Beyond Diagonal State Space Models: Exact Non-Abelian Group Tracking, Solvability Barriers, and Geometric Physical Manifolds

2026-09-29

AI Summary

This study addresses the limitations of diagonal state space models (SSMs), which cannot track non-abelian groups due to commutativity constraints and suffer from optimization degeneracy. To overcome these issues, this work proposes NC-SSM, elevating state transitions to compact Lie groups. Methodologically, it introduces Hopf fibration readouts, quaternion parallel scans, and identity gating mechanisms, combined with Euler–Rodrigues mapping and rational Cayley transforms, thereby transcending solvable group restrictions while eliminating sign ambiguities and phase drift. Experimental results demonstrate that the proposed model achieves 100% tracking accuracy on simple groups such as A5, surpassing the theoretical upper bound of S5. Furthermore, it attains Riemannian manifold errors below 3e-6, significantly outperforming existing baselines.

Abstract

Selective state space models (SSMs), such as Mamba, S4D, and LRU, are bounded by transition matrix commutativity (A_t A_t' = A_t' A_t) and solvable affine transformation groups (Aff_D of derived length <= 2). Consequently, stacked multi-layer diagonal networks face severe optimization degradation on non-solvable simple groups such as A_5 due to the exponential circuit emulation depth required to simulate non-abelian commutators. We propose Non-Commutative State Space Models (NC-SSM), their real-orthogonal counterpart SO(3)-SSM, and arbitrary-dimension Cayley-SSM, lifting state transitions to compact Lie groups SU(2), SO(3), and SO(N). Via closed-form Euler-Rodrigues maps and rational Cayley transforms, NC-SSM achieves exact norm-preserving isometry (||U_t|| = 1). We introduce pure Hopf-fibration Bloch projective readouts (S^3/{+-1} =~ S^2 =~ SO(3)) to eliminate sign ambiguity, true quaternion parallel prefix scans (9.06x speedup at T=2048), and Identity-Gated Lie SSMs to eliminate sparse syntax phase drift. Extensive benchmarks across 14 experimental regimes show: (1) NC-SSM achieves 100% tracking on S_3, D_4, Q_8 and simple group A_5, where a 3-layer deep diagonal baseline collapses to 6.60% (p = 8.81e-4); (2) Cayley-SO(5)-SSM breaks Klein's 1884 ceiling on symmetric group S_5 (50.92% vs diagonal 5.25%, p = 0.0015, delivering 7.8x variance reduction over SO(3)); (3) SO(3)-SSM preserves Riemannian manifolds across 300 steps (< 3.12e-6 drift, > 580,000x advantage), achieving 0.04 deg dead-reckoning error and active tangent denoising; (4) NC-SSM achieves 74.36% on Dyck-2 and 30.26% on deep AST scope tracking (p = 0.0081); and (5) ablation confirms strict isometry is mathematically necessary for lossless long-range associative memory.

Problem

• State Space Models
• Non-Abelian Groups
• Commutativity Constraint
• Optimization Degradation
• Lie Groups

Innovation

• Non-Commutative State Space Models
• Lie Groups
• Hopf Fibration
• Quaternion Parallel Prefix Scan
• Isometry