Add files using upload-large-folder tool

parse/train/H1lj0nNFwB/H1lj0nNFwB.md

@@ -584,7 +584,7 @@ $$
 
 While we were able to generalize our result to gradient descent for $N = 2$ , our proof technique relies on the ability to get a non-implicit solution for $\sigma ( t )$ which we discretized and bounded. This is harder to generalize to larger values of $N$ , where the solution is implicit. Still, we can informally illustrate the effect of depth on the dynamics of gradient descent by approximating the update rule of the values.
 
-We start by reminding ourselves of the gradient descent update rule for $\sigma$ , for a learning rate $\eta =$ c  1σ ∗ 
 2 − 2N :
+We start by reminding ourselves of the gradient descent update rule for $\sigma$ , for a learning rate $\eta =$ c  1σ ∗ 
 2 − 2N :
 
 $$
 \sigma _ { i } ( t + 1 ) = \sigma _ { i } ( t ) \Big ( 1 + \frac { c } { N } \big ( \frac { 1 } { \sigma _ { 1 } ^ { * } } \big ) ^ { 2 - \frac { 2 } { N } } \sigma _ { i } ( t ) ^ { 1 - \frac { 2 } { N } } \big ( \sigma _ { i } ^ { * } - \sigma _ { i } ( t ) \big ) \Big ) ^ { N }

parse/train/ryeK6nNFDr/ryeK6nNFDr.md

@@ -36,7 +36,7 @@ $$
 \mathcal { P } _ { \mathcal { X } } ( x ) = A \cdot e ^ { - \lvert \beta x \rvert ^ { c } } ,
 $$
 
-where β = 1σ  Γ(3/c)Γ(1/c) 
 and $\begin{array} { r } { A = \frac { \beta c } { 2 \Gamma ( 1 / c ) } } \end{array}$ , with $\Gamma ( \cdot )$ being the Gamma function. Note that the mean parameter $\mu$ is omitted above, as $\mu$ has no relation with the shape of distribution and we set it as 0 without loss of generality. A nice characteristic of GGD is that it covers many popular distributions
+where β = 1σ  Γ(3/c)Γ(1/c) 
 and $\begin{array} { r } { A = \frac { \beta c } { 2 \Gamma ( 1 / c ) } } \end{array}$ , with $\Gamma ( \cdot )$ being the Gamma function. Note that the mean parameter $\mu$ is omitted above, as $\mu$ has no relation with the shape of distribution and we set it as 0 without loss of generality. A nice characteristic of GGD is that it covers many popular distributions
 
 with varied shape factors. For example, when $c = 1$ , then it becomes the Laplacian distribution; when $c = 2$ , then it is the Gaussian distribution with a variance of √ √ $\sigma ^ { 2 }$ ; when $c \to + \infty$ , then it is specified as a uniform distribution on $( - \sqrt { 2 } \sigma , \sqrt { 2 } \sigma )$ .
 

parse/train/v_4XcXsAZUn/v_4XcXsAZUn.md

@@ -92,7 +92,7 @@ Figure 3: Influence of block size $b$ on PR-BCD (dashed $L _ { 0 }$ PGD $\mathbb
 12: pt[maskres.] 0   
 13: Resample it[maskres.]   
 14: P ⇠ Bernoulli(pE) s.t. $\sum P \leq \Delta$   
-15: Return A  P
+15: Return A  P
 
 PR-BCD. For $L _ { 0 }$ -norm PGD we relax the discrete edge perturbations $_ { P }$ from $\{ 0 , 1 \} ^ { ( n \times n ) }$ to $[ 0 , 1 ] ^ { ( n \times n ) }$ as proposed by $\mathrm { X u }$ et al. [43]. Each entry of $_ { r }$ denotes the probability for flipping it. In each epoch we only look at a randomly sampled, non-contiguous block of $_ { r }$ of size $b$ (line 3, line 10-13) and additionally ignore the diagonal elements (i.e. self-loops). If using an undirected graph, the potential edges are restricted to the upper/lower triangular $n \times n$ matrix. In each epoch $t \in \{ \bar { 1 } , \bar { 2 } , \dots \}$ , $\pmb { p }$ is added to $/$ subtracted from the discrete edge weight (line 6). Note, we overload $\oplus$ s.t. $\pmb { A } _ { i j } \oplus p _ { i j } = A _ { i j } + p _ { i j }$ if $A _ { i j } = 0$ and $A _ { i j } - p _ { i j }$ otherwise. We use $\pmb { p }$ and $_ { P }$ interchangeably while $\pmb { p }$ only corresponds to the current subset/block of $\bar { P } _ { i _ { t } }$ . After each gradient update (line 7), the projection $\Pi _ { \mathbb { E } [ \mathrm { B e r n o u l l i } ( p ) ] \leq \Delta } ( p )$ adjusts the probability mass such that $\begin{array} { r } { \mathbb { E } [ \bar { \mathrm { B e r n o u l l i } } ( \bar { p } ) ] = \sum _ { i \in b } p _ { i } \le \bar { \Delta } } \end{array}$ and that $\pmb { p } \in [ 0 , 1 ]$ (line 8). In the end we draw $b$ sample s.t. $P \in \{ 0 , 1 \} ^ { ( n \times n ) }$ via $P \sim \mathrm { B e r n o u l l i } ( p )$ (line 14).