Permutation-Equivariant 2D State Space Models: Theory and Canonical Architecture for Multivariate Time Series

TL;DR

It is proved 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, validating the theoretical necessity of symmetry-preserving 2D modeling.

Abstract

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.

Figures (4)

• Figure 1: Left: 1D SSM models only along the temporal axis, overlooking dependencies across variables. Middle: Conventional 2D SSM captures inter-variable correlations through sequential scans, but still imposes artificial ordering and struggles with distant relationships. Right: Our method performs global aggregation over variables, enabling simultaneous and order-free modeling of inter-variable relationships over time.
• Figure 2: Overview of the proposed Variable-Invariant 2D Mamba. Left: Variable-Invariant 2D SSM with global coupling. A permutation-invariant set aggregator enforces invariance to variable ordering, after which the 2D state space model captures jointly coupled dynamics along temporal and variable dimensions. The globally coupled state transitions enable structured long-range dependency modeling while preserving symmetry across variables. Right: Variable-Invariant 2D Mamba architecture. The framework decomposes representation learning into complementary long-term, short-term, and spectral components. These components are fused to form a unified representation that simultaneously captures global temporal dynamics, local variations, and frequency-domain structure under a variable-invariant formulation.
• Figure 3: Results of classification (accuracy) and anomaly detection (F1 score).
• Figure 4: Efficiency analysis with respect to the number of variables. The left $y$-axis reports the training time in sec/epoch, while the right $y$-axis reports throughput in transactions per second.

Theorems & Definitions (11)

• Definition 1: Variable-Axis Exchangeability
• Theorem 1: Characterization of Permutation-Commuting Matrices
• Theorem 2: Reduction of Variable-Axis Dependency Depth
• Theorem 3: Discrete Stability under ZOH
• Proposition 1: Permutation invariance of the pooled summary
• Proposition 2
• proof
• proof
• proof
• proof