# Universal Local Attractors on Graphs
## Appendix B Theoretical Proofs
Let us also consider the posteriors  
\[r(t)=F(\hat{M})h(t)\]1  
arising from the graph signal generation function \(\mathcal{H}_{\theta(t)}\) at times \(t\) for parameters \(\theta(t)\), as well as the least squares problem of minimizing the projection of the loss's gradient to the row space of \(\mathbb{J}_{\mathcal{H}_{\theta}}\big{(}\theta(t)\big{)}\):  
\[r(t)=F(\hat{M})h(t)\]2  
The closed form solution to this problem can be found by  
\[r(t)=F(\hat{M})h(t)\]3  
Thus, as long as \(h(t)=\mathcal{H}_{\theta(t)}\) (we will call this Proposition P.I), this lemma's first precondition can be written for posteriors \(r(t)=F(\hat{M})h(t)\) and \(\
abla_{r}\mathcal{L}\big{(}r(t)\big{)}\
eq\mathbf{0}\) as  
\[r(t)=F(\hat{M})h(t)\]4  
We now set \(x(t)\) as the derivatives of some parameters across their trajectory, that is,  
\[r(t)=F(\hat{M})h(t)\]5Therefore, from Lemma A1 we obtain that our system asymptotically arrives at locally optimal posteriors, i.e., with \(\
abla_{r}\mathcal{L}(r_{\infty})=0\) for \(r_{\infty}=\lim_{t\to\infty}r(t)\). As a final step, we now investigate the priors signal editing converges at. For this, we can see that the update rule leads to the selection of priors \(h(t)\) at times \(t\) for which  
\[r(t)=F(\hat{M})h(t)\]6  
For some parameters \(\theta_{\infty}=\lim_{t\to\infty}\theta(t)\). We can similarly show that Proposition P.I holds true. Hence, there exists an optimization path of prior editing parameters (not necessarily the same as the one followed by the optimization algorithm used in practice) that arrives at the edited priors \(\mathcal{H}_{\theta_{\infty}}\) that let posteriors exhibit local optimality. Take the setting of Lemma A2, where the loss's Hessian matrix \(\mathbb{H}_{\mathcal{L}}(r)\) linearly approximates the gradients \(\
abla_{r}\mathcal{L}(r)\) within the domain \(r\in\mathcal{R}\) with error at most \(\epsilon_{\mathbb{H}}\). Let us construct a node feature matrix \(H^{(0)}\) whose columns are graph signals (its rows \(H^{(0)}[v]\) correspond to nodes \(v\)). If \(h_{0}\), \(F(\hat{M})h_{0}\), and all graph signals other than \(r\) involved in the calculation of \(\mathcal{L}(r)\) are columns of \(H^{(0)}\), and it holds that  
\[r(t)=F(\hat{M})h(t)\]7  
for any \(r\in\mathcal{R}\setminus\{r_{\infty}\}\), then for any \(\epsilon_{\infty}>0\) there exists \(MLP_{\theta}\) with relu activation, for which  
\[r(t)=F(\hat{M})h(t)\]8  
Let \(\mathbb{H}_{\mathcal{L}}(r)\) be the Hessian matrix of \(\mathcal{L}(r)\) with respect to node values of the graph signal \(r\in\mathcal{R}\) of a domain \(\mathcal{R}\). Given that its rank is \(K(r)\), there exists a decomposition \(\mathbb{H}_{\mathcal{L}}(r)=J(r)D(r)\) with fixed inner dimensions \(K_{\max}=\sup_{r\in\mathcal{R}}K(r)\). That is, \(J(r)\) has dimensions \(|\mathcal{V}|\times K_{\max}\) and \(D(r)\) has dimensions \(K_{\max}\times dim(\Theta)\), where \(dim(\Theta)\) counts the dimensions of the parameter space \(\Theta\) (i.e., is the number of parameters). When \(K(r)=K_{\max}\), the decomposition even becomes unique for the particular \(r\). We now select a prior generation model \(\mathcal{H}_{\theta}\) with linearly independent rows and a Jacobian \(\mathbb{J}_{\mathcal{H}_{\theta}}(\theta)\) that is a function (e.g., an MLP) approximating well the first of the decomposition elements. That is, for any chosen small-enough _but non-negligible_ constant \(\epsilon_{\mathbb{J}}>0\) within the domain \(\mathcal{R}\) it holds that  
\[r(t)=F(\hat{M})h(t)\]9  
Throughout this proof, \(r\) is a function of \(\theta\), but we do not write \(r(\theta)\) for simplicity. The Jacobian does not need to have linearly independent rows; small perturbations (with the cumulative effect that when added to the actual approximation error make it smaller than \(\epsilon_{\mathbb{J}}\)) can be injected in each row to make the rows of its approximation linearly independent.