Eigenvalues as a Metric for Memory Dynamics in Sequence Models

Abstract

While softmax attention drives state-of-the-art performance in sequence modeling, its quadratic complexity motivates linear alternatives such as state space models (SSMs). Structural differences between the two model classes, however, hinder direct comparisons of their memory dynamics, creating the need for a common metric to analyze, interpret, and improve their information processing capabilities. Inspired by recent advances in SSM performance driven by eigenvalue-guided insights, we leverage the dynamical systems framework to bring attention models into a unified analytical framework with SSMs. This allows us to perform a structured analysis, which investigates the applicability of an eigenvalue-spectrum memory dynamics metric to attention models. To this end, we first conduct an extensive empirical study across diverse attention-based models and SSMs on a range of benchmarks. We show that, for both model classes, eigenvalues influence key aspects of memory and long-range dependency modeling, revealing spectral signatures that align with task requirements. Building on these findings, we show how spectral signatures can motivate architectural modifications, how they can be guided through the training process, and how they can provide information about feature importance. The results thereby enable and emphasize the role of eigenvalue analysis as a principled metric for interpreting, explaining, and ultimately improving the capabilities of sequence models.