0,
+ 1870.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 293.0,
+ 1863.0,
+ 473.0,
+ 1863.0,
+ 473.0,
+ 1898.0,
+ 293.0,
+ 1898.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 567.0,
+ 1863.0,
+ 615.0,
+ 1863.0,
+ 615.0,
+ 1898.0,
+ 567.0,
+ 1898.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 706.0,
+ 1863.0,
+ 1319.0,
+ 1863.0,
+ 1319.0,
+ 1898.0,
+ 706.0,
+ 1898.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 293.0,
+ 1907.0,
+ 1406.0,
+ 1907.0,
+ 1406.0,
+ 1945.0,
+ 293.0,
+ 1945.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 295.0,
+ 1943.0,
+ 1406.0,
+ 1943.0,
+ 1406.0,
+ 1976.0,
+ 295.0,
+ 1976.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 293.0,
+ 1973.0,
+ 1404.0,
+ 1973.0,
+ 1404.0,
+ 2006.0,
+ 293.0,
+ 2006.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 293.0,
+ 2000.0,
+ 1406.0,
+ 2000.0,
+ 1406.0,
+ 2037.0,
+ 293.0,
+ 2037.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 294.0,
+ 1512.0,
+ 1404.0,
+ 1512.0,
+ 1404.0,
+ 1546.0,
+ 294.0,
+ 1546.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 294.0,
+ 1541.0,
+ 1405.0,
+ 1541.0,
+ 1405.0,
+ 1577.0,
+ 294.0,
+ 1577.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 293.0,
+ 1570.0,
+ 1349.0,
+ 1570.0,
+ 1349.0,
+ 1609.0,
+ 293.0,
+ 1609.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 294.0,
+ 1058.0,
+ 491.0,
+ 1058.0,
+ 491.0,
+ 1094.0,
+ 294.0,
+ 1094.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 523.0,
+ 1058.0,
+ 552.0,
+ 1058.0,
+ 552.0,
+ 1094.0,
+ 523.0,
+ 1094.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 585.0,
+ 1058.0,
+ 1065.0,
+ 1058.0,
+ 1065.0,
+ 1094.0,
+ 585.0,
+ 1094.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1396.0,
+ 1058.0,
+ 1403.0,
+ 1058.0,
+ 1403.0,
+ 1094.0,
+ 1396.0,
+ 1094.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 294.0,
+ 1085.0,
+ 371.0,
+ 1085.0,
+ 371.0,
+ 1126.0,
+ 294.0,
+ 1126.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 391.0,
+ 1085.0,
+ 1368.0,
+ 1085.0,
+ 1368.0,
+ 1126.0,
+ 391.0,
+ 1126.0
+ ],
+ "score": 1.0,
+ "text": ""
+ }
+ ],
+ "page_info": {
+ "page_no": 7,
+ "width": 1700,
+ "height": 2200
+ }
+ },
+ {
+ "layout_dets": [
+ {
+ "category_id": 1,
+ "poly": [
+ 297,
+ 933,
+ 1404,
+ 933,
+ 1404,
+ 1391,
+ 297,
+ 1391
+ ],
+ "score": 0.983
+ },
+ {
+ "category_id": 5,
+ "poly": [
+ 300,
+ 1474,
+ 1395,
+ 1474,
+ 1395,
+ 1731,
+ 300,
+ 1731
+ ],
+ "score": 0.982,
+ "html": "| Methods | Task Units | α a | △ε meV | εHOMO meV | εLUMO meV | μ D | Cv cal/mol K | G meV | H meV | R a | U meV | U meV | ZPVE meV |
| NMP(Gilmer et al., 2017)t | | .092 | 69 | 43 | 38 | .030 | .040 | 19 | 17 | .180 | 20 | 20 | 1.50 |
| SchNet (Schutt et al.,2017) | | .235 | 63 | 41 | 34 | .033 | .033 | 14 | 14 | .073 | 19 | 14 | 1.70 |
| Cormorant (Anderson et al., 2019)† | | .085 | 61 | 34 | 38 | .038 | .026 | 20 | 21 | .961 | 21 | 22 | 2.03 |
| LieConv (Finzi et al.,020)t | .084 | 49 | | 30 | 25 | .032 | .038 | 22 | 24 | .800 | 19 | 19 | 2.28 |
| DimeNet++ (Gasteiger et al., 2020a) | | .044 | 33 | 25 | 20 | .030 | .023 | 8 | 7 | .331 | 6 | 6 | 1.21 |
| TFN (Thomas et al., 2018)† | | .223 | 58 | 40 | 38 | .064 | .101 | - | - | - | - | - | - |
| SE(3)-Transformer(Fuchs etal.,2020)† | | .142 | 53 | 35 | 33 | .051 | .054 | - | - | - | - | - | - |
| EGNN (Satorras et al.,2021)† | .071 | 48 | | 29 | 25 | .029 | .031 | 12 | 12 | .106 | 12 | 11 | 1.55 |
| PaiNN (Schutt et al.,2021) | .045 | | 46 | 28 | 20 | .012 | .024 | 7.35 | 5.98 | .066 | 5.83 | 5.85 | 1.28 |
| TorchMD-NET(Tholke & Fabritiis,2022) | | .059 | 36 | 20 | 18 | .011 | .026 | 7.62 | 6.16 | .033 | 6.38 | 6.15 | 1.84 |
| SphereNet (Liu et al.,2022) | .046 | 32 | | 23 | 18 | .026 | .021 | 8 | 6 | .292 | 7 | 6 | 1.12 |
| SEGNN (Brandstetter et al.,2022)† | .060 | 42 | | 24 | 21 | .023 | .031 | 15 | 16 | .660 | 13 | 15 | 1.62 |
| EQGAT (Le et al.,2022) | .053 | 32 | | 20 | 16 | .011 | .024 | 23 | 24 | .382 | 25 | 25 | 2.00 |
| Equiformer | | .046 | 30 | 15 | 14 | .011 | .023 | 7.63 | 6.63 | .251 | 6.74 | 6.59 | 1.26 |
+
+| Hyper-parameters | Value or description |
| Optimizer | AdamW |
| Learning rate scheduling Warmup epochs | Cosine learning rate with linear warmup 5 |
| Maximum learning rate | 1.5 × 10-4,5 × 10-4 |
| Batch size | 64,128 |
| |
| Number of epochs | 300,600 |
| Weight decay | 0,5×10-3 |
| Dropout rate | 0.0,0.1, 0.2 |
| Cutoff radius (A) | 5 |
| Number of radial bases | 128 for Gaussian radial basis,8 for radial bessel basis |
| Hidden sizes of radial functions | 64 |
| Number of hidden layers in radial functions | 2 |
| Number of Transformer blocks Embedding dimension dembed | Equiformer 6 [(128,0),(64,1),(32,2)] |
| Spherical harmonics embedding dimension dsh Numberof attention heads h Attention head dimension dhead | [(1,0),(1,1),(1,2)] 4 [(32,0),(16,1),(8,2)] |
| Hidden dimension in feed forward networks dffn Output feature dimension d feature | (384,0),(192,1),(96,2)] [(512,0)] |
| E(3)-Equiformer |
| |
| |
| |
| 6 |
| Number of Transformer blocks | |
| Embedding dimension dembed | |
| [(128,0,e),(32,0,0),(32,1,e),(32,1,0),(16,2,e),(16,2,0)] |
| Spherical harmonics embedding dimension dsh | [(1,0,e),(1,1,0),(1,2,e)] |
| Number of attention heads h | 4 |
| |
| Attention head dimension dhead | [(32,0,e),(8,0,0),(8,1,e),(8,1,0),(4,2,e),(4,2,0)] |
| Hidden dimension in feed forward networks df fn | |
| (384,0,e),(96,0,0),(96,1,e),(96,1,0),(48,2,e),(48,2,0)] |
| |
| Output feature dimension dfeature | [(512,0,e)] |
| |
| |
+
+Table 8: Hyper-parameters for QM9 dataset. We denote $C _ { L }$ type- $L$ vectors as $( C _ { L } , L )$ and $C _ { ( L , p ) }$ type- $( L , p )$ vectors as $( C _ { ( L , p ) } , L , p )$ and use brackets to represent concatenations of vectors.
+
+Second, compared to linear messages, using non-linear messages increases the number of tensor product operations from 1 to 2. Since tensor products are compute-intensive, this inevitably increases training and inference time.
+
+Please refer to Sec. D.1 and Sec. F.2 for the exact numbers of training time on QM9 and OC20.
+
+# D DETAILS OF EXPERIMENTS ON QM9
+
+# D.1 TRAINING DETAILS
+
+We use the same data partition as TorchMD-NET (Thölke & Fabritiis, 2022). For the task of $U$ , $U _ { 0 } , G$ , and $H$ , where single-atom reference values are available, we subtract those reference values from ground truth. For other tasks, we normalize ground truth by subtracting mean and dividing by standard deviation.
+
+We train Equiformer with 6 blocks with $L _ { m a x } = 2$ . We choose Gaussian radial basis (Schütt et al., 2017; Shuaibi et al., 2021; Klicpera et al., 2021; Shi et al., 2022) for the first six tasks in Table 1 and radial Bessel basis (Gasteiger et al., 2020b;a) for the others. We apply dropout (Srivastava et al., 2014) to attention weights $a _ { i j }$ . The dropout rate is 0.0 for the tasks of $G$ , $H$ , $U$ and $U _ { 0 }$ , is 0.1 for the task of $R ^ { 2 }$ and is 0.2 for others. Since the tasks of $G$ , $H$ , $U$ , and $U _ { 0 }$ require longer training, we use slightly different hyper-parameters. The number of epochs is 600 for the tasks of $G$ $\ d s _ { r } , H , U$ , and $U _ { 0 }$ and is 300 for others. The learning rate is $1 . 5 \times 1 0 ^ { - 4 }$ for the tasks of $G$ , $H , U$ , and $U _ { 0 }$ and is $5 \times 1 0 ^ { - 4 }$ for others. The batch size is 64 for the tasks of $G$ ${ \mathrm { ? } } , H , U ,$ and $U _ { 0 }$ and is 128 for others. The weight decay is 0 for the tasks of $G$ , $H$ , $U$ , and $U _ { 0 }$ and is $5 \times 1 0 ^ { - 3 }$ for others. Table 8 summarizes the hyper-parameters for the QM9 dataset. The detailed description of architectural hyper-parameters can be found in Sec. C.2.
+
+We use one A6000 GPU with 48GB to train each model and summarize the computational cost of training for one epoch as follows. Training $E ( 3 )$ -Equiformer in Table 9 for one epoch takes about 16.3 minutes. The time of training Equiformer, Equiformer with linear messages (indicated by Index 2 in Table 6), and Equiformer with linear messages and dot product attention (indicated by Index 3 in Table 6) for one epoch is 12.1 minutes, 7.2 minutes and 7.8 minutes, respectively.
+
+# D.2 COMPARISON BETWEEN $S E ( 3 )$ AND $E ( 3 )$ EQUIVARIANCE
+
+We train two versions of Equiformers, one with $S E ( 3 )$ -equivariant features denoted as “Equiformer” and the other with $E ( 3 )$ -equivariant features denoted as “ $E ( 3 )$ -Equiformer”, and we compare them in Table 9. As for Table 1, we compare “Equiformer” with other works since most of them do not include equivariance to inversion.
+
+Table 9: Ablation study of $S E ( 3 ) / E ( 3 )$ equivariance on QM9 testing set. “Equiformer” operates on $S E ( 3 )$ -equivariant features while “ $E ( 3 )$ -Equiformer” uses $E ( 3 )$ -equivariant features. Including inversion achieves similar performance.
+
+| Hyper-parameters | Value or description |
| Optimizer | AdamW |
| Learning rate scheduling | Cosine learning rate with linear warmup |
| Warmup epochs | 10 1 × 10-4,2 × 10-4,5× 10-4 |
| Maximum learning rate Batch size | 5,8 |
| Number of epochs | 1500,2000 |
| 1×10-6 |
| Weight decay Dropout rate | 0.0 |
| |
| Weight for energy loss Weight for force loss | 1 80 |
| |
| Cutoff radius (A) | 5 |
| Number of radial bases | 32 |
| Hidden sizes of radial functions | 64 2 |
| Number of hidden layers in radial functions Equiformer (Lmax = 2) |
| Embedding dimension dembed Spherical harmonics embedding dimension d sh Number of attention heads h 4 Attention head dimension dhead Hidden dimension in feed forward networks d f fn | [(128,0),(64,1),(32,2)] [(1,0),(1,1),(1,2)] [(32,0),(16,1),(8,2)] [(384,0),(192,1),(96,2)] |
| Output feature dimension dfeature Equiformer (Lmax = 3) |
| [(512,0)] |
| |
| Number of Transformer blocks | 6 |
|
| Embedding dimension dembed | [(128,0),(64,1),(64,2),(32,3)] |
| Spherical harmonics embedding dimension d sh | [(1,0),(1,1),(1,2),(1,3)] |
| Number of attention heads h | 4 |
| |
| Attention head dimension dhead | [(32,0),(16,1),(16,2),(8,3)] |
| Hidden dimension in feed forward networks dffn | (384,0),(192,1),(192,2),(96,3)] |
+
+We use one A5000 GPU with 24GB to train different models for each molecule. Training Equiformer with $L _ { m a x } = 2$ takes about 15.4 hours, and training Equiformer with $L _ { m a x } = 3$ takes about 54.4 hours.
+
+# E.2 ADDITIONAL COMPARISON TO TORCHMD-NET
+
+Since TorchMD-NET (Thölke & Fabritiis, 2022) is also an equivariant Transformer but uses dot product attention instead of the proposed equivariant graph attention, we provide additional comparisons in Table 12. For each molecule, we adjust the ratio of the weight for force loss to the weight for energy loss so that Equiformer with $L _ { m a x } = 2$ can achieve lower MAE for both energy and forces.
+
+# E.3 COMPARISON OF TRAINING TIME AND NUMBERS OF PARAMETERS
+
+We compare training time and number of parameters between NequIP (Batzner et al., 2022) and Equiformer and summarize the results in Table 13. Since NequIP does not report the number of epochs, we compare the time spent for each epoch and note that NequIP is trained for more than 1000 epochs. Equiformer with $L _ { m a x } = 2$ is faster than NequIP with $L _ { m a x } = 3$ since smaller $L _ { m a x }$ is used. Moreover, Equiformer with $L _ { m a x } = 2$ achieves overall better results as we use equivariant graph attention instead of linear convolution used by NequIP. When increasing $L _ { m a x }$ from 2 to 3,
+
+Table 12: Additional comparison to TorchMD-Net (Thölke & Fabritiis, 2022) on MD17 dataset. Energy and force are in units of meV and $\mathrm { m e V } / \mathring { \mathrm { A } }$ .
+
+| Hyper-parameters | Value or description |
| Optimizer | AdamW |
| Learning rate scheduling | Cosine learning rate with linear warmup |
| Warmup epochs | 2 |
| Maximum learning rate | 2×10-4 |
| Batch size | 32 |
| Number of epochs | 20 |
| Weight decay | 1×10-3 |
| Dropout rate | 0.2 |
| Cutoff radius (A) | 5 |
| Number of radial basis | 128 |
| Hidden size of radial function | 64 |
| Numberofhiddenlayers in radial function | 2 |
| Equiformer |
| Numberof Transformer blocks Embedding dimension dembed | 6 |
| Spherical harmonics embedding dimension dsh | [(256,0),(128,1)] [(1,0),(1,1)] |
| Number of attention heads h | 8 |
| Attention head dimension dhead | [(32,0),(16,1)] |
| Hidden dimension in feed forward networks d f fn | (768,0),(384,1)] |
| Output feature dimension d feature | [(512,0)] |
| |
| E(3)-Equiformer |
| Number of Transformer blocks Embedding dimension dembed | 6 [(256,0,e),(64,0,0),(64,1,e),(64,1,0)] |
| Spherical harmonics embedding dimension dsh | [(1,0,e),(1,1,0)] |
| Number of attention heads h | 8 |
| Attention head dimension dhead | [(32,0,e),(8,0,0),(8,1,e),(8,1,0)] |
| Hidden dimension in feed forward networks d ffn | [(768,0,e),(192,0,0),(192,1,e),(192,1,0)] |
| Output feature dimension d feature | [(512,0,e)] |
+
+| Methods | Task Units | α a | △ε meV | εHOMO meV | εLUMO meV | μ D | Cv cal/mol K | G meV | H meV | R a | U meV | U meV | ZPVE meV |
| NMP(Gilmer et al., 2017)t | | .092 | 69 | 43 | 38 | .030 | .040 | 19 | 17 | .180 | 20 | 20 | 1.50 |
| SchNet (Schutt et al.,2017) | | .235 | 63 | 41 | 34 | .033 | .033 | 14 | 14 | .073 | 19 | 14 | 1.70 |
| Cormorant (Anderson et al., 2019)† | | .085 | 61 | 34 | 38 | .038 | .026 | 20 | 21 | .961 | 21 | 22 | 2.03 |
| LieConv (Finzi et al.,020)t | .084 | 49 | | 30 | 25 | .032 | .038 | 22 | 24 | .800 | 19 | 19 | 2.28 |
| DimeNet++ (Gasteiger et al., 2020a) | | .044 | 33 | 25 | 20 | .030 | .023 | 8 | 7 | .331 | 6 | 6 | 1.21 |
| TFN (Thomas et al., 2018)† | | .223 | 58 | 40 | 38 | .064 | .101 | - | - | - | - | - | - |
| SE(3)-Transformer(Fuchs etal.,2020)† | | .142 | 53 | 35 | 33 | .051 | .054 | - | - | - | - | - | - |
| EGNN (Satorras et al.,2021)† | .071 | 48 | | 29 | 25 | .029 | .031 | 12 | 12 | .106 | 12 | 11 | 1.55 |
| PaiNN (Schutt et al.,2021) | .045 | | 46 | 28 | 20 | .012 | .024 | 7.35 | 5.98 | .066 | 5.83 | 5.85 | 1.28 |
| TorchMD-NET(Tholke & Fabritiis,2022) | | .059 | 36 | 20 | 18 | .011 | .026 | 7.62 | 6.16 | .033 | 6.38 | 6.15 | 1.84 |
| SphereNet (Liu et al.,2022) | .046 | 32 | | 23 | 18 | .026 | .021 | 8 | 6 | .292 | 7 | 6 | 1.12 |
| SEGNN (Brandstetter et al.,2022)† | .060 | 42 | | 24 | 21 | .023 | .031 | 15 | 16 | .660 | 13 | 15 | 1.62 |
| EQGAT (Le et al.,2022) | .053 | 32 | | 20 | 16 | .011 | .024 | 23 | 24 | .382 | 25 | 25 | 2.00 |
| Equiformer | | .046 | 30 | 15 | 14 | .011 | .023 | 7.63 | 6.63 | .251 | 6.74 | 6.59 | 1.26 |
",
+ "bbox": [
+ 187,
+ 101,
+ 805,
+ 253
+ ],
+ "page_idx": 6
+ },
+ {
+ "type": "table",
+ "img_path": "images/f6bd800e064be95eb07dc9b9156a5764ddf22a6190a4b9e97cbe5cc5d1bb94bb.jpg",
+ "table_caption": [],
+ "table_footnote": [
+ "Table 2: MAE results on MD17 testing set. Energy and force are in units of meV and meV/Å. "
+ ],
+ "table_body": "| Hyper-parameters | Value or description |
| Optimizer | AdamW |
| Learning rate scheduling Warmup epochs | Cosine learning rate with linear warmup 5 |
| Maximum learning rate | 1.5 × 10-4,5 × 10-4 |
| Batch size | 64,128 |
| |
| Number of epochs | 300,600 |
| Weight decay | 0,5×10-3 |
| Dropout rate | 0.0,0.1, 0.2 |
| Cutoff radius (A) | 5 |
| Number of radial bases | 128 for Gaussian radial basis,8 for radial bessel basis |
| Hidden sizes of radial functions | 64 |
| Number of hidden layers in radial functions | 2 |
| Number of Transformer blocks Embedding dimension dembed | Equiformer 6 [(128,0),(64,1),(32,2)] |
| Spherical harmonics embedding dimension dsh Numberof attention heads h Attention head dimension dhead | [(1,0),(1,1),(1,2)] 4 [(32,0),(16,1),(8,2)] |
| Hidden dimension in feed forward networks dffn Output feature dimension d feature | (384,0),(192,1),(96,2)] [(512,0)] |
| E(3)-Equiformer |
| |
| |
| |
| 6 |
| Number of Transformer blocks | |
| Embedding dimension dembed | |
| [(128,0,e),(32,0,0),(32,1,e),(32,1,0),(16,2,e),(16,2,0)] |
| Spherical harmonics embedding dimension dsh | [(1,0,e),(1,1,0),(1,2,e)] |
| Number of attention heads h | 4 |
| |
| Attention head dimension dhead | [(32,0,e),(8,0,0),(8,1,e),(8,1,0),(4,2,e),(4,2,0)] |
| Hidden dimension in feed forward networks df fn | |
| (384,0,e),(96,0,0),(96,1,e),(96,1,0),(48,2,e),(48,2,0)] |
| |
| Output feature dimension dfeature | [(512,0,e)] |
| |
| |
",
+ "bbox": [
+ 184,
+ 102,
+ 810,
+ 478
+ ],
+ "page_idx": 24
+ },
+ {
+ "type": "text",
+ "text": "Second, compared to linear messages, using non-linear messages increases the number of tensor product operations from 1 to 2. Since tensor products are compute-intensive, this inevitably increases training and inference time. ",
+ "bbox": [
+ 174,
+ 546,
+ 825,
+ 588
+ ],
+ "page_idx": 24
+ },
+ {
+ "type": "text",
+ "text": "Please refer to Sec. D.1 and Sec. F.2 for the exact numbers of training time on QM9 and OC20. ",
+ "bbox": [
+ 174,
+ 594,
+ 795,
+ 609
+ ],
+ "page_idx": 24
+ },
+ {
+ "type": "text",
+ "text": "D DETAILS OF EXPERIMENTS ON QM9 ",
+ "text_level": 1,
+ "bbox": [
+ 174,
+ 638,
+ 514,
+ 656
+ ],
+ "page_idx": 24
+ },
+ {
+ "type": "text",
+ "text": "D.1 TRAINING DETAILS ",
+ "text_level": 1,
+ "bbox": [
+ 176,
+ 676,
+ 354,
+ 691
+ ],
+ "page_idx": 24
+ },
+ {
+ "type": "text",
+ "text": "We use the same data partition as TorchMD-NET (Thölke & Fabritiis, 2022). For the task of $U$ , $U _ { 0 } , G$ , and $H$ , where single-atom reference values are available, we subtract those reference values from ground truth. For other tasks, we normalize ground truth by subtracting mean and dividing by standard deviation. ",
+ "bbox": [
+ 174,
+ 707,
+ 825,
+ 762
+ ],
+ "page_idx": 24
+ },
+ {
+ "type": "text",
+ "text": "We train Equiformer with 6 blocks with $L _ { m a x } = 2$ . We choose Gaussian radial basis (Schütt et al., 2017; Shuaibi et al., 2021; Klicpera et al., 2021; Shi et al., 2022) for the first six tasks in Table 1 and radial Bessel basis (Gasteiger et al., 2020b;a) for the others. We apply dropout (Srivastava et al., 2014) to attention weights $a _ { i j }$ . The dropout rate is 0.0 for the tasks of $G$ , $H$ , $U$ and $U _ { 0 }$ , is 0.1 for the task of $R ^ { 2 }$ and is 0.2 for others. Since the tasks of $G$ , $H$ , $U$ , and $U _ { 0 }$ require longer training, we use slightly different hyper-parameters. The number of epochs is 600 for the tasks of $G$ $\\ d s _ { r } , H , U$ , and $U _ { 0 }$ and is 300 for others. The learning rate is $1 . 5 \\times 1 0 ^ { - 4 }$ for the tasks of $G$ , $H , U$ , and $U _ { 0 }$ and is $5 \\times 1 0 ^ { - 4 }$ for others. The batch size is 64 for the tasks of $G$ ${ \\mathrm { ? } } , H , U ,$ and $U _ { 0 }$ and is 128 for others. The weight decay is 0 for the tasks of $G$ , $H$ , $U$ , and $U _ { 0 }$ and is $5 \\times 1 0 ^ { - 3 }$ for others. Table 8 summarizes the hyper-parameters for the QM9 dataset. The detailed description of architectural hyper-parameters can be found in Sec. C.2. ",
