# Maslov Phase Budget for Symplectic Recurrence

- ID: 3122
- Canonical URL: https://synthcore.org/idea/3122/maslov-phase-budget-for-symplectic-recurrence
- API JSON: https://synthcore.org/api/idea/3122.json
- API Markdown: https://synthcore.org/api/idea/3122.md
- Verification status: unverified
- Source: [arXiv:2609.03061](https://arxiv.org/abs/2609.03061)
- Category: dynamics
- Solves: stability, accuracy, generalization
- ML areas: rnn, ssm, regularization, training-dynamics
- Math tags: symplectic, geometry, dynamical-systems, lie-groups
- Ratings: usefulness 7/10; difficulty 5/10; novelty 7/10

## Idea description

Use the paper's explicit compact factor of a symplectic state-transition matrix to measure aggregate rotation speed in hidden-state dynamics. Penalize excessive or rapidly varying angular velocity rather than penalizing the full recurrent matrix, preserving nontrivial Hamiltonian rotations while suppressing phase drift that can destabilize long sequences.

## Mathematical statement

For a real symplectic matrix M = [[a,b],[c,d]] in Sp(2n,R), define D[M] = a^T a + c^T c, which is symmetric positive definite, and define the unitary representative G[M] = (a+i c)D[M]^{-1/2}. Here a,b,c,d are n-by-n real blocks, i is the imaginary unit, and D[M]^{-1/2} is the positive-definite inverse square root. The paper states that G[M] is unitary and induces the same Lagrangian-subspace image as M. For a differentiable compact path U(t) = K(V(t) Diag(t) V(t)^{-1}), where K embeds a complex unitary matrix into a real symplectic matrix and Diag(t) contains planar rotations R(theta_j(t)), the paper derives Trace(J U'(t) U(t)^{-1}) = 2 sum_j d theta_j(t)/dt. We use q_t = Trace(J (U_t-U_{t-1}) U_{t-1}^{-1})/(2 Delta t) as a discrete aggregate angular velocity and add lambda_phase q_t^2 plus lambda_tv(q_t-q_{t-1})^2 to the sequence loss. J is the standard 2n-by-2n symplectic matrix, Delta t is the step size, and lambda_phase and lambda_tv are nonnegative hyperparameters.

## Key formulas

- $$D[M]=a^{t}a+c^{t}c,\qquad G[M]=(a+i\,c)\,(a^{t}a+c^{t}c)^{-1/2}$$
- $$\operatorname{Trace}\!\left(J\,K(Diag(t))'K(Diag(t))^{-1}\right)=2\sum_{j=1}^{n}\frac{d\theta_j(t)}{dt}$$
- $$q_t=\frac{1}{2\Delta t}\operatorname{Trace}\!\left(J\,(U_t-U_{t-1})U_{t-1}^{-1}\right),\qquad \mathcal L_{\mathrm{phase}}=\lambda_{\mathrm{phase}}q_t^2+\lambda_{\mathrm{tv}}(q_t-q_{t-1})^2$$

## Implementation notes

Integrate this at the recurrent transition, not at the input projection. Build a symplectic hidden-state matrix M_t for each sequence step, for example from a Hamiltonian exponential M_t = exp(JH_t) with H_t symmetric, or from a learned symplectic matrix shared across time. Partition M_t into n-by-n blocks a_t,b_t,c_t,d_t. Compute D_t = a_t^T a_t + c_t^T c_t, obtain D_t^{-1/2} by an eigendecomposition for the first MVP or by Newton-Schulz iterations for larger models, and form the complex matrix G_t = (a_t+i c_t)D_t^{-1/2}. Convert G_t to its real symplectic embedding U_t = K(G_t). For t greater than zero, calculate q_t = Trace(J (U_t-U_{t-1}) U_{t-1}^{-1})/(2 Delta t); use a linear solve instead of explicitly forming U_{t-1}^{-1}. Add lambda_phase times q_t squared to the task loss and optionally lambda_tv times (q_t-q_{t-1}) squared. Do not detach U_t so gradients reach the recurrent parameters. The paper supplies the compact factor and trace identity; the engineer estimates q_t by finite differences and computes matrix functions numerically. Start with a 64-dimensional symplectic RNN on sequential MNIST and a copy-memory task of lengths 100, 500, and 1000. Compare vanilla tanh, orthogonal RNN, unregularized symplectic RNN, and the proposed loss at equal parameter count. Track gradient norm, hidden-state norm, q_t variance, and accuracy versus sequence length. Success means lower long-horizon gradient variance and at least 5 percentage points higher copy accuracy at length 1000 without more than 10 percent training slowdown.

## Disclaimer

AI-generated research hypothesis, automatically tested. Not peer-reviewed.