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parse/train/N5hQI_RowVA/N5hQI_RowVA.md

@@ -47,7 +47,7 @@ $$
 { \bf R } { \bf z } _ { x } + { \bf t } , { \bf z } _ { h } = f ( { \bf R } { \bf x } + { \bf t } , { \bf h } )
 $$
 
-$\mathbf { E ( n ) }$ Equivariant Graph Neural Networks (EGNN) (Satorras et al., 2021) consider a graph $\mathcal { G } =$ $( \nu , \mathcal { E } )$ with nodes $v _ { i } \in \mathcal V$ and edges $e _ { i j }$ . Each node $v _ { i }$ is associated with a position vector $\mathbf { x } _ { i }$ and node features $\mathbf { h } _ { i }$ as the ones defined in previous paragraphs. Then, an $\operatorname { E } ( n )$ Equivariant Graph Convolutional Layer (EGCL) takes as input the set of node embeddings $\mathbf { h } ^ { l } = \{ \mathbf { h } _ { 0 } ^ { l } , \dots , \mathbf { h } _ { M - 1 } ^ { l } \}$ , coordiante embeddings $\mathbf { x } ^ { l } = \{ \mathbf { x } _ { 0 } ^ { l } , \dots , \mathbf { x } _ { M - 1 } ^ { l } \}$ at layer $l$ and edge information $\mathcal { E } = ( e _ { i j } )$ and outputs a transformation on $\mathbf { h } ^ { l + 1 }$ and $\mathbf { x } ^ { l + 1 }$ . Concisely: $\mathbf { h } ^ { l + 1 } , \mathbf { x } ^ { l + 1 } = \operatorname { E G C L } [ \mathbf { h } ^ { l } , \mathbf { x } ^ { l } , \boldsymbol { \mathcal { E } } ]$ . This layer satisfies the equivariant constraint defined in Equation $\textcircled { 3 }$ The equations that define this layer are the following:
+$\mathbf { E ( n ) }$ Equivariant Graph Neural Networks (EGNN) (Satorras et al., 2021) consider a graph $\mathcal { G } =$ $( \nu , \mathcal { E } )$ with nodes $v _ { i } \in \mathcal V$ and edges $e _ { i j }$ . Each node $v _ { i }$ is associated with a position vector $\mathbf { x } _ { i }$ and node features $\mathbf { h } _ { i }$ as the ones defined in previous paragraphs. Then, an $\operatorname { E } ( n )$ Equivariant Graph Convolutional Layer (EGCL) takes as input the set of node embeddings $\mathbf { h } ^ { l } = \{ \mathbf { h } _ { 0 } ^ { l } , \dots , \mathbf { h } _ { M - 1 } ^ { l } \}$ , coordiante embeddings $\mathbf { x } ^ { l } = \{ \mathbf { x } _ { 0 } ^ { l } , \dots , \mathbf { x } _ { M - 1 } ^ { l } \}$ at layer $l$ and edge information $\mathcal { E } = ( e _ { i j } )$ and outputs a transformation on $\mathbf { h } ^ { l + 1 }$ and $\mathbf { x } ^ { l + 1 }$ . Concisely: $\mathbf { h } ^ { l + 1 } , \mathbf { x } ^ { l + 1 } = \operatorname { E G C L } [ \mathbf { h } ^ { l } , \mathbf { x } ^ { l } , \boldsymbol { \mathcal { E } } ]$ . This layer satisfies the equivariant constraint defined in Equation $\textcircled { 3 }$ The equations that define this layer are the following:
 
