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parse/train/HJxiMAVtPH/HJxiMAVtPH.md

@@ -83,7 +83,7 @@ Interpretation: Assuming $\mathcal { G }$ is ergodic: $\begin{array} { r } { p (
 
 # 4.1 MULTI-SCALE CASE (MUSAE)
 
-We know that the SGNS aspect of MUSAE (Algorithm 2, Line 17) is minimized when the learned embeddings grv , hrf satisfy grw>hrf ≈ log  #(w,f )r|Dr|#(w)r#(f )r 
 − log b ∀w ∈ V, f ∈ F. Our aim is to express this factorization in terms of known properties of the graph $\mathcal { G }$ and its features.
+We know that the SGNS aspect of MUSAE (Algorithm 2, Line 17) is minimized when the learned embeddings grv , hrf satisfy grw>hrf ≈ log  #(w,f )r|Dr|#(w)r#(f )r 
 − log b ∀w ∈ V, f ∈ F. Our aim is to express this factorization in terms of known properties of the graph $\mathcal { G }$ and its features.
 
 Lemma 1. The empirical statistics of node-feature pairs obtained from random walks give unbiased estimates of joint probabilities of observing feature $f \in \mathbb { F } r$ steps $( i )$ after; or $( i i )$ before node $v \in \mathbb { V }$ , as given by:
 

parse/train/HkCsm6lRb/HkCsm6lRb.md

@@ -357,7 +357,7 @@ As explained in the main paper, we fit the joint graphical model $p ( x , y , z
 
 • Image and Label encoder, $q ( z | x , y )$ : Our architecture (Figure 9) for the image-label encoder first separately processes the images and the labels, and then concatenates them downstream in the network and then passes the concatenated features through a multi-layered perceptron. More specifically, we have convolutional layers which process image into 32, 64, 128, 16 feature maps with strides $1 , 2 , 2 , 2$ in the corresponding layers. We use batch normalization in the convolutional layers before applying the ELU non-linearity. On the label encoder side, we first encode the each attribute label into a 32d continuous vector and then pass each individual attribute vector through a 2-layered MLP with 512 hidden dimensions each. For example, for MNIST-A we have 4 attributes, which gives us 4 vectors of 512d. We then concatenate these vectors and pass it through a two layer MLP. Finally we concatenate this label feature with the image feature after the convolutional layers (after flattening the conv-features) and then pass the result through a 2 layer MLP to predict the mean $( \mu )$ and standard deviation $( \sigma )$ for the latent space gaussian. Following standard practice, we predict $\log \sigma$ for the standard deviation in order to get values which are positive.
 
-# µ 
+# µ 
 
 ![](images/3c8666b4b928e365d467fb0e5f9ee430477aaa538630478bba8cd6275380d78a.jpg)  
 Figure 9: Architecture for the $q ( z | x , y )$ network in our JVAE models for MNIST-A. Images are ( $6 4 \times 6 4 \times 1 )$ , class has 10 possible values, scale has 2 possible values, orientation has 3 possible values, and location has 4 possible values.

parse/train/ZKbZ4mebI9l/ZKbZ4mebI9l.md

@@ -123,7 +123,7 @@ $$
 \widehat { \mathcal { R } } _ { k } ( \mathcal { D } ^ { ( \mathrm { t r } ) } ; \mathbf { W } _ { 1 } , \mathbf { W } _ { 2 } , \mathbf { W } _ { 3 } ) = \frac { 1 } { N ^ { \mathrm { t r } } } \sum _ { i = 1 } ^ { N ^ { \mathrm { t r } } } l \big ( r _ { \mathbf { W } } ^ { ( k , \mathrm { G N N } ) } ( G _ { i } ) , y _ { i } \big ) .
 $$
 
