Add files using upload-large-folder tool

parse/train/_IY3_4psXuf/_IY3_4psXuf.md

@@ -92,7 +92,7 @@ Corollary 3.2.1. Consider EXTRA $C T _ { 1 }$ , where $\forall v \in \mathcal {
 
 Corollary 3.2.2. Consider EXTRA $C T _ { 2 }$ , where $\forall u , v \in \mathcal { V }$ , $\mathcal { V } _ { [ v ] } \neq \mathcal { V } _ { [ u ] }$ . Then $| \mathcal { M } | = | \mathcal { V } | a . s$
 
-Theorem 3.2 proves SHADOW-GCN does not oversmooth: 1. A normal GCN pushes the aggregation of same-degree nodes to the same point, while SHADOW-GCN with $\mathtt { E X T R A C T } _ { 2 }$ ensures any two nodes (even with the same degree) have different aggregation. 2. A normal GCN wipes out all information in $\boldsymbol { X }$ after many times of aggregation, while SHADOW-GCN always preserves feature information. Particularly, with φG (v) = δ[v](v)
−1/2, a normal GCN generates only one unique value of for all $v$ . By contrast, SHADOW-GNN generates $| \nu |$ different values for any $\phi _ { \mathcal { G } }$ function.
+Theorem 3.2 proves SHADOW-GCN does not oversmooth: 1. A normal GCN pushes the aggregation of same-degree nodes to the same point, while SHADOW-GCN with $\mathtt { E X T R A C T } _ { 2 }$ ensures any two nodes (even with the same degree) have different aggregation. 2. A normal GCN wipes out all information in $\boldsymbol { X }$ after many times of aggregation, while SHADOW-GCN always preserves feature information. Particularly, with φG (v) = δ[v](v)
−1/2, a normal GCN generates only one unique value of for all $v$ . By contrast, SHADOW-GNN generates $| \nu |$ different values for any $\phi _ { \mathcal { G } }$ function.
 
 We compare the expressivity by showing 1. SHADOW-SAGE can express all functions GraphSAGE can, and 2. SHADOW-SAGE can express some function GraphSAGE cannot. Recall, a GraphSAGE layer performs the following: $\begin{array} { r } { \dot { \overline { { h _ { v } ^ { ( \ell ) } } } } = \sigma \left( \left( W _ { 1 } ^ { ( \ell ) } \right) ^ { \top } h _ { v } ^ { ( \ell - 1 ) } + \left( W _ { 2 } ^ { ( \ell ) } \right) ^ { \top } \left( \frac { 1 } { | \mathcal { N } _ { v } | } \sum _ { u \in \mathcal { N } _ { v } } \dot { h _ { u } ^ { ( \ell - 1 ) } } \right) \right) . } \end{array}$ We can prove Point 1 by making an $L ^ { \prime }$ -layer SHADOW-SAGE identical to an $L$ -layer GraphSAGE with the following steps: 1. let EXTRACT return the full $L$ -hop neighborhood, and 2. set $\bar { \boldsymbol { W } } _ { 1 } ^ { ( \ell ) } = \boldsymbol { I }$ , $W _ { 2 } ^ { ( \ell ) } = { \bf 0 }$ for $L + 1 \leq \ell \leq L ^ { \prime }$ . For point 2, we consider a target function: $\tau \left( \boldsymbol { X } , \boldsymbol { \mathcal { G } } _ { [ v ] } \right) =$ $\begin{array} { r } { C \cdot \sum _ { u \in \mathcal { V } _ { [ v ] } } \delta _ { [ v ] } \left( u \right) \cdot \pmb { x } _ { u } } \end{array}$ for some neighborhood $\mathcal { G } _ { [ v ] }$ , scaling constant $C$ and $\delta _ { [ v ] } \left( u \right)$ as defined in Proposition 3.1. An expressive model should be able to learn well this simple linear function $\tau$ .