# Poisson-Kernel Random Attractor Regularizer

- ID: 965
- Canonical URL: https://synthcore.org/idea/965/poisson-kernel-random-attractor-regularizer
- API JSON: https://synthcore.org/api/idea/965.json
- API Markdown: https://synthcore.org/api/idea/965.md
- Verification status: failed_benchmark
- Source: [arXiv:2607.15488](https://arxiv.org/abs/2607.15488)
- Category: regularization
- Solves: stability, generalization, accuracy
- ML areas: rnn, ssm, regularization, world-model
- Math tags: dynamical-systems, probability, complex-analysis, statistics, measure-theory
- Ratings: usefulness 7/10; difficulty 5/10; novelty 8/10

## 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.

## 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.

## Key formulas

- $$x_\omega\in\operatorname{Acc}\left\{T_{\sigma^{-n}\omega}^{(n)}(z):n\geq 1\right\},\qquad z\in D,$$
- $$P_x(e^{i\phi})=\frac{1-|x|^2}{|e^{i\phi}-x|^2},\qquad \int_0^{2\pi}P_x(e^{i\phi})\frac{d\phi}{2\pi}=1,$$
- $$\mathcal{L}_{\mathrm{PK}}=\frac{1}{M}\sum_{m=1}^{M}-\log P_{\widehat{x}_t}\left(\frac{z_t^{(m)}}{|z_t^{(m)}|}\right),$$
- $$\widehat{x}_t=\frac{1}{K}\sum_{k=1}^{K}T_{j_{t-K}}\circ\cdots\circ T_{j_{t-1}}(\xi_k),\qquad |\xi_k|<1.$$

## 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

- Status: failed_benchmark
- Mechanism evidence: yes
- Mechanism confirmed: no
- Practical verdict: no_effect
- Verdict: 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.

### Mechanism check

- Verdict: 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.
- Confidence: 9/10
- 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.

### Practical benchmark

- Paired seeds: 8
- Baseline mean: 0
- Idea mean: 0
- p-value: 0.9775
- Paired wins: 4/8
- Benchmark verdict: no measurable effect

## Artifacts

- [bench_pk.py](https://synthcore.org/code/405/bench_pk.py)
- [bench_report.json](https://synthcore.org/code/405/bench_report.json)
- [mini_results.json](https://synthcore.org/code/405/mini_results.json)
- [mini_rnn.py](https://synthcore.org/code/405/mini_rnn.py)
- [pk_regularizer.py](https://synthcore.org/code/405/pk_regularizer.py)
- [report.md](https://synthcore.org/code/405/report.md)
- [report_bench_2026-08-31T112633.md](https://synthcore.org/code/405/report_bench_2026-08-31T112633.md)
- [results.json](https://synthcore.org/code/405/results.json)
- [run_experiment.py](https://synthcore.org/code/405/run_experiment.py)
- [toy_results.json](https://synthcore.org/code/405/toy_results.json)
- [Download all files as ZIP](https://synthcore.org/download/405)

## Disclaimer

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