{"ID":2826042,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2512.18965","arxiv_id":"2512.18965","title":"Lag Operator SSMs: A Geometric Framework for Structured State Space Modeling","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.","short_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 int...","url_abs":"https://arxiv.org/abs/2512.18965","url_pdf":"https://arxiv.org/pdf/2512.18965v1","authors":"[\"Sutashu Tomonaga\",\"Kenji Doya\",\"Noboru Murata\"]","published":"2025-12-22T02:25:26Z","proceeding":"cs.LG","tasks":"[\"cs.LG\"]","methods":"[]","has_code":false}