Add files using upload-large-folder tool

parse/train/BkgrBgSYDS/BkgrBgSYDS.md

@@ -637,7 +637,7 @@ Now, we consider the process of transforming $\mathbf { P }$ into a modular-bala
 ![](images/a11fbc418e35ae750d3922a0ae3f160b759972fa724560634f39157fd9066c93.jpg)  
 Figure 8: First step of balancing $8 \times 8$ bit reversal permutation (a component of the $8 \times 8$ DFT). Red signifies edges that must be flipped.
 
-Lemma G.2. Let M be a $k \times k$ matrix with $I$ non-zero entry per column, such that for each $\textstyle 0 \leq m < { \frac { k } { 2 } }$ , there are exactly 2 columns with non-zero entry in a row with index $\equiv m$ mod $\frac { k } { 2 }$ 
 . Then, there is a butterfly factor $\mathbf { B } _ { k }$ such that $\mathbf { M } \mathbf { B } _ { k } = \mathbf { M } ^ { \prime }$ , where $\mathbf { M } ^ { \prime }$ meets the $\frac { k } { 2 }$ balance condition.
+Lemma G.2. Let M be a $k \times k$ matrix with $I$ non-zero entry per column, such that for each $\textstyle 0 \leq m < { \frac { k } { 2 } }$ , there are exactly 2 columns with non-zero entry in a row with index $\equiv m$ mod $\frac { k } { 2 }$ 
 . Then, there is a butterfly factor $\mathbf { B } _ { k }$ such that $\mathbf { M } \mathbf { B } _ { k } = \mathbf { M } ^ { \prime }$ , where $\mathbf { M } ^ { \prime }$ meets the $\frac { k } { 2 }$ balance condition.
 
 Proof. We construct a directed graph $G$ with nodes in $\left[ \frac { k } { 2 } \right]$ . For each $\begin{array} { r } { 0 \leq i < \frac { k } { 2 } } \end{array}$ we add a directed edge from node $\left( s \mod \frac { k } { 2 } \right)$ 
 to node $\left( t \mod \frac { k } { 2 } \right)$ if $\mathbf { M } [ : , i ] = \mathbf { e } _ { s }$ and $\begin{array} { r } { \mathbf { M } \left[ : , i + \frac { k } { 2 } \right] = \mathbf { e } _ { t } } \end{array}$ . Each node has (undirected) degree exactly 2 by the structure of M. Hence, $G$ is a union of disjoint (undirected) cycles.
 

parse/train/v8b3e5jN66j/v8b3e5jN66j.md

@@ -99,7 +99,7 @@ i x ${ \bf \Pi } _ { - } 1 = 1 * 1 \mathbf { e } \mathbf { n }$ ( q u e u e )
 i x $\mathbf { u = u * l e n }$ ( q u e u e )   
 r i n g d p $s =$ a l l d p s [ : , i x l : i x u ]   
 # n o n p a r a m e t r i c s o f t m a x   
-l o s s=dps + logsumexp ( r i n g d p s )   
+l o s s=dps + logsumexp ( r i n g d p s )   
 l o s s . b a c k w a r d ( )   
 s t e p ( g q . p a r a m s )   
 # moco u p d a t e s