Beyond Diagonal State Space Models: Exact Non-Abelian Group Tracking, Solvability Barriers, and Geometric Physical Manifolds
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 Cayl...
| Authors | Zeyu Jia |
|---|---|
| Primary category | cs.LG |
| Published | 29 September 2026 |
| Last updated at source | 29 September 2026 |
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