+ "bbox": [
+ 173,
+ 768,
+ 826,
+ 922
+ ],
+ "page_idx": 24
+ },
+ {
+ "type": "text",
+ "text": "We use one A6000 GPU with 48GB to train each model and summarize the computational cost of training for one epoch as follows. Training $E ( 3 )$ -Equiformer in Table 9 for one epoch takes about 16.3 minutes. The time of training Equiformer, Equiformer with linear messages (indicated by Index 2 in Table 6), and Equiformer with linear messages and dot product attention (indicated by Index 3 in Table 6) for one epoch is 12.1 minutes, 7.2 minutes and 7.8 minutes, respectively. ",
+ "bbox": [
+ 174,
+ 103,
+ 825,
+ 174
+ ],
+ "page_idx": 25
+ },
+ {
+ "type": "text",
+ "text": "D.2 COMPARISON BETWEEN $S E ( 3 )$ AND $E ( 3 )$ EQUIVARIANCE ",
+ "text_level": 1,
+ "bbox": [
+ 174,
+ 194,
+ 619,
+ 209
+ ],
+ "page_idx": 25
+ },
+ {
+ "type": "text",
+ "text": "We train two versions of Equiformers, one with $S E ( 3 )$ -equivariant features denoted as “Equiformer” and the other with $E ( 3 )$ -equivariant features denoted as “ $E ( 3 )$ -Equiformer”, and we compare them in Table 9. As for Table 1, we compare “Equiformer” with other works since most of them do not include equivariance to inversion. ",
+ "bbox": [
+ 174,
+ 222,
+ 826,
+ 279
+ ],
+ "page_idx": 25
+ },
+ {
+ "type": "table",
+ "img_path": "images/352d82f045c85929ca3479948aee8ea6e472d8e5e1f9fd9c246c9816540f85da.jpg",
+ "table_caption": [
+ "Table 9: Ablation study of $S E ( 3 ) / E ( 3 )$ equivariance on QM9 testing set. “Equiformer” operates on $S E ( 3 )$ -equivariant features while “ $E ( 3 )$ -Equiformer” uses $E ( 3 )$ -equivariant features. Including inversion achieves similar performance. "
+ ],
+ "table_footnote": [],
+ "table_body": "| Hyper-parameters | Value or description |
| Optimizer | AdamW |
| Learning rate scheduling | Cosine learning rate with linear warmup |
| Warmup epochs | 10 1 × 10-4,2 × 10-4,5× 10-4 |
| Maximum learning rate Batch size | 5,8 |
| Number of epochs | 1500,2000 |
| 1×10-6 |
| Weight decay Dropout rate | 0.0 |
| |
| Weight for energy loss Weight for force loss | 1 80 |
| |
| Cutoff radius (A) | 5 |
| Number of radial bases | 32 |
| Hidden sizes of radial functions | 64 2 |
| Number of hidden layers in radial functions Equiformer (Lmax = 2) |
| Embedding dimension dembed Spherical harmonics embedding dimension d sh Number of attention heads h 4 Attention head dimension dhead Hidden dimension in feed forward networks d f fn | [(128,0),(64,1),(32,2)] [(1,0),(1,1),(1,2)] [(32,0),(16,1),(8,2)] [(384,0),(192,1),(96,2)] |
| Output feature dimension dfeature Equiformer (Lmax = 3) |
| [(512,0)] |
| |
| Number of Transformer blocks | 6 |
|
| Embedding dimension dembed | [(128,0),(64,1),(64,2),(32,3)] |
| Spherical harmonics embedding dimension d sh | [(1,0),(1,1),(1,2),(1,3)] |
| Number of attention heads h | 4 |
| |
| Attention head dimension dhead | [(32,0),(16,1),(16,2),(8,3)] |
| Hidden dimension in feed forward networks dffn | (384,0),(192,1),(192,2),(96,3)] |
",
+ "bbox": [
+ 243,
+ 184,
+ 750,
+ 588
+ ],
+ "page_idx": 26
+ },
+ {
+ "type": "text",
+ "text": "We use one A5000 GPU with 24GB to train different models for each molecule. Training Equiformer with $L _ { m a x } = 2$ takes about 15.4 hours, and training Equiformer with $L _ { m a x } = 3$ takes about 54.4 hours. ",
+ "bbox": [
+ 173,
+ 648,
+ 825,
+ 690
+ ],
+ "page_idx": 26
+ },
+ {
+ "type": "text",
+ "text": "E.2 ADDITIONAL COMPARISON TO TORCHMD-NET ",
+ "text_level": 1,
+ "bbox": [
+ 174,
+ 710,
+ 550,
+ 724
+ ],
+ "page_idx": 26
+ },
+ {
+ "type": "text",
+ "text": "Since TorchMD-NET (Thölke & Fabritiis, 2022) is also an equivariant Transformer but uses dot product attention instead of the proposed equivariant graph attention, we provide additional comparisons in Table 12. For each molecule, we adjust the ratio of the weight for force loss to the weight for energy loss so that Equiformer with $L _ { m a x } = 2$ can achieve lower MAE for both energy and forces. ",
+ "bbox": [
+ 174,
+ 737,
+ 825,
+ 794
+ ],
+ "page_idx": 26
+ },
+ {
+ "type": "text",
+ "text": "E.3 COMPARISON OF TRAINING TIME AND NUMBERS OF PARAMETERS ",
+ "text_level": 1,
+ "bbox": [
+ 176,
+ 813,
+ 679,
+ 828
+ ],
+ "page_idx": 26
+ },
+ {
+ "type": "text",
+ "text": "We compare training time and number of parameters between NequIP (Batzner et al., 2022) and Equiformer and summarize the results in Table 13. Since NequIP does not report the number of epochs, we compare the time spent for each epoch and note that NequIP is trained for more than 1000 epochs. Equiformer with $L _ { m a x } = 2$ is faster than NequIP with $L _ { m a x } = 3$ since smaller $L _ { m a x }$ is used. Moreover, Equiformer with $L _ { m a x } = 2$ achieves overall better results as we use equivariant graph attention instead of linear convolution used by NequIP. When increasing $L _ { m a x }$ from 2 to 3, ",
+ "bbox": [
+ 174,
+ 840,
+ 825,
+ 924
+ ],
+ "page_idx": 26
+ },
+ {
+ "type": "table",
+ "img_path": "images/8cdc9e5e5aeb8f9a9d0d0fda9a15496439050f4e589c4ce359f31e05bde83777.jpg",
+ "table_caption": [
+ "Table 12: Additional comparison to TorchMD-Net (Thölke & Fabritiis, 2022) on MD17 dataset. Energy and force are in units of meV and $\\mathrm { m e V } / \\mathring { \\mathrm { A } }$ . "
+ ],
+ "table_footnote": [],
+ "table_body": "| Hyper-parameters | Value or description |
| Optimizer | AdamW |
| Learning rate scheduling | Cosine learning rate with linear warmup |
| Warmup epochs | 2 |
| Maximum learning rate | 2×10-4 |
| Batch size | 32 |
| Number of epochs | 20 |
| Weight decay | 1×10-3 |
| Dropout rate | 0.2 |
| Cutoff radius (A) | 5 |
| Number of radial basis | 128 |
| Hidden size of radial function | 64 |
| Numberofhiddenlayers in radial function | 2 |
| Equiformer |
| Numberof Transformer blocks Embedding dimension dembed | 6 |
| Spherical harmonics embedding dimension dsh | [(256,0),(128,1)] [(1,0),(1,1)] |
| Number of attention heads h | 8 |
| Attention head dimension dhead | [(32,0),(16,1)] |
| Hidden dimension in feed forward networks d f fn | (768,0),(384,1)] |
| Output feature dimension d feature | [(512,0)] |
| |
| E(3)-Equiformer |
| Number of Transformer blocks Embedding dimension dembed | 6 [(256,0,e),(64,0,0),(64,1,e),(64,1,0)] |
| Spherical harmonics embedding dimension dsh | [(1,0,e),(1,1,0)] |
| Number of attention heads h | 8 |
| Attention head dimension dhead | [(32,0,e),(8,0,0),(8,1,e),(8,1,0)] |
| Hidden dimension in feed forward networks d ffn | [(768,0,e),(192,0,0),(192,1,e),(192,1,0)] |
| Output feature dimension d feature | [(512,0,e)] |
",
+ "bbox": [
+ 230,
+ 102,
+ 764,
+ 478
+ ],
+ "page_idx": 28
+ },
+ {
+ "type": "table",
+ "img_path": "images/24e3e0655d9e071e6c2e2ebc61db502863d45640b4a3b5a62b3b751799d29ad9.jpg",
+ "table_caption": [],
+ "table_footnote": [
+ "Table 15: Results on OC20 IS2RE validation set. $\\dagger$ denotes results reported by Liu et al. (2022). "
+ ],
+ "table_body": "| Methods | Task Units | α a | △ε meV | εHOMO meV | εLUMO meV | μ D | Cv cal/mol K | G meV | H meV | R a | U meV | U meV | ZPVE meV |
| NMP(Gilmer et al., 2017)t | | .092 | 69 | 43 | 38 | .030 | .040 | 19 | 17 | .180 | 20 | 20 | 1.50 |
| SchNet (Schutt et al.,2017) | | .235 | 63 | 41 | 34 | .033 | .033 | 14 | 14 | .073 | 19 | 14 | 1.70 |
| Cormorant (Anderson et al., 2019)† | | .085 | 61 | 34 | 38 | .038 | .026 | 20 | 21 | .961 | 21 | 22 | 2.03 |
| LieConv (Finzi et al.,020)t | .084 | 49 | | 30 | 25 | .032 | .038 | 22 | 24 | .800 | 19 | 19 | 2.28 |
| DimeNet++ (Gasteiger et al., 2020a) | | .044 | 33 | 25 | 20 | .030 | .023 | 8 | 7 | .331 | 6 | 6 | 1.21 |
| TFN (Thomas et al., 2018)† | | .223 | 58 | 40 | 38 | .064 | .101 | - | - | - | - | - | - |
| SE(3)-Transformer(Fuchs etal.,2020)† | | .142 | 53 | 35 | 33 | .051 | .054 | - | - | - | - | - | - |
| EGNN (Satorras et al.,2021)† | .071 | 48 | | 29 | 25 | .029 | .031 | 12 | 12 | .106 | 12 | 11 | 1.55 |
| PaiNN (Schutt et al.,2021) | .045 | | 46 | 28 | 20 | .012 | .024 | 7.35 | 5.98 | .066 | 5.83 | 5.85 | 1.28 |
| TorchMD-NET(Tholke & Fabritiis,2022) | | .059 | 36 | 20 | 18 | .011 | .026 | 7.62 | 6.16 | .033 | 6.38 | 6.15 | 1.84 |
| SphereNet (Liu et al.,2022) | .046 | 32 | | 23 | 18 | .026 | .021 | 8 | 6 | .292 | 7 | 6 | 1.12 |
| SEGNN (Brandstetter et al.,2022)† | .060 | 42 | | 24 | 21 | .023 | .031 | 15 | 16 | .660 | 13 | 15 | 1.62 |
| EQGAT (Le et al.,2022) | .053 | 32 | | 20 | 16 | .011 | .024 | 23 | 24 | .382 | 25 | 25 | 2.00 |
| Equiformer | | .046 | 30 | 15 | 14 | .011 | .023 | 7.63 | 6.63 | .251 | 6.74 | 6.59 | 1.26 |
",
+ "type": "table",
+ "image_path": "99449f7c420a58ee605341ce298638673593e231af3640cfcdd9d40c37f10b68.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 1,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 115,
+ 80,
+ 493,
+ 120.33333333333334
+ ],
+ "spans": [],
+ "index": 0
+ },
+ {
+ "bbox": [
+ 115,
+ 120.33333333333334,
+ 493,
+ 160.66666666666669
+ ],
+ "spans": [],
+ "index": 1
+ },
+ {
+ "bbox": [
+ 115,
+ 160.66666666666669,
+ 493,
+ 201.00000000000003
+ ],
+ "spans": [],
+ "index": 2
+ }
+ ]
+ },
+ {
+ "type": "table_caption",
+ "bbox": [
+ 132,
+ 203,
+ 474,
+ 214
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 135,
+ 202,
+ 475,
+ 216
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 135,
+ 202,
+ 312,
+ 216
+ ],
+ "score": 1.0,
+ "content": "Table 1: MAE results on QM9 testing set.",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 313,
+ 203,
+ 319,
+ 215
+ ],
+ "score": 0.61,
+ "content": "\\dagger",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 320,
+ 202,
+ 475,
+ 216
+ ],
+ "score": 1.0,
+ "content": "denotes using different data partitions.",
+ "type": "text"
+ }
+ ],
+ "index": 3
+ }
+ ],
+ "index": 3
+ }
+ ],
+ "index": 2.0
+ },
+ {
+ "type": "table",
+ "bbox": [
+ 70,
+ 217,
+ 541,
+ 293
+ ],
+ "blocks": [
+ {
+ "type": "table_body",
+ "bbox": [
+ 70,
+ 217,
+ 541,
+ 293
+ ],
+ "group_id": 1,
+ "lines": [
+ {
+ "bbox": [
+ 70,
+ 217,
+ 541,
+ 293
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 70,
+ 217,
+ 541,
+ 293
+ ],
+ "score": 0.975,
+ "html": "| Methods | Task Units | α a | △ε meV | εHOMO meV | εLUMO meV | μ D | Cv cal/mol K | G meV | H meV | R a | U meV | U meV | ZPVE meV |
| NMP(Gilmer et al., 2017)t | | .092 | 69 | 43 | 38 | .030 | .040 | 19 | 17 | .180 | 20 | 20 | 1.50 |
| SchNet (Schutt et al.,2017) | | .235 | 63 | 41 | 34 | .033 | .033 | 14 | 14 | .073 | 19 | 14 | 1.70 |
| Cormorant (Anderson et al., 2019)† | | .085 | 61 | 34 | 38 | .038 | .026 | 20 | 21 | .961 | 21 | 22 | 2.03 |
| LieConv (Finzi et al.,020)t | .084 | 49 | | 30 | 25 | .032 | .038 | 22 | 24 | .800 | 19 | 19 | 2.28 |
| DimeNet++ (Gasteiger et al., 2020a) | | .044 | 33 | 25 | 20 | .030 | .023 | 8 | 7 | .331 | 6 | 6 | 1.21 |
| TFN (Thomas et al., 2018)† | | .223 | 58 | 40 | 38 | .064 | .101 | - | - | - | - | - | - |
| SE(3)-Transformer(Fuchs etal.,2020)† | | .142 | 53 | 35 | 33 | .051 | .054 | - | - | - | - | - | - |
| EGNN (Satorras et al.,2021)† | .071 | 48 | | 29 | 25 | .029 | .031 | 12 | 12 | .106 | 12 | 11 | 1.55 |
| PaiNN (Schutt et al.,2021) | .045 | | 46 | 28 | 20 | .012 | .024 | 7.35 | 5.98 | .066 | 5.83 | 5.85 | 1.28 |
| TorchMD-NET(Tholke & Fabritiis,2022) | | .059 | 36 | 20 | 18 | .011 | .026 | 7.62 | 6.16 | .033 | 6.38 | 6.15 | 1.84 |
| SphereNet (Liu et al.,2022) | .046 | 32 | | 23 | 18 | .026 | .021 | 8 | 6 | .292 | 7 | 6 | 1.12 |
| SEGNN (Brandstetter et al.,2022)† | .060 | 42 | | 24 | 21 | .023 | .031 | 15 | 16 | .660 | 13 | 15 | 1.62 |
| EQGAT (Le et al.,2022) | .053 | 32 | | 20 | 16 | .011 | .024 | 23 | 24 | .382 | 25 | 25 | 2.00 |
| Equiformer | | .046 | 30 | 15 | 14 | .011 | .023 | 7.63 | 6.63 | .251 | 6.74 | 6.59 | 1.26 |
",
+ "type": "table",
+ "image_path": "99449f7c420a58ee605341ce298638673593e231af3640cfcdd9d40c37f10b68.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 1,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 115,
+ 80,
+ 493,
+ 120.33333333333334
+ ],
+ "spans": [],
+ "index": 0
+ },
+ {
+ "bbox": [
+ 115,
+ 120.33333333333334,
+ 493,
+ 160.66666666666669
+ ],
+ "spans": [],
+ "index": 1
+ },
+ {
+ "bbox": [
+ 115,
+ 160.66666666666669,
+ 493,
+ 201.00000000000003
+ ],
+ "spans": [],
+ "index": 2
+ }
+ ]
+ },
+ {
+ "type": "table_caption",
+ "bbox": [
+ 132,
+ 203,
+ 474,
+ 214
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 135,
+ 202,
+ 475,
+ 216
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 135,
+ 202,
+ 312,
+ 216
+ ],
+ "score": 1.0,
+ "content": "Table 1: MAE results on QM9 testing set.",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 313,
+ 203,
+ 319,
+ 215
+ ],
+ "score": 0.61,
+ "content": "\\dagger",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 320,
+ 202,
+ 475,
+ 216
+ ],
+ "score": 1.0,
+ "content": "denotes using different data partitions.",
+ "type": "text"
+ }
+ ],
+ "index": 3
+ }
+ ],
+ "index": 3
+ }
+ ],
+ "index": 2.0
+ },
+ {
+ "type": "table",
+ "bbox": [
+ 70,
+ 217,
+ 541,
+ 293
+ ],
+ "blocks": [
+ {
+ "type": "table_body",
+ "bbox": [
+ 70,
+ 217,
+ 541,
+ 293
+ ],
+ "group_id": 1,
+ "lines": [
+ {
+ "bbox": [
+ 70,
+ 217,
+ 541,
+ 293
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 70,
+ 217,
+ 541,
+ 293
+ ],
+ "score": 0.975,
+ "html": "| Hyper-parameters | Value or description |
| Optimizer | AdamW |
| Learning rate scheduling Warmup epochs | Cosine learning rate with linear warmup 5 |
| Maximum learning rate | 1.5 × 10-4,5 × 10-4 |
| Batch size | 64,128 |
| |
| Number of epochs | 300,600 |
| Weight decay | 0,5×10-3 |
| Dropout rate | 0.0,0.1, 0.2 |
| Cutoff radius (A) | 5 |
| Number of radial bases | 128 for Gaussian radial basis,8 for radial bessel basis |
| Hidden sizes of radial functions | 64 |
| Number of hidden layers in radial functions | 2 |
| Number of Transformer blocks Embedding dimension dembed | Equiformer 6 [(128,0),(64,1),(32,2)] |
| Spherical harmonics embedding dimension dsh Numberof attention heads h Attention head dimension dhead | [(1,0),(1,1),(1,2)] 4 [(32,0),(16,1),(8,2)] |
| Hidden dimension in feed forward networks dffn Output feature dimension d feature | (384,0),(192,1),(96,2)] [(512,0)] |
| E(3)-Equiformer |
| |
| |
| |
| 6 |
| Number of Transformer blocks | |
| Embedding dimension dembed | |
| [(128,0,e),(32,0,0),(32,1,e),(32,1,0),(16,2,e),(16,2,0)] |
| Spherical harmonics embedding dimension dsh | [(1,0,e),(1,1,0),(1,2,e)] |
| Number of attention heads h | 4 |
| |
| Attention head dimension dhead | [(32,0,e),(8,0,0),(8,1,e),(8,1,0),(4,2,e),(4,2,0)] |
| Hidden dimension in feed forward networks df fn | |
| (384,0,e),(96,0,0),(96,1,e),(96,1,0),(48,2,e),(48,2,0)] |
| |
| Output feature dimension dfeature | [(512,0,e)] |
| |
| |
",
+ "type": "table",
+ "image_path": "03578d6670ec62325bd270b8703e94123f4e5a7faf6fef8c8d3a5bac027459ea.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 1,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 113,
+ 81,
+ 496,
+ 180.33333333333331
+ ],
+ "spans": [],
+ "index": 0
+ },
+ {
+ "bbox": [
+ 113,
+ 180.33333333333331,
+ 496,
+ 279.66666666666663
+ ],
+ "spans": [],
+ "index": 1
+ },
+ {
+ "bbox": [
+ 113,
+ 279.66666666666663,
+ 496,
+ 378.99999999999994
+ ],
+ "spans": [],
+ "index": 2
+ }
+ ]
+ },
+ {
+ "type": "table_footnote",
+ "bbox": [
+ 107,
+ 381,
+ 502,
+ 406
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 104,
+ 377,
+ 504,
+ 398
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 377,
+ 340,
+ 398
+ ],
+ "score": 1.0,
+ "content": "Table 8: Hyper-parameters for QM9 dataset. We denote",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 340,
+ 381,
+ 354,
+ 393
+ ],
+ "score": 0.87,
+ "content": "C _ { L }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 354,
+ 377,
+ 376,
+ 398
+ ],
+ "score": 1.0,
+ "content": "type-",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 376,
+ 382,
+ 384,
+ 391
+ ],
+ "score": 0.75,
+ "content": "L",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 384,
+ 377,
+ 427,
+ 398
+ ],
+ "score": 1.0,
+ "content": "vectors as",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 427,
+ 381,
+ 460,
+ 393
+ ],
+ "score": 0.91,
+ "content": "( C _ { L } , L )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 460,
+ 377,
+ 478,
+ 398
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 478,
+ 382,
+ 504,
+ 394
+ ],
+ "score": 0.89,
+ "content": "C _ { ( L , p ) }",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 3
+ },
+ {
+ "bbox": [
+ 105,
+ 392,
+ 476,
+ 407
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 392,
+ 127,
+ 407
+ ],
+ "score": 1.0,
+ "content": "type-",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 127,
+ 393,
+ 152,
+ 405
+ ],
+ "score": 0.92,
+ "content": "( L , p )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 153,
+ 392,
+ 196,
+ 407
+ ],
+ "score": 1.0,
+ "content": "vectors as",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 196,
+ 393,
+ 250,
+ 406
+ ],
+ "score": 0.93,
+ "content": "( C _ { ( L , p ) } , L , p )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 251,
+ 392,
+ 476,
+ 407
+ ],
+ "score": 1.0,
+ "content": "and use brackets to represent concatenations of vectors.",
+ "type": "text"
+ }
+ ],
+ "index": 4
+ }
+ ],
+ "index": 3.5
+ }
+ ],
+ "index": 2.25
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 433,
+ 505,
+ 466
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 432,
+ 506,
+ 446
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 432,
+ 506,
+ 446
+ ],
+ "score": 1.0,
+ "content": "Second, compared to linear messages, using non-linear messages increases the number of tensor",
+ "type": "text"
+ }
+ ],
+ "index": 5
+ },
+ {
+ "bbox": [
+ 106,
+ 445,
+ 505,
+ 457
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 445,
+ 505,
+ 457
+ ],
+ "score": 1.0,
+ "content": "product operations from 1 to 2. Since tensor products are compute-intensive, this inevitably increases",
+ "type": "text"
+ }
+ ],
+ "index": 6
+ },
+ {
+ "bbox": [
+ 106,
+ 456,
+ 218,
+ 467
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 456,
+ 218,
+ 467
+ ],
+ "score": 1.0,
+ "content": "training and inference time.",
+ "type": "text"
+ }
+ ],
+ "index": 7
+ }
+ ],
+ "index": 6
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 471,
+ 487,
+ 483
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 470,
+ 488,
+ 485
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 470,
+ 488,
+ 485
+ ],
+ "score": 1.0,
+ "content": "Please refer to Sec. D.1 and Sec. F.2 for the exact numbers of training time on QM9 and OC20.",
+ "type": "text"
+ }
+ ],
+ "index": 8
+ }
+ ],
+ "index": 8
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 107,
+ 506,
+ 315,
+ 520
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 506,
