ONN Canonical Specification

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1. #1

   Ontology graph. The target ontology is a weighted directed graph

   $$\mathcal{O}=(V_{\mathcal{O}},E_{\mathcal{O}},w_{\mathcal{O}})$$
2. #2

   Latent state. Each vertex carries a latent fiber. The global latent state is the concatenation

   $$z=(z_v{)}_{v\in V_{\mathcal{O}}}\in {\mathbb{R}}^{|V_{\mathcal{O}}|\times d_{\mathrm{fiber}}}$$
3. #3

   ONN dynamics. The composite operator defining a single update step is

   $$T=T_{\mathrm{proj}}\circ T_{\mathrm{consensus}}.$$
4. #4

   Constraint Satisfaction Rate (CSR). Given a set of predicates bound to edges , the CSR is the fraction of predicates satisfied at convergence:

   $$\mathrm{CSR}=\frac{1}{N}\sum _{i=1}^{N}1[p_i(z_{e_i}^{(\infty )})].$$
5. #5

   Total loss as Lyapunov certificate. The ONN total loss

   $${\mathcal{L}}_{\mathrm{total}}(S,A)={\mathcal{L}}_{\mathrm{semantic}}+{\mathcal{L}}_{\mathrm{topo}}+{\mathcal{L}}_{\mathrm{context}}$$