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@@ -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>