Chapter 16 — Open Problems

Open source document →

1. #1

   If and are two global minima of with , then:

   $${\rho }_{F^{\circ }}^{(1)}(i)={\rho }_{F^{\circ }}^{(2)}(i)\forall i\in F^{\circ }.$$
2. #2

   Energy-based doors and holonomy-based doors are related:

   $${\Sigma }_{\tau }^{\mathrm{hol}}\subset {\Sigma }_{{\tau }^{\prime }}\text{for some }{\tau }^{\prime }={\tau }^{\prime }(\tau ,G).$$
3. #3

   The mutual information between a fruit and its exterior is bounded by door energy:

   $$I(F;\bar{F})\le C\sum _{p\in \Sigma }{e}_p\log \frac{e_p}{\tau }.$$
4. #4

   The evidence extraction functional:

   $$\mathrm{Ev}(s_t,s_{t+1},e_t)=(e_t,\Delta W,\Delta \Omega ,\Delta \mathfrak{F},\Delta \Sigma )$$
5. #5

   The fruit set is derived from the weights via the Cheeger conductance:

   $${\mathfrak{F}}_t=\{\text{subsets }F:{\varphi }_t(F)<\theta ,\text{ maximal},\text{ non-overlapping}\}.$$