发表机构
Tianjin Medical University; Tianjin University（天津医科大学; 天津大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出非交换状态空间模型（NC-SSM）系列，通过李群提升和等距变换，解决了对角SSM在非可解群上的优化退化问题，实现了精确群跟踪、流形保持和长程记忆，并在多项基准上显著超越现有方法。
AI 中文摘要
选择性状态空间模型（SSM），如Mamba、S4D和LRU，受限于转移矩阵的交换性（A_t A_t' = A_t' A_t）以及可解的仿射变换群（派生长度不超过2的Aff_D）。因此，堆叠的多层对角网络在不可解的单群（如A_5）上会遭受严重的优化退化，这是因为模拟非阿贝尔交换子所需的指数级电路仿真深度。我们提出了非交换状态空间模型（NC-SSM）、其实正交对应模型SO(3)-SSM以及任意维度的Cayley-SSM，将状态转移提升到紧致李群SU(2)、SO(3)和SO(N)。通过闭式Euler-Rodrigues映射和有理Cayley变换，NC-SSM实现了精确的保范等距（||U_t|| = 1）。我们引入了纯Hopf纤维化Bloch射影读出（S^3/{±1} ≅ S^2 ≅ SO(3)）以消除符号模糊性，真正的四元数并行前缀扫描（在T=2048时加速9.06倍），以及恒等门控Lie SSM以消除稀疏语法相位漂移。在14个实验场景中的广泛基准测试显示：（1）NC-SSM在S_3、D_4、Q_8和单群A_5上实现了100%的跟踪，而3层深对角基线崩溃至6.60%（p = 8.81e-4）；（2）Cayley-SO(5)-SSM打破了Klein在1884年对对称群S_5的上限（50.92%对比对角的5.25%，p = 0.0015，相对于SO(3)实现了7.8倍的方差缩减）；（3）SO(3)-SSM在300步内保持黎曼流形（漂移小于3.12e-6，优势超过580,000倍），实现0.04度的航位推算误差和主动切空间去噪；（4）NC-SSM在Dyck-2上达到74.36%，在深度AST作用域跟踪上达到30.26%（p = 0.0081）；（5）消融实验证实严格等距对于无损长程关联记忆在数学上是必要的。
英文摘要
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.
Comments26 pages, 1 table, 5 theorems. Source code and reproducible benchmarks available
