New Geometry Method Analyzes Recurrent Neural Network Dynamics

Recurrent representations, often seen as trajectories, are typically analyzed using static snapshots, which can miss their dynamic nature. This research introduces "finite-lag operator geometry" to specifically analyze recurrent hidden states based on observed source-successor pairs. The core idea involves estimating a conditional transport law using a dense Gaussian source-smoothing operator. From this directed finite-lag law, the method derives a source-centered transport tensor that precisely decomposes into conditional spread and coherent displacement. It also yields an antisymmetric coordinate circulation, which summarizes directed lagged flow. The authors prove affine covariance, estimator stability, and a finite-lag separation result, demonstrating that this geometry can detect deterministic recurrent motion not captured by infinitesimal methods. Controlled experiments validate the decomposition and reveal architecture-dependent differences in total transport scale and coherent displacement, offering a deeper understanding of how recurrent networks process information over time.