The Expressive Limits of Diagonal SSMs for State-Tracking

Abstract:State-Space Models (SSMs) have recently been shown to achieve strong empirical performance on a variety of long-range sequence modeling tasks while remaining efficient and highly-parallelizable. However, the theoretical understanding of their expressive power remains limited. In this work, we study the expressivity of input-Dependent Complex-valued Diagonal (DCD) SSMs on sequential state-tracking tasks. We show that single-layer DCD SSMs cannot express state-tracking of any non-Abelian group at finite precision. More generally, we show that $k$-layer DCD SSMs can express state-tracking of a group if and only if that group has a subnormal series of length $k$, with Abelian factors. That is, we identify the precise expressivity range of $k$-layer DCD SSMs within the solvable groups. Empirically, we find that multi-layer models often fail to learn state-tracking for non-Abelian groups, highlighting a gap between expressivity and learnability.

18 pages, 5 figures, 4 tables. Accepted at ICLR 2026

Machine Learning (cs.LG)

arXiv:2603.01959 [cs.LG]

(or arXiv:2603.01959v2 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2603.01959 Focus to learn more arXiv-issued DOI via DataCite

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