mlsanityData to 7 Oct 2026

Finite-Lag Operator Geometry of Recurrent Representations

arXiv 2607.01746 ↗First posted 2 Jul 2026cs.LGcited by 0

Not on the map. Why?

Abstract (arXiv)

Recurrent representations are trajectories, but representation geometry is often measured from static snapshots. We develop finite-lag operator geometry for recurrent hidden states from observed source-successor pairs (Xt,Xt+Delta). The primitive is the conditional transport law Q_Delta(dymid x), estimated by a dense Gaussian source-smoothing operator. From this directed finite-lag law we derive a source-centered transport tensor G_Delta, which decomposes exactly into conditional spread and coherent displacement, and an antisymmetric coordinate circulation W_Deltaρ, which summarizes directed lagged flow. We prove affine covariance with explicit metric dependence of scalar summaries, dense estimator stability on bounded trajectory clouds, and a finite-lag separation result showing that source-centered transport detects deterministic recurrent motion not recorded by infinitesimal carre-du-champ geometry. A linear-Gaussian closed form calibrates the quantities in terms of the update A_Delta, source covariance, and innovation covariance. Controlled experiments validate the decomposition, circulation, covariance, and stability predictions. In performance matched repeat-copy networks, the framework reveals architecture dependent differences in total transport scale and coherent displacement trace, while coherent displacement fraction is metric and resolution dependent.

Key facts. No citing papers in the map yet (data through 7 Oct 2026). Builds on 17 of its 43 references that are in the map. First posted on arXiv 2 Jul 2026.

Builds on

17 of 43 references.

References · 17

  1. 2014Learning Phrase Representations using RNN Encoder-Decoder for Statistical Machine Translationcited by 4,016
  2. 2019Similarity of Neural Network Representations Revisitedcited by 1,695
  3. 2020Prevalence of Neural Collapse during the terminal phase of deep learning trainingcited by 713
  4. 2019Intrinsic dimension of data representations in deep neural networkscited by 276
  5. 2017VAMPnets: Deep learning of molecular kineticscited by 135
  6. 2019Universality and individuality in neural dynamics across large populations of recurrent networkscited by 61
  7. 2017Variational approach for learning Markov processes from time series datacited by 51
  8. 2017Linking connectivity, dynamics and computations in low-rank recurrent neural networkscited by 51
  9. 2017Eigendecompositions of Transfer Operators in Reproducing Kernel Hilbert Spacescited by 48
  10. 2019Reverse engineering recurrent networks for sentiment classification reveals line attractor dynamicscited by 38
  11. 2020Lipschitz regularity of graph Laplacians on random data cloudscited by 26
  12. 2021Reverse engineering recurrent neural networks with Jacobian switching linear dynamical systemscited by 17
5 more
  1. 2023Assessing Neural Network Representations During Training Using Noise-Resilient Diffusion Spectral Entropycited by 13
  2. 2020Shaping dynamics with multiple populations in low-rank recurrent networkscited by 10
  3. 2024Hidden Traveling Waves bind Working Memory Variables in Recurrent Neural Networkscited by 5
  4. 2025Dynamical phases of short-term memory mechanisms in RNNscited by 1
  5. 2026Diffusion Operator Geometry of Feedforward Representationscited by 1

Built on by

Not cited yet.