提出一种几何化框架，直接构建离散状态空间模型。
Lag Operator SSMs: A Geometric Framework for Structured State Space Modeling
- 用新定义的滞后算子，从几何角度推导离散递推关系。
- 单个内积即可计算状态矩阵，支持灵活组合基函数与时间变换。
- 可精确复现HiPPO模型，为序列建模提供新理论工具。
结构化状态空间模型（SSMs）是近期流行的Mamba架构的核心，虽在序列建模中表现强大，但其理论基础依赖于复杂的连续时间建模与后续离散化过程，易造成理解困难。本文提出一种基于第一性原理的直接离散时间SSM构建框架，兼具灵活性与模块性。该方法引入一种新型滞后算子，通过几何方式刻画系统基函数在相邻时间步间的域扩展行为，从而推导出离散时间递推关系。所得状态矩阵仅需一次内积运算即可确定，允许通过灵活组合不同基函数与时间扭曲方案来设计新型SSM。我们验证了某一具体实例可精确恢复经典HiPPO模型的递推形式。数值模拟结果证实了推导正确性，为设计灵活且鲁棒的序列模型提供了新的理论工具。
原文摘要 · Abstract (English)
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, multistage 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 undergo what we call a domain expansion from one timestep to the next. The resulting state matrices are computed via a single inner product involving this operator, enabling a modular design space for creating novel SSMs by flexibly combining different basis functions and time-warping schemes. To validate our framework, 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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