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parse/dev/0IywQ8uxJx/0IywQ8uxJx.md

@@ -394,7 +394,7 @@ $$
 271 The residual connection from a spectral perspective. Given a sufficiently small step-size so   
 272 that the right hand side of inequality 13 is satisfied, $\mathbf { F } ( t + \tau ) = \mathbf { F } ( t ) + \tau \bar { \mathbf { A } } \mathbf { F } ( \dot { t } ) \mathbf { W }$ is HFD for a.e.   
 273 $\mathbf F ( 0 )$ if $| \lambda _ { - } ^ { \mathbf { \tilde { W } } } | ( \rho _ { \Delta } - 1 ) > \lambda _ { + } ^ { \mathbf { W } }$ , i.e. ‘there is more mass’ in the negative spectrum of W than in the   
-274 positive one. This means that differently from $\pm \varTheta \left| 3 0 \right| \bigotimes 1$ , there is no requirement on the minimal   
+274 positive one. This means that differently from $\pm \varTheta \left| 3 0 \right| \bigotimes 1$ , there is no requirement on the minimal   
 275 magnitude of the spectral radius of W coming from the graph topology as long as $\lambda _ { + } ^ { \mathbf { w } }$ is small   
 276 enough. Conversely, without a residual term, the dynamics is LFD for a.e. ${ \bf \ddot { F } } ( 0 )$ independently of the   
 277 sign and magnitude of the eigenvalues of W. This is also confirmed by the GCN-curve in Figure 2.   

parse/dev/siCt4xZn5Ve/siCt4xZn5Ve.md

@@ -1169,7 +1169,7 @@ In the following subsections, we provide the proofs of all the components used i
 
 # D.2 PROOF OF LEMMA 6.2
 
-Recall that for each $i \in [ n ] f _ { i } ( x ) = f ( u , v ) = z _ { i } ^ { \top } ( u ^ { \odot 2 } - v ^ { \odot 2 } ) , \nabla f _ { i } ( x ) = 2 \binom { z _ { i } \odot u } { z _ { i } \odot v }$ 2ziuziv
, and K(x) = $( K _ { i j } ( x ) ) _ { i , j \in [ n ] }$ where each $K _ { i j } ( x ) = \langle \nabla f _ { i } ( x ) , \nabla f _ { j } ( x ) \rangle$ . Then
+Recall that for each $i \in [ n ] f _ { i } ( x ) = f ( u , v ) = z _ { i } ^ { \top } ( u ^ { \odot 2 } - v ^ { \odot 2 } ) , \nabla f _ { i } ( x ) = 2 \binom { z _ { i } \odot u } { z _ { i } \odot v }$ 2ziuziv
, and K(x) = $( K _ { i j } ( x ) ) _ { i , j \in [ n ] }$ where each $K _ { i j } ( x ) = \langle \nabla f _ { i } ( x ) , \nabla f _ { j } ( x ) \rangle$ . Then
 
 $$
 \begin{array} { r } { \nabla ^ { 2 } \ell _ { i } ( x ) = 2 \left( \begin{array} { l } { z _ { i } \odot u } \\ { - z _ { i } \odot v } \end{array} \right) \left( ( z _ { i } \odot u ) ^ { \top } \quad - ( z _ { i } \odot v ) ^ { \top } \right) + ( f _ { i } ( u , v ) - y _ { i } ) \cdot \mathrm { d i a g } ( z _ { i } , z _ { i } ) . } \end{array}
@@ -1798,7 +1798,7 @@ $$
 \nabla \varphi ( \boldsymbol { x } ) = \sum _ { j = q + 1 } ^ { d } 2 w _ { j } ^ { * } \left[ \frac { \mathbb { 1 } \{ w _ { j } ^ { * } > 0 \} } { u _ { j } } \cdot e _ { j } - \frac { \mathbb { 1 } \{ w _ { j } ^ { * } < 0 \} } { v _ { j } } \cdot e _ { D + j } \right]
 $$
 
-where $e _ { j }$ is the $j$ -th canonical base of $\mathbb { R } ^ { d }$ . Recall that $\nabla f _ { i } ( x ) = 2 { \left( \begin{array} { l l } { z _ { i } \odot u } \\ { - z _ { i } \odot v } \end{array} \right) }$  ziu−ziv
, and we further have
+where $e _ { j }$ is the $j$ -th canonical base of $\mathbb { R } ^ { d }$ . Recall that $\nabla f _ { i } ( x ) = 2 { \left( \begin{array} { l l } { z _ { i } \odot u } \\ { - z _ { i } \odot v } \end{array} \right) }$  ziu−ziv
, and we further have
 
 $$
 \begin{array} { l } { \displaystyle \mathcal { Z } _ { 2 } = \sum _ { i = q ^ { \prime } + 1 } ^ { n } \lambda _ { i } ( x ( t ) ) \sum _ { j = q + 1 } ^ { d } w _ { j } ^ { * } \left[ \frac { \mathbb { 1 } \{ w _ { j } ^ { * } > 0 \} } { u _ { j } } \langle e _ { j } , z _ { i } \odot u \rangle + \frac { \mathbb { 1 } \{ w _ { j } ^ { * } < 0 \} } { v _ { j } } \langle e _ { j } , z _ { i } \odot v \rangle \right] } \\ { \displaystyle = \sum _ { i = q ^ { \prime } + 1 } ^ { n } \lambda _ { i } ( x ( t ) ) \sum _ { j = q + 1 } ^ { d } w _ { j } ^ { * } \left[ \frac { \mathbb { 1 } \{ w _ { j } ^ { * } > 0 \} } { u _ { j } } z _ { i , j } u _ { j } + \frac { \mathbb { 1 } \{ w _ { j } ^ { * } < 0 \} } { v _ { j } } z _ { i , j } v _ { j } \right] } \\ { \displaystyle = \sum _ { i = q ^ { \prime } + 1 } ^ { n } \lambda _ { i } ( x ( t ) ) \sum _ { j = q + 1 } ^ { d } w _ { j } ^ { * } z _ { i , j } = \sum _ { i = q ^ { \prime } + 1 } ^ { n } \lambda _ { i } ( x ( t ) ) \langle z _ { i , ( q + 1 ) : d } , w _ { ( q + 1 ) : d } ^ { * } \rangle = 0 } \end{array}