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md/train/ByxGkySKwH/ByxGkySKwH.md

@@ -313,7 +313,7 @@ $$
 \frac { \sum _ { k } \alpha _ { k } } { \beta _ { l ^ { * } } } \exp \left( - d ( x , \mu _ { k ^ { * } } ) ^ { 2 } \left( \frac { 1 } { 2 \sigma _ { k ^ { * } } ^ { 2 } } - \frac { 1 } { 2 \theta _ { l ^ { * } } ^ { 2 } } \right) + \frac { d ( \mu _ { k ^ { * } } , \nu _ { l ^ { * } } ) d ( x , \mu _ { k ^ { * } } ) } { \theta _ { l ^ { * } } ^ { 2 } } + \frac { d ( \mu _ { k ^ { * } } , \nu _ { l ^ { * } } ) ^ { 2 } } { 2 \theta _ { l ^ { * } } ^ { 2 } } \right) \leq \frac { \epsilon \lambda } { M - 1 }
 $$
 
-Thus we get with a =  12σ2k∗ $\begin{array} { r } { a = \big ( \frac { 1 } { 2 \sigma _ { k ^ { * } } ^ { 2 } } - \frac { 1 } { 2 \theta _ { l ^ { * } } ^ { 2 } } \big ) } \end{array}$ , $\begin{array} { r } { b = \frac { d ( \mu _ { k ^ { * } } , \nu _ { l ^ { * } } ) } { \theta _ { l ^ { * } } ^ { 2 } } } \end{array}$ and $\begin{array} { r } { c = \frac { d ( \mu _ { k ^ { * } } , \nu _ { l ^ { * } } ) ^ { 2 } } { 2 \theta _ { l ^ { * } } ^ { 2 } } } \end{array}$ , $\begin{array} { r } { d = \log \left( \frac { \epsilon \lambda } { M - 1 } \frac { \beta _ { l ^ { * } } } { \sum _ { k } \alpha _ { k } } \right) } \end{array}$ , the quadratic inequality
+Thus we get with a =  12σ2k∗ $\begin{array} { r } { a = \big ( \frac { 1 } { 2 \sigma _ { k ^ { * } } ^ { 2 } } - \frac { 1 } { 2 \theta _ { l ^ { * } } ^ { 2 } } \big ) } \end{array}$ , $\begin{array} { r } { b = \frac { d ( \mu _ { k ^ { * } } , \nu _ { l ^ { * } } ) } { \theta _ { l ^ { * } } ^ { 2 } } } \end{array}$ and $\begin{array} { r } { c = \frac { d ( \mu _ { k ^ { * } } , \nu _ { l ^ { * } } ) ^ { 2 } } { 2 \theta _ { l ^ { * } } ^ { 2 } } } \end{array}$ , $\begin{array} { r } { d = \log \left( \frac { \epsilon \lambda } { M - 1 } \frac { \beta _ { l ^ { * } } } { \sum _ { k } \alpha _ { k } } \right) } \end{array}$ , the quadratic inequality
 
 $$
 - d ( x , \mu _ { k ^ { * } } ) ^ { 2 } a + d ( x , \mu _ { k ^ { * } } ) b + c \leq d ,

md/train/d0MtHWY0NZ/d0MtHWY0NZ.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 $\tau$ .
 

md/train/rHCzkRd0UK/rHCzkRd0UK.md

@@ -365,7 +365,7 @@ Linear time GW. Finally all the quadratic operations appearing in Algorithm (3)
 
 Our goal in this section is to demonstrate that, for a far smaller computational budget, the GW-LR approach is competitive with the direct entropic approach on datasets that are either synthesized to exhibit local clusters, or directly validated on a real high-dimensional dataset as well. Because both approaches have different hyperparameters, our goal is to stick to a realistic evaluation that stresses both optimality of solutions as a function of computational effort, as well as performance in real life applications. We start by investigating the sensitivity of hyperparamaters $\varepsilon$ and $\gamma$ on our method. Since GW is not convex, these may interact in unexpected ways. Experiments were run on a personal MacBook Pro 2019 laptop. We reused code from github.com/meyerscetbon/LOT, and downloaded genomics data from github.com/rsinghlab/SCOT.
 
-Benchmarks. We consider three synthetic problems and one real world problem to evaluate timeaccuracy trade-offs, and also compare the couplings obtained by our method and that of the entropic version [27]. More precisely, we compare the quadratic approach in GW-LR computed with algorithm (3) (and its linear time counterpat, Lin GW-LR as presented in $\ S 5$ ), with EntropicGW, the cubic implementation of [27] (as well as its quadratic counterpart, Quad Entropic-GW presented in Algo. 2). For GW-LR and Lin GW-LR, and in all experiments, we set the lower bound on entries of g to ↵ = 1010 .
+Benchmarks. We consider three synthetic problems and one real world problem to evaluate timeaccuracy trade-offs, and also compare the couplings obtained by our method and that of the entropic version [27]. More precisely, we compare the quadratic approach in GW-LR computed with algorithm (3) (and its linear time counterpat, Lin GW-LR as presented in $\ S 5$ ), with EntropicGW, the cubic implementation of [27] (as well as its quadratic counterpart, Quad Entropic-GW presented in Algo. 2). For GW-LR and Lin GW-LR, and in all experiments, we set the lower bound on entries of g to ↵ = 1010 .
 
 Initialization To initialize all algorithms with a common strategy, we adapted the first lower bound of [26, Def. 6.1] to the entropic case. In all experiments showing time-accuracy tradeoffs, we choose to use number of operations to provide platform independent quantities. Accuracy is measured by evaluating the ground-truth energy $\mathcal { E } _ { , B }$ (even in scenarios when the method uses a low rank approximation for $A , B$ at optimization time).