# Lie-Group Lyapunov Recall Dynamics

- ID: 1978
- Canonical URL: https://synthcore.org/idea/1978/lie-group-lyapunov-recall-dynamics
- API JSON: https://synthcore.org/api/idea/1978.json
- API Markdown: https://synthcore.org/api/idea/1978.md
- Verification status: mech_ok_no_baseline
- Source: [arXiv:2608.08226](https://arxiv.org/abs/2608.08226)
- Category: dynamics
- Solves: stability, memory, accuracy
- ML areas: rnn, optimizer, training-dynamics, memory
- Math tags: dynamical-systems, control-theory, geometry, linear-algebra
- Ratings: usefulness 7/10; difficulty 6/10; novelty 7/10

## Idea description

Use dissipative dynamics directly on the SU(d) manifold instead of unconstrained Euclidean recurrent updates. A Riemannian gradient or damped Landau-Lifshitz-Gilbert-like flow preserves the unitary constraint and supplies an explicit Lyapunov certificate: the associative-memory energy should decrease monotonically until the state reaches a recalled attractor.

## Mathematical statement

For U_i in SU(d), define the local field H_i = sum_j J_ij(U_j). The tangent-space projection uses skew_0(A) = 0.5(A - A dagger) - Tr(A - A dagger) I/(2d), which is traceless and skew-Hermitian. Let Omega_i be the projected tangent force derived from the negative energy gradient. The continuous update dU_i/dt = U_i Omega_i remains on SU(d), because U_i dagger dU_i/dt belongs to the Lie algebra su(d). For gradient descent with learning rate eta, the energy obeys dE/dt = -eta sum_i ||grad_i E||_F^2, which is nonpositive. A discrete exponential-map update preserves the group constraint much better than an unconstrained Euler step.

## Key formulas

- H_i = sum_j J_ij(U_j),   skew_0(A) = 0.5(A - A dagger) - Tr(A - A dagger) I/(2d)
- dU_i/dt = -eta grad_i E = U_i Omega_i,   Omega_i in su(d)
- dE/dt = -eta sum_i ||grad_i E||_F^2 <= 0
- U_i(k+1) = U_i(k) exp(-eta Omega_i(k))

## Implementation notes

Integration point: use this as the recurrent inference solver for the SU(d) memory layer, or as a constrained optimizer for a trainable associative-memory energy. Do not update real and imaginary matrix entries independently; update each state with a matrix exponential so unitarity is preserved. Pseudocode: compute H_i = N^(-1) sum over j and mu of M_i^mu (M_j^mu) dagger U_j; calculate the Euclidean energy gradient; multiply by U_i dagger; project the result with skew_0; set U_i to U_i exp(-eta Omega_i); then evaluate energy, overlap, and the constraint error ||U_i dagger U_i - I||_F. If energy increases, halve eta and retry. The Lie-group flow and Lyapunov identity come from the paper; estimate the local gradient Lipschitz constant L from successive gradients and use eta below 2/L as a conservative discrete-time ceiling. First cheap experiment: run N = 128, d = 3, and P = 0.5N on random memories and corrupted RGB-image patches. Compare exponential-map, naive matrix Euler, and Euclidean projected-gradient updates. The predicted signature is monotonic energy decay and machine-precision unitarity for the exponential-map method; fixed-step instability should emerge near eta = 2/L, while backtracking should remove that instability and converge to a high-overlap attractor.

## Verification

- Status: mech_ok_no_baseline
- Mechanism evidence: yes
- Mechanism confirmed: yes
- Practical verdict: inconclusive
- Verdict: Built an SU(3) associative-memory solver using projected su(d) forces and exponential-map updates, with Euler and QR-projected baselines. The three Stage-1 mechanism predictions were confirmed: the Lyapunov finite-difference ratio approached 1.00000002, coupling scaling followed the predicted c^4 law within 7e-8 relative error, and exponential-map unitarity stayed at 1.7e-15–2.9e-15 while Euler error grew to 1.50. The proposed local 2/L discrete ceiling was not quantitatively reliable: monotonicity failed around 0.35*(2/L), but backtracking restored monotonic descent.

### Mechanism check

- Verdict: Built an SU(3) associative-memory solver using projected su(d) forces and exponential-map updates, with Euler and QR-projected baselines. The three Stage-1 mechanism predictions were confirmed: the Lyapunov finite-difference ratio approached 1.00000002, coupling scaling followed the predicted c^4 law within 7e-8 relative error, and exponential-map unitarity stayed at 1.7e-15–2.9e-15 while Euler error grew to 1.50. The proposed local 2/L discrete ceiling was not quantitatively reliable: monotonicity failed around 0.35*(2/L), but backtracking restored monotonic descent.
- Confidence: 9/10
- Limitations: This is a small N=32,d=3,P=16 synthetic random-memory test, not the requested N=128 RGB-patch experiment. No learned memory, CUDA benchmark, multi-seed statistics, or rigorous global Lipschitz estimate was tested; the local 2/L estimate did not predict the observed transition.

### Practical benchmark

- Paired seeds: 8
- Baseline mean: 0
- Idea mean: 0
- p-value: 0.61035
- Paired wins: 3/8
- Benchmark verdict: no significant win

## Artifacts

- [bench_report.json](https://synthcore.org/code/753/bench_report.json)
- [lie_group_recall.py](https://synthcore.org/code/753/lie_group_recall.py)
- [report.md](https://synthcore.org/code/753/report.md)
- [report_bench_2026-09-01T024741.md](https://synthcore.org/code/753/report_bench_2026-09-01T024741.md)
- [results.json](https://synthcore.org/code/753/results.json)
- [stage1_results.json](https://synthcore.org/code/753/stage1_results.json)
- [stage1_verify.py](https://synthcore.org/code/753/stage1_verify.py)
- [stage2_bench.py](https://synthcore.org/code/753/stage2_bench.py)
- [Download all files as ZIP](https://synthcore.org/download/753)

## Disclaimer

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