✗ Failed on benchmark 2026

Poisson-Kernel Random Attractor Regularizer

Implementation & benchmark of arXiv:2607.15488 — The transition between synchronization and chaos for random Blaschke products

Usefulness7/10

Difficulty5/10

Novelty8/10

Regularization Stability Generalization Accuracy Rnn Ssm Regularization World model Dynamical systems Probability Complex analysis Statistics Measure theory

Source paper: The transition between synchronization and chaos for random Blaschke products arXiv:2607.15488 ⓘ · analyzed Aug 30, 2026

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 Not peer-reviewed.

Idea description

Use the paper's random fixed-point attractor and associated Poisson-kernel invariant density as an explicit distributional target for an ensemble of recurrent latent states. Instead of forcing hidden states toward zero, estimate the attractor induced by the recent random map sequence and regularize the ensemble toward its analytically specified angular density.

Formulas

$$x_\omega\in\operatorname{Acc}\left\{T_{\sigma^{-n}\omega}^{(n)}(z):n\geq 1\right\},\qquad z\in D,$$

$$P_x(e^{i\phi})=\frac{1-|x|^2}{|e^{i\phi}-x|^2},\qquad \int_0^{2\pi}P_x(e^{i\phi})\frac{d\phi}{2\pi}=1,$$

$$\mathcal{L}_{\mathrm{PK}}=\frac{1}{M}\sum_{m=1}^{M}-\log P_{\widehat{x}_t}\left(\frac{z_t^{(m)}}{|z_t^{(m)}|}\right),$$

$$\widehat{x}_t=\frac{1}{K}\sum_{k=1}^{K}T_{j_{t-K}}\circ\cdots\circ T_{j_{t-1}}(\xi_k),\qquad |\xi_k|<1.$$

Mathematical statement

For a backward random composition T_{sigma^{-n}omega}^{(n)} applied to any z in the disk, the paper identifies an accumulation point x_omega in the disk in the attracting regime. The corresponding random invariant measure has angular density P_x(exp(i phi))=(1-|x|^2)/|exp(i phi)-x|^2 with respect to normalized Lebesgue measure. In a neural ensemble, x_hat_t is estimated by applying the recent composition of random maps to several probe points and averaging the results. The empirical phases of the ensemble states are then penalized by negative log likelihood under P_xhat. The mechanism predicts concentration around x_hat in the synchronized regime and a broader stationary Poisson-shaped distribution in the expanding regime.

Implementation notes

1. Integration point: maintain M parallel latent states for each sequence, all receiving the same input but having independent small initial perturbations or map draws. Their complex phases form an empirical angular distribution. Keep a queue of the last K randomly selected Blaschke maps. Estimate x_hat_t by applying that queued composition to K probe points xi_k in the disk and averaging the outputs, followed by exponential smoothing. 2. Pseudocode: compute ensemble states z_t^(m); calculate phi_m=atan2(Im(z_m),Re(z_m)); evaluate p_m=(1-|x_hat|^2)/(1+|x_hat|^2-2 Re(conjugate(x_hat) exp(i phi_m))+epsilon); add gamma times the mean of -log(p_m) to the task loss; update network parameters. Stop gradients through x_hat in the first implementation. 3. Taken from the paper: the random attractor construction and Poisson-kernel density. Estimated empirically: x_hat from finite backward depth K, the phase histogram, and stationarity. Disable the loss when |x_hat| is too close to one, where the density becomes numerically singular. 4. First experiment: use a small complex RNN on sequential MNIST or noisy sine-wave forecasting with M=8 ensemble states and K in {4,8,16}. Compare no regularizer, isotropic phase entropy regularization, and the Poisson-kernel loss at equal coefficient strength. Measure hidden-state pairwise distance, validation negative log likelihood, and long-rollout calibration. The predicted signature is that KL(empirical phase distribution || P_xhat) decreases as K increases and is lower than with isotropic regularization. In attracting settings, pairwise distances should decay with the negative derivative exponent estimated from the maps. Accept the mechanism if KL falls by at least 20% without increasing one-step validation loss.

Verification

Failed on benchmark

Stage 1 · Toy mechanism gate: Passed ✓

Stage 2 · Mechanism transferred to benchmark: Not reproduced ✗

Stage 2 · Practical benchmark result: No measurable effect

Methodology: 8 paired seeds, permutation test (p<0.05) against a learning-rate-tuned baseline. How verification works

Stage 1 — Mechanism check agent confidence 9/10

Built a Poisson-kernel regularizer module, exact toy verification, and a matched small recurrent forecasting experiment. The mechanism manifested: Poisson normalization error was below 2.3e-14, finite-composition attractor error followed the predicted q^K law with relative slope errors below 6e-10, and empirical phase KL to P_x remained about 1e-4 while KL to uniform increased from 0.00012 to 1.015 as |x| grew. In the RNN, the regularizer reduced latent pairwise distance from 0.7544 to 0.7215 and PK NLL from 0.1882 to -0.0391, but validation MSE slightly worsened from 0.02756 to 0.02763, so this supports the mechanism rather than demonstrating a task-performance win.

Agent confidence: 9/10

Baseline: RNN validation MSE 0.027555; latent pairwise distance 0.754402; PK NLL 0.188178
Idea: RNN validation MSE 0.027630; latent pairwise distance 0.721493; PK NLL -0.039106; toy normalization max error 2.20e-14; contraction relative slope errors for q=(0.2,0.5,0.8): (5.3e-10,1.3e-14,5.0e-16)

Limitations: The toy contraction uses affine disk maps rather than full random Blaschke products, and the neural test used one fixed attractor, one regularization coefficient, one K-like depth, and one seed. No sequential-MNIST benchmark, K sweep in the RNN, long-rollout calibration, or statistically repeated runs were tested.

How to run: python3 run_experiment.py && /home/maxwelhelp/main/bin/python3 mini_rnn.py

Stage 2 — Benchmark vs. tuned baseline (arXiv:2607.15488) Sep 2, 2026

Ahead of the baseline (−0.1%) but not statistically significant (p=0.9775, wins 4 / 8 paired seeds).

Benchmark: Dynamics forecasting
Model: rnn_small
Paired seeds: 8
Baseline mean: 0.0005
Idea mean: 0.0005
Effect (Δ): -0 (−0.1%; negative = idea better)
Wins: 4 / 8 paired seeds
p-value: 0.9775 (permutation test, 20 000 shuffles)
Smallest detectable effect: ±4.7%
Mechanism: Not confirmed ✗
Practical effect: No effect
Baseline tuning: swept over 6 configs

Limitations:

Only the registered dynamics track was tested. The estimator uses finite deterministic affine disk maps rather than full random Blaschke products; no long-rollout calibration or larger ensembles were tested.

How to run: python3 bench_pk.py

Verdict computed by deterministic test code from paired-seed statistics — not by the language model.

Stage 2 — Benchmark Aug 31, 2026 latest Failed ✗

Agent confidence: 9/10

Implemented the Poisson-Kernel hidden-state regularizer on the registered dynamics/rnn_small benchmark. The best idea setting had mean test MSE 0.000519053 versus the tuned baseline's 0.000519486, but paired delta was -4.33e-7 with permutation p=0.9775, so there was no significant task-metric win. Hidden-state spread decreased slightly, while phase KL to the estimated Poisson density worsened, so the predicted mechanism was not confirmed.

Baseline: mean test MSE 0.0005194859513721894, std 0.0001693891077925739, best_cfg [0.006,0.0]
Idea: mean test MSE 0.0005190527626837138, std 0.00016528570799671346, best_cfg [0.006,0.02], paired delta -4.3318868847563863e-07, p=0.9775

Benchmark result