-Equation (1) is impractical for all but the smallest graphs, since $r _ { \mathbf { W } } ^ { ( k , \mathrm { G N N } ) }$ s a sum overusing a samall k-vertex induced subgraphs S(k) of G. Hence, we approximate r(k,GW $r _ { \mathbf { W } } ^ { ( k , \mathrm { G N N } ) }$ S (k)B ${ \mathcal S } _ { B } ^ { ( k ) } ~ \subset ~ { \mathcal S } ^ { ( k ) }$ at every gradient step, i.e.. Due to non-linearities in $\widehat { r } _ { \mathbf { W } } ^ { ( k , \mathrm { G N N } ) } ( G ) \ =$ $\begin{array} { r } { \rho _ { \mathbf { W } _ { 1 } } \Big ( | S ^ { ( k ) } | / | S _ { B } ^ { ( k ) } | \sum _ { S \in S _ { B } ^ { ( k ) } } \phi _ { \mathbf { W } _ { 2 } } \big ( h _ { \mathbf { W } _ { 3 } } ^ { \mathrm { ( G N N ) } } ( G [ S ] ) \big ) \Big ) } \end{array}$ $\rho _ { \mathbf { W } _ { 1 } }$ $l$ ging $\widehat { r } _ { \mathbf { W } } ^ { ( k , \mathrm { G N N } ) }$ B into Equatio (1) does not provide us with an unbiased estimate of $\widehat { \mathcal { R } } _ { k }$ . However, if i. $l ( \rho _ { \mathbf { W } _ { 1 } } ( a ) , y )$ $a$ per upper bound of our loss,. In practice, many models e., 1/N tr PN tri=1 l $\begin{array} { r } { 1 / N ^ { \mathrm { t r } } \sum _ { i = 1 } ^ { N ^ { \mathrm { t r } } } l \big ( r _ { \mathbf { W } } ^ { ( k , \mathrm { G N N } ) } ( G _ { i } ) , y _ { i } \big ) \leq 1 / N ^ { \mathrm { t r } } \sum _ { i = 1 } ^ { N ^ { \mathrm { t r } } } l \big ( \widehat { r } _ { \mathbf { W } } ^ { ( k , \mathrm { G N N } ) } ( G _ { i } ) , y _ { i } \big ) . } \end{array}$ tr1 lr (k,GNN)W (Gi), yi
 brely on this approximation and provide scalable and reliable training procedures, cf. [48, 79, 80, 112].
+Equation (1) is impractical for all but the smallest graphs, since $r _ { \mathbf { W } } ^ { ( k , \mathrm { G N N } ) }$ s a sum overusing a samall k-vertex induced subgraphs S(k) of G. Hence, we approximate r(k,GW $r _ { \mathbf { W } } ^ { ( k , \mathrm { G N N } ) }$ S (k)B ${ \mathcal S } _ { B } ^ { ( k ) } ~ \subset ~ { \mathcal S } ^ { ( k ) }$ at every gradient step, i.e.. Due to non-linearities in $\widehat { r } _ { \mathbf { W } } ^ { ( k , \mathrm { G N N } ) } ( G ) \ =$ $\begin{array} { r } { \rho _ { \mathbf { W } _ { 1 } } \Big ( | S ^ { ( k ) } | / | S _ { B } ^ { ( k ) } | \sum _ { S \in S _ { B } ^ { ( k ) } } \phi _ { \mathbf { W } _ { 2 } } \big ( h _ { \mathbf { W } _ { 3 } } ^ { \mathrm { ( G N N ) } } ( G [ S ] ) \big ) \Big ) } \end{array}$ $\rho _ { \mathbf { W } _ { 1 } }$ $l$ ging $\widehat { r } _ { \mathbf { W } } ^ { ( k , \mathrm { G N N } ) }$ B into Equatio (1) does not provide us with an unbiased estimate of $\widehat { \mathcal { R } } _ { k }$ . However, if i. $l ( \rho _ { \mathbf { W } _ { 1 } } ( a ) , y )$ $a$ per upper bound of our loss,. In practice, many models e., 1/N tr PN tri=1 l $\begin{array} { r } { 1 / N ^ { \mathrm { t r } } \sum _ { i = 1 } ^ { N ^ { \mathrm { t r } } } l \big ( r _ { \mathbf { W } } ^ { ( k , \mathrm { G N N } ) } ( G _ { i } ) , y _ { i } \big ) \leq 1 / N ^ { \mathrm { t r } } \sum _ { i = 1 } ^ { N ^ { \mathrm { t r } } } l \big ( \widehat { r } _ { \mathbf { W } } ^ { ( k , \mathrm { G N N } ) } ( G _ { i } ) , y _ { i } \big ) . } \end{array}$ tr1 lr (k,GNN)W (Gi), yi
 brely on this approximation and provide scalable and reliable training procedures, cf. [48, 79, 80, 112].
 
 # 4.1 Expressive power