Recurrent Self-Attention Dynamics: An Energy-Agnostic Perspective from Jacobians

Standard self-attention compares tokens by similarity but does not account for how much of a past state survives to the present. We introduce Dynamical Kernel Attention (DKA), which derives attention from robust estimation of a latent state evolving under stochastic dynamics. As a past state decays toward equilibrium under the dynamics, the information it carries about the present vanishes, bounding the total evidence a head can draw. Unlike recency biases, DKA can abstain when relevant evidence is weak and guarantees a bounded error when old tokens are discarded. DKA preserves the parallel structure and computational cost of standard attention. We present the model and its theory; empirical evaluation is deferred to a subsequent version.

Transformer self-attention can be interpreted as a gradient flow on the unit sphere, in which tokens evolve under softmax interaction potentials and tend to form clusters. While prior work has established clustering behavior for single-head attention, the multi-head setting remains less understood due to geometric interference between heads, which invalidates standard monotonicity arguments. In this work, we develop a theoretical framework for multi-head self-attention dynamics and resolve several open questions. We show that, under suitable conditions on the score matrices, a natural multi-head energy functional is non-decreasing along both flat and spherical dynamics. We identify the key obstruction to per-head monotonicity as radial shadow terms, which are projections of each head's output onto token directions, persisting even under orthogonality assumptions. We introduce a sufficient condition ensuring monotonicity and establish robustness to approximate orthogonality. In a simplified scalar-head regime with equiangular token configurations, we derive a closed-form expression for the critical inverse temperature governing clustering behavior, and show that heterogeneous heads exhibit super-additive clustering rates. In this regime, we also prove a separation in clustering time between ReLU and softmax attention in the linearized dynamics. Finally, we establish an entropy production identity and show that attention entropy increases monotonically toward equilibrium as clustering progresses. Our results provide a unified perspective on the dynamics of multi-head attention and clarify the mechanisms underlying clustering and stability in transformer models.

Softmax attention maps every query--key interaction into a probability distribution, but the underlying structure remains largely unexplored. We define the energy field, the row-centered attention logit, and show that it exhibits invariant properties across models, architectures, and inputs. Two classes of invariants emerge. Mechanism-level invariants follow from the algebraic structure of softmax attention. They include a per-row zero-sum constraint, a rank bound determined by the head dimension, and spectral signatures that follow from them. Model-level regularities are not required by the mechanism, yet hold in every autoregressive language model we test, spanning several architecture families. The energy field distributes its variance over key positions without concentrating at a few. This delocalization traces to a property of the key matrix we call key incoherence. These invariants have practical consequences. The rank bound confines the energy field to a low-dimensional subspace. Key incoherence yields a per-head training monitor. All results are verified at multiple context lengths and input texts.