+ 316,
+ 522
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 506,
+ 316,
+ 522
+ ],
+ "score": 1.0,
+ "content": "D DETAILS OF EXPERIMENTS ON QM9",
+ "type": "text"
+ }
+ ],
+ "index": 9
+ }
+ ],
+ "index": 9
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 108,
+ 536,
+ 217,
+ 548
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 536,
+ 218,
+ 549
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 536,
+ 218,
+ 549
+ ],
+ "score": 1.0,
+ "content": "D.1 TRAINING DETAILS",
+ "type": "text"
+ }
+ ],
+ "index": 10
+ }
+ ],
+ "index": 10
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 560,
+ 505,
+ 604
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 560,
+ 506,
+ 572
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 560,
+ 493,
+ 572
+ ],
+ "score": 1.0,
+ "content": "We use the same data partition as TorchMD-NET (Thölke & Fabritiis, 2022). For the task of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 493,
+ 561,
+ 502,
+ 570
+ ],
+ "score": 0.79,
+ "content": "U",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 503,
+ 560,
+ 506,
+ 572
+ ],
+ "score": 1.0,
+ "content": ",",
+ "type": "text"
+ }
+ ],
+ "index": 11
+ },
+ {
+ "bbox": [
+ 106,
+ 571,
+ 505,
+ 583
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 571,
+ 132,
+ 583
+ ],
+ "score": 0.27,
+ "content": "U _ { 0 } , G",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 132,
+ 571,
+ 152,
+ 583
+ ],
+ "score": 1.0,
+ "content": ", and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 153,
+ 572,
+ 163,
+ 581
+ ],
+ "score": 0.78,
+ "content": "H",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 163,
+ 571,
+ 505,
+ 583
+ ],
+ "score": 1.0,
+ "content": ", where single-atom reference values are available, we subtract those reference values",
+ "type": "text"
+ }
+ ],
+ "index": 12
+ },
+ {
+ "bbox": [
+ 105,
+ 581,
+ 505,
+ 595
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 581,
+ 505,
+ 595
+ ],
+ "score": 1.0,
+ "content": "from ground truth. For other tasks, we normalize ground truth by subtracting mean and dividing by",
+ "type": "text"
+ }
+ ],
+ "index": 13
+ },
+ {
+ "bbox": [
+ 106,
+ 594,
+ 184,
+ 604
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 594,
+ 184,
+ 604
+ ],
+ "score": 1.0,
+ "content": "standard deviation.",
+ "type": "text"
+ }
+ ],
+ "index": 14
+ }
+ ],
+ "index": 12.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 609,
+ 506,
+ 731
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 609,
+ 506,
+ 623
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 609,
+ 269,
+ 623
+ ],
+ "score": 1.0,
+ "content": "We train Equiformer with 6 blocks with",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 269,
+ 610,
+ 312,
+ 621
+ ],
+ "score": 0.91,
+ "content": "L _ { m a x } = 2",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 312,
+ 609,
+ 506,
+ 623
+ ],
+ "score": 1.0,
+ "content": ". We choose Gaussian radial basis (Schütt et al.,",
+ "type": "text"
+ }
+ ],
+ "index": 15
+ },
+ {
+ "bbox": [
+ 106,
+ 621,
+ 505,
+ 632
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 621,
+ 505,
+ 632
+ ],
+ "score": 1.0,
+ "content": "2017; Shuaibi et al., 2021; Klicpera et al., 2021; Shi et al., 2022) for the first six tasks in Table 1",
+ "type": "text"
+ }
+ ],
+ "index": 16
+ },
+ {
+ "bbox": [
+ 106,
+ 631,
+ 505,
+ 644
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 631,
+ 505,
+ 644
+ ],
+ "score": 1.0,
+ "content": "and radial Bessel basis (Gasteiger et al., 2020b;a) for the others. We apply dropout (Srivastava et al.,",
+ "type": "text"
+ }
+ ],
+ "index": 17
+ },
+ {
+ "bbox": [
+ 105,
+ 642,
+ 506,
+ 656
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 642,
+ 216,
+ 656
+ ],
+ "score": 1.0,
+ "content": "2014) to attention weights",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 216,
+ 644,
+ 229,
+ 655
+ ],
+ "score": 0.88,
+ "content": "a _ { i j }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 230,
+ 642,
+ 394,
+ 656
+ ],
+ "score": 1.0,
+ "content": ". The dropout rate is 0.0 for the tasks of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 394,
+ 644,
+ 403,
+ 653
+ ],
+ "score": 0.51,
+ "content": "G",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 403,
+ 642,
+ 407,
+ 656
+ ],
+ "score": 1.0,
+ "content": ",",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 407,
+ 644,
+ 417,
+ 653
+ ],
+ "score": 0.45,
+ "content": "H",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 417,
+ 642,
+ 421,
+ 656
+ ],
+ "score": 1.0,
+ "content": ",",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 422,
+ 644,
+ 431,
+ 653
+ ],
+ "score": 0.53,
+ "content": "U",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 431,
+ 642,
+ 449,
+ 656
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 450,
+ 643,
+ 462,
+ 654
+ ],
+ "score": 0.87,
+ "content": "U _ { 0 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 462,
+ 642,
+ 506,
+ 656
+ ],
+ "score": 1.0,
+ "content": ", is 0.1 for",
+ "type": "text"
+ }
+ ],
+ "index": 18
+ },
+ {
+ "bbox": [
+ 105,
+ 654,
+ 506,
+ 667
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 654,
+ 151,
+ 667
+ ],
+ "score": 1.0,
+ "content": "the task of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 151,
+ 654,
+ 164,
+ 665
+ ],
+ "score": 0.88,
+ "content": "R ^ { 2 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 164,
+ 654,
+ 325,
+ 667
+ ],
+ "score": 1.0,
+ "content": "and is 0.2 for others. Since the tasks of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 325,
+ 655,
+ 334,
+ 665
+ ],
+ "score": 0.48,
+ "content": "G",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 334,
+ 654,
+ 338,
+ 667
+ ],
+ "score": 1.0,
+ "content": ",",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 338,
+ 655,
+ 348,
+ 665
+ ],
+ "score": 0.38,
+ "content": "H",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 348,
+ 654,
+ 351,
+ 667
+ ],
+ "score": 1.0,
+ "content": ",",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 352,
+ 655,
+ 361,
+ 665
+ ],
+ "score": 0.43,
+ "content": "U",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 361,
+ 654,
+ 381,
+ 667
+ ],
+ "score": 1.0,
+ "content": ", and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 381,
+ 655,
+ 394,
+ 666
+ ],
+ "score": 0.88,
+ "content": "U _ { 0 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 394,
+ 654,
+ 506,
+ 667
+ ],
+ "score": 1.0,
+ "content": "require longer training, we",
+ "type": "text"
+ }
+ ],
+ "index": 19
+ },
+ {
+ "bbox": [
+ 106,
+ 666,
+ 506,
+ 678
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 666,
+ 448,
+ 678
+ ],
+ "score": 1.0,
+ "content": "use slightly different hyper-parameters. The number of epochs is 600 for the tasks of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 449,
+ 666,
+ 457,
+ 676
+ ],
+ "score": 0.51,
+ "content": "G",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 454,
+ 666,
+ 484,
+ 676
+ ],
+ "score": 0.38,
+ "content": "\\ d s _ { r } , H , U",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 484,
+ 666,
+ 506,
+ 678
+ ],
+ "score": 1.0,
+ "content": ", and",
+ "type": "text"
+ }
+ ],
+ "index": 20
+ },
+ {
+ "bbox": [
+ 107,
+ 676,
+ 506,
+ 689
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 677,
+ 119,
+ 688
+ ],
+ "score": 0.86,
+ "content": "U _ { 0 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 119,
+ 676,
+ 293,
+ 689
+ ],
+ "score": 1.0,
+ "content": "and is 300 for others. The learning rate is",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 293,
+ 676,
+ 340,
+ 687
+ ],
+ "score": 0.92,
+ "content": "1 . 5 \\times 1 0 ^ { - 4 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 340,
+ 676,
+ 406,
+ 689
+ ],
+ "score": 1.0,
+ "content": "for the tasks of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 406,
+ 677,
+ 415,
+ 687
+ ],
+ "score": 0.48,
+ "content": "G",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 415,
+ 676,
+ 417,
+ 689
+ ],
+ "score": 1.0,
+ "content": ",",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 418,
+ 677,
+ 443,
+ 687
+ ],
+ "score": 0.29,
+ "content": "H , U",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 443,
+ 676,
+ 464,
+ 689
+ ],
+ "score": 1.0,
+ "content": ", and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 464,
+ 677,
+ 477,
+ 688
+ ],
+ "score": 0.88,
+ "content": "U _ { 0 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 477,
+ 676,
+ 506,
+ 689
+ ],
+ "score": 1.0,
+ "content": "and is",
+ "type": "text"
+ }
+ ],
+ "index": 21
+ },
+ {
+ "bbox": [
+ 107,
+ 686,
+ 506,
+ 700
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 687,
+ 145,
+ 698
+ ],
+ "score": 0.91,
+ "content": "5 \\times 1 0 ^ { - 4 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 146,
+ 686,
+ 331,
+ 700
+ ],
+ "score": 1.0,
+ "content": "for others. The batch size is 64 for the tasks of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 331,
+ 688,
+ 340,
+ 698
+ ],
+ "score": 0.64,
+ "content": "G",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 336,
+ 688,
+ 368,
+ 698
+ ],
+ "score": 0.35,
+ "content": "{ \\mathrm { ? } } , H , U ,",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 369,
+ 686,
+ 387,
+ 700
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 387,
+ 688,
+ 400,
+ 699
+ ],
+ "score": 0.89,
+ "content": "U _ { 0 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 400,
+ 686,
+ 506,
+ 700
+ ],
+ "score": 1.0,
+ "content": "and is 128 for others. The",
+ "type": "text"
+ }
+ ],
+ "index": 22
+ },
+ {
+ "bbox": [
+ 105,
+ 697,
+ 506,
+ 711
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 697,
+ 241,
+ 711
+ ],
+ "score": 1.0,
+ "content": "weight decay is 0 for the tasks of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 241,
+ 699,
+ 250,
+ 709
+ ],
+ "score": 0.54,
+ "content": "G",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 250,
+ 697,
+ 254,
+ 711
+ ],
+ "score": 1.0,
+ "content": ",",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 254,
+ 699,
+ 264,
+ 709
+ ],
+ "score": 0.45,
+ "content": "H",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 264,
+ 697,
+ 268,
+ 711
+ ],
+ "score": 1.0,
+ "content": ",",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 268,
+ 699,
+ 277,
+ 709
+ ],
+ "score": 0.42,
+ "content": "U",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 277,
+ 697,
+ 298,
+ 711
+ ],
+ "score": 1.0,
+ "content": ", and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 298,
+ 699,
+ 310,
+ 710
+ ],
+ "score": 0.89,
+ "content": "U _ { 0 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 311,
+ 697,
+ 338,
+ 711
+ ],
+ "score": 1.0,
+ "content": "and is",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 338,
+ 698,
+ 376,
+ 709
+ ],
+ "score": 0.92,
+ "content": "5 \\times 1 0 ^ { - 3 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 377,
+ 697,
+ 506,
+ 711
+ ],
+ "score": 1.0,
+ "content": "for others. Table 8 summarizes",
+ "type": "text"
+ }
+ ],
+ "index": 23
+ },
+ {
+ "bbox": [
+ 105,
+ 709,
+ 506,
+ 722
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 709,
+ 506,
+ 722
+ ],
+ "score": 1.0,
+ "content": "the hyper-parameters for the QM9 dataset. The detailed description of architectural hyper-parameters",
+ "type": "text"
+ }
+ ],
+ "index": 24
+ },
+ {
+ "bbox": [
+ 105,
+ 719,
+ 209,
+ 732
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 719,
+ 209,
+ 732
+ ],
+ "score": 1.0,
+ "content": "can be found in Sec. C.2.",
+ "type": "text"
+ }
+ ],
+ "index": 25
+ }
+ ],
+ "index": 20
+ }
+ ],
+ "page_idx": 24,
+ "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 2023",
+ "type": "text"
+ }
+ ]
+ }
+ ]
+ },
+ {
+ "type": "discarded",
+ "bbox": [
+ 300,
+ 751,
+ 311,
+ 760
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 298,
+ 750,
+ 313,
+ 764
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 298,
+ 750,
+ 313,
+ 764
+ ],
+ "score": 1.0,
+ "content": "",
+ "type": "text",
+ "height": 14,
+ "width": 15
+ }
+ ]
+ }
+ ]
+ }
+ ],
+ "para_blocks": [
+ {
+ "type": "table",
+ "bbox": [
+ 113,
+ 81,
+ 496,
+ 379
+ ],
+ "blocks": [
+ {
+ "type": "table_body",
+ "bbox": [
+ 113,
+ 81,
+ 496,
+ 379
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 113,
+ 81,
+ 496,
+ 379
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 113,
+ 81,
+ 496,
+ 379
+ ],
+ "score": 0.98,
+ "html": "| Hyper-parameters | Value or description |
| Optimizer | AdamW |
| Learning rate scheduling Warmup epochs | Cosine learning rate with linear warmup 5 |
| Maximum learning rate | 1.5 × 10-4,5 × 10-4 |
| Batch size | 64,128 |
| |
| Number of epochs | 300,600 |
| Weight decay | 0,5×10-3 |
| Dropout rate | 0.0,0.1, 0.2 |
| Cutoff radius (A) | 5 |
| Number of radial bases | 128 for Gaussian radial basis,8 for radial bessel basis |
| Hidden sizes of radial functions | 64 |
| Number of hidden layers in radial functions | 2 |
| Number of Transformer blocks Embedding dimension dembed | Equiformer 6 [(128,0),(64,1),(32,2)] |
| Spherical harmonics embedding dimension dsh Numberof attention heads h Attention head dimension dhead | [(1,0),(1,1),(1,2)] 4 [(32,0),(16,1),(8,2)] |
| Hidden dimension in feed forward networks dffn Output feature dimension d feature | (384,0),(192,1),(96,2)] [(512,0)] |
| E(3)-Equiformer |
| |
| |
| |
| 6 |
| Number of Transformer blocks | |
| Embedding dimension dembed | |
| [(128,0,e),(32,0,0),(32,1,e),(32,1,0),(16,2,e),(16,2,0)] |
| Spherical harmonics embedding dimension dsh | [(1,0,e),(1,1,0),(1,2,e)] |
| Number of attention heads h | 4 |
| |
| Attention head dimension dhead | [(32,0,e),(8,0,0),(8,1,e),(8,1,0),(4,2,e),(4,2,0)] |
| Hidden dimension in feed forward networks df fn | |
| (384,0,e),(96,0,0),(96,1,e),(96,1,0),(48,2,e),(48,2,0)] |
| |
| Output feature dimension dfeature | [(512,0,e)] |
| |
| |
",
+ "type": "table",
+ "image_path": "03578d6670ec62325bd270b8703e94123f4e5a7faf6fef8c8d3a5bac027459ea.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 1,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 113,
+ 81,
+ 496,
+ 180.33333333333331
+ ],
+ "spans": [],
+ "index": 0
+ },
+ {
+ "bbox": [
+ 113,
+ 180.33333333333331,
+ 496,
+ 279.66666666666663
+ ],
+ "spans": [],
+ "index": 1
+ },
+ {
+ "bbox": [
+ 113,
+ 279.66666666666663,
+ 496,
+ 378.99999999999994
+ ],
+ "spans": [],
+ "index": 2
+ }
+ ]
+ },
+ {
+ "type": "table_footnote",
+ "bbox": [
+ 107,
+ 381,
+ 502,
+ 406
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 104,
+ 377,
+ 504,
+ 398
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 377,
+ 340,
+ 398
+ ],
+ "score": 1.0,
+ "content": "Table 8: Hyper-parameters for QM9 dataset. We denote",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 340,
+ 381,
+ 354,
+ 393
+ ],
+ "score": 0.87,
+ "content": "C _ { L }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 354,
+ 377,
+ 376,
+ 398
+ ],
+ "score": 1.0,
+ "content": "type-",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 376,
+ 382,
+ 384,
+ 391
+ ],
+ "score": 0.75,
+ "content": "L",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 384,
+ 377,
+ 427,
+ 398
+ ],
+ "score": 1.0,
+ "content": "vectors as",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 427,
+ 381,
+ 460,
+ 393
+ ],
+ "score": 0.91,
+ "content": "( C _ { L } , L )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 460,
+ 377,
+ 478,
+ 398
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 478,
+ 382,
+ 504,
+ 394
+ ],
+ "score": 0.89,
+ "content": "C _ { ( L , p ) }",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 3
+ },
+ {
+ "bbox": [
+ 105,
+ 392,
+ 476,
+ 407
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 392,
+ 127,
+ 407
+ ],
+ "score": 1.0,
+ "content": "type-",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 127,
+ 393,
+ 152,
+ 405
+ ],
+ "score": 0.92,
+ "content": "( L , p )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 153,
+ 392,
+ 196,
+ 407
+ ],
+ "score": 1.0,
+ "content": "vectors as",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 196,
+ 393,
+ 250,
+ 406
+ ],
+ "score": 0.93,
+ "content": "( C _ { ( L , p ) } , L , p )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 251,
+ 392,
+ 476,
+ 407
+ ],
+ "score": 1.0,
+ "content": "and use brackets to represent concatenations of vectors.",
+ "type": "text"
+ }
+ ],
+ "index": 4
+ }
+ ],
+ "index": 3.5
+ }
+ ],
+ "index": 2.25
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 433,
+ 505,
+ 466
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 432,
+ 506,
+ 446
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 432,
+ 506,
+ 446
+ ],
+ "score": 1.0,
+ "content": "Second, compared to linear messages, using non-linear messages increases the number of tensor",
+ "type": "text"
+ }
+ ],
+ "index": 5
+ },
+ {
+ "bbox": [
+ 106,
+ 445,
+ 505,
+ 457
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 445,
+ 505,
+ 457
+ ],
+ "score": 1.0,
+ "content": "product operations from 1 to 2. Since tensor products are compute-intensive, this inevitably increases",
+ "type": "text"
+ }
+ ],
+ "index": 6
+ },
+ {
+ "bbox": [
+ 106,
+ 456,
+ 218,
+ 467
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 456,
+ 218,
+ 467
+ ],
+ "score": 1.0,
+ "content": "training and inference time.",
+ "type": "text"
+ }
+ ],
+ "index": 7
+ }
+ ],
+ "index": 6,
+ "bbox_fs": [
+ 105,
+ 432,
+ 506,
+ 467
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 471,
+ 487,
+ 483
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 470,
+ 488,
+ 485
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 470,
+ 488,
+ 485
+ ],
+ "score": 1.0,
+ "content": "Please refer to Sec. D.1 and Sec. F.2 for the exact numbers of training time on QM9 and OC20.",
+ "type": "text"
+ }
+ ],
+ "index": 8
+ }
+ ],
+ "index": 8,
+ "bbox_fs": [
+ 105,
+ 470,
+ 488,
+ 485
+ ]
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 107,
+ 506,
+ 315,
+ 520
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 506,
+ 316,
+ 522
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 506,
+ 316,
+ 522
+ ],
+ "score": 1.0,
+ "content": "D DETAILS OF EXPERIMENTS ON QM9",
+ "type": "text"
+ }
+ ],
+ "index": 9
+ }
+ ],
+ "index": 9
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 108,
+ 536,
+ 217,
+ 548
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 536,
+ 218,
+ 549
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 536,
+ 218,
+ 549
+ ],
+ "score": 1.0,
+ "content": "D.1 TRAINING DETAILS",
+ "type": "text"
+ }
+ ],
+ "index": 10
+ }
+ ],
+ "index": 10
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 560,
+ 505,
+ 604
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 560,
+ 506,
+ 572
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 560,
+ 493,
+ 572
+ ],
+ "score": 1.0,
+ "content": "We use the same data partition as TorchMD-NET (Thölke & Fabritiis, 2022). For the task of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 493,
