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

2026-03-06

AI Summary

This work addresses the structural bias introduced by arbitrary variable ordering in multivariate time series modeling, which violates the inherent exchangeability of variables in real-world systems. To resolve this, the authors propose VI 2D SSM, a two-dimensional state space model that respects permutation equivariance by decomposing dynamics into local self-evolution and global aggregated interactions, thereby eliminating dependence on variable order. They provide the first theoretical characterization of the canonical form for permutation-equivariant linear coupling, proving that ordered recursive architectures are structurally suboptimal and reducing inter-variable dependency depth from O(C) to O(1). By further integrating multi-scale temporal dynamics and spectral representations, they develop a unified architecture, VI 2D Mamba, which achieves state-of-the-art performance across forecasting, classification, and anomaly detection tasks, demonstrating the advantages of symmetry-preserving modeling in both structural scalability and empirical effectiveness.

Technology Category

• Machine Learning: Structured Learning
• Computer Vision: Multi-modal Vision
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

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.

Problem

• multivariate time series
• permutation symmetry
• variable ordering
• exchangeability
• state-space models

Innovation

• Permutation Equivariance
• 2D State Space Models
• Variable-Invariant Architecture
• Multivariate Time Series
• Symmetry-Preserving Modeling