提出可处理变量无序性的二维状态空间模型，提升多变量时间序列建模的稳定性与效率。
Permutation-Equivariant 2D State Space Models: Theory and Canonical Architecture for Multivariate Time Series
- 基于变量置换等变性设计新架构，用全局聚合替代传统顺序依赖
- 将变量间依赖深度从O(C)降至O(1)，稳定分析简化为两个标量模式
- 适用于需要高可扩展性的多变量时序预测、分类与异常检测任务
多变量时间序列（MTS）建模常隐式施加变量顺序，违背许多真实系统中变量无序的交换对称性。本文将此问题形式化为置换对称性违反，并要求状态空间动态在变量轴上保持置换等变。理论上，我们刻画了满足该对称性约束下的线性变量耦合完整规范形式：任意置换等变的线性2D状态空间系统自然分解为局部自动力学和全局池化交互，表明有序递归不仅不必要，且结构上次优。基于此理论基础，我们提出变量不变二维状态空间模型（VI 2D SSM），通过置换不变聚合实现规范等变形式。该设计消除了变量轴上的序列依赖链，使依赖深度由$/mathcal{O}(C)$降至$/mathcal{O}(1)$，并将稳定性分析简化为两个标量模式。进一步提出VI 2D Mamba，融合多尺度时间动态与谱表示的统一架构。大量实验在预测、分类与异常检测基准上验证，本模型达到先进性能并具备优异结构可扩展性，证实对称性保持的2D建模具有理论必要性。
原文摘要 · Abstract (English)
Multivariate time series (MTS) modeling often implicitly imposes an artificial ordering over variables, violating the inherent exchangeability found in many real-world systems where no canonical variable axis exists. We formalize this limitation as a violation of the permutation symmetry principle and require state-space dynamics to be permutation-equivariant along the variable axis. In this work, we theoretically characterize the complete canonical form of linear variable coupling under this symmetry constraint. We prove that any permutation-equivariant linear 2D state-space system naturally decomposes into local self-dynamics and a global pooled interaction, rendering ordered recurrence not only unnecessary but structurally suboptimal. Motivated by this theoretical foundation, we introduce the Variable-Invariant Two-Dimensional State Space Model (VI 2D SSM), which realizes the canonical equivariant form via permutation-invariant aggregation. This formulation eliminates sequential dependency chains along the variable axis, reducing the dependency depth from $\mathcal{O}(C)$ to $\mathcal{O}(1)$ and simplifying stability analysis to two scalar modes. Furthermore, we propose VI 2D Mamba, a unified architecture integrating multi-scale temporal dynamics and spectral representations. Extensive experiments on forecasting, classification, and anomaly detection benchmarks demonstrate that our model achieves state-of-the-art performance with superior structural scalability, validating the theoretical necessity of symmetry-preserving 2D modeling.
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