+ 561,
+ 502,
+ 570
+ ],
+ "score": 0.79,
+ "content": "U",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 503,
+ 560,
+ 506,
+ 572
+ ],
+ "score": 1.0,
+ "content": ",",
+ "type": "text"
+ }
+ ],
+ "index": 11
+ },
+ {
+ "bbox": [
+ 106,
+ 571,
+ 505,
+ 583
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 571,
+ 132,
+ 583
+ ],
+ "score": 0.27,
+ "content": "U _ { 0 } , G",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 132,
+ 571,
+ 152,
+ 583
+ ],
+ "score": 1.0,
+ "content": ", and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 153,
+ 572,
+ 163,
+ 581
+ ],
+ "score": 0.78,
+ "content": "H",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 163,
+ 571,
+ 505,
+ 583
+ ],
+ "score": 1.0,
+ "content": ", where single-atom reference values are available, we subtract those reference values",
+ "type": "text"
+ }
+ ],
+ "index": 12
+ },
+ {
+ "bbox": [
+ 105,
+ 581,
+ 505,
+ 595
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 581,
+ 505,
+ 595
+ ],
+ "score": 1.0,
+ "content": "from ground truth. For other tasks, we normalize ground truth by subtracting mean and dividing by",
+ "type": "text"
+ }
+ ],
+ "index": 13
+ },
+ {
+ "bbox": [
+ 106,
+ 594,
+ 184,
+ 604
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 594,
+ 184,
+ 604
+ ],
+ "score": 1.0,
+ "content": "standard deviation.",
+ "type": "text"
+ }
+ ],
+ "index": 14
+ }
+ ],
+ "index": 12.5,
+ "bbox_fs": [
+ 105,
+ 560,
+ 506,
+ 604
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 609,
+ 506,
+ 731
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 609,
+ 506,
+ 623
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 609,
+ 269,
+ 623
+ ],
+ "score": 1.0,
+ "content": "We train Equiformer with 6 blocks with",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 269,
+ 610,
+ 312,
+ 621
+ ],
+ "score": 0.91,
+ "content": "L _ { m a x } = 2",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 312,
+ 609,
+ 506,
+ 623
+ ],
+ "score": 1.0,
+ "content": ". We choose Gaussian radial basis (Schütt et al.,",
+ "type": "text"
+ }
+ ],
+ "index": 15
+ },
+ {
+ "bbox": [
+ 106,
+ 621,
+ 505,
+ 632
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 621,
+ 505,
+ 632
+ ],
+ "score": 1.0,
+ "content": "2017; Shuaibi et al., 2021; Klicpera et al., 2021; Shi et al., 2022) for the first six tasks in Table 1",
+ "type": "text"
+ }
+ ],
+ "index": 16
+ },
+ {
+ "bbox": [
+ 106,
+ 631,
+ 505,
+ 644
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 631,
+ 505,
+ 644
+ ],
+ "score": 1.0,
+ "content": "and radial Bessel basis (Gasteiger et al., 2020b;a) for the others. We apply dropout (Srivastava et al.,",
+ "type": "text"
+ }
+ ],
+ "index": 17
+ },
+ {
+ "bbox": [
+ 105,
+ 642,
+ 506,
+ 656
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 642,
+ 216,
+ 656
+ ],
+ "score": 1.0,
+ "content": "2014) to attention weights",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 216,
+ 644,
+ 229,
+ 655
+ ],
+ "score": 0.88,
+ "content": "a _ { i j }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 230,
+ 642,
+ 394,
+ 656
+ ],
+ "score": 1.0,
+ "content": ". The dropout rate is 0.0 for the tasks of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 394,
+ 644,
+ 403,
+ 653
+ ],
+ "score": 0.51,
+ "content": "G",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 403,
+ 642,
+ 407,
+ 656
+ ],
+ "score": 1.0,
+ "content": ",",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 407,
+ 644,
+ 417,
+ 653
+ ],
+ "score": 0.45,
+ "content": "H",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 417,
+ 642,
+ 421,
+ 656
+ ],
+ "score": 1.0,
+ "content": ",",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 422,
+ 644,
+ 431,
+ 653
+ ],
+ "score": 0.53,
+ "content": "U",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 431,
+ 642,
+ 449,
+ 656
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 450,
+ 643,
+ 462,
+ 654
+ ],
+ "score": 0.87,
+ "content": "U _ { 0 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 462,
+ 642,
+ 506,
+ 656
+ ],
+ "score": 1.0,
+ "content": ", is 0.1 for",
+ "type": "text"
+ }
+ ],
+ "index": 18
+ },
+ {
+ "bbox": [
+ 105,
+ 654,
+ 506,
+ 667
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 654,
+ 151,
+ 667
+ ],
+ "score": 1.0,
+ "content": "the task of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 151,
+ 654,
+ 164,
+ 665
+ ],
+ "score": 0.88,
+ "content": "R ^ { 2 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 164,
+ 654,
+ 325,
+ 667
+ ],
+ "score": 1.0,
+ "content": "and is 0.2 for others. Since the tasks of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 325,
+ 655,
+ 334,
+ 665
+ ],
+ "score": 0.48,
+ "content": "G",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 334,
+ 654,
+ 338,
+ 667
+ ],
+ "score": 1.0,
+ "content": ",",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 338,
+ 655,
+ 348,
+ 665
+ ],
+ "score": 0.38,
+ "content": "H",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 348,
+ 654,
+ 351,
+ 667
+ ],
+ "score": 1.0,
+ "content": ",",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 352,
+ 655,
+ 361,
+ 665
+ ],
+ "score": 0.43,
+ "content": "U",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 361,
+ 654,
+ 381,
+ 667
+ ],
+ "score": 1.0,
+ "content": ", and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 381,
+ 655,
+ 394,
+ 666
+ ],
+ "score": 0.88,
+ "content": "U _ { 0 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 394,
+ 654,
+ 506,
+ 667
+ ],
+ "score": 1.0,
+ "content": "require longer training, we",
+ "type": "text"
+ }
+ ],
+ "index": 19
+ },
+ {
+ "bbox": [
+ 106,
+ 666,
+ 506,
+ 678
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 666,
+ 448,
+ 678
+ ],
+ "score": 1.0,
+ "content": "use slightly different hyper-parameters. The number of epochs is 600 for the tasks of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 449,
+ 666,
+ 457,
+ 676
+ ],
+ "score": 0.51,
+ "content": "G",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 454,
+ 666,
+ 484,
+ 676
+ ],
+ "score": 0.38,
+ "content": "\\ d s _ { r } , H , U",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 484,
+ 666,
+ 506,
+ 678
+ ],
+ "score": 1.0,
+ "content": ", and",
+ "type": "text"
+ }
+ ],
+ "index": 20
+ },
+ {
+ "bbox": [
+ 107,
+ 676,
+ 506,
+ 689
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 677,
+ 119,
+ 688
+ ],
+ "score": 0.86,
+ "content": "U _ { 0 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 119,
+ 676,
+ 293,
+ 689
+ ],
+ "score": 1.0,
+ "content": "and is 300 for others. The learning rate is",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 293,
+ 676,
+ 340,
+ 687
+ ],
+ "score": 0.92,
+ "content": "1 . 5 \\times 1 0 ^ { - 4 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 340,
+ 676,
+ 406,
+ 689
+ ],
+ "score": 1.0,
+ "content": "for the tasks of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 406,
+ 677,
+ 415,
+ 687
+ ],
+ "score": 0.48,
+ "content": "G",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 415,
+ 676,
+ 417,
+ 689
+ ],
+ "score": 1.0,
+ "content": ",",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 418,
+ 677,
+ 443,
+ 687
+ ],
+ "score": 0.29,
+ "content": "H , U",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 443,
+ 676,
+ 464,
+ 689
+ ],
+ "score": 1.0,
+ "content": ", and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 464,
+ 677,
+ 477,
+ 688
+ ],
+ "score": 0.88,
+ "content": "U _ { 0 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 477,
+ 676,
+ 506,
+ 689
+ ],
+ "score": 1.0,
+ "content": "and is",
+ "type": "text"
+ }
+ ],
+ "index": 21
+ },
+ {
+ "bbox": [
+ 107,
+ 686,
+ 506,
+ 700
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 687,
+ 145,
+ 698
+ ],
+ "score": 0.91,
+ "content": "5 \\times 1 0 ^ { - 4 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 146,
+ 686,
+ 331,
+ 700
+ ],
+ "score": 1.0,
+ "content": "for others. The batch size is 64 for the tasks of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 331,
+ 688,
+ 340,
+ 698
+ ],
+ "score": 0.64,
+ "content": "G",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 336,
+ 688,
+ 368,
+ 698
+ ],
+ "score": 0.35,
+ "content": "{ \\mathrm { ? } } , H , U ,",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 369,
+ 686,
+ 387,
+ 700
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 387,
+ 688,
+ 400,
+ 699
+ ],
+ "score": 0.89,
+ "content": "U _ { 0 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 400,
+ 686,
+ 506,
+ 700
+ ],
+ "score": 1.0,
+ "content": "and is 128 for others. The",
+ "type": "text"
+ }
+ ],
+ "index": 22
+ },
+ {
+ "bbox": [
+ 105,
+ 697,
+ 506,
+ 711
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 697,
+ 241,
+ 711
+ ],
+ "score": 1.0,
+ "content": "weight decay is 0 for the tasks of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 241,
+ 699,
+ 250,
+ 709
+ ],
+ "score": 0.54,
+ "content": "G",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 250,
+ 697,
+ 254,
+ 711
+ ],
+ "score": 1.0,
+ "content": ",",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 254,
+ 699,
+ 264,
+ 709
+ ],
+ "score": 0.45,
+ "content": "H",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 264,
+ 697,
+ 268,
+ 711
+ ],
+ "score": 1.0,
+ "content": ",",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 268,
+ 699,
+ 277,
+ 709
+ ],
+ "score": 0.42,
+ "content": "U",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 277,
+ 697,
+ 298,
+ 711
+ ],
+ "score": 1.0,
+ "content": ", and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 298,
+ 699,
+ 310,
+ 710
+ ],
+ "score": 0.89,
+ "content": "U _ { 0 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 311,
+ 697,
+ 338,
+ 711
+ ],
+ "score": 1.0,
+ "content": "and is",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 338,
+ 698,
+ 376,
+ 709
+ ],
+ "score": 0.92,
+ "content": "5 \\times 1 0 ^ { - 3 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 377,
+ 697,
+ 506,
+ 711
+ ],
+ "score": 1.0,
+ "content": "for others. Table 8 summarizes",
+ "type": "text"
+ }
+ ],
+ "index": 23
+ },
+ {
+ "bbox": [
+ 105,
+ 709,
+ 506,
+ 722
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 709,
+ 506,
+ 722
+ ],
+ "score": 1.0,
+ "content": "the hyper-parameters for the QM9 dataset. The detailed description of architectural hyper-parameters",
+ "type": "text"
+ }
+ ],
+ "index": 24
+ },
+ {
+ "bbox": [
+ 105,
+ 719,
+ 209,
+ 732
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 719,
+ 209,
+ 732
+ ],
+ "score": 1.0,
+ "content": "can be found in Sec. C.2.",
+ "type": "text"
+ }
+ ],
+ "index": 25
+ }
+ ],
+ "index": 20,
+ "bbox_fs": [
+ 105,
+ 609,
+ 506,
+ 732
+ ]
+ }
+ ]
+ },
+ {
+ "preproc_blocks": [
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 82,
+ 505,
+ 138
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 82,
+ 506,
+ 95
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 82,
+ 506,
+ 95
+ ],
+ "score": 1.0,
+ "content": "We use one A6000 GPU with 48GB to train each model and summarize the computational cost of",
+ "type": "text"
+ }
+ ],
+ "index": 0
+ },
+ {
+ "bbox": [
+ 106,
+ 93,
+ 505,
+ 106
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 94,
+ 282,
+ 106
+ ],
+ "score": 1.0,
+ "content": "training for one epoch as follows. Training",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 282,
+ 93,
+ 304,
+ 106
+ ],
+ "score": 0.88,
+ "content": "E ( 3 )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 304,
+ 94,
+ 505,
+ 106
+ ],
+ "score": 1.0,
+ "content": "-Equiformer in Table 9 for one epoch takes about",
+ "type": "text"
+ }
+ ],
+ "index": 1
+ },
+ {
+ "bbox": [
+ 105,
+ 104,
+ 506,
+ 117
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 104,
+ 506,
+ 117
+ ],
+ "score": 1.0,
+ "content": "16.3 minutes. The time of training Equiformer, Equiformer with linear messages (indicated by Index",
+ "type": "text"
+ }
+ ],
+ "index": 2
+ },
+ {
+ "bbox": [
+ 105,
+ 115,
+ 505,
+ 127
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 115,
+ 505,
+ 127
+ ],
+ "score": 1.0,
+ "content": "2 in Table 6), and Equiformer with linear messages and dot product attention (indicated by Index 3 in",
+ "type": "text"
+ }
+ ],
+ "index": 3
+ },
+ {
+ "bbox": [
+ 105,
+ 125,
+ 434,
+ 140
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 125,
+ 434,
+ 140
+ ],
+ "score": 1.0,
+ "content": "Table 6) for one epoch is 12.1 minutes, 7.2 minutes and 7.8 minutes, respectively.",
+ "type": "text"
+ }
+ ],
+ "index": 4
+ }
+ ],
+ "index": 2
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 107,
+ 154,
+ 379,
+ 166
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 153,
+ 380,
+ 168
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 153,
+ 237,
+ 168
+ ],
+ "score": 1.0,
+ "content": "D.2 COMPARISON BETWEEN",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 237,
+ 154,
+ 265,
+ 167
+ ],
+ "score": 0.88,
+ "content": "S E ( 3 )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 266,
+ 153,
+ 288,
+ 168
+ ],
+ "score": 1.0,
+ "content": "AND",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 288,
+ 154,
+ 309,
+ 167
+ ],
+ "score": 0.87,
+ "content": "E ( 3 )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 310,
+ 153,
+ 380,
+ 168
+ ],
+ "score": 1.0,
+ "content": "EQUIVARIANCE",
+ "type": "text"
+ }
+ ],
+ "index": 5
+ }
+ ],
+ "index": 5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 176,
+ 506,
+ 221
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 175,
+ 507,
+ 188
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 175,
+ 295,
+ 188
+ ],
+ "score": 1.0,
+ "content": "We train two versions of Equiformers, one with",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 295,
+ 176,
+ 324,
+ 188
+ ],
+ "score": 0.91,
+ "content": "S E ( 3 )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 324,
+ 175,
+ 507,
+ 188
+ ],
+ "score": 1.0,
+ "content": "-equivariant features denoted as “Equiformer”",
+ "type": "text"
+ }
+ ],
+ "index": 6
+ },
+ {
+ "bbox": [
+ 105,
+ 187,
+ 506,
+ 200
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 187,
+ 181,
+ 200
+ ],
+ "score": 1.0,
+ "content": "and the other with",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 181,
+ 187,
+ 203,
+ 199
+ ],
+ "score": 0.89,
+ "content": "E ( 3 )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 203,
+ 187,
+ 335,
+ 200
+ ],
+ "score": 1.0,
+ "content": "-equivariant features denoted as “",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 335,
+ 187,
+ 357,
+ 199
+ ],
+ "score": 0.85,
+ "content": "E ( 3 )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 357,
+ 187,
+ 506,
+ 200
+ ],
+ "score": 1.0,
+ "content": "-Equiformer”, and we compare them",
+ "type": "text"
+ }
+ ],
+ "index": 7
+ },
+ {
+ "bbox": [
+ 105,
+ 198,
+ 505,
+ 210
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 198,
+ 505,
+ 210
+ ],
+ "score": 1.0,
+ "content": "in Table 9. As for Table 1, we compare “Equiformer” with other works since most of them do not",
+ "type": "text"
+ }
+ ],
+ "index": 8
+ },
+ {
+ "bbox": [
+ 106,
+ 209,
+ 243,
+ 221
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 209,
+ 243,
+ 221
+ ],
+ "score": 1.0,
+ "content": "include equivariance to inversion.",
+ "type": "text"
+ }
+ ],
+ "index": 9
+ }
+ ],
+ "index": 7.5
+ },
+ {
+ "type": "table",
+ "bbox": [
+ 155,
+ 234,
+ 453,
+ 272
+ ],
+ "blocks": [
+ {
+ "type": "table_body",
+ "bbox": [
+ 155,
+ 234,
+ 453,
+ 272
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 156,
+ 234,
+ 450,
+ 272
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 156,
+ 234,
+ 450,
+ 272
+ ],
+ "score": 0.968,
+ "html": "| Hyper-parameters | Value or description |
| Optimizer | AdamW |
| Learning rate scheduling | Cosine learning rate with linear warmup |
| Warmup epochs | 10 1 × 10-4,2 × 10-4,5× 10-4 |
| Maximum learning rate Batch size | 5,8 |
| Number of epochs | 1500,2000 |
| 1×10-6 |
| Weight decay Dropout rate | 0.0 |
| |
| Weight for energy loss Weight for force loss | 1 80 |
| |
| Cutoff radius (A) | 5 |
| Number of radial bases | 32 |
| Hidden sizes of radial functions | 64 2 |
| Number of hidden layers in radial functions Equiformer (Lmax = 2) |
| Embedding dimension dembed Spherical harmonics embedding dimension d sh Number of attention heads h 4 Attention head dimension dhead Hidden dimension in feed forward networks d f fn | [(128,0),(64,1),(32,2)] [(1,0),(1,1),(1,2)] [(32,0),(16,1),(8,2)] [(384,0),(192,1),(96,2)] |
| Output feature dimension dfeature Equiformer (Lmax = 3) |
| [(512,0)] |
| |
| Number of Transformer blocks | 6 |
|
| Embedding dimension dembed | [(128,0),(64,1),(64,2),(32,3)] |
| Spherical harmonics embedding dimension d sh | [(1,0),(1,1),(1,2),(1,3)] |
| Number of attention heads h | 4 |
| |
| Attention head dimension dhead | [(32,0),(16,1),(16,2),(8,3)] |
| Hidden dimension in feed forward networks dffn | (384,0),(192,1),(192,2),(96,3)] |
",
+ "type": "table",
+ "image_path": "88f2ba9303d4afe3f93464c2ec9204befeefe568084e0fc9a152478e1b2979e5.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 5,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 149,
+ 146,
+ 459,
+ 252.66666666666669
+ ],
+ "spans": [],
+ "index": 4
+ },
+ {
+ "bbox": [
+ 149,
+ 252.66666666666669,
+ 459,
+ 359.33333333333337
+ ],
+ "spans": [],
+ "index": 5
+ },
+ {
+ "bbox": [
+ 149,
+ 359.33333333333337,
+ 459,
+ 466.00000000000006
+ ],
+ "spans": [],
+ "index": 6
+ }
+ ]
+ },
+ {
+ "type": "table_caption",
+ "bbox": [
+ 105,
+ 469,
+ 505,
+ 492
+ ],
+ "group_id": 1,
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 467,
+ 505,
+ 481
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 467,
+ 362,
+ 481
+ ],
+ "score": 1.0,
+ "content": "Table 11: Hyper-parameters for MD17 dataset. We denote",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 362,
+ 469,
+ 376,
+ 480
+ ],
+ "score": 0.86,
+ "content": "C _ { L }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 377,
+ 467,
+ 399,
+ 481
+ ],
+ "score": 1.0,
+ "content": "type-",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 400,
+ 469,
+ 407,
+ 479
+ ],
+ "score": 0.76,
+ "content": "L",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 408,
+ 467,
+ 453,
+ 481
+ ],
+ "score": 1.0,
+ "content": "vectors as",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 453,
+ 469,
+ 486,
+ 481
+ ],
+ "score": 0.92,
+ "content": "( C _ { L } , L )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 486,
+ 467,
+ 505,
+ 481
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ }
+ ],
+ "index": 7
+ },
+ {
+ "bbox": [
+ 107,
+ 478,
+ 504,
+ 493
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 480,
+ 133,
+ 493
+ ],
+ "score": 0.91,
+ "content": "C _ { ( L , p ) }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 133,
+ 478,
+ 155,
+ 493
+ ],
+ "score": 1.0,
