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parse/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 ,

parse/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).
 

parse/train/rJgYxn09Fm/rJgYxn09Fm.md

@@ -56,7 +56,7 @@ In convolutional neural networks (CNNs) and variants such as residual CNNs (ResN
 As an alternative to learning hard parameter sharing schemes – which correspond to the strict structure present in RNNs – our method consists of learning soft sharing schemes through a relaxation of this structure. We accomplish this by expressing each layer’s parameters $\boldsymbol { \mathsf { W } } ^ { ( i ) }$ as a linear combination of parameter templates $\mathbf { \overline { { I } } } ^ { ( 1 ) } , \ldots , \mathbf { \overline { { I } } } ^ { ( k ) }$ , each with the same dimensionality as $\boldsymbol { \mathsf { W } } ^ { ( i ) }$ :
 
 ![](images/30aa8071d0e5ad98b194b3c861782577cf515638614a5ad4b9997a5766ff7be7.jpg)  
-Figure 2: Connection between the LSM matrix $S$  where $\begin{array} { r } { S _ { i , j } = \frac { | \langle { \pmb { \alpha } } ^ { ( i ) } , { \pmb { \alpha } } ^ { ( j ) } \rangle | } { \| { \pmb { \alpha } } ^ { ( i ) } \| \| { \pmb { \alpha } } ^ { ( j ) } \| } \ ) } \end{array}$ and the structure of the network. White and black entries correspond to maximum and minimum similarities $( S _ { i , j } = 1$ and $S _ { i , j } = 0$ , respectively). Left: Empirically, CNNs present no similarity between parameters of different layers. Middle: Trained with our method, the layer similarity matrix (LSM) captures similarities between different layers, including pairs with close to maximum similarity. Such pairs (depicted by same-colored coefficients and weights, and by white entries in the LSM) perform similar operations on their inputs. Right: We can tie together parameters of similar layers, creating a hard parameter sharing scheme. The network can then be folded, creating self-loops and revealing an explicit recurrent computation structure.
+Figure 2: Connection between the LSM matrix $S$  where $\begin{array} { r } { S _ { i , j } = \frac { | \langle { \pmb { \alpha } } ^ { ( i ) } , { \pmb { \alpha } } ^ { ( j ) } \rangle | } { \| { \pmb { \alpha } } ^ { ( i ) } \| \| { \pmb { \alpha } } ^ { ( j ) } \| } \ ) } \end{array}$ and the structure of the network. White and black entries correspond to maximum and minimum similarities $( S _ { i , j } = 1$ and $S _ { i , j } = 0$ , respectively). Left: Empirically, CNNs present no similarity between parameters of different layers. Middle: Trained with our method, the layer similarity matrix (LSM) captures similarities between different layers, including pairs with close to maximum similarity. Such pairs (depicted by same-colored coefficients and weights, and by white entries in the LSM) perform similar operations on their inputs. Right: We can tie together parameters of similar layers, creating a hard parameter sharing scheme. The network can then be folded, creating self-loops and revealing an explicit recurrent computation structure.
 
 $$
 \pmb { \mathsf { W } } ^ { ( i ) } : = \sum _ { j = 1 } ^ { k } \alpha _ { j } ^ { ( i ) } \pmb { \mathsf { T } } ^ { ( j ) }