Assume (at any $t$) the $m$ points are sampled from some manifold $\mathbb{M}_t$. Then, the sum of \textcolor{red}{diffusion distances} can be interpreted as a \textcolor{red}{Monte Carlo approximation} [4] s.t. $S_{\epsilon, t} :=\sum_{i,j}\mathbf{K}_{\epsilon, t}(i,j)\approx\frac{m^2}{\mathrm{vol}(\mathbb{M}_t)^2}\int_{\mathbb{M}_t}\int_{\mathbb{M}_t}\exp(-D(x, y)/\epsilon)\dd x \dd y \approx\frac{m^2}{\mathrm{vol}(\mathbb{M}_t)}(2\pi\epsilon)^{d/2}$, where $\mathbb{R}^d$ is the tangential space of $\mathbb{M}_t$. $S_{\epsilon, t}$ is a measure of the \textcolor{red}{density} of points, with $\lim_{{\epsilon\to0}} S_{\epsilon, t}= m$ and $\lim_{{\epsilon\to\infty}}S_{\epsilon, t}= m^2$. Also $\log(S_{\epsilon, t})\approx\frac{d}{2}\log(\epsilon)+\log(\frac{(2\pi)^{d/2} m^2}{\mathrm{vol}(\mathbb{M}_t)})$.
Assume (at any $t$) the $m$ points are sampled from some manifold $\mathbb{M}_t$. Then, the sum of \textcolor{red}{diffusion distances} can be interpreted as a \textcolor{red}{Monte Carlo approximation} [4] s.t. $S_{\epsilon, t} :=\sum_{i,j}\mathbf{K}_{\epsilon, t}(i,j)\approx\frac{m^2}{\mathrm{vol}(\mathbb{M}_t)^2}\int_{\mathbb{M}_t}\int_{\mathbb{M}_t}\exp(-D(x, y)/\epsilon)\dd x \dd y \approx\frac{m^2}{\mathrm{vol}(\mathbb{M}_t)}(2\pi\epsilon)^{d/2}$, where $\mathbb{R}^d$ is the tangential space of $\mathbb{M}_t$. $S_{\epsilon, t}$ is a measure of the \textcolor{red}{density} of points, with $\lim_{{\epsilon\to0}} S_{\epsilon, t}= m$ and $\lim_{{\epsilon\to\infty}}S_{\epsilon, t}= m^2$. Also $\log(S_{\epsilon, t})\approx\frac{d}{2}\log(\epsilon)+\log(\frac{(2\pi)^{d/2} m^2}{\mathrm{vol}(\mathbb{M}_t)})$.
\vspace{8in}\\
\vspace{6.5in}\\
We obtain $d =\dim(\mathbb{M}_t)$ by maximizing $\frac{\dd\log(S_{\epsilon, t})}{\dd\log(\epsilon)}$ and calculate the point density measure $\rho(t) :=\frac{1}{d}\log(\frac{m}{\mathrm{vol}(\mathbb{M}_t)})$.
We obtain $d =\dim(\mathbb{M}_t)$ by maximizing $\frac{\dd\log(S_{\epsilon, t})}{\dd\log(\epsilon)}$ and calculate the point density measure $\rho(t) :=\frac{1}{d}\log(\frac{m}{\mathrm{vol}(\mathbb{M}_t)})$.
\vspace{5.9in}\\
\vspace{5.4in}\\
The trajectories tend to be \textcolor{red}{denser after passage} through the QSAS, which is in line with the established notion of blockings as regions characterized by high stability.
The trajectories tend to be \textcolor{red}{denser after passage} through the QSAS, which is in line with the established notion of blockings as regions characterized by high stability.
}
}
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%\begin{tikzpicture}
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% First image (bottom layer)
% First image (bottom layer)
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@@ -89,8 +89,8 @@ The trajectories tend to be \textcolor{red}{denser after passage} through the QS
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@@ -89,8 +89,8 @@ The trajectories tend to be \textcolor{red}{denser after passage} through the QS
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Flow across the boundary of $\mathbb{M}_t$ is possible. Therefore, \textcolor{red}{Dirichlet boundary conditions}$\mathbf{P}_{\epsilon, t}(i, j)=0\forall x^i_t \text{ or } x^j_t \in\partial\mathbb{M}_t$ have to be applied. Boundary points are detected with $\alpha$\texttt{-shapes} [3] using $\alpha\approx\mathcal{O}(\epsilon^{-1/2})$.
Flow across the boundary of $\mathbb{M}_t$ is possible. Therefore, \textcolor{red}{Dirichlet boundary conditions}$\mathbf{P}_{\epsilon, t}(i, j)=0\forall x^i_t \text{ or } x^j_t \in\partial\mathbb{M}_t$ have to be applied. Boundary points are detected with $\alpha$\texttt{-shapes} [3] using $\alpha\approx\mathcal{O}(\epsilon^{-1/2})$.
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@@ -100,7 +100,7 @@ Flow across the boundary of $\mathbb{M}_t$ is possible. Therefore, \textcolor{re
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@@ -100,7 +100,7 @@ Flow across the boundary of $\mathbb{M}_t$ is possible. Therefore, \textcolor{re
Eigenvaluedecomposition of $\mathbf{Q}_{\epsilon}$ results in
Eigenvaluedecomposition of $\mathbf{Q}_{\epsilon}$ results in
