wasserstein.tex (1710B)
1 \begin{frame}{1-Wasserstein-based Use of Topological Features} 2 \begin{columns}% 3 \begin{column}{6.5cm}% 4 \begin{block}{Earth Mover Distance} 5 \centering% 6 \input{mainmatter/graph/Wasserstein.tex} 7 \end{block} 8 \begin{block}{Compare Topological Features} 9 Skip recoloring, directly compare neighborhoods in \(\symbb{R}^d\): 10 11 \smallskip 12 13 \(S(x, k) = \text{samples at distance \(k\) of \(x\)}\) 14 15 \smallskip 16 17 \(\symfrak{S}(x, k) =\)\\ 18 \hfill \(\{\,\bertcoder(y)\in\symbb{R}^d \mid y\in S(x, k)\,\}\)\\ 19 20 \smallskip 21 22 \centering% 23 \fbox{\(W_1(\symfrak{S}(x, 1), \symfrak{S}(x', 1))\)} 24 \end{block} 25 \end{column}% 26 \begin{column}{6.5cm}% 27 \begin{algorithmic} 28 \Function{Weisfeiler--Leman}{} 29 \FunctionInputs{} \(G=(V, E)\) graph 30 \FunctionInputs*{} \(k\) dimensionality 31 \FunctionOutput{} \(\chi_\infty\) coloring of \(k\)-tuples 32 \State 33 \State \(\chi_0(\vctr{x}) \gets \operatorname{iso}(\vctr{x}) \quad \forall \vctr{x}\in V^k\) 34 \For{\(\ell=1,2,\dotsc\)} 35 \State \(\symfrak{I}_\ell\gets \text{new color index}\) 36 \ForAll{\(\vctr{x}\in V^k\)} 37 \State \(c_\ell(\vctr{x}) \mathop{\raisebox{-1mm}{\(\Lsh\)}}\) 38 \State \hspace{2mm}\(\lMultiBrace\,\chi_{\ell-1}(\vctr{y}) \middlerel{|} \vctr{y}\in\gfneighbors^k(\vctr{x})\,\rMultiBrace\) 39 \State \(\chi_\ell(\vctr{x}) \mathop{\raisebox{-1mm}{\(\Lsh\)}}\) 40 \State \hspace{10mm}\((\chi_{\ell-1}(\vctr{x}), c_\ell(\vctr{x})) \text{ in } \symfrak{I}_\ell\) 41 \EndFor 42 \EndFor 43 \State \textbf{until} \(\chi_\ell = \chi_{\ell-1}\) 44 \State \Output \(\chi_\ell\) 45 \EndFunction 46 \end{algorithmic} 47 \end{column}% 48 \end{columns}% 49 \end{frame}