Bilinear-Form Structured Transition
Implementation & benchmark of arXiv:2607.10956 — Based maps to Lagrangian Grassmannians, Quivers, and Bott Periodicity
Source paper: Based maps to Lagrangian Grassmannians, Quivers, and Bott Periodicity arXiv:2607.10956 ⓘ · analyzed Aug 30, 2026
AI-generated research hypothesis, automatically tested. Not peer-reviewed.
Idea description
Construct the latent transition from a nondegenerate bilinear form phi and a form-compatible operator instead of from an unconstrained dense matrix. The resulting SSM has an exact orthogonal or symplectic algebraic structure, reducing transition parameter redundancy and testing whether preservation of a latent pairing improves extrapolation on reversible, parity-sensitive, or Hamiltonian-like sequence tasks.
Formulas
Mathematical statement
The paper uses an isomorphism phi:A to A^vee defining a nondegenerate orthogonal or symplectic pairing <v,w>, and imposes alpha^vee phi=phi alpha. In coordinates this is <v,alpha w>=<alpha v,w>, so alpha is self-adjoint with respect to the pairing. The extracted parameter-space identification is V_+=Sym^2 A^vee direct-sum Hom(A,U) in the symmetric case and V_-=Lambda^2 A^vee direct-sum Hom(A,U) in the skew case. Here A is the latent state space, U is the feature or output space, phi is invertible, alpha^vee is the transpose/dual map, and Sym^2 or Lambda^2 selects symmetric or skew-symmetric matrices. The transferable construction is exact parameter tying: choose a free symmetric matrix S or skew-symmetric matrix K and set alpha=phi^{-1}S or alpha=phi^{-1}K. This guarantees alpha^T phi=phi alpha by construction. For a symplectic pairing use even d and the canonical matrix J=[[0,I],[-I,0]].
Implementation notes
(1) Integration point: replace the transition matrix in an SSM or recurrent block. Let h_t be d-dimensional, with even d for the skew case. Fix phi=I for the symmetric pairing, or phi=J=[[0,I],[-I,0]] for a canonical symplectic pairing. Learn an unconstrained matrix R and derive S=(R+R^T)/2 or K=(R-R^T)/2, then set alpha=solve(phi,S) or alpha=solve(phi,K). Use h_{t+1}=rho(alpha h_t+B u_t), with ordinary learned input B and readout C. Initially remove the recurrent nonlinearity or put it only in a residual branch so the algebraic effect can be isolated. (2) Pseudocode: R is a trainable d-by-d matrix; if symmetric, S=0.5*(R+R.T), alpha=linear_solve(phi,S); if skew, K=0.5*(R-R.T), alpha=linear_solve(phi,K); optionally rescale alpha by c/max(1,rho(alpha)) to control the discrete-time spectral radius; then run the recurrence. (3) Computed exactly from the paper’s math: symmetry or skew-symmetry of the free parameter and the identity alpha^T phi=phi alpha. Estimated empirically: spectral radius, long-horizon gradient norm, pairing error ||alpha^T phi-phi alpha||_F, and task error. Compare fixed phi against a learnable invertible phi parameterized by a matrix exponential. (4) First experiment: compare dense, orthogonal, symmetric-pairing, and symplectic-pairing SSMs with width 64 on Copy, reverse-copy, parity, and synthetic Hamiltonian trajectory prediction. Match parameter counts and training FLOPs. Measure validation accuracy, long-horizon prediction error, gradient norm, spectral radius, pairing error, and wall-clock throughput. Success is lower extrapolation error or better reversal/parity accuracy at equal parameters, with pairing error near machine precision and no substantial per-step slowdown.
Verification
This idea has not been verified yet.
Verification happens in two stages: Stage 1 — a mechanism check on a toy system confirms the claimed mathematical phenomenon reproduces; Stage 2 — a benchmark implements the idea on a real (small) neural network task and compares it against a tuned baseline over 8 paired seeds with a permutation test.
Artifacts
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