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md/dev/dFbKQaRk15w/dFbKQaRk15w.md

@@ -648,7 +648,7 @@ $c _ { v , S } ^ { 0 }  \bar { c }$ $\iota ( v )$
 Proposition 6. Let $G ^ { 1 } , G ^ { 2 }$ be any pair   
 of finite graphs. There exists $\bar { t } < \infty$ such that the DSS-WL (DS-WL) test converges when run on ${ \dot { G } } ^ { 1 } , G ^ { 2 }$ with any subgraph selection policy $\pi$ .
 
-Proof. Let $\mathcal { G } ^ { 1 } , \mathcal { G } ^ { 2 }$ be the bags induced by $\pi$ on $G ^ { 1 } , G ^ { 2 }$ . The two tests converge when, on all subgraph, the node partitioning induced by the coloring does not vary across two consecutive steps. Due to Proposition 5, we know that their colorings are vertex refinements on each of the subgraphs. This implies that, at iteration $t + 1$ , the partitioning on a subgraph is either as fine as, or finer than the one at iteration $t$ . In other words, if $N _ { t , S }$ is the number of color classes at round $t$ on subgraph $S$ , we have that $N _ { t + 1 , S } \ \geq \ N _ { t , S }$ . At the same time, the value $N _ { t , S } , \forall t \geq 0$ is upper bounded by $N _ { S } < \infty$ , i.e. the cardinality of the vertex set of $S$ . This implies that the number of possible iterations before convergence must be upper-bounded by the value $\nu = \mathrm { m a x } _ { i }$  $\scriptstyle \sum _ { S \in { \mathcal { G } } ^ { i } } N _ { S } )$ , which is, itself, finite. □
+Proof. Let $\mathcal { G } ^ { 1 } , \mathcal { G } ^ { 2 }$ be the bags induced by $\pi$ on $G ^ { 1 } , G ^ { 2 }$ . The two tests converge when, on all subgraph, the node partitioning induced by the coloring does not vary across two consecutive steps. Due to Proposition 5, we know that their colorings are vertex refinements on each of the subgraphs. This implies that, at iteration $t + 1$ , the partitioning on a subgraph is either as fine as, or finer than the one at iteration $t$ . In other words, if $N _ { t , S }$ is the number of color classes at round $t$ on subgraph $S$ , we have that $N _ { t + 1 , S } \ \geq \ N _ { t , S }$ . At the same time, the value $N _ { t , S } , \forall t \geq 0$ is upper bounded by $N _ { S } < \infty$ , i.e. the cardinality of the vertex set of $S$ . This implies that the number of possible iterations before convergence must be upper-bounded by the value $\nu = \mathrm { m a x } _ { i }$  $\scriptstyle \sum _ { S \in { \mathcal { G } } ^ { i } } N _ { S } )$ , which is, itself, finite. □
 
 The concept of vertex color refinement is linked to the ability to distinguish between non-isomorphic graphs. This is a consequence of the following Lemma, whose proof is adapted from Bodnar et al. (2021b).
 

md/dev/kroqZZb-6s/kroqZZb-6s.md

@@ -78,7 +78,7 @@ Given, $\hat { \mathbf { o } } _ { n , t }$ , a similar procedure is used to com
 
 # 3.4 ATTENTION MAP VISUALISATION
 
-Given cluster representation vectors, $\mathbf { c } _ { k }$ , we can compute a cluster-wise feature-time attention visualisation map as follows. First, we normalise feature-weights $\widehat { \mathbf { a } } _ { t }$ according to a softmax function, $\mathbf { s } _ { t } = \sigma ( \widehat { \pmb { \alpha } } _ { t } ) \in \mathbb { R } ^ { D _ { f } }$ , where $\sigma$ is the softmax function: $\begin{array} { r } { \sigma ( \mathbf { x } ) = \frac { \exp { | \mathbf { x } | } } { \| \exp { | \mathbf { x } | } \| _ { 1 } } } \end{array}$ . Secondly, we compute tsolved as before. We similarly normalise cluster-wise weights, $\gamma _ { t } ^ { k }$ according to a least-square approximation of $\gamma _ { n , t } ^ { k }$ tto obtain cluster temporal scores, $\begin{array} { r } { \dot { \mathbf { c } } _ { k } \approx \sum _ { t = 1 } ^ { T _ { n } } \widehat { \mathbf { o } } _ { n , t } \gamma _ { n , t } ^ { k } } \end{array}$ $e _ { n , t } ^ { k } = \sigma ( \gamma _ { n , t } ^ { k } )$ , and . Finally, we can compute $K$ scoring matrices, $M _ { n } ^ { 1 } , . . . , M _ { n } ^ { K } \in \mathbb { R } _ { T _ { n } \times D _ { f } }$ :  M kn 
 t,f $\left( M _ { n } ^ { k } \right) _ { t , f } = e _ { n , t } ^ { k } \pmb { s } _ { t } ^ { f }$ . Note that: $\begin{array} { r } { \| M _ { n } ^ { k } \| _ { 1 } = \sum _ { t } e _ { n , t } ^ { k } \sum _ { f } s _ { t } ^ { f } = \sum _ { t } e _ { n , t } ^ { k } = 1 } \end{array}$ . Given that Matrices $M _ { n } ^ { k }$ are normalised, they may be consequently, visualised as a normalised feature-time map for cluster assignment relevance and provide further model interpretability.
+Given cluster representation vectors, $\mathbf { c } _ { k }$ , we can compute a cluster-wise feature-time attention visualisation map as follows. First, we normalise feature-weights $\widehat { \mathbf { a } } _ { t }$ according to a softmax function, $\mathbf { s } _ { t } = \sigma ( \widehat { \pmb { \alpha } } _ { t } ) \in \mathbb { R } ^ { D _ { f } }$ , where $\sigma$ is the softmax function: $\begin{array} { r } { \sigma ( \mathbf { x } ) = \frac { \exp { | \mathbf { x } | } } { \| \exp { | \mathbf { x } | } \| _ { 1 } } } \end{array}$ . Secondly, we compute tsolved as before. We similarly normalise cluster-wise weights, $\gamma _ { t } ^ { k }$ according to a least-square approximation of $\gamma _ { n , t } ^ { k }$ tto obtain cluster temporal scores, $\begin{array} { r } { \dot { \mathbf { c } } _ { k } \approx \sum _ { t = 1 } ^ { T _ { n } } \widehat { \mathbf { o } } _ { n , t } \gamma _ { n , t } ^ { k } } \end{array}$ $e _ { n , t } ^ { k } = \sigma ( \gamma _ { n , t } ^ { k } )$ , and . Finally, we can compute $K$ scoring matrices, $M _ { n } ^ { 1 } , . . . , M _ { n } ^ { K } \in \mathbb { R } _ { T _ { n } \times D _ { f } }$ :  M kn 
 t,f $\left( M _ { n } ^ { k } \right) _ { t , f } = e _ { n , t } ^ { k } \pmb { s } _ { t } ^ { f }$ . Note that: $\begin{array} { r } { \| M _ { n } ^ { k } \| _ { 1 } = \sum _ { t } e _ { n , t } ^ { k } \sum _ { f } s _ { t } ^ { f } = \sum _ { t } e _ { n , t } ^ { k } = 1 } \end{array}$ . Given that Matrices $M _ { n } ^ { k }$ are normalised, they may be consequently, visualised as a normalised feature-time map for cluster assignment relevance and provide further model interpretability.
 
 The model is optimised through consideration of three distinct loss functions. We introduce a weighted cross-entropy loss function: