arXiv：cs.LG（机器学习，全量分类）· Zeyu Jia (School of Biomedical Engineering,Technology, Tianjin Medical University, Medical School, Tianjin University)·· 14 小时前AI 评分36
超越对角状态空间模型：非阿贝尔群精确追踪、可解性壁垒与几何物理流形
Beyond Diagonal State Space Models: Exact Non-Abelian Group Tracking, Solvability Barriers, and Geometric Physical Manifolds
AI 导读
研究提出 NC-SSM，突破 Mamba、S4D、LRU 等选择性状态空间模型受转移矩阵可交换性与可解仿射变换群限制的瓶颈，实现非阿贝尔群精确追踪。NC-SSM 在 Dyck-2 上达 74.36%，深度 AST 作用域追踪达 30.26%（p = 0.0081），并取得 580,000x 优势、0.04 度航位推算误差。消融实验证实严格等距是无损长程联想记忆的数学必要条件。
正文
Authors:Zeyu Jia (School of Biomedical Engineering and Technology, Tianjin Medical University, Medical School, Tianjin University)
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.
| Comments: | 26 pages, 1 table, 5 theorems. Source code and reproducible benchmarks available |
| Subjects: | Machine Learning (cs.LG) |
| Report number: | TMU-TJU-NCSSM-2026 |
| Cite as: | arXiv:2610.00329 [cs.LG] |
| (or arXiv:2610.00329v1 [cs.LG] for this version) | |
| https://doi.org/10.48550/arXiv.2610.00329 arXiv-issued DOI via DataCite (pending registration) |
Submission history
From: Zeyu Jia [view email]
[v1]
Tue, 29 Sep 2026 11:14:24 UTC (37 KB)
来源：arXiv：cs.LG（机器学习，全量分类） · arxiv.org