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

TL;DR

This paper introduces a permutation-equivariant 2D state space model for multivariate time series that respects inherent exchangeability, providing a theoretically grounded, scalable, and state-of-the-art architecture for various time series tasks.

Contribution

The work formalizes the canonical form of permutation-equivariant linear 2D state-space systems and proposes the VI 2D SSM architecture that enforces this symmetry, improving scalability and interpretability.

Findings

Achieves state-of-the-art results on forecasting, classification, and anomaly detection.

Reduces variable dependency depth from O(C) to O(1).

Validates the importance of symmetry-preserving modeling in multivariate time series.

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.