Blind directions of physical learning networks: where to measure and what to measure

Abstract

A physical learning network is read at a few accessible nodes. Every task and learning rule that uses only the steady voltages and currents there, at a fixed operating point, acts through the boundary response map. Changes in the kernel of its Jacobian are blind to first order. A walk along one fiber ran to a 600-step cap, one edge then at 15.2 times its start. A decomposition theorem splits the response Jacobian over the hidden components. The blind dimension adds over components whose surviving slots are disjoint. A boundary-to-boundary edge adds one parameter and deletes one slot. Exposing a hidden node changes only its component. For one hidden node the contribution counts the bipartite components of the non-adjacency graph of its neighbors. For a pocket of h hidden nodes a factor-analysis bound caps what outside electrodes can expose. It is attained when every pocket node meets every neighbor and no edge joins two of those neighbors, so past a threshold further electrodes outside such a pocket expose nothing. An electrode inside such a pocket, when its hidden nodes form a clique, is worth h-1 directions where one outside is worth none. Read as vector displacements rather than potentials, the same nodes left no deficit beyond counting in all 90 spring networks tested, each a pocket fully joined to five or more accessible nodes, and nothing blind in 88. What limits the reading is the quantity measured as much as the number of contacts. Maximum-weight spanning forests give a proved upper bound on the blind dimension. The matching equality is proved for one hidden node and conjectured beyond. It holds in all 5,343 components whose maximum could be attained and certified, and the forest count matched the certified blind dimension in 1,448 of 1,500 held-out networks, where the maximum must be searched. A self-learning circuit shows the split on hardware.