# Nonlinear Fourier amplitude budget regularizer

- ID: 2375
- Canonical URL: https://synthcore.org/idea/2375/nonlinear-fourier-amplitude-budget-regularizer
- API JSON: https://synthcore.org/api/idea/2375.json
- API Markdown: https://synthcore.org/api/idea/2375.md
- Verification status: unverified
- Source: [arXiv:2608.15895](https://arxiv.org/abs/2608.15895)
- Category: regularization
- Solves: stability, generalization
- ML areas: ssm, regularization, training-dynamics
- Math tags: harmonic-analysis, functional-analysis, lie-groups, inequalities
- Ratings: usefulness 6/10; difficulty 4/10; novelty 8/10

## Idea description

Use the logarithmic transmission amplitude of an SU(1,1) scan as a differentiable spectral penalty. The paper's constant-one nonlinear Hausdorff–Young inequality provides a principled upper budget for this amplitude in terms of the input L^p norm, replacing an arbitrary spectral-weight penalty with a scale-aware constraint.

## Mathematical statement

For the scattering solution initialized at (a,b)(-infinity,xi)=(1,0), a_f(xi) is the terminal transmission coefficient and the paper defines H_f(xi)=(log|a_f(xi)|^2)^{1/2}. The SU(1,1) identity |a_f(xi)|^2-|b_f(xi)|^2=1 implies |a_f|>=1 and makes H_f real. The main continuous inequality is ||H_f||_{L^{p'}(R)}<=||f||_{L^p(R)} for 1<=p<2, where p'=p/(p-1), f is the input potential, and L^q norms are ||g||_{L^q}=(integral |g(xi)|^q dxi)^{1/q}. For a discrete neural implementation with frequencies xi_j and quadrature weight Delta_xi, use the adapted budget (Delta_xi sum_j H_j^{p'})^{1/p'}<=(h sum_k |f_k|^p)^{1/p}, where h is the position step and H_j=sqrt(log(|a_j|^2+epsilon)). Penalize only violations with a squared hinge, preserving the inequality's one-sided meaning.

## Key formulas

- $$H_f(\xi):=\bigl(\log|a_f(\xi)|^2\bigr)^{1/2}$$
- $$\big\|H_f\big\|_{L^{p'}(\mathbb{R})}\leq\|f\|_{L^{p}(\mathbb{R})},\qquad 1\leq p<2$$
- $$|a_f(\xi)|^2-|b_f(\xi)|^2=1$$
- $$\mathcal L_{\mathrm{budget}}=\left[\max\left(0,\left(\Delta\xi\sum_jH_j^{p'}\right)^{1/p'}-\left(h\sum_k|f_k|^p\right)^{1/p}\right)\right]^2$$

## Implementation notes

Attach the regularizer to the terminal transmission coefficients of the SU(1,1) scan described above, or to any differentiable discretized scattering layer. For each batch example, run the scan over positions k and frequencies j, obtain a_j, and compute H_j=sqrt(clamp(log(abs(a_j)^2+epsilon), min=0)). Choose p=1.5 initially, so p'=3, and use uniformly spaced frequencies with quadrature weight Delta_xi. Compute input_budget=(h*sum_k(abs(f_k)^p))^(1/p) and spectral_budget=(Delta_xi*sum_j(H_j^pprime))^(1/pprime). Add lambda*relu(spectral_budget-input_budget)^2 to the task loss; detach neither quantity, so the network learns to reduce excess spectral concentration or increase the appropriate input representation norm. Use the identity residual invariant_error=abs(abs(a)^2-abs(b)^2-1) as a separate diagnostic, not as the main loss. The exact mathematics supplies the one-sided budget and the logarithmic amplitude; finite frequency truncation, quadrature error, epsilon, and lambda are empirical approximations. Start with a small character-level language model or sequential-MNIST classifier using 8-16 frequencies and a 64-dimensional potential generator. Compare no regularizer, ordinary L2 on f, and this hinge penalty at matched training loss. Look for reduced validation loss, less sensitivity to sequence length and learning rate, and lower out-of-band spectral spikes at equal accuracy. A negative result is clear if the constraint activates frequently, harms training loss, or gives no generalization or stability improvement over L2.

## Disclaimer

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