 $$
 \begin{array} { r l } { \displaystyle \mathbf { m } _ { i j } = \phi _ { e } \left( \mathbf { h } _ { i } ^ { l } , \mathbf { h } _ { j } ^ { l } , \left. \mathbf { x } _ { i } ^ { l } - \mathbf { x } _ { j } ^ { l } \right. ^ { 2 } \right) } & { ~ \mathrm { a n d } \quad \mathbf { m } _ { i } = \displaystyle \sum _ { j \neq i } e _ { i j } \mathbf { m } _ { i j } , } \\ { \displaystyle \mathbf { x } _ { i } ^ { l + 1 } = \mathbf { x } _ { i } ^ { l } + \displaystyle \sum _ { j \neq i } \left( \mathbf { x } _ { i } ^ { l } - \mathbf { x } _ { j } ^ { l } \right) \phi _ { x } \left( \mathbf { m } _ { i j } \right) } & { ~ \mathrm { a n d } \quad \mathbf { h } _ { i } ^ { l + 1 } = \phi _ { h } \left( \mathbf { h } _ { i } ^ { l } , \mathbf { m } _ { i } \right) . } \end{array}
@@ -180,7 +180,7 @@ Implementation details: As in the previous experiment, we compare our E-NF to it
 ![](images/dd7850e4f484ece0b6b6347b9d3a1daf192a15e2bf380795a45b6cdb48c37c85.jpg)  
 Figure 4: The table on the left presents the Negative Log Likelihood $( \mathrm { N L L } ) - \log p _ { V } ( \mathbf { x } )$ for the QM9 Positional dataset on the test data. The figure on the right shows the training curves for all methods.
 
-Results: In the table from Figure 4 we report the cross validated Negative Log Likelihood  $\log p _ { V } ( \mathbf { x } )$ for the test data and the Jensen-Shannon divergence. Our E-NF outperforms all other algorithms in terms of NLL of the dataset. Additionally, the optimization curve with respect to the number of iterations converges much quicker for our E-NF than for the other methods as shown on the right in Figure 4. Regarding the JS divergence, the E-NF and GNF-att-aug achieve the best performance.
+Results: In the table from Figure 4 we report the cross validated Negative Log Likelihood  $\log p _ { V } ( \mathbf { x } )$ for the test data and the Jensen-Shannon divergence. Our E-NF outperforms all other algorithms in terms of NLL of the dataset. Additionally, the optimization curve with respect to the number of iterations converges much quicker for our E-NF than for the other methods as shown on the right in Figure 4. Regarding the JS divergence, the E-NF and GNF-att-aug achieve the best performance.
 
 # 5.3 QM9 Molecules
 
@@ -192,7 +192,7 @@ Implementation details: We compare our E-NF to the non-equivariant GNF-att and G
 
 Results (quantitative): Results are reported in Table $2 .$ As in previous experiments, our E-NF significantly outperforms the non-equivariant models GNF and GNF-aug. In terms of negative log-likelihood, the E-NF performs much better than its non-equivariant counterparts. One factor that increases this difference is the E-NFs ability to capture the very specific distributions of interatomic distances. Since the E-NF is able to better capture these sharp peaks in the distribution, the negative log-likelihood becomes much lower. This effect is also seen when studying the number of stable atoms and molecules, which is very sensitive to the inter-atomic distances. This stablity metric was computed over 10.000 samples from the model, for a detailed explanation of stability see Appendix $\bar { \mathbf { A . } 3 } .$ Observe that it might also be possible to utilize post-processing to increase the molecule stability using prior knowledge. However, here we are mainly using this metric to see how many stable atoms and molecules the E-NF is able to generate in one shot, only by learning the molecule distribution. The E-NF on average produces $85 \%$ valid atoms, whereas the best baseline only produces $7 5 \%$ valid atoms. An even stronger improvement is visible when comparing molecule stability: where the E-NF produces $4 . 9 \%$ stable molecules versus $0 . 5 \%$ by the best baseline. Interestingly, the percentage of stable molecules is much lower than that of atoms. This is not unexpected: if even one atom in a large molecule is unstable, the entire molecule is considered to be unstable.
 
-Table 2: Neg. log-likelihood  log pV (x, h, M ), atom stability and mol stability for the QM9 dataset.   
+Table 2: Neg. log-likelihood  log pV (x, h, M ), atom stability and mol stability for the QM9 dataset.   
 