+ "content": "type-",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 156,
+ 480,
+ 180,
+ 492
+ ],
+ "score": 0.92,
+ "content": "( L , p )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 181,
+ 478,
+ 224,
+ 493
+ ],
+ "score": 1.0,
+ "content": "vectors as",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 225,
+ 479,
+ 279,
+ 492
+ ],
+ "score": 0.94,
+ "content": "( C _ { ( L , p ) } , L , p )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 279,
+ 478,
+ 504,
+ 493
+ ],
+ "score": 1.0,
+ "content": "and use brackets to represent concatenations of vectors.",
+ "type": "text"
+ }
+ ],
+ "index": 8
+ }
+ ],
+ "index": 7.5
+ }
+ ],
+ "index": 6.25
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 514,
+ 505,
+ 547
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 514,
+ 505,
+ 527
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 514,
+ 505,
+ 527
+ ],
+ "score": 1.0,
+ "content": "We use one A5000 GPU with 24GB to train different models for each molecule. Training Equiformer",
+ "type": "text"
+ }
+ ],
+ "index": 9
+ },
+ {
+ "bbox": [
+ 104,
+ 524,
+ 506,
+ 539
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 524,
+ 127,
+ 539
+ ],
+ "score": 1.0,
+ "content": "with",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 127,
+ 526,
+ 171,
+ 537
+ ],
+ "score": 0.92,
+ "content": "L _ { m a x } = 2",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 171,
+ 524,
+ 391,
+ 539
+ ],
+ "score": 1.0,
+ "content": "takes about 15.4 hours, and training Equiformer with",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 392,
+ 526,
+ 435,
+ 537
+ ],
+ "score": 0.91,
+ "content": "L _ { m a x } = 3",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 435,
+ 524,
+ 506,
+ 539
+ ],
+ "score": 1.0,
+ "content": "takes about 54.4",
+ "type": "text"
+ }
+ ],
+ "index": 10
+ },
+ {
+ "bbox": [
+ 105,
+ 536,
+ 135,
+ 549
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 536,
+ 135,
+ 549
+ ],
+ "score": 1.0,
+ "content": "hours.",
+ "type": "text"
+ }
+ ],
+ "index": 11
+ }
+ ],
+ "index": 10
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 107,
+ 563,
+ 337,
+ 574
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 562,
+ 339,
+ 577
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 562,
+ 339,
+ 577
+ ],
+ "score": 1.0,
+ "content": "E.2 ADDITIONAL COMPARISON TO TORCHMD-NET",
+ "type": "text"
+ }
+ ],
+ "index": 12
+ }
+ ],
+ "index": 12
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 584,
+ 505,
+ 629
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 585,
+ 506,
+ 596
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 585,
+ 506,
+ 596
+ ],
+ "score": 1.0,
+ "content": "Since TorchMD-NET (Thölke & Fabritiis, 2022) is also an equivariant Transformer but uses dot prod-",
+ "type": "text"
+ }
+ ],
+ "index": 13
+ },
+ {
+ "bbox": [
+ 106,
+ 596,
+ 505,
+ 608
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 596,
+ 505,
+ 608
+ ],
+ "score": 1.0,
+ "content": "uct attention instead of the proposed equivariant graph attention, we provide additional comparisons",
+ "type": "text"
+ }
+ ],
+ "index": 14
+ },
+ {
+ "bbox": [
+ 105,
+ 606,
+ 505,
+ 619
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 606,
+ 505,
+ 619
+ ],
+ "score": 1.0,
+ "content": "in Table 12. For each molecule, we adjust the ratio of the weight for force loss to the weight for",
+ "type": "text"
+ }
+ ],
+ "index": 15
+ },
+ {
+ "bbox": [
+ 105,
+ 618,
+ 504,
+ 631
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 618,
+ 251,
+ 631
+ ],
+ "score": 1.0,
+ "content": "energy loss so that Equiformer with",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 252,
+ 618,
+ 294,
+ 629
+ ],
+ "score": 0.92,
+ "content": "L _ { m a x } = 2",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 294,
+ 618,
+ 504,
+ 631
+ ],
+ "score": 1.0,
+ "content": "can achieve lower MAE for both energy and forces.",
+ "type": "text"
+ }
+ ],
+ "index": 16
+ }
+ ],
+ "index": 14.5
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 108,
+ 644,
+ 416,
+ 656
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 643,
+ 419,
+ 657
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 643,
+ 419,
+ 657
+ ],
+ "score": 1.0,
+ "content": "E.3 COMPARISON OF TRAINING TIME AND NUMBERS OF PARAMETERS",
+ "type": "text"
+ }
+ ],
+ "index": 17
+ }
+ ],
+ "index": 17
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 666,
+ 505,
+ 732
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 665,
+ 505,
+ 678
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 665,
+ 505,
+ 678
+ ],
+ "score": 1.0,
+ "content": "We compare training time and number of parameters between NequIP (Batzner et al., 2022) and",
+ "type": "text"
+ }
+ ],
+ "index": 18
+ },
+ {
+ "bbox": [
+ 106,
+ 677,
+ 505,
+ 689
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 677,
+ 505,
+ 689
+ ],
+ "score": 1.0,
+ "content": "Equiformer and summarize the results in Table 13. Since NequIP does not report the number of",
+ "type": "text"
+ }
+ ],
+ "index": 19
+ },
+ {
+ "bbox": [
+ 105,
+ 687,
+ 505,
+ 700
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 687,
+ 505,
+ 700
+ ],
+ "score": 1.0,
+ "content": "epochs, we compare the time spent for each epoch and note that NequIP is trained for more than",
+ "type": "text"
+ }
+ ],
+ "index": 20
+ },
+ {
+ "bbox": [
+ 105,
+ 698,
+ 504,
+ 713
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 698,
+ 230,
+ 713
+ ],
+ "score": 1.0,
+ "content": "1000 epochs. Equiformer with",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 231,
+ 699,
+ 273,
+ 710
+ ],
+ "score": 0.9,
+ "content": "L _ { m a x } = 2",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 273,
+ 698,
+ 380,
+ 713
+ ],
+ "score": 1.0,
+ "content": "is faster than NequIP with",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 381,
+ 699,
+ 424,
+ 710
+ ],
+ "score": 0.92,
+ "content": "L _ { m a x } = 3",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 424,
+ 698,
+ 479,
+ 713
+ ],
+ "score": 1.0,
+ "content": "since smaller",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 479,
+ 699,
+ 504,
+ 710
+ ],
+ "score": 0.9,
+ "content": "L _ { m a x }",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 21
+ },
+ {
+ "bbox": [
+ 105,
+ 709,
+ 506,
+ 723
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 709,
+ 252,
+ 723
+ ],
+ "score": 1.0,
+ "content": "is used. Moreover, Equiformer with",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 253,
+ 710,
+ 295,
+ 721
+ ],
+ "score": 0.91,
+ "content": "L _ { m a x } = 2",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 296,
+ 709,
+ 506,
+ 723
+ ],
+ "score": 1.0,
+ "content": "achieves overall better results as we use equivariant",
+ "type": "text"
+ }
+ ],
+ "index": 22
+ },
+ {
+ "bbox": [
+ 105,
+ 720,
+ 507,
+ 734
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 720,
+ 429,
+ 734
+ ],
+ "score": 1.0,
+ "content": "graph attention instead of linear convolution used by NequIP. When increasing",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 430,
+ 721,
+ 454,
+ 732
+ ],
+ "score": 0.91,
+ "content": "L _ { m a x }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 455,
+ 720,
+ 507,
+ 734
+ ],
+ "score": 1.0,
+ "content": "from 2 to 3,",
+ "type": "text"
+ }
+ ],
+ "index": 23
+ }
+ ],
+ "index": 20.5
+ }
+ ],
+ "page_idx": 26,
+ "page_size": [
+ 612,
+ 792
+ ],
+ "discarded_blocks": [
+ {
+ "type": "discarded",
+ "bbox": [
+ 107,
+ 27,
+ 293,
+ 37
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 26,
+ 293,
+ 38
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 26,
+ 293,
+ 38
+ ],
+ "score": 1.0,
+ "content": "Published as a conference paper at ICLR 2023",
+ "type": "text"
+ }
+ ]
+ }
+ ]
+ },
+ {
+ "type": "discarded",
+ "bbox": [
+ 300,
+ 751,
+ 311,
+ 760
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 298,
+ 750,
+ 313,
+ 765
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 298,
+ 750,
+ 313,
+ 765
+ ],
+ "score": 1.0,
+ "content": "",
+ "type": "text",
+ "height": 15,
+ "width": 15
+ }
+ ]
+ }
+ ]
+ }
+ ],
+ "para_blocks": [
+ {
+ "type": "table",
+ "bbox": [
+ 176,
+ 80,
+ 431,
+ 119
+ ],
+ "blocks": [
+ {
+ "type": "table_body",
+ "bbox": [
+ 176,
+ 80,
+ 431,
+ 119
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 176,
+ 80,
+ 431,
+ 119
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 176,
+ 80,
+ 431,
+ 119
+ ],
+ "score": 0.955,
+ "html": "| Hyper-parameters | Value or description |
| Optimizer | AdamW |
| Learning rate scheduling | Cosine learning rate with linear warmup |
| Warmup epochs | 10 1 × 10-4,2 × 10-4,5× 10-4 |
| Maximum learning rate Batch size | 5,8 |
| Number of epochs | 1500,2000 |
| 1×10-6 |
| Weight decay Dropout rate | 0.0 |
| |
| Weight for energy loss Weight for force loss | 1 80 |
| |
| Cutoff radius (A) | 5 |
| Number of radial bases | 32 |
| Hidden sizes of radial functions | 64 2 |
| Number of hidden layers in radial functions Equiformer (Lmax = 2) |
| Embedding dimension dembed Spherical harmonics embedding dimension d sh Number of attention heads h 4 Attention head dimension dhead Hidden dimension in feed forward networks d f fn | [(128,0),(64,1),(32,2)] [(1,0),(1,1),(1,2)] [(32,0),(16,1),(8,2)] [(384,0),(192,1),(96,2)] |
| Output feature dimension dfeature Equiformer (Lmax = 3) |
| [(512,0)] |
| |
| Number of Transformer blocks | 6 |
|
| Embedding dimension dembed | [(128,0),(64,1),(64,2),(32,3)] |
| Spherical harmonics embedding dimension d sh | [(1,0),(1,1),(1,2),(1,3)] |
| Number of attention heads h | 4 |
| |
| Attention head dimension dhead | [(32,0),(16,1),(16,2),(8,3)] |
| Hidden dimension in feed forward networks dffn | (384,0),(192,1),(192,2),(96,3)] |
",
+ "type": "table",
+ "image_path": "88f2ba9303d4afe3f93464c2ec9204befeefe568084e0fc9a152478e1b2979e5.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 5,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 149,
+ 146,
+ 459,
+ 252.66666666666669
+ ],
+ "spans": [],
+ "index": 4
+ },
+ {
+ "bbox": [
+ 149,
+ 252.66666666666669,
+ 459,
+ 359.33333333333337
+ ],
+ "spans": [],
+ "index": 5
+ },
+ {
+ "bbox": [
+ 149,
+ 359.33333333333337,
+ 459,
+ 466.00000000000006
+ ],
+ "spans": [],
+ "index": 6
+ }
+ ]
+ },
+ {
+ "type": "table_caption",
+ "bbox": [
+ 105,
+ 469,
+ 505,
+ 492
+ ],
+ "group_id": 1,
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 467,
+ 505,
+ 481
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 467,
+ 362,
+ 481
+ ],
+ "score": 1.0,
+ "content": "Table 11: Hyper-parameters for MD17 dataset. We denote",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 362,
+ 469,
+ 376,
+ 480
+ ],
+ "score": 0.86,
+ "content": "C _ { L }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 377,
+ 467,
+ 399,
+ 481
+ ],
+ "score": 1.0,
+ "content": "type-",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 400,
+ 469,
+ 407,
+ 479
+ ],
+ "score": 0.76,
+ "content": "L",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 408,
+ 467,
+ 453,
+ 481
+ ],
+ "score": 1.0,
+ "content": "vectors as",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 453,
+ 469,
+ 486,
+ 481
+ ],
+ "score": 0.92,
+ "content": "( C _ { L } , L )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 486,
+ 467,
+ 505,
+ 481
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ }
+ ],
+ "index": 7
+ },
+ {
+ "bbox": [
+ 107,
+ 478,
+ 504,
+ 493
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 480,
+ 133,
+ 493
+ ],
+ "score": 0.91,
+ "content": "C _ { ( L , p ) }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 133,
+ 478,
+ 155,
+ 493
+ ],
+ "score": 1.0,
+ "content": "type-",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 156,
+ 480,
+ 180,
+ 492
+ ],
+ "score": 0.92,
+ "content": "( L , p )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 181,
+ 478,
+ 224,
+ 493
+ ],
+ "score": 1.0,
+ "content": "vectors as",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 225,
+ 479,
+ 279,
+ 492
+ ],
+ "score": 0.94,
+ "content": "( C _ { ( L , p ) } , L , p )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 279,
+ 478,
+ 504,
+ 493
+ ],
+ "score": 1.0,
+ "content": "and use brackets to represent concatenations of vectors.",
+ "type": "text"
+ }
+ ],
+ "index": 8
+ }
+ ],
+ "index": 7.5
+ }
+ ],
+ "index": 6.25
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 514,
+ 505,
+ 547
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 514,
+ 505,
+ 527
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 514,
+ 505,
+ 527
+ ],
+ "score": 1.0,
+ "content": "We use one A5000 GPU with 24GB to train different models for each molecule. Training Equiformer",
+ "type": "text"
+ }
+ ],
+ "index": 9
+ },
+ {
+ "bbox": [
+ 104,
+ 524,
+ 506,
+ 539
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 524,
+ 127,
+ 539
+ ],
+ "score": 1.0,
+ "content": "with",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 127,
+ 526,
+ 171,
+ 537
+ ],
+ "score": 0.92,
+ "content": "L _ { m a x } = 2",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 171,
+ 524,
+ 391,
+ 539
+ ],
+ "score": 1.0,
+ "content": "takes about 15.4 hours, and training Equiformer with",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 392,
+ 526,
+ 435,
+ 537
+ ],
+ "score": 0.91,
+ "content": "L _ { m a x } = 3",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 435,
+ 524,
+ 506,
+ 539
+ ],
+ "score": 1.0,
+ "content": "takes about 54.4",
+ "type": "text"
+ }
+ ],
+ "index": 10
+ },
+ {
+ "bbox": [
+ 105,
+ 536,
+ 135,
+ 549
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 536,
+ 135,
+ 549
+ ],
+ "score": 1.0,
+ "content": "hours.",
+ "type": "text"
+ }
+ ],
+ "index": 11
+ }
+ ],
+ "index": 10,
+ "bbox_fs": [
+ 104,
+ 514,
+ 506,
+ 549
+ ]
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 107,
+ 563,
+ 337,
+ 574
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 562,
+ 339,
+ 577
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 562,
+ 339,
+ 577
+ ],
+ "score": 1.0,
+ "content": "E.2 ADDITIONAL COMPARISON TO TORCHMD-NET",
+ "type": "text"
+ }
+ ],
+ "index": 12
+ }
+ ],
+ "index": 12
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 584,
+ 505,
+ 629
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 585,
+ 506,
+ 596
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 585,
+ 506,
+ 596
+ ],
+ "score": 1.0,
+ "content": "Since TorchMD-NET (Thölke & Fabritiis, 2022) is also an equivariant Transformer but uses dot prod-",
+ "type": "text"
+ }
+ ],
+ "index": 13
+ },
+ {
+ "bbox": [
+ 106,
+ 596,
+ 505,
+ 608
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 596,
+ 505,
+ 608
+ ],
+ "score": 1.0,
+ "content": "uct attention instead of the proposed equivariant graph attention, we provide additional comparisons",
+ "type": "text"
+ }
+ ],
+ "index": 14
+ },
+ {
+ "bbox": [
+ 105,
+ 606,
+ 505,
+ 619
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 606,
+ 505,
+ 619
+ ],
+ "score": 1.0,
+ "content": "in Table 12. For each molecule, we adjust the ratio of the weight for force loss to the weight for",
+ "type": "text"
+ }
+ ],
+ "index": 15
+ },
+ {
+ "bbox": [
+ 105,
+ 618,
+ 504,
+ 631
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 618,
+ 251,
+ 631
+ ],
+ "score": 1.0,
+ "content": "energy loss so that Equiformer with",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 252,
+ 618,
+ 294,
+ 629
+ ],
+ "score": 0.92,
+ "content": "L _ { m a x } = 2",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 294,
+ 618,
+ 504,
+ 631
+ ],
+ "score": 1.0,
+ "content": "can achieve lower MAE for both energy and forces.",
+ "type": "text"
+ }
+ ],
+ "index": 16
+ }
+ ],
+ "index": 14.5,
+ "bbox_fs": [
+ 105,
+ 585,
+ 506,
+ 631
+ ]
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 108,
+ 644,
+ 416,
+ 656
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 643,
+ 419,
+ 657
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 643,
+ 419,
+ 657
+ ],
+ "score": 1.0,
+ "content": "E.3 COMPARISON OF TRAINING TIME AND NUMBERS OF PARAMETERS",
+ "type": "text"
+ }
+ ],
+ "index": 17
+ }
+ ],
+ "index": 17
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 666,
+ 505,
+ 732
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 665,
+ 505,
+ 678
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 665,
+ 505,
+ 678
+ ],
+ "score": 1.0,
+ "content": "We compare training time and number of parameters between NequIP (Batzner et al., 2022) and",
+ "type": "text"
+ }
+ ],
+ "index": 18
+ },
+ {
+ "bbox": [
+ 106,
+ 677,
+ 505,
+ 689
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 677,
+ 505,
+ 689
+ ],
+ "score": 1.0,
+ "content": "Equiformer and summarize the results in Table 13. Since NequIP does not report the number of",
+ "type": "text"
+ }
+ ],
+ "index": 19
+ },
+ {
+ "bbox": [
+ 105,
+ 687,
+ 505,
+ 700
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 687,
+ 505,
+ 700
+ ],
+ "score": 1.0,
+ "content": "epochs, we compare the time spent for each epoch and note that NequIP is trained for more than",
+ "type": "text"
+ }
+ ],
+ "index": 20
+ },
+ {
+ "bbox": [
+ 105,
+ 698,
+ 504,
+ 713
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 698,
+ 230,
+ 713
+ ],
+ "score": 1.0,
+ "content": "1000 epochs. Equiformer with",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 231,
+ 699,
+ 273,
+ 710
+ ],
+ "score": 0.9,
+ "content": "L _ { m a x } = 2",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 273,
+ 698,
+ 380,
+ 713
+ ],
+ "score": 1.0,
+ "content": "is faster than NequIP with",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 381,
+ 699,
+ 424,
+ 710
+ ],
+ "score": 0.92,
+ "content": "L _ { m a x } = 3",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 424,
+ 698,
+ 479,
+ 713
+ ],
+ "score": 1.0,
+ "content": "since smaller",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 479,
+ 699,
+ 504,
+ 710
+ ],
+ "score": 0.9,
+ "content": "L _ { m a x }",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 21
+ },
+ {
+ "bbox": [
+ 105,
+ 709,
+ 506,
+ 723
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 709,
+ 252,
+ 723
+ ],
+ "score": 1.0,
+ "content": "is used. Moreover, Equiformer with",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 253,
+ 710,
+ 295,
+ 721
+ ],
+ "score": 0.91,
+ "content": "L _ { m a x } = 2",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 296,
+ 709,
+ 506,
+ 723
+ ],
+ "score": 1.0,
+ "content": "achieves overall better results as we use equivariant",
+ "type": "text"
+ }
+ ],
+ "index": 22
+ },
+ {
+ "bbox": [
+ 105,
+ 720,
+ 507,
+ 734
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 720,
+ 429,
+ 734
+ ],
+ "score": 1.0,
+ "content": "graph attention instead of linear convolution used by NequIP. When increasing",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 430,
+ 721,
+ 454,
+ 732
+ ],
+ "score": 0.91,
+ "content": "L _ { m a x }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 455,
