The Solution of the Deep Boltzmann Machine on the Nishimori Line

The deep Boltzmann machine on the Nishimori line with a finite number of layers is exactly solved by a theorem that expresses its pressure through a finite dimensional variational problem of min–max type. In the absence of magnetic fields the order parameter is shown to exhibit a phase transition whose dependence on the geometry of the system is investigated.

SYMMETRY BREAKING RESULTS FOR PROBLEMS WITH EXPONENTIAL GROWTH IN THE UNIT DISK

We investigate some asymptotic properties of extrema uα to the two-dimensional variational problem [Formula: see text] as α → +∞. Here B is the unit disk of ℝ2 and 0 < γ ≤ 4π is a given parameter. We prove that in a certain range of γ′s, the maximizers are not radial for α large.