Iterative graph lifting for automatic design of path-complete stability certificates

What the math gives to ML

The paper provides a constructive way to refine a finite-state stability certificate when a single path-complete graph is too coarse: inspect which graph-node inequalities are active at the optimum, split bottleneck nodes, and re-solve the resulting certificate problem. The transferable asset is not the switched-system application itself, but the combination of graph-indexed Lyapunov functions, semidefinite contraction inequalities, and parsimonious node splitting that preserves feasibility while increasing certificate expressivity. A promising neural-network use is a routed state-space backbone whose transition matrix changes with tokens or experts; the graph-lifting loop can automatically discover a small mode-history-dependent quadratic certificate and enforce stability under arbitrary routing.