+ 720,
+ 507,
+ 734
+ ],
+ "score": 1.0,
+ "content": "from 2 to 3,",
+ "type": "text"
+ }
+ ],
+ "index": 23
+ }
+ ],
+ "index": 20.5,
+ "bbox_fs": [
+ 105,
+ 665,
+ 507,
+ 734
+ ]
+ }
+ ]
+ },
+ {
+ "preproc_blocks": [
+ {
+ "type": "table",
+ "bbox": [
+ 82,
+ 80,
+ 529,
+ 124
+ ],
+ "blocks": [
+ {
+ "type": "table_body",
+ "bbox": [
+ 82,
+ 80,
+ 529,
+ 124
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 82,
+ 80,
+ 529,
+ 124
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 82,
+ 80,
+ 529,
+ 124
+ ],
+ "score": 0.97,
+ "html": "| Hyper-parameters | Value or description |
| Optimizer | AdamW |
| Learning rate scheduling | Cosine learning rate with linear warmup |
| Warmup epochs | 2 |
| Maximum learning rate | 2×10-4 |
| Batch size | 32 |
| Number of epochs | 20 |
| Weight decay | 1×10-3 |
| Dropout rate | 0.2 |
| Cutoff radius (A) | 5 |
| Number of radial basis | 128 |
| Hidden size of radial function | 64 |
| Numberofhiddenlayers in radial function | 2 |
| Equiformer |
| Numberof Transformer blocks Embedding dimension dembed | 6 |
| Spherical harmonics embedding dimension dsh | [(256,0),(128,1)] [(1,0),(1,1)] |
| Number of attention heads h | 8 |
| Attention head dimension dhead | [(32,0),(16,1)] |
| Hidden dimension in feed forward networks d f fn | (768,0),(384,1)] |
| Output feature dimension d feature | [(512,0)] |
| |
| E(3)-Equiformer |
| Number of Transformer blocks Embedding dimension dembed | 6 [(256,0,e),(64,0,0),(64,1,e),(64,1,0)] |
| Spherical harmonics embedding dimension dsh | [(1,0,e),(1,1,0)] |
| Number of attention heads h | 8 |
| Attention head dimension dhead | [(32,0,e),(8,0,0),(8,1,e),(8,1,0)] |
| Hidden dimension in feed forward networks d ffn | [(768,0,e),(192,0,0),(192,1,e),(192,1,0)] |
| Output feature dimension d feature | [(512,0,e)] |
",
+ "type": "table",
+ "image_path": "c50d7c02fb63abd9dacfb09debf66eb69839a4091800debba15edf4bdaf4d099.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 1,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 141,
+ 81,
+ 468,
+ 180.33333333333331
+ ],
+ "spans": [],
+ "index": 0
+ },
+ {
+ "bbox": [
+ 141,
+ 180.33333333333331,
+ 468,
+ 279.66666666666663
+ ],
+ "spans": [],
+ "index": 1
+ },
+ {
+ "bbox": [
+ 141,
+ 279.66666666666663,
+ 468,
+ 378.99999999999994
+ ],
+ "spans": [],
+ "index": 2
+ }
+ ]
+ },
+ {
+ "type": "table_caption",
+ "bbox": [
+ 106,
+ 381,
+ 507,
+ 415
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 380,
+ 506,
+ 394
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 380,
+ 506,
+ 394
+ ],
+ "score": 1.0,
+ "content": "Table 14: Hyper-parameters for OC20 dataset under the setting of training without IS2RS aux-",
+ "type": "text"
+ }
+ ],
+ "index": 3
+ },
+ {
+ "bbox": [
+ 105,
+ 390,
+ 505,
+ 406
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 390,
+ 197,
+ 406
+ ],
+ "score": 1.0,
+ "content": "iliary task. We denote",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 198,
+ 392,
+ 212,
+ 403
+ ],
+ "score": 0.87,
+ "content": "C _ { L }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 212,
+ 390,
+ 233,
+ 406
+ ],
+ "score": 1.0,
+ "content": "type-",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 234,
+ 393,
+ 241,
+ 402
+ ],
+ "score": 0.71,
+ "content": "L",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 241,
+ 390,
+ 283,
+ 406
+ ],
+ "score": 1.0,
+ "content": "vectors as",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 284,
+ 392,
+ 316,
+ 404
+ ],
+ "score": 0.93,
+ "content": "( C _ { L } , L )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 317,
+ 390,
+ 334,
+ 406
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 334,
+ 392,
+ 361,
+ 405
+ ],
+ "score": 0.92,
+ "content": "C _ { ( L , p ) }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 361,
+ 390,
+ 383,
+ 406
+ ],
+ "score": 1.0,
+ "content": "type-",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 383,
+ 392,
+ 407,
+ 404
+ ],
+ "score": 0.91,
+ "content": "( L , p )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 407,
+ 390,
+ 450,
+ 406
+ ],
+ "score": 1.0,
+ "content": "vectors as",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 450,
+ 392,
+ 505,
+ 405
+ ],
+ "score": 0.92,
+ "content": "( C _ { ( L , p ) } , L , p )",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 4
+ },
+ {
+ "bbox": [
+ 106,
+ 403,
+ 330,
+ 416
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 403,
+ 330,
+ 416
+ ],
+ "score": 1.0,
+ "content": "and use brackets to represent concatenations of vectors.",
+ "type": "text"
+ }
+ ],
+ "index": 5
+ }
+ ],
+ "index": 4
+ }
+ ],
+ "index": 2.5
+ },
+ {
+ "type": "table",
+ "bbox": [
+ 99,
+ 426,
+ 513,
+ 509
+ ],
+ "blocks": [
+ {
+ "type": "table_body",
+ "bbox": [
+ 99,
+ 426,
+ 513,
+ 509
+ ],
+ "group_id": 1,
+ "lines": [
+ {
+ "bbox": [
+ 99,
+ 426,
+ 513,
+ 509
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 99,
+ 426,
+ 513,
+ 509
+ ],
+ "score": 0.974,
+ "html": "| Hyper-parameters | Value or description |
| Optimizer | AdamW |
| Learning rate scheduling | Cosine learning rate with linear warmup |
| Warmup epochs | 2 |
| Maximum learning rate | 2×10-4 |
| Batch size | 32 |
| Number of epochs | 20 |
| Weight decay | 1×10-3 |
| Dropout rate | 0.2 |
| Cutoff radius (A) | 5 |
| Number of radial basis | 128 |
| Hidden size of radial function | 64 |
| Numberofhiddenlayers in radial function | 2 |
| Equiformer |
| Numberof Transformer blocks Embedding dimension dembed | 6 |
| Spherical harmonics embedding dimension dsh | [(256,0),(128,1)] [(1,0),(1,1)] |
| Number of attention heads h | 8 |
| Attention head dimension dhead | [(32,0),(16,1)] |
| Hidden dimension in feed forward networks d f fn | (768,0),(384,1)] |
| Output feature dimension d feature | [(512,0)] |
| |
| E(3)-Equiformer |
| Number of Transformer blocks Embedding dimension dembed | 6 [(256,0,e),(64,0,0),(64,1,e),(64,1,0)] |
| Spherical harmonics embedding dimension dsh | [(1,0,e),(1,1,0)] |
| Number of attention heads h | 8 |
| Attention head dimension dhead | [(32,0,e),(8,0,0),(8,1,e),(8,1,0)] |
| Hidden dimension in feed forward networks d ffn | [(768,0,e),(192,0,0),(192,1,e),(192,1,0)] |
| Output feature dimension d feature | [(512,0,e)] |
",
+ "type": "table",
+ "image_path": "c50d7c02fb63abd9dacfb09debf66eb69839a4091800debba15edf4bdaf4d099.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 1,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 141,
+ 81,
+ 468,
+ 180.33333333333331
+ ],
+ "spans": [],
+ "index": 0
+ },
+ {
+ "bbox": [
+ 141,
+ 180.33333333333331,
+ 468,
+ 279.66666666666663
+ ],
+ "spans": [],
+ "index": 1
+ },
+ {
+ "bbox": [
+ 141,
+ 279.66666666666663,
+ 468,
+ 378.99999999999994
+ ],
+ "spans": [],
+ "index": 2
+ }
+ ]
+ },
+ {
+ "type": "table_caption",
+ "bbox": [
+ 106,
+ 381,
+ 507,
+ 415
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 380,
+ 506,
+ 394
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 380,
+ 506,
+ 394
+ ],
+ "score": 1.0,
+ "content": "Table 14: Hyper-parameters for OC20 dataset under the setting of training without IS2RS aux-",
+ "type": "text"
+ }
+ ],
+ "index": 3
+ },
+ {
+ "bbox": [
+ 105,
+ 390,
+ 505,
+ 406
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 390,
+ 197,
+ 406
+ ],
+ "score": 1.0,
+ "content": "iliary task. We denote",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 198,
+ 392,
+ 212,
+ 403
+ ],
+ "score": 0.87,
+ "content": "C _ { L }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 212,
+ 390,
+ 233,
+ 406
+ ],
+ "score": 1.0,
+ "content": "type-",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 234,
+ 393,
+ 241,
+ 402
+ ],
+ "score": 0.71,
+ "content": "L",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 241,
+ 390,
+ 283,
+ 406
+ ],
+ "score": 1.0,
+ "content": "vectors as",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 284,
+ 392,
+ 316,
+ 404
+ ],
+ "score": 0.93,
+ "content": "( C _ { L } , L )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 317,
+ 390,
+ 334,
+ 406
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 334,
+ 392,
+ 361,
+ 405
+ ],
+ "score": 0.92,
+ "content": "C _ { ( L , p ) }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 361,
+ 390,
+ 383,
+ 406
+ ],
+ "score": 1.0,
+ "content": "type-",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 383,
+ 392,
+ 407,
+ 404
+ ],
+ "score": 0.91,
+ "content": "( L , p )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 407,
+ 390,
+ 450,
+ 406
+ ],
+ "score": 1.0,
+ "content": "vectors as",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 450,
+ 392,
+ 505,
+ 405
+ ],
+ "score": 0.92,
+ "content": "( C _ { ( L , p ) } , L , p )",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 4
+ },
+ {
+ "bbox": [
+ 106,
+ 403,
+ 330,
+ 416
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 403,
+ 330,
+ 416
+ ],
+ "score": 1.0,
+ "content": "and use brackets to represent concatenations of vectors.",
+ "type": "text"
+ }
+ ],
+ "index": 5
+ }
+ ],
+ "index": 4
+ }
+ ],
+ "index": 2.5
+ },
+ {
+ "type": "table",
+ "bbox": [
+ 99,
+ 426,
+ 513,
+ 509
+ ],
+ "blocks": [
+ {
+ "type": "table_body",
+ "bbox": [
+ 99,
+ 426,
+ 513,
+ 509
+ ],
+ "group_id": 1,
+ "lines": [
+ {
+ "bbox": [
+ 99,
+ 426,
+ 513,
+ 509
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 99,
+ 426,
+ 513,
+ 509
+ ],
+ "score": 0.974,
+ "html": "| Methods | Task Units | α a | △ε meV | εHOMO meV | εLUMO meV | μ D | Cv cal/mol K | G meV | H meV | R a | U meV | U meV | ZPVE meV |
| NMP(Gilmer et al., 2017)t | | .092 | 69 | 43 | 38 | .030 | .040 | 19 | 17 | .180 | 20 | 20 | 1.50 |
| SchNet (Schutt et al.,2017) | | .235 | 63 | 41 | 34 | .033 | .033 | 14 | 14 | .073 | 19 | 14 | 1.70 |
| Cormorant (Anderson et al., 2019)† | | .085 | 61 | 34 | 38 | .038 | .026 | 20 | 21 | .961 | 21 | 22 | 2.03 |
| LieConv (Finzi et al.,020)t | .084 | 49 | | 30 | 25 | .032 | .038 | 22 | 24 | .800 | 19 | 19 | 2.28 |
| DimeNet++ (Gasteiger et al., 2020a) | | .044 | 33 | 25 | 20 | .030 | .023 | 8 | 7 | .331 | 6 | 6 | 1.21 |
| TFN (Thomas et al., 2018)† | | .223 | 58 | 40 | 38 | .064 | .101 | - | - | - | - | - | - |
| SE(3)-Transformer(Fuchs etal.,2020)† | | .142 | 53 | 35 | 33 | .051 | .054 | - | - | - | - | - | - |
| EGNN (Satorras et al.,2021)† | .071 | 48 | | 29 | 25 | .029 | .031 | 12 | 12 | .106 | 12 | 11 | 1.55 |
| PaiNN (Schutt et al.,2021) | .045 | | 46 | 28 | 20 | .012 | .024 | 7.35 | 5.98 | .066 | 5.83 | 5.85 | 1.28 |
| TorchMD-NET(Tholke & Fabritiis,2022) | | .059 | 36 | 20 | 18 | .011 | .026 | 7.62 | 6.16 | .033 | 6.38 | 6.15 | 1.84 |
| SphereNet (Liu et al.,2022) | .046 | 32 | | 23 | 18 | .026 | .021 | 8 | 6 | .292 | 7 | 6 | 1.12 |
| SEGNN (Brandstetter et al.,2022)† | .060 | 42 | | 24 | 21 | .023 | .031 | 15 | 16 | .660 | 13 | 15 | 1.62 |
| EQGAT (Le et al.,2022) | .053 | 32 | | 20 | 16 | .011 | .024 | 23 | 24 | .382 | 25 | 25 | 2.00 |
| Equiformer | | .046 | 30 | 15 | 14 | .011 | .023 | 7.63 | 6.63 | .251 | 6.74 | 6.59 | 1.26 |
"
+ },
+ {
+ "category_id": 1,
+ "poly": [
+ 299,
+ 1676,
+ 1403,
+ 1676,
+ 1403,
+ 1829,
+ 299,
+ 1829
+ ],
+ "score": 0.974
+ },
+ {
+ "category_id": 1,
+ "poly": [
+ 297,
+ 1147,
+ 1404,
+ 1147,
+ 1404,
+ 1268,
+ 297,
+ 1268
+ ],
+ "score": 0.971
+ },
+ {
+ "category_id": 1,
+ "poly": [
+ 299,
+ 1911,
+ 1404,
+ 1911,
+ 1404,
+ 2034,
+ 299,
+ 2034
+ ],
+ "score": 0.97
+ },
+ {
+ "category_id": 1,
+ "poly": [
+ 295,
+ 1278,
+ 1400,
+ 1278,
+ 1400,
+ 1342,
+ 295,
+ 1342
+ ],
+ "score": 0.94
+ },
+ {
+ "category_id": 1,
+ "poly": [
+ 298,
+ 1839,
+ 1401,
+ 1839,
+ 1401,
+ 1902,
+ 298,
+ 1902
+ ],
+ "score": 0.936
+ },
+ {
+ "category_id": 2,
+ "poly": [
+ 300,
+ 76,
+ 814,
+ 76,
+ 814,
+ 105,
+ 300,
+ 105
+ ],
+ "score": 0.885
+ },
+ {
+ "category_id": 0,
+ "poly": [
+ 298,
+ 866,
+ 541,
+ 866,
+ 541,
+ 900,
+ 298,
+ 900
+ ],
+ "score": 0.873
+ },
+ {
+ "category_id": 0,
+ "poly": [
+ 298,
+ 1113,
+ 432,
+ 1113,
+ 432,
+ 1143,
+ 298,
+ 1143
+ ],
+ "score": 0.859
+ },
+ {
+ "category_id": 0,
+ "poly": [
+ 298,
+ 1643,
+ 445,
+ 1643,
+ 445,
+ 1672,
+ 298,
+ 1672
+ ],
+ "score": 0.851
+ },
+ {
+ "category_id": 7,
+ "poly": [
+ 303,
+ 820,
+ 1379,
+ 820,
+ 1379,
+ 851,
+ 303,
+ 851
+ ],
+ "score": 0.769
+ },
+ {
+ "category_id": 2,
+ "poly": [
+ 841,
+ 2088,
+ 857,
+ 2088,
+ 857,
+ 2111,
+ 841,
+ 2111
+ ],
+ "score": 0.74
+ },
+ {
+ "category_id": 6,
+ "poly": [
+ 367,
+ 565,
+ 1317,
+ 565,
+ 1317,
+ 596,
+ 367,
+ 596
+ ],
+ "score": 0.725
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 738,
+ 1912,
+ 875,
+ 1912,
+ 875,
+ 1943,
+ 738,
+ 1943
+ ],
+ "score": 0.91,
+ "latex": "L _ { m a x } \\ = \\ 2"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 1361,
+ 1532,
+ 1396,
+ 1532,
+ 1396,
+ 1563,
+ 1361,
+ 1563
+ ],
+ "score": 0.87,
+ "latex": "R ^ { 2 }"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 1290,
+ 1538,
+ 1311,
+ 1538,
+ 1311,
+ 1567,
+ 1290,
+ 1567
+ ],
+ "score": 0.83,
+ "latex": "\\mu"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 264,
+ 742,
+ 340,
+ 742,
+ 340,
+ 761,
+ 264,
+ 761
+ ],
+ "score": 0.76,
+ "latex": "( L _ { m a x } = 3 )"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 870,
+ 566,
+ 888,
+ 566,
+ 888,
+ 598,
+ 870,
+ 598
+ ],
+ "score": 0.61,
+ "latex": "\\dagger"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 551,
+ 1177,
+ 609,
+ 1177,
+ 609,
+ 1206,
+ 551,
+ 1206
+ ],
+ "score": 0.28,
+ "latex": "1 3 4 \\mathrm { k }"
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 295.0,
+ 72.0,
+ 817.0,
+ 72.0,
+ 817.0,
+ 108.0,
+ 295.0,
+ 108.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 291.0,
+ 861.0,
+ 545.0,
+ 861.0,
+ 545.0,
+ 906.0,
+ 291.0,
+ 906.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 290.0,
+ 1107.0,
+ 437.0,
+ 1107.0,
+ 437.0,
+ 1153.0,
+ 290.0,
+ 1153.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 292.0,
+ 1636.0,
+ 450.0,
+ 1636.0,
+ 450.0,
+ 1679.0,
+ 292.0,
+ 1679.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 315.0,
+ 814.0,
+ 1382.0,
+ 814.0,
+ 1382.0,
+ 856.0,
+ 315.0,
+ 856.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 839.0,
+ 2086.0,
+ 860.0,
+ 2086.0,
+ 860.0,
+ 2118.0,
+ 839.0,
+ 2118.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 377.0,
+ 562.0,
+ 869.0,
+ 562.0,
+ 869.0,
+ 601.0,
+ 377.0,
+ 601.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 889.0,
+ 562.0,
+ 1322.0,
+ 562.0,
+ 1322.0,
+ 601.0,
+ 889.0,
+ 601.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 295.0,
+ 1350.0,
+ 1403.0,
+ 1350.0,
+ 1403.0,
+ 1386.0,
+ 295.0,
+ 1386.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 294.0,
+ 1378.0,
+ 1405.0,
+ 1378.0,
+ 1405.0,
+ 1417.0,
+ 294.0,
+ 1417.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 294.0,
+ 1408.0,
+ 1406.0,
+ 1408.0,
+ 1406.0,
+ 1451.0,
+ 294.0,
+ 1451.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 294.0,
+ 1442.0,
+ 1405.0,
+ 1442.0,
+ 1405.0,
+ 1478.0,
+ 294.0,
+ 1478.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 294.0,
+ 1472.0,
+ 1408.0,
+ 1472.0,
+ 1408.0,
+ 1509.0,
+ 294.0,
+ 1509.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 294.0,
+ 1500.0,
+ 1406.0,
+ 1500.0,
+ 1406.0,
+ 1538.0,
+ 294.0,
+ 1538.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 291.0,
+ 1532.0,
+ 1289.0,
+ 1532.0,
+ 1289.0,
+ 1569.0,
+ 291.0,
+ 1569.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1312.0,
+ 1532.0,
+ 1360.0,
+ 1532.0,
+ 1360.0,
+ 1569.0,
+ 1312.0,
+ 1569.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1397.0,
+ 1532.0,
+ 1406.0,
+ 1532.0,
+ 1406.0,
+ 1569.0,
+ 1397.0,
+ 1569.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 294.0,
+ 1563.0,
+ 1405.0,
+ 1563.0,
+ 1405.0,
+ 1599.0,
+ 294.0,
+ 1599.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 295.0,
+ 1594.0,
+ 1391.0,
+ 1594.0,
+ 1391.0,
+ 1630.0,
+ 295.0,
+ 1630.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 294.0,
+ 909.0,
+ 1410.0,
+ 909.0,
+ 1410.0,
+ 946.0,
+ 294.0,
+ 946.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 295.0,
+ 944.0,
+ 1405.0,
+ 944.0,
+ 1405.0,
+ 976.0,
+ 295.0,
+ 976.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 295.0,
+ 973.0,
+ 1403.0,
+ 973.0,
+ 1403.0,
+ 1009.0,
+ 295.0,
+ 1009.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 294.0,
+ 1002.0,
+ 1406.0,
+ 1002.0,
+ 1406.0,
+ 1039.0,
+ 294.0,
+ 1039.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 295.0,
+ 1034.0,
+ 1408.0,
+ 1034.0,
+ 1408.0,
+ 1070.0,
+ 295.0,
+ 1070.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 295.0,
+ 1066.0,
+ 1178.0,
+ 1066.0,
+ 1178.0,
+ 1098.0,
+ 295.0,
+ 1098.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 297.0,
+ 1677.0,
+ 1403.0,
+ 1677.0,
+ 1403.0,
+ 1706.0,
+ 297.0,
+ 1706.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 293.0,
+ 1704.0,
+ 1407.0,
+ 1704.0,
+ 1407.0,
+ 1742.0,
+ 293.0,
+ 1742.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 293.0,
+ 1737.0,
+ 1407.0,
+ 1737.0,
+ 1407.0,
+ 1772.0,
+ 293.0,
+ 1772.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 294.0,
+ 1766.0,
+ 1407.0,
+ 1766.0,
+ 1407.0,
+ 1803.0,
+ 294.0,
+ 1803.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 293.0,
+ 1797.0,
+ 1292.0,
+ 1797.0,
+ 1292.0,
+ 1832.0,
+ 293.0,
+ 1832.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 295.0,
+ 1145.0,
+ 1405.0,
+ 1145.0,
+ 1405.0,
+ 1180.0,
+ 295.0,
+ 1180.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 292.0,
+ 1174.0,
+ 550.0,
+ 1174.0,
+ 550.0,
+ 1212.0,
+ 292.0,
+ 1212.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 610.0,
+ 1174.0,
+ 1405.0,
+ 1174.0,
+ 1405.0,
+ 1212.0,
+ 610.0,
+ 1212.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 296.0,
+ 1205.0,
+ 1404.0,
+ 1205.0,
+ 1404.0,
+ 1239.0,
+ 296.0,
+ 1239.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 296.0,
+ 1239.0,
+ 1265.0,
+ 1239.0,
+ 1265.0,
+ 1270.0,
+ 296.0,
+ 1270.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 292.0,
+ 1907.0,
+ 737.0,
+ 1907.0,
+ 737.0,
+ 1950.0,
+ 292.0,
+ 1950.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 876.0,
+ 1907.0,
+ 1413.0,
+ 1907.0,
+ 1413.0,
+ 1950.0,
+ 876.0,
+ 1950.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 295.0,
+ 1942.0,
+ 1408.0,
+ 1942.0,
+ 1408.0,
+ 1975.0,
+ 295.0,
+ 1975.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 295.0,
+ 1974.0,
+ 1408.0,
+ 1974.0,
+ 1408.0,
+ 2006.0,
+ 295.0,
+ 2006.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 293.0,
+ 2003.0,
+ 1405.0,
+ 2003.0,
+ 1405.0,
+ 2039.0,
+ 293.0,
+ 2039.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 293.0,
+ 1274.0,
+ 1405.0,
+ 1274.0,
+ 1405.0,
+ 1316.0,
+ 293.0,
+ 1316.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 296.0,
+ 1310.0,
+ 500.0,
+ 1310.0,
+ 500.0,
+ 1343.0,
+ 296.0,
+ 1343.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 293.0,
+ 1834.0,
+ 1406.0,
+ 1834.0,
+ 1406.0,
+ 1876.0,
+ 293.0,
