Attention as an Adaptive Filter

We introduce Adaptive Filter Attention (AFA), a novel attention mechanism that incorporates a learnable dynamics model directly into the computation of attention weights. Rather than comparing queries and keys directly, we model the input sequence as discrete observations of a linear stochastic differential equation (SDE). By assuming a continuous-time linear time-invariant system with simultaneously-diagonalizable state matrices and noise covariances, we can make use of a closed-form solution of the differential Lyapunov equation to efficiently propagate uncertainties through the dynamics from keys to queries. A generalization of attention naturally arises as the maximum likelihood solution for filtering the trajectory of this linear SDE, with attention weights corresponding to robust residual-based reweightings of the propagated query-key precisions. We further constrain the system dynamics and noise in order to obtain a simplified variant with the same computational and memory complexity as standard attention. In the limit of zero decay and process noise, and using a small-angle approximation, we recover a complex-valued generalization of ordinary dot-product attention with rotary positional encodings.