Based maps to Lagrangian Grassmannians, Quivers, and Bott Periodicity
arXiv:2607.10956
2026
Dynamics
2 ideas extracted · analyzed Aug 30, 2026
What the math gives to ML
The paper turns based maps into explicit finite-dimensional quiver data: an internal state space A, an endomorphism alpha, and input/output maps gamma and j, with a precise controllability condition for every eigenvalue of alpha. This is directly transferable to recurrent and state-space neural layers because it provides an algebraic test for whether every latent mode is reachable from the input and visible at the output. The most actionable adaptation is a controllability-aware regularizer for SSMs. A second adaptation uses the paper’s orthogonal/symplectic bilinear-form constraints to parameterize structured recurrent transitions with exact algebraic symmetries.
Ideas from this paper
✗ Failed on benchmark
2026
Replace an unconstrained latent transition in an SSM or recurrent block by quiver data (alpha,gamma), where alpha evolves the latent state and gamma injects token or feature inputs. Add a differentiable penalty that detects eigenmodes of alpha not reached from gamma, preventing dead latent directions and improving long-context signal propagation. The paper’s exact open condition becomes a practical regularizer rather than a hard architectural constraint.
Useful7/10
Difficulty5/10
Novelty6/10
Unverified
2026
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.
Useful5/10
Difficulty5/10
Novelty5/10