+ 1876.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 296.0,
+ 1871.0,
+ 501.0,
+ 1871.0,
+ 501.0,
+ 1904.0,
+ 296.0,
+ 1904.0
+ ],
+ "score": 1.0,
+ "text": ""
+ }
+ ],
+ "page_info": {
+ "page_no": 6,
+ "width": 1700,
+ "height": 2200
+ }
+ },
+ {
+ "layout_dets": [
+ {
+ "category_id": 1,
+ "poly": [
+ 297,
+ 1335,
+ 1405,
+ 1335,
+ 1405,
+ 1641,
+ 297,
+ 1641
+ ],
+ "score": 0.98
+ },
+ {
+ "category_id": 1,
+ "poly": [
+ 297,
+ 982,
+ 1405,
+ 982,
+ 1405,
+ 1288,
+ 297,
+ 1288
+ ],
+ "score": 0.979
+ },
+ {
+ "category_id": 1,
+ "poly": [
+ 298,
+ 1850,
+ 1404,
+ 1850,
+ 1404,
+ 2033,
+ 298,
+ 2033
+ ],
+ "score": 0.976
+ },
+ {
+ "category_id": 1,
+ "poly": [
+ 297,
+ 1649,
+ 1404,
+ 1649,
+ 1404,
+ 1771,
+ 297,
+ 1771
+ ],
+ "score": 0.973
+ },
+ {
+ "category_id": 1,
+ "poly": [
+ 299,
+ 1780,
+ 1399,
+ 1780,
+ 1399,
+ 1843,
+ 299,
+ 1843
+ ],
+ "score": 0.945
+ },
+ {
+ "category_id": 5,
+ "poly": [
+ 313,
+ 224,
+ 1379,
+ 224,
+ 1379,
+ 449,
+ 313,
+ 449
+ ],
+ "score": 0.928,
+ "html": "| Hyper-parameters | Value or description |
| Optimizer | AdamW |
| Learning rate scheduling Warmup epochs | Cosine learning rate with linear warmup 5 |
| Maximum learning rate | 1.5 × 10-4,5 × 10-4 |
| Batch size | 64,128 |
| |
| Number of epochs | 300,600 |
| Weight decay | 0,5×10-3 |
| Dropout rate | 0.0,0.1, 0.2 |
| Cutoff radius (A) | 5 |
| Number of radial bases | 128 for Gaussian radial basis,8 for radial bessel basis |
| Hidden sizes of radial functions | 64 |
| Number of hidden layers in radial functions | 2 |
| Number of Transformer blocks Embedding dimension dembed | Equiformer 6 [(128,0),(64,1),(32,2)] |
| Spherical harmonics embedding dimension dsh Numberof attention heads h Attention head dimension dhead | [(1,0),(1,1),(1,2)] 4 [(32,0),(16,1),(8,2)] |
| Hidden dimension in feed forward networks dffn Output feature dimension d feature | (384,0),(192,1),(96,2)] [(512,0)] |
| E(3)-Equiformer |
| |
| |
| |
| 6 |
| Number of Transformer blocks | |
| Embedding dimension dembed | |
| [(128,0,e),(32,0,0),(32,1,e),(32,1,0),(16,2,e),(16,2,0)] |
| Spherical harmonics embedding dimension dsh | [(1,0,e),(1,1,0),(1,2,e)] |
| Number of attention heads h | 4 |
| |
| Attention head dimension dhead | [(32,0,e),(8,0,0),(8,1,e),(8,1,0),(4,2,e),(4,2,0)] |
| Hidden dimension in feed forward networks df fn | |
| (384,0,e),(96,0,0),(96,1,e),(96,1,0),(48,2,e),(48,2,0)] |
| |
| Output feature dimension dfeature | [(512,0,e)] |
| |
| |
"
+ },
+ {
+ "category_id": 1,
+ "poly": [
+ 299,
+ 1204,
+ 1404,
+ 1204,
+ 1404,
+ 1296,
+ 299,
+ 1296
+ ],
+ "score": 0.976
+ },
+ {
+ "category_id": 1,
+ "poly": [
+ 299,
+ 1556,
+ 1405,
+ 1556,
+ 1405,
+ 1679,
+ 299,
+ 1679
+ ],
+ "score": 0.975
+ },
+ {
+ "category_id": 1,
+ "poly": [
+ 298,
+ 1311,
+ 1354,
+ 1311,
+ 1354,
+ 1344,
+ 298,
+ 1344
+ ],
+ "score": 0.918
+ },
+ {
+ "category_id": 0,
+ "poly": [
+ 298,
+ 1408,
+ 876,
+ 1408,
+ 876,
+ 1445,
+ 298,
+ 1445
+ ],
+ "score": 0.907
+ },
+ {
+ "category_id": 0,
+ "poly": [
+ 300,
+ 1490,
+ 604,
+ 1490,
+ 604,
+ 1524,
+ 300,
+ 1524
+ ],
+ "score": 0.897
+ },
+ {
+ "category_id": 2,
+ "poly": [
+ 300,
+ 76,
+ 814,
+ 76,
+ 814,
+ 104,
+ 300,
+ 104
+ ],
+ "score": 0.892
+ },
+ {
+ "category_id": 2,
+ "poly": [
+ 835,
+ 2088,
+ 864,
+ 2088,
+ 864,
+ 2112,
+ 835,
+ 2112
+ ],
+ "score": 0.854
+ },
+ {
+ "category_id": 7,
+ "poly": [
+ 298,
+ 1060,
+ 1395,
+ 1060,
+ 1395,
+ 1128,
+ 298,
+ 1128
+ ],
+ "score": 0.611
+ },
+ {
+ "category_id": 6,
+ "poly": [
+ 298,
+ 1060,
+ 1395,
+ 1060,
+ 1395,
+ 1128,
+ 298,
+ 1128
+ ],
+ "score": 0.485
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 546,
+ 1093,
+ 697,
+ 1093,
+ 697,
+ 1130,
+ 546,
+ 1130
+ ],
+ "score": 0.93,
+ "latex": "( C _ { ( L , p ) } , L , p )"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 355,
+ 1093,
+ 424,
+ 1093,
+ 424,
+ 1127,
+ 355,
+ 1127
+ ],
+ "score": 0.92,
+ "latex": "( L , p )"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 816,
+ 1880,
+ 946,
+ 1880,
+ 946,
+ 1911,
+ 816,
+ 1911
+ ],
+ "score": 0.92,
+ "latex": "1 . 5 \\times 1 0 ^ { - 4 }"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 940,
+ 1941,
+ 1047,
+ 1941,
+ 1047,
+ 1972,
+ 940,
+ 1972
+ ],
+ "score": 0.92,
+ "latex": "5 \\times 1 0 ^ { - 3 }"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 298,
+ 1911,
+ 405,
+ 1911,
+ 405,
+ 1941,
+ 298,
+ 1941
+ ],
+ "score": 0.91,
+ "latex": "5 \\times 1 0 ^ { - 4 }"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 749,
+ 1697,
+ 868,
+ 1697,
+ 868,
+ 1726,
+ 749,
+ 1726
+ ],
+ "score": 0.91,
+ "latex": "L _ { m a x } = 2"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 1188,
+ 1061,
+ 1278,
+ 1061,
+ 1278,
+ 1094,
+ 1188,
+ 1094
+ ],
+ "score": 0.91,
+ "latex": "( C _ { L } , L )"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 1329,
+ 1062,
+ 1402,
+ 1062,
+ 1402,
+ 1097,
+ 1329,
+ 1097
+ ],
+ "score": 0.89,
+ "latex": "C _ { ( L , p ) }"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 1077,
+ 1912,
+ 1112,
+ 1912,
+ 1112,
+ 1942,
+ 1077,
+ 1942
+ ],
+ "score": 0.89,
+ "latex": "U _ { 0 }"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 829,
+ 1944,
+ 863,
+ 1944,
+ 863,
+ 1973,
+ 829,
+ 1973
+ ],
+ "score": 0.89,
+ "latex": "U _ { 0 }"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 1061,
+ 1821,
+ 1095,
+ 1821,
+ 1095,
+ 1851,
+ 1061,
+ 1851
+ ],
+ "score": 0.88,
+ "latex": "U _ { 0 }"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 1291,
+ 1883,
+ 1326,
+ 1883,
+ 1326,
+ 1912,
+ 1291,
+ 1912
+ ],
+ "score": 0.88,
+ "latex": "U _ { 0 }"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 601,
+ 1791,
+ 638,
+ 1791,
+ 638,
+ 1821,
+ 601,
+ 1821
+ ],
+ "score": 0.88,
+ "latex": "a _ { i j }"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 421,
+ 1819,
+ 457,
+ 1819,
+ 457,
+ 1848,
+ 421,
+ 1848
+ ],
+ "score": 0.88,
+ "latex": "R ^ { 2 }"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 946,
+ 1061,
+ 985,
+ 1061,
+ 985,
+ 1092,
+ 946,
+ 1092
+ ],
+ "score": 0.87,
+ "latex": "C _ { L }"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 1250,
+ 1788,
+ 1284,
+ 1788,
+ 1284,
+ 1819,
+ 1250,
+ 1819
+ ],
+ "score": 0.87,
+ "latex": "U _ { 0 }"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 298,
+ 1883,
+ 331,
+ 1883,
+ 331,
+ 1912,
+ 298,
+ 1912
+ ],
+ "score": 0.86,
+ "latex": "U _ { 0 }"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 729,
+ 994,
+ 776,
+ 994,
+ 776,
+ 1021,
+ 729,
+ 1021
+ ],
+ "score": 0.81,
+ "latex": "d _ { f f n }"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 1372,
+ 1559,
+ 1397,
+ 1559,
+ 1397,
+ 1586,
+ 1372,
+ 1586
+ ],
+ "score": 0.79,
+ "latex": "U"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 425,
+ 1589,
+ 453,
+ 1589,
+ 453,
+ 1615,
+ 425,
+ 1615
+ ],
+ "score": 0.78,
+ "latex": "H"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 721,
+ 690,
+ 755,
+ 690,
+ 755,
+ 714,
+ 721,
+ 714
+ ],
+ "score": 0.77,
+ "latex": "d _ { s h }"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 721,
+ 922,
+ 755,
+ 922,
+ 755,
+ 945,
+ 721,
+ 945
+ ],
+ "score": 0.75,
+ "latex": "d _ { s h }"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 729,
+ 763,
+ 776,
+ 763,
+ 776,
+ 789,
+ 729,
+ 789
+ ],
+ "score": 0.75,
+ "latex": "d _ { f f n }"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 1046,
+ 1063,
+ 1068,
+ 1063,
+ 1068,
+ 1088,
+ 1046,
+ 1088
+ ],
+ "score": 0.75,
+ "latex": "L"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 298,
+ 1588,
+ 331,
+ 1588,
+ 331,
+ 1619,
+ 298,
+ 1619
+ ],
+ "score": 0.71,
+ "latex": "U _ { 0 }"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 566,
+ 970,
+ 620,
+ 970,
+ 620,
+ 994,
+ 566,
+ 994
+ ],
+ "score": 0.68,
+ "latex": "d _ { h e a d }"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 922,
+ 1913,
+ 946,
+ 1913,
+ 946,
+ 1940,
+ 922,
+ 1940
+ ],
+ "score": 0.64,
+ "latex": "G"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 537,
+ 896,
+ 603,
+ 896,
+ 603,
+ 921,
+ 537,
+ 921
+ ],
+ "score": 0.64,
+ "latex": "d _ { e m b e d }"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 566,
+ 739,
+ 620,
+ 739,
+ 620,
+ 763,
+ 566,
+ 763
+ ],
+ "score": 0.59,
+ "latex": "d _ { h e a d }"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 564,
+ 1018,
+ 642,
+ 1018,
+ 642,
+ 1046,
+ 564,
+ 1046
+ ],
+ "score": 0.55,
+ "latex": "d _ { f e a t u r e }"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 672,
+ 1944,
+ 696,
+ 1944,
+ 696,
+ 1970,
+ 672,
+ 1970
+ ],
+ "score": 0.54,
+ "latex": "G"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 343,
+ 1588,
+ 368,
+ 1588,
+ 368,
+ 1616,
+ 343,
+ 1616
+ ],
+ "score": 0.54,
+ "latex": "G"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 1173,
+ 1789,
+ 1198,
+ 1789,
+ 1198,
+ 1816,
+ 1173,
+ 1816
+ ],
+ "score": 0.53,
+ "latex": "U"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 537,
+ 665,
+ 603,
+ 665,
+ 603,
+ 690,
+ 537,
+ 690
+ ],
+ "score": 0.53,
+ "latex": "d _ { e m b e d }"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 1097,
+ 1789,
+ 1121,
+ 1789,
+ 1121,
+ 1816,
+ 1097,
+ 1816
+ ],
+ "score": 0.51,
+ "latex": "G"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 1248,
+ 1852,
+ 1272,
+ 1852,
+ 1272,
+ 1878,
+ 1248,
+ 1878
+ ],
+ "score": 0.51,
+ "latex": "G"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 996,
+ 1912,
+ 1021,
+ 1912,
+ 1021,
+ 1939,
+ 996,
+ 1939
+ ],
+ "score": 0.5,
+ "latex": "U"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 767,
+ 823,
+ 815,
+ 823,
+ 815,
+ 850,
+ 767,
+ 850
+ ],
+ "score": 0.5,
+ "latex": "E ( 3 )"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 1130,
+ 1882,
+ 1154,
+ 1882,
+ 1154,
+ 1909,
+ 1130,
+ 1909
+ ],
+ "score": 0.48,
+ "latex": "G"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 564,
+ 787,
+ 643,
+ 787,
+ 643,
+ 815,
+ 564,
+ 815
+ ],
+ "score": 0.48,
+ "latex": "d _ { f e a t u r e }"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 904,
+ 1821,
+ 928,
+ 1821,
+ 928,
+ 1849,
+ 904,
+ 1849
+ ],
+ "score": 0.48,
+ "latex": "G"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 708,
+ 1944,
+ 734,
+ 1944,
+ 734,
+ 1970,
+ 708,
+ 1970
+ ],
+ "score": 0.45,
+ "latex": "H"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 1133,
+ 1789,
+ 1160,
+ 1789,
+ 1160,
+ 1816,
+ 1133,
+ 1816
+ ],
+ "score": 0.45,
+ "latex": "H"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 978,
+ 1821,
+ 1003,
+ 1821,
+ 1003,
+ 1849,
+ 978,
+ 1849
+ ],
+ "score": 0.43,
+ "latex": "U"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 746,
+ 1944,
+ 771,
+ 1944,
+ 771,
+ 1970,
+ 746,
+ 1970
+ ],
+ "score": 0.42,
+ "latex": "U"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 1206,
+ 1883,
+ 1231,
+ 1883,
+ 1231,
+ 1910,
+ 1206,
+ 1910
+ ],
+ "score": 0.41,
+ "latex": "U"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 940,
+ 1821,
+ 967,
+ 1821,
+ 967,
+ 1849,
+ 940,
+ 1849
+ ],
+ "score": 0.38,
+ "latex": "H"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 1263,
+ 1851,
+ 1346,
+ 1851,
+ 1346,
+ 1880,
+ 1263,
+ 1880
+ ],
+ "score": 0.38,
+ "latex": "\\ d s _ { r } , H , U"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 1322,
+ 1852,
+ 1346,
+ 1852,
+ 1346,
+ 1879,
+ 1322,
+ 1879
+ ],
+ "score": 0.38,
+ "latex": "U"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 573,
+ 945,
+ 589,
+ 945,
+ 589,
+ 966,
+ 573,
+ 966
+ ],
+ "score": 0.36,
+ "latex": "h"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 935,
+ 1912,
+ 1024,
+ 1912,
+ 1024,
+ 1941,
+ 935,
+ 1941
+ ],
+ "score": 0.35,
+ "latex": "{ \\mathrm { ? } } , H , U ,"
+ },
+ {
+ "category_id": 14,
+ "poly": [
+ 793,
+ 870,
+ 1366,
+ 870,
+ 1366,
+ 1046,
+ 793,
+ 1046
+ ],
+ "score": 0.33,
+ "latex": "\\begin{array} { l } { 6 } \\\\ { \\left[ ( 1 2 8 , 0 , e ) , ( 3 2 , 0 , o ) , ( 3 2 , 1 , e ) , ( 3 2 , 1 , o ) , ( 1 6 , 2 , e ) , ( 1 6 , 2 , o ) \\right] } \\\\ { \\left[ ( 1 , 0 , e ) , ( 1 , 1 , o ) , ( 1 , 2 , e ) \\right] } \\\\ { 4 } \\\\ { \\left[ ( 3 2 , 0 , e ) , ( 8 , 0 , o ) , ( 8 , 1 , e ) , ( 8 , 1 , o ) , ( 4 , 2 , e ) , ( 4 , 2 , o ) \\right] } \\\\ { \\left[ ( 3 8 4 , 0 , e ) , ( 9 6 , 0 , o ) , ( 9 6 , 1 , e ) , ( 9 6 , 1 , o ) , ( 4 8 , 2 , e ) , ( 4 8 , 2 , o ) \\right] } \\\\ { \\left[ ( 5 1 2 , 0 , e ) \\right] } \\end{array}"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 573,
+ 714,
+ 589,
+ 714,
+ 589,
+ 735,
+ 573,
+ 735
+ ],
+ "score": 0.33,
+ "latex": "h"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 1162,
+ 1881,
+ 1231,
+ 1881,
+ 1231,
+ 1911,
+ 1162,
+ 1911
+ ],
+ "score": 0.29,
+ "latex": "H , U"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 958,
+ 1912,
+ 985,
+ 1912,
+ 985,
+ 1939,
+ 958,
+ 1939
+ ],
+ "score": 0.28,
+ "latex": "H"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 297,
+ 1587,
+ 368,
+ 1587,
+ 368,
+ 1620,
+ 297,
+ 1620
+ ],
+ "score": 0.27,
+ "latex": "U _ { 0 } , G"
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 294.0,
+ 1407.0,
+ 880.0,
+ 1407.0,
+ 880.0,
+ 1451.0,
+ 294.0,
+ 1451.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 296.0,
+ 1490.0,
+ 608.0,
+ 1490.0,
+ 608.0,
+ 1527.0,
+ 296.0,
+ 1527.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 295.0,
+ 73.0,
+ 817.0,
+ 73.0,
+ 817.0,
+ 108.0,
+ 295.0,
+ 108.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 829.0,
+ 2084.0,
+ 871.0,
+ 2084.0,
+ 871.0,
+ 2124.0,
+ 829.0,
+ 2124.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 291.0,
+ 1048.0,
+ 945.0,
+ 1048.0,
+ 945.0,
+ 1106.0,
+ 291.0,
+ 1106.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 986.0,
+ 1048.0,
+ 1045.0,
+ 1048.0,
+ 1045.0,
+ 1106.0,
+ 986.0,
+ 1106.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1069.0,
+ 1048.0,
+ 1187.0,
+ 1048.0,
+ 1187.0,
+ 1106.0,
+ 1069.0,
+ 1106.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1279.0,
+ 1048.0,
+ 1328.0,
+ 1048.0,
+ 1328.0,
+ 1106.0,
+ 1279.0,
+ 1106.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 294.0,
+ 1090.0,
+ 354.0,
+ 1090.0,
+ 354.0,
+ 1131.0,
+ 294.0,
+ 1131.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 425.0,
+ 1090.0,
+ 545.0,
+ 1090.0,
+ 545.0,
+ 1131.0,
+ 425.0,
+ 1131.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 698.0,
+ 1090.0,
+ 1323.0,
+ 1090.0,
+ 1323.0,
+ 1131.0,
+ 698.0,
+ 1131.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 291.0,
+ 1048.0,
+ 945.0,
+ 1048.0,
+ 945.0,
+ 1106.0,
+ 291.0,
+ 1106.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 986.0,
+ 1048.0,
+ 1045.0,
+ 1048.0,
+ 1045.0,
+ 1106.0,
+ 986.0,
+ 1106.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1069.0,
+ 1048.0,
+ 1187.0,
+ 1048.0,
+ 1187.0,
+ 1106.0,
+ 1069.0,
+ 1106.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1279.0,
+ 1048.0,
+ 1328.0,
+ 1048.0,
+ 1328.0,
+ 1106.0,
+ 1279.0,
+ 1106.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 294.0,
+ 1090.0,
+ 354.0,
+ 1090.0,
+ 354.0,
+ 1131.0,
+ 294.0,
+ 1131.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 425.0,
+ 1090.0,
+ 545.0,
+ 1090.0,
+ 545.0,
+ 1131.0,
+ 425.0,
+ 1131.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 698.0,
+ 1090.0,
+ 1323.0,
+ 1090.0,
+ 1323.0,
+ 1131.0,
+ 698.0,
+ 1131.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 294.0,
+ 1693.0,
+ 748.0,
+ 1693.0,
+ 748.0,
+ 1731.0,
+ 294.0,
+ 1731.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 869.0,
+ 1693.0,
+ 1408.0,
+ 1693.0,
+ 1408.0,
+ 1731.0,
+ 869.0,
+ 1731.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 296.0,
+ 1726.0,
+ 1404.0,
+ 1726.0,
+ 1404.0,
+ 1757.0,
+ 296.0,
+ 1757.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 295.0,
+ 1755.0,
+ 1404.0,
+ 1755.0,
+ 1404.0,
+ 1790.0,
+ 295.0,
+ 1790.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 294.0,
+ 1785.0,
+ 600.0,
+ 1785.0,
+ 600.0,
+ 1824.0,
+ 294.0,
+ 1824.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 639.0,
+ 1785.0,
+ 1096.0,
+ 1785.0,
+ 1096.0,
+ 1824.0,
+ 639.0,
+ 1824.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1122.0,
+ 1785.0,
+ 1132.0,
+ 1785.0,
+ 1132.0,
+ 1824.0,
+ 1122.0,
+ 1824.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1161.0,
+ 1785.0,
+ 1172.0,
+ 1785.0,
+ 1172.0,
+ 1824.0,
+ 1161.0,
+ 1824.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1199.0,
+ 1785.0,
+ 1249.0,
+ 1785.0,
+ 1249.0,
+ 1824.0,
+ 1199.0,
+ 1824.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1285.0,
+ 1785.0,
+ 1407.0,
+ 1785.0,
+ 1407.0,
+ 1824.0,
+ 1285.0,
+ 1824.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 294.0,
+ 1817.0,
+ 420.0,
+ 1817.0,
+ 420.0,
+ 1855.0,
+ 294.0,
+ 1855.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 458.0,
+ 1817.0,
+ 903.0,
+ 1817.0,
+ 903.0,
+ 1855.0,
+ 458.0,
+ 1855.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 929.0,
+ 1817.0,
+ 939.0,
+ 1817.0,
+ 939.0,
+ 1855.0,
+ 929.0,
+ 1855.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 968.0,
+ 1817.0,
+ 977.0,
+ 1817.0,
+ 977.0,
+ 1855.0,
+ 968.0,
+ 1855.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1004.0,
+ 1817.0,
+ 1060.0,
+ 1817.0,
+ 1060.0,
+ 1855.0,
+ 1004.0,
+ 1855.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1096.0,
+ 1817.0,
+ 1406.0,
+ 1817.0,
+ 1406.0,
+ 1855.0,
+ 1096.0,
+ 1855.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 295.0,
+ 1850.0,
+ 1247.0,
+ 1850.0,
+ 1247.0,
+ 1885.0,
+ 295.0,
+ 1885.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1347.0,
+ 1850.0,
+ 1406.0,
+ 1850.0,
+ 1406.0,
+ 1885.0,
+ 1347.0,
+ 1885.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 332.0,
+ 1880.0,
+ 815.0,
+ 1880.0,
+ 815.0,
+ 1915.0,
+ 332.0,
+ 1915.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 947.0,
+ 1880.0,
+ 1129.0,
+ 1880.0,
+ 1129.0,
+ 1915.0,
+ 947.0,
+ 1915.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1155.0,
+ 1880.0,
+ 1161.0,
+ 1880.0,
+ 1161.0,
+ 1915.0,
+ 1155.0,
+ 1915.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1232.0,
+ 1880.0,
+ 1290.0,
+ 1880.0,
+ 1290.0,
+ 1915.0,
+ 1232.0,
+ 1915.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1327.0,
+ 1880.0,
+ 1406.0,
+ 1880.0,
+ 1406.0,
+ 1915.0,
+ 1327.0,
+ 1915.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 292.0,
+ 1907.0,
+ 297.0,
+ 1907.0,
+ 297.0,
+ 1945.0,
+ 292.0,
+ 1945.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 406.0,
+ 1907.0,
+ 921.0,
+ 1907.0,
+ 921.0,
+ 1945.0,
+ 406.0,
+ 1945.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1025.0,
+ 1907.0,
+ 1076.0,
+ 1907.0,
+ 1076.0,
+ 1945.0,
+ 1025.0,
+ 1945.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1113.0,
+ 1907.0,
+ 1406.0,
+ 1907.0,
+ 1406.0,
+ 1945.0,
+ 1113.0,
+ 1945.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 292.0,
+ 1938.0,
+ 671.0,
+ 1938.0,
+ 671.0,
+ 1977.0,
+ 292.0,
+ 1977.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 697.0,
+ 1938.0,
+ 707.0,
+ 1938.0,
+ 707.0,
+ 1977.0,
+ 697.0,
+ 1977.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 735.0,
+ 1938.0,
+ 745.0,
+ 1938.0,
+ 745.0,
+ 1977.0,
+ 735.0,
+ 1977.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 772.0,
+ 1938.0,
+ 828.0,
+ 1938.0,
+ 828.0,
+ 1977.0,
+ 772.0,
+ 1977.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 864.0,
+ 1938.0,
+ 939.0,
+ 1938.0,
+ 939.0,
+ 1977.0,
+ 864.0,
+ 1977.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1048.0,
+ 1938.0,
+ 1406.0,
+ 1938.0,
+ 1406.0,
+ 1977.0,
+ 1048.0,
+ 1977.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 294.0,
+ 1972.0,
+ 1406.0,
+ 1972.0,
+ 1406.0,
+ 2007.0,
+ 294.0,
+ 2007.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 294.0,
+ 1999.0,
+ 582.0,
+ 1999.0,
+ 582.0,
+ 2034.0,
+ 294.0,
+ 2034.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 293.0,
+ 1202.0,
+ 1406.0,
+ 1202.0,
+ 1406.0,
+ 1240.0,
+ 293.0,
+ 1240.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 295.0,
+ 1237.0,
+ 1404.0,
+ 1237.0,
+ 1404.0,
+ 1270.0,
+ 295.0,
+ 1270.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 297.0,
+ 1268.0,
+ 608.0,
+ 1268.0,
+ 608.0,
+ 1298.0,
+ 297.0,
+ 1298.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 295.0,
+ 1557.0,
+ 1371.0,
+ 1557.0,
+ 1371.0,
+ 1590.0,
+ 295.0,
+ 1590.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1398.0,
+ 1557.0,
+ 1407.0,
+ 1557.0,
+ 1407.0,
+ 1590.0,
+ 1398.0,
+ 1590.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 369.0,
+ 1587.0,
+ 424.0,
+ 1587.0,
+ 424.0,
+ 1620.0,
+ 369.0,
+ 1620.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 454.0,
+ 1587.0,
+ 1405.0,
+ 1587.0,
+ 1405.0,
+ 1620.0,
+ 454.0,
+ 1620.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 293.0,
+ 1616.0,
+ 1405.0,
+ 1616.0,
+ 1405.0,
+ 1655.0,
+ 293.0,
+ 1655.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 297.0,
+ 1650.0,
+ 513.0,
+ 1650.0,
+ 513.0,
+ 1680.0,
+ 297.0,
+ 1680.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 294.0,
+ 1307.0,
+ 1356.0,
+ 1307.0,
+ 1356.0,
+ 1348.0,
+ 294.0,
+ 1348.0
+ ],
+ "score": 1.0,
+ "text": ""
+ }
+ ],
+ "page_info": {
+ "page_no": 24,
+ "width": 1700,
+ "height": 2200
+ }
+ },
+ {
+ "layout_dets": [
+ {
+ "category_id": 1,
+ "poly": [
+ 298,
+ 1075,
+ 1403,
+ 1075,
+ 1403,
+ 1291,
+ 298,
+ 1291
+ ],
+ "score": 0.981
+ },
+ {
+ "category_id": 1,
+ "poly": [
+ 298,
+ 1305,
+ 1405,
+ 1305,
+ 1405,
+ 1490,
+ 298,
+ 1490
+ ],
+ "score": 0.979
+ },
+ {
+ "category_id": 1,
+ "poly": [
+ 298,
+ 1785,
+ 1404,
+ 1785,
+ 1404,
+ 2033,
+ 298,
+ 2033
+ ],
+ "score": 0.978
+ },
+ {
+ "category_id": 1,
+ "poly": [
+ 298,
+ 228,
+ 1404,
+ 228,
+ 1404,
+ 384,
+ 298,
+ 384
+ ],
+ "score": 0.975
+ },
+ {
+ "category_id": 1,
+ "poly": [
+ 299,
+ 1678,
+ 1404,
+ 1678,
+ 1404,
+ 1771,
+ 299,
+ 1771
+ ],
+ "score": 0.971
+ },
+ {
+ "category_id": 5,
+ "poly": [
+ 434,
+ 651,
+ 1251,
+ 651,
+ 1251,
+ 756,
+ 434,
+ 756
+ ],
+ "score": 0.968,
+ "html": "| Hyper-parameters | Value or description |
| Optimizer | AdamW |
| Learning rate scheduling | Cosine learning rate with linear warmup |
| Warmup epochs | 10 1 × 10-4,2 × 10-4,5× 10-4 |
| Maximum learning rate Batch size | 5,8 |
| Number of epochs | 1500,2000 |
| 1×10-6 |
| Weight decay Dropout rate | 0.0 |
| |
| Weight for energy loss Weight for force loss | 1 80 |
| |
| Cutoff radius (A) | 5 |
| Number of radial bases | 32 |
| Hidden sizes of radial functions | 64 2 |
| Number of hidden layers in radial functions Equiformer (Lmax = 2) |
| Embedding dimension dembed Spherical harmonics embedding dimension d sh Number of attention heads h 4 Attention head dimension dhead Hidden dimension in feed forward networks d f fn | [(128,0),(64,1),(32,2)] [(1,0),(1,1),(1,2)] [(32,0),(16,1),(8,2)] [(384,0),(192,1),(96,2)] |
| Output feature dimension dfeature Equiformer (Lmax = 3) |
| [(512,0)] |
| |
| Number of Transformer blocks | 6 |
|
| Embedding dimension dembed | [(128,0),(64,1),(64,2),(32,3)] |
| Spherical harmonics embedding dimension d sh | [(1,0),(1,1),(1,2),(1,3)] |
| Number of attention heads h | 4 |
| |
| Attention head dimension dhead | [(32,0),(16,1),(16,2),(8,3)] |
| Hidden dimension in feed forward networks dffn | (384,0),(192,1),(192,2),(96,3)] |
"
+ },
+ {
+ "category_id": 5,
+ "poly": [
+ 490,
+ 223,
+ 1199,
+ 223,
+ 1199,
+ 333,
+ 490,
+ 333
+ ],
+ "score": 0.955,
+ "html": "| Hyper-parameters | Value or description |
| Optimizer | AdamW |
| Learning rate scheduling | Cosine learning rate with linear warmup |
| Warmup epochs | 2 |
| Maximum learning rate | 2×10-4 |
| Batch size | 32 |
| Number of epochs | 20 |
| Weight decay | 1×10-3 |
| Dropout rate | 0.2 |
| Cutoff radius (A) | 5 |
| Number of radial basis | 128 |
| Hidden size of radial function | 64 |
| Numberofhiddenlayers in radial function | 2 |
| Equiformer |
| Numberof Transformer blocks Embedding dimension dembed | 6 |
| Spherical harmonics embedding dimension dsh | [(256,0),(128,1)] [(1,0),(1,1)] |
| Number of attention heads h | 8 |
| Attention head dimension dhead | [(32,0),(16,1)] |
| Hidden dimension in feed forward networks d f fn | (768,0),(384,1)] |
| Output feature dimension d feature | [(512,0)] |
| |
| E(3)-Equiformer |
| Number of Transformer blocks Embedding dimension dembed | 6 [(256,0,e),(64,0,0),(64,1,e),(64,1,0)] |
| Spherical harmonics embedding dimension dsh | [(1,0,e),(1,1,0)] |
| Number of attention heads h | 8 |
| Attention head dimension dhead | [(32,0,e),(8,0,0),(8,1,e),(8,1,0)] |
| Hidden dimension in feed forward networks d ffn | [(768,0,e),(192,0,0),(192,1,e),(192,1,0)] |
| Output feature dimension d feature | [(512,0,e)] |
"
+ },
+ {
+ "category_id": 1,
+ "poly": [
+ 297,
+ 1501,
+ 1404,
+ 1501,
+ 1404,
+ 1624,
+ 297,
+ 1624
+ ],
+ "score": 0.972
+ },
+ {
+ "category_id": 1,
+ "poly": [
+ 302,
+ 1721,
+ 1261,
+ 1721,
+ 1261,
+ 1754,
+ 302,
+ 1754
+ ],
+ "score": 0.901
+ },
+ {
+ "category_id": 0,
+ "poly": [
+ 301,
+ 1663,
+ 810,
+ 1663,
+ 810,
+ 1695,
+ 301,
+ 1695
+ ],
+ "score": 0.895
+ },
+ {
+ "category_id": 2,
+ "poly": [
+ 301,
+ 76,
+ 814,
+ 76,
+ 814,
+ 104,
+ 301,
+ 104
+ ],
+ "score": 0.894
+ },
+ {
+ "category_id": 6,
+ "poly": [
+ 296,
+ 1060,
+ 1409,
+ 1060,
+ 1409,
+ 1153,
+ 296,
+ 1153
+ ],
+ "score": 0.888
+ },
+ {
+ "category_id": 7,
+ "poly": [
+ 294,
+ 1425,
+ 1394,
+ 1425,
+ 1394,
+ 1457,
+ 294,
+ 1457
+ ],
+ "score": 0.859
+ },
+ {
+ "category_id": 2,
+ "poly": [
+ 835,
+ 2088,
+ 864,
+ 2088,
+ 864,
+ 2112,
+ 835,
+ 2112
+ ],
+ "score": 0.847
+ },
+ {
+ "category_id": 0,
+ "poly": [
+ 298,
+ 1791,
+ 1046,
+ 1791,
+ 1046,
+ 1826,
+ 298,
+ 1826
+ ],
+ "score": 0.667
+ },
+ {
+ "category_id": 1,
+ "poly": [
+ 298,
+ 1791,
+ 1046,
+ 1791,
+ 1046,
+ 1826,
+ 298,
+ 1826
+ ],
+ "score": 0.247
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 789,
+ 1091,
+ 880,
+ 1091,
+ 880,
+ 1124,
+ 789,
+ 1124
+ ],
+ "score": 0.93,
+ "latex": "( C _ { L } , L )"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 1009,
+ 1947,
+ 1068,
+ 1947,
+ 1068,
+ 1976,
+ 1009,
+ 1976
+ ],
+ "score": 0.92,
+ "latex": "E ( 3 )"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 823,
+ 1854,
+ 901,
+ 1854,
+ 901,
+ 1884,
+ 823,
+ 1884
+ ],
+ "score": 0.92,
+ "latex": "S E ( 3 )"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 1251,
+ 1090,
+ 1403,
+ 1090,
+ 1403,
+ 1127,
+ 1251,
+ 1127
+ ],
+ "score": 0.92,
+ "latex": "( C _ { ( L , p ) } , L , p )"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 929,
+ 1091,
+ 1003,
+ 1091,
+ 1003,
+ 1127,
+ 929,
+ 1127
+ ],
+ "score": 0.92,
+ "latex": "C _ { ( L , p ) }"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 1065,
+ 1091,
+ 1132,
+ 1091,
+ 1132,
+ 1124,
+ 1065,
+ 1124
+ ],
+ "score": 0.91,
+ "latex": "( L , p )"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 1341,
+ 2003,
+ 1403,
+ 2003,
+ 1403,
+ 2036,
+ 1341,
+ 2036
+ ],
+ "score": 0.91,
+ "latex": "( 1 , e )"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 654,
+ 1794,
+ 731,
+ 1794,
+ 731,
+ 1827,
+ 654,
+ 1827
+ ],
+ "score": 0.9,
+ "latex": "S E ( 3 )"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 505,
+ 1881,
+ 565,
+ 1881,
+ 565,
+ 1915,
+ 505,
+ 1915
+ ],
+ "score": 0.9,
+ "latex": "E ( 3 )"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 795,
+ 1794,
+ 854,
+ 1794,
+ 854,
+ 1827,
+ 795,
+ 1827
+ ],
+ "score": 0.9,
+ "latex": "E ( 3 )"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 930,
+ 1886,
+ 993,
+ 1886,
+ 993,
+ 1915,
+ 930,
+ 1915
+ ],
+ "score": 0.89,
+ "latex": "^ { \\bullet } E ( 3 )"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 550,
+ 1091,
+ 589,
+ 1091,
+ 589,
+ 1121,
+ 550,
+ 1121
+ ],
+ "score": 0.87,
+ "latex": "C _ { L }"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 803,
+ 996,
+ 850,
+ 996,
+ 850,
+ 1021,
+ 803,
+ 1021
+ ],
+ "score": 0.8,
+ "latex": "d _ { f f n }"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 795,
+ 690,
+ 828,
+ 690,
+ 828,
+ 714,
+ 795,
+ 714
+ ],
+ "score": 0.76,
+ "latex": "d _ { s h }"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 795,
+ 922,
+ 828,
+ 922,
+ 828,
+ 945,
+ 795,
+ 945
+ ],
+ "score": 0.76,
+ "latex": "d _ { s h }"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 803,
+ 764,
+ 849,
+ 764,
+ 849,
+ 790,
+ 803,
+ 790
+ ],
+ "score": 0.74,
+ "latex": "d _ { f f n }"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 650,
+ 1092,
+ 671,
+ 1092,
+ 671,
+ 1119,
+ 650,
+ 1119
+ ],
+ "score": 0.71,
+ "latex": "L"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 878,
+ 1427,
+ 897,
+ 1427,
+ 897,
+ 1458,
+ 878,
+ 1458
+ ],
+ "score": 0.71,
+ "latex": "\\dagger"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 441,
+ 1506,
+ 465,
+ 1506,
+ 465,
+ 1530,
+ 441,
+ 1530
+ ],
+ "score": 0.71,
+ "latex": "^ +"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 767,
+ 824,
+ 815,
+ 824,
+ 815,
+ 850,
+ 767,
+ 850
+ ],
+ "score": 0.7,
+ "latex": "E ( 3 )"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 640,
+ 970,
+ 693,
+ 970,
+ 693,
+ 994,
+ 640,
+ 994
+ ],
+ "score": 0.68,
+ "latex": "d _ { h e a d }"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 640,
+ 739,
+ 694,
+ 739,
+ 694,
+ 763,
+ 640,
+ 763
+ ],
+ "score": 0.67,
+ "latex": "d _ { h e a d }"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 451,
+ 1537,
+ 475,
+ 1537,
+ 475,
+ 1561,
+ 451,
+ 1561
+ ],
+ "score": 0.62,
+ "latex": "^ +"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 611,
+ 896,
+ 677,
+ 896,
+ 677,
+ 921,
+ 611,
+ 921
+ ],
+ "score": 0.61,
+ "latex": "d _ { e m b e d }"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 638,
+ 1018,
+ 716,
+ 1018,
+ 716,
+ 1046,
+ 638,
+ 1046
+ ],
+ "score": 0.55,
+ "latex": "d _ { f e a t u r e }"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 612,
+ 665,
+ 677,
+ 665,
+ 677,
+ 690,
+ 612,
+ 690
+ ],
+ "score": 0.54,
+ "latex": "d _ { e m b e d }"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 638,
+ 787,
+ 716,
+ 787,
+ 716,
+ 815,
+ 638,
+ 815
+ ],
+ "score": 0.47,
+ "latex": "d _ { f e a t u r e }"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 647,
+ 945,
+ 663,
+ 945,
+ 663,
+ 966,
+ 647,
+ 966
+ ],
+ "score": 0.4,
+ "latex": "h"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 647,
+ 714,
+ 663,
+ 714,
+ 663,
+ 735,
+ 647,
+ 735
+ ],
+ "score": 0.34,
+ "latex": "h"
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 292.0,
+ 1660.0,
+ 817.0,
+ 1660.0,
+ 817.0,
+ 1701.0,
+ 292.0,
+ 1701.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 295.0,
+ 73.0,
+ 817.0,
+ 73.0,
+ 817.0,
+ 108.0,
+ 295.0,
+ 108.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 293.0,
+ 1058.0,
+ 1407.0,
+ 1058.0,
+ 1407.0,
+ 1096.0,
+ 293.0,
+ 1096.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 292.0,
+ 1086.0,
+ 549.0,
+ 1086.0,
+ 549.0,
+ 1129.0,
+ 292.0,
+ 1129.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 590.0,
+ 1086.0,
+ 649.0,
+ 1086.0,
+ 649.0,
+ 1129.0,
+ 590.0,
+ 1129.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 672.0,
+ 1086.0,
+ 788.0,
+ 1086.0,
+ 788.0,
+ 1129.0,
+ 672.0,
+ 1129.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 881.0,
+ 1086.0,
+ 928.0,
+ 1086.0,
+ 928.0,
+ 1129.0,
+ 881.0,
+ 1129.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1004.0,
+ 1086.0,
+ 1064.0,
+ 1086.0,
+ 1064.0,
+ 1129.0,
+ 1004.0,
+ 1129.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1133.0,
+ 1086.0,
+ 1250.0,
+ 1086.0,
+ 1250.0,
+ 1129.0,
+ 1133.0,
+ 1129.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1404.0,
+ 1086.0,
+ 1407.0,
+ 1086.0,
+ 1407.0,
+ 1129.0,
+ 1404.0,
+ 1129.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 295.0,
+ 1122.0,
+ 918.0,
+ 1122.0,
+ 918.0,
+ 1156.0,
+ 295.0,
+ 1156.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 300.0,
+ 1420.0,
+ 877.0,
+ 1420.0,
+ 877.0,
+ 1462.0,
+ 300.0,
+ 1462.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 898.0,
+ 1420.0,
+ 1397.0,
+ 1420.0,
+ 1397.0,
+ 1462.0,
+ 898.0,
+ 1462.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 828.0,
+ 2084.0,
+ 871.0,
+ 2084.0,
+ 871.0,
+ 2123.0,
+ 828.0,
+ 2123.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 291.0,
+ 1788.0,
+ 653.0,
+ 1788.0,
+ 653.0,
+ 1832.0,
+ 291.0,
+ 1832.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 732.0,
+ 1788.0,
+ 794.0,
+ 1788.0,
+ 794.0,
+ 1832.0,
+ 732.0,
+ 1832.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 855.0,
+ 1788.0,
+ 1050.0,
+ 1788.0,
+ 1050.0,
+ 1832.0,
+ 855.0,
+ 1832.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 293.0,
+ 1849.0,
+ 822.0,
+ 1849.0,
+ 822.0,
+ 1884.0,
+ 293.0,
+ 1884.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 902.0,
+ 1849.0,
+ 1408.0,
+ 1849.0,
+ 1408.0,
+ 1884.0,
+ 902.0,
+ 1884.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 296.0,
+ 1880.0,
+ 504.0,
+ 1880.0,
+ 504.0,
+ 1916.0,
+ 296.0,
+ 1916.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 566.0,
+ 1880.0,
+ 929.0,
+ 1880.0,
+ 929.0,
+ 1916.0,
+ 566.0,
+ 1916.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 994.0,
+ 1880.0,
+ 1406.0,
+ 1880.0,
+ 1406.0,
+ 1916.0,
+ 994.0,
+ 1916.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 295.0,
+ 1910.0,
+ 1406.0,
+ 1910.0,
+ 1406.0,
+ 1945.0,
+ 295.0,
+ 1945.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 293.0,
+ 1940.0,
+ 1008.0,
+ 1940.0,
+ 1008.0,
+ 1977.0,
+ 293.0,
+ 1977.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1069.0,
+ 1940.0,
+ 1406.0,
+ 1940.0,
+ 1406.0,
+ 1977.0,
+ 1069.0,
+ 1977.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 296.0,
+ 1973.0,
+ 1404.0,
+ 1973.0,
+ 1404.0,
+ 2005.0,
+ 296.0,
+ 2005.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 292.0,
+ 1999.0,
+ 1340.0,
+ 1999.0,
+ 1340.0,
+ 2041.0,
+ 292.0,
+ 2041.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1404.0,
+ 1999.0,
+ 1408.0,
+ 1999.0,
+ 1408.0,
+ 2041.0,
+ 1404.0,
+ 2041.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 291.0,
+ 1500.0,
+ 440.0,
+ 1500.0,
+ 440.0,
+ 1537.0,
+ 291.0,
+ 1537.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 466.0,
+ 1500.0,
+ 1406.0,
+ 1500.0,
+ 1406.0,
+ 1537.0,
+ 466.0,
+ 1537.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 295.0,
+ 1533.0,
+ 450.0,
+ 1533.0,
+ 450.0,
+ 1566.0,
+ 295.0,
+ 1566.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 476.0,
+ 1533.0,
+ 1405.0,
+ 1533.0,
+ 1405.0,
+ 1566.0,
+ 476.0,
+ 1566.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 295.0,
+ 1564.0,
+ 1405.0,
+ 1564.0,
+ 1405.0,
+ 1596.0,
+ 295.0,
+ 1596.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 290.0,
+ 1593.0,
+ 363.0,
+ 1593.0,
+ 363.0,
+ 1626.0,
+ 290.0,
+ 1626.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 295.0,
+ 1720.0,
+ 1267.0,
+ 1720.0,
+ 1267.0,
+ 1759.0,
+ 295.0,
+ 1759.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 291.0,
+ 1788.0,
+ 653.0,
+ 1788.0,
+ 653.0,
+ 1832.0,
+ 291.0,
+ 1832.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 732.0,
+ 1788.0,
+ 794.0,
+ 1788.0,
+ 794.0,
+ 1832.0,
+ 732.0,
+ 1832.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 855.0,
+ 1788.0,
+ 1050.0,
+ 1788.0,
+ 1050.0,
+ 1832.0,
+ 855.0,
+ 1832.0
+ ],
+ "score": 1.0,
+ "text": ""
+ }
+ ],
+ "page_info": {
+ "page_no": 28,
+ "width": 1700,
+ "height": 2200
+ }
+ },
+ {
+ "layout_dets": [
+ {
+ "category_id": 1,
+ "poly": [
+ 298,
+ 1259,
+ 1405,
+ 1259,
+ 1405,
+ 1687,
+ 298,
+ 1687
+ ],
+ "score": 0.983
+ },
+ {
+ "category_id": 1,
+ "poly": [
+ 298,
+ 1849,
+ 1403,
+ 1849,
+ 1403,
+ 2033,
+ 298,
+ 2033
+ ],
+ "score": 0.977
+ },
+ {
+ "category_id": 1,
+ "poly": [
+ 299,
+ 667,
+ 1405,
+ 667,
+ 1405,
+ 822,
+ 299,
+ 822
+ ],
+ "score": 0.974
+ },
+ {
+ "category_id": 1,
+ "poly": [
+ 298,
+ 993,
+ 1404,
+ 993,
+ 1404,
+ 1117,
+ 298,
+ 1117
+ ],
+ "score": 0.973
+ },
+ {
+ "category_id": 5,
+ "poly": [
+ 252,
+ 225,
+ 1446,
+ 225,
+ 1446,
+ 336,
+ 252,
+ 336
+ ],
+ "score": 0.96,
+ "html": "| BAIR(64 × 64) [past p → pred ; trained on k]| | p | k | pred | FVD↓ | PSNR↑ | SSIM↑ |
| LVT [Rakhimov et al., 2020] | 1 | 15 | 15 | 125.8 | 1 | 1 |
| DVD-GAN-FP [Clark et al., 2019] | 1 | 15 | 15 | 109.8 | 1 | 1 |
| MCVD spatin (Ours) | 1 | 5 | 15 | 103.8 | 18.8 | 0.826 |
| TrIVD-GAN-FP [Luc et al., 2020] | 1 | 15 | 15 | 103.3 | 1 | 1 |
| VideoGPT[Yan et al., 2021] | 1 | 15 | 15 | 103.3 | 1 | 1 |
| CCVS [Le Moing et al., 2021] | 1 | 15 | 15 | 99.0 | 1 | 一 |
| MCVD concat (Ours) | 1 | 5 | 15 | 98.8 | 18.8 | 0.829 |
| MCVD spatin past-mask (Ours) | 1 | 5 | 15 | 96.5 | 18.8 | 0.828 |
| MCVD concat past-mask (Ours) | 1 | 5 | 15 | 95.6 | 18.8 | 0.832 |
| Video Transformer [Weissenborn et al., 2019] | 1 | 15 | 15 | 94-96a | 1 | 1 |
| FitVid [Babaeizadeh et al., 2021] | 1 | 15 | 15 | 93.6 | | 一 |
| MCVD concat past-future-mask (Ours) | 1 | 5 | 15 | 89.5 | 16.9 | 0.780 |
| SAVP [Lee et al., 2018] | 2 | 14 | 14 | 116.4 | 1 | 1 |
| MCVD spatin (Ours) | | 5 | 14 | 94.1 | 19.1 | 0.836 |
| MCVD spatin past-mask (Ours) | 22222 | 5 | 14 | 90.5 | 19.2 | 0.837 |
| MCVD concat (Ours) | | 5 | 14 | 90.5 | 19.1 | 0.834 |
| MCVD concat past-future-mask (Ours) | | 5 | 14 | 89.6 | 17.1 | 0.787 |
| MCVD concat past-mask (Ours) | | 5 | 14 | 87.9 | 19.1 | 0.838 |
| SAVP [Lee et al., 2018] | 2 | 10 | 28 | 143.4 | 1 | 0.795 |
| Hier-vRNN [Castrej6n et al., 2019] | | 10 | 28 | 143.4 | 1 | 0.822 |
| MCVD spatin (Ours) | | 5 | 28 | 132.1 | 17.5 | 0.779 |
| MCVD spatin past-mask (Ours) | ２２２２２2 | 5 | 28 | 127.9 | 17.7 | 0.789 |
| MCVD concat (Ours) | | 5 | 28 | 120.6 | 17.6 | 0.785 |
| MCVD concat past-mask (Ours) | | 5 | 28 | 119.0 | 17.7 | 0.797 |
| MCVD concat past-future-mask (Ours) | | 5 | 28 | 118.4 | 16.2 | 0.745 |
"
+ },
+ {
+ "category_id": 1,
+ "poly": [
+ 299,
+ 1893,
+ 1405,
+ 1893,
+ 1405,
+ 2016,
+ 299,
+ 2016
+ ],
+ "score": 0.979
+ },
+ {
+ "category_id": 1,
+ "poly": [
+ 297,
+ 202,
+ 1404,
+ 202,
+ 1404,
+ 326,
+ 297,
+ 326
+ ],
+ "score": 0.973
+ },
+ {
+ "category_id": 5,
+ "poly": [
+ 327,
+ 1517,
+ 1372,
+ 1517,
+ 1372,
+ 1741,
+ 327,
+ 1741
+ ],
+ "score": 0.966,
+ "html": "