Neural-network complexity conjecture for C-GNP latent domains

Let $4\Omega4$ be the latent space of a deep neural network satisfying the C-GNP property. Denote by $4\tau_{\Omega}4$ its thickness function and by $4\gamma(\Omega)4$ its convexity gap.

Neural-network complexity conjecture. The complexity of the architecture, measured by the number of layers and its width, is controlled by $4\|\tau_{\Omega}\|_{\infty}4$ and $4\gamma(\Omega)4$. More precisely, a bound on these two measures implies a bound on the approximation capacity of the network.

This conjecture proposes a link between geometric control of a C-GNP latent domain and the expressive complexity of the corresponding neural network. The statement is presented as a perspective in the source, with no evidence of resolution supplied.