Centralizer-Constrained Hyperbolic Dynamics

Integrate this into a hyperbolic embedding model or recurrent/neural-ODE block operating on 2D tangent coordinates. (1) Store known symmetry generators \(g_1,\ldots,g_r\in\operatorname{PSL}(2,\mathbb R)\) as determinant-one matrices, and let the trainable infinitesimal update be \(A=\begin{bmatrix}a&b\\c&-a\end{bmatrix}\), which automatically lies in \(\mathfrak{sl}(2,\mathbb R)\). Use \(h_{t+1}=\exp(\Delta tA)h_t\) in tangent coordinates, or apply \(\exp(\Delta tA)\) as a Möbius transformation to an upper-half-plane embedding. (2) Start with the soft version by adding \(\rho\sum_i\|Ag_i-g_iA\|_F^2\) to the task loss. Then implement the exact version: vectorize \((a,b,c)\), construct the linear equations \(Ag_i-g_iA=0\), compute their SVD nullspace once, and parameterize \(A=N\theta\), where columns of \(N\) span the nullspace. (3) Track the spectral radius of \(\exp(\Delta tA)\), hidden-state norm growth, and equivariance error \(\|F(gx)-gF(x)\|\). If generators are unknown, estimate them from paired augmented examples and compare learned commutator residuals with the soft penalty. (4) Test on a toy classification or sequence task generated from upper-half-plane points, using random \(\operatorname{PSL}(2,\mathbb R)\) transformations as nuisance symmetries. Compare an unconstrained MLP or ODE, ordinary augmentation, soft centralizer regularization, and exact nullspace parameterization. Success is lower symmetry-held-out error and bounded hidden-state norms at equal parameter count, with reduced equivariance error and no loss of in-distribution accuracy.