# Beurling–Kato Polynomial Stability Regularizer

- ID: 693
- Canonical URL: https://synthcore.org/idea/693/beurling-kato-polynomial-stability-regularizer
- API JSON: https://synthcore.org/api/idea/693.json
- API Markdown: https://synthcore.org/api/idea/693.md
- Verification status: unverified
- Source: [arXiv:2607.10507](https://arxiv.org/abs/2607.10507)
- Category: dynamics
- Solves: stability, generalization, accuracy
- ML areas: rnn, ssm, training-dynamics, regularization
- Math tags: functional-analysis, complex-analysis, dynamical-systems, linear-algebra
- Ratings: usefulness 6/10; difficulty 5/10; novelty 6/10

## Idea description

Add a functional-calculus regularizer to the transition operator of an RNN, linear state-space model, or deep-equilibrium layer. The regularizer uses polynomial probes to detect non-normal transient amplification that ordinary eigenvalue-radius penalties can miss.

## Mathematical statement

The paper recalls the Beurling–Kato polynomial criterion: if there exists a polynomial w such that limsup_{t\downarrow 0} ||w(e^{-tA})|| < ||w||_infty, where A is the generator of a bounded semigroup, e^{-tA} is its time-t operator, ||·|| is the operator norm, and ||w||_infty := sup_{|z|=1}|w(z)| is the polynomial boundary norm, then the semigroup is holomorphic. The simpler zero–two criterion is limsup_{t\downarrow 0} ||I-e^{-tA}||<2. For an ML transition matrix T, use a finite-step discrete analogue: choose polynomials w_j(z)=sum_{k=0}^{K_j} a_{jk}z^k, evaluate q_j(T)=w_j(T)=sum_{k=0}^{K_j}a_{jk}T^k, estimate its spectral norm ||q_j(T)||_2, and penalize violation of ||q_j(T)||_2 <= (1-delta)||w_j||_infty. Here T is the learned d-by-d transition matrix and delta>0 is a chosen margin. Also penalize excessive sampled powers ||T^n||_2 to control finite-horizon transient growth. The transferable property is an operator-polynomial norm condition that is sensitive to non-normal dynamics, not only eigenvalue magnitude.

## Key formulas

- $$\limsup_{t\downarrow 0}\|I-e^{-tA}\|<2,$$
- $$\limsup_{t\downarrow 0}\|w(e^{-tA})\|<\|w\|_{\infty},\qquad \|w\|_{\infty}:=\sup\{|w(z)|:\ |z|=1\},$$
- $$q_j(T)=w_j(T)=\sum_{k=0}^{K_j}a_{jk}T^k,\qquad \mathcal{L}_{\mathrm{BK}}(T)=\sum_j\left[\max\left(0,\|q_j(T)\|_2-(1-\delta)\|w_j\|_\infty\right)\right]^2+\lambda_P\sum_{n\in\mathcal{N}}\left[\max(0,\|T^n\|_2-B)\right]^2,$$
- $$\|M\|_2\approx\max_{r=1,\ldots,R}\frac{\|Mv_r\|_2}{\|v_r\|_2},\qquad v_r\leftarrow\frac{M^\top Mv_r}{\|M^\top Mv_r\|_2}.$$

## Implementation notes

Integrate the regularizer into the recurrent transition or SSM state-update matrix, rather than the input or output projections. For an RNN with h_{t+1}=phi(T h_t+B x_t), apply it to T. For a residual or deep-equilibrium block, apply it to the linear transition reused during iteration. Begin with three fixed real probes: w_1(z)=1-z, w_2(z)=1+z, and w_3(z)=1-z^2. Estimate each ||w_j||_infty by evaluating the polynomial on a dense unit-circle grid z_m=exp(2 pi i m/M), using max_m |w_j(z_m)|. During each optimizer step, construct P_0=I and P_k=P_{k-1}T, then form q_j(T)=sum_k a_{jk}P_k. Estimate ||q_j(T)||_2 with one or two differentiable power iterations: initialize a random vector v, compute u=Mv/(||Mv||_2+epsilon), then v=M^T u/(||M^T u||_2+epsilon), and use ||Mv||_2 as the estimate. Add L_BK to the task loss with a small coefficient, using the hinge-square penalty in the displayed formula. Every 100 steps, separately measure ||T^n||_2 for n in {1,2,4,8,16}; these measurements may initially be detached from backpropagation. The polynomial bank, margin delta, horizon set, and bound B are empirical choices; the mathematical ingredients are the polynomial operator evaluation and norm-gap target. First test a one-layer tanh RNN and a two-layer linear SSM on sequential MNIST and the Adding Problem. Compare no regularizer, spectral-radius regularization, and this method at equal hidden size and FLOPs. Record validation accuracy, training loss, gradient norms, maximum sampled power norm, and exploding or vanishing-gradient frequency. A successful result is reduced transient amplification and fewer long-sequence failures at equal or better accuracy, with less than 10 percent training overhead.

## Disclaimer

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