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

Open source document →

1. #1

   The energy is minimized on the constraint manifold

   $$\Sigma_m = \Big\{u \in [0,1]^n : \sum_{x \in X_t} u_t(x) = m\Big\}$$
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

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

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

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

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

   The adjacency structure is instantiated through a local relation kernel satisfying

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

   The associated aggregation operator is defined as

   $$(P_t f)(x) = \frac{\sum_{y} K_t(x,y)\, f(y)}{\sum_{y} K_t(x,y) + \varepsilon}$$
9. #9

   The currently adopted closure operator is

   $$\mathrm{Cl}_t(u)(x) = \sigma\!\Big( a_{\mathrm{cl}} \big( (1 - \eta_{\mathrm{cl}})\, u(x) + \eta_{\mathrm{cl}}\, (P_t u)(x) - \tau_{\mathrm{cl}} \big) \Big)$$
10. #10

    The currently adopted distinction operator is

    $$\mathbf{D}_t(x; 1-u) = \sigma\!\Big( a_D \big( (P_t u)(x) - \lambda_D\, (P_t(1-u))(x) \big) - \tau_D \Big)$$
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

    $$\mathbf{M}_{t \to s}(x,y) = \frac{ \exp\!\Big( -\dfrac{\|y - \Psi_{t \to s}(x)\|^2}{2\sigma_M^2} - \gamma_M \|\varphi_t(x) - \varphi_s(y)\|^2 \Big) }{ \sum_{y'} \exp\!\Big( -\dfrac{\|y' - \Psi_{t \to s}(x)\|^2}{2\sigma_M^2} - \gamma_M \|\varphi_t(x) - \varphi_s(y')\|^2 \Big) + \varepsilon }$$