On the Number of Observation Nodes in Recurrent Neural Networks with Linear Threshold and ReLU Functions
This paper investigates how node update rules and admissible state domains affect the minimum number of observation nodes required for global finite-horizon observability in recurrent neural networks with linear-threshold and ReLU update functions. Over a common binary state domain, we construct a class of $K$-linear-threshold ($K$-LT) networks whose initial states can be uniquely reconstructed from the finite output trajectory of a single observation node. We further establish a dynamical equivalence between binary-valued $K$-ReLU networks and $K$-AND Boolean networks, which transfers existing class-level observation-node bounds to binary-valued $K$-ReLU networks. For nonnegative-valued ReLU networks, the observability problem reduces to the classical linear-system setting whenever the relevant pre-activations remain nonnegative. For general real-valued ReLU networks, we prove that global finite-horizon observability requires at least $n/2$ observation nodes when no restriction is imposed on the number of state variables involved in each node update. This lower bound is tight when $n=2K$, for which we construct a $K$-ReLU network observable from exactly $K$ nodes. These results show that both update rules and state domains fundamentally affect extremal observation requirements: temporal evolution can concentrate finite-state information into a single measured trajectory, whereas activation-induced rank loss creates an intrinsic sensor lower bound in continuous-state ReLU networks.