Lag Operator SSMs: A Geometric Framework for Structured State Space Modeling

• URL: http://arxiv.org/abs/2512.18965v1
• Date: Mon, 22 Dec 2025 02:25:26 GMT
• Title: Lag Operator SSMs: A Geometric Framework for Structured State Space Modeling
• Authors: Sutashu Tomonaga, Kenji Doya, Noboru Murata,
• Abstract summary: We introduce a framework for constructing discrete-time Structured State Space Models (SSMs) that is both flexible and modular.<n>Our approach is based on a novel lag operator, which geometrically derives the discrete-time recurrence by measuring how the system's basis functions "slide"
• Score: 3.3864018929063477
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Structured State Space Models (SSMs), which are at the heart of the recently popular Mamba architecture, are powerful tools for sequence modeling. However, their theoretical foundation relies on a complex, multi-stage process of continuous-time modeling and subsequent discretization, which can obscure intuition. We introduce a direct, first-principles framework for constructing discrete-time SSMs that is both flexible and modular. Our approach is based on a novel lag operator, which geometrically derives the discrete-time recurrence by measuring how the system's basis functions "slide" and change from one timestep to the next. The resulting state matrices are computed via a single inner product involving this operator, offering a modular design space for creating novel SSMs by flexibly combining different basis functions and time-warping schemes. To validate our approach, we demonstrate that a specific instance exactly recovers the recurrence of the influential HiPPO model. Numerical simulations confirm our derivation, providing new theoretical tools for designing flexible and robust sequence models.

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