While I was reading the paper "Geometric deep learning on graphs and manifolds using mixture model CNNs", I didn't understand the figure of "patch operator weighting functions". Can someone explain me clearly how these red curves relates to graphs and manifolds? Thank you.
Weighting functions in the local polar system of coordinates.
geometric-topologygeometrygraph theorymanifolds
Related Solutions
Yes. What you wrote is correct. Instead of viewing $S^2\subset\Bbb{R}^3$ as you did, let $$S^3=\{(\alpha,\beta)\in\Bbb{C}^2: |\alpha|^2+|\beta|^2=1\}$$ $$S^2=\{(x,z)\in\Bbb{R}\times\Bbb{C}: x^2+|z|^2=1\}$$ $$S^1=\{z\in\Bbb{C}: |z|^2=1\}$$ Then $h:S^3\rightarrow S^2$ can be written slightly more compactly as $$h(\alpha,\beta)=(|\alpha|^2-|\beta|^2,2\alpha\bar{\beta})$$
Now let's come to the fibers $h^{-1}(p)$.
Claim: $\, h(\alpha_1,\beta_1)=h(\alpha_2,\beta_2)\iff (\alpha_1,\beta_1)=(z\alpha_2,z\beta_2) \ \mbox{ for some } z\in S^1$.
Consequence: each fiber $h^{-1}(p)\simeq S^1$ is a circle!
Proof of the Claim: If $z\in S^1$ then $|z|^2=z\bar{z}=1$, so $$h(z\alpha,z\beta)=(|z|^2|\alpha|^2-|z|^2|\beta|^2,2z\bar{z}\alpha\bar{\beta})=h(\alpha,\beta)$$
Viceversa, suppose $h(\alpha_1,\beta_1)=h(\alpha_2,\beta_2)$. In polar form we can write $$\alpha_k=r_ke^{i\theta_k}, \ \beta_k=s_ke^{i\phi_k} \quad (k=1,2)$$ Imposing the two conditions $(\alpha_k,\beta_k)\in S^3$ and $h(\alpha_1,\beta_1)=h(\alpha_2,\beta_2)$ we get $$\begin{cases} r_1^2+s_1^2=r_2^2+s_2^2=1 \\ r_1^2-r_2^2=s_1^2-s_2^2 \end{cases} \Rightarrow \begin{cases} r_1=r_2 \\ s_1=s_2 \end{cases}$$ Moreover, it has to be $\theta_1-\theta_2=\phi_1-\phi_2$, which implies $\phi_1-\theta_1=\phi_2-\theta_2=\vartheta$ for some fixed $\vartheta$. Conclusion: $(\alpha_1,\beta_1)=(z\alpha_2,z\beta_2)$, where $z=e^{i\vartheta}$.
Edit: Some fibers stereographically projected on $\Bbb{R}^3$, as explained in the comments below:
The diagram in correct proportion. To get the square edge length $10,$ multiply all lengths by $$ \frac{10}{8} = \frac{5}{4} = 1.25 $$
Best Answer
Any differentiable manifold is locally homeomorphic to Euclidean space. In other words, if we select a point on the manifold, then over very small distances the manifold can be approximated by Euclidean space. It is then possible to parameterise the manifold with local polar coordinates $(\rho,\theta)$ which behave like polar coordinates in an infinitesimal region around the selected point.
The models GCNN, ACNN and MonNet each use a differentiable manifold parameterised by local polar coordinates. They have a weighting function, called the patch operator weighting function $w_i(\rho,\theta)$. Table $1$ in the paper gives $w_i(\rho,\theta)$ for ACNN and GCNN.
The red curves are $0.5$ level sets. That is to say, $w_i(\rho,\theta)=0.5$ along the red curves.
Edit: The OP asked about the definition of MoNet
In section 4, the paper mentions using a weighting function of the form $w_j({\bf{u}})=\exp\left(-\frac{1}2(\bf{\mu}-\bf{\mu}_j)^T\bf{\Sigma}_j(\bf{\mu}-\bf{\mu}_j)\right)$ with $\bf{\Sigma}_j$ and $\bf{\mu}_j$ learnable (formula 11 in the paper). $\bf{\Sigma}_j$ is restricted to being a diagonal matrix.
The paper then describes the neural network used to learn $\bf{\Sigma}_j$ and $\bf{\mu}_j$ and the procedure used to train it. The Adam method is explained by the following paper: https://arxiv.org/abs/1412.6980