发表机构
TU Wien; MIT CSAIL; Liquid AI（维也纳工业大学; 麻省理工学院计算机科学与人工智能实验室; Liquid AI）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出 RiccatiSSM，一种状态更新遵循 Riccati 微分方程的非线性状态空间模型，通过 Möbius 变换的复合封闭性实现单次并行扫描精确评估，在长序列任务中性能与 LrcSSM 相当且运行时间降低 22-33%。
AI 中文摘要
状态空间模型（SSM）之所以能实现高效的序列处理，是因为其仿射状态更新在复合运算下封闭，因此可以通过结合并行扫描（associative parallel scan）进行评估。非线性循环模型可以提供更丰富的、依赖于状态的动态特性，但通常会失去这种复合结构：此时并行评估需要采用迭代方法，反复对循环进行线性化和扫描。我们提出的问题是：可以设计怎样的依赖于状态的非线性动态特性，使其保持精确的可复合性？我们通过引入 RiccatiSSM 来回答这一问题，它是一种非线性 SSM，其中每个状态维度遵循一个输入条件化的 Riccati 微分方程。其二次状态依赖性使得局部雅可比矩阵显式地依赖于状态，而其在分段常数输入下的精确每步流（per-step flow）则是一个 Möbius 变换。由于 Möbius 映射在复合运算下封闭，并通过 $2\ imes 2$ 矩阵乘法进行复合，因此完整的非线性状态轨迹可以通过一次结合并行扫描精确评估，无需迭代线性化。我们进一步推导了一种约束参数化，以确保有界、收缩的动态特性，并避免分式线性状态更新中出现极点。在长序列分类、回归和预测任务中，RiccatiSSM 在匹配架构下与非线性 LrcSSM 相比，实现了具有竞争力的预测性能，同时将运行时间减少了 $22{-}33\%$。这些结果表明，依赖于状态的非线性动态特性可以保持精确的可复合性，并能在单次并行扫描中高效评估。
英文摘要
State space models (SSMs) achieve efficient sequence processing because their affine state updates are closed under composition and can therefore be evaluated with an associative parallel scan. Nonlinear recurrent models can provide richer, state-dependent dynamics, but generally lose this compositional structure: parallel evaluation then requires iterative methods that repeatedly linearize and scan the recurrence. We ask, what state-dependent nonlinear dynamics can be designed to remain exactly composable? We answer by introducing RiccatiSSM, a nonlinear SSM, in which each state dimension follows an input-conditioned Riccati differential equation. Its quadratic state dependence makes the local Jacobian explicitly state-dependent, while its exact per-step flow under piecewise-constant inputs is a Möbius transformation. Since Möbius maps are closed under composition and compose through $2\times 2$ matrix multiplication, the complete nonlinear state trajectory can be evaluated exactly with a single associative parallel scan, without iterative linearization. We further derive a constrained parameterization that ensures bounded, contractive dynamics, and avoids poles in the fractional-linear state update. Across long-sequence classification, regression, and forecasting tasks, RiccatiSSM achieves competitive predictive performance while reducing runtime by $22{-}33\%$ compared to the nonlinear LrcSSM under matched architectures. These results demonstrate that state-dependent nonlinear dynamics can retain exact composability and be evaluated efficiently within a single parallel scan.
