SCC Canonical Spec — Part 3: Energy Principle & Provisional Operators

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1. #1

   The energy is minimized on the constraint manifold

   $${\Sigma }_m=\{u\in [0,1{]}^n:\sum _{x\in X_t}{u}_t(x)=m\}$$
2. #2

   The canonical energy over a temporal window is:

   $$\mathcal{E}(\mathbf{u})={\lambda }_{\mathrm{cl}}{\mathcal{E}}_{\mathrm{cl}}+{\lambda }_{\mathrm{sep}}{\mathcal{E}}_{\mathrm{sep}}+{\lambda }_{\mathrm{bd}}{\mathcal{E}}_{\mathrm{bd}}+{\lambda }_{\mathrm{tr}}{\mathcal{E}}_{\mathrm{tr}}$$
3. #3

   $${\mathcal{E}}_{\mathrm{cl}}(u_t)=\sum _{x\in X_t}(u_t(x)-{\mathrm{Cl}}_t(u_t)(x){)}^2$$
4. #4

   $${\mathcal{E}}_{\mathrm{sep}}(u_t)=\sum _{x\in X_t}{u}_t(x)(1-{\mathbf{D}}_t(x;1-u_t))$$
5. #5

   $${\mathcal{E}}_{\mathrm{bd}}(u_t)=\alpha \sum _{x,y\in X_t}{\mathbf{N}}_t(x,y)(u_t(x)-u_t(y){)}^2+\beta \sum _{x\in X_t}{u}_t(x{)}^2(1-u_t(x){)}^2$$
6. #6

   $${\mathcal{E}}_{\mathrm{tr}}(u_t,u_s)=\sum _{x\in X_t}\sum _{y\in X_s}{\mathbf{M}}_{t\to s}(x,y)\omega (u_t(x),u_s(y))(u_s(y)-u_t(x){)}^2$$
7. #7

   The adjacency structure is instantiated through a local relation kernel satisfying

   $$K_t(x,y)\ge 0,K_t(x,y)=K_t(y,x).$$
8. #8

   The associated aggregation operator is defined as

   $$(P_{t}f)(x)=\frac{{\sum }_y{K}_t(x,y)f(y)}{{\sum }_y{K}_t(x,y)+\epsilon }$$
9. #9

   The currently adopted closure operator is

   $${\mathrm{Cl}}_t(u)(x)=\sigma \left(a_{\mathrm{cl}}\right((1-{\eta }_{\mathrm{cl}})u(x)+{\eta }_{\mathrm{cl}}(P_{t}u)(x)-{\tau }_{\mathrm{cl}}\left)\right)$$
10. #10

    The currently adopted distinction operator is

    $${\mathbf{D}}_t(x;1-u)=\sigma \left(a_D\right((P_{t}u)(x)-{\lambda }_D(P_t(1-u))(x))-{\tau }_D)$$
11. #11

    The currently adopted co-belonging operator is the resolvent form

    $${\mathbf{C}}_t=(I-\alpha W_{\mathrm{sym}}{)}^{-1}$$
12. #12

    where is the symmetrized cohesion-weighted adjacency matrix. The resolvent can be expanded as a Neumann series:

    $${\mathbf{C}}_t=\sum _{k=0}^{\infty }{\alpha }^k{W}_{\mathrm{sym}}^k$$
13. #13

    The currently adopted temporal transport kernel is

    $${\mathbf{M}}_{t\to s}(x,y)=\frac{\exp (-\frac{\Vert y-{\Psi }_{t\to s}(x){\Vert }^2}{2{\sigma }_M^2}-{\gamma }_M\Vert {\phi }_t(x)-{\phi }_s(y){\Vert }^2)}{{\sum }_{y^{\prime }}\exp (-\frac{\Vert y^{\prime }-{\Psi }_{t\to s}(x){\Vert }^2}{2{\sigma }_M^2}-{\gamma }_M\Vert {\phi }_t(x)-{\phi }_s(y^{\prime }){\Vert }^2)+\epsilon }$$