Add files using upload-large-folder tool

md/train/PxMfDdPnTfV/PxMfDdPnTfV.md

@@ -156,7 +156,7 @@ Whereas the $( \alpha , \theta )$ -diatomic LJSDs explicitly enforce the trace n
 
 Deriving conditions on whether a shift will hurt or help a model’s performance is crucial to building an understanding of covariate shift. Motivated in part by the above results for linear regression, and in part by the results for random feature regression that we present in $\mathrm { S e c . } \underline { {   \bar { 5 . 3 }  } }$ we introduce the following definition of shift strength, which is a direct generalization of the conditions of Prop. 3.2:
 
-Definition 4.1. Let $\mu _ { 1 }$ and $\mu _ { 2 }$ be LJSDs with the same marginal distribution of $\lambda ,$ , denoted $\mu _ { t r a i n }$ . If the asymptotic overlap coefficients are such that $\mathbb { E } _ { \mu _ { 1 } } [ r | \lambda ] / \mathbb { E } _ { \mu _ { 2 } } [ r | \lambda ]$ is nondecreasing as a function of $\lambda$ on the support of $\mu _ { t r a i n }$ and $\mathbb { E } _ { \mu _ { 1 } } [ r ] \le \mathbb { E } _ { \mu _ { 2 } } [ r ]$ , we say $\mu _ { 1 }$ is easier than $\mu _ { 2 }$ (or $\mu _ { 2 }$ is harder than $\mu _ { 1 . }$ ), and write $\mu _ { 1 } \leq \mu _ { 2 }$ . Comparing against the case of no shift $\mu _ { \emptyset }$ , we say $\mu _ { 1 }$ is easy when $\mu _ { 1 } \leq \mu _ { \emptyset }$ and hard when µ1  µ .
+Definition 4.1. Let $\mu _ { 1 }$ and $\mu _ { 2 }$ be LJSDs with the same marginal distribution of $\lambda ,$ , denoted $\mu _ { t r a i n }$ . If the asymptotic overlap coefficients are such that $\mathbb { E } _ { \mu _ { 1 } } [ r | \lambda ] / \mathbb { E } _ { \mu _ { 2 } } [ r | \lambda ]$ is nondecreasing as a function of $\lambda$ on the support of $\mu _ { t r a i n }$ and $\mathbb { E } _ { \mu _ { 1 } } [ r ] \le \mathbb { E } _ { \mu _ { 2 } } [ r ]$ , we say $\mu _ { 1 }$ is easier than $\mu _ { 2 }$ (or $\mu _ { 2 }$ is harder than $\mu _ { 1 . }$ ), and write $\mu _ { 1 } \leq \mu _ { 2 }$ . Comparing against the case of no shift $\mu _ { \emptyset }$ , we say $\mu _ { 1 }$ is easy when $\mu _ { 1 } \leq \mu _ { \emptyset }$ and hard when µ1  µ .
 
 A priori, there is little reason to hope that such a model-independent definition of shift strength would adequately characterize a shift’s impact on the total error, bias, or variance of a given model. Even for the relatively simple case of random feature kernel regression, the nonlinear feature maps of Eq. $\mathbb { Q } )$ would seem to inextricably couple the covariance distribution to the model.
 
@@ -193,7 +193,7 @@ $$
 \begin{array} { l } { { \displaystyle B _ { \mu } = \phi \mathcal { T } _ { 1 , 2 } ^ { * } } } \\ { { \displaystyle V _ { \mu } = - \rho \frac { \psi } { \phi } \frac { \partial x } { \partial \gamma } \bigg ( \mathcal { T } _ { 1 , 1 } ( \omega + \phi \mathcal { T } _ { 1 , 2 } ) ( \omega + \mathcal { T } _ { 1 , 1 } ^ { * } ) + \frac { \phi ^ { 2 } } { \psi } \gamma \bar { \tau } \mathcal { T } _ { 1 , 2 } \mathcal { T } _ { 2 , 2 } ^ { * } } } \\ { { \displaystyle \quad \quad + \gamma \tau \mathcal { T } _ { 2 , 2 } ( \omega + \phi \mathcal { T } _ { 1 , 2 } ^ { * } ) + \sigma _ { \varepsilon } ^ { 2 } \Big ( ( \omega + \phi \mathcal { T } _ { 1 , 2 } ) ( \omega + \mathcal { T } _ { 1 , 1 } ^ { * } ) + \frac { \phi } { \psi } \gamma \bar { \tau } \mathcal { T } _ { 2 , 2 } ^ { * } \Big ) \bigg ) , } } \end{array}
 $$
 
-where x is the unique nonnegative real root of x = 1⌧!+I1,1 , @x@ $\begin{array} { r } { \frac { 1 - \gamma \tau } { \omega + \mathcal { T } _ { 1 , 1 } } , \frac { \partial x } { \partial \gamma } = - \frac { x } { \gamma + \rho \gamma ( \tau \psi / \phi + \bar { \tau } ) ( \omega + \phi \mathcal { T } _ { 1 , 2 } ) } } \end{array}$ , and
+where x is the unique nonnegative real root of x = 1⌧!+I1,1 , @x@ $\begin{array} { r } { \frac { 1 - \gamma \tau } { \omega + \mathcal { T } _ { 1 , 1 } } , \frac { \partial x } { \partial \gamma } = - \frac { x } { \gamma + \rho \gamma ( \tau \psi / \phi + \bar { \tau } ) ( \omega + \phi \mathcal { T } _ { 1 , 2 } ) } } \end{array}$ , and
 
 $$
 \tau = \frac { \sqrt { ( \psi - \phi ) ^ { 2 } + 4 x \psi \phi \gamma / \rho } + \psi - \phi } { 2 \psi \gamma } a n d \bar { \tau } = \frac { 1 } { \gamma } + \frac { \psi } { \phi } \bigl ( \tau - \frac { 1 } { \gamma } \bigr ) .

md/train/dvyUaK4neD0/dvyUaK4neD0.md

@@ -145,7 +145,7 @@ $$
 \Phi _ { X , \alpha , \Delta } ( \mu ) \leq \frac { \alpha ( \alpha - 1 ) \Delta ^ { 2 } } { 2 \mu } + \operatorname* { m i n } \left( \frac { ( 2 \alpha - 1 ) ( \alpha - 1 ) \Delta ^ { 2 } } { 4 \mu ^ { 2 } } + \frac { 3 ( \alpha - 1 ) | \Delta | } { 2 \mu ^ { 2 } } , \frac { 3 ( \alpha - 1 ) | \Delta | } { 2 \mu } \right) .
 $$
 
-Note that in contrast if we consider the analogous notion of $\Phi$ for the Gaussian mechanism (replacing IX(µ) with the Gaussian density eX2/2µ), we readily get the bound ↵(↵1)22µ , which is the same as our bound up to lower order terms. We now provide the proof of Theorem 3.5.
+Note that in contrast if we consider the analogous notion of $\Phi$ for the Gaussian mechanism (replacing IX(µ) with the Gaussian density eX2/2µ), we readily get the bound ↵(↵1)22µ , which is the same as our bound up to lower order terms. We now provide the proof of Theorem 3.5.
 
 Proof of Theorem 3.5. By RDP definition (2.2), we need to bound the following for any $\Delta , \alpha \geq 1$ ,
 

md/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 ) }