arXiv:2601.19070 · v3
Critical Organization of p-Adic Deep Neural Networks
摘要
We study continuous neural networks whose neurons are the points of the ring of p-adic integers, a fractal tree-like space that encodes hierarchical architectures. The hidden states solve a nonlinear integral equation, which describes the equilibria of recurrent networks and deep equilibrium models; feedforward architectures appear as the special case of kernels without feedback. We prove that the state is unique and depends Lipschitz-continuously on the parameters whenever the product of the Lipschitz constant of the activation function and the operator norm of the connectivity is less than one, and that, when the parameters are locally constant, every state is exactly the state of a finite network with a tree-like topology. We introduce p-adic discrete deep networks, whose layers are the levels of the tree and whose weights are coarse-grainings of a single kernel, and show that, in the stable region, they converge to the continuous network as the depth tends to infinity. Feedforward networks always have a unique state; hence a multiplicity of states, and therefore critical organization in our sense, requires recurrence. For recurrent networks with radial kernels we prove a sharp transition: the zero state is the only state if and only if the total coupling does not exceed one. For hierarchical kernels, above the threshold, the states are indexed by sign patterns on the leaves of the tree, their number is computed exactly, and they form a tree-indexed partially ordered set graded by instability. Numerical experiments illustrate these results and test conjectures on general radial kernels and on random recurrent networks on the tree. Finally, we interpret the critical organization hypothesis for deep neural networks, originally formulated for brain organization, through the lens of multistability, and we discuss the continuous limits of families of finite networks.
作者备注：43 pages, 4 figures. Fully revised version. The paper now focuses only on the critical organization of p-adic neural networks; for this reason, the part on statistical field theories (SFTs) was removed. The title was changed, new references were added, and the new version includes numerical simulations. The simulation code is available as ancillary files