 <table><tr><td># Metrics</td><td>NLL</td><td>Atom stability</td><td>Mol stable</td></tr><tr><td>GNF-attention</td><td>-28.2</td><td>72%</td><td>0.3%</td></tr><tr><td rowspan="2">GNF-attention-augmentation E-NF (ours)</td><td>-29.3</td><td>75%</td><td>0.5%</td></tr><tr><td>-59.7</td><td>85%</td><td>4.9%</td></tr><tr><td>Data</td><td>=</td><td>99%</td><td>95.2%</td></tr></table>
 

parse/train/tGZu6DlbreV/tGZu6DlbreV.md

@@ -155,7 +155,7 @@ For more detailed analysis of the EM algorithm for optimizing $\mathcal { O } (
 
 Datasets. We choose four datasets for evaluation, including FB15k-237 (Toutanova & Chen, 2015), WN18RR (Dettmers et al., 2018), Kinship and UMLS (Kok & Domingos, 2007). For Kinship and UMLS, there are no standard data splits, so we randomly sample $30 \%$ of all the triplets for training, $20 \%$ for validation, and the rest $50 \%$ for testing. The detailed statistics are summarized in the App. D.
 
-Compared Algorithms. We compare the following algorithms in experiment:  Rule learning methods. For traditional statistical relational learning methods, we choose Markov logic networks (Richardson & Domingos, 2006), boosted relational dependency networks (Natarajan et al., 2010) and path ranking (Lao & Cohen, 2010). We also consider neural logic programming methods, including NeuralLP (Yang et al., 2017), DRUM (Sadeghian et al., 2019) and NLIL (Yang & Song, 2020). In addition, we compare against CTP (Minervini et al., 2020), a differentiable method based on neural theorem provers. Besides, we consider three reinforcement learning methods, which are MINERVA (Das et al., 2018), MultiHopKG (Lin et al., 2018) and M-Walk (Shen et al., 2018).  Other methods. We also compare with some embedding methods, including TransE (Bordes et al., 2013), DistMult (Yang et al., 2015), ComplEx (Trouillon et al., 2016), ComplEx-N3 (Lacroix et al., 2018), ConvE (Dettmers et al., 2018), TuckER (Balazevic et al., 2019) and RotatE (Sun et al., 2019).  RNNLogic. For RNNLogic, we consider two model variants. The first variant assigns a constant score to different grounding paths in the reasoning predictor, i.e., $\phi _ { w } ( p a t h ) = 1 $ in Eq. (4), and we denote this variant as w/o emb.. The second variant leverages entity embeddings and relation embeddings to compute the path score $\phi _ { w } ( p a t h )$ , and we denote the variant as with emb..
+Compared Algorithms. We compare the following algorithms in experiment:  Rule learning methods. For traditional statistical relational learning methods, we choose Markov logic networks (Richardson & Domingos, 2006), boosted relational dependency networks (Natarajan et al., 2010) and path ranking (Lao & Cohen, 2010). We also consider neural logic programming methods, including NeuralLP (Yang et al., 2017), DRUM (Sadeghian et al., 2019) and NLIL (Yang & Song, 2020). In addition, we compare against CTP (Minervini et al., 2020), a differentiable method based on neural theorem provers. Besides, we consider three reinforcement learning methods, which are MINERVA (Das et al., 2018), MultiHopKG (Lin et al., 2018) and M-Walk (Shen et al., 2018).  Other methods. We also compare with some embedding methods, including TransE (Bordes et al., 2013), DistMult (Yang et al., 2015), ComplEx (Trouillon et al., 2016), ComplEx-N3 (Lacroix et al., 2018), ConvE (Dettmers et al., 2018), TuckER (Balazevic et al., 2019) and RotatE (Sun et al., 2019).  RNNLogic. For RNNLogic, we consider two model variants. The first variant assigns a constant score to different grounding paths in the reasoning predictor, i.e., $\phi _ { w } ( p a t h ) = 1 $ in Eq. (4), and we denote this variant as w/o emb.. The second variant leverages entity embeddings and relation embeddings to compute the path score $\phi _ { w } ( p a t h )$ , and we denote the variant as with emb..
 
 Table 1: Results of reasoning on FB15k-237 and WN18RR. $\mathrm { H @ } k$ is in $\%$ . $[ ^ { * } ]$ means the numbers are taken from the original papers. [†] means we rerun the methods with the same evaluation process.