The paper investigates the discrete-time self‑attention dynamics of transformers, providing a full classification of equilibria and their stability for both single‑head and multi‑head cases. It shows that the main attractor is a consensus equilibrium aligned with the principal eigenvector of the value matrix, while locally asymptotically stable bipartite equilibria also exist in the single‑head scenario. The theoretical findings are corroborated by examining learned weight matrices from an actual transformer.
arXiv:2609.36230v1 Announce Type: new Abstract: We study the dynamical behavior of tokens in transformers from a control-theoretic perspective. Our model includes the feed-forward layer present after...
Selective state space models (SSMs) use a recurrence to mix token information, a process analogous to attention in transformers. By modeling token evolution as an ordinary differential equation and applying input‑to‑state stability, the study proves that SSMs exhibit local exponential stability of consensus equilibria and delineates their domain of attraction for time‑varying weight matrices. Experiments on a pretrained Mamba‑2 model reveal that the output gate controls the degree of consensus, preventing tokens from fully converging.
arXiv:2512. 21113v2 Announce Type: replace Abstract: Transformers are increasingly adopted for modeling and forecasting time-series, yet their internal mechanisms remain poorly understood from a dynamical systems perspective.
The paper introduces a Mixture‑Gated Theory of Anisotropic Representation Dynamics that decomposes Transformer self‑attention into value transport, query gating, and key gating. It shows that the geometry of attended representations can cause a representation difference to contract or expand, a phenomenon termed gating‑induced regime switching. The authors derive a spectral criterion to predict switching, quantify its strength, and identify subspaces where switching occurs, validating these predictions experimentally across four Transformer architectures.
arXiv:2610.23242v2 Announce Type: replace Abstract: Small differences between internal representations can shrink or grow as they pass through a Transformer layer, affecting how information is preser...
arXiv:2602. 19143v2 Announce Type: replace Abstract: This paper studies simple transformers trained on a high-order Markov chain, where the model must incorporate information from multiple past positions, each with different statistical importance.
arXiv:2501. 18322v2 Announce Type: replace Abstract: Transformers, which are state-of-the-art in most machine learning tasks, represent the data as sequences of vectors called tokens.
arXiv:2606. 07600v1 Announce Type: cross Abstract: We formulate data propagation through the Transformer, the machine learning architecture powering large language models, as a nonlinear control system on the space of probability measures.
arXiv:2609.28448v1 Announce Type: cross Abstract: We study the nonequilibrium dynamics of a minimal recurrent transformer with $N$ normalized tokens, $Q=K=I$, and a negative value map $V=-I$. Similar...
arXiv:2608. 08922v1 Announce Type: cross Abstract: Transformer layers generate state-dependent interaction networks: token representations determine the attention matrix, which in turn updates the representations.
The paper investigates the training dynamics of attention mechanisms in high-dimensional settings, focusing on attention-indexed models that encompass multi-layer and multi-head architectures. It shows that while the loss landscape can be described by a finite set of trace order parameters, the online stochastic gradient descent dynamics involve an infinite hierarchy of matrix moments that can be accurately approximated by a finite truncated system. The study further reveals that the choice of attention parameterization acts as an implicit bias: untied attention can get trapped in uninformative states, whereas tied attention induces symmetry breaking and enables weak recovery with θ(d² log d) samples, and untied attention exhibits a fast-slow dynamic leading to weak recovery when symmetry is broken.
arXiv:2609.01231v1 Announce Type: new Abstract: We prove that a multi-head scaled dot product attention can be viewed as a parameter identification strategy. The ratio of unidentified parameters to t...