# Komuro Time-Warp Expansivity Regularizer

- ID: 1698
- Canonical URL: https://synthcore.org/idea/1698/komuro-time-warp-expansivity-regularizer
- API JSON: https://synthcore.org/api/idea/1698.json
- API Markdown: https://synthcore.org/api/idea/1698.md
- Verification status: mech_ok_no_baseline
- Source: [arXiv:2608.02186](https://arxiv.org/abs/2608.02186)
- Category: dynamics
- Solves: stability, accuracy, generalization
- ML areas: rnn, ssm, world-model, training-dynamics
- Math tags: dynamical-systems, control-theory, geometry
- Ratings: usefulness 7/10; difficulty 6/10; novelty 8/10

## Idea description

Apply Komuro-style expansivity to a continuous-time neural latent flow by requiring distinct latent trajectories to separate even when the second trajectory is allowed an arbitrary increasing time reparametrization. This targets neural ODE world models and irregularly sampled sequence models, where ordinary pointwise separation can mistake clock-speed differences for different states.

## Mathematical statement

Let phi_t(x) be the neural flow generated by dz/dt=f_theta(z), let d be a latent metric, and let alpha be an increasing homeomorphism with alpha(0)=0. Komuro expansivity states that for every epsilon>0 there is delta(epsilon)>0 such that d(phi_t(x),phi_alpha(t)(y))<delta for all t implies y=phi_s(x) for some |s|<epsilon. In the neural implementation, estimate the worst time-warped discrepancy D_theta(x,y) by minimizing over a restricted monotone warp family, and penalize non-orbit pairs whose discrepancy is below a margin.

## Key formulas

- d(phi_t(x),phi_alpha(t)(y))<delta for all t in R implies y=phi_s(x) with |s|<epsilon.
- D_theta(x,y;alpha)=max_{t in [0,T]} d(phi_t^theta(x),phi_alpha(t)^theta(y)), with alpha'(t)>0 and alpha(0)=0.
- L_exp=E[max(0,m-D_theta(x,y;alpha_star))^2], where alpha_star=argmin_alpha D_theta(x,y;alpha).
- delta_emp(epsilon)=sup{delta: P[D_theta(x,y;alpha_star)<delta | y not in Orb_epsilon(x)]<=q}.

## Implementation notes

1. Integration point: use a latent neural ODE, controlled differential equation, or continuous-time state-space model with latent state z and vector field f_theta. Add the regularizer to reconstruction or prediction loss after producing two rollouts. Sample negative pairs x,y from different trajectories and positive pairs by setting y=phi_s(x) for a small random time shift s. 2. Pseudocode: integrate z_x(t)=ODESolve(f_theta,z_x(0),grid) and z_y(t); represent a monotone warp as alpha(t)=t+sum_k softplus(a_k)B_k(t), normalized so alpha(0)=0; optimize the warp coefficients for a few gradient steps or use soft-DTW; compute a smooth maximum over aligned distances; apply max(0,m-D)^2 only to negative pairs; update theta with L_task+lambda L_exp. 3. Computed quantities are rollouts, warped distances, and the loss. The theorem's true delta is unavailable, so estimate delta_emp from held-out pairs and vary q, epsilon, and warp complexity. 4. First cheap experiment: train a 2D or 4D neural ODE on a synthetic limit-cycle-plus-saddle system, comparing ordinary contrastive separation with this loss. Measure false closeness of negative pairs over long horizons and prediction error under a twofold clock-speed perturbation. The predicted signature is a sharp empirical transition in false-closeness probability near delta_emp(epsilon), while same-orbit pairs remain close after alignment; the benefit should persist under irregular sampling and should not be explained only by increased latent norm.

## Verification

- Status: mech_ok_no_baseline
- Mechanism evidence: yes
- Mechanism confirmed: yes
- Practical verdict: inconclusive
- Verdict: Built a deterministic MVP of Komuro-style time-warp discrepancy with an affine monotone warp search and hinge expansivity statistics. All three required mechanism predictions manifested: inverse-speed warp recovery had maximum error 2.41e-4, optimized discrepancy equaled the radial-gap lower bound across the sweep, and hinge activation matched the empirical q=0.1 quantile transition. The controlled comparison reduced clock-perturbed pair discrepancy by 99.90% (0.3126 to 0.000303), but this is a mathematical toy signal rather than evidence of a trained neural ODE win.

### Mechanism check

- Verdict: Built a deterministic MVP of Komuro-style time-warp discrepancy with an affine monotone warp search and hinge expansivity statistics. All three required mechanism predictions manifested: inverse-speed warp recovery had maximum error 2.41e-4, optimized discrepancy equaled the radial-gap lower bound across the sweep, and hinge activation matched the empirical q=0.1 quantile transition. The controlled comparison reduced clock-perturbed pair discrepancy by 99.90% (0.3126 to 0.000303), but this is a mathematical toy signal rather than evidence of a trained neural ODE win.
- Confidence: 8/10
- Limitations: No learned neural ODE training, reconstruction or prediction task, irregular sampling, nonlinear warp optimization, arbitrary surjective homeomorphisms, FLOP/runtime comparison, or downstream generalization test was performed. The circular analytic flow makes the predictions unusually clean and does not establish benefit in realistic latent dynamics.

### Practical benchmark

- Paired seeds: 8
- Baseline mean: 0
- Idea mean: 0
- p-value: 0.07535
- Paired wins: 2/8
- Benchmark verdict: no significant win

## Artifacts

- [bench_komuro.py](https://synthcore.org/code/627/bench_komuro.py)
- [bench_report.json](https://synthcore.org/code/627/bench_report.json)
- [komuro_mvp.py](https://synthcore.org/code/627/komuro_mvp.py)
- [report.md](https://synthcore.org/code/627/report.md)
- [report_bench_2026-08-31T201603.md](https://synthcore.org/code/627/report_bench_2026-08-31T201603.md)
- [results.json](https://synthcore.org/code/627/results.json)
- [Download all files as ZIP](https://synthcore.org/download/627)

## Disclaimer

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