Chapter 16 — Open Problems

Open source document →

1. #1

   If and are two global minima of with , then:

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

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

   $$\Sigma^{\mathrm{hol}}_\tau\subset\Sigma_{\tau'}\quad\text{for some }\tau'=\tau'(\tau,G